Polarization of an Electron–Positron Vacuum by a Strong Magnetic Field

Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 395Ð402. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 453Ð461. Original Russian Text Copyright © 2004 by Rodionov. NUCLEI, PARTICLES, AND THEIR INTERACTION

Polarization of an ElectronÐPositron Vacuum by a Strong Magnetic Field with an Allowance Made for the Anomalous Magnetic Moments of Particles V. N. Rodionov Moscow State Geological Prospecting University, Moscow, 118873 Russia e-mail: [email protected] Received June 10, 2003

Abstract—Given the anomalous magnetic moments of electrons and positrons in the one-loop approximation, we calculate the exact Lagrangian of an intense constant magnetic field that replaces the HeisenbergÐEuler Lagrangian in traditional quantum electrodynamics (QED). We have established that the derived generalization of the Lagrangian is real for arbitrary magnetic fields. In a weak field, the calculated Lagrangian matches the standard HeisenbergÐEuler formula. In extremely strong fields, the field dependence of the Lagrangian completely disappears and the Lagrangian tends to a constant determined by the anomalous magnetic moments of the particles. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION for large momentum transfers and the processes in intense electromagnetic fields [2–17]. In fact, the over- The quantum corrections to the Maxwellian lapping of seemingly distinctly different areas of phys- Lagrangian of a constant electromagnetic field were ics is not accidental and is suggested by simple dimen- first calculated by Heisenberg and Euler [1] in 1936. sion considerations. Radiative corrections corresponding to polarization of Allowance for the electromagnetic field intensity an electronÐpositron vacuum by external electromag- based on the exact integrability of the equations of netic fields with diagrams containing different numbers motion is known to play an important role in studying of electron loops are still the focus of attention [2Ð4]. the quantum effects of the interaction between charged Estimates suggest that the quantum (radiative) correc- particles and the electromagnetic field. In particular, the tions could reach the Maxwellian energy density of the standard Schwinger correction to the Bohr magneton electromagnetic field only in exponentially strong elec- 1 π α e tromagnetic fields (Fc ~ exp(3 / )Hc) [5]. The calcu- µ = ------, lations by Heisenberg and Euler are known to contain 0 2m no approximations in the intensity of external electro- which is called the anomalous magnetic moment of a magnetic fields, and their results have been repeatedly particle, confirmed by calculations performed in terms of differ- α ent approaches. On this basis, several authors have ∆µ µ = 0------, identified the field intensity Fc with the validity bound- 2π ary of universally accepted QED. However, it is clear that, although such quantities were greatly overesti- manifests itself only in the nonrelativistic limit for mated because the corresponding scale lengths are weak quasi-static fields [7]. Indeed, when the influence many orders of magnitude smaller than not only the of an intense external field is accurately taken into scale on which weak interactions manifest themselves, account, the anomalous magnetic moment of a particle but also the Planck length, determining the validity calculated in QED as a one-loop radiative correction range of traditional QED is currently of fundamental decreases with increasing field intensity and increasing importance. While on the subject of the physics of energy of the moving particles from the Schwinger extremely small distances, we cannot but say that there value to zero. In particular, for magnetic fields H ~ Hc, is a strong analogy2 between the phenomena that arise the anomalous magnetic moment of an electron is described by the asymptotic formula [7, 10] 1 ប 2 | | Here, we use a system of units with = c = 1, Hc = m / e = α Hc 2H 13 ∆µ()H = µ------ln------. (1) 4.41 × 10 G is the characteristic scale of the electromagnetic 02π H H field intensity in QED, e and m are the electron charge and mass, c and α = e2 = 1/137 is the fine-structure constant. It follows from (1) that ∆µ(H) becomes zero only at one 2 This analogy was first pointed out by Migdal [6] and Ritus [2]. point while decreasing with increasing field. A similar

396 RODIONOV expression for the anomalous magnetic moment of an terms. Thus, an electron in modified QED has an anom- electron in an intense constant crossed field E ⊥ H alous magnetic moment from the outset: ӷ (E = H) at Hp⊥ mHc, where p⊥ is the electron momentum component perpendicular to E × H, can be 2 ∆µ µ µ µ m represented as [11] ==Ð 0 0 1 +1------Ð . (4) M2 αΓ() Ð2/3 ∆µ() µ 1/3 3 p⊥ H E = 0------------. (2) An important aspect of the problem under consider- 93 mHc ation is that the current state of the art in the develop- ∆µ ≠ ment of laser physics [14] allows one to carry out a Note that (E) 0 in the entire range of parameters. number of optical experiments to directly measure the Numerous calculations of the Lagrangian for an contributions from the nonlinear vacuum effects pre- electromagnetic field (see, e.g., [1–3, 5–7, 11]) have dicted by various generalizations of Maxwellian elec- been performed by assuming that the magnetic moment trodynamics [15]. Therefore, it should be emphasized of electrons is exactly equal to the Bohr magneton,3 i.e., that experimental verification of the nonlinear vacuum at ∆µ = 0. However, the following question is of consid- effects with a high accuracy in the presence of rela- erable importance in elucidating the internal closeness tively weak electromagnetic fields can also provide of QED: What effects will allowance for the anomalous valuable information about the validity of QED predic- magnetic moments of electrons and positrons produce tions at small distances [16, 17]. Note in passing that when calculating the polarization of an electronÐ precision measurements of various quantities (e.g., the positron vacuum by intense electromagnetic fields? anomalous magnetic moments of an electron and a muon) at nonrelativistic energies, together with studies Thus, it is of interest to take the radiative corrections of the particle interaction at high energies, are of cur- to the Maxwellian Lagrangian of a constant field calcu- rent interest in the same sense. lated by the traditional method and compare them with the results that can be obtained from similar calculations by taking into account the nonzero anomalous 2. CORRECTION TO THE LAGRANGIAN magnetic moments of particles. The fact that the OF AN ELECTROMAGNETIC FIELD Lagrangian replacing the HeisenbergÐEuler Lagran- WITH ALLOWANCE gian with nonzero anomalous magnetic moments can FOR ANOMALOUS MAGNETIC MOMENTS be calculated by retaining the method of exact solutions OF PARTICLES of the Dirac equation in arbitrarily intense electromagnetic fields also deserves serious attention. In the Let us consider the correction to the Lagrangian of approach under development, the suggested theoretical an electromagnetic field attributable to the polarization generalization initially contains no constraints on the of an electronÐpositron vacuum in the presence of an electromagnetic field intensity. arbitrarily strong constant magnetic field by taking into account the nonzero anomalous magnetic moments of It should be noted that nonzero anomalous magnetic the particles. To solve this problem, it is convenient, as moments also appear in some of the modified quantum in the standard approach [5], to represent the electronÐ field theories (QFTs) that also describe the electromag- positron vacuum as a system of electrons that fill “neg- netic interactions. In particular, this is true for a gener- ative” energy levels. For a constant uniform magnetic alization of the traditional QFT known as the theory field, the Dirac–Pauli equation containing the interac- with “fundamental mass” (see, [12, 13] and references tion of a charged lepton with the field (including the therein). The starting point of this theory is the condi- anomalous magnetic moment of the particle) has an tion that the mass spectrum of elementary particles is exact solution [18]. In this case, the energy eigenvalues limited. This condition can be represented as explicitly depend on the spin orientation with respect to ≤ the axis of symmetry specified by the magnetic field mM, (3) direction. Thus, the energy spectrum of an electron that where the new universal parameter M is called the fun- moves in an arbitrarily intense constant uniform mag- damental mass. Relation (3) is used as an additional netic field is fundamental physical principle that underlies the new 2 (),,ξ p QFT. A significant deviation from traditional calcula- En pH = m ------tions is the fact that the charged leptons in QED with m2 fundamental mass have magnetic moments that are not (5) µµÐ H 2 1/2 equal to the Bohr magneton. This is because, apart from + H ()12++n ξ + 1 + ξ------0 ------, ------2µ H the traditional “minimal” term, the new Lagrangian of Hc 0 c the electromagnetic interaction includes “nonminimal” where p is the electron momentum component along 3 The authors of [4] took into account the anomalous magnetic the external field H; n = 0, 1, 2, … is the quantum num- moments of particles when analyzing the equilibrium processes ber of the Landau levels; and ξ = ±1 characterizes the in a degenerate electronÐnucleon gas in a strong magnetic field. electron spin component along the magnetic field.

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POLARIZATION OF AN ELECTRONÐPOSITRON VACUUM 397

Noting that the radiative correction to the classical ral to call the quantity density of the Lagrangian is equal, to within the sign, to 2 2 the total energy density of the electronÐpositron vac- M M Hc* ==------Hc (10) uum in the presence of an external field [5], e m2 a fundamental field. ᏸ' = ÐW H, Passing to the limits of integration over x from zero to ∞ in (8) and using the evenness of the function F (z), H 1 2 let us calculate W in a constant magnetic field by tak- we obtain ing into account the anomalous magnetic moment of ∞ the electron. Without dwelling on the details of stan- 4γ m b dη Ðη dard calculations, we represent WH as ᏸ' = Ð------1∫------e 8π2 η2 0 H eH (11) W = Ð------∞ 2 ()γ ()2π × 2 dx sin 2 x cosxxy+ sin sinh () sinhb +,---∫------12F z ∞ ∞ (6) π x cosh()2y Ð cos() 2γx 0 × dp Ð ε+()p +,[]εÐ()p + ε+()p ∫ 0 ∑ n n ηγ 4 2 where y = /b1 and z = Ðb /64x . Ð∞ n = 0 In particular, it immediately follows from (11) that where Imᏸ'0.= 2 ᏸ ε± 2 2H ± H The fact that the Lagrangian ' is real for all possible n = p +.m 12+ ------n ------(7) Hc 4H* field intensities suggests the absence of unstable c modes; i.e., the vacuum in a constant uniform magnetic Using the Laplace and Fourier integral transforms field in the case under consideration, as in traditional for the functions that define (6) and performing summa- QED, is stable against the spontaneous production of tion over Landau levels, we can obtain the following electronÐpositron pairs. formula for ᏸ': Next, let us separate out the integral over x in expression (11). After several obvious substitutions, it ∞ 4γ η reduces to ᏸ m b1 d Ðη ' = Ð------2 -∫------2 e ∞ 8π η du[]2a sinubcos()u + () 1 Ð a2 sin()b u 0 I = ------2 2 2 2 - ∞ ∫ 2 Ðix u[]1 + a Ð 2a cosu (12) γη 0 2 2 × 1 e dx i γ sinhb + π--- ∫ ------cotÐ ------+ x (8) × () x b1 12F z1 , Ð∞ where 1 3 b4 {},,, 4 2 + 12F 1 ------2 , Ð2y 1 b ()2γ 4 4 64x a2 ===e , b2 ------, z1 Ð------. 2γ 64u2 where we use the notation Since the expansions η 2 a sinu H H 1 γ H ------a1 = ------, b1 = 1 +------, b = -----, = ------; 2 2Hc* 4Hc* b1 Hc 1 + a2 Ð 2a2 cosu () 2 ()… 1F2(z) is the generalized hypergeometric function. For- = sinua+++2 sin 2u a2 sin 3u , mula (8) is an exact expression for the Lagrangian with 2 allowance for the anomalous magnetic moment calcu- 1 Ð a2 2 ()… ------12= + a2 cosu + 2a2 cos 2u + lated in the one-loop approximation in an arbitrarily 1 + a2 Ð 2a cosu intense magnetic field. An important deviation from the 2 2 HeisenbergÐEuler Lagrangian is that expression (8) are valid, we obtain for (12) contains the additional field scale ∞ ∞ du k m () []() * I = ∫------Ð2sin b2u + ∑ a2 sin uk+ b2 Hc = ------∆µ. (9) u (13) 4 0 k = 0 × () In the theory with fundamental mass, it would be natu- 12F z1 .

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 398 RODIONOV 4 It is easy to see that the following expansion of the ing diverging integral, we obtain function 1F2(z1) at zero may be used in a field that is ∞ 4 []()ηγ weak compared to the fundamental field H* : m dη Ðη cosh 1 + a /b c ᏸ' = Ð------e ηb γ ------1 2∫ 3 1 ()ηγ 8π η sinh /b1 16 256 2 0 (20) F ()z = 1 ++------z ------z +…. (14) 12 1 3 1 105 1 η2 Ð b 2 Ð,----- γ 2()26++a 3a2 Hence, we obtain for (13) 1 6 π1 + a π where a = H /.2H* I ==------2 --- cothy, (15) c c 2 1 Ð a2 2 where y = ηγ. 3. ASYMPTOTIC RESULTS Ӷ Substituting (15) into (11) and performing standard In weak fields (H Hc) and in the limit of very regularization of the derived integral [5] yields ӷ 2 strong magnetic fields (H 16Hc* /Hc), the integrand ∞ 4 Ðη η2γ 2 in (20) admits an expansion into a series. In the first ᏸ m e ηγ() ηγ η approximation, ' = Ð------2∫------3- cothÐ 1 Ð ------d . (16) 8π η 3 ∞ 0 4 4 m γ 2 2 Ðη Thus, it follows from (16) that in the limit of a weak ᏸ' = ------[]ηη815Ð a ()2 + a ∫d e , 2880π2b2 field, formula (11) matches the HeisenbergÐEuler 1 0 Lagrangian [1] for an arbitrarily intense constant uni- whence it follows that form magnetic field. 4 4 * m γ 2 2 Next, let us consider H > 4Hc . It is easy to verify ᏸ' = ------[]815Ð a ()2 + a . (21) ӷ π2 2 that in the limit of extremely strong fields, H 2880 b1 2 16Hc* /Hc, we may again use expansion (14) and can Thus, the quantum correction to the Maxwellian Ӷ obtain the following expression for integral (13): Lagrangian in the limit of a weak field (H Hc) can be π represented as I = --- cothy, (17) 4 4 2 m H ᏸ' = ------360π2 H4 η 2 c where y = 16 Hc* /(HcH). Results (15) and (17) have (22) 15 2 15 3 15 4 2 2 2 a simple meaning: for a sufficiently wide energy gap × 1 Ð ------a Ð ------a Ð ------a +,ᏻ()γ ; γ a that separates the electron and positron states, the terms 2 2 8 with large numbers k make the largest contribution to integral (13). However, for magnetic fields close to the where the first term matches the standard HeisenbergÐ * Euler formula. The first correction to it attributable to fundamental field, H ~ 4Hc , i.e., when the gap width the anomalous magnetic moments of the particles is is close to zero, the term with k = 0 makes the largest negative and quadratic in a. contribution to the sum in the integrand of (13). In this ӷ 2 case, integral (13) can be calculated exactly. Our calcu- In an extremely strong field (H 16Hc* /Hc), we lations yield can also obtain from (21) π 4 η H 1 m 2 3 4 I = --- cosh ------. (18) ᏸ' = ------()860Ð60a Ð15a Ð a 2 2H* ()2 2 4 c 1 + H/4Hc* 180π a (23) The estimates of integral (12) in the three ranges of 8 × 1 Ð ------. Ӷ ӷ 2 2 magnetic fields (H Hc* , H ~ 4Hc* , and H Hc* ) can γ a be represented as a single formula: 4 π H 1 First, as usual [5], the part of the integral that does not contain the I = --- cosh η------magnetic field intensity and that represents the energy of the free 2 2H* ()2 vacuum electrons should be discarded. Second, it is necessary to c 1 + H/4Hc* (19) subtract the contribution proportional to H2 that has already been ηγ × included in the unperturbed field energy. Discarding this term is coth ------2 . related to renormalizing the field intensity and, hence, the charge. () 4 4 1 + H/4Hc* Finally, subtracting a contribution on the order of H /H* basically corresponds to renormalizing an additional parameter of the Substituting (19) into (11) and regularizing the remain- theory—the anomalous magnetic moment of the particle.

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According to (23), in the limit of extremely strong ( '/m4) × 10–11 fields, the Lagrangian ᏸ' ceases to depend on the field; i.e., as the field grows, the quantum correction to the 7 density of the Lagrangian in the case under consideration asymptotically approaches the constant 6 3 α2 5 ᏸ∞' = ------. (24) 360π2()∆µ 4 2 4 In a certain sense, the result obtained may be compared with the situation observed in the standard model, 3 1 where the cross sections for several processes cease to increase with energy if, apart from a photon, vector W±- 2 and Z0 bosons, an additional diagram with a Higgs H boson is included in the analysis. Allowance made 1 for this diagram reduces the increasing terms in amplitude and leads to a behavior of the cross sections con- 0 sistent with the unitary limit. Since the standard model 1 2 3 4 5 6 does not predict the mass of the H-boson, it may well γ × 10–6 be that this particle is much heavier than the t quark, the 4 heaviest known elementary particle. Thus, MH ~ 1 TeV Fig. 1. Normalized Lagrangian ᏸ'/m versus magnetic field γ may prove to be the critical mass that limits the mass intensity = H/Hc for various anomalous magnetic ∆µ µ Ð3 ∆µ µ spectrum of elementary particles, i.e., acting as the fun- moments of particles: 1— 1/ 0 = 10 , 2— 2/ 0 = 5 −3.05 ∆µ µ Ð3.1 damental mass (see (3)). 10 , and 3— 3/ 0 = 10 . By comparing the correction ᏸ' with the Lagrangian of the Maxwellian field, we can determine the field Ӷ Ӷ 2 Ӷ Ӷ intensity For Hc H 16Hc* /Hc, the range 1 x 2 2 2 16H* /H is important in integral (26). In this case, αH* c c * 256 c the hyperbolic functions may be substituted with expo- Fc = ------π------, (25) 45 Hc nentials and the integrand in (26) becomes []()αγ,, [](),,γ ᏸ exp f 1 x exp f 2 a x at which 0 becomes equal to (24). For H = Fc* , the ------Ð ------ᏸ x2 x3 quantum correction ' does not yet reach its asymp- (27) [](),,γ totic value of ᏸ∞' . A comparison of ᏸ and ᏸ' in other exp f 2 a x 3 2 0 Ð ------13++a ---a , field ranges clearly shows that the corrections ᏸ' are 3x 2 always small compared to the Lagrangian ᏸ . The rel- 0 where ative corrections ᏸ'/m4 derived from (20) for the anom- ∆µ µ Ð3 1 2 alous magnetic moments of particles 1/ 0 = 10 , f ()a,,γ x = Ð------()2 Ð aγ x, ∆µ µ Ð3.05 ∆µ µ Ð3.1 1 γ 2/ 0 = 10 , and 3/ 0 = 10 are plotted against 4 γ magnetic field intensity = H/Hc in Fig. 1. 2 1a 2 f ()a,,γ x = Ð--- 1 + -----γ x. We estimate the Lagrangian for strong magnetic 2 γ 4 fields by using formula (20), which we will represent γη Ӷ Ӷ γ γ after the substitution /b1 x as For Hc H < 4Hc* (1 < 2/a), f1 ~ f2 = Ðx/

∞ and we ﬁnd from (26) with logarithmic accuracy that 4 2 ()γ m γ exp Ðb1 x/ 4 2 ᏸ' = Ð------m γ 3 2 2 ∫ 3 ᏸ' = ------13++a ---a lnγ . (28) 8π x 22 0 (26) 24π 2 cosh[]x()1 + a x 3 2 For a 0, this formula matches the HeisenbergÐ × x------Ð 1 Ð ----- 13++a ---a dx. sinh()x 3 2 Euler Lagrangian in the limit of strong magnetic ﬁelds, ӷ H Hc [5]. 5 2 Note in this connection that the chief goal in the research at the If 4H* < H Ӷ 16H* /H or 2/a < γ Ӷ 4/a2, then Large Hadron Collider (LHC) at CERN is the search for Higgs c c c γ 2 Ӷ Ӷ 2γ bosons in a mass range up to 1 TeV. f1 ~ f2 = Ð a x/4. The range 1 x 4/(a ) gives the

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'/m4 Neglecting the ﬁrst and the second terms in (27), we ﬁnd from (26) that 134.375 2 m4 3a2 2 ᏸ' = ------13++a ------ln---. 1 2 2 134.365 6π a 2 a

This result agrees with formulas (28) and (29) from which it can be obtained by the substitution γ = 2/a. 134.355 Thus, these functions are continuously joined at H =

4Hc* . 134.345 Finally, let us estimate the Lagrangian ᏸ' in terms of 1000.00 1000.01 1000.02 1000.03 1000.04 1000.05 traditional QED, i.e., by taking into account the non- γ zero anomalous magnetic moments of particles in γ intense electromagnetic fields attributable to radiative Fig. 2. Lagrangian versus magnetic field intensity = H/Hc effects. Substituting ∆µ from (1) into the expression with (curve 1) and without (curve 2) an allowance for the ∆µ 2 ᏸ anomalous magnetic moment of an electron. b1 = 1 + ( H/m) yields an estimate of ' (26) in the limit of extremely strong fields. For a constant mag- γ ӷ µ ∆µ netic field, 0/ , largest contribution to the integral. In this case, we find from (26) that ∼ α 2 ()γ b1 2 ln 2 , 4γ 2 ᏸ m 3 2 2 γ ' = ------13++a ---a 2ln--- Ð ln . (29) where 24π2 2 a α α2 π2 ӷ 2 2 = /16 . For H 16Hc* /Hc, we return to the cases considered above (see (23)) where the range x Ӷ 1 gives the largest In this case, the exponent in (26) is contribution to integral (26). 2 If aγ = 2, i.e., for H = 4H* , the exponent (f ) in one b α ln ()2γ c 1 f ==Ðx-----1 Ð------2 x. of the exponentials in (27) becomes zero. It is easy to 3 γ γ verify that this term is attributable to the contribution from the ground energy state ε (see formula (7)) in γ ӷ α 2 γ Ӷ Ӷ γ α 2 γ 0 If 2ln (2 ), then the range 1 x / 2ln (2 ) which the dependence on particle mass completely is important in (26). Thus, we can find from (26) that drops out at this field intensity. There is no such state with a “dropping” mass in the structure of the Heisen- 4γ 2 berg–Euler Lagrangian for a fixed field. However, if we ᏸ m []γα ()γ ' = ------2 ln Ð22ln 2 Ð ln ln . (30) consider the passage to the limit m2 0, then the 24π HeisenbergÐEuler Lagrangian can simulate such an effect. It is easy to see that the total contribution of the The first term in (30) is identical to its estimate in the ground state is small compared to the contribution of HeisenbergÐEuler theory [5]. The relative effective the last term in (27), which owes its origin to the field Lagrangian ᏸ'/m4 derived from the integral representa- α ≈ × Ð7 renormalization in expression (11). A similar conclu- tion (26) (with 2 3.4 10 ) is plotted against the sion can also be reached by considering integral (16). magnetic field intensity in Fig. 2 (curve 1). For com- The following should be emphasized when comparison, the same figure also shows a plot for ∆µ = 0 menting on the analogy between the limits m2 0 in (curve 2). the HeisenbergÐEuler Lagrangian (see formula (16)) Note that allowance for the anomalous magnetic and H 4H* in (26). As we showed above, for H = moments of vacuum particles in terms of universally c accepted QED leads to a decrease in the radiative cor- * 4Hc in the modified Lagrangian, just as in the Heisen- rection to the field energy density. Recall that we bergÐEuler Lagrangian for m2 = 0, the exponent in the reached a similar conclusion by considering the static terms whose contributions are vanishingly small anomalous magnetic moment that arises, in particular, against the background of the contributions from the in the modified field theory. Thus, irrespective of the renormalization procedure becomes zero. In other nature of the anomalous magnetic moment attributable words, in both cases, the ground states in the structure to the dynamic or static types of interaction, we obtain of the integrand are equally preferential, but their con- consistent results. Our conclusions are also important tribution to the integral is not dominant. in studying the anomalous magnetic moment as the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 POLARIZATION OF AN ELECTRONÐPOSITRON VACUUM 401 most accurately calculable and measurable (in numer- of the lowest-order hadron polarization of a vacuum ous precision experiments) characteristic of particles. that generalizes the data on hadron τ-decay and e+eÐ annihilation appears as follows [23, 24]:6 4. CONCLUSIONS had Ð10 aµ = 692() 6 × 10 . Our results can be of considerable importance in constructing astrophysical models, in particular, in The theoretical anomalous magnetic moment of a studying extremely magnetized neutron stars—magne- muon in the standard model takes the form [28] tars; interest in the existence of the latter objects has SM Ð10 aµ = 11 659177() 7 × 10 . increased appreciably in recent years (see, e.g., [4] and references therein). According to models for the macro- The results of one of the most recent (g Ð 2) experi- scopic magnetization of bodies composed mostly of ments aimed at measuring the anomalous magnetic neutrons, the intensity of the magnetic fields frozen into moments of positive polarized muons carried out on a them increases from the surface to the central regions storage ring with superconducting magnets at and can reach 1015Ð1017 G [19]. Brookhaven National Laboratory (BNL) can be repre- Note also that the radiative effects can be enhanced sented as by external intense electromagnetic fields not only in exp Ð10 Abelian, but also in non-Abelian quantum field theo- aµ = 11659204() 7 ()5 × 10 . (31) ries. For example, allowance for the influence of an external field on such parameters as the lepton mass and (Both statistical and systematic errors are included magnetic moment in terms of the standard model leads here.) The data obtained yield the difference to nontrivial results. In this case, apart from the electrodynamic contribution, the one-loop mass operator of a ∆ exp SM × Ð10 charged lepton also contains the contributions from the µ ==aµ Ð2710aµ , (32) interaction of W±-, Z0-, and H bosons with a vacuum. It is easy to see that, in the absence of an external field, the which exceeds the total measurement errors and the contribution from weak interactions to the radiative uncertainties of the theoretical estimates. According to shift of the lepton mass m is suppressed by a factor of the most recent reports from the BNL muon (g Ð 2) col- (m/M )2 (i = W, Z, H) compared to the electrodynamic laboration [29], the relative value of this excess is 2.6. i A twofold increase in this accuracy is expected in the contribution. However, the contributions of weak cur- immediate future. Clearly, the solution of the muon rents in the ultrarelativistic limit can dominate in (g − 2) problem may lead to the appearance of a new intense external fields, as was first noted in [20] (see theory beyond the scope of the standard theory. also [21]). Recall in this connection that the anomalous mag- In close analogy with the quantum corrections to the netic moment of a muon in the modified theory con- particle masses, the anomalous magnetic moments of tains a contribution attributable to the new universal charged leptons in the standard model are attributable parameter M from the outset. According to (4), to the vacuum radiative effects of electromagnetic and weak interactions and contain the contribution from the 2 hadron polarization of the vacuum. For example, for the mµ aµ()M = ------, (33) anomalous magnetic moment of a muon, 2M2

SM QED weak had aµ = aµ ++aµ aµ . where mµ is the muon mass. It is easy to see that aµ(M) is equal in order of magnitude to (32) at M ~ 1 TeV. According to recent theoretical estimates made in the The principal conclusion drawn from a comparison standard model [22], the contributions from electro- of the above estimates is that we cannot rule out the magnetic and weak interactions can be written as possibility that the observed difference between the theoretical and experimental values for ∆µ is equal to QED ()× Ð10 aµ = 11 658470.57 0.29 10 , aµ(M). As was pointed out above, the parameter M in the new theory may be related to the Higgs boson mass weak ()× Ð10 exp SM aµ = 15.1 0.4 10 . MH. In this case, the difference between aµ and aµ can provide valuable information about a particle Although the calculations of the contributions from the SM 6 hadron polarization of a vacuum to aµ have a history See, however, [25], where the contribution of the highest orders of hadron polarization of a vacuum were calculated, and the had that spans almost forty years, aµ is currently under- recent papers [26, 27], in which the contribution from the third- had stood with the largest uncertainty (see, e.g., [23Ð28]). order diagrams to aµ attributable to photonÐphoton scattering One of the most reliable estimates for the contributions was taken into account.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 402 RODIONOV whose mass has not been determined in the standard 10. V. N. Baœer, V. M. Katkov, and V. M. Strakhovenko, Yad. model. Substituting mµ and the anomalous magnetic Fiz. 24, 379 (1976) [Sov. J. Nucl. Phys. 24, 197 (1976)]. moment of a muon into (33), we can easily impose the 11. V. I. Ritus, Zh. Éksp. Teor. Fiz. 57, 2176 (1969) [Sov. following constraints on the H-boson mass: Phys. JETP 30, 1181 (1969)]. ≤≤ 12. V. G. Kadyshevskiœ, Fiz. Élem. Chastits At. Yadra 11, 5 1.2 TeV MH 1.8 TeV. (1980) [Sov. J. Part. Nucl. 11, 1 (1980)]. The standard model with the Higgs boson mass 13. V. G. Kadyshevskiœ, Fiz. Élem. Chastits At. Yadra 29, ≥ 563 (1998) [Phys. Part. Nucl. 29, 227 (1998)]. MH 1 TeV entails several additional features, in particular, the impossibility of describing the weak interac- 14. G. E. Stedman et al., Phys. Rev. A 51, 4944 (1995). tions in the sector of H-, W-, and Z particles in terms of 15. V. I. Denisov and I. P. Denisova, Teor. Mat. Fiz. 129, 131 perturbation theory [30]. Naturally, the need for con- (2001); Dokl. Akad. Nauk 378, 4 (2001) [Dokl. Phys. 46, structing a new nonperturbative theory arises in this 377 (2001)]. case. Apart from the condition for the mass spectrum 16. V. G. Kadyshevski and V. N. Rodionov, in Proceedings ≤ œ being limited, m MH (see (2)), the Higgs mechanisms of Seminar on Symmetry and Integrated Systems, Ed. by of mass formation and compensation for the discrepan- A. N. Sisakyan (Ob. Inst. Yad. Issled., Dubna, 1999), cies can become integral elements of one of the most p. 103. promising versions of the modiﬁed theory—the stan- 17. V. G. Kadyshevskiœ and V. N. Rodionov, Teor. Mat. Fiz. dard model with fundamental mass. 125, 432 (2000). 18. I. M. Ternov, V. G. Bagrov, and V. Ch. Zhukovskiœ, Vestn. Mosk. Univ., Ser. 3: Fiz., Astron., No. 1, 30 (1966). ACKNOWLEDGMENTS 19. R. C. Duncan, astro-ph/0002442. I am grateful to V.G. Kadyshevsky for helpful dis- 20. I. M. Ternov, V. N. Rodionov, and A. I. Studenikin, in cussions and valuable remarks and to V.R. Khalilov and Abstracts of All-Union Meeting on Quantum Metrology A.E. Lobanov for attention to the work and fruitful dis- and Fundamental Physical Constants (Leningrad, cussions. This study was supported by the Russian Foun- 1982), p. 47. dation for Basic Research (project no. 02-02-16784). 21. I. M. Ternov, V. N. Rodionov, and A. I. Studenikin, Yad. Fiz. 37, 1270 (1983) [Sov. J. Nucl. Phys. 37, 755 (1983)]. REFERENCES 22. A. Czarnecki and W. Marciano, Phys. Rev. D 64, 013014 1. W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936). (2001). 2. V. I. Ritus, Zh. Éksp. Teor. Fiz. 69, 1517 (1975) [Sov. 23. M. Davier and A. Hocker, Phys. Lett. B 435, 427 (1998). Phys. JETP 42, 774 (1975)]; Zh. Éksp. Teor. Fiz. 73, 807 (1977) [Sov. Phys. JETP 46, 423 (1977)]. 24. M. Davier, S. Eidelman, A. Hocker, and Z. Zhang, hep- ph/0208177. 3. V. I. Ritus, hep-th/9812124. 25. B. Krause, Phys. Lett. B 390, 392 (1997). 4. V. R. Khalilov, Teor. Mat. Fiz. 133, 103 (2002). 26. E. Bartos, A. Z. Dubnichkova, S. Dubnichka, et al., 5. V. B. Berestetskiœ, E. M. Lifshitz, and L. P. Pitaevskiœ, Nucl. Phys. B 632, 330 (2002). Quantum Electrodynamics, 3rd ed. (Nauka, Moscow, 27. E. Bartos, S. Dubnichka, A. Z. Dubnichkova, et al., hep- 1989; Pergamon Press, Oxford, 1982). ph/0305051. 6. A. B. Migdal, Zh. Éksp. Teor. Fiz. 62, 1621 (1972) [Sov. 28. C. S. Ozben, G. W. Bennett, B. Bousquet, et al., hep- Phys. JETP 35, 845 (1972)]. ex/0211044. 7. I. M. Ternov, V. R. Khalilov, and V. N. Rodionov, Inter- 29. G. W. Bennett et al. (Muon g-2 Collaboration), Phys. action of Charged Particles with Strong Electromag- Rev. Lett. 89, 101804 (2002). netic Field (Mosk. Gos. Univ., Moscow, 1982). 30. L. B. Okun’, Leptons and Quarks, 2nd ed. (Nauka, Mos- 8. V. N. Rodionov, Zh. Éksp. Teor. Fiz. 111, 3 (1997) [JETP cow, 1981; North-Holland, Amsterdam, 1984). 84, 1 (1997)]. 9. V. N. Rodionov, Zh. Éksp. Teor. Fiz. 113, 21 (1998) [JETP 86, 11 (1998)]. Translated by V. Astakhov

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 403Ð416. From Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 462Ð475. Original English Text Copyright © 2004 by Afanasev, Akushevich, Merenkov. NUCLEI, PARTICLES, AND THEIR INTERACTION

QED Correction to Asymmetry for Polarized ep Scattering from the Method of Electron Structure Functions¦ A. V. Afanaseva,b, I. Akushevicha,b, and N. P. Merenkovc aNorth Carolina Central University, Durham, NC 27707, USA bTJNAF, Newport News, VA 23606, USA cNational Scientiﬁc Center Kharkov Institute of Physics and Technology, Kharkov, 61108 Ukraine e-mail: [email protected] Received August 18, 2003

Abstract—The electron structure function method is applied to calculate model-independent QED radiative corrections to the asymmetry of electronÐproton scattering. Representations for both spin-independent and spin-dependent parts of the cross section are derived. Master formulas include the leading corrections in all orders and the main contribution of the second order and provide accuracy of the QED corrections at the level of one per mill. Numerical calculations illustrate our analytic results for both elastic and deep inelastic events. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION leading log approach [15]. The recent developments are reviewed in [16]. Precise polarization measurements in both inclusive [1, 2] and elastic [3, 4] scattering are crucial for The resolution can be found in applying the formal- understanding the structure and fundamental properties ism of electron structure functions (ESFs). Within this of a nucleon. approach, such processes as the electronÐpositron annihilation into hadrons and the deep inelastic electronÐ One important component of the precise data analy- proton scattering in the one-photon exchange approxi- sis is radiative effects, which always accompany the mation can be considered as the DrellÐYan process [17] processes of electron scattering. The first calculation of in the annihilation or scattering channel, respectively. radiative corrections to polarized deep inelastic scatter- Therefore, the QED radiative corrections to the corre- ing was done by Kukhto and Shumeiko [5], who sponding cross sections can be written as a contraction applied the covariant method of extraction of the infra- of two electron structure functions and the hard part of red divergence [6, 7] to this process. The polarization the cross section (see [18, 19]). Traditionally, these states were described by 4-vectors, which were kept in radiative corrections include effects caused by loop cor- their general forms during the calculation. This rections and soft and hard collinear radiation of photons required a tedious procedure of tensor integration over and e+eÐ pairs. But it was shown in [19] that this method photonic phase space and, as a result, led to a very com- can be improved by also including effects due to radia- plicated structure of the final formulas for the radiative tion of one noncollinear photon. The corresponding corrections. The next step was taken in [8], where addi- procedure results in a modification of the hard part of tional covariant expansion of polarization 4-vectors the cross section, which takes the lowest-order correc- over a certain basis allowed one to simplify the calculation into account exactly and allows going beyond the tion and final results. It resulted in producing the For- leading approximation. We applied this approach to the tran code POLRAD [9] and Monte Carlo generator recoil proton polarization in elastic electron scattering RADGEN [10]. These tools are widely used in all cur- in [20]. In the present paper, we calculate radiative cor- rent experiments in polarized deep inelastic. Later, the rections to polarized deep inelastic and elastic scatter- calculation was applied to collider experiments on deep ing following [20]. inelastic scattering [11, 12]. We also applied this method to the elastic processes in [13, 14]. Section 2 gives a short introduction to the structure function method. There, we present two known forms However, the method of covariant extraction of the of the electron structure functions, iterative and analyt- infrared divergence is essentially restricted by the low- ical, which resume singular infrared terms in all orders est-order radiative corrections. All attempts to go into the exponent. In this section, we also obtain master beyond the lowest order lead to very unwieldy formu- formulas for observed cross sections. Leading log las, which are difficult to cross check, or to a simple results are presented in Section 3. These results are valid for both deep inelastic and elastic cases. We also ¦This article was submitted by the authors in English. use the iterative form of electron structure functions to

404 AFANASEV et al.

+ Ð γ e e extract the lowest-order correction, which can provide we here use the form given in [18] for D , DN , and a crosscheck via comparison with the known results. In e+eÐ Sections 4 and 5, we describe the procedure of general- DS , izing the results to the next-to-leading order in the deep γ 2 1 β/2Ð 1 inelastic and elastic cases. Numerical analysis is pre- D ()zQ, = ---β()1 Ð z sented in Section 6. We consider kinematical conditions 2 of current polarization experiments at fixed targets and β2 β collider kinematics. Some conclusions are given in Sec- × 3β 1 π2 47 () 1 + --- Ð ---------L + Ð ------Ð --- 1 + z tion 7. 8 48 3 8 4 (5) β2 13+ z2 + ------Ð 41()+ z ln()1 Ð z Ð,------lnz Ð 5 Ð z 2. ELECTRON STRUCTURE FUNCTIONS 32 1 Ð z A straightforward calculation based on the quasi- 2α real electron method [21] can be used to write the β = ------()L Ð 1 , invariant cross section of the deep inelastic scattering π process + Ð e e (), 2 DN zQ eÐ()k ++Pp() eÐ()k Xp() (1) 1 1 2 x α2 1 2m β/25 2 = ------1 Ð z Ð ------L Ð --- as π2 12() 1 Ð z ε 1 3 (6) 1 1 β dσ()k , k × 2 5 θ2m 1 2 (), 1 ++z --- L1 Ð --- 1 Ð z Ð ------, ------= ∫ dz1 ∫ dz2 Dz1 L 63 ε dQ2dy z1m z2m (2) 2 3 2 ˜ ˜ 2 e+eÐ α 2 21()Ð z 1 d σ ()k1, k2 Q D = ------L ------× ----Dz(), L ------hard -, L = ln------, S π2 3z 2 2 ˜ 2 2 4 z2 dQ dy˜ m (7) 1 2m where m is the electron mass and + ---()1 Ð z + ()1 + z lnz θ 1 Ð z Ð ------, 2 ε 2 p ()k Ð k Q2 ==Ð()k Ð k 2, y ------1 1 2 , V =2 p k . where ε is the energy of the parent electron and 1 2 V 1 1 () L1 = L + 21ln Ð z . The reduced variables that define the hard cross section in the integrand are We note that the above form of the structure function e+eÐ ˜ ˜ k2 DN includes effects due to real pair production only. k1 ==z1k1, k2 ---- , z2 The correction caused by the virtual pair is included in (3) Dγ. Terms containing a contribution of the order α2L3 2 z 2 1 Ð y + Ð Q˜ ==----1Q , y˜ 1 Ð ------. γ e e are canceled in the sum D + DN . z2 z1z2 The electron structure function D(z, L) includes con- Instead of the photon structure function given by tributions due to photon emission and pair production, Eqs. (5)Ð(7), one can use their iterative form [23] ∞ k γ + Ð + Ð γ 1 αL ⊗k DD= ++De e De e , (4) D ()δzL, = ()1 Ð z + ∑ ------P ()z , N S k!2π 1 k = 1 where Dγ is responsible for the photon radiation and ⊗k + Ð + Ð ()⊗⊗ … () () e e e e P1 z P1 z = P1 z , DN and DS describe pair production in nonsinglet k (by single photon mechanism) and singlet (by double photon mechanism) channels, respectively. 1 z dt The structure functions on the right-hand side of P ()z ⊗ P ()z = P ()t P - ---- , (8) 1 1 ∫ 1 1t t Eq. (4) satisfy the DGLAP equations [22] (see z also [18]). The respective functions D(z1, L) and D(z , L) are responsible for radiation of the initial and 1 + z2 3 2 P ()z = ------θ()δ1 Ð z Ð ∆ + ()1 Ð z 2ln∆ + --- , final electrons. 1 1 Ð z 2 There exist different representations for the photon contribution to the structure function [18, 23, 24], but ∆ Ӷ 1.

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The iterative form (8) of Dγ does not include any effects by the double-photon mechanism [28]. Therefore, rep- caused by pair production. The corresponding nons- resentation (2), being slightly modiﬁed, can be used for inglet part of the structure due to real and virtual pair the calculation of radiative corrections to cross sections production can be included into the iterative form of of different processes with a longitudinally polarized Dγ(z, L) by replacing αL/2π on the right-hand side of electron beam. Eq. (8) with the effective electromagnetic coupling In our recent paper [20], we applied the electron structure function method to compute radiative correc- α αL eff 3 αL tions to the ratio of the recoil proton polarizations ------= Ð--- ln 1 Ð ------, (9) measured at CEBAF by Jefferson Lab Hall A Collab- 2π 2π 2 3π oration [3]. The aim of this high-precision experiment is the measurement of the proton electric form factor which (within the leading accuracy) is the integral of GE. In the present work, we use this method for calcu- the running electromagnetic constant. lation of the model-independent part of the radiative corrections to the asymmetry in the scattering of longi- The lower limits of integration with respect to z1 and tudinally polarized electrons on polarized protons at the z2 in the master equation (2) can be obtained from the level of per mill accuracy for elastic and deep inelastic condition for the existence of inelastic hadronic events, hadronic events. The cross section of the scattering of the longitudi- ()2 > 2 ˜ ˜ p1 + q˜ Mth, q˜ ==k1 Ð k2, Mth Mm+ π, (10) nally polarized electron by the proton with given longitudinal (||) or transverse (⊥) polarizations for both elastic and deep inelastic events can be written as a sum of where mπ is the pion mass. This constraint can be the spin-independent and spin-dependent parts, rewritten in terms of dimensionless variables as σ(),, σ(), σ||, ⊥(),, ≥ d k1 k2 S d k1 k2 d k1 k2 S z1z2 + y Ð 1 Ð xyz1 z2zth, ------= ------+ η------, (12) dQ2dy dQ2dy dQ2dy 2 2 2 (11) Q Mth Ð M x ==------(), zth ------, 2 p1 k1 Ð k2 V where S is the 4-vector of the target proton polarization and η is the product of the electron and proton polarization degrees. Hereafter, we assume η = 1. which leads to Master equation (2) describes the radiative corrections to the spin-independent part of the cross section 1 Ð y + xyz1 1 + zth Ð y on the right-hand side of Eq. (12), and the correspond- z2m ==------, z1m ------. z1 Ð zth 1 Ð xy ing equation for the spin-dependent part is given by

The squared matrix element of the considered pro- σ||, ⊥(),, 1 1 d k1 k2 S ()p (), cess in the one-photon exchange approximation is pro------= ∫ dz1 ∫ dz2 D z1 L dQ2dy portional to the contraction of the leptonic and hadronic z z tensors. Representation (2) reflects the properties of the 1m 2m (13) 2 ||, ⊥ ˜ ˜ leptonic tensor. Therefore, it has a universal nature 1 d σ ()k1,,k2 S × ----Dz(), L ------hard -, (because of the universality of the leptonic tensor) and 2 2 ˜ 2 can be applied to processes with different final hadronic z2 dQ dy˜ states. In particular, we can use the electron structure function method to compute radiative corrections to the where elastic and deep inelastic (inclusive and semi-inclusive) electronÐproton scattering cross sections. ()p γ e+eÐ e+eÐ()p D = D ++DN DS , On the other hand, straightforward calculations in the first order in α [5, 8, 21] and the recent calculations and [28] of the leptonic current tensor in the second order [25Ð 2 28] for the longitudinally polarized initial electron + Ð α () e e ()p 251Ð z () demonstrate that, in the leading approximation, the DS = ------L ------1+ + z lnz spin-dependent and spin-independent parts of this ten- 4π2 2 sor are the same for the nonsinglet channel contribu- (14) 2m tion. The latter corresponds to photon radiation and ×θ 1 Ð z Ð ------− ε e+e -pair production through the single-photon mechanism. The difference appears in the second order due to the possibility of pair production in the singlet channel describes the radiation of the initial polarized electron.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 406 AFANASEV et al. This representation is valid if radiation of collinear Radiative corrections to the spin-independent part particles does not change the polarizations S|| and S⊥. were calculated (within the electron structure function Such stabilized 4-vectors of the proton polarization can approach) in [19]. In the present work, we compute the be written as [8] radiative corrections to the spin-dependent parts for longitudinal and transverse polarizations of the target 2 proton and longitudinally polarized electron beam. For || 2M k1µ Ð Vp1µ Sµ = ------, completeness, we briefly recall the result for the unpo- MV larized case. []τ () (15) ⊥ up1µ + Vk2µ Ð 2u + V 1 Ð y k1µ Sµ = ------, 2 2 Ð uV ()1 Ð y Ð u M2 3. THE LEADING APPROXIMATION where Within the leading accuracy (with the terms of the order (αL)n taken into account), the electron structure 2 τ 2 function can be computed, in principle, in all orders of uQ==Ð , M /V. the perturbation theory. In this approximation, we have to take the Born cross section as a hard part on the right- It can be verified that, in the laboratory system, the hand sides of Eqs. (2) and (13). 4-vector S|| has the components (0, n), where the 3-vector n has the orientation of the initial electron We express the Born cross section in terms of lep- ⊥ || tonic and hadronic tensors as 3-momentum k1. It can also be verified that S S = 0 and that in the laboratory system, 2 2 dσ 4πα ()Q B ⊥ 2 ------= ------LµνHµν, (17) (), ⋅ 2 4 S ===0 n⊥ , n⊥ 1, nn⊥ 0, dQ dy VQ where the 3-vector n⊥ belongs to the plane (k , k ). 1 2 where α(Q2) is the running electromagnetic constant, If the longitudinal direction L is chosen along the which accounts for the effects of vacuum polarization, 3-momentum k1 Ð k2 in the laboratory system, which and coincides with the direction of the 3-vector q for nonra- diative process, and we choose the transverse direction F2 T in the plane (k1, k2), then we have the relations Hµν = Ð F1g˜ µν + ------p˜ 1µ p˜ 1ν p1q || ⊥ dσL dσ dσ ------= cosθ------+ sinθ------, ⑀ 2 2 2 M µνλρqλ () Sq dQ dy dQ dy dQ dy Ð i------g1 + g2 Sρ Ð,g2------p1ρ p1q p1q (18) σT σ|| σ⊥ 2 d θ d θ d B Q ε ------= Ðsin ------+ cos ------, Lµν = Ð ------gµν +++k1µk2ν k1νk2µ i µνλρqλk1ρ, dQ2dy dQ2dy dQ2dy 2

y + 2xyτ xyτ()1 Ð y Ð xyτ qµqν p1q θ ------θ g˜ µν ==gµν Ð ------, p˜ µ p µ Ð ------qµ. cos = , sin = Ð 2 ------2 , 2 1 1 2 y2 + 4xyτ y + 4xyτ q q and the master formula (13) for dσ|| and dσ⊥. In Eq. (18), we assume the proton and electron polarization degrees equal to 1. The spin-independent The asymmetry in elastic scattering and deep inelas- (F , F ) and spin-dependent (g , g ) proton structure tic scattering processes is deﬁned as the ratio 1 2 1 2 functions depend on the two variables ||, ⊥ ||, ⊥ dσ ()k ,,k S A = ------1 2 -, (16) 2 σ(), Ðq 2 ()2 d k1 k2 x' ==------, q px Ð p1 . 2 p1q and therefore, calculating the radiative corrections to the asymmetry requires knowing radiative corrections In the Born approximation, x' = x, but these variables to both spin-independent and spin-dependent parts of differ in the general case, when radiation of photons the cross section. and electronÐpositron pairs is allowed.

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Because the normalization is chosen, the elastic For this purpose, we can use the iterative form of the γ 2 2 photon structure function D with L L Ð 1 and limit (px = M ) can be obtained by simply substituting 2()∆ε 2 1 2 2 ∆∆ τ F ()x', q ---δ()1 Ð x' G ()q , 1 = ------+,z+ 1 2 M V()1 Ð xy 2 ()λ2 2 ()2 2 GE q + GM q ()∆ε F ()δx', q ()1 Ð x' ------, ()2 Ӷ 2 1 + λ z+ = y 1 Ð x , ------1 V  (), 2 1δ()()2 ()2 for D(z1, L) and g1 x' q --- 1 Ð x' GM q GE q --- 2  2()∆ε ∆∆= ------τ + z 2 () + λ 2 2 2  (19) V 1 Ð z+ + ------[]G ()q Ð G ()q G ()q , 1 + λ M E M  ∆ε for D(z2, L), where ( ) is the minimal energy of a hard 2 1 λ collinear photon in the special system (k Ð k + p = 0). g ()xq, Ð ---δ()1 Ð x' ------1 2 1 2 2 1 + λ Straightforward calculations yield the expression

2 2 2 σ()1 (), × []G ()q Ð G ()q G ()q , d k1 k2 α()L Ð 1 M E M ------= ------2 π 2 dQ dy 2 λ q = Ð------2 4M () dσ B ()k , k 4()∆ε 2()z + τ × ------1 2 32+ ln------+ in the hadronic tensor, where GM and GE are the mag- 2 ()()  V 1 Ð z+ 1 Ð xy netic and electric proton form factors. dQ dy z Ð ρ (23) A simple calculation gives the spin-independent and + 2 ()B 1 + z dσ ()z k , k spin-dependent parts of the well-known Born cross sec- + dz ------1 ------1 1 2- ∫ ()1 Ð xy ()1 Ð z 2 tion in the form 1 dQt dyt zth dσB 4πα2()Q2 ------= ------2 ()B 2 4 1 + z dσ ()k , k /z  dQ dy Q y (20) + ------2 ------1 2 2- , ()1 Ð z ()1 Ð z 2 × []()τ (), 2 2 (), 2 + 2 dQs dys  1 Ð y Ð xy F2 xQ + xy F1 xQ ,

B 2 2 where dσ|| 8πα ()Q ------2 - = 2 dQ dy V y 2 2 (21) Mx Ð M 1 Ð y + z z ==------, z1 ------, 2 Ð y 2 2τ 2 ×τÐ ------g ()xQ, +,----- g ()xQ, V 1 Ð xy 2xy 1 y 2 ()∆ε 1 Ð z+ ρ 2 τ B 2 2 2 z2 ==------, ------+,z+ dσ⊥ 8πα ()Q M 1 Ð z V ------= Ð------()1 Ð y Ð xyτ dQ2dy V 2 y Q2 (22) 2 2 2 2 2 2 Q 2 2 2 Q ==Ðq z Q , Q ==Ðq ------, × g ()xQ, +.---g ()xQ, t t 1 s s z 1 y 2 2

Thus, within the leading accuracy, the radiatively cor- 1 Ð y yts, = 1 Ð ------. rected cross section of process (1) is defined by Eq. (2) z12, (for its spin-independent part) with (20) as the hard part of the cross section and by Eq. (13) (for its spin-depen- Similar equations can be derived for the first-order dent part) with (21) or (22) as the hard part. correction to the spin-dependent part of the cross sec- It is useful to extract the first-order correction to the tion for both longitudinal and transverse polarizations Born approximation, as defined by master equation (2). of the target proton.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 408 AFANASEV et al. 4. DEEP INELASTIC SCATTERING we use the corresponding leptonic tensor in the form CROSS SECTION BEYOND THE LEADING ACCURACY 3 γ α Hun() H d k Lµν = ------()Lµν + Lµν ------, To go beyond the leading accuracy, we have to 4π2 ω improve the expressions for hard parts of the cross sections in master equations (2) and (13) in order to H include effects caused by radiation of a hard noncol- Lµν = 2iεµνλρqλ()k ρ R + k ρ R , linear photon. In principle, we can also improve the 1 t 2 s expression for the D function in order to take collinear ut+ 21 1 (27) Rt = ------2Ð m ---- + --- , next-to-leading effects in the second order of perturba- st s2 t2 tion theory into account. The essential part of these effects is included in our D functions due to the replacement L L Ð 1. The rest can be written using the us+ 2 st Ðu() u+ VyÐ Vz Rs ==,------Ð 2 m ------, st ------results of the corresponding calculations for the double st ut2 uV+ photon emission [27, 30], pair production [28, 31, 32], one-loop corrected Compton tensor [25, 26, 33], and vir- Hun() tual correction [34]. But we restrict ourselves here to the where ω is the energy of the radiated photon; Lµν is D functions given above in Eqs. (5), (6), (7), and (14). the leptonic tensor for unpolarized particles (see [33]); To compute the improved hard cross section, we and we use the notation must find the full first-order radiative corrections to the cross section of process (1) and subtract from it (to 2 avoid double counting) its leading part defined by s ==2kk2, t Ð2kk1, q =ust++ Eq. (23) (for the unpolarized case). Therefore, the improved hard part can be written as for kinematic invariants. The result for the unpolarized case was derived in [19], and here we rewrite it using () () dσ dσB dσ SV+ dσH dσ 1 standard notation as ------hard- = ------++------Ð ------, (24) dQ2dy dQ2dy dQ2dy dQ2dy dQ2dy σ σB α α d hard d δ ------= ------1 + ------+ ------where dσ(S + V) is the correction to the cross section of dQ2dy dQ2dy 2π VQ2 process (1) due to virtual and soft photon emission and z+ dσH is the cross section of the radiative process  × 1 Ð r1 ˆ 1 Ð r2 ˆ ∫ dz------Pt N Ð ------Ps N 1 Ð xy 1 Ð z+ Ð() () Ð()γ() () zth e k1 +++Pp1 e k2 k Xpx . (25)

r+ The virtual and soft corrections are factored in a sim- 2W ilar way for both polarized and unpolarized cases [19] + ∫ dr------(28) and can be written as y2 + 4xyτ rÐ

B ()SV+ B dσ dσ dσ α r+ ------+ ------= ------1 + ------ˆ ()2 2 2 2 2π dr 1 Ð Pt 1 + r N () dQ dy dQ dy dQ dy + P∫ ------------+ r1 Ð r T t 1 Ð r rrÐ 1 1 Ð xy rÐ 2 ρ 2  2 2 ×δ+ ()L Ð 1 32+ ln------, 1 Ð Pˆ s ()1 + r N α ()rQ ()() () 1 Ð xy 1 Ð z+ Ð ------------+ r2 Ð r T s ------2 -, (26) rrÐ 2 1 Ð z+  r π2 τ δ 1 Ð y Ð xy 2 1 Ð xy = Ð 1 Ð2----- Ð f ()------()- Ð ln ------, 3 1 Ð xy 1 Ð z+ 1 Ð z+ where r = Ðq2/Q2 and the limits of integration with respect to r are x dt fx()= ---- ln()1 Ð t . ∫ t () 1 r± z = ------()τ - 0 2xy + z+ × []()τ ()()± 2 τ To calculate the cross section of radiative process (25), 2xy + z + z+ Ð z yy+ 4xy .

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Here, we used the notation r+ dr 1 Ð Pˆ s + P∫ ------()- 2x' 1 Ð y 1 Ð r rrÐ12 Ð z+ N = 2F ()x', r + ------Ð τ F ()x', r , r 1 rxyxy 2 Ð (30) () 2 ||, ⊥ 2 r Ð r ||, ⊥ 1 Ð Pˆ 2x'τ × ()1 + r N + ------2 T Ð ------t W = 2F ()x', r Ð ------F ()x', r , s r s rrÐ 1 rxy 2 2 1 2 ||, ⊥ ()1 + r N ||, ⊥ []() × t () 2x'1Ð r 1 Ð y (), ------+ 2 rr1 Ð r T t T t = Ð------F2 x' r , (29) 1 Ð xy x2 y2r r  + ||, ⊥ () 2W x'α2()Q2r 2x'1Ð y Ð r (), + dr------------, T s = Ð------F2 x' r , ∫ 3 x2 y2r y2 + 4xyτ  r rÐ  1 xyr r1 ===z1, r2 ----, x' ------. where z2 xyr+ z

2 || ⊥ M Ð1 The action of the operators Pˆ t and Pˆ s is defined as U ==1, U ------()1 Ð y Ð xyτ , Q2 ˆ (), (), ˆ (), (), Pt frx' ==fr1 xt , Ps frx' fr2 xs , || ⊥ W ==4yτW, W 2y2()12+ xτ W, xyr1 xyr2 () xt ==------, xs ------. W = 1 + r xg1 + x'g2, xyr1 + z xyr2 + z || []()τ τ The hard cross section (29) has neither collinear nor Nt = 22rzÐ Ð xy r + 2 g1 Ð 8x' g2, infrared singularities. The different terms on the right- || N = 22[]Ð z Ð xyr()12+ τ g Ð 8x'τg , hand side of Eq. (29) have singularities at r = r1, r = r2, s 1 2 and r = 1. Singularities at the first two points are col- ⊥ []()τ () linear, and the one at the third point, nonphysical, Nt = 21Ð y Ð zrxyr+ Ð + 2 xyg1 + 2x'g2 , arising at integration. Collinear singularities vanish ⊥ 1 Ð z because of the action of the operators Pˆ and Pˆ on the N = 21Ð y + ------Ð xy()12+ τ ()xyrg + 2x'g , t s s r 1 2 terms containing N. The nonphysical singularity cancels because, in the limiting case r 1, we have || τ || ()()τ T t ==2rg1 Ð24x' g2, T s z Ð 1 g1 Ð 2x' g2 , r Ð r r Ð r ------2 == 1 , ------1 Ð0.1, T + T = ⊥ ()τ t s T t = 2xyrg1 + 2x'1Ð y + r Ð 2xy g2, r2 Ð r r1 Ð r ⊥ ()[]()τ To derive the hard cross section for the polarized T s = 2 z Ð 1 xyg1 + x'1Ð1/y + r Ð 2xy g2 . case, we have to use the analogue of Eq. (24) for dσ|| The polarized hard cross section defined by Eq. (30) σ⊥ σV + S σ(1) and d . Taking into account that d and d are is also free from collinear singularities due to the action the same in the polarized and unpolarized cases and of the operators 1 Ð Pˆ and 1 Ð Pˆ . The nonphysical sin- using expression (27) for the antisymmetric part of the t s gularity at r = 1 on the right-hand side of Eq. (30) can- σH leptonic tensor to compute d in the polarized case, cels because we arrive at ||, ⊥ 1 ||, ⊥ ||, ⊥ B T t = ------T s dσ dσ||, ⊥ α α ||, ⊥ z Ð 1 ------hard- = ------1 + ------δ + ------U 2 2 2π 4 dQ dy dQ dy Q in this limit. We note that radiation of a photon at large angles by the initial and final electrons increases the z+   range of r in (28) and (30), because r1 < r < r2 for col- × 1 Ð r1 ˆ ||, ⊥ 1 Ð r2 ˆ ||, ⊥ ∫ dz------Pt Nt + ------Ps Ns ∫ linear radiation, and now rÐ < r1 and r+ > r2. This may 1 Ð xy 1 Ð z+ be important if the hadron structure functions are large z th  in these additional regions.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 410 AFANASEV et al. 5. HARD CROSS SECTION Eq. (32) is defined as FOR ELASTIC HADRONIC EVENTS dσB 4πα2()Q2 ------To describe the hard cross section for elastic had------2 - = 2 ronic events, we use the replacement defined by (19) in dQ dy V Eqs. (28) and (30). We refer to Eqs. (21)Ð(23) for the (33) 2 λ 2 2 1 2 GE + GM Q Born cross sections that enter these equations. The × --- []()τ ------δ δ GM + 1 Ð y 1 + 2 y Ð ------. function (1 Ð x') is used to integrate with respect to the 2 y ()1 + λ V inelasticity z, In writing this last equation, we took into account that ∫dzδ()1 Ð x' = xyr. (31) Q2 δ()1 Ð x = yδ y Ð ------. V The final result for the unpolarized case is given by (we do not introduce special notation for the elastic cross The spin-dependent hard cross section for elastic section) hadronic events can be written in a form very similar to (32), dσ dσB α α ------hard- = ------1 + ------δ + ------σ||, ⊥ σB α α 2 2 2π 2 d hard d ||, ⊥ δ ||, ⊥ dQ dy dQ dy V ------= ------1 + ------+ ------U dQ2dy dQ2dy 2π V 2

  1 Ð r ||, ⊥ 1 Ð r ||, ⊥ r+ × 1 ˆ 2 ˆ  ------Pt Nt + ------Ps Ns 1 Ð r1 1 Ð r2 2W × ˆ ˆ 1 Ð xy 1 Ð z+ ------Pt N Ð ------Ps N + ∫ dr------1 Ð xy 1 Ð z 2  + y + 4xyτ  rÐ (32) r+ r+ ||, ⊥ ˆ r W dr 1 Ð Pt + 2 + ∫ dr------+ P∫ ------Ð------dr 1 Ð Pˆ 1 + r 2 1 Ð r rrÐ + P ------t ------Nr+ ()Ð r T y + 4xyτ 1 ∫ 1 t rÐ rÐ (34) 1 Ð r rrÐ 1 1 Ð xy 2 rÐ ˆ × 1 + r ||, ⊥ ()||, ⊥ 1 Ð Ps ------Nt + 2rr1 Ð r T t + ------()- 1 Ð xy rrÐ12 Ð z+  2  2 2 1 Ð Pˆ s 1 + r α ()Q r 2 2 () ||, ⊥ 2()r Ð r ||, ⊥  α () Ð ------------Nr+ 2 Ð r T s ------, × ()2 2 Q r rrÐ 1 Ð z r 1 + r Ns + ------T s ------, 2 +  r 2 2 2  2 ()4M + Q r r where where || ⊥ W ==4yτW, W 2y2()12+ xτ W, 2 λ 2 2 2 1 Ð y GE + GM 2 4τ NG= + ------Ð τ ------, Wrx= []()1 + r Ð 1 G + r + ----- ()1 + r G G , M xyrxy 1 + λ M y M E

|| ()τ ()2 τ 1 τ 2 λ 2 Nt = r 2 + r 2 Ð xy GM + 8 r-----Ð 1 Ð GMGE, 2 2τ GE + GM xy WG==------, M xyr 1 + λ || 2 1 N = r()2τ + 1 ()2 Ð xyr G + 8τ ----- Ð r()1 + τ G G , s M xy M E 2 λ 2 2 GE + GM T = Ð------[]1 Ð r()1 Ð y ------, ⊥ t 2 2 λ []()τ x y r 1 + Nt = 1 Ð y + rxyrÐ + 2 × []()2 ()τ Ð2r 2 Ð xy GM + r + 2 GMGE , 2 λ 2 2 GE + GM T = Ð------()1 Ð r Ð y ------. ⊥ 1 s 2 2 1 + λ N = 1 Ð y + --- Ð xy()12+ τ x y r s r × []()2 ()τ The Born cross section on the right-hand side of Ð2r 2 Ð xyr GM + r 12+ GMGE ,

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|| 2 2 6. NUMERICAL ESTIMATIONS T = rr()+ 2τ G +,2τ -----Ð 1 G G t M xy M E The formulas obtained in the last section include some operators that emphasize the physical meaning of || 2 2 the transformations performed, but they are not conve- T = Ð1r()+ 2τ G Ð 2τ ----- Ð r G G , s M xy M E nient in numerical analysis. Here, we present a uniﬁed version of the formulas without any operators. For ⊥ { []() τ 2 example, the symbol P is explicitly treated as T t = rrÐ 1 Ð xy + 1 Ð2y Ð xy GM []τ τ } r+ +1Ð2y Ð xy ++r 4 GMGE , dr P ------Fr() ∫ 1 Ð r ⊥ 1 2 T = r --- Ð1xy()12+ τ + Ð y G rÐ s r M r+ []τ ()() dr ()() () () 1 Ð rÐ Ð2Ð12 Ð xyr ++r 1 Ð y GMGE. = ∫ ------Fr Ð F 1 + F 1 ln------. 1 Ð r r+ Ð 1 We note that the argument of the electromagnetic form rÐ 2 factors in Eqs. (32) and (34) is ÐQ r. Therefore, all cross sections given by Eqs. (28), The Born cross sections on the right-hand side of (30), (32), and (34) can be written by means of the uni- Eq. (34) are given by ﬁed formula

B 2 2 i B dσ|| πα () σ σ 4 Q ττ 1 d hard d i α ------= ------4 1 + Ð --- GMGE ------= ------1 + ------δ + αU dQ2dy V()4M2 + Q2 y 2 2 2π i (35) dQ dy dQ dy 2 ()τ y 2 δQ z+ Ð1+ 2 1 Ð --- GM y Ð ------,  2 V × dzL{}i N ()r + Li N ()r ----- ∫  1 i 1 2 i 2 for the longitudinal polarization of the target proton zth and by r+  1 (37) + drW ++T ------N ()r Ð N ()r --- σB πα2()2 2 ∫ i i i 1 i 2 d ⊥ 8 Q M []()τ  1 Ð r ------= ------1 Ð y 1 + rÐ dQ2dy V()4M2 + Q2 Q2 (36) 1 Ð r1 []() () 2 + ------Ni r Ð Ni r1 y 2 Q × 1 Ð --- G Ð ()12+ τ G G δ y Ð ------, rrÐ 1 2 M M E V  1 Ð r2 for the transverse one. The argument of the form factors + ------[]N ()r Ð N ()r  , rrÐ i i 2  in (35) and (36) is ÐQ2. 2  The results in this section can be generalized to elas- where tic electronÐdeuteron scattering in both polarized and unpolarized cases in a very simple way, because the 2 ()1 Ð r , 1 Ð r d i +− 12 +− Ð respective deuteron tensors Hµν are connected with the L12, = bi------2 - ln------, r+ Ð 1 p 1 + r12, proton ones Hµν by the relations bu ==Ð1.1, blt, dun() 4τ + xyr pun() Hµν = ------Hµν 4τ The index i runs over all polarization states (i = u, l, t). The functions N (r) and T are given by 2 2 2 i i 2 2 ()d 2 2 2 8x y r 2 × G ---G , G G + ------G , M 3 M E C ()τ 2 Q 2 α2 94 () 1 + r x' Ni r = ------Ni------3 -, z+ Ð z d()||, ⊥ 4τ + xyr p()||, ⊥ r Hµν = Ð------Hµν 8τ  2 T i1 x'α ()d xyr ±------, rr< , rr> × G G , G 2G + ------G , 3 1 2 M M E C 6τ Q  1 Ð r r T =  i 2 ()  α where G d , G , and G are the magnetic, charged, and x' << M C Q T i2------3 -, r1 rr2. quadrupole deuteron form factors, respectively.  r

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Unpolarized Longitudinal Transverse x' 1.1 + ()y Ð 2z + yr()12Ð x --- g r 2

1.0  -+ x'y()12+ xτ ()1 + r g , δ δ 2 t t  δ 0.9 t and δ i δ ||, ⊥ i 2W α2 2W xα2 Wu ==------, Wlt, ------, 0.8 δ 2 2 2 3 i y + 4xyτ r y + 4xyτ r

||, ⊥ 1 U˜ 0.7 Uu ==------Ult, ------. 0.1 0.2 0.5 0.1 0.2 0.5 0.1 0.2 0.5 y VQ2 Q4

Fig. 1. Radiative correction to unpolarized and polarized The same formulas can be used in the elastic case. (both longitudinal and transverse) parts of the cross section Only Eqs. (19) and (31) are needed here. In the elastic for kinematics close to JLab experiments, V = 10 GeV2, x = 0.5. case, we must therefore substitute ∫dz xyr, The pole at r = 1 can be reached only in the region r1 < r < r2, and hence there is no singularity in the terms involving T . For T , this pole cancels explicitly: set x' = 1 and z = 0, and replace the proton structure i1 i2 functions in accordance with (19). () It is believed that the formulas obtained within the 22Ð y F2 () τ T u2 ==------2 2 -, T l2 Ð841+ r g1 + x' g2, presented formalism are not convenient for numerical x y analysis. There are two reasons for such an opinion. First, the electron structure function in form (5), (6) has 1 a very sharp peak as z tends to unity. Second, because T = Ð441()+ r xyg Ð x' r ++---2Ð2y Ð xyτ g . l2 1 r 2 absolute values appear in denominators, the integrand cannot be a continuous function of the integration vari- In the unpolarized case, N = rN/x' with N from (28). In ables. This produces obstacles for numerical analysis if u it is carried out in the traditional style based on adaptive other cases, they are methods of numerical integration, which is used in such programs as TERAD/HECTOR [36] or POLRAD [9]. y()12+ xτ []1 Ð z + r()1 Ð xy N = 21Ð Ð r + ------g But it is possible to perform numerical analysis if l 2 Ð y 1 Monte Carlo integration is used instead of adaptive τ integration and the regions with sharp peaks are +8x' g2, extracted into separate integration subregions. Based on these ideas, we developed the Fortran code 41[]Ð z + r()1 Ð xy 1 N = Ð------[xyr()1 Ð y Ð xyτ g ESFRAD, which allows one to perform the numerical t r()2 Ð y 1 analysis without any serious difﬁculties. ()()] ()τ + x'1Ð y ++zr1 Ð y + xy g2 + 4x'y 12+ x g2, We considered two radiative processes. In the ﬁrst case, the continuum of hadrons is produced, and in the 21()+ r F second case, the proton remains in the ground state. T = Ð------2, Both of the effects considered contribute to the experi- u1 2 2 x y mentally observed cross section of deep inelastic scattering. They are usually called the radiative tails from y()1 + r2 ()12+ xτ the continuous spectrum and the elastic peak, or simply T = 4------g + 8x'1()τ+ r g , l1 2 Ð y 1 2 the inelastic and elastic radiative tails. Below, we study the contributions of the tails numerically within kine-

1 1 + r2 Electron Structure Function method for RADiative corrections. T = 4------Ð2xy()1 Ð y Ð xyτ g 2 σ t1 2 Ð y 1 Here and below, we mean double differential cross section =  dσ/dydQ2.

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Unpolarized Longitudinal Transverse correction factor for all polarization states (unpolar- 3.0 ized, longitudinal, or transverse) δ t δ obs t δt σ δ , = ------. (38) 2.5 it σB

The observed double differential cross section is given 2.0 by master formulas (2) and (13), and the Born cross section is calculated via (20), (21), and (22). Both elas- σ tic and inelastic contributions must be taken for hard. In this case, we obtain the total radiative corrections factor 1.5 δ ( t). The subscripts i and t correspond to the cases where the elastic radiative tail is included in the total δ correction ( t) or the inelastic radiative tail contributes δ δ δ δ 1.0 i i i only ( i). The elastic radiative tail may optionally not be included, because there sometimes exist experimental 0.5 1.00.5 1.0 0.5 1.0 methods to separate this contribution. We note that, for y the HERA kinematics, we do not include it because it is usually separated experimentally. Also, we can extract Fig. 2. Radiative correction to unpolarized and polarized a one-loop contribution in order to study the effect of (both longitudinal and transverse) parts of the cross section higher-order corrections. The observed cross section in for kinematics close to HERMES experiments, V = this case is given by the sum of the cross sections in 2 δ δ 50 GeV , x = 0.1, t = Ð t. Eqs. (23) and (37). We note that this can provide an additional cross check by comparison with POLRAD. We use rather simple models for spin-averaged and matical conditions of the current experiments on deep spin-dependent structure functions. It allows us not to inelastic scattering. mix the pure radiative effects, which are of interest, We take three typical values of V equal to 10, 50, and with the effects due to hadron structure functions. Spe- 105 GeV2. They correspond to JLab, HERMES, and ciﬁcally, we use the so-called D8 model for the spin- HERA measurements. Figures 1Ð3 give the radiative average structure function [35] (see also discussion

Unpolarized Longitudinal Transverse 1.8

1.6

1.4

1.2

1.0

0.8 0.5 1.0 0.5 1.0 0.5 1.0 y

Fig. 3. One-loop and total radiative corrections (dashed and solid lines) for collider kinematics (HERA); V = 105 GeV2. Lines from top to bottom correspond to different values of x = 0.001, 0.01, and 0.1.

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A1, % AT, % 68 2 x = 0.5, V = 10 GeV2 x = 0.5, V = 10 GeV2 0

64 –2

–4 60 –6

–8 56 –10 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 y y A1, % AT, % 4.0 0.15

3.6 0.10

3.2 0.05 0 2.8 –0.05 2.4 2 –0.10 x = 0.01, V = 50 GeV x = 0.01, V = 50 GeV2 2.0 0.2 0.4 0.6 0.8 1.0 –0.15 0.2 0.4 0.6 0.8 1.0 y y Fig. 4. Radiative correction to asymmetries for the HERMES (lower plots) and JLab (upper plots) kinematics. The dotted line shows the Born asymmetry. The full and dashed lines correspond to the total and one-loop contributions. Asymmetries with the elastic contribution taken into account are marked by dots at the end.

0.725 in [9]), and A1(x) = x , suggested in [37]; we set kinematical depolarization factor depending on the g2 = 0 (the definition of A1(x) is given below). ratio R of the longitudinal and transverse photoabsorp- From these plots, we can see that the total radiative tion cross sections, correction is basically determined by the one-loop correction with some important effect around kinematical 2 y()2 Ð y ()1 + γ y/2 boundaries. The sign and value of the higher-order D = ------, effects are in agreement with the leading log estima- y2()1 + γ 2 + 21()Ð y Ð γ 2 y2/4 ()1 + R tions and calculations of the correction to the elastic radiative tail in [38, 39]. Two regions require special σ MQ()2 + ν2 F consideration: the region of higher y for the HERMES R ==-----L------2 Ð 1 , σ 2 and JLab kinematics and the region near the pion T Q ν F1 threshold at JLab. We define the polarization asymmetries in the stan- ν γ2 2 ν2 dard way, where = yV/2M and = Q / . For fixed x, A1 is a constant within our model, and it is therefore very conve- σ|| σ⊥ nient for graphical presentation and analysis of differ- A ==-----, A ------. (39) L σ T σ ent radiative effects. Figure 4 gives the asymmetries A1 and AT for the kinematics of HERMES and JLab up to We can also define the spin asymmetry A1 as AL = DA1 y = 0.95. The influence of higher-order and elastic radi- (for the chosen model where g2 = 0), where D is the ative effects can be seen. Figure 5 gives the total correc-

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AL, % A , % radiation of one noncollinear photon, which enlarges T the range of the hadron structure function arguments. It 28 8 may be important that these functions are sufficiently sharp. In this case, the loss in the radiation probability 24 4 (the loss of L) can be compensated by the increase in the value of the hard cross section. 20 0 We note that we extracted the explicit formulas for 16 –4 the first-order contribution at both leading and next-to- 0.9 1.0 1.1 1.2 1.3 0.9 1.0 1.1 1.2 1.3 leading order levels. We found analytic agreement between these results for the one-loop correction with , GeV W W, GeV the previous results in [8], which provides the most σ 2 d /dydQ , nb important test of the total correction. 103 On the basis of the analytic results, we constructed the Fortran code ESFRAD.3 Because of several known reasons discussed in Section 6, results obtained by the 102 electron structure function method are usually not very convenient for precise numerical analysis. But we believe that our numerical procedure based on Monte Carlo integration allows us to overcome the obstacles. 0.95 1.00 1.05 1.10 1.15 1.20 1.25 Using the ESFRAD code, we performed numerical W, GeV analysis for kinematical conditions of the current and Fig. 5. The cross section (lower plot) and polarization future polarization experiments. We found two kine- asymmetries (both longitudinal and transverse) for the JLab matical regions where the higher-order radiative cor- kinematics (Q2 = 1 GeV2) near the pion threshold. The dot- rection can be important. These are the traditional ted line shows the Born cross section and asymmetry. Full region of high y and the region around the pion thresh- and dashed lines correspond to the total and one-loop con- old. We gave a detailed analysis of the effects within tributions. these regions and presented numerical results within one of the simplest possibilities for modeling the deep inelastic scattering structure functions. Model depen- tions to the cross sections and asymmetries for the dence of the result is certainly an important issue threshold region of JLab. requiring separate investigation for specific applications within the experimental data analysis. 7. CONCLUSIONS We have considered model-independent QED radi- ACKNOWLEDGMENTS ative correction to the polarized deep inelastic and elas- We thank our colleagues at the Jefferson Lab for use- tic electronÐproton scattering. Together with the ana- ful discussions. We thank the US Department of Energy lytic expression for the radiative corrections, we give its for support under contract no. DE-AC05-84ER40150. numerical values for different experimental situations. The work of N.M. was in addition supported by Rutgers Our analytic calculations are based on the electron University through the NSF grant no. PHY-9803860 structure function method, which allows us to write and by Ukrainian DFFD. A.A. acknowledges additional both the spin-independent and spin-dependent parts of support through the NSF grant no. PHY-0098642. the cross section with the radiative corrections to the leptonic part of interaction taken into account in the form of the well-known DrellÐYan representation. The REFERENCES corresponding radiative corrections explicitly include 1. B. W. Filippone and X. Ji, hep-ph/0101224. the first-order correction as well as the leading-log con- 2. M. Anselmino, A. Efremov, and E. Leader, Phys. Rep. tribution in all orders of the perturbation theory and the 261, 1 (1995); Erratum: Phys. Rep. 281, 399 (1997). main part of the second-order next-to-leading-log con- 3. M. K. Jones et al. (Jefferson Lab Hall A Collaboration), tribution. Moreover, any model-dependent radiative Phys. Rev. Lett. 84, 1398 (2000). correction to the hadronic part of the interaction can be 4. L. D. van Buuren (97-01 Collaboration), Nucl. Phys. A included in our analytic result by inserting it as an addi- 684, 324 (2001). tive part of the hard cross section in the integrand sign 5. T. V. Kukhto and N. M. Shumeiko, Nucl. Phys. B 219, in master formulas (2) and (13). 412 (1983). To derive the radiative corrections, we take into 6. D. Y. Bardin and N. M. Shumeiko, Nucl. Phys. B 127, account the radiation of photons and e+eÐ pairs in col- 242 (1977). linear kinematics, which produces a large logarithm L in the radiation probability (in D functions), and the 3 Fortran code ESFRAD is available at http://www.jlab.org/~aku/RC.

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7. A. A. Akhundov, D. Y. Bardin, L. Kalinovskaya, and 24. M. Skrzypek, Acta Phys. Pol. B 23, 135 (1992). T. Riemann, Fortschr. Phys. 44, 373 (1996). 25. I. Akushevich, A. Arbuzov, and E. Kuraev, Phys. Lett. B 8. I. V. Akushevich and N. M. Shumeiko, J. Phys. G 20, 513 432, 222 (1998). (1994). 26. G. I. Gakh, M. I. Konchatnij, and N. P. Merenkov, Pis’ma 9. I. Akushevich, A. Ilichev, N. Shumeiko, et al., Comput. Zh. Éksp. Teor. Fiz. 71, 328 (2000) [JETP Lett. 71, 227 Phys. Commun. 104, 201 (1997). (2000)]. 10. I. Akushevich, H. Bottcher, and D. Ryckbosch, hep- 27. M. I. Konchatnij and N. P. Merenkov, Pis’ma Zh. Éksp. ph/9906408. Teor. Fiz. 69, 845 (1999) [JETP Lett. 69, 893 (1999)]. 11. D. Y. Bardin, J. Bluemlein, P. Christova, and L. Kali- 28. M. I. Konchatnij, N. P. Merenkov, and O. N. Shekhov- novskaya, Nucl. Phys. B 506, 295 (1997). zova, Zh. Éksp. Teor. Fiz. 118, 5 (2000) [JETP 91, 1 12. I. V. Akushevich, A. N. Ilyichev, and N. M. Shumeiko, (2000)]. J. Phys. G 24, 1995 (1998). 29. A. V. Afanasev, I. V. Akushevich, G. I. Gakh, and 13. A. Afanasev, I. Akushevich, and N. Merenkov, Phys. N. P. Merenkov, Zh. Éksp. Teor. Fiz. 120, 515 (2001) Rev. D 64, 113009 (2001). [JETP 93, 449 (2001)]. 14. A. V. Afanasev, I. Akushevich, A. Ilyichev, and 30. N. P. Merenkov, Yad. Fiz. 48, 1782 (1988) [Sov. J. Nucl. N. P. Merenkov, Phys. Lett. B 514, 269 (2001). Phys. 48, 1073 (1988)]. 15. I. V. Akushevich, A. N. Ilichev, and N. M. Shumeiko, 31. N. P. Merenkov, Yad. Fiz. 50, 750 (1989) [Sov. J. Nucl. Phys. At. Nucl. 61, 2154 (1998). Phys. 50, 469 (1989)]. 16. I. Akushevich, E. Kuraev, and B. Shaikhatdenov, Phys. 32. A. B. Arbuzov, E. A. Kuraev, N. P. Merenkov, and Part. Nucl. 32, 257 (2001). L. Trentadue, Zh. Éksp. Teor. Fiz. 108, 1164 (1995) 17. S. D. Drell and T. Yan, Phys. Rev. Lett. 25, 316 (1970); [JETP 81, 638 (1995)]. Erratum: Phys. Rev. Lett. 25, 902 (1970). 33. E. A. Kuraev, N. P. Merenkov, and V. S. Fadin, Yad. Fiz. 18. E. A. Kuraev and V. S. Fadin, Yad. Fiz. 41, 733 (1985) 45, 782 (1987) [Sov. J. Nucl. Phys. 45, 486 (1987)]. [Sov. J. Nucl. Phys. 41, 466 (1985)]. 34. R. Barbieri, J. A. Mignaco, and E. Remiddi, Nuovo 19. E. A. Kuraev, N. P. Merenkov, and V. S. Fadin, Yad. Fiz. Cimento A 11, 824 (1972). 47, 1593 (1988) [Sov. J. Nucl. Phys. 47, 1009 (1988)]. 35. P. Amaudruz et al. (New Muon Collaboration), Nucl. 20. A. Afanasev, I. Akushevich, and N. Merenkov, hep- Phys. B 371, 3 (1992). ph/0009273; Phys. Rev. D 65, 013006 (2002). 21. V. N. Baier, V. S. Fadin, and V. A. Khoze, Nucl. Phys. B 36. A. Arbuzov, D. Y. Bardin, J. Bluemlein, et al., Comput. 65, 381 (1973). Phys. Commun. 94, 128 (1996). 22. V. N. Gribov and L. N. Lipatov, Yad. Fiz. 15, 781 (1972) 37. A. P. Nagaitsev, V. G. Krivokhijine, I. A. Savin, and [Sov. J. Nucl. Phys. 15, 438 (1972)]; L. N. Lipatov, Yad. G. I. Smirnov, JINR Rapid Commun. (Dubna), Fiz. 20, 181 (1974) [Sov. J. Nucl. Phys. 20, 94 (1975)]; No. 3[71]-95, 59 (1995). Y. L. Dokshitzer, Zh. Éksp. Teor. Fiz. 73, 1216 (1977) 38. A. A. Akhundov, D. Y. Bardin, and N. M. Shumeiko, Yad. [Sov. Phys. JETP 46, 641 (1977)]; G. Altarelli and Fiz. 44, 1517 (1986) [Sov. J. Nucl. Phys. 44, 988 G. Parisi, Nucl. Phys. B 126, 298 (1977). (1986)]. 23. S. Jadach, M. Skrzypek, and B. F. Ward, Phys. Rev. D 47, 39. I. Akushevich, E. A. Kuraev, and B. G. Shaikhatdenov, 3733 (1993). Phys. Rev. D 62, 053016 (2000).

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 417Ð426. From Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 476Ð485. Original English Text Copyright © 2004 by Kerbikov, Kudryavtsev, Lensky. NUCLEI, PARTICLES, AND THEIR INTERACTION

NeutronÐAntineutron Oscillations in a Trap Revisited¦ B. O. Kerbikov, A. E. Kudryavtsev, and V. A. Lensky State Research Center Institute of Theoretical and Experimental Physics, Moscow, 117218 Russia e-mail: [email protected], [email protected], [email protected] Received September 9, 2003

Abstract—We reexamine the problem of nÐn oscillations for ultracold neutrons conﬁned within a trap. We show that, for up to 103 collisions with the walls, the process can be described in terms of wave packets. The n component grows linearly with time, with the enhancement factor depending on the reﬂection properties of the walls. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION tance between adjacent levels (see below). The true rea- For quite a long time, physics beyond the Standard son due to which the above treatment is of little physi- Model has continued to be an intriguing subject. Sev- cal relevance is as follows. The spectrum of the neu- eral reactions that may serve as signatures for the new trons provided to the trap by the source is continuous physics have been discussed. One of the most elegant and a certain time is needed for rearrangement of the initial wave function into standing waves correspond- proposals is to look for nÐn oscillations [1] (see ing to the trap eigenstates. As is shown below, this time also [2]). There are three possible experimental settings interval appears to be of the order of the β-decay time, aimed at observation of this process. The first is to and, therefore, the standing wave regime, being inter- establish a limit on nuclear instability because n pro- esting by itself, can hardly be reached in the real phys- duced inside a nucleus will blow it up. The second is to ical situation. use a neutron beam from a reactor. This beam propa- The second approach [3, 6, 7] treats the neutrons gates a long distance to the target in which the possible and antineutrons inside a trap as freely moving particles n component would annihilate and, thus, is detected. that undergo reflections from the trap walls. Collisions The third option, which we discuss in the present paper, with the walls result in a reduction of the n component is to use ultracold neutrons (UCNs) confined in a trap. compared to the case of the free-space evolution. This The main question is to what extent generation of the n suppression is due to two factors. The first is the anni- component is reduced by the interaction with the trap hilation inside the walls. The second is the phase deco- walls. This subject was addressed by several authors herence of the n and n components induced by the dif- [3Ð8]. In our opinion, a thorough investigation of the problem is still lacking. ference of the wall potentials acting on n and n . Reflections of antineutrons from the trap walls were, First of all, a clear formulation of the problem of for the first time, considered in [3]. The purpose of that n−n oscillations in a cavity has been hitherto missing. paper was to investigate the principal possibility of Two different approaches were used without presenting observing nÐn oscillations in a trap, and the authors sound arguments in favor of their applicability and estimated the reflection coefficient for antineutrons without tracing connections between them. without paying attention to the decoherence phenom- In the first approach [4, 5], nÐn oscillations are con- ena. Only a single collision with the trap wall was considered in the basis of the discrete eigenstates of the sidered in [3]. A comprehensive study of nÐn oscilla- trap potential, with the splitting between n and n levels tions in a trap was presented in [6, 7]. Decoherence and and n annihilation taken into account. The density of multiple reflections and the influence of gravitational the trap eigenstates, which is proportional to the macro- and magnetic fields were included. The approximate scopic trap volume, is huge and the states cluster equation for the annihilation probability after N collisions obtained in [7, Eq. (3.8)] coincides with the exact together extremely thickly. But these arguments do not ӷ suffice to discard the discrete-state approach, because formula (59) in the present paper when N 1. As we show below, the N-independent asymptotic regime set- the nÐn mixing parameter is much smaller than the dis- tles at N տ 10. Derivation of the exact equation for the annihilation ¦This article was submitted by the authors in English. probability with an arbitrary number of collisions is not

418 KERBIKOV et al. the only purpose of the present work. We already men- Evolution of the time-dependent part of Ψˆ ()xt, is then tioned the problem of the relation between the eigen- described by the equation value and the wave-packet approaches. Within the wave-packet approach, some basic notions such as the  Γβ time between successive collisions and the collision E Ð i----- ⑀ ∂ ψ ()t n ψ ()t time itself can be defined in a clear and rigorous way. i-----n = 2 n . (3) Another question within the wave-packet formalism is ∂tψ () Γ ψ n t β n()t the independence of the reflection coefficient from the ⑀ E Ð i----- n 2 width of the wave packet and the applicability of the stationary formalism to calculate reflections from the trap walls. These and some other principal points are, The difference between En and En due to the Earth’s for the first time, considered in detail in the present magnetic field is paper. ω µ ≈ × Ð12 We also mention that an alternative approach to the ==En Ð2En n B 610 eV. (4) evaluation of the reflection coefficients for n and n was Diagonalizing the matrix in (3), we find ψ (t) and outlined in [8]. It is based on the time-dependent n ψ ()t in terms of their values at t = 0, Hamilton formalism for the interaction of n and n with n the trap walls. This subject remains outside the scope of iω i⑀ the present paper. ψ ()t = ψ()0 cos νt + ------sinνt Ð ψ ()0 ---- sinνt n n 2ν n ν The paper is organized as follows. In Section 2, we recall the basic equations describing nÐn oscillations 1 (5) × exp Ð,---()iΩΓ+ β t in free space. Section 3 is devoted to the optical poten- 2 tial approach to the interaction of n and n with the trap walls. In Section 4, we analyze the two formalisms pro- i⑀ iω ψ ()t = Ð ψ()0 ---- sinνt + ψ ()0 cos νt Ð ------sinνt posed to treat nÐn oscillations in the cavity, namely, n n ν n 2ν box eigenstates and wave packets. In Section 5, reflection from the trap walls is considered. Section 6 contains 1 (6) × exp Ð,---()iΩΓ+ β t the main result in this work, the time dependence of the 2 n component production probability. In Section 7, con- Ω ν ω2 ⑀2 1/2 ω clusions are formulated and problems to be solved out- where = En + En , = ( /4 + ) , and = En Ð En. lined. ψ ψ () In particular, if n(0) = 1 and n 0 = 0, we have

⑀2 2. OSCILLATIONS IN FREE SPACE ψ ()2 4 ()Γ n t = ------exp Ð βt ω2 + 4⑀2 We start by recalling the basic equations describing (7) nÐn oscillations in free space. The phenomenological 2 1 2 2 × × sin --- ω + 4⑀ t . Hamiltonian is a 2 2 matrix in the basis of the two- 2 component nÐn wave function (we set ប = 1), The use of this equation to test fundamental symmetries Γβ is discussed in [9]. H = H Ð i----- δ + ⑀()σ , (1) ω jl j 2 jl x jl Without the magnetic ﬁeld, i.e., for = 0, and for t Ӷ ⑀Ð1, Eq. (7) yields 2 µ µ where j, l = n, n , Hj = k /2m Ð jB, j is the magnetic 2 2 2 ψ ()t ≈ ⑀ t exp()ÐΓβt . (8) moment, B is the external (e.g., the Earth’s) magnetic n Γ β ⑀ ﬁeld, β is the -decay width, is the nÐn mixing Ð1 σ This law (for t Ӷ Γβ ) has been used to establish the parameter (see below), and x is the Pauli matrix. τ ⑀Ð1 Assuming the n and n wave functions to be plane lower limit on the oscillation time = . According to waves, we write the two-component wave function of the ILL-Grenoble experiment [10], the nÐn system as 8 τ > 0.86× 10 s. (9)  The corresponding value of the mixing parameter is ⑀ ≈ ψ ()t ikx Ψˆ ()xt, = n e . (2) 10Ð23 eV. This number is used in obtaining numerical ψ () n t results presented below.

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The Earth’s magnetic field leads to a strong suppres- of wave packets, and, hence, the above values must be sion of the nÐn oscillations. With the value of ω given attributed to the center of the packet. by (4), Eq. (7) leads to We treat the interaction of n and n with the trap walls in terms of an energy-independent optical poten- ⑀2 ψ ()2 ≈ 4 ()Γ 2 τ tial. The validity of this approach to UCN has been jus- n t ------exp Ð βt sin t/ B ω2 (10) tified in a number of papers (see, e.g., [11, 12, 14]). There is still an open question concerning the discrep- ≈ Ð23 2 τ ancy between theoretical prediction and experimental 10 sin t/ B, data on the UCN absorption. Interesting by itself, this τ |µ | Ð1 ≈ × Ð4 problem is outside the scope of our work because, as where B = ( n B) 2 10 s. In what follows, we assume that the magnetic field is screened. was already mentioned, absorption of neutrons may be ignored in the nÐn oscillation process. The low-energy For ω = 0 but for arbitrary initial conditions, Eqs. (5) and (6) take the form optical potential is given by 2π ψ ()t = () ψ()0 cos⑀tiÐ ψ ()0 sin⑀t U = ------Na , (14) n n n jA m jA Γβ (11) × exp Ð iE + ----- t , 2 where j = n, n ; m is the neutron mass; N is the number of nuclei in a unit volume; and ajA is the jÐA scattering ψ () ()ψ () ⑀ ψ () ⑀ n t = Ði n 0 sin t + n 0 cos t length, which is real for n and complex for n . For neutrons, the scattering lengths a are accurately known Γ (12) nA × β for various materials [12]. For antineutrons, the situa- exp ÐiE + ----- t , 2 tion is different. Experimental data on low-energy n ÐA interaction are absent. Only some indirect information where E = En = En . may be gained from level shifts in antiprotonic atoms,

and, therefore, the values of anA used in [3, 6, 8, 15] as 3. OPTICAL POTENTIAL MODEL an input in the nÐn oscillation problem are similar but FOR THE TRAP WALL not the same. We consider the set of anA calculated We remind the reader that neutrons with the energy in [16] within the framework of the internuclear cas- E < 10Ð7 eV are called ultracold. An excellent review of cade model as most reliable. Even this particular model UCN physics was given in [11] (see also [12]). leads to several solutions, and the one that we have cho- 12 v sen for C (graphite and diamond) may be called A useful relation connecting the neutron velocity “motivated” by [16]. To estimate the dependence on the in cm/s and E in eV is given by material of the walls and to compare our results with

2 9 1/2 those in [3], we also performed calculations for Cu. v []cm/s = 10 ()10 E[]eV /5.22 . (13) Scattering lengths for Cu are not given in [16], and we used the solution proposed in [3]. Our calculations For E = 10Ð7 eV, the velocity is v ≈ 4.4 × 102 cm/s. were thus performed with the n ÐA scattering lengths A less formal definition of UCN involves the notion () () of the real part of the optical potential corresponding to anC ==3 Ð i1 fm, anCu 5 Ð i0.5 fm. (15) the trap material (see below). Neutrons with energies less than the height of this potential are called ultracold. The scattering lengths for neutrons are [12] The two definitions are essentially equivalent because, as we see in what follows, the real part of the optical anC ==6.65 fm, anCu 7.6 fm. (16) potential is of the order 10Ð7 eV for most materials. Our main interest is in strongly absorptive interac- The concentrations of atoms N entering (14) are as tion of the n component with the trap walls. We there- follows: fore ignore very weak absorption of UCN on the walls × Ð16 Ð3 [11, 12]. Due to complete reflection from the trap walls, NC() graphite = 1.13 10 fm , UCN can be stored for about 103 s (β-decay time), as × Ð16 Ð3 was first pointed out in [13]. NC() diamond = 1.63 10 fm , To be specific, we consider UCN with E = 0.8 × −7 v × 2 × Ð16 Ð3 10 eV, which corresponds to = 3.9 10 cm/s NCu = 0.84 10 fm . (see (13)), k = 12.3 eV, and de Broglie wavelength λ ≈ 10Ð5 cm. In the next section, we describe UCN in terms In accordance with (14), the optical potentials are

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 420 KERBIKOV et al. then given by discarded without further analysis, because the nÐn mixing parameter ⑀ ≈ 10Ð23 eV is much smaller than δE. × Ð7 UnCgr()= 1.95 10 eV, To understand the relation between the two × Ð7 (17) approaches, we note that the initial conditions corre- UnC() diam = 2.8 10 eV, spond to a beam of UCN provided by a source. The U = 1.66× 10Ð7 eV; momentum spectrum of UCN depends on the speciﬁc nCu experimental conditions. In order to stay on general grounds and, at the same time, to simplify the problem, ()× Ð7 UnCgr()= 0.9 Ð i0.3 10 eV, we assume that the UCN beam entering the trap has the ()× Ð7 form of a Gaussian wave packet. We suppose that, at UnC() diam = 1.3 Ð i0.4 10 eV, (18) t = 0, the center of the wave packet is at x = x0, and, ()× Ð7 hence, UnCu = 2 Ð i0.2 10 eV. ψ ()π, ()2 Ð1/4 In this paper, we consider particles (n and n ) with ener- k xt= 0 = a gies below the potential barrier formed by the real part 2 12 ()xxÐ (21) of the potential. For n and C, the limiting velocity is × expÐ ------0 + ikx , v = 4.15 × 102 cm/s. 2a2

4. WAVE PACKET VERSUS STANDING WAVES where a is the width of the wave packet in coordinate space. The normalization of wave function (21) corre- It is convenient to use the short notation sponds to one particle in the entire one-dimensional δ space, U j = V j Ð iW j jn (19) +∞ for optical potentials (17) and (18), where j = n, n and 2 dx ψ ()xt, = 0 = 1. (22) the wall material is not indicated explicitly. We con- ∫ k sider the following model for the trap in which nÐn Ð∞ oscillations may possibly be observed. We imagine two × Ð7 2 For E = 0.8 10 eV and the beam resolution walls of type (19) separated by a distance of L ~ 10 cm, ∆E/E = 10Ð3, we have i.e., the one-dimensional potential well of the form (){} θ()θ(){}δ k ==12.3 eV, a 3.2× 10Ð3 cm. (23) U j x = Ð x Ð L + x V j Ð iW j jn , (20) with θ(x) being the step function. Our goal is to follow The width of wave packet (21) increases with time the time evolution of the n component in such a trap, according to assuming that the initial state is a pure n one. 2 1/2 t ≈ t The first question to be answered is how to describe a' = a 1 + ------2 ------(24) the wave function of the system. Two different ma ma approaches seem to be feasible, and both were discussed in the literature [4, 6, 8]. The first is to consider and becomes comparable with the trap size L for t ~ oscillations occurring in the wave packet and to inves- 103 s. For the wave hitting the wall and the reflected Ӷ τ tigate to what extent reflections from the walls distort wave to be clearly resolved, the condition a'/v L or Ӷ τ the picture compared to the free-space regime. The sec- a' L must be satisfied, where L ~ 1 s is the time ond approach is to consider the eigenvalue problem in between two consecutive collisions with the trap walls. potential well (20), to find energy levels for n and n , Reflection of the wave packet from the walls is consid- and to consider oscillations in this basis. Because of ered in detail in the next section. Here, we show that t ~ different interactions with the walls, the levels of n and 103 s is the characteristic time needed for the rearrange- n are split and the n levels acquire annihilation widths. ment of the initial wave packet into stationary states of the trapping box. At first glance, this approach might seem inadequate because, in a trap with L ~ 102 cm, the density of states We consider the eigenvalue problem for potential is very high, the characteristic quantum number corre- well (20). The parameters of potential (20) for neutrons ≈ × Ð7 ≈ 2 sponding to the UCN energy is very large, and the split- are Vn 2 10 eV and L 10 cm. The number of lev- ting δE between adjacent n levels (or between the levels els is of the n and n spectra) is extremely small. The values L 2mV 108 of all these quantities are given below, and it follows M ≈≈------. (25) that δE < 10Ð14 eV. However, this approach cannot be π π

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According to (23), the center of wave packet (21) has tation for G, we can write the momentum k = 12.3 eV, which corresponds to a ≈ × 7 ≈ × ÐiE t state with the number of nodes j 2 10 and kjL 6 ψ(), j ϕ () ϕ ()ϕ(), xt = ∑e j x ∫dx' *j x' k x'0. (29) 107 ӷ 1. Positions of such highly excited levels in a finite-depth potential are indistinguishable from the j spectrum in a potential box with infinite walls. There- In the semiclassical approximation, the distance fore, δ π τ between the adjacent levels is E = / L, and, therefore, τ π one may think that, at t = L, i.e., already at the first col- ϕ ()≈ 2 ω ω j j x --- sin j x, j = -----. (26) lision, the neighboring terms in (29) would cancel each L L other. But this is not the case. Indeed, Wave functions (26) describe semiclassical states with ϕ () ()ϕ () () j ӷ 1 in a potential well with sharp edges. The “fre- j + 1 x exp ÐiEj + 1t + j x exp ÐiEjt ω quency” j is very high compared to the width of the () π wave packet in momentum space, exp ÐiEjt ()ω () = ------exp i j x 1 + expi--- x Ð vt i 2L L ω ≈ × 5 Ð1 ӷ ν 1 ≈ × 2 Ð1 j 610 cm = ------210 cm . 2a π ()ω () Ð exp Ði j x 1 + expÐi--- x + vt . This implies that the wave packet spans over a large L number of levels. To determine this number, we note that the distance between adjacent levels around the Therefore, there is a constructive interference at x = ±vt center of the wave packet is either in the first or in the second term, respectively. This is true with the whole sum of terms in (29) taken δ ∼ Ð14 EE= j + 1 Ð E j 10 eV. into account, and, hence, we can pass from summation to integration in (29). The overlap of the wave functions The highly excited levels within the energy band entering (29) can be easily evaluated, provided the center of the wave packet x0 is not within the bandwidth ∆E = 10Ð3E ∼ 10Ð10 eV distance a' from the trap walls. The overlap is given by the integral corresponding to wave packet (21) are to a high accu- i racy equidistant, as they should be in the semiclassical ϕ*()ψ(), ≈ ------∆ ∫dx' j x' k x'0 1/2 regime. The number of states within E is ()2 πLa 4 ∆j = ∆E/δE ∼ 10 0 ()x' Ð x 2 × dx'expÐ ------0 + ik()Ð ω x' and their density in momentum space is ∫ 2 j 2a ÐL (30) ρω()= a∆jL≈ /π ∼ 106 eVÐ1. (27) i = ------()π 1/2 We can now answer the question formulated at the 2 La beginning of this section, namely, whether the nÐn L oscillations in the trap should be described in terms of ()x'' + x 2 × dx''expÐ ------0 - Ð ik()Ð ω x'' . the wave packet or in terms of the stationary eigenfunc- ∫ 2 j 2a tions. At t = 0, the wave function has the form of the 0 wave packet (21) provided by the UCN source. Due to collisions with the trap walls, transitions from the initial At this step, we have omitted the exponential with the ω state (21) into discrete (or quasi-discrete for n ) eigen- high frequency (k + j). We next take (x'' + x0)/(2a ) as | | ӷ | | ӷ states (26) occur. a new variable and assume that x0 a, L Ð x0 a (we The time evolution of the initial wave function (21) recall that x0 is negative because ÐL < x < 0). The result proceeds according to is that

ψ(), ()ψ, , (), ϕ ()ψ(), xt = ∫dx'Gxt; x'0 k x'0, (28) ∫dx' *j x' k x'0

1/2 2 (31) πa a 2 where G(x, t; x', 0) is the time-dependent Green’s func- ≈ i ------exp Ð -----()k Ð ω + ik()Ð ω x . tion for potential well (20). Using the spectral represen- L 2 j j 0

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 422 KERBIKOV et al. Corrections to (31) are of the order of a/L. We now we have the estimate consider frequency summation in (29). This summation ω can be replaced by integration over , because the den- δψ(), ()ω ωρ() ω ()ω sity of semiclassical states ρ(ω) is very high. We thus xt = ∑ f n Ð ∫d f arrive at n

1 ω ψ()xt, = ------n + 1 t 1/2 ωρ() ω ()()ω ()ω π = Ð∑ d f Ð f n a1 + i------2 ∫ ma n ω n ω (36)  n + 1  α()xt, ≈ωρωÐ d ()f '()ωωω ()Ð ×  exp Ð------(32) ∑ ∫ n n 2 n ω  2t n  2a 1 + ------m2a4 1 = Ð---∑ f '()ωω ()Ð ω .  2 n n + 1 n α(),  n Ðxt  + exp Ð------2 , 2t  From (35), we find that 2a 1 + ------ m2a4 f '()ω = g()ω f ()ω , ()2 (37) α(), ()2 xxÐ 0 ()ω ()v ()ω 2 xt = xxÐ 0 Ð v 0t Ð it------g = ixÐ x0 Ð t Ð k0 Ð a . ma2 (33) 2 2 Because f(ω) is a narrow Gaussian peak, we can substi- k a 2 2()0 2 t tute g(ω) by g(k ), and then (36) results in Ð2ik0a xxÐ 0 ++i------t 2ik0 x0a + ------. 0 m m2a2 π The second term in Eq. (32) describes the reflected δψ()xt, ≈ ------()ψxxÐ Ð v t ()xt, . (38) wave packet (see the next section). According to (21), 2L 0 0 int (28), and (32), all that happens to the wave packet in the trap is broadening and reflections. This is true during From (34) and (38), we have some initial period of its life history at least. How long does this period last? The answer to this question may +∞ π be obtained by estimating the accuracy of performing δw ≈ ------dxxÐ x Ð v t ψ ()xt, 2 frequency integration instead of summation over dis- 2L ∫ 0 0 int crete states in (29). Ð∞ (39) a' t t To estimate the time scale for the rearrangement of ∝ ---- ∼∼------, initial wave packet (21) into trap standing waves (26), L maL 103 s it is convenient to introduce the difference δψ(), ψ()ψ, (), where a' is given by (23). xt = sum xt Ð int xt Roughly speaking, the time t ~ 103 s needed for the between the “exact” wave function (29) and the approx- neutron wave function to rearrange into the trap eigen- imate integral representation (32). Whenever state is comparable to the neutron lifetime, and the neutron would rather “die” than adjust to the new boundary δ () ()ψ 2 ψ 2 wt = ∫dx sum Ð int conditions. The wave packet formalism is therefore (34) used in what follows. Some additional subtleties arising ℜψ()δψ Ӷ from the quantization of levels in the trapping box are = 2∫dx int 1, discussed in Section 7. we can consider oscillations as proceeding in the wave packet basis. With 5. REFLECTION FROM THE TRAP WALLS πa f ()ω = ------We return to one-dimensional trap (20). Let the par- 2 ticle moving from x = Ð∞ enter the trap at t = 0 through 2L τ (35) the window at x = ÐL. At t = L, it reaches the wall at 2 2 a 2 ω x = 0, the n component is reflected from the wall, and × exp Ð -----()k Ð ω Ð i------t ++iω()xxÐ ikx , 2 0 2m 0 0 the n component is partly reflected and partly

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 NEUTRONÐANTINEUTRON OSCILLATIONS IN A TRAP REVISITED 423 α (), α()φ , absorbed. The wave packet describing the interaction refl xt = inc Ð x + ' t with the wall has the form 2 (45) +∞ φ 2 t +2ik0 'a + ------. Ð3/4 a 2 2 ψ()xt, = π --- dkψ ()kx, m a 2 ∫ j Ð∞ (40) From (43)Ð(45), we see that the essence of R(k) in 2 the wave packet formalism is the same as in the time- a 2 t 2 × exp Ð -----()kkÐ + iL() kÐ k Ð i------k , independent approach. Therefore, imposing standard 2 0 0 2m boundary conditions at x = 0, we obtain the reflection coefficients where j = n, n and κ () ρ() ()φ () kiÐ j ψ (), ikx ()Ðikx R j k ==j k exp i j k ------, (46) j kx = e + Rke ki+ κ (41) j φ ()k = eikx + ρ ()k e j eÐikx. κ []()1/2 j n = 2mVn Ð E , (47) For the n component, ρ (k) = 1, because we neglect 1/2 n κ ==[]2mV()Ð iW Ð E κ' Ð iκ'', very weak absorption of neutrons at the surface. The n n n n n integral (40) with the first term in (41) is trivial. To inte- Ð2kκ Ð2kκ' grate the second term in (41), we note that, due to the tanφ ==------n , tanφ ------n -, (48) n 2 2 n 2 3 ӷ κ 2 ()κ 2 ()κ Gaussian form factor with ak0 ~ 10 1, the dominant k Ð n k Ð n' Ð n'' contribution to integral (40) comes from a narrow inter- 4kκ'' val of k around k0. Expanding Rj(k) at k Ð k0 and keeping ρ ρ2 n n ==1, n 1 Ð ------. (49) the leading term, we obtain ()κ 2 ()κ 2 k + n'' + n' R ()k ≈ ρ()k exp()iφ ()k j j 0 j 0 For 12C (graphite), in particular, ρ'()k × φ'()()δ------j 0-() ρ θφ≡ φ 1 ++i j k0 kkÐ 0 jnρ ()kkÐ 0 (42) ==0.56, n Ð n 0.72. (50) j k0 ≈ρ()k exp()iφ ()k + iφ'()k ()kkÐ . The first term on the right-hand side of (45) can be j 0 j 0 j 0 0 φ 2 written as [Ðx + L Ð v0(t Ð '/v0)] . Hence, the collision The validity of the last step for n becomes clear from time or time delay is [17, 18] the explicit expressions for ρ ()k and φ ()k presented n n φ'() below. τ j k0 2m j, coll ==------R e ------κ-. (51) Integration in (40) can now be easily performed, v 0 k j with the result [17] κ For neutrons, i.e., for real n, Eq. (51) gives the well- 1 ψ ()xt, = ------known result j t 1/2 π Ð1/2 a1 + i------2 τ []() ma n, coll = EVn Ð E . τ  This result is in line with the naive estimate n, coll ~ Ð8 Շ λ  α ()xt, l/v0 ~ 10 s [8], where l is the penetration depth. ×  exp Ð------inc (43) 2 For 12C (graphite), Eq. (51) yields  2t  2a 1 + ------m2a4 τ × Ð8 τ × Ð8 n, coll ==0.7 10 s, n, coll 1.1 10 s. (52)  Equations (43)Ð(45) supplemented by the above α ()xt,  () refl  inequality make it possible to follow the time evolution + R j k0 exp Ð------2 , 2t  of the beam inside the trap. We imagine an observer 2a 1 + ------2 4  placed at a bandwidth distance from the wall, i.e., at m a x = Ða. According to (43)Ð(45), such an observer con- 2 cludes that the incident wave (the first term in (43)) 2 ()xL+ ≤ τ τ α ()xt, = ()xL+ Ð v t Ð it------dominates at times t L Ð a, while the reflected wave inc 0 2 ≥ τ τ ma prevails at t L + a. With this splitting of the time (44) τ 2 2 2 interval around N L, N = 1, 2, …, we use the notation 2 k0 a 2 t τ τ () (N LÐ) and (N L+) for the moments before and after the Ð2ik0a xL+ ++i------t 2ik0 La+ ------, m m2a2 Nth collision. Thus, we can calculate the n production

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 424 KERBIKOV et al. ψ τ rate, because we have rigorous definitions of the colli- The answer for (N LÐ) now seems evident: sion time and the time interval between the two subsequent collisions. θ ρN iN ψ ()τ ⑀τ 1 Ð e n N LÐ = L------θ 1 Ð ρei (56) 6. ANNIHILATION RATE IN A TRAP × []()τ φ π exp ÐiNEL Ð n + /2 . We can now inquire into the problem of time-dependence of the n production probability. In free space, it This conjecture is easy to verify by induction. For t = 2 τ is given by ψ ()t = ⑀2t2 (see (2)), while in a trap with (2 LÐ), the result was derived explicitly in (55). Evolv- n τ the complete annihilation or total loss of coherence at ing (56) through one reflection at t = N L and free propagation from t = (Nτ +) to t = ((N + 1)τ Ð), we arrive ψ ()2 L L each collision, it has a linear time dependence n t = at (56) with (N + 1) instead of N. This completes the ⑀2τ Lt [8]. proof. To avoid cumbersome equations and because we Therefore, the admixture of n before the Nth colli- Ð1 τ consider the time interval t Ӷ Γβ , we omit exp(ÐΓβt) sion, i.e., at t = N LÐ, is factors. Production of n during the collision can also be ρ2N ρN θ neglected [8]. The difference in collision times (52) for ψ ()τ 2 ⑀2τ2 1 + Ð 2 cos N n N LÐ = L------. (57) n and n may also be ignored. In the previous section, 1 + ρ2 Ð 2ρθcos we have seen that the interaction of the wave packet with the wall is described in terms of reflection coefficients (46).1 The annihilation probability at the jth collision is

We assume that, at t = 0, a pure-n beam enters the () ()ψρ2 ()τ 2 τ Pa j = 1 Ð n j LÐ . (58) trap at x = ÐL. After crossing the trap, i.e., at t = ( LÐ), the time-dependent parts of the wave functions are given by (12),2 The total annihilation probability after N collisions is therefore given by ψ ()τ ()⑀τ ()τ n LÐ = cos L exp ÐiE L , (53) N ψ ()τ ()⑀τ []()τ π n LÐ = sin L exp ÐiEL + /2 . () ()ψρ2 ()τ 2 Pa N = 1 Ð ∑ n k L After the ﬁrst reﬂection at t = (τ +), we have k = 1 L 2 2 2 ⑀ τ ()1 Ð ρ  ρ2()1 Ð ρ2N = ------L N + ------(59) ψ ()τ ()⑀τ []()τ φ 2  2 n L+ = cos L exp ÐiEL Ð n , 1 + ρ Ð 2ρθcos 1 Ð ρ (54) ψ ()ρτ ()⑀τ []()τ φ π N n L+ = n sin L exp ÐiEL Ð n + /2 . θ ρ ρ []()θρθ ρ cos Ð Ð cos N + 1 + cos N  Ð2 ------2 - . τ τ 1 + ρ Ð 2ρθcos Evolution from t = ( L+) to t = (2 LÐ) again proceeds in accordance with (12), After several collisions, the terms proportional to ρN, ρ2N, and ρN + 1 may be dropped, because ρ ~ 0.5 ψ 1 ()⑀τ ()ρ iθ n = --- sin 2 L 1 + e (see (50)). Then, (59) takes the form 2 × []()τ φ π (55) exp Ði 2E L Ð n + /2 ⑀2τ2 L iθ ()≈ ≈ ⑀τ ()ρ []()τ φ π Pa N ------2 L 1 + e exp Ði 2E L Ð n + /2 , 1 + ρ Ð 2ρθcos (60) 2 2 where θ = φ Ð φ is the decoherence phase and ρ ≡ ρ . 2 ()1 Ð ρ n n n × N()1 Ð ρ + 1 Ð ------. 1 + ρ2 Ð 2ρθcos 1 An alternative description using time-evolution operators was proposed in [8]. 2 We state this although the Gaussian form factor in (43) also Three different regimes may be inferred from (60). depends on time, the corresponding terms in the time-dependent For a very strong annihilation, i.e., ρ Ӷ 1, Schrödinger equation are of the order of 1/ak0 compared to the derivative of the exponent exp(ÐiEt); we also note that the form () ⑀2τ2 ⑀2τ factors are the same for n and n up to a constant multiplier. Pa N ==L N Lt. (61)

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Qa Neff 2.0

800 1.5 600

1.0 400

0.5 200

01020304050 0 0.2 0.4 0.6 0.8 ρ N Fig. 1. Plot of the Q (N) = (⑀2τ t)Ð1P (N) dependence vs. a L a Fig. 2. Plot of the N dependence vs. ρ at θ = 0. The solid 12 eff N. The solid line corresponds to C (graphite); the dashed line is for the number of collisions N = 50; the dashed line one, to 12C (diamond); and the dotted one, to Cu. corresponds to N = 10.

For the complete decoherence at each collision, i.e., for can be expected from the trap experiments in the most θ = π, favorable, although hardly realistic, scenario. ρ ρ()ρ ρ () ⑀2τ2 1 Ð 2 Ð ≈ 1 Ð ⑀2τ Pa N = LN------+ ------Lt. (62) 7. CONCLUDING REMARKS 1 + ρ ()1 + ρ 2 1 + ρ We have reexamined the problem of nÐn oscilla- For the (unrealistic) situation where θ = 0, tions for UCN in a trap. Our aim was to present a clear formulation of the problem, to calculate the amplitude ρ ρ()ρ ρ () ⑀2τ2 1 + 2 + ≈ 1 + ⑀2τ of the n component for an arbitrary observation time Pa N = LN------Ð ------Lt. (63) 1 Ð ρ ()1 Ð ρ 2 1 Ð ρ and for any given reflection properties of the trap walls. We have shown that, for the physically relevant obser- For the values of ρ and θ corresponding to optical vation time (i.e., for the time interval less than the potentials (17) and (18), the quantity β-decay time), the process of nÐn oscillations is described in terms of wave packets, while the standing- () ()⑀2τ2 Ð1 ()⑀2τ Ð1 () Qa N ==L N Lt Pa N wave regime may settle only at later times. By calculating the difference between the n and n collision times, calculated in accordance with the exact equation (59) is new light has been shed on the decoherence phenom- displayed in Fig. 1. This figure shows that the linear ena. For the first time, an exact equation has been time dependence settles after about 10 collisions with derived for the annihilation probability for an arbitrary the trap walls. The asymptotic value of Qa(N), which number of collisions with the trap walls. In line with the may be called the enhancement factor, is 1.5Ð2, conclusions of the previous authors on the subject, this depending on the wall material. probability grows linearly with time. We have calcu- Proposals have been discussed in the literature [6, 19] lated the enhancement factor entering this linear time to compensate the decoherence phase θ by applying the dependence and found this factor to be 1.5Ð2, depend- external magnetic field. Assuming the ideal situation ing on the reflection properties of the wall material. that the regime θ = 0 may be achieved in such a way and Despite the extensive investigations reviewed in this also assuming that the reflection coefficient ρ can be article and the results of the present paper, the list of varied in the whole range by varying the trap material, problems for further work is large. The central and most ρ we plot the quantity Neff( ) defined as difficult task is to obtain reliable parameters of the optical potential for antineutrons. The beam of n with the () ⑀2τ2 ()ρ Ð7 Pa N = L Neff (64) energy in the range of 10 eV will hardly be accessible in the near future. Therefore, work has to be continued ρ in Fig. 2. Thus defined, Neff( ) obviously depends also along the two lines mentioned above: to deduce the on the number of collisions N; the results for N = 10 and parameters of the optical potential the level shifts in N = 50 are presented in Fig. 2. This figure shows what antiprotonic atoms and to construct reliable optical

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 426 KERBIKOV et al. models that can be confronted with the available exper- 3. K. G. Chetyrkin, M. V. Kazarnovsky, V. A. Kuzmin, and imental data on n Ðnuclear interaction at higher ener- M. E. Shaposhnikov, Phys. Lett. B 99B, 358 (1981). gies. In a forthcoming publication, we plan to present 4. S. Marsh and K. W. McVoy, Phys. Rev. D 28, 2793 numerical calculation of the time evolution of a wave (1983). packet into standing waves and to discuss some features 5. M. Baldo Ceolin, in Festschrift for Val Telegdi, Ed. by K. Winter (Elsevier, Amsterdam, 1988), p. 17. of nÐn oscillations in the eigenfunction basis, which 6. R. Golub and H. Yoshiki, Nucl. Phys. A 501, 869 (1989). were not discussed in [4]. Another task is to perform calculation for the specific geometry of the trap and a 7. H. Yoshiki and R. Golub, Nucl. Phys. A 536, 648 (1992). realistic spectrum of the neutron beam. This requires an 8. B. O. Kerbikov, Phys. At. Nucl. 66, 2178 (2003). input corresponding to a specific experimental setting. 9. Yu. G. Abov, F. S. Dzheparov, and L. B. Okun, Pis’ma Zh. Éksp. Teor. Fiz. 39, 493 (1984) [JETP Lett. 39, 599 (1984)]. ACKNOWLEDGMENTS 10. M. Baldo-Ceolin et al., Z. Phys. C 63, 409 (1994). 11. I. M. Frank, Usp. Fiz. Nauk 161, 109 (1991) [Sov. Phys. The authors are grateful to Yu.A. Kamyshkov for Usp. 34, 988 (1991)]. interest in this work and discussions. Useful remarks 12. V. K. Ignatovich, The Physics of Ultracold Neutrons and suggestions from L.N. Bogdanova, F.S. Dzheparov, (Nauka, Moscow, 1986; Clarendon Press, Oxford, A.I. Frank, V.D. Mur, V.S. Popov, and G.V. Danilian are 1990). gratefully acknowledged. 13. Ya. B. Zel’dovich, Zh. Éksp. Teor. Fiz. 36, 1952 (1959) One of the authors (B.O.K.) expresses his gratitude [Sov. Phys. JETP 9, 1389 (1959)]. for financial support to V.A. Novikov, L.B. Okun, 14. M. Lax, Rev. Mod. Phys. 23, 287 (1951); Phys. Rev. 85, INTAS (grant no. 110), and K.A. Ter-Martirosian sci- 621 (1952). entific school (grant no. NSchool-1774). 15. J. Hufber and B. Z. Kopelovich, hep-ph/9807210. 16. Ye. S. Golubeva and L. A. Kondratyuk, Nucl. Phys. B (Proc. Suppl.) 56, 103 (1997). REFERENCES 17. V. M. Galitsky, B. M. Karnakov, and V. I. Kogan, Prob- 1. V. A. Kuz’min, Pis’ma Zh. Éksp. Teor. Fiz. 13, 335 lems in Quantum Mechanics (Nauka, Moscow, 1992). (1970) [JETP Lett. 13, 238 (1970)]. 18. C. J. Goebel and K. W. McVoy, Ann. Phys. (N.Y.) 37, 62 2. R. N. Mohapatra and R. E. Marshak, Phys. Rev. Lett. 44, (1966). 1316 (1980). 19. V. K. Ignatovich, Phys. Rev. D 67, 016004 (2002).

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 427Ð437. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 486Ð498. Original Russian Text Copyright © 2004 by Rozanov, Fedorov, Shatsev. ATOMS, SPECTRA, RADIATION

Energy-Flow Patterns and Bifurcations of Two-Dimensional Cavity Solitons N. N. Rozanov, S. V. Fedorov, and A. N. Shatsev Research Institute for Laser Physics, Vavilov State Optical Institute, St. Petersburg, 199034 Russia e-mail: [email protected] Received May 27, 2003

Abstract—An analysis of two-dimensional spatial optical solitons in a large-aperture class A laser with a saturable absorber is developed. New types of rotating asymmetric solitons are found by computing the governing equation. The existence of weakly and strongly coupled solitons is demonstrated. Essential distinctions between them manifest themselves in the pattern of energy flows. A strongly coupled state evolves with time from an initial superposition of the fields of two solitons via successive bifurcations (topological changes in the energy-flow pattern). © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION response is determined by instantaneous values of the electric field envelope. In other words, we neglect the Optical solitons are blobs of light whose linear (dif- effects due to finite relaxation times (examined in [11, fractive or dispersive) spread is balanced by nonlinear 14–16]). Under this assumption, field dynamics are compression. There exist optical solitons of two types. governed by the complex GinzburgÐLandau equation One is the conservative soliton developing and pro- for the envelope of electric field. The governing equa- pagating in a transparent medium with negligible losses tion and its key properties are considered in Section 2. [1, 2]. The basic characteristics of conservative solitons Numerical solution of the governing equation reveals a have continuous spectra. In other words, the corre- number of interesting structures that can be interpreted sponding peak intensities or blob widths can vary con- as strongly coupled states of cavity solitons (see [11, 17] tinuously (within certain limits). The other type is the and references cited therein). In this paper, we also dissipative soliton (also called autosoliton) [3]. Origi- compute several new structures of this kind. However, nally, dissipative optical solitons were predicted and the present analysis is focused on the elucidation of the analyzed theoretically in large-aperture nonlinear inter- “internal structure” of cavity solitons, which is most ferometers [4, 5] and lasers with saturable absorbers [6]. obviously manifested in the pattern of energy flow. We By virtue of an additional requirement of energy bal- apply some well-known methods of the theory of non- ance (not imposed in the case of conservative solitons), linear oscillations to analyze the pattern of Poynting- the basic characteristics of dissipative solitons have dis- vector streamlines (Section 3). The relatively simple crete spectra. This implies substantial difference in patterns corresponding to single solitons with axially properties between the two types of solitons. Dissipa- symmetric intensity distributions are discussed in Sec- tive optical solitons are characterized by an excitation tion 4. In Section 5, we consider a pair of interacting threshold. In conjunction with the discreteness of their solitons and expose qualitative (topological) difference spectra, this property enhances their potential utility for in energy-flow pattern between widely separated data-processing technologies. Progress in experiment (weakly interacting) and closely spaced (strongly inter- and application was stimulated by the introduction of acting) solitons. This analysis indicates that the pattern semiconductor microcavities [7, 8]. The current status of changes via bifurcations as the distance between the theory and experiment in studies of optical solitons was solitons decreases. The main results are summarized in discussed in reviews [9, 10] and a monograph [11]. the Conclusions Section. One-, two-, and three-dimensional dissipative optical solitons can be implemented in certain lasers with 2. MODEL OF A LASER and without feedback [11, 12]. In this paper, we con- AND THE BASIC EQUATIONS sider the spatial solitons that arise in a large-aperture laser with nonlinear (saturable) gain and absorption. Consider a large-aperture laser with saturable gain Since the longitudinal variation of the electric-field and absorption in the optical cavity between plane par- envelope is slow, we can use a paraxial equation aver- allel mirrors. In the case of fast optical nonlinearity (for aged over this direction [13], i.e., solitons of this kind a class A laser), the paraxial equation for the envelope are essentially two-dimensional. We restrict our analy- E(r⊥, t) averaged over the longitudinal coordinate z (in sis to media with fast optical nonlinearity, whose the mean-field approximation valid when the electric-

428 ROZANOV et al. field envelope changes weakly over the cavity length) is we used the function f corresponding to a two-level the generalized complex GinzburgÐLandau equation medium with saturable gain and absorption: (see [11]) g a ()2 0 ------0 fE = Ð 1 + ------2 Ð 2, (2.9) ∂E 2 1 + E 1 + bE ------= ()∆id+ ⊥EfE+ ()E. (2.1) ∂t where g0 and a0 are the linear gain and absorption coefficients, respectively; b is the ratio of gain and absorp- We use the following dimensionless notation here: t is tion saturation intensities; and nonresonant losses are the time measured in units of field decay time in an represented by the first term on the right-hand side empty cavity, d is the effective diffusion coefficient (equal to Ð1 on the time scale used here). characterizing a weakly dispersive medium (0 < d Ӷ 1), 2 2 2 2 and ∆⊥ = ∂ /∂x + ∂ /∂y is the transverse Laplace operator. The transverse coordinates x and y are measured in 3. FLOWS OF RADIANT ENERGY | | 1/2 units of Fresnel zone width wF = [Lc/2k(1 Ð R )] , In the paraxial approximation employed here, the where Lc is the cavity length, k is the wavenumber in a Poynting vector S averaged over the period of an elec- linear medium, and R is the product of the amplitude tromagnetic wave with constant (e.g., almost linear) reflectivities of the cavity mirrors. The function f(|E|2) polarization is related to the envelope E, the real ampli- characterizes gain and absorption saturation and tude A = |E|, and phase Ψ = argE as follows [18, 19]: includes constant (nonresonant) losses. Without specifying this function, we note here that stable localized 2∇Ψ ()∇ structures (with field amplitude rapidly decreasing S ==A Im E* E . (3.1) toward periphery) can exist only if In accordance with the laser model, we treat the z axis ()< as the predominant wave propagation direction, and Re f 0 = Re f 0 0 . (2.2) governing equation (2.1) involves only the transverse coordinates r⊥ = (x, y). Therefore, the vector S⊥ = Otherwise, the peripheral structure would be unstable (S , S ) at a time t can be expressed as with respect to small perturbations. x y Equation (2.1) is invariant under phase shift of the 2 S⊥ ==A ∇⊥ψ Im()E*∇⊥E . (3.2) field ()Φ EEiexp 0 , (2.3) The corresponding streamlines (curves with tangents parallel to S⊥ at every point) are conveniently parame- shift in the transverse coordinates terized by equations written in terms of a function τ of the arclength (cf. the ray equation in geometrical (),, (), , Exyt Ex+ X0 yY+ 0 t (2.4) optics):

(Φ , X , and Y are constants), and inversion of either dx dy 0 0 0 ------==S ()xy, , ------S ()xy, . (3.3) transverse coordinate, such as dτ x dτ y

Exyt(),, Ex(), Ð yt, . (2.5) Note that these streamlines would be almost everywhere described by Eq. (3.3) with the Poynting vector Furthermore, Eq. (2.1) entails the following integral bal- replaced by the phase gradient. However, the Poynting ance energy relation valid for localized structures [12]: vector is more suitable in most cases. Indeed, whereas the phase is not defined at the points of screw wavefront d 2 () -----∫ E dr⊥ = 2W' E , (2.6) dislocations (where the field amplitude vanishes), the dt Poynting vector vanishes at these points. Note also that where the functional W'(E) is defined as both energy flow and Poynting vector have well- defined physical meaning in the paraxial approximation. The transverse distribution of the Poynting vector () []2 ()∇2 2 W' E = ∫ E Re fE Ð d ⊥E dr⊥. (2.7) combined with the transverse intensity distribution provides an unambiguous description of the field (up to an For steady structures (including those moving or rotat- insignificant constant). ing as a whole), it holds that Equations similar to (3.3) have been analyzed in detail in the theory of nonlinear oscillations [20, 21]. W'()E = 0. (2.8) The degenerate case corresponds to conservative systems almost everywhere satisfying the relation In the case of a negligible frequency detuning, f is a real div⊥S⊥ = 0. Conservative systems do not have isolated function (Imf = 0). In the calculations presented below, closed orbits and are not robust. The phase portrait of

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ENERGY-FLOW PATTERNS AND BIFURCATIONS 429 such a system consists of “cells” occupied by trajecto- energy-flow streamlines are described by the equations ries of similar type that can be altered by an arbitrarily small change in Sx(x, y) and Sy(x, y) [20, 21]. Examples dx ------Im= ()E*E()xqp+ ()+ y, of Poynting-vector streamlines for conservative optical dτ 0 xx systems can be found in [22]. (3.8) dy () () ------= pqÐ x + Im E0*E()yy y, Lasers are dissipative systems (div⊥S⊥ ≠ 0) in which dτ the field exchanges energy with active and absorbing media. To divide a phase portrait into cells of similar where p = Im(E0* E(xy)). The type of a singular point is τ τ behavior of the trajectories described by x( ) and y( ), determined by the roots of the quadratic equation one should find the singular (fixed) points (x0, y0) at which Sx(x0, y0) = 0 and Sy(x0, y0) = 0. According λ2 +0,σλ+ ∆ = (3.9) to (3.1), the singular points are of two types, since the Poynting vector vanishes with field amplitude or phase with gradient. In the former case, the expansion of the complex field envelope in terms of small deviations of x and σ ()() = ÐImIm E0*E()xx Ð E0*E()yy , y from x0 and y0 about a singular point starts from linear (3.10) ∆ ()()2 2 terms: = Im E0*E()xx Im E0*E()yy Ð p +.q

When σ ≠ 0 and ∆ ≠ 0, the singular point is robust 1 2 Exy(), = E()xE++()y ---E()x (node, focus, or saddle point). In the general case, to x y 2 xx (3.4) divide a phase portrait into cells, one must know not 1 2 only the singular points and their types, but also nonlo- + E()xy ++---E()y …, yy 2 yy cal elements (periodic orbits and saddle separatrices) [20, 21]. where the derivatives, E = ∂E/∂x| , etc., are cal- Energy-flow streamlines change with system (x) x = y = 0 parameters, initial conditions, or time. Of particular culated at the singular point. Then, interest are bifurcations, i.e., topological changes in the partition of the phase portrait into cells. All types Sx ==qy, Sy Ðqx, (3.5) of bifurcations admitted by Eqs. (3.3) have been well studied [20, 21], which facilitates analysis of energy- flow patterns in two-dimensional solitons. where q = Im(E(x) E()*y ), and Eqs. (3.3) become 4. SOLITONS dx dy WITH AXIALLY SYMMETRIC INTENSITY ------==qy, ------Ðqx. (3.6) dτ dτ DISTRIBUTIONS For a steady soliton, the time dependence of the ν Solutions to these equations correspond to trajectories envelope has the form exp(Ði t), where the frequency ν x2 + y2 = R2 in the phase plane xy. In this case, the point shift is the eigenvalue of the problem. Solitons with axially symmetric intensity distributions should be con- is a center (i.e., not a robust one) and its type may ϕ ϕ change when higher order terms of expansion (3.4) are sidered in polar coordinates r and (x = rcos , y = rsinϕ). Writing taken into account. The analysis below shows that the point is actually a focus in the case of a cavity soliton EFr= ()exp()imϕ exp()Ðiνt , (4.1) with an axially symmetric intensity distribution. The expansion of the field about a singular point of where the integer m = 0, ±1, ±2, … is a topological the other type contains a constant term E ≠ 0: index, we obtain an equation for the complex radial 0 function F(r):

1 2 2 2 Exy(), = E +++E()xE()y ---E()x d F 1dF m 1 2 0 x y 2 xx ------+ ------==------F +0.------[]iν + fF()F (4.2) (3.7) dr2 r dr r2 id+ 1 2 + E()xy + ---E()y + … xy 2 yy The “instantaneous” phase Ψ = argE is related to the Ψ radial phase 0(r) = argF : Here, the Poynting vector vanishes at the singular point Ψ()Ψ, ϕ () ϕ (x = y = 0) if Im(E0* E(x)) = Im(E0* E(y)) = 0, and the r = 0 r + m . (4.3)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 430 ROZANOV et al. Ψ It is assumed that time t is fixed and 0(0) = 0 since the Accordingly, Eqs. (3.3) for energy-flow streamlines are constant component of the phase is of no importance. rewritten as When the field amplitude is small, Eq. (4.2) reduces to a linear equation: Ψ dr 2d 0 dϕ m 2 ----- ==A ------, ------A , (4.11) τ τ 2 2 2 d dr d r d F 1dF m 2 ------+0,------Ð ------F Ð p F = (4.4) 2 r dr 2 dr r and trajectories are described by the equations where r ϕ d m ϕϕ dr ν ------==------, Ð 0 m∫ ------. (4.12) 2 i + f 0 dr 2 Ψ 2 Ψ p = Ð.------(4.5) r d 0/dr r d 0/dr id+ r0

Without loss of generality, we can assume that Rep > 0. The first equation in (4.11) is strictly radial (does not | |2 ϕ τ ±∞ τ When m = 0, we can use (4.5) with f0 f( F0 ), contain ). As , the radius r( ) tends to a con- | |2 stant R0, where R0 is a root of the equation where F0 is the intensity at the center of a fundamental soliton. The solution to (4.4) can be expressed in terms of cylinder functions. However, only their asymp- dΨ ------0()R = 0, totics can be used at r ∞ (at the periphery of a soli- dr 0 ton) and r 0 (when m ≠ 0, since the field vanishes ∞ at the center of the soliton in this case). As r , the whose roots include R = 0. When m = 0, the trajectories complex amplitude F, the real amplitude A = |F|, and the 0 Ψ are radial line segments with endpoints on circles of radial phase 0 are ≠ radius R0. When m 0, these circles are limit cycles, and the state point moves along them with a constant FF≈ ∞ exp()Ðpr , AA≈ ∞ exp()Ðp'r , (4.6) angular velocity Ω = mA2(R )/R2 . We should note that Ψ = const Ð p''r, 0 0 0 stability analysis of energy-flow patterns developed here has nothing to do with time evolution, because a where p' = Rep > 0 and p'' = Imp. fixed point in time is considered. However, one can As r 0, the power series expansion of the solu- consider “stability” with respect to increase in τ (hence tion to (4.4) with m ≠ 0 or the solution to (4.2) with the quotation marks). Note that the singular points m = 0 has the form (nodes or foci) and limit cycles that are “unstable” in the limit of r ∞ become “stable” as the sign of τ m ()2 … FF= 0r 1 ++F2r . (4.7) reverses. This property can be used in calculations, in particular, to find “unstable” limit cycles. Substituting (4.7) into (4.4), we find Since Eq. (4.2) for an axially symmetric localized distribution is a complex one, it is equivalent to a non- p2 linear system of four first-order ordinary differential F = ------, (4.8) 2 4()m + 1 equations. The equations for the real amplitude A and Ψ the radial phase 0 entailed by (4.2) are which yields the lowest order term in the radial phase 2 2 2 dΨ 2 Ψ () r 2 d A 1dA m 0 0 r = ------Im p ------+ ------Ð ------A Ð A------4()m + 1 2 r dr r2 dr (4.9) dr 2 ν 2 r f 0 + i iν + fA() = Ð------Im------. +Re------A = 0, 4()m + 1 id+ id+ (4.13) 2 Note that the asymptotic behavior of the radial phase is d Ψ 1 dAdΨ 1dΨ determined by the same complex quantity p both as ------0 ++ 2 ------0 ------0 r ∞ and as r 0. dr2 A dr dr r dr In polar coordinates (with basis vectors er and eϕ), iν + fA()2 Poynting vector (3.2) for solitons of this type has the +Im------= 0. form id+ Ψ 2 d 0 m Since Eqs. (4.13) do not contain the phase, the order of S⊥ = A ()r ------e + ----eϕ . (4.10) dr r r the system can be reduced to three. The equivalent sys-

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v v 0.3 0.080 (a)(3, 1) (b) (1, 1) 0.2 0.075 S S1 2 (3, 3) (3, 1) 0.1 0.070 (3, 1)

0 0.065 2.00 2.08 2.16 2.12 2.14 2.16 v g0 v g0 0.4 (c) (d)

0.080 (1, 1) (3, 1)

(3, 3) 0.2 S2 S1 0.072

(3, 1) (3, 1) 0 –0.4 0 0.4 –0.2 –0.1 0 0.1 d d

ν Fig. 1. The eigenvalue for solitons with topological charge m = 1 versus (a, b) gain g0 for d = 0 and (c, d) diffusion coefﬁcient d for g0 = 2.13. The second loop of the spiral is shown in more detail in panels (b) and (d) as compared to (a) and (c). The numbers at the spiral illustrate the change in (NQ, NK) across the points SN with vertical tangents. Unstable solitons are marked with circles. The dots with arrows near the self-intersection in panel (d) (d = 0.04) correspond to the fundamental and excited solitons depicted in Figs. 2c and 2d, respectively.

Ψ tem of equations for A(r), Q(r) = d 0/dr, and K(r) = tons with axially symmetric intensity distributions con- (1/A)dA/dr is sidered here. The calculations presented below were performed dA dQ Q ------Ð0,KA = ------++ 2 KQ ---- for a0 = 2 and b = 10. When all parameters are held con- dr dr r stant, every solution to Eq. (4.14) with appropriate asymptotic behavior corresponds to an eigenvalue ν iν + fA()2 +Im------= 0, (4.14) belonging to a discrete set. The dependence of ν on a id+ control parameter, such as g0 or d, is graphically repre- 2 2 sented by a self-intersecting spiral with several loops dK 2 2 KmÐ iν + fA() ------+0.K Ð Q ++------R e ------= (see Fig. 1). Different branches of the spiral are charac- dr r id+ terized by different numbers of zeros NQ and NK of the functions Q(r) and K(r) at r > 0. The numbers N and ∞ Q As r 0 and r , the asymptotics of these func- NK change by 1 (when m = 0) or 2 (when m is any inte- tions can easily be determined from expressions (4.6) ger) at the points S with vertical tangents in Fig. 1 ∞ N and (4.7). Speciﬁcally, as r , we ﬁnd that K (N = 1, 2). Ðp' < 0 and Q Ðp'' > 0. As r 0, both Q(r) and K(r) Ð |m|/r must vanish simultaneously. These asymp- If m = 0, then NQ = NK = 0 for the outer loop of the totics can be used to calculate characteristics of the soli- spiral (above S1) and the corresponding localized struc-

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Ψ Ψ Ä 0 Ä 0 3 3 0.8 0.8 (a) (b)

2 2 0

–0.8 1 1

0 –1.6

02468100 5 10 15 20 25 rrΨ Ψ ÄÄ0 0 3 0.4 3 1 (c) (d)

0.2 0 2 2

0 –1 1 1

–0.2 –2

04812160102030 rr Ψ Fig. 2. Radial profiles of amplitude A(r) and “radial” phase 0(r) for stable fundamental solitons with (a) m = 0 (g0 = 2.11, d = ν ν 0.01, = 0.12036) and (c) m = 1 (g0 = 2.13, d = 0.04, = 0.070658) and for excited solitons with (d) m = 1 (g0 = 2.13, d = 0.04, ν ν = 0.070998) and (b) m = 2 (g0 = 2.10, d = 0.06, = 0.0809). tures are unstable. Here, stability (without quotation and the radial phase is an oscillating function of radius marks) is interpreted in terms of time evolution and is with number of oscillations depending on |m| (see determined by applying the conventional linear analy- Fig. 2). Note that the fundamental and excited states of sis [11]. There exist stable solitons corresponding to the a stable soliton (which correspond to different loops of segments of the next loop where NQ = 1 and NK = 0. If spirals similar to those in Fig. 1) differ only quantita- m = 1 or 2, then there exist no stable solitons corre- tively (cf. Figs. 2b and 2c). sponding to the outer loop, where NQ = NK = 1. Stable According to the foregoing analysis, the pattern of solitons correspond to the segments of the next loops energy flows in the phase plane is as shown in Fig. 3. where NQ = 3 and NK = 1 (fundamental soliton) and The central point, r = 0, is a “stable” node when m = 0 NQ = 3 and NK = 3 (excited soliton). The dots with and a focus when m ≠ 0. In the latter case, this point is arrows in Fig. 1d represent two close values of ν corre- encompassed by an odd number of limit cycles, i.e., cir- sponding to the same value of g0 and lying on different cles of radius R0 defined by the condition loops of the spiral near its self-intersection. These points are associated with the fundamental and excited Ψ () d 0() solitons with the radial amplitude and phase distribu- QR0 ==------R0 0 tions illustrated by Figs. 2c and 2d, respectively. dr The analysis that follows is focused on stable solitons. When m = 0, the field amplitude is a monotoni- (see above). The cycles have alternating “stability” cally decreasing function of radius. When m ≠ 0, the properties: the cycle closest to the central point is the radial profile of the amplitude has a single maximum “unstable” boundary of the basin of attraction of the

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y y 40 55 (a) (b)

30 45

20 35

10 20 30 40 25 35 45 55 x x Fig. 3. Phase portraits of single stable solitons with (a) m = 0 and (b) m = 1. The ﬁxed points at r = 0 are node and focus, respectively. Cells occupied by trajectories with different behavior are separated by thick solid (“stable”) and dashed (“unstable”) limit cycles (m ≠ 0) or a set of ﬁxed points (in the degenerate case of m = 0). The thin curves with arrows are Poynting-vector streamlines. The phase portraits with m = 1 and m = 2 are topologically equivalent.

(a) (c)

y y (b) (d)

40 40

30 30 S S N N N N S N N S 20 S S 20 N

10 10

010203040x 010203040x

Fig. 4. Bound states of solitons with m1 = m2 = 0, g0 = 2.11, and d = 0.06: (a, c) instantaneous transverse intensity distributions I(x, y) and (b, d) phase portraits for (a, b) weakly and (c, d) strongly coupled solitons. Dashed curves are separatrices associated with saddle points S. Arrows indicate the direction of the Poynting vector. focus, the next cycle is a “stable” one that attracts tra- system is not robust (degenerate). In this case, the circle jectories as τ ∞, and the outermost cycle is “unsta- encompassing the “stable” node is a set of individual ble” since the trajectories lying outside it tend to inﬁn- ﬁxed points rather than a limit cycle. A small perturba- ity as r ∞. Figures 1Ð3 illustrate cases when m ≥ 0. tion, such as a distant soliton, gives rise to two pairs of Those corresponding to m < 0 can readily be obtained “unstable” nodes N and saddle points S. The four circu- by coordinate inversion, as in (2.5). When m = 0, the lar arcs that connect them are separatrices incident to

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 434 ROZANOV et al. the saddle points (see Fig. 4b). This pattern is robust of time (not control parameter). The corresponding (not degenerate). field dynamics are found numerically by solving Eq. (2.1) with function (2.9) by a splitting method with the use of fast Fourier transform. 5. WEAK AND STRONG INTERACTION BETWEEN CAVITY SOLITONS Figures 4c and 4d show the instantaneous intensity distribution and energy-flow pattern in a steady two- Suppose that the distance L between the centers of hump structure rotating with a constant angular veloc- two solitons is much greater than the size of a single ity. This structure can be interpreted as a strongly cou- soliton, which is determined by the radius of the outer- pled state of two fundamental cavity solitons (with zero most limit cycle. Then, the distortion of their structure topological charges). Note that the phase portrait in caused by the overlap of soliton tails is weak and the Fig. 4d contains only “unstable” fixed points. In other instantaneous field can be expressed as words, the two nodes that are “stable” in the case of () []Ψ() ϕ weakly coupled solitons (see Fig. 4b) transform into EAr= 1 exp i r1 + im1 1 (5.1) “unstable” nodes in the case of strongly coupled soli- () []Ψ () ϕ ϑ + Ar2 exp i 0 r2 ++im2 2 i . tons (see Fig. 4d). Another manifestation of strong coupling is the absence of “individual” closed orbits For simplicity, we consider solitons whose topological encompassing the nodes (characteristic of weak cou- | | | | charges are equal in absolute value ( m1 = m2 ). This pling), which explains the change in “stability” of the implies that the solitons are associated with equal nodes. Almost every trajectory tends to infinity as eigenvalues ν and their amplitudes A and radial phases τ ∞. The separatrices of the saddle point S lying Ψ 0 are similar functions of radius. However, the argu- between the nodes partition the plane into cells occu- ϕ ments of these functions are different: rn and n (n = 1, 2) pied by trajectories going to infinity along different are polar coordinates with origin at the center of the nth directions. The bound solitons rotate as a whole without soliton: any dislocation of the wavefront. This demonstrates that field asymmetry is the key condition for rotation [11]. 2 ± 2 ± L 2 ϕ xL/2 Now, consider the interaction between two cavity rn ==x --- + y , cos n ------, 2 rn solitons with topological charges m = m = 1 (see (5.2) 1 2 Fig. 5). The corresponding strongly coupled state ϕ y sin n = ----. evolves in time from an initial superposition (5.1) of rn two independent fields with certain ϑ and L at t = 0. In The constant phase difference ϑ affects the interaction Fig. 5a, the individual solitons that make up the pair are between the solitons [11]. represented by the “stable” foci corresponding to wave- Since the degree of overlap is low, the energy flows front dislocations. (The field amplitude vanishes at near the solitons’ centers change insignificantly. these points and remains finite at other fixed points.) Accordingly, the pattern of Poynting-vector stream- The final steadily rotating structure of a strongly cou- lines determined by Eqs. (3.3) preserves all closed pled state (see Fig. 5f) preserves only one of the three orbits that encompass the fixed points representing individual limit cycles associated with each of the two individual solitons (stable and unstable limit cycles, as dislocations (the inner “unstable” one, see Fig. 3b). well as closed orbits consisting of individual separa- However, the system has two “common” limit cycles trices). Additional fixed (typically, saddle) points may encompassing both foci, one of which is “stable” and arise in the intermediate regions between them. The the other is “unstable.” The trajectories lying outside separatrices emanating from these points partition the the outer limit cycle go to infinity, whereas those inside plane into cells occupied by trajectories moving away it either approach the “stable” common limit cycle or from individual solitons. The intensity distribution and tend to the stable foci (they are partitioned by the sepa- energy-flow pattern corresponding to a steady pair of ratrices of the saddle point S lying between the foci). relatively weakly coupled solitons with zero topologi- The separatrices incident to the saddle point unwind off cal charges are depicted in Figs. 4a and 4b, respectively. the individual “unstable” limit cycles, while the outgoing separatrices wind to the “stable” common limit Bound state (5.1) with a constant phase difference ϑ cycle. The overall partition of the phase portrait into and an arbitrary distance L between the solitons may cells of steady-state trajectories characterized by differ- ӷ vary with time. The constant distances L R0 corre- ent behavior is obvious from Fig. 5f. Actually, transi- sponding to steady (stable or unstable) pairs of weakly tion between the initial and final states involves a time interacting solitons can be found by substituting (5.1) sequence of bifurcations of the energy-flow pattern. into (2.8). As the distance between the solitons Some of these are illustrated by Fig. 5. In particular, the ≤ decreases to L R0 in the course of time, the interaction transition between the phase portraits shown in Figs. 5a becomes stronger and the phase portrait becomes qual- and 5b involves a bifurcation in which the outer two itatively (topologically) different from that in the case limit cycles encompassing the left focus coalesce and ӷ of L R0. Note that this change in internal soliton disappear. The transition from Fig. 5b to Fig. 5c can be structure is due to bifurcations associated with variation associated with a bifurcation at the central saddle point

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y y 80 (a) (b) 60 S S

0 20 40 60 80 020406080 x x 80

(e) (f) 60 N S S S 40

0 20 40 60 80 020406080 x x

Fig. 5. Instantaneous phase portraits of energy ﬂows for an evolving, strongly coupled, rotating pair of solitons with m1 = m2 = 1, ∞ g0 = 2.11, and d = 0.06 at t = 0 (a), 5.6 (b), 6 (c), 30 (d), 59 (e), and (f).

S that gives rise to a homoclinic orbit encompassing the is more complicated. This rotating and slowly moving right outer limit cycle. Several bifurcations, including structure evolves from superposition (5.1) in a rela- the last one (in transition from Fig. 5e to Fig. 5f), cor- tively long time interval. The wavefront dislocations respond to the appearance and coalescence of two ﬁxed (centers of individual solitons) are represented by the points, a node N and a noncentral saddle point S (see “stable” foci in Fig. 6. In the ﬁnal state, both foci are Figs. 5d and 5e). encompassed by single “unstable” limit cycles (with opposite senses of rotation corresponding to opposite The phase portrait of a strongly coupled pair of soli- charges). The trajectories unwinding off these limit − tons with opposite topological charges (m1 = 1, m2 = 1) cycles include separatrices of the adjacent saddle

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 436 ROZANOV et al.

(a) (c) y y

60 60

40 40

20 20

0 20 40 60 x 0 20 40 60 x y (b)y (d)

60 60

S N S 40 S 40

N S 20 20

0 20 40 60 x 0 20 40 60 x

Fig. 6. Instantaneous (a, c) transverse distributions I(x, y) and (b, c) phase portraits for pairs of solitons with m1 = 1 and m2 = Ð1 at (a, b) t = 0 and (c, d) t = 10000. points. Each of the three saddle points emits a separatrix structure” of cavity solitons. Even in the case of an indi- going to infinity. The separatrices emanating from differ- vidual soliton with an axially symmetric intensity dis- ent saddle points asymptotically converge. In addition to tribution, the energy-flow plane is partitioned by con- those from the two foci and three saddle points, there are centric circles into several cells occupied by topologi- two “unstable” nodes in the phase plane. The rays ema- cally equivalent trajectories (energy-flow streamlines). nating from the nodes include saddle separatrices. The radius of each circle corresponds to an extremum Interactions between cavity solitons can give rise to in the radial phase distribution. The phase portrait of a strongly coupled states of other types. In particular, pair of weakly interacting solitons retains some “indi- trains of several solitons with m = 1 and two- and three- vidual” features (fixed points and weakly distorted cir- wave trains including both fundamental and excited cular periodic orbits). The additional “collective” fea- solitons were computed by solving Eq. (2.1) (see tures typically include a saddle point with two incident Figs. 1 and 2 in [23]). All of these states are rotating, separatrices coming from closed orbits and two separa- and their phase portraits are analogous to those dis- trices going to infinity. The interaction is weak if the cussed above. distance between the solitons is much greater than the diameter of the outermost “individual” closed orbit. As the distance between the solitons decreases, a sequence 6. CONCLUSIONS of bifurcations results in a qualitatively (topologically) The phase portraits of energy flows (Poynting vec- different phase portrait corresponding to strongly inter- tor) provide important information about the “internal acting solitons. Individual elements (outer trajectories

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ENERGY-FLOW PATTERNS AND BIFURCATIONS 437 in the first place) are replaced by “collective” ones, 8. M. Brambilla, L. A. Lugiato, F. Prati, et al., Phys. Rev. such as limit cycles encompassing both solitons. Both Lett. 79, 2042 (1997). “stability” properties and number of fixed points may 9. V. B. Taranenko and C. O. Weiss, IEEE J. Sel. Top. change. Even though the bifurcations alter the distribu- Quantum Electron. 8, 488 (2002). tion of light intensity, the phase portraits of energy 10. IEEE J. Quantum Electron. 39 (2) (2003). flows appear to be more informative. 11. N. N. Rosanov, Spatial Hysteresis and Optical Patterns The variety of strongly coupled soliton structures (Springer, Berlin, 2002). turns out to be very wide. They include trains and pairs 12. N. N. Rozanov, Optical Bistability and Hysteresis in characterized by strongly asymmetric fields. Analysis Distributed Nonlinear Systems (Nauka, Moscow, 1997) of the corresponding energy flows is also an important [in Russian]. task. Note that analysis of energy diagrams (broadly 13. A. F. Suchkov, Zh. Éksp. Teor. Fiz. 49, 1495 (1965) [Sov. analogous to Poynting-vector patterns) would also be Phys. JETP 22, 1026 (1966)]. useful in studies of dissipative solitons of different (not 14. S. V. Fedorov, A. G. Vladimirov, G. V. Khodova, and only optical) nature. N. N. Rosanov, Phys. Rev. E 61, 5814 (2000). 15. N. N. Rozanov, S. V. Fedorov, and G. V. Khodova, Opt. Spektrosk. 88, 869 (2000) [Opt. Spectrosc. 88, 790 ACKNOWLEDGMENTS (2000)]. 16. N. N. Rozanov, S. V. Fedorov, and A. N. Shatsev, Opt. We thank A.M. Kokushkin for assistance in preparing Spektrosk. 91, 252 (2001) [Opt. Spectrosc. 91, 232 the graphs. This work was supported by the Russian (2001)]. Foundation for Basic Research, project no. 02-02-17242. 17. S. V. Fedorov, N. N. Rosanov, A. N. Shatsev, et al., IEEE J. Quantum Electron. 39, 197 (2003). 18. M. B. Vinogradova, O. V. Rudenko, and A. P. Sukho- REFERENCES rukov, The Theory of Waves, 2nd ed. (Nauka, Moscow, 1. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and 1990) [in Russian]. L. P. Pitaevskiœ, Theory of Solitons: the Inverse Scatter- 19. S. N. Vlasov and V. I. Talanov, Wave Self-Focusing (Inst. ing Method (Nauka, Moscow, 1980; Consultants Bureau, Prikl. Fiz. Ross. Akad. Nauk, Nizhni Novgorod, 1997) New York, 1984). [in Russian]. 2. E. M. Dianov, P. V. Mamyshev, and A. M. Prokhorov, 20. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and Kvantovaya Élektron. (Moscow) 15, 5 (1988). A. G. Maœer, Theory of Bifurcation of Dynamic Systems on Surface (Nauka, Moscow, 1967) [in Russian]. 3. B. S. Kerner and V. V. Osipov, Autosolitons (Nauka, Moscow, 1991; Kluwer, Dordrecht, 1994). 21. N. V. Butenin, Yu. I. Neœmark, and N. A. Fufaev, Intro- duction to Theory of Nonlinear Oscillations (Nauka, 4. N. N. Rozanov and G. V. Khodova, Opt. Spektrosk. 65, Moscow, 1976) [in Russian]. 1399 (1988) [Opt. Spectrosc. 65, 828 (1988)]. 22. M. S. Soskin and M. V. Vasnetsov, Prog. Opt. 42, 219 5. N. N. Rosanov, A. V. Fedorov, and G. V. Khodova, Phys. (2001). Status Solidi B 150, 545 (1988). 23. N. N. Rozanov, S. V. Fedorov, and A. N. Shatsev, Opt. 6. N. N. Rozanov and S. V. Fedorov, Opt. Spektrosk. 72, Spektrosk. 95, 902 (2003) [Opt. Spectrosc. 95, 843 1394 (1992) [Opt. Spectrosc. 72, 782 (1992)]. (2003)]. 7. D. Michaelis, U. Peschel, and F. Lederer, Phys. Rev. A 56, R3366 (1997). Translated by A. Betev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 438Ð454. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 499Ð517. Original Russian Text Copyright © 2004 by Prudnikov, Taichenachev, Tumaœkin, Yudin. ATOMS, SPECTRA, RADIATION

Kinetics of Atoms in the Field Produced by Elliptically Polarized Waves O. N. Prudnikov, A. V. Taichenachev, A. M. Tumaœkin, and V. I. Yudin* Novosibirsk State University, Institute of Laser Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia *e-mail: [email protected] Received July 31, 2003

Abstract—The kinetics of atoms with degenerate energy levels in the field produced by elliptically polarized waves is considered in the semiclassical approximation. Analytic expressions for the force acting on an atom and for the diffusion coefficient in the momentum space are derived for the optical transition Jg = 1/2 Je = 1/2 in the slow atom approximation. These expressions are valid for an arbitrary one-dimensional configuration of the light field and for an arbitrary intensity. The peculiarities of the atomic kinetics are investigated in detail; these peculiarities are associated with ellipticity of light waves and are absent in particular configurations formed by circularly or linearly polarized waves, which were considered earlier. © 2004 MAIK “Nauka/Inter- periodica”.

1. INTRODUCTION on the kinetics of atoms in the field produced by elliptically polarized waves is interesting in itself. For At the initial stage (up to 1988), the mechanical example, as was proved in our earlier publication [8], action of resonance radiation on atoms and, in particu- ellipticity leads to qualitative differences in the kinetics lar, the motion of atoms in a light field were studied as compared to the results obtained by using the two- comprehensively in the framework of the simplest level model even in the simple case of a uniformly model of a two-level atom [1, 2]. This description has polarized field. made it possible to explain the physical mechanisms and the origin of forces acting on an atom in a light field Here, the motion of atoms in the field of a one- and to predict the minimal temperature of laser-assisted dimensional configuration formed by waves with arbi- បγ γ trary elliptical polarizations is studied theoretically. In cooling of atoms (Doppler limit) kBTD ~ , where is the natural width of an excited level. Experimental the framework of the semiclassical approach, the kinetics of atoms is described by the FokkerÐPlanck equa- observation of temperatures below TD [3, 4] stimulated further development of the theory. Detailed analysis tion. Analytic expressions are derived for the force act- proved that sub-Doppler cooling is associated with the ing on an atom as well as for the diffusion and friction degeneracy of the ground atomic state in the projection coefficients in the slow atom approximation for the of the angular momentum. It should be noted that, in optical transition Jg = 1/2 Je = 1/2 (where Jg and Je spite of a large number of publications concerning the are total angular momenta of the ground (g) state and an theory of sub-Doppler cooling [5, 6], only a limited excited (e) state). These expressions are analyzed for a class of laser field configurations formed by linearly or number of specific field configurations; in each case, circularly polarized waves was studied. Nevertheless, new terms associated with the ellipticity of waves are these simple examples have made it possible to single separated and analyzed. The physical mechanisms out two main mechanisms of friction leading to sub- leading to these new contributions are interpreted qual- Doppler cooling. In the case of counterpropagating lin- itatively. early polarized waves, we are dealing with the Sisyphean mechanism of friction, associated with the action of induced light pressure, while in the case of 2. FORMULATION OF THE PROBLEM σ σ counterpropagating circularly polarized waves ( +Ð Ð configuration), the mechanism is of the orientation type We consider the motion of atoms undergoing the and is associated with the action of spontaneous light optical transition Jg Je in a resonance monochro- pressure [7]. However, the analysis of these particular matic field cases is not exhaustive in view of the nonlinear nature ω of interaction of atoms with the resonance field and Er(), t = Er()eÐi t + c.c. gives no idea about the motion of particles in a field with a more general configuration. Thus, the problem We represent the spatially inhomogeneous vector

KINETICS OF ATOMS IN THE FIELD PRODUCED BY ELLIPTICALLY POLARIZED WAVES 439 amplitude in the form this equation is a complicated problem. However, in the iΦ()r semiclassical approximation, rapid processes of order- Er()= E()r er()e , (1) ing in internal degrees of freedom can be separated where E(r) is the real-valued amplitude and e(r) is the from the slow processes associated with translational unit complex polarization vector. The field phase Φ(r) motion of an atom if the following three basic condi- is defined so that e(r) á e(r) = cos(2ε(r)) is a real-valued tions are satisfied. quantity determining the local ellipticity of the light 1. The dispersion ∆p of an atomic momentum con- ប field (ε(r) is the ellipticity parameter and tanε is siderably exceeds the recoil momentum k of a photon: equal to the ratio of the semiminor axis of the polariza- បk ------Ӷ 1. (7) tion ellipse to its semimajor axis). The Hamiltonian of ∆p a free atom in a rotating (in the energy pseudospin space) basis has the form 2. The time of interaction between an atom and the field must exceed the characteristic time of evolution of pˆ 2 the internal degrees of freedom of the atom, Hˆ = ------Ð បδΠˆ , (2) 0 2M e Ð1 Ð1 δ ω ω t ӷ τ = max{}γ , ()γG , (8) where = Ð 0 is the field frequency detuning from ω the atomic transition frequency 0, M is the mass of the where atom, and the projection operator Ω 2 Πˆ e = ∑|〉J , µ 〈|J , µ (3) G = ------e e e e ()γ 2 δ2 µ 3 /4 + e is constructed from the wave functions of the Zeeman is the saturation parameter. | µ 〉 sublevels Je, e of the excited state. In the dipole 3. The recoil energy ប2k2/2M is much smaller than approximation, the operator of resonant interaction បτÐ1, with field (1) can be written in the form

q 2 ˆ បΩ ˆ បk Ð1 V()r = ()r ∑ Dqe (r )+ H.c., (4) ------Ӷ τ . (9) q = 01, ± M Ω ប where = ÐdE/ is the Rabi frequency, d is the reduced It was shown in [1, 2, 9, 10] that, under the above con- matrix element, and eq(r) are contravariant components ditions, Eq. (6) can be reduced to the equation of the polarization vector in the cyclic basis ᐃ(r, p, t) = Tr{ρˆ (r, p, t)} for the atomic distribution − {}e = e , e± = +()e ± ie /2. function in the phase space (the trace is taken over 0 z 1 x y internal states of the density matrix in the Wigner rep-

Operator Dˆ q can be expressed in terms of the 3jm resentation): symbols:  ∂ pi J Ð µ + ∑-----∇ ᐃ e e ∂ i Dˆ q = |〉J , µ ()Ð1 t M ∑ e e i µ , µ (10) e g (5) ∂ ∂2  , ᐃ , ᐃ J 1 J = Ð ∑∂ Fi()rp + ∑∂------∂ Dij()rp . × e g 〈|J , µ . pi pi p j µ µ g g i ij Ð e q g The kinetic coefficients Fi(r, p) and Dij(r, p) are the The state of an atomic ensemble is described by the components of the force and the diffusion tensor in the one-particle density matrix ρˆ ; the quantum-mechani- Cartesian basis. cal kinetic equation for this matrix has the form 3. KINETIC COEFFICIENTS ∂ ρ i []ˆ , ρ i []Γˆ , ρ ˆ {}ρ FOR SLOW ATOMS ∂ ˆ = Ðប--- H0 ˆ Ð ប--- V()rˆ ˆ Ð ˆ , (6) t The kinetic coefficients of the FokkerÐPlanck equation can be determined from the equations derived by where Γˆ {}ρˆ is the radiation relaxation operator. Inter- the reduction of Eq. (6) to Eq. (10) and depend on the atomic interactions will be disregarded. Since Eq. (6) distribution of atoms over internal states. These coeffi- describes the evolution of external as well as internal cients are often determined using the approximation of (translational) degrees of freedom of an atom, solving slow atoms, which are displaced over a distance much

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 440 PRUDNIKOV et al. shorter than the wavelength of light during the relaxation with the normalization condition Tr{σˆ } = 1 and time for internal degrees of freedom; i.e., vτ Ӷ λ or describes the steady-state distribution of atoms over magnetic sublevels in the zeroth approximation in the kv Ӷ γγ, G. (11) recoil parameter and the velocity of atoms. It should be noted that the explicit analytic form of σˆ for all dipole In this case, to take into account dissipative processes transitions was determined in [11, 12]. correctly, it is sufficient to confine the analysis to the linear approximation (in velocity v = p/M) for the force, The coefficient of friction is proportional to the spatial gradient of σˆ : , ≈ ξ ξ ប {}ϕˆ ∇ σˆ Fi()rv Fi()r + ∑ ij()r v j, (12) ij()r = Ð kTr i j . (16) j The diffusion coefficient can be represented as the sum of two terms, and to the zeroth approximation for diffusion, ()s ()i , ≈ Dij()r = Dij ()r + Dij (),r (17) D()rvij D()r ij. (13) Quantity F(r) is the force of light pressure acting on a the first of which, ξ stationary atom. The asymmetric part of tensor ij cor- γ ()s ()ប 2 responds to the effective Lorentz force, while its sym- Dij ()r = --- k metric part defines friction. 5 (18) The general procedure for deriving the FokkerÐ  × δ Πˆ 2Je + 1()ˆ ˆ † ˆ ˆ † σ Planck equation via the semiclassical expansion in Trij e Ð ------Di D j + D j Di ˆ , small parameter បk/∆p is described in detail in [1, 2, 9, 4 10]. However, the determination of the explicit form of the kinetic coefficients for atoms with energy levels is due to recoil in the case of spontaneous emission, degenerate in the angular momentum projection in a while the second term, field of the most general form is a complex mathematical problem even for small values of angular momenta ()i បk J and J . For example, in order to determine the fric- Dij ()r = ------g e 4 (19) tion and diffusion coefficients, it is necessary to calcu- 〈 〈 〈 〈 × {}ϕ ()σσδ δ σ ()ϕδ δ σ late and integrate a matrix exponential [9]. The com- Tr ˆ i ˆ F j + F j ˆ + ˆ Fi + Fi ˆ ˆ j , plexity of such calculations increases rapidly with the angular momentum of atomic levels. It is probably for is determined by the operator of force fluctuation, this reason that the atomic kinetics was analyzed only 〈 for several particular light field configurations formed δ ˆ by linearly or circularly polarized traveling waves. F()r = F()r Ð F().r (20) An alternative method for determining the kinetic Matrix ϕˆ used in formulas (16) and (19) satisfies the coefficients was proposed (without derivation) in [8]. This method, which is in line with the methods devel- equation (see Appendix A) oped earlier, is more convenient in our opinion since the search for the coefficients of friction and diffusion is γ γ i --- + iδ Πˆ eϕˆ + --- Ð iδ ϕˆ Πˆ e Ð ---[]Vˆ , ϕˆ reduced to solving a single algebraic equation for an 2 i 2 i ប i auxiliary matrix ϕˆ . Here, we will use these results (the 〈 (21) δ derivation is given in Appendix A). † Fi Ð γ ()2J + 1 ∑Dˆ qϕˆ Dˆ q = ------e i បk The gradient force acting on a stationary atom is a q quantum-mechanical mean of the force operator Fˆ , and makes it possible to write the expressions for the {}ˆ σ ˆ ∇ ˆ friction and diffusion coefficients in a universal form. Fi()r == Tr Fi()r ˆ ()r , Fi()r Ð iV(),r (14) σˆ where the density matrix ()r is the solution to the 4. ANALYTIC EXPRESSIONS optical Bloch equation FOR KINETIC COEFFICIENTS γ γ FOR TRANSITION Jg = 1/2 Je = 1/2 --- Ð iδ Πˆ σˆ + --- Ð iδ σˆ Πˆ 2 e 2 e We will consider here one-dimensional configura- (15) tions of a light field, when the vector amplitude is a i † function of only one coordinate z. In the general case, []γˆ , σˆ ()ˆ σˆ ˆ + ប--- V Ð02Je + 1 ∑Dq Dq = such a field is formed by two counterpropagating trav- q eling plane waves with arbitrary intensities and ellipti-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 KINETICS OF ATOMS IN THE FIELD PRODUCED BY ELLIPTICALLY POLARIZED WAVES 441 cal polarizations. In this case, the local vector field tons. It should be noted that expression (23) does not ∇ φ amplitude (1) can be characterized by four real-valued contain the contribution from the gradient z of orien- quantities: the real amplitude, phase, ellipticity, and the tation of the polarization ellipse. This is due to the spe- angle of orientation of the polarization ellipse. In the σ cific form of matrix ˆ for optical transitions Jg = J coordinate system where the z axis is orthogonal to the J = J (J is a half-integer) [12, 13], for which the mag- polarization plane, vector e(z) in a cyclic basis can be e ε netic sublevels of the excited state are populated uni- written in terms of the local ellipticity (z) of the light formly (isotropically). field and the angle of rotation ϕ(z) of the polarization ellipse: 4.2. Coefficients of Friction and Diffusion π π ε Ðiφ ε Ðiφ Coefficients of friction, ξ = ξ , and induced diffu- e()z = cos + --- e eÐ1 Ð cos Ð --- e e+1. (22) zz   () 4 4 (i) i sion, D = Dzz , can be written in the general form, For simplicity, we will henceforth consider a one- ξ ប χ ∇ β∇ β dimensional problem. It should be noted that, in the = ∑ ββ' z z ', (24) general case, the kinetic problem can be reduced to a ββ' one-dimensional problem only approximately, since recoil processes occurring during spontaneous reemis- ()i ប2 Ᏸ ∇ β∇ β sion of photons of the field result in the formation of a D = ∑ ββ' z z ', (25) coupling between translational degrees of freedom of ββ' an atom in all three directions. Coupling in the longitu- β β Λ ε dinal and transverse directions relative to the z axis can where summation indices and ' correspond to { , , Φ φ χ Ᏸ be neglected when the width ∆p⊥ of the momentum dis- , }, and components ββ' and ββ' can be expressed tribution in the transverse direction considerably in the parameters of the light field. Since in the problem exceeds the width ∆p of the momentum distribution in under investigation we are dealing only with gradients z along the z axis, the coefficients of friction and induced the longitudinal direction (∆p⊥ ӷ ∆p ) and the recoil z diffusion contain only symmetric combinations of non- effect in the transverse direction does not lead to a β ≠ β χ χ Ᏸ Ᏸ noticeable change in the momentum distribution. Such diagonal ( ') terms: ββ' + β'β and ββ' + β'β. It χ Ᏸ a model is applicable, for example, for a problem on should be noted that expressions ββ' and ββ' are also interaction of an atomic beam with a transverse light applicable in the 3D formulation of the problem. In this field. Henceforth, we will assume that the force and the case, it remains for us to calculate the required gradi- diffusion tensor are functions only of coordinate z and ents of the light field parameters to find the tensors of the corresponding velocity component, which will be friction and diffusion. Note that, in the most general 3D denoted by v. In the notation adopted here, it is conve- formulation of the problem, we must take into account, nient to express kinetic coefficients in terms of the grain addition to the available gradients of {Λ, ε, Φ, φ}, the ∇ Φ ∇ φ ∇ ε dients of light field parameters: z , z , and z (gra- two angles describing the rotation drawing from the dients of the phase, the angle of rotation of the polariza- plane of the polarization ellipse. However, for 3D prob- ∇ Λ Λ lems in which the plane of the polarization ellipse tion ellipse, and the ellipticity) as well of z ( = lnE) (gradient of the logarithm of the real part of the field remains unchanged, the above results are sufficient. amplitude). The expressions for the components of the coefficients χ Ᏸ of friction ββ' and diffusion ββ' are given in Appen- dix B. 4.1. Force () Spontaneous diffusion coefficient D(s) = D s is pro- σ zz Using the explicit form for matrix ˆ (15), we obtain portional to the total population of the excited level and the following expression for force in the zeroth order in is independent of the external field gradients: velocity: 2 ()s γ ()បk Gε ប γ D = ------. (26) Gε δ()()∇ε ε∇Λ ∇ Φ Fz = ------tan 2 z Ð z +.--- z (23) 12 1 + Gε 1 + Gε 2 The expressions for the force and the coefficients of For the sake of simplicity, we introduced here the effec- friction and diffusion derived here make it possible to 2 tive saturation parameter Gε = Gcos (2ε). The first describe the kinetics of slow atoms with the optical term, proportional to δ, is the force of induced light transition Jg = 1/2 Je = 1/2, which is a simple model pressure emerging from the gradients of ellipticity and for atoms with energy levels degenerate in the angular light field amplitude, while the gradient of phase makes momentum projection. It should be noted that simple a contribution to the force of spontaneous light pres- configurations of light field exist, which are character- sure, associated with spontaneous rescattering of pho- ized by the gradient of only one parameter. For exam-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 442 PRUDNIKOV et al. ple, a standing uniformly polarized wave is character- atoms in a traveling wave is associated with fluctuation ∇ Λ ized by intensity gradient z , a traveling uniformly of the number of photons scattered from an atom [1, 14]. ∇ Φ polarized wave is characterized by phase gradient z , This quantity is determined by the statistics of the num- σ σ the field of the +Ð Ð configuration is characterized by ber of scattered photons; for a two-level atom in a low- ∇ φ intensity field (G Ӷ 1), the distribution is of the Pois- gradient z of the orientation of the polarization vector, while the field of the lin ⊥ lin configuration is char- son type. The presence of the additional term in ∇ ε expression (28) indicates a deviation from the Poisson acterized by ellipticity gradient z . For these configurations, the coefficients of friction and diffusion are distribution in the statistics of the number of photons scattered in a low-intensity field. These deviations are determined by the corresponding diagonal elements χββ due to correlation of processes of emission of photons and Ᏸββ. σ σ with the + and Ð polarization in an elliptically polarized field. Indeed, the scattering probability σ± of a cir- 5. UNIFORMLY POLARIZED FIELD cularly polarized quantum of the external field in a small time interval of ∆t, In a uniformly polarized field, the ellipticity and orientation of the polarization ellipse are independent of 12± sin()ε p± = γG------π− ∆t, the coordinate; i.e., such a field is characterized only by 2 + the gradients of phase and intensity. On the other hand, precisely these gradients appear in the description of depends on the degree of ellipticity and the probability the atomic kinetics in the nondegenerate two-level π± of finding an atom in the ground state with an angu- model of an atom [1, 2]. For this reason, it would be µ ± π π lar momentum projection of g = 1/2 ( + + Ð = 1). Let especially important to consider this type of light fields σ us suppose, for example, that a + photon was scattered separately and compare the expressions for the kinetic π π coefficients (force and coefficients of friction and diffu- in a certain interval; then, + = 1/3 and Ð = 2/3 for a sion) obtained in this way with the available results for subsequent interval in accordance with the relative the two-level model of an atom. This will enable us to probabilities of spontaneous decay via the channels |µ 〉 |µ 〉 |µ 〉 |µ 〉 determine the differences associated with the degener- e = 1/2 g = 1/2 and e = 1/2 g = Ð1/2 . σ acy of atomic levels in the angular momentum projec- On the contrary, after the scattering of a Ð photon, we π π ε ≠ tion and to single out the effects determined by the have + = 2/3 and Ð = 1/3. Consequently, for 0, the ellipticity of a light field. scattering probability for a photon (of any type), G p + p = γ ----[]∆1 Ð ()π Ð π sin()2ε t, 5.1. Elliptically Polarized Traveling Wave + Ð 2 + Ð A simple example of a uniformly polarized configuration of a light field is a traveling plane monochro- depends on the polarization of the quantum scattered matic wave, before this event; i.e., the scheme of independent trials is violated. The average number of scattered quanta is , ikz Ðiωt determined by the mean values E()zt = E0ee e + c.c. (27) ± ()ε Light field amplitude E0 and polarization vector e are 12sin π± = ------. spatially homogeneous, while phase Φ = kz is a func- 2 tion of the coordinate. The force acting on an atom in this field is proportional to phase gradient (23) and is Consequently, the statistics of the number of photons is a force of spontaneous light pressure by nature. not Poissonian any longer; we can conclude from rela- Expressions (B.3) for the friction coefficient, as well as tion (28) that the statistics is super-Poissonian by the expressions for the force, coincide in form with the nature. results for a two-level atom with an effective saturation parameter of Gε . The most significant difference 5.2. Uniformly Polarized Standing Wave from the model of a two-level atom appears in expression (B.19) for the diffusion coefficient, In a uniformly polarized standing wave, the real amplitude (1) is a periodic function of coordinate ()i 2 2 γ Gε ប (E(z) = E0cos(kz)) and polarization vector e is spatially D = k ------3 4()1 + Gε homogeneous. In such a field, only the gradient of (28) Λ 2 intensity differs from zero; consequently, the force 2 2 γ acting on a stationary atom, × ()ε ()ε ε 1 + G ++32sin G 1 Ð ------2 -2 ; γ /4 + δ បδ ∇ Λ Gε z 2 ε F(z) = Ð------, (29) namely, an additional term proportional to sin (2 ) 1 + Gε appears, which is significant in a low-intensity field (Gε Ӷ 1). It is well known that induced diffusion of is the force of induced light pressure. The coefficients

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 KINETICS OF ATOMS IN THE FIELD PRODUCED BY ELLIPTICALLY POLARIZED WAVES 443 〈ξ 〉 of friction and induced diffusion have the form Contribution 1 corresponds to the well-known Dop- pler mechanism of friction [1, 2]. For large detunings ξ បχ ∇ Λ 2 ()z = ΛΛ()z , (30) (δ ӷ γ), the relation ()i 2 2 〈〉ξ 2 D ()z = ប ᏰΛΛ()∇ Λ . (31) 2 δ z ------2 ≈ 62sin ()ε ----- 〈〉ξ 2 The case of a homogeneously polarized standing wave 1 γ ε with an arbitrary analyticity was considered in our holds, which shows that, in an elliptically polarized earlier publication [8]. Henceforth, we will consider field (ε ≠ 0), the additional contribution can exceed the only some interesting features in the kinetics which had result for the two-level model by several orders of mag- not been analyzed earlier. In particular, we will find the nitude. difference between our results and the results obtained ӷ in the nondegenerate model of an atom on the basis of In intense fields (Sε 1), the expressions for the spatially averaged expressions for the coefficients of coefficients assume the form friction and diffusion. δ ()γ 2 δ2 〈〉ξ ប 2 Ð 12 Sε The gradient force averaged over the spatial period 1 = k ------, (38) of a light field vanishes (〈F(z)〉 = 0); the friction coeffi- 4γ 4δ2 + γ 2 cient can be represented as the sum of two contribu- 2 tions: 92sin ()ε δ Sε 〈〉ξ = បk2------. (39) 〈〉ξ 〈〉 ξ 〈〉 ξ 2 4γ = 1 + 2 . (32) The first term, It should be noted that the sign of the friction coefficient in the two-level model of an atom in a high- 2 intensity field is determined not only by the sign of បk δγSε()2 + Sε 〈〉ξ detuning, but also by its magnitude (38). The inclusion 1 = ------3/2 2 2 4 ()1 + Sε ()δ + γ /4 of degeneracy of atomic levels leads to the following (33) interesting effect: for field ellipticities sin2(2ε) > 1/3, in 3 2 5/2 δ()83++Sε 15Sε +20Sε Ð 81()+ Sε view of additional contribution (39), the sign of the Ð ------γ 5/2 total friction coefficient (32) is determined only by the ()1 + Sε sign of detuning: heating for δ > 0 and cooling for δ < 0. corresponds to the expression in the two-level model of The averaged coefficient of induced diffusion, as an atom [1] with the effective saturation parameter well as the friction coefficient, can be presented as the sum of two terms: 2 2 E d cos ()2ε () () () 0 〈〉i 〈〉i 〈〉i Sε = ------. (34) D = D1 + D2 . (40) ប23()δ2 + γ 2/4 The first term represented the familiar result [1] for a The second term, two-level atom with a new parameter Sε: 2 2 3δsin ()2ε Sε()43+ Sε 2 2 2 2 2 〈〉ξ ប () ប k γ Sε ប δ Sε 2 = k ------(35) 〈〉i k γ ()3/2 D1 = ------+ ------γ - 3 + ----- 4 1 + Sε 8 2 (41) is an additional contribution to friction and may be inter- 2 2 2 2 2 Ð1 preted in terms of the probabilities of transitions between 15Sε +++40Sε 24 γ Sε ()δ + γ /4 Ð ------. dressed states of the atom (Sisyphus effect) [8, 15]. For 3/2 81()+ Sε small saturation parameters (Sε < 1), the main contribution to this effect comes from transitions between The second term, dressed states, which correspond to the Zeeman sublevels of the ground state; this leads to an additional con- ប2 2 2 ()ε 〈〉()i k sin 2 tribution to friction. D2 = ------γ Ӷ In the low-intensity limit of the light field (Sε 1), (42) 〈ξ 〉 〈ξ 〉 2 2 coefficients 1 and 2 have the form 3δ Sε()43+ Sε 2 γ 1 + Sε/2Ð 1 + Sε × ------+ δ + ------2/3 4 2 ប 2 δγ 81()+ Sε 32cos ()ε 〈〉ξ k Sε 1 = ------2 2 -, (36) 2 δ + γ /4 is additional as compared to the two-level model of the atom and describes the contributions from the diffusion δSε 〈〉ξ = បk232sin2 ()ε ------. (37) processes associated with Zeeman degeneracy of 2 γ energy levels.

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T 〈ξ〉 < 0 and the expression for temperature has sense 1.0 only for δ < 0. For the linear ﬁeld polarization, the result can be reduced to the known expression for the Doppler temperature of laser cooling in the model of a 0.8 two-level atom,

0.6 ប()4δ2 + γ 2 k T()ε = 0 = Ð------; (44) B 12δ 0.4 the minimum of this temperature is attained for δ = Ðγ/2 បγ and amounts to kBTD = /3. In the general case, the 0.2 inclusion of the field polarization leads to a slightly different dependence of temperature on detuning (Fig. 1). 0 It can be seen that the temperature can assume values –2.0 –1.5 –1.0 –0.5 0 smaller than in the case of the linear polarization of the δ field. It can be verified that the lowest temperature value is reached for a field ellipticity of |ε| π/4 (i.e., បγ Fig. 1. Temperature in units of /kB as a function of detun- for the circular polarization). ing for various ellipticities of the light field: the solid curve In the other limiting case of a high-intensity light corresponds to the linear polarization of the field, the ӷ dashed curve corresponds to a field ellipticity of sin2(2ε) = field (Sε 1), the atomic temperature can be estimated 1/3, and the dotted curve is the formal limit for the circular using the following dependence: field polarization. ប k T = Ð--- T B δ (45)

20 2 2 2 2 2 ()32cos ()ε + sin ()2ε ()4δ + γ Sε × ------. 2 2 2 2 2 2 2 16 62cos ()ε ()36sin ()δ 2ε ++92sin ()ε γ γ Ð 12δ

12 Here, we can separate two different cases. In the first case, when the light field ellipticity is sin2(2ε) ≥ 1/3, the direction of the kinetic process is determined only by 8 the sign of δ (cooling for δ < 0). For a field ellipticity of sin2(2ε) < 1/3, cooling takes place for detunings 4 belonging to the following two intervals:

0 ()39Ð sin2 () 2ε ()92sin2 ()ε + 1 –1.0 –0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Ðγ ------<<δ 0, δ 618Ð sin2 () 2ε (46) 2 2 Fig. 2. Temperature in units of បγS1/2/k as a function of ()39Ð sin () 2ε ()92sin ()ε + 1 B δγ> ------. detuning for various ellipticities of the light field: the solid 2 ()ε curve corresponds to the linear polarization of the field and 618Ð sin 2 the dashed curve corresponds to a field ellipticity of ε = π/12. In particular, for the linear field polarization, the expression for temperature assumes the familiar form [1] According to [1], the expressions written above for the coefficients of friction and diffusion averaged over 2 2 2 ប ()4δ + γ Sε the space period make it possible to estimate the tem- k T()ε = 0 = ------. (47) 〈 〉 〈ξ〉 B 2δ δ2 γ 2 perature of the atomic ensemble, kBT = Ð D / , with- 12 Ð out taking into account localization as a function of the field ellipticity. In a weak field (Sε Ӷ 1), we obtain The dependence of temperature on the light field detuning for light field ellipticities of sin2(2ε) < 1/3 is shown 2 2 2 2 2 in Fig. 2. ប ()92sin ()ε δ + γ ()4δ + γ k T = Ð------. (43) B δ 2 2 2 2 2 It should be emphasized once again that the estimate 612()sin () 2ε δ ++32sin ()ε γ 2γ for temperature is the ratio of the friction and diffusion coefficients; i.e., this estimate is based on slow atom The diffusion coefficient is positive; consequently, the approximation (11). It is well known [1], however, that steady-state distribution of atoms is possible only for the applicability of this approximation is restricted by

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 KINETICS OF ATOMS IN THE FIELD PRODUCED BY ELLIPTICALLY POLARIZED WAVES 445 σ σ rather severe conditions even in the two-level model of for the +Ð Ð configuration, which are diagonal in gra- an atom in a standing wave in view of the presence of dients χφφ and Ᏸφφ, respectively. “Simple” configura- light field nodes. In the case of a nonzero ellipticity of tions of a nonuniformly polarized field are undoubtedly the light field, this problem requires additional analysis, convenient for analysis of the atomic kinetics and make which will not be carried out here. it possible to explain the main mechanisms of sub-Dop- pler cooling in low-intensity fields [7]. However, these 5.3. Uniformly Polarized Field of a “Mixed” Type mechanisms do not exhaust the atomic kinetics in fields with nonuniform polarization since, as follows from the In the general case, a uniformly polarized field may general results for the friction and diffusion coeffi- simultaneously contain the amplitude gradient and the cients, the presence in the field of several gradients phase gradient. A situation of this kind arises, for exam- simultaneously leads to new interesting effects and new ple, in the field formed by counterpropagating waves mechanisms of sub-Doppler cooling. We will consider with identical polarization ellipses, but with different these effects in greater detail. amplitudes. The friction and diffusion coefficients in such a field contain, in addition contributions diagonal in gradients, crossed contributions χΛΦ and ᏰΛΦ. It 6.1. The εÐθÐε Field Configuration should be noted that the additional contribution to the A simple example of a field containing all gradients friction coefficient exhibits an even dependence on (of intensity, ellipticity, orientation, and phase) is the detuning δ, while the additional contribution to the dif- δ field formed by counterpropagating plane waves of the fusion coefficient is an odd function of . Thus, in a same intensity and modulo equal ellipticities, but with “mixed”-type field, an “abnormal” dependence of the opposite directions of rotation: friction and diffusion coefficients on the detuning is observed (in particular, the friction coefficient does not Ðiωt E()z = E %()z e + c.c. (50) vanish in the case of exact resonance, δ = 0, owing to 0 additional contributions χΛΦ and χΦΛ; see Appendix B). Here, E0 is the amplitude of each of the counterpropagating waves and vector %(z) = a+e+1 + aÐeÐ1 with cyclic 6. NONUNIFORMLY POLARIZED FIELDS components a+(z) and aÐ(z) defines the local polariza- It is well known that sub-Doppler cooling of atoms tion ellipse, amplitude, and phase of the field, is possible in fields with nonuniform polarization. Sim- π π ε ikz ε Ðikz Ðiθ ple examples of this kind are fields with an ellipticity a+ = Ðcos0 + --- e Ð cos0 + --- e e , gradient (lin ⊥ lin field configuration) and fields with an 4 4 σ σ (51) orientation gradient ( +Ð Ð field configuration). In the π π ε ikz ε Ðikz iθ former case, the field is a superposition of counterprop- aÐ = cos0 + --- e + cos0 Ð --- e e , agating plane waves with linear polarizations oriented 4 4 at right angles to each other: where θ is the angle between the principal semiaxes of , ()ikz Ðikz Ðiωt the polarization ellipses of the counterpropagating E()zt = E0 exe + eye e + c.c. (48) ε waves (Fig. 3) and parameter 0 characterizes the Here, E0 is the amplitude of each wave, while vectors ex degree of ellipticity of the counterpropagating waves. and ey describe linear polarization of the counterpropa- For field configuration (51), we will use the notation gating waves along the x and y axes, respectively. This ε−θÐε . It should be noted that the familiar configura- expression can be reduced to formula (1) with parame- tions lin ⊥ lin (ε = 0, θ = π/2) and σ Ðσ (ε = π/4) are ε 0 + Ð 0 ters E(z) = 2E0 and ellipticity (z) = kz. The phase and special cases of the given field configuration. the angle of orientation are independent of coordinate z. σ σ 6.1.1. Gradient force. For low field intensities A field of the +Ð Ð configuration is a superposition (G Ӷ 1), the kinetic coefficients assume a simple form. of counterpropagating waves of circular polarizations For example, the gradient force splits into the sum of with opposite directions of rotation: two contributions: ω E()zt, = E ()e eikz + e eÐikz eÐi t + c.c. (49) 0 +1 Ð1 ()i ()s ε Fz()= F ()z + F (),z As we pass to formula (1), this gives E(z) = 2E0, (z) = 0, Φ φ ()i ប δ (z) = 0, and (z) = kz. The motion along the z axis F ()z = 2 k S0 changes only the orientation of the polarization vector × []()ε ()()θ ()ε () of the light field. cos2 0 sin 2kz cos + cos 2 0 cos 2kz Ð1 (52) It should be recalled that the expressions for the × ()12+ cos()θε cos cos()2kz , kinetic coefficients in these configurations are deter- 0 ប γ ()ε ()θ mined by contributions (∇ε)2 (B.2) and (∇φ)2 (B.18) for ()s k S0 sin4 0 sin 2kz sin ⊥ χ Ᏸ F ()z = ------()θε ()-. the lin lin configuration and εε (B.4) and εε (B.20) 2 12+ cos 0 cos cos 2kz

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Ðε

x k

ε z λ/2

0 y

Fig. 3. Spatial configuration εÐθÐε of a light field. The field is produced by counterpropagating waves of elliptic polarization with ellipticity parameters of ε and Ðε.

θ (s) δ ε Here, the saturation parameter is given by , while F , on the contrary, is even in and odd in 0 and θ. Ω 2 6.1.2. Friction coefficient. Like the gradient force, 0 the friction coefficient slits onto two terms of different S0 = ------, (53) 3()δ2 + γ 2/4 origin: () () ξ()z = ξ i ()z + ξ s (),z where Ω = ÐE d/ប, E being the amplitude of the coun- 0 0 0 2 (i) ()i 6បk δ 2 2 terpropagating waves. The first term, F , is the force of ξ ()z = Ð------sin θcos ()2ε induced light pressure (dipole force), while the second γ 0 (s) term, F , is the force of spontaneous light pressure. It × () () θ ()ε ] can be seen that the force of spontaneous light pressure cos 2kz [2cos kz + cos cos 2 0 induces in this case a periodic optical potential if the × []()ε ()θ Ð3 ellipticities of the light waves differ from linear or cir- 12+ cos 0 cos 2kz cos , (54) cular ellipticities (ε ≠ 0, π/4) and also if angle θ ≠ 0; in 0 ប 2 ξ()s 3 k θ ()ε other words, the potential of familiar configurations ()z = Ð------sin sin 4 0 with linear or circular polarizations is equal to zero. 2 The mechanism of formation of the force of spontane- × [2cos()kz + cosθcos()2ε ] ous light pressure is as follows: spatially inhomoge- 0 neous anisotropy of atoms is created by a nonuniformly × []()ε ()θ Ð3 polarized field; as a result of this anisotropy, atoms res- 12+ cos 0 cos 2kz cos . catter photons of counterpropagating waves with differ- After averaging over the spatial period, we obtain ent probabilities, which leads to disbalance of the 2 2 θ 2 ()ε forces of spontaneous light pressure. It should be ()i 3បk δ sin cos 2 〈〉ξ = Ð------0 , emphasized that a spontaneous force in the given field γ 2 2 3/2 ()12Ð cos ()ε cos θ leads to a periodic optical potential that does not van- 0 (55) ish in the case of exact resonance (δ = 0). It should also 2 ()θ ()ε ()ε ()s 3បk sin 2 cos 2 sin 4 be observed that F(i) is an odd function of detuning δ 〈〉ξ = ------0 0 . ε 8 ()2 ()ε 2 θ 3/2 and an even function of polarization parameters 0 and 12Ð cos 0 cos

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It is worth noting that average friction coefficients 〈ξ(i)〉 changes strongly over distances much smaller than and 〈ξ(s)〉 cannot be expressed in analytic form at point wavelength λ). Analysis of situations in these regions θ ε = 0, 0 = 0, in the vicinity of which we have requires more exact expressions for the force of friction, which would take into account all orders in the θ2 〈〉ξ()i ∼ velocity of atoms. ------3/2 ()θ2 + 4ε2 6.1.3. Diffusion coefficient. The spontaneous diffu- 0 sion coefficient has the form and 2 2 2 θε ()s γ ប k S ()s 0 0 ()ε ()θ 〈〉ξ ∼ ------. D = ------12+ cos 0 cos 2kz cos ()θ2 ε2 3/2 6 + 4 0 (56) cos2 ()2ε sin2 θsin2 ()2kz θ 0 Using the linÐ –lin configuration as an example, it was Ð ------()ε ()θ- . noted in [16] that such a behavior of the friction coeffi- 12+ cos 0 cos 2kz cos cient is associated with inapplicability of the slow atom approximation in the vicinity of nodes of the field, It is convenient to decompose the induced diffusion where, on the one hand, the local saturation parameter coefficients into the terms corresponding to different (i) is small and, on the other hand, noticeable gradients of degrees of detuning, D = D0 + D1 + D2, where D0 is the field polarization arise (i.e., the field polarization the term independent of detuning,

γ ប2 2 2 ()ε ()2 ()2 θ 2 ()2 θ 2 ()ε k S0 12Ð cos 0 cos 2kz cos Ð sin 2kz sin + sin 2 0 D0()z = ------()ε ()θ ------2 12+ cos 0 cos 2kz cos (57) 3γ ប2k2S sin2 ()4ε sin2 θsin2 ()2kz Ð ------0 ------0 -. 8 ()()ε ()θ 3 12+ cos0 cos 2kz cos

The term with an abnormal dependence on detuning 6δ2ប2k2S cos2 ()2ε sin2 θcos2 ()2kz has the form 0 0 D2()z = ------γ ()()ε ()θ- 12+ cos 0 cos 2kz cos (59) cos2 ()2ε sin2 θsin2 ()2kz 2 2 × 1 Ð.------0 - δប k S sin()θ4ε sin cos()2kz 2 0 0 ()12+ cos()ε cos()θ2kz cos D1()z = ------()ε ()θ 0 12+ cos 0 cos 2kz cos (58) After averaging over the spatial period, we obtain the 32cos2 ()ε sin2 θsin2 ()2kz following expressions for the diffusion coefficient. The × 2 Ð,------0 - ()()ε ()θ2 average spontaneous diffusion coefficient has the form 12+ cos 0 cos 2kz cos ប2 γ 2 ()s k S Qsinθ + cos()2θ 〈〉D = ------0 ------. (60) 6 cos2 θ while the part of the diffusion coefficient quadratic in The average induced diffusion coefficient is given by 〈 (i)〉 〈 〉 〈 〉 〈 〉 detuning is given by D = D0 + D1 + D2 , where

ប2 γ ()2 θ 2 ()ε ()2 ()ε 2 ()ε 2 θ 〈〉 k S0 21Ð Qsin sin 2 0 22sin 0 + cos 2 0 cos Ð 1 D0 = ------+,------4 cos2 θ Q3 ប2 δ θ ()ε ()[]2()2 θ ()θ2 〈〉 k S0 sin sin 2 0 Q Ð 1 2Q 2 + sin Ð 3 Q + 1 sin (61) D1 = ------, Q3 cos3 θ ប2 δ2 2 3 kS ()θQ Ð 1 sin 3 2 2 2 2 〈〉D = ------0 ------()2Q sin θ Ð 2Q ()sin θ + 1 + ()θQ + 1 sin . 2 γ 3 4 Q cos θ

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T perature minimum is attained for wave ellipticities 4 close to linear. The characteristic increase in the tem- (a) perature for ellipticities of light waves close to circular is associated with the specific features of the optical 3 transition Jg = 1/2 Je = 1/2, in which the friction coefficient is equal to zero in the limiting case of the σ Ðσ configuration of the light field and, hence, there 2 + Ð is no cooling. Note that friction is present for optical transition with larger values of angular moments in the σ σ 1 +Ð Ð configuration also (e.g., for the transition Jg = 1 Je = 2) [7]. 0 6.2. The εÐθÐε Configuration of the Light Field 4 (b) In this section, we consider an example of the light field configuration formed by elliptically polarized 3 waves with equal ellipticities and the same directions of rotation for the polarization vectors. In contrast to the previous case, a field with the εÐθÐε configuration con- 2 tains only the gradients of intensity and ellipticity of the light field (Fig. 5). In particular cases, when angle θ 1 between the principal axes of the ellipses is zero, we return to the case of a uniformly polarized standing θ ±π ε wave; for = /2 and 0 = 0, we have a field of the 0 ⊥ π π lin lin configuration, in which only the ellipticity gra- – /4 0 /4 dient differs from zero. ε 0 In the first order in the field intensity, the explicit បγ ε Fig. 4. Temperature in units of S0/kB as a function of dependence of the force on polarization parameters 0 ε θ π θ ellipticity 0 of light waves for different angles: = /3 and has the form (bold curves), π/5 (fine curves), and π/10 (dashed curves); light field detuning δ = 2γ (a) and 0 (b). ប δ Fz()= 2 k S0 ()2 ()ε ()θ () (62) × sin 2kz cos 2 0 cos+ cos 2kz For brevity, we have introduced the notation ------()θ ()θε ()-, 12+ coskz cos Ð sin 2 0 sin sin 2kz 2 ()ε 2 θ Q = 12Ð.cos 0 cos where the saturation parameter S0 is defined in terms of amplitude (53) of the counterpropagating waves. It fol- It should be noted that the coefficients of friction lows hence that the force averaged over a spatial period and diffusion in the εÐθÐε field configuration contain differs from zero (effect of rectification of the dipole force): ξ(s) contributions and D1, which are anomalous in detuning; among other things, this may lead to cooling 〈〉F = បkδS in the case of exact resonance. The direction of the 0 (63) 2 ()ε ()ε ()θ ()θ ()ε 2 kinetic process (heating or cooling) in this case is deter- cos 2 0 sin2 0 sin 2 1 Ð sincos 2 0 mined by the sign of ellipticity ε of counterpropagat- × ------. 0 ()2 θ 2 ()ε 2 θ 2 ing light waves and by angle θ determining the mutual cos + sin 2 0 sin orientation of polarization ellipses. The abnormal contribution to friction remains significant for a nonzero The mean force is an odd function of the light field detuning as well. Consequently, heating can be detuning; in addition, it is an odd function of angle θ ε replaced by cooling or vice versa by choosing an appro- and ellipticity 0 of the light waves, which is also priate ellipticity of the counterpropagating waves and observed in the general case of an arbitrary field inten- angle θ. The temperature in the cooling region is a func- sity. It can be proved rigorously proceeding from the ε θ tion of ellipticity 0 of the light waves and angle . Fig- general symmetry relations for the optical Bloch equa- ure 4 shows the dependence of the atomic temperature tion that such a form of the dependence of the force on δ ε θ ε θ ε on the ellipticity of light waves for certain angles θ. It parameters , 0, and in the Ð Ð field configuration can be seen that the shape of the curves is asymmetric takes place for arbitrary optical transitions J J . ε g e in ellipticity parameter 0; however, for a large detun- Note that the rectified force vanishes in the case of lin- ε ±π θ ing, this effect becomes less pronounced and the tem- ear and circular polarizations 0 = 0, /4 for any as

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ε z λ/2

Fig. 5. Spatial configuration εÐθÐε of a light field. The field is produced by counterpropagating waves of the same ellipticity.

θ ±π ε well as for angles = 0, /2 in the case of arbitrary 0. The linear (in velocity) correction to force (62) can This becomes obvious if we use the symmetry relations be defined in terms of the friction coefficient; in the Ӷ for the force acting on a stationary atom. Reflection in case of a low intensity of the light field (S0 1), the the xy plane gives the following relation for the force: friction coefficient has the form

〈〉ε(), θ 〈〉ε(), θ F 0 = Ð F 0 Ð . (64) ប 2δ ξ 6 k θ 2 ()ε ()() θ ()z = Ð------γ - sin cos 2 0 cos kz + cos It can be seen that, for the circular polarization of the ε ±π θ ±π × ()() θ θ ()ε () (66) waves ( 0 = /4) or for angles = 0, /2, the reflec- cos kz sin + cos sin 2 0 sin kz tion in the xy plane leads to the initial configuration of × []θ () θ ()ε ()Ð3 the light field and, hence, 〈F〉 = 0. Another symmetry 1 + cos cos kz Ð sin sin 2 0 sin kz . relation for the force can be obtained using two consecutive reflections in the xy and xz planes: The friction coefficient averaged over the spatial period for θ ≠ 0, π and for field polarizations other than circu- 〈〉εF (), θ =.Ð 〈〉F ()Ð ε, θ (65) ε ≠ ±π 0 0 lar, 0 /4, If the field is produced by waves with the linear polar- ε 3បk2δ ization ( 0 = 0), after the spatial reflections mentioned 〈〉ξ = Ðγθ------()ε -, (67) above, we return to the initial configuration of the light sin cos 2 0 〈 〉 ε θ θ field; i.e., F ( 0 = 0, ) = 0 for any . The effect of rectification of the dipole force and the determines the correction to the force for atoms per- mechanism for the emergence of this effect was consid- forming above-barrier motion in the potential created ered in detail in our earlier publication [17]. We only by force (62). Figure 6 shows that the rectified force note here that the mechanism for the emergence of the produces an optical potential in which cold atoms can constant component of the dipole force in an εÐθÐε be localized. It is important to note that, with the field is associated with spatially inhomogeneous optical appropriate choice of the sign of detuning δ, the force pumping and with the presence of the gradients of the of friction in the vicinity of the minima of potential light field intensity and ellipticity. Ueff can lead to cooling of atoms. The latter circum-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 450 PRUDNIKOV et al.

Ueff two-level atom in the case when a uniform polarization of the light field does not coincide with the linear polar- 0 ization. For light fields of the second type, the effects (a) resulting from the ellipticity of traveling waves that –10 produce the field were demonstrated. The two most interesting configurations were singled out: the εÐθÐε –20 field created by counterpropagating waves of the opposite ellipticities and the εÐθÐε field created by counter- –30 propagating waves with the same ellipticity (θ is the angle of mutual orientation of the polarization ellipses –40 for the counterpropagating waves). For example, in the ε θ ε –50 Ð Ð field, the ellipticity of traveling waves may lead to sub-Doppler cooling of atoms even in the case of ξ exact resonance (δ = 0); in addition, anomalous (rela- 0 tive to conventional sub-Doppler cooling) regions of heating and cooling are formed. It was shown that, in –2000 (b) the field of the εÐθÐε configuration, the ellipticity of traveling waves leads to rectification of the dipole –4000 force. Note that the optical transition Jg = 1/2 Je = 1/2 chosen for our analysis is a simple example of tran- –6000 sition with energy levels degenerate in the angular momentum projection. It should also be observed that –8000 the effects described in Section 6 take place for transi- –10000 tions with larger values of angular momenta Jg and Je as well. 0 λ/2 λ

Fig. 6. Spatial dependence of effective potential Ueff in ACKNOWLEDGMENTS units of បγ (a) and the friction coefficient in units of បk2 (b) Ω γ for 0 = 100 and for parameters optimal for rectification This study was supported by the Russian Founda- ε θ δ γ ( 0 = 0.46, = 0.51, = 62 ). tion for Basic Research (project nos. 01-02-17036, 01-02-17744, 03-02-06047-mas, and 04-02-16488), the Ministry of Education of the Russian Federation stance plays an important role for a stable localization (grant no. UR.01.01.060), and the Special Federal Sci- of atoms. ence and Engineering Program “Research and Develop- ment Works in Priority Trends of Science and Engineer- 6. CONCLUSIONS ing” for 2002–2006 (state contract no. 01-40-01-06-05). We have derived analytic expressions for the kinetic coefficients (the force acting on a stationary atom and APPENDIX A coefficients of friction and diffusion) of the FokkerÐ Planck equation for atoms performing the optical transition Jg = 1/2 Je = 1/2 in a field of an arbitrary con- Procedure of Reduction figuration and intensity. The expressions are analyzed of the Quantum-Mechanical Kinetic Equation for the fields produced by elliptically polarized waves. for the Density Matrix to the Fokker-Planck Equation We have considered two important types of light fields: for Slow Atoms (i) fields with a uniform polarization, in which the The expansion of the kinetic equation for the density gradients of ellipticity and spatial orientation are equal matrix in recoil parameter បk/∆p has the form to zero and

(ii) nonuniformly polarized fields produced by ellip- () ∂ () d ρ , ˆ 0 {}ρ , ប ˆ 1 {}ρ , tically polarized waves. ˆ()rp = L ˆ()rp + k∑∂ Li ˆ()rp dt pi It was shown that, in the first case, in the absence of i (A.1) 2 gradients of the light field polarization, the kinetics of ∂ () atoms resembles the results obtained in the two-level ()ប 2 ˆ 2 {}…ρ , + k ∑∂------∂ Lij ˆ()rp + , pi p j model of atoms (in particular, our results coincide with ij, the results of this model in the case of the linear ellipticity of the light field). In addition, it was found that where (r, p) are the coordinates of a point in the phase our results differ from the predictions of the model of a space. In the zeroth order in recoil effects, the evolution

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 KINETICS OF ATOMS IN THE FIELD PRODUCED BY ELLIPTICALLY POLARIZED WAVES 451 of the density matrix is determined by the operator tion. Consequently, in the zeroth order in recoil, we have ()0 γ γ Lˆ {}ρˆ = Ð --- Ð iδ Πˆ eρˆ + --- + iδ ρˆ Πˆ e   d d ()0 2 2 Tr{}ρˆ ==ᐃ Tr{}Lˆ {}ρˆ =0, (A.7) (A.2) dt dt i []γˆ , ρ ()ˆ †ρ ˆ Ð --- V()r ˆ + 2Je + 1 ∑Dq ˆ Dq, ប i.e., the derivative of the distribution function is at least q of the first order of smallness in recoil, and the zeroth () corresponding to the optical Bloch equation (15). The order ρˆ 0 ()rp,,t can be written in the form first order terms, () () ρˆ 0 ()rp,,t = σˆ ()rp, ᐃ 0 ().rp,,t (A.8) ()1 1 Lˆ {}ρˆ = Ð------()Fˆ ()r ρˆ + ρˆ Fˆ ()r , (A.3) i 2បk i i Thus, σˆ Ᏺ is the part of the density matrix adiabatically can be expressed in terms of the force operator (14). tracing the motion and χˆ is a small nonadiabatic cor- Second-order corrections contain both induced terms rection. Indeed, in the zeroth order in the recoil param- proportional to the second derivative of operator Vˆ ()r eter, the equation for the density matrix has the form with respect to the coordinate and the term associated with the recoil effect in spontaneous emission: d ()0 d ()0 ()0 ()0 ()0 σˆ ᐃ + χˆ = Lˆ {}σˆ ᐃ + Lˆ {}χˆ . (A.9) dt dt γ () ˆ ()2 {}ρ i []∇ ˆ , ρ 2Je + 1 Lij ˆ = Ð------iF j()r ˆ + ------8បk2 5 In accordance with Eqs. (A.6) and (A.7), we find that, (A.4) ()0 in the zeroth order in recoil, χˆ satisfies the equation 1 † ×δ∑ δ Ð ---()δ δ + δ δ Dˆ mρˆ Dˆ n. ij nm 4 in jm im jn mn, = 123,, d ()0 ()0 ()0 χˆ ()rp,,t = Lˆ {}χˆ ()rp,,t (A.10) In order to derive Eq. (10), we make use of the exist- dt ence of characteristic time τ. Over time intervals t ≤ τ, () only the change in the internal degrees of freedom of an with the normalization condition Tr{}χˆ 0 ()rp,,t = 0. atom, which strongly depend on the initial conditions, ӷ τ The solution to this equations over time periods t corre- is significant, while time intervals t correspond to ()0 the kinetic stage of evolution; accordingly, a change in sponding to the kinetic stage of the evolution is χˆ = 0; the internal states of an atom is matched with the thus, nonadiabatic correction χˆ contains terms starting change in distribution function ᐃ [2]. Thus, at the only from the first order in recoil. kinetic stage of evolution, the density matrix is a linear functional of the distribution function and can be repre- Taking the trace of expression (A.1) and retaining in sented in the form it only the terms up to the second order in recoil, we obtain ρˆ()rp,,t = σˆ ()rp, ᐃ()rp,,t + χˆ().rp,,t (A.5) ∂ () d ᐃ ប {}ˆ 1 {}σ ᐃ = k∑∂ Tr Li ˆ Here, σˆ ()rp, is the reduced density matrix describing dt pi i the stationary state of the internal degrees of freedom of 2 ∂ () an atom moving with velocity v, ()ប 2 {}ˆ 2 {}σ ᐃ + k ∑∂------∂ Tr L j ˆ (A.11) pi p j ij, d ()0 σˆ ()rp, = Lˆ {}σˆ ()rp, , ∂ () dt ប {}ˆ 1 {}χ (A.6) + k∑∂ Tr Li ˆ . ∂ pi σˆ ()0,rp, = i ∂t The last term has a smallness of at least the second with the normalization condition Tr{σˆ ()rp, } = 1. order in recoil and contributes to the diffusion coefficient. We will not seek these nonadiabatic corrections Closed optical transitions exhibit the property to the density matrix as was done, for example, in [15], ()0 Tr{Lˆ ()ρˆ } = 0, indicating the conservation of popula- but will use an alternative approach. To this end, we

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 452 PRUDNIKOV et al. supplement Eq. (A.11) with the linear combination of This condition leads to the equation for density matrix Eq. (A.1), ϕ ϕ ˆ i (21). Matrix ˆ is a solution to nonhomogeneous linear equation (21) with a source in the form of the fluc- ∂  ∂ () ប ϕ d ρ ប {}ϕ ˆ 0 {}ρ tuation of the force operator (20). Since the force acting k∑∂ Trˆ i ˆ = k∑∂ Tr ˆ i L ˆ pi dt pi on an atom should be defined in the zeroth and first i i (A.12) orders in velocity, while the diffusion coefficient should 2 ∂ () be defined in the zeroth approximation, it is sufficient to ()ប 2 {}…ϕ ˆ 1 {}ρ + k ∑∂------∂ Tr ˆ i L j ˆ + , retain in expression (A.14) only the zeroth order (in pi p j ij, velocity) of matrix σˆ (v = 0, r) satisfying optical Bloch ϕ (0) σˆ where ˆ i is a matrix of dimension (2Je + 1) + (2Jg + 1). equation (15) for stationary density matrix L { } = 0. The left-hand side of this equality has the form APPENDIX B ∂  ∂  ប ϕ d ρ ប ϕ d σ ᐃ k∑∂ Trˆ i ˆ = k∑∂ Trˆ i ˆ pi dt pi dt i i Components of Friction and Diffusion Coefficients (A.13) The diagonal elements of the friction coefficient com- ∂ ∂  ប {}ϕ σ d ᐃ ប ϕ d χ ponents are functions of only odd powers of detuning δ, + k∑∂ Tr ˆ i ˆ + k∑∂ Trˆ i ˆ . pi dt pi dt i i 2δ Gε χΛΛ = ------γ ()3 The last term contains the second order in recoil. Over 1 + Gε (B.1) time periods t ӷ τ corresponding to the kinetic stage of 2 γ 1 Ð Gε 2 2 evolution, dχˆ /dt = v á ∇ χˆ ; consequently, the contribu- × ------Ð Gε + 32sin ()ε , r γ 2 δ2 2 tion from this term to the diffusion coefficient can be /4 + neglected since only zero-order terms in velocity are retained in the diffusion coefficient for slow atoms. 2δ 1  2 χεε = Ð------32cos ()ε - Subtracting expression (A.11) from (A.12) and retain- γ 3 ()1 + Gε  ing terms up to the second order in recoil, we ultimately obtain 2 2 γ 2 + Gε 10Ð 3cos () 2ε Ð ------tan ()2ε ∂ 2 2 d ᐃ ប {}ϕ ⋅ ∇σ ᐃ γ /4 + δ Ð k∑∂ Tr ˆ iv ˆ (B.2) dt pi 2 2 i 2 62Ð cos ()ε γ 2 + Gε ------+ ------tan ()2ε 2 ()ε γ 2 δ2 ∂ ∂ () cos 2 /4 + ប {}ϕ σ d ᐃ ប {}ˆ 1 {}σ ᐃ Ð k∑∂ Tr ˆ i ˆ = k∑∂ Tr Li ˆ pi dt pi 2 i i 322Ð cos ()ε   + Gε ------2 , ∂ () cos ()2ε  ប {}ϕ ˆ 0 {}σ ᐃ Ð k∑∂ Tr ˆ i L ˆ pi i δγ Gε (A.14) χΦΦ = ------, (B.3) 2 2 2 2 ∂ () γ /4 + δ ()1 + Gε ()ប 2 {}ˆ 2 {}σ ᐃ + k ∑∂------∂ Tr Lij ˆ pi p j χ ij, φφ = 0. (B.4)

2 ∂ () We will write the nondiagonal elements associated ()ប 2 {}ϕ ˆ 1 {}σ ᐃ with correlations of gradients of the amplitude and Ð k ∑∂------∂ Tr ˆ i L j ˆ pi p j ij, phase, 2 2 Gε γ δ ∂ ()1 ∂ ()0 χ /4 Ð ប {}{}χ ប {}ϕ {}χ ΛΦ = ------Gε + ------, (B.5) + k∑ Tr Lˆ i ˆ Ð k∑ Tr ˆ i Lˆ ˆ . 22 2 ∂ ∂ ()ε γ δ pi pi 1 + G /4 + i i ϕ Gε Note that by choosing matrix ˆ i appropriately, we can χ ΦΛ = ------3 ensure the compensation of the contributions contain- ()1 + Gε ing nonadiabatic correction χˆ so that (B.6) 2 2 2 δ Ð γ /4 Gεγ /2 2 × ------+ ------3Ð sin () 2ε , {}ϕ ()0 {}χ {}()1 {}χ γ 2 δ2 γ 2 δ2 Tr ˆ i Lˆ ˆ = Tr Lˆ i ˆ . (A.15) /4 + /4 +

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 KINETICS OF ATOMS IN THE FIELD PRODUCED BY ELLIPTICALLY POLARIZED WAVES 453 ellipticity and phase, the form

2 2 Gε tan()2ε δ γ Ð /4 Gε  χεΦ = ------Ð Gε , (B.7) 1 2 2 2 2 2 2 ᏰΛΛ = ------γ + 12δ sin ()2ε --- ()1 + Gε δ γ γ 3 + /4 4 ()1 + Gε  tan()2ε γ 4 χΦε = ------γ 2 ()δ2 γ 2 2 ()ε ()3 + Gε Ð ++------4+ tan 2 (B.17) 1 + Gε γ 2/4 + δ2 (B.8) γ 2 2 2 Gε 2 2 2 2 2 2 2()4δ + γ  × ε ()ε ()ε []γ ()δ γ ()ε 1 ++G 1 Ð G ------2 -32 Ð sin 3 , + Gε 3 + 24 + tan 2 + Gε ------, γ /4 + δ cos2 ()2ε  ellipticity and angle, 2 1 Gε  222Ð cos ()ε 2 Gε Ᏸεε = ------γ ------+ 12δ cos()2ε χ 4γ ()3 cos()2ε εφ = ------2 1 + Gε  cos()2ε ()1 + Gε (B.9) 2 2 ()ε 2 δ ()γδ2 γ 2 215Ð cos 2 × 22sin ()ε ------Ð 1 Ð Gε , + Gε 74 + + ------2 γ 2 δ2 cos ()2ε /4 + (B.18) 4 2 ()ε γ 2 2 2 2 32+ cos () 2ε cos 2 2Gε + ------tan ()2ε + Gε ()4δ + γ ------χφε = 3------Gε Ð 1 Ð ------, (B.10) 2 2 2 22 γ /4 + δ cos ()2ε ()1 + Gε cos ()2ε

2 2 2 ellipticity and amplitude, 213Ð cos () 2ε 3 4δ + γ  + γ ------+ Gε ------, δ ()ε cos2 ()2ε cos2 ()2ε  χ 2 Gε tan 2 εΛ = ------3 - γ ()1 + Gε γ Gε (B.11) ᏰΦΦ = ------2 2 4 3 2 2 3δ Gε + ()γ5Gε Ð 2 /4 ()1 + Gε × 32cos ()ε ++Gε ------, (B.19) γ 2/4 + δ2 γ 2 × ()2 2 ()ε  1 + Gε ++32sin Gε1 Ð ------, γ 2/4 + δ2 δ ()ε  χ 2 tan 2  2 2 ()ε () Λε = ------3 Gε Ð 32cos 1 Ð Gε γ Gε γ ()1 + Gε  Ᏸφφ = ------. (B.20) (B.12) 21 + Gε 2 2 2 2 2 2Gε ()γ /4 Ð δ Ð Gε()7δ + 11γ /4  Ᏸ + ------ , It was noted above that components ββ' are sym- 2 2 ββ γ /4 + δ  metric in indices ' and, hence, the expression for diffusion coefﬁcient (25) will contain only symmetric Ᏸ Ᏸ amplitude and angle, sums ββ' + β'β. These are two normal nondiagonal elements even in detuning, 2 sin()2ε Gε δ χ ()ε Λφ = Ð2------2------2 2-, (B.13) sin 4 Gε ()()γ δ ᏰεΛ + ᏰΛε = ------1 + Gε /4 + γ 3 4 ()1 + Gε sin()2ε Gε 2 χφΛ = 3------, (B.14) 2 γ 2 2 × 12δ Ð ------+ Gε------(B.21) ()1 + Gε 2 2 cos ()2ε cos ()2ε and phase and angle, 4 2 2 2 2 γ /2 2 8δ Ð γ × 8δ + γ Ð ------+ Gε ------, ()ε δγ 2 2 2 χ sin 2 Gε γ /4 + δ cos ()2ε Φφ = ------2------2 -2 , (B.15) ()1 + Gε γ /4 + δ γ sin()2ε Gε ᏰΦφ ᏰφΦ + = Ð------2 . (B.22) χφΦ = 0. (B.16) ()1 + Gε The diagonal elements of the diffusion coefﬁcient Finally, we write four anomalous nondiagonal elements Ᏸ δ components ββ', which are even functions of , have leading to asymmetry in the dependence of the diffu-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 454 PRUDNIKOV et al. sion coefﬁcient on δ: 6. Laser Cooling and Trapping, Special issue of Laser Phys. 4, 5 (1994). δGε ᏰΦΛ ᏰΛΦ 7. J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B + = ------3 ()1 + Gε 6, 2023 (1989). (B.23) 8. O. N. Prudnikov, A. V. Taœchenachev, A. M. Tumaœkin, γ 2 × 2 ()ε 2 and V. I. Yudin, Zh. Éksp. Teor. Fiz. 115, 791 (1999) Ð 32sin ++Gε------Gε , [JETP 88, 433 (1999)]. γ 2/4 + δ2 9. J. Dalibard and C. Cohen-Tannoudji, J. Phys. B 18, 1661 2δsin()2ε Gε (1985). ᏰφΛ ᏰΛφ + = ------2 , (B.24) 10. J. Javanainen, Phys. Rev. A 44, 5857 (1991). ()1 + Gε 11. V. S. Smirnov, A. M. Tumaœkin, and V. I. Yudin, Zh. Éksp. Teor. Fiz. 96, 1613 (1989) [Sov. Phys. JETP Ðδtan()2ε Gε ᏰΦε ᏰεΦ 69, 913 (1989)]; A. V. Taœchenachev, A. M. Tumaœkin, + = ------3 - ()1 + Gε and V. I. Yudin, Zh. Éksp. Teor. Fiz. 110, 1727 (1996) (B.25) [JETP 83, 949 (1996)]; G. Nienhuis, A. V. Taichenachev, 2 2 γ 2 A. M. Tumaikin, and V. I. Yudin, Europhys. Lett. 44, 20 × ()ε ε ε 32cos ++,G 3 + ------2 -2 G (1998); A. V. Taœchenachev, A. M. Tumaœkin, and γ /4 + δ V. I. Yudin, Zh. Éksp. Teor. Fiz. 118, 77 (2000) [JETP 91, 67 (2000)]. δ 2 ()ε 2 Gε cos 2 + Gε 12. A. V. Taœchenachev, A. M. Tumaœkin, V. I. Yudin, and Ᏸφε + Ᏸεφ = ------. (B.26) 2 ()ε G. Nienhuis, Zh. Éksp. Teor. Fiz. 108, 415 (1995) [JETP ()1 + Gε cos 2 81, 224 (1995)]. 13. A. V. Bezverbnyi, G. Nienhuis, and A. M. Tumaikin, REFERENCES Opt. Commun. 148, 151 (1998). 14. A. Yu. Pusep, Zh. Éksp. Teor. Fiz. 70, 851 (1976) [Sov. 1. A. P. Kazantsev, G. I. Surdutovich, and V. P. Yakovlev, Phys. JETP 43, 441 (1976)]. The Mechanical Action of Light on Atoms (Nauka, Mos- cow, 1991). 15. J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707 (1985). 2. V. G. Minogin and V. S. Letokhov, The Pressure of Laser Radiation on Atoms (Nauka, Moscow, 1986). 16. V. Finkelstein, P. R. Berman, and J. Guo, Phys. Rev. A 3. P. D. Lett, R. N. Watts, S. I. Westbrook, et al., Phys. Rev. 45, 1829 (1992). Lett. 61, 169 (1988). 17. O. N. Prudnikov, A. V. Taœchenachev, A. M. Tumaœkin, 4. A. Aspect, E. Arimondo, R. Kaiser, et al., Phys. Rev. and V. I. Yudin, Zh. Éksp. Teor. Fiz. 120, 76 (2001) Lett. 61, 826 (1988). [JETP 93, 63 (2001)]. 5. Laser Cooling and Trapping of Atoms, Special issue of J. Opt. Soc. Am. B 6, 11 (1989). Translated by N. Wadhwa

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 455Ð462. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 518Ð526. Original Russian Text Copyright © 2004 by Shpatakovskaya. ATOMS, SPECTRA, RADIATION

Spherical ElectronÐIon Systems in Semiclassical Approximation: Atoms and Atomic Clusters G. V. Shpatakovskaya Institute of Mathematical Modeling, Russian Academy of Sciences, Moscow, 125047 Russia e-mail: [email protected] Received August 15, 2003

Abstract—A semiclassical method for calculating the total energy and spatial distribution of electron density in spherically symmetric electronÐion systems is applied to atoms and both solid and hollow atomic clusters. Both exchangeÐcorrelation interaction and second-order gradient correction are taken into account. The contribution due to the fourth-order gradient correction is discussed. An expression is proposed for the oscillating correction to the averaged electron density. An expression is obtained for the equilibrium radius of a hollow cluster. The dependence of the equilibrium radius of an endohedral cluster on the valence of the central atom is analyzed. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION perature. Its extension to nonzero temperatures was presented in [11]. Spherically symmetric electronÐion systems with The semiclassical energy functional on a class of fixed distributions of ions can be used to model atoms, trial density functions has also been applied to describe ions, neutral and charged atomic clusters, etc. Their atomic clusters [12, 13]. In [13], the fourth-order gradi- characteristics are calculated in the Hartree or HartreeÐ ent correction was included in the expression for Fock approximation or by the KohnÐSham method in kinetic energy. The authors claimed (referring also to density functional theory [1]. However, the complexity a numerical comparison made in computations of of these calculations increases with the number of par- nuclei [14]) that the use of this particular correction ticles in the system, whereas the simpler and more made it possible to attain the best accuracy of mean explicit semiclassical methods become more accurate. electron density. In particular, semiclassical approximations have been The simplest semiclassical functional based on the successfully applied in calculations of thermodynamic ThomasÐFermi theory was used in [15] to analyze a properties [2Ð5] and in analytical treatments of shell hollow cluster as a model of the fullerene C . effects in the mass spectra of medium-sized and large 60 atomic clusters at zero and finite temperatures [6–8]. In the present study, a unified semiclassical ITF model is applied to analyze the properties of these and In this paper, an improved ThomasÐFermi (ITF) other systems. In Section 2, the general model equation model and corrections to it are used as a basis for a uni- for electron density is written out, including the fied algorithm for computing local characteristics (den- exchange and correlation interactions and the second- sity and potential) and electron energy in a spherically order gradient correction. The model describes average symmetric electronÐion system. The efficiency of the characteristics of the system. However, it can also be proposed algorithm is demonstrated by applying it to used to calculate the oscillations in electron density due atoms and atomic clusters. to the discrete nature of the electron spectrum. The oscillatory effects are discussed in Section 3, where a Originally, the semiclassical generalization of the corresponding correction to the mean electron density ThomasÐFermi model based on an energy functional is derived. The results of computations performed for allowing for the exchange interaction and the lowest particular atoms, solid and hollow clusters, and order (gradient) correction was applied to calculate fullerenes are presented and discussed in Section 4. the energy of the electron shell of a free atom in [9]. Finally, the principal conclusions are summarized. The energy functional was minimized over the sim- Some preliminary results of this study were pub- plest class of trial functions. To develop the method, lished in [16, 17]. the authors of [10] proposed a quantum-statistical model in which the electron density was calculated by solving the EulerÐLagrange equation. This model was 2. EQUATION FOR ELECTRON DENSITY used to compute an equation of state in the cell In the ITF model, the system of Ne electrons occu- approximation for a wide range of density at zero tem- pying a volume V in the field Ui(r) generated by ions at

456 SHPATAKOVSKAYA zero temperature is described by the following energy ν(x) = xn , one obtains a nonlinear integrodifferential functional of density n = ne(r) (in atomic units) equation for this function and the chemical potential:

E []n = dr 2ν ()π2 2/3 e ∫ 1 d 3 Ð4/3ν7/3 ()νµ ------2 ------2 Ð0,------x + Ð U = (1) 18R dx 2 1 × tn()+++nU()r + ---U ()r e ()n e ()n , i 2 e ex corr ν()0 == ν()L 0, L (5) where t(n) is the kinetic energy density including the sec- π 3 ν2() ond-order correction to the ThomasÐFermi model [2], Ne = 4 R ∫dx' x' , 0 () () δ() tn = tTF n + 2tn L L 2 2 ()∇ π 2 2 3 2 2/3 5/3 1 n () 4 R ν () x ν () = ------()3π n Ð ------; Ue x = ------∫dx' x' Ð ∫dx'1Ð --- x' . 10 72 n (2) x x' 0 x n ()r' U ()r = dr'------e - e ∫ rrÐ ' It can be solved by Newton’s method in a ﬁnite-difference approximation for 0 ≤ x ≤ L ӷ 1. is the potential generated by electrons; The ITF model can be used to evaluate the contributions due to effects ignored in the model (see [7]). In 1/3 1/3 33 4/3 3 1/3 e ()n ==Ð------n , U Ð --- n , particular, the contribution to energy due to the fourth- ex 4π ex π order gradient correction is expressed as follows [9]

1/3 rs 3 1 1/3 X ==------, rs ------, ∆ 11.4 4πn 4E = ------2/3∫drn 540() 3π2 (6) e ()n = Ð0.033n (3) ∆n 2 9∇n 2∆n 1∇n 4 corr × ------Ð.------+ ------n 8n n 3n 3 Ð1 X 2 1 × ()1 + X ln()1 + X +,---- Ð X Ð --- 2 3 3. OSCILLATING ELECTRON DENSITY ()Ð1 Ucorr = Ð0.033ln 1 + X IN A SHELL are the exchange and correlation energy densities [18] Solving the Schrödinger equation with the potential and the corresponding potentials. U(r) obtained in the ITF approximation, one can find the spectrum and wavefunctions of electrons and use The extremum condition for the functional subject the results to calculate the electron density for a discrete to conservation of the number of particles Ne = spectrum. In what follows, this is done analytically in a drn ()r entails the EulerÐLagrange equation for semiclassical approximation, and the expression for the ∫ e oscillating correction to electron density obtained density in [19] for an infinite atom is extended to the case of a finite system. The analysis relies on a semiclassical 2 1 2 2/3 1 ∇n 1 ∆n solution to the Schrödinger equation for the radial ---()3π n ++0,------Ð ------U()r Ð µ= (4) 2 72n 36 n wavefunction, the BohrÐSommerfeld quantization condition, and Poisson’s summation formula (see [2, 19] where the Lagrange multiplier µ is the chemical poten- for details). The resulting expression for the oscillating tial of the system and the effective potential U(r) is the correction is sum of external, electrostatic, exchange, and correla- µ 2 ()2 tion terms, pE r r 1 2 n ()r = ------dE dλ () () () () () osc 2 2 ∫ ∫ U r = Ui r ++Ue r Uex r +Ucorr r . 2π r Ur() 0 (7) ∞ Using spherical symmetry and changing from the []σ () σ0 π λ × ()ks+ sin 2 Eλ r ++2s Eλ 2 k radius to the dimensionless variable x = r/R (R is a ref- ∑ Ð1 ------() . pEλ r erence length) and from the density to the function ks,∞= Ð

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Here, t 1 (λ 0). Therefore, only the terms with s = k Ð 1 and s = 2k are summed for atoms and clusters, 2 respectively. 2 λ 2 p λ()r ==p ()r Ð,----- p ()r 2()EUrÐ (), E E 2 E Sums of the form r ∞ 0 R λ () E ()ik cos xkx+ ∑ Ð1 ------α - σ λ()r = dr' p λ()r' + k E ∫ E k = Ð∞ r can be calculated analytically (see [20]), with i = 0 and are the radial momentum and action, respectively; 1 for atoms and clusters, respectively. Performing the σ0 σ Eλ = Eλ(R'Eλ ); R'Eλ and REλ are the left- and right- integral by parts and retaining the integrated term to the hand turning points, respectively; and λ = l + 1/2 (l is leading order in the semiclassical parameter at t = 1, the orbital quantum number). one obtains an oscillating correction to the electron density in an atom: Performing the integral by parts with respect to energy, retaining the integrated term to the leading 0 µ sin[]2σ˜ µ()r + α˜ ()()π2n + 1 Ð 2σµ order in the semiclassical parameter at E = , and n ()r ≈ ------, (9) changing to the variable osc 2 0 ˜ 4πr pµ()τr µδµ()r sin() πα˜

2 λ pµλ()r t ==1 Ð ------where 2 2 () pµ()r r pµ r r σ () () σ0 σ () in the integral with respect to λ2, one obtains ˜ µ r ==∫dr' pµ r' , µ ˜ µ Rµ , 0 1 ∞ pµ()r n ()r Ӎ Ð------dt ()Ð1 ks+ 0 osc 2 ∫ ∑ σµ τ˜ µ()r 2π n ≤≤------n + 1, α˜ = ------, 0 ks,∞= Ð π 0 (8) τµ []σ () σ0 π () 2 cos 2 µt r ++2s µt 2 kpµ r r 1 Ð t × ------, r τ () τ0 µt r + s µt τ () dr' τ0 τ () ˜ µ r ==∫------, µ ˜ µ Rµ , pµ()r' where 0

R ∂σ () µt r µt r dr' 0 ˜ dr' 1 1 2 τµ ()r ==------, τµ =τµ ()Rµ' δµ()r = ------Ð ------Ð ------. t ∂µ ∫ () t t t ∫ 2 () pµt r' r' pµ r' 2Z/r' 2Zr r 0 is the classical time. Note that a similar expression for In the ThomasÐFermi model of a free atom, the radius σ τ the oscillating correction is obtained if µt(r) and µt(r) 0 is inﬁnite, τµ = ∞, and (9) is identical to the expression are replaced by the “complementary” quantities obtained in [19]. In the case of a cluster, the oscillating σ () σ0 σ () τ () τ0 τ () correction is ˜ µt r ==µt Ð µt r , ˜ µt r µt Ð µt r . 0 These formulas are used in the computations for atoms sin[]2σµ()r + α()nπ Ð 2σµ n ()r ≈ Ð------, (10) presented below. osc 2 0 8πr pµ()τr µδµ()r sin()0.5πα Expression (8) for nosc(r) contains a double sum and an integral of an oscillating function. Their values are where mainly determined by the contributions of the integration limits and the regions of slow variation of the argu- 0 1 2σµ 1 τµ()r ment of an oscillating function. An analysis shows that n Ð --- ≤≤------n + ---, α = ------, π 0 the argument of the cosine in (8) varies rapidly when 2 2 τµ s ≠ k Ð 1 for attractive potentials of atomic type

(U(r 0) ÐZ/r, where Z is the charge number) Rµ and when s ≠ 2k for anharmonic attractive potentials of dr' δ () ------cluster type (U(r 0) U(0) = const < 0). In par- µ r = ∫ 2 , r' pµ()r' ticular, its derivative with respect to t goes to inﬁnity as r

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D/Z4/3 δ˜ () atom, r = 0, r = Rµ, and r = r0 (µ r0 = 0), which Ӷ Ӷ 0.4 0.4 restricts the applicability of expression (9) to 0 r r0. In the case of a cluster, there are two such points, r = 0 0.3 and r = Rµ, and the interval of applicability is wider. 0.3 0.2

0.2 0.1 4. ATOMS, ATOMIC CLUSTERS, HOLLOW CLUSTERS, AND FULLERENES 0 0.1 0.2 0.3 0.4 0.5 In this section, the formulas obtained above are 0.1 applied to speciﬁc spherically symmetric systems: medium-sized and large atoms, solid sodium clusters, hollow clusters, and endohedrally doped hollow clus- 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 ters. The possibility of using a hollow spherical cluster ⁄ rZ13 as a model of the fullerene C60 is also discussed. The only difference between the electronÐion sys- 2 Fig. 1. Radial electron-density distribution D(r) = 4πr n(r) tems considered in this section lies in the potential Ui(r) in a free atom predicted by the ThomasÐFermi model (solid generated by ions. curves) and by the ITF model with Z = 10 (circles) and 80 (triangles). 4.1. Atom

D(r) The external field in an atom is generated by a nucleus with charge number Z (Z = Ne in a neutral atom): 140 () 120 Ui r = ÐZ/r. 100 Figure 1 compares the radial electron-density distribu- 80 tions calculated by solving Eq. (5) for neon (Z = 10) and mercury (Z = 80) with those predicted by the ThomasÐ 60 Fermi model. It is well known that the model yields a ∝ Ð3/2 40 divergent density at the origin, nTF r , and the cor- 20 responding radial density distribution behaves as a ∝ 0 square root, D(r 0) r . In the ITF model, as 0.1 0.2 0.3 0.4 0.5 in [10], the effects of second order in the semiclassical r parameter are treated in a self-consistent manner, electron density is constant at the origin, and the radial den- Fig. 2. Radial electron-density distribution in the central sity distribution is a quadratic function of radius region of the mercury atom (without exchange and correla- (D(r 0) ∝ r2). In the system of units employed tion interactions) predicted by the ITF model (dotted curve), Hartree model (solid curve), and the ITF model with oscil- here, the ThomasÐFermi model yields the same curve lating correction (9) (thin curve). for any Z. The figure demonstrates that the ITF results tend to this universal curve as the charge number Z increases, while their correct behavior near the origin is the integrals in σµ(r) and τµ(r) with respect to r' are preserved (see the enlarged inset). This is a good illus- 0 tration of the well-known validity conditions ZÐ1/3 Ӷ 1 taken from r to the right-hand turning point Rµ, σµ = and r > 1/Z for the semiclassical description as applied 0 σµ(0), and τµ = τµ(0). to an atom. Figure 2 compares the electron density in the central Let us briefly discuss the applicability of the expres- region of the mercury atom predicted by the ITF model sions obtained here. Integrating by parts and retaining (without exchange and correlation terms) with and the integrated term, one obtains an accurate estimate for without the use of oscillating correction (9) against that the integral if the derivative of the argument of the calculated in the Hartree approximation [21]. Within its oscillating function (cosine) in (8) with respect to t is limits of applicability, the oscillating correction accu- sufficiently large. In the present case, this derivative (at rately describes the oscillation associated with the con- 2 2 ˜ 2 2 t = 1) is 2 pµ()r r δµ()r for an atom and 2 pµ()r r δµ()r centration of electrons in the K, L, and M shells. for a cluster. Accordingly, the domains where the esti- The top row in Table 1 shows the results of a self- mation procedure used here is not valid are determined consistent calculation of the total electron-shell energy by the points where the factors in these products vanish. for the mercury atom and some of its components, There exist exactly three such points in the case of an including the exchange energy, the correlation energy,

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Table 1. Energy of the electron shell of the mercury atom according to the ITF model δ δ δ EEex Ecorr 2E 4EE Ð Ecorr + 4EEstat Ð1.9616 × 104 Ð3.3196 × 102 Ð1.0086 × 101 1.2772 × 103 7.8599 × 102 Ð1.8820 × 104 Ð1.8400 × 104 Ð3.2560 × 102 Ð9.9442 × 100 1.2770 × 103 7.8632 × 102 and the second-order gradient correction. The bottom obtained in the present calculations (fourth column). row presents the exchange and correlation energies cal- This is explained by recalling that the Scott correction culated as additive corrections (not self-consistently) was shown in [23] to arise from the summation of a by solving Eq. (5) without exchange and correlation divergent series of quantum-mechanical corrections to terms. It is obvious that the self-consistent and non- energy. The divergence is due to the inapplicability of self-consistent results are very close. This agrees with the semiclassical approximation in the central region of δ δ the assertion in [7] about the energy correction associ- an atom. Thus, the large values of 2E and 4E given by ated with the small terms ignored in functional (1). the ITF model are explained by the contributions of this As noted in the Introduction, the calculations per- region that are responsible for the Scott correction. formed for nuclei and clusters in [13, 14] took into account the fourth-order gradient correction in a self- consistent manner. This correction can also be esti- 4.2. Atomic Cluster mated for an atom by using expression (6) with the den- In the jellium model of an atomic cluster, Ni ions sity obtained by solving Eq. (5). The resulting estimates uniformly distributed over the volume of radius R gen- based on the ITF model (with and without exchange erate the potential and correlation terms) are also presented in Table 1. First, note that the fourth-order correction is smaller  than the second-order one by only a factor of 1.5, which N r 2 Ð------e- 3 Ð,--- rR≤ , implies slow convergence of the expansion. Second, the  2R R fourth-order gradient correction cannot be used sepa- () Ui r =  rately, since it must be combined with quantum-  N Ð------e, rR> , mechanical corrections to the exchange energy, which R have yet to be found (see [9, 22]). Third, the self-con-  sistent treatment of the exchangeÐcorrelation interaction weakly affects the values of gradient corrections; where Ne = wNi (w is the valence of an atom). The however, this is not true for clusters (see below). radius of a cluster is related to the number of electrons The last two columns in Table 1 present the values in it: R = r N1/3 , where r is the WignerÐSeitz radius. of electron energy obtained by using the ITF model s e s (without the correlation term and with fourth-order gra- Figure 3 takes the electron density distribution in the dient correction (6) added) and the well-known semi- cluster Na58 (rs = 3.92, R(N = 58) = 15.17) predicted classical formula by the ITF model with and without oscillating correction (10) and compares it with the results obtained by 7/3 2 5/3 Estat = Ð0.768745Z + 0.5Z Ð 0.2699Z . (11) the KohnÐSham method borrowed from [24]. Expres- sion (10) provides a good approximation of the oscilla- For Z > 4, expression (11) corresponds to the depen- tion amplitude and phase outside the neighborhoods of dence of the electron-shell energy on Z in the HartreeÐ r = 0 and r = Rµ = 16.5, in agreement with the analysis Fock model within fractions of a percent. A comparison presented in Section 3. of the present results with Estat shows that the error in Figure 4 compares the total energy per atom the calculated energy is reduced from 6.5% to 2.3% by δ taking into account 4E. E E + E ---- = ------e -i The correspondence between the corrections calcu- N N lated in this study and the summands in (11) can be explained as follows. The first term in (11) is the energy for a sodium cluster with its analog for bulk metal. The predicted by the ThomasÐFermi model, the second one energy of ions uniformly distributed over a ball of is the Scott correction, and the third one is the sum of 2 the exchange energy and a finite part of the second- radius R is Ei = 0.6Ne /R. The energy calculated in this order gradient correction. Subtracting the exchange study agrees (within 1Ð5%) with the semiclassical energy from the last term in (11) for Z = 80, one finds results obtained in [13]. The agreement improves with that the finite part of the second-order quantum- increasing number of particles, i.e., with a decreasing mechanical correction is much smaller than the value semiclassical parameter (which is proportional to

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ne/ni tively, the results predicted by the complete ITF model 1.4 and by the ITF model without exchangeÐcorrelation terms. Additive fourth-order correction (6) calculated 1.2 for clusters according to the complete ITF model is by 1.0 a factor of 1.5 greater than the contribution of the second-order correction (top row). However, both gradient 0.8 corrections are reduced and their ratio changes (bottom 0.6 row) by dropping the exchangeÐcorrelation terms in Eq. (5), whereas the exchange and correlation terms 0.4 calculated self-consistently (top row) and as additive 0.2 corrections (bottom row) are similar in both cases. Thus, exchange and correlation effects are much 0 2 4 6 8 10 12 14 16 18 20 more important for clusters as compared to free atoms. r When they are taken into account, the values of gradient terms are substantially modiﬁed and the use of the Fig. 3. Relative electron-density distribution in Na58 pre- fourth-order correction leads to unsatisfactory results. dicted by the ITF models with and without oscillating cor- Since this correction has no theoretical justiﬁcation rection (10) (thick and thin solid curves, respectively) and by the KohnÐSham method in [24] (symbols). The dotted (see Section 4.1), the good results of the self-consistent curve is the distribution of ions. treatment of the fourth-order correction in [13, 14] should be interpreted as accidental and attributed to the use of a more suitable class of trial functions. E/N(|ε∞|) –0.7 Finally, note that the contribution of the shell- related oscillatory effects to the energy of cluster electrons can be calculated by the method described in [7].

–0.8 4.3. Hollow Cluster and Fullerene The calculated characteristics of the hollow cluster proposed in [15] as a model of the fullerene C will –0.9 60 now be presented. The charge of Ni ions in a hollow cluster is “smeared out” over a spherical shell of radius R. This distribution of ions generates the potential –1.0 0 2 4 6 8 10 N 1/3  e ≤ N Ð------, rR, () R Fig. 4. Total energy per atom in sodium clusters in units of Ui r =  the absolute value of energy per atom in a metal (|ε | =  Ne ∞ Ð------, rR> , 2.252 eV). The dotted and solid curves correspond to the r ITF models with and without the fourth-order gradient correction (6), respectively. The symbols were obtained in self- which acts on mutually interacting valence electrons consistent calculations using correction (6) [13]. (Ne = wNi, where w = 1 and Ne = Ni = 60). The energy of ions uniformly distributed over the sphere is Ei = Ð1/3 2 Ne ). Furthermore, Fig. 4 demonstrates that the addi- 0.5Ne /R. According to [15], the ThomasÐFermi model tive contribution of correction (6) is too large and a bet- of a hollow cluster does not admit any ﬁnite equilib- ter result is obtained without it. rium radius R0. The ﬁnite value R0 = 7.36 close to ()exp The results presented in Table 2 for the cluster Na100 R0 = 6.75 measured for the fullerene C60 was illustrate the contributions to the energy of electrons obtained in [15] by using the ion energy due to the exchange, correlation, and gradient terms. As in Table 1, the top and bottom rows here show, respec- N2 E = 0.4311------e , (12) i R Table 2. Energies of electrons and ions in the Na100 cluster which corresponds to the real C molecule with ions δ δ 60 Ee Eex Ecorr 2E 4EEi located at the vertices of the truncated icosahedron of Ð337.71 Ð11.254 Ð3.6959 0.2126 0.36 329.76 radius R. The calculations performed for a hollow cluster in Ð10.577 Ð3.5387 0.10552 0.0478 this study show that the equilibrium radius R0 predicted

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D(r) 2, and 3. The resulting monotonically decreasing func- 5 tion is accurately approximated by the expression Z = 0 4 () () ()2 () R0 Z ==R0 0 expÐ0.04Z , R0 0 29. 3 Figure 5 shows the radial electron-density distribution D(r) = 4πr2n (r) calculated for a hollow cluster (Z = 0) 2 e and an endohedrally doped hollow cluster with a trivalent central atom (Z = 3) having the corresponding equi- 1 Z = 3 librium radii.

0 5 10 15 20 25 30 35 5. CONCLUSIONS r Fig. 5. Radial electron-density distributions in a hollow Based on an improved semiclassical model, a uni- cluster (Z = 0, solid curve) and in a hollow cluster with a fied efficient algorithm is proposed for calculating the trivalent central atom (Z = 3, dotted curve) (Ne Ð Z = 60). characteristics of a spherically symmetric many-body electronÐion system. The EulerÐLagrange equation for density is derived by minimizing the semiclassical by the ITF model is finite. Its value was determined by energy functional including the exchangeÐcorrelation minimizing the total energy of the cluster, E = Ee + Ei, interaction and second-order gradient terms. This equa- as a function of R. As a result, a reasonable square-root tion is applied to calculate averaged characteristics of dependence of the equilibrium radius on the number of various physical systems, such as atoms and solid or α 1/2 α hollow clusters. particles was obtained: R0 = Ne with = 3.743. When this formula is applied to the fullerene, the result The proposed model can be used to calculate the is too large (R0 = 29), because the energy of ions is sub- total energy of electrons and their local characteristics stantially overestimated by replacing their actual distri- (density and potential distributions). The radial bution with a spherical shell. The corresponding radial Schrödinger equation with this potential is solved ana- electron-density distribution is shown in Fig. 5. When lytically in the semiclassical approximation. The result- the actual (not self-consistent) energy of ions given ing expression is used to obtain an oscillating correc- by (12) is used, the ITF model yields R = 5 [16]. tion to density in a finite system and calculate electron- 0 density distributions for atoms and atomic clusters Analysis of spherical hollow clusters based on self- reflecting the discreteness of their electronic spectra. consistent modeling is of special interest because it can The contributions of the fourth-order corrections based on the averaged electron densities to the energies of elucidate, in particular, the dependence of R0 on the valence Z of the atom located at the center of the cluster. these systems are evaluated. The use of these correc- The real system corresponding to this model is a highly tions in calculations is shown to be unjustified. symmetric fullerene with a central atom, such as LaC60. An analysis of the properties of a hollow cluster In this case, the potential generated by the ions and their with ions uniformly “smeared” over a sphere is pre- energy are expressed as sented. In particular, the spatial density distribution is calculated for the valence electrons, and the total N Ð Z energy of a hollow cluster is determined as a function  e ≤ of radius for a given number of atoms in the cluster. It ------, rR () Z R is shown that the resulting curve has a minimum at a Ui r = Ð --- Ð  r N Ð Z finite (equilibrium) value of radius. A reasonable ------e , rR> ,  r square-root dependence of the equilibrium radius on the number of particles in a cluster is obtained. The dependence of the equilibrium radius of an endohe- 2 2 drally doped hollow cluster on the valence of the atom Ne Ð Z Ei = ------, located at its center is analyzed. The possibility of using 2R hollow spherical clusters as models of the fullerenes C60 and LaC60 is also discussed. It is shown that an ade- where Z is the charge number of the central ion and the quate quantitative description cannot be developed, number Ne of electrons includes the Z valence electrons because the energy of ions is highly overestimated of the central atom. Calculations were performed for when their actual distribution in a fullerene is approxi- Ne Ð Z = 60 electrons in a spherical shell and Z = 0, 1, mated by a spherical shell.

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ACKNOWLEDGMENTS 10. N. N. Kalitkin and L. V. Kuz’mina, Fiz. Tverd. Tela (Leningrad) 13, 2314 (1971) [Sov. Phys. Solid State 13, This work was supported in part by the Russian 1938 (1971)]. Foundation for Basic Research, project nos. 03-01-00438 11. F. Perro, Phys. Rev. A 30, 586 (1979). and 02-01-00185, and by the Ministry of Industry 12. M. Membrado, A. F. Pacheco, and J. Saudo, Phys. Rev. and Science of the Russian Federation, grant B 41, 5643 (1990). NSh-2060.2003.2. 13. C. Yannouleas and U. Landman, Phys. Rev. B 48, 8376 (1993). REFERENCES 14. M. Brack and P. Quentin, Nucl. Phys. A 361, 35 (1981). 15. D. P. Clougherty and X. Zhu, Phys. Rev. A 56, 632 1. Theory of the Inhomogeneous Electron Gas, Ed. by (1997). S. Lundqvist and N. H. March (Plenum, New York, 16. V. Ya. Karpov and G. V. Shpatakovskaya, Élektron. Zh. 1983; Mir, Moscow, 1987). Issled. Rossii 191, 2118 (2002), http://zhur- 2. D. A. Kirzhnits, Yu. E. Lozovik, and G. V. Shpatak- nal.ape.relarn.ru/article/2002/191.pdf. ovskaya, Usp. Fiz. Nauk 111, 3 (1975) [Sov. Phys. Usp. 17. G. V. Shpatakovskaya, in Physics of Extremal States, Ed. 16, 587 (1975)]. by V. E. Fortov et al. (Inst. Probl. Khim. Fiz. Ross. Akad. 3. G. V. Shpatakovskaya, Teploﬁz. Vys. Temp. 23, 42 Nauk, Chernogolovka, 2003), p. 119. (1985). 18. L. Hedin and B. I. Lundqvist, J. Phys. C 4, 2064 (1971). 4. E. A. Kuz’menkov and G. V. Shpatakovskaya, Int. 19. D. A. Kirzhnits and G. V. Shpatakovskaya, Zh. Éksp. J. Thermophys. 13, 315 (1992). Teor. Fiz. 62, 2082 (1972) [Sov. Phys. JETP 35, 1088 (1972)]. 5. D. A. Kirzhnits and G. V. Shpatakovskaya, Preprint 20. I. S. Gradshteœn and I. M. Ryzhik, Tables of Integrals, No. 33, FI RAN (Physical Inst., Russian Academy of Sums, Series, and Products, 4th ed. (Gostekhizdat, Mos- Sciences, Moscow, 1998). cow, 1962; Academic, New York, 1980). 6. G. V. Shpatakovskaya, Pis’ma Zh. Éksp. Teor. Fiz. 70, 21. D. R. Hartree and W. Hartree, Proc. R. Soc. London 149, 333 (1999) [JETP Lett. 70, 334 (1999)]; cond- 210 (1935). mat/0001116. 22. D. A. Kirzhnits, Zh. Éksp. Teor. Fiz. 34, 1625 (1958) 7. G. V. Shpatakovskaya, Zh. Éksp. Teor. Fiz. 118, 87 [Sov. Phys. JETP 7, 1116 (1958)]. (2000) [JETP 91, 76 (2000)]. 23. D. A. Kirzhnits and G. V. Shpatakovskaya, Zh. Éksp. 8. G. V. Shpatakovskaya, Pis’ma Zh. Éksp. Teor. Fiz. 73, Teor. Fiz. 108, 1238 (1995) [JETP 81, 679 (1995)]. 306 (2001) [JETP Lett. 73, 268 (2001)]. 24. M. Brack, Rev. Mod. Phys. 65, 677 (1993). 9. D. A. Kirzhnits, Zh. Éksp. Teor. Fiz. 32, 115 (1957) [Sov. Phys. JETP 5, 64 (1957)]. Translated by A. Betev

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 463Ð477. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 527Ð542. Original Russian Text Copyright © 2004 by Martemyanov, Dolgova, Fedyanin. ATOMS, SPECTRA, RADIATION

Optical Third-Harmonic Generation in One-Dimensional Photonic Crystals and Microcavities M. G. Martemyanov*, T. V. Dolgova, and A. A. Fedyanin** Moscow State University, Moscow, 119992 Russia e-mail: *[email protected]; **URL:http://www.shg.ru Received August 25, 2003

Abstract—The formalism of nonlinear transfer matrices is used to develop a phenomenological model of a cubic nonlinear-optical response of one-dimensional photonic crystals and microcavities. It is shown that third- harmonic generation can be resonantly enhanced by frequency-angular tuning of the fundamental wave to the photonic band-gap edges and the microcavity mode. The positions and amplitudes of third-harmonic resonances at the edges of a photonic band gap strongly depend on the value and sign of the dispersion of refractive indexes of the layers that constitute the photonic crystal. Model calculations elucidate the role played by phase matching and spatial localization of the fundamental and third-harmonic ﬁelds inside a photonic crystal as the main mechanisms of enhancement of third-harmonic generation. The experimental spectrum of third-harmonic intensity of a porous silicon microcavity agrees with the calculated results. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION susceptibility. The first process was considered for an infinite photonic crystal in [12], where it was shown Photonic crystals have been extensively studied in that there were structure parameters at which phase recent years because of their unique dispersion proper- matching conditions were simultaneously satisfied for ties and the possibility of modulating the spectral den- the fundamental and second-harmonic waves and the sity of optical field modes, which manifests itself by the fundamental and third-harmonic waves. With these formation of photonic band gaps [1]. Fundamental parameters, the time evolution of second- and third-har- interest in photonic crystals, in particular, stems from monic intensities was studied, and it was shown that the peculiar nonlinear optical effects, such as bistability [2] pump energy could not be completely transfered to the and optical switching [3] due to modulation of the refractive index of one-dimensional photonic crystal second or third harmonic. In cascade THG by a one- layers in a high-intensity field. This modulation causes dimensional photonic crystal with quadratic suscepti- a dynamic or quasi-stationary shift of the photonic band bility [13, 14], simultaneous phase matching of the gap in a photonic crystal with a cubic nonlinearity. In pump with the second and third harmonics can also be such crystals, one can observe four-wave mixing and achieved by adjusting the optical thicknesses of photo- excitation of the waveguide mode at the anti-Stokes fre- nic crystal layers. The calculations reported in [13] quency which propagates along interfaces [4]. In media were performed for a photonic crystal with an infinite with modulation only of nonlinear susceptibility with a number of layers, and only phase matching effects were period of the order of the wavelength, nonlinear diffrac- therefore studied. In [14], a photonic crystal with a tion effects are observed [5]. finite number of layers was considered, and effects of the spatial localization of fields related to its finite The use of photonic crystals for effectively generat- dimensions were taken into account. Pumping field ing radiation at the second harmonic frequency was localization effects can be enhanced by the introduction suggested in [6] and experimentally implemented for of a defect into a photonic crystal. In such a microcavity the first time in [7]. Phase mismatch between the fun- with distributed mirrors, the electromagnetic field reso- damental and second harmonic waves is minimized by nant to the microcavity mode is effectively localized, adding the reciprocal lattice vector of the periodic which enhances second [15, 16] or third [17] harmonic medium to the wave vectors of the interacting waves. generation. The enhancement of harmonic wave gener- When one of the frequencies is tuned to the edge of a ation at the photonic band gap edge and in the micro- photonic band gap, the phase matching condition for cavity mode is a result of the combined effects of phase pumping and second harmonic waves is satisfied, matching due to the periodicity of Bragg reflectors and which results in the enhancement of second harmonic field localization caused by the presence of a microcav- generation in photonic crystals [8Ð11]. ity spacer and finite photonic-crystal dimensions [11]. Third-harmonic generation (THG) in a photonic A key parameter that determines the enhancement of crystal can occur either directly due to cubic suscepti- third harmonic generation is the dispersion of refractive bility or in a cascade manner as a result of quadratic indexes of layers in a photonic crystal or microcavity.

464 MARTEMYANOV et al. Its compensation is the essence of the phase matching across the photonic crystal. There are two equivalent at the photonic band gap edges in a photonic crystal and approaches. The first one, described in [21], uses the in the microcavity mode. However, the dependence of formalism of Green functions suggested by Sipe [22]. the magnitude and spectral position of third harmonic The second approach given in [23] is based on the for- resonances on the magnitude and sign of the dispersion malism of coupled and free harmonic waves introduced of layers constituting a photonic crystal or microcavity by Bloembergen and Pershan [24]. Both rely on direct has not been studied yet. solution of an inhomogeneous wave equation with the In this work, we study THG in one-dimensional use of the Green functions of a photonic crystal and finite photonic crystals and microcavities characterized provide additional information about the nonlinear ()3 optical processes under consideration, such as contribu- by a cubic nonlinearity χˆ . The formalism of nonlin- tions of separate layers to the resulting third-harmonic ear transfer matrices is used to elucidate the role played wave. by each of the mechanisms of the enhancement of third The method of nonlinear transfer matrices sug- harmonic generation, that is, phase matching and gested in [23] can conveniently be used to calculate pumping and third harmonic field localization when the nonlinear-optical effects in one-dimensional photonic pumping wave is in resonance with the microcavity crystals because of its simplicity and form optimal for mode or photonic band gap edge. The spectra of third numerical calculations. The problem of THG in photo- harmonic intensity in the spectral range containing a nic crystals can be decomposed into three sequential photonic band gap with a microcavity mode and a pass- stages. First, the fundamental wave propagation in a band region are calculated in the approximation of a multilayer linear structure is described taking into given pumping field. The dependence of third harmonic account multiple-reflection interference, and the spatial generation enhancement on the dispersion of the refrac- pumping field distribution within the photonic crystal is tive indexes of the layers that constitute a photonic calculated. At the second stage, cubic polarization crystal is studied. This dispersion determines the () mutual arrangement of photonic band gaps and micro- induced in a medium with nonzero χˆ 3 is determined. cavity modes at the pump and third-harmonic wave- Lastly, the linear problem of third harmonic wave prop- lengths. The calculation results are compared with the agation in a layered structure is solved taking into experimental third harmonic spectrum generated in a account coupled and homogeneous waves, and the microcavity made from mesoporous silicon. intensity of the third harmonic wave that emerges from the photonic crystal is found. 2. NONLINEAR TRANSFER-MATRIX METHODS Let the z axis be perpendicular to the surface of the FOR CALCULATING THIRD-HARMONIC photonic crystal and xz be the plane of pumping wave GENERATION IN PHOTONIC CRYSTALS incidence (Fig. 1a). A monochromatic linearly polarized fundamental wave with frequency ω, wave vector 2.1. Nonlinear Transfer-Matrix Method + + k1 , and amplitude E1 propagates in half-space 1 at There are several approaches to calculating optical θ angle 1 to the normal to the surface of the photonic harmonic generation in one-dimensional photonic crys- crystal. We assume that the photonic crystal is optically tals. One of these is via solving a system of reduced inactive and nonmagnetic; s- and p-polarized waves equations obtained in the method of slowly varying will therefore be considered separately. The electro- amplitudes [12Ð14, 18]. This approach can be used to magnetic field in the jth layer is a superposition of two take into account energy transfer from the pump to the plane forward waves (propagating in the positive direc- generated harmonic; analyze simultaneous generation tion along the z axis) and backward, of the second and third harmonics; and examine the time evolution of the fundamental and the second- and ω(), + []ω ()()ω ω third-harmonic waves, which is important for studying E j zt = E j exp ikzj, zdÐ ij + ikxj, xiÐ t harmonic generation by femtosecond pulses. In the (1) Ð []ω ()()ω ω approximation of constant fundamental field, THG is + E j exp Ð ikzj, zdÐ ij + ikxj, xiÐ t , described by a single inhomogeneous equation, which can be directly solved using the Green function of a mul- where tilayer structure [4, 19]; the solution is constructed based on linearly independent solutions to the homoge- ω θ ω θ neous wave equation, which can be found using the for- kzj, ==k j cos j, kxj, k j sin j, malism of transfer matrices [20]. Lastly, a convenient method is the extension of the formalism of transfer dij is the z coordinate of the boundary between the ith matrices to harmonic generation. This method is appli- θ and jth layers, kj is the wave vector, and j is the angle cable in the approximation of constant fundamental of refraction of the pumping wave in the jth layer. The wave and when the fields are stationary, that is, under ω ω the condition that the pumping pulse width is much exp(ikxj, x Ð i t) term will further be omitted because greater than the time of fundamental-wave propagation of the translational invariance of the problem in the xy

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OPTICAL THIRD-HARMONIC GENERATION IN ONE-DIMENSIONAL PHOTONIC CRYSTALS 465 plane and its stationary character. The field at the left The third stage of solving the problem can be con- boundary of the jth layer is represented by the two- siderably simplified. In the approximation of a given component vector E ≡ (E+ + EÐ ); the first component pumping field, cubic polarization in each layer is a third j j j harmonic source independent of the other layers. Thus, of this vector is the amplitude of the forward wave and we can solve the problem of third-harmonic generation the second one, that of the backward wave. The relation and propagation for a single photonic crystal layer with between Ej and Ek in the kth layer at its left boundary is a cubic susceptibility. The third-harmonic field gener- given by the two 2 × 2 matrices ated by the photonic crystal can then be obtained by summing such partial third-harmonic outputs of all lay-  ers and taking into account their phases. +  Ek 1/tkj rkj/tkj =  Let the jth layer be nonlinear. The interference of the Ð r /t 1/t (s) Ek kj kj kj coupled E and free Ej third harmonic waves is (2) included in the boundary conditions, which, for the ijth  and jkth boundaries, have the form ()ω + × expikzj, d j 0 E j . ()ω Ð ()s ()s 0 exp Ðikzj, d j E j FiEi = MijE j + Mi E , (5) ()s ()s ()s MkjF jE j + Mk F E = Ek. Here, the first matrix contains the Fresnel reflectivity rkj and transmissivity tkj for the wave incident from the kth to the jth layer and relates fields to the left and right of The amplitude of the inhomogeneous third-harmonic the kjth interface. In what follows, this matrix is wave in (5) is calculated in the jth nonlinear layer at its Φ left boundary, and all homogeneous waves are third- denoted by Mkj. The second j(dj) matrix describes field propagation in the jth layer of thickness d from the harmonic waves. Matrices M with index (s) are con- j structed similarly to the usual transfer matrices, but the left to right boundary. The fields in half-spaces 1 and l Fresnel coefficients in their elements contain refractive can therefore be related. Under the assumption that the indexes for the inhomogeneous third-harmonic wave in backward wave is absent in half-space l and that a wave the nonlinear layer and the free third harmonic in the with unit amplitude is incident on the 1Ð2 boundary, we layer whose number equals the lower index of the obtain (s) Φ matrix. The F matrix is similar to j and is obtained from the latter by replacing the wave vectors of the free  T T  third harmonic with the wave vectors of the inhomoge- t = 11 12 1 . (3)   neous wave. System (5) yields 0 T 21 T 22 r E = M F ()M F E + S , (6) Here, r and t are the reflectivity and transmissivity of k kj j ji i i j the photonic crystal as a whole. They are determined from (3), which gives where the vector

T T T Ð T T ()()s Φ()s ()s ()s r ==Ð------21, t ------11 22 12 21-, S j = F jM j Ð M j E (7) T 22 T 22

is singled out for convenience. The F j matrix is where Tαβ are the elements of the transfer matrix for the ≡ inverse to Fj. Equation (6) determines the third-har- photonic crystal as a whole, T Ml(l Ð 1)F(l Ð 1) … M21. The fundamental field distribution within the photonic monic field in the kth layer as a superposition of the crystal is given by waves transmitted from the ith layer and generated in the jth layer by nonlinear sources. Equation (7) contains the contribution of the nonlinear jth layer to the ω() E j z third-harmonic wave; the term in parentheses takes into account the interference of the homogeneous and inho-  (4) () … 1 mogeneous waves. Under the assumption that no exter- = F j zdÐ jj()Ð 1 M jj()Ð 1 F()j Ð 1 M21. r nal field with the third-harmonic wavelength is incident on the photonic crystal, (6) can be rewritten as Spatial distribution (4) can be used to calculate the spatial distribution of the cubic polarization wave and +() 0 R El j Ð L = S , (8) inhomogeneous third harmonic can be calculated; this jl j1 Ð() j is done in the next Section. 0 E1 j

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 466 MARTEMYANOV et al. dipole surface and bulk quadrupole sources; con- . . . (s) . . . 1 2 i j E⊥ k l – 1 l versely, the generation of the bulk dipole third har- E(s) (a) || monic is allowed. As a result, the cascade process k– 1 E– E(s) becomes less effective. 1 y ks θ + A nonlinear photonic crystal layer can be treated as + kj 1 – Ej E+ ki a medium rotationally isotropic in the plane of the k+ 1 layer. Such a medium is characterized by the symmetry 1 ∞ ∞ ∞ + – groups /mm and 2 (cylinders) and m (a cone with ki E – j kj a symmetry axis of an inﬁnite order perpendicular to the surface of the layer). Equations for inhomogeneous i PII II k x θII ks j third-harmonic waves for other symmetry groups can I (b) θII Px be obtained similarly. The tensor of dipole cubic sus- θI I () x θI ks χ 3 ∞ ∞ II II ceptibility ˆ invariant with respect to the m, 2, Ps Pz I I ∞ θII Ps Pz and /mm groups and symmetrical with respect to per- yz mutations of the last three indices has four independent θI nonzero components [25],

Ex Ep 1 1 θ χ ≡χχ ==---χ =---χ , 1 xxyy yyxx 3 xxxx 3 yyyy Es Ez kω χ ≡χχ = , θ 2 yyzz xxzz (9) χ ≡ χ 3 zzzz, Fig. 1. (a) Scheme of a one-dimensional photonic crystal χ ≡χχ = . (layers 2 … l Ð 1) (half-spaces 1 and l denote the vacuum 4 zzxx zzyy and substrate, s-polarized fundamental wave is shown) and (b) THG scheme in the jth nonlinear layer. Let the jth layer with cubic susceptibility be situated between two linear layers i and k (Fig. 1). Let us determine the amplitude of the inhomogeneous third har- where the matrices monic E(s) on the jith interface from the fundamental field amplitude E on the same boundary. The angle ≡ …Φ j R jl F jM jkFkMkk()+ 1 ()l Ð 1 M()l Ð 1 l, between the fundamental wave vector ≡ … ω L j1 M jiFiMii()Ð 1 F2M21 kω = nω----, c characterize the propagation of homogeneous third- harmonic waves from half-spaces 1 and l to the jth where nω is the refractive index at the fundamental fre- layer. It follows that, given Sj, we can find the ampli- quency, and the z axis is θ. Here and throughout, index tude and phase of the third-harmonic fields EÐ()j and j numbering layers is omitted. The dipole cubic polar- 1 ization is given by the convolution +() El j generated in the jth photonic crystal layer and emerging from the photonic crystal into the vacuum 3ω ()3 + ω Ð ω 3 P = χˆ ()E exp()ik z + E exp()Ðik z (half-space 1) and substrate (half-space l). The total … z z third-harmonic field in the substrate or vacuum is the ≡ I+ ()sI, IÐ ()sI, P exp ikz z + P exp Ðikz z (10) sum of E+()j or EÐ()j taken over all layers. l 1 II+ ()sII, IIÐ ()sII, + P exp ikz z + P exp Ðikz z . 2.2. Inhomogeneous Third-Harmonic Waves ω θ in a Photonic Crystal Here, kz = kωcos is the projection of the fundamental Let us obtain equations describing inhomogeneous wave vector onto the normal to the interface. In third-harmonic waves for direct THG by means of a Eq. (10), all cubic polarization terms are divided into χ()3 two types. The terms of the first type have the z wave cubic susceptibility ˆ . Equations for cascade THG sI, ω () vector component k = 3k and are obtained by the χˆ 2 z z caused by quadratic susceptibility can be obtained convolution of three fundamental waves propagating in similarly. The cascade process can be ignored when the the same direction. The other terms have the normal nonlinear medium has an inversion center. The genera- sII, ω tion of the bulk dipole second harmonic is then forbid- projection of the wave vector kz = kz and are den, and the second harmonic is only generated by obtained by the convolution of three fundamental

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 OPTICAL THIRD-HARMONIC GENERATION IN ONE-DIMENSIONAL PHOTONIC CRYSTALS 467 waves one of which has the z component of the wave + χ ()E E E sin2 θ + 2E E E sin2 θ , vector opposite to the projections of the other two 2 p p s s p p waves. The propagation of cubic polarization and inho- PII = Ðχ E3 sin3 θ mogeneous third-harmonic waves is determined by the z 3 p χ θ()3 2 θ effective permittivity calculated as Ð 4 sin 3E p cos ++EsEsE p 2E pEsEs

sIII, (), 3ω III, II k = ------⑀ . for P . Here, we use the equation c E ==E , E E cosθ, E =ÐE sinθ. For the waves of type I, y s x p z p The product of three two-component vectors is given sI, 3ω k ==3kω ------⑀()ω ; by the rule c  that is, + ⋅⋅+ + Ea Eb Ec EaEbEc = (13) ⑀I ⑀()ω Ð ⋅⋅Ð Ð = Ea Eb Ec and the polarization wave propagates in the medium collinearly to the fundamental wave, θI = θ. Similar cal- for the waves of type I and by the rule culations for the polarization waves of type II give  + ⋅⋅+ Ð ⑀II ⑀()ω ()2 θ Ea Eb Ec = 18+ sin /9. EaEbEc = (14) EÐ ⋅⋅EÐ E+ The angle between axis z and the propagation direction a b c of the polarization waves of type II is different from θ; it is given by for the waves of type II. Cubic polarization can conveniently be decomposed into components with polari- ω ω θII = arctan()3k /k . zation directions normal and parallel to the wave vector x z of the inhomogeneous wave and the s-polarized com- Taking into account nonzero components (9) of the ponent, ()3 tensor χˆ , the projections of the two-component vec- III, III, III, III, III, P|| = P sin()θ + P cos()θ , tors PI = (PI+, PIÐ) and PII = (PII+, PIIÐ) onto the coordi- x z nate axes expressed in terms of the s- and p-polarized III, III, III, III, III, P⊥ = P cos()θ Ð P sin()θ , (15) fundamental wave components can be written as x z III, Py . I χ ()3 3 θ 2 θ Px = 1 E p cos + Es E p cos For such components, the transition to third-harmonic χ 3 θθ2 + 2E p cos sin , inhomogeneous waves is simple [24]:

I 3 2 2 2 2 χ ()χθ θ ()s III, 4π III, III, Py = 1 Es + EsE p cos + 2EsE p sin , (11) () E = ------, - Py + P⊥ ⑀IIIÐ ⑀()3ω (16) I 1 3 3 P = Ð---χ E sin θ 4π III, z 3 p Ð ------P|| . 3 ⑀()3ω χ ()2 θ 3 θθ2 Ð 4 E pEs sin + E p sin cos Equations (16) for third-harmonic inhomogeneous for PI and waves are substituted into (7). The reduction of the general problem of THG in a photonic crystal to THG in a II χ ()3 3 θ θ θ photonic crystal containing a single layer with cubic Px = 1 3E p cos ++2E pEsEs cos EsEsE p cos susceptibility allows partial contributions of each pho- +3χ E3 sin2 θθcos , tonic crystal layer to the total third-harmonic wave to 2 p be calculated. In combination with the feasibility of PII = χ (3E 2 + 2E E E cos2 θ calculating the fundamental ﬁeld at each point in the y 1 s s p p photonic crystal, this is a convenient tool for analyzing the mechanisms of nonlinear-optical phenomena in 2 θ ) + E pE pEs cos (12) one-dimensional photonic crystals. Variation of the ini-

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Rs of porous silicon layers (porous silicon was used to fab- 1.0 ricate the microcavity studied experimentally in this 20 pairs (‡) work). The microcavity is obtained by doubling the 0.8 thickness of the central twenty-first layer. It is assumed 0.6 Si(001) that the photonic crystal and microcavity are placed between two half-spaces, air and a linear medium with 0.4 a refractive index equal to the refractive index of sili- 0.2 con. THG from the silicon substrate is ignored. Funda- mental radiation comes from air. The mechanisms of 0 3ωω2ω THG enhancement are studied in the frequency 1.0 domain, that is, under fundamental wavelength varia- cavity spacer (b) 0.8 10 pairs 9.5 pairs tions, which allows both photonic band gap edges to be observed for the same sample. This cannot be done by 0.6 Si(001) means of angular third-harmonic spectroscopy with a 0.4 fixed fundamental wavelength and normal component of the fundamental wave vector tuned by changing the 0.2 angle of incidence. All frequency spectra are calculated θ ° 0 at the incidence angle 1 = 45 . The fundamental wave- 300 500 700 900 1100 length λω is varied from 730 to 1100 nm. The refractive λ, nm indexes of photonic crystal layers are assumed to be constant in the spectral ranges of the fundamental and Fig. 2. Dependence of the linear reflection coefficient of the s-polarized fundamental wave on its wavelength calculated third harmonic wavelengths. for (a) a photonic crystal and (b) a microcavity. Photonic The spectra of the linear reflection coefficient of the crystal and microcavity schemes are shown in the insets. photonic crystal and microcavity are shown in Fig. 2. The same figure contains the spectra of the linear reflec- tial calculation parameters, such as the fundamental- tion coefficient in the wavelength ranges of the second radiation wavelength or the corresponding incidence (360Ð560 nm, this spectrum is further denoted by R2ω) angle, allows us to obtain the frequency and angular and third (240Ð370 nm, spectrum R3ω) harmonics. The spectra of the linear reflection coefficient and the inten- reflection coefficient Rω is close to unity from 820 to sities and phases of third harmonic waves. 940 nm, which is a manifestation of the photonic band gap. The R ω spectrum also contains a wavelength Note that (11), (12), and (15) specify possible polar- 3 izations of third-harmonic waves. If the fundamental region with the reflection coefficient close to unity wave is s-polarized, only an s-polarized third harmonic (photonic band gap at third-harmonic wavelengths), whereas there is no photonic band gap in the R2ω spec- is generated (the first term of the Py component), and if the fundamental wave is p-polarized, the third har- trum, because the phase difference between the waves reflected from layers with optical thicknesses λ/4 or monic is also p-polarized (the first terms of the Px and λ π P components). 3 /4 is n, where n is an even integer, and the waves are z added in phase. If the optical thickness of layers is λ/2, the phase difference between the waves reflected from 3. THIRD-HARMONIC GENERATION such layers is πm, where m is an odd integer, and the IN PHOTONIC CRYSTALS waves interfere destructively. The presence of a resonant layer manifests itself in the spectrum of the linear 3.1. The Absence of Dispersion reflection coefficient by a dip within the photonic band Calculations of the enhancement of third-harmonic gap corresponding to the microcavity mode. The micro- generation at the edge of the photonic band gap are per- cavity mode is present in both Rω and R3ω spectra. formed for a photonic crystal. For a microcavity, we The spatial distributions of the fundamental field study THG enhancement effects that appear when the amplitude shown in Fig. 3 were calculated at the wave- fundamental radiation is tuned across the region in the lengths corresponding to the minima in the spectrum of frequency-angle space that corresponds to the micro- the linear reflection coefficient that are closest to pho- cavity mode. The influence of the microcavity layer is tonic band gap (Figs. 3b, 3c), in the region within the weak at the edge of the photonic band gap, and third- photonic band gap of the photonic crystal (Fig. 3d), and harmonic enhancement is similar to that characteristic in the microcavity mode (Fig. 3a). The fundamental of photonic crystals. The model photonic crystal con- | + Ð | + Ð sists of 20 pairs of alternating layers with refractive field amplitude is Eω = E j + E j , where E j and E j are H L ω indexes nω = 1.93 and nω = 1.61 and optical thick- the complex components of the E j two-component λ λ λ nesses 0/4, where 0 = 960 nm (Fig. 2). The refractive pumping field vector given by (4). When ω is tuned indexes selected are close to the real refractive indexes across the photonic band gap, the fundamental field

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Eω λ3ω, nm 4.0 250 275 300 325 350 375 3.5 (a) (b) 10 1.0 3.0 2.5 8 0.8 2.0 1.5 6 0.6 s ω 3 R 1.0 I 0.5 4 0.4 0 3.5 (c) (d) 2 0.2 3.0 2.5 0 0 750 800 850 900 950 1000 1050 1100 2.0 λω, nm 1.5 Fig. 4. The third harmonic intensity spectrum of a photonic 1.0 crystal calculated for s-polarized fundamental and third harmonic waves (SS geometry) (the thick solid line); the reﬂec- 0.5 tion coefﬁcient spectra of an s-polarized wave calculated in the regions of fundamental (solid thin line) and third-har- 0 20004000 0 2000 4000 monic (dashed line) wavelengths. The third-harmonic 1000 3000 5000 1000 3000 5000 intensity I3ω is given in arbitrary units. d, nm

Fig. 3. Fundamental field amplitude distributions inside a The third-harmonic intensity spectrum I3ω generated photonic crystal and microcavity: (a) for the microcavity λ in a photonic crystal in the absence of refractive index mode, fundamental wavelength is ω = 877 nm; (b) at the dispersion is shown in Fig. 4. Multiple peaks located short-wavelength edge of the photonic crystal band gap, both within the photonic band gap and near every Rω λω = 806 nm; (c) at the long-wavelength edge of the photonic band gap, λω = 966 nm; and (d) within the photonic spectrum minimum to the left and right outside the pho- band gap, λω = 870 nm. The dashed line is the incident wave tonic band gap are observed in the third-harmonic spec- amplitude equal to one. trum. Third-harmonic enhancement is less pronounced at the edge of the photonic band gap, but the amplitude of third-harmonic peaks increases in the next linear exponentially decays as the depth of penetration into reflection coefficient minima. the photonic crystal increases (Fig. 3d). The fundamen- When the fundamental wavelength is tuned across tal wave which is in resonance with the microcavity the photonic band gap, the amplitude of the fundamen- mode is strongly localized inside the resonant layer. At tal field decreases exponentially as the depth of pene- the chosen microcavity parameters, Eω increases tration into the photonic crystal increases and the approximately fourfold. At the short-wavelength and source of third-harmonic generation is several photonic long-wavelength edges of the photonic band gap, the crystal layers near surface. Upon tuning λω to the min- fundamental field is localized less strongly and is imum of the reflection coefficient, the fundamental enhanced 2.1Ð2.3 times. This effect is caused by the field effectively penetrates deep into the photonic crys- finite photonic crystal length; with a larger number of tal and all its layers become sources of third-harmonic layers, the fundamental field is distributed more evenly. waves. At the same time, the amplitudes of the peaks of The amplitude of the pumping field resonant to the third-harmonic intensity are commensurate no matter microcavity mode reaches a maximum in the microcav- whether λω is tuned across the photonic band gap or the ity layer and sharply decreases in several neighboring minima of the reflection coefficient spectrum. This is layers, whereas in a photonic crystal, the fundamental evidence of dephasing of third-harmonic waves coming field tuned to photonic band gap edges is more evenly from different photonic crystal layers; these waves distributed over the photonic crystal. This means that destructively interfere with each other. It can be for effective THG in a photonic crystal, phase matching expected that the inclusion of dispersion into calcula- of the homogeneous third harmonic waves generated tions (when the refractive indexes of photonic crystal by various layers and having amplitudes of the same layers have different values at the fundamental and order is important. third-harmonic wavelengths) will change the phases

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λ λ 3ω, nm 3ω, nm 250 300 350 250 300 350 10

(‡) (b) 8

ω 6 3 I

1.0

0.6 s R

0.4

0.2

0 800 900 1000 1100 800 900 1000 1100

λω, nm λω, nm

Fig. 5. (a, b) Third-harmonic intensity spectra of a photonic crystal calculated for the SS geometry in the presence of dispersion of photonic crystal layers and (c, d) s-polarized wave reﬂection coefﬁcient spectra calculated in the wavelength ranges of fundamental waves (solid line) and the third harmonic (dashed line). Third-harmonic intensity units in Figs. 5a, 5b and 4 are incommensurate.

L − ∆ L L of partial third-harmonic waves coming from different nω = 0.045, and Figs 5b and 5d, to n3ω = n3ω Ð nω = layers and the peaks corresponding to the minimum of 0.051. The refractive index at the fundamental wave- Rω will gain in third-harmonic intensity. L length is taken to be nω = 1.610 + i × 0.00003; that is, absorption is virtually absent. The dispersion of the 3.2. Enhancement of Third-Harmonic Generation refractive indexes of optically denser photonic crystal in the Presence of Dispersion ∆ H L λ layers is set equal to n3ω nω /nω . The I3ω( ω) spectra Characteristic third-harmonic intensity spectra cal- show that the third-harmonic intensity resonantly culated in the presence of dispersion of the refractive increases in the spectral region of photonic band gap ∆ indexes of photonic crystal layers are shown in Figs. 5a edges. At n3ω < 0, the intensity of the third harmonic and 5b. The bottom panels show the linear reflection peak at the first Rω spectrum minimum to the left of the coefficient spectra Rω and R3ω. Figures 5a and 5c corre- photonic band gap (the short-wavelength edge of the spond to the dispersion of the refractive indexes of opti- photonic band gap) increases by no less than two orders ∆ L cally less dense photonic crystal layers n3ω = n3ω Ð of magnitude as compared to the intensity of the peaks

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 OPTICAL THIRD-HARMONIC GENERATION IN ONE-DIMENSIONAL PHOTONIC CRYSTALS 471 within the photonic band gap. At ∆n ω > 0, the third har- max 3 I ω monic intensity at the long-wavelength edge of the pho- 3 10 tonic band gap increases by approximately an order of 0.5 magnitude. –0.114 0.4 The correspondence between the sign of dispersion 8 and the spectral position of the third-harmonic intensity 0.3 ω max 3

peak with respect to the photonic band gap is unambig- I uous; namely, at ∆n ω < 0 and ∆n ω > 0, third-harmonic 0.2 3 3 6 intensity resonances are observed only at the short- and long-wavelength photonic band gap edges, respec- 0.1 tively. This is clearly seen in Fig. 6, where the depen- 4 dence of the amplitude of the third-harmonic intensity –0.045 0 0.05 0.10 0.15 0.20 ∆ max ∆ n3ω peaks I3ω on dispersion n3ω are shown at the short- and long-wavelength photonic band gap edges. The 2 main conclusion is that we do not observe simultaneous –0.131 enhancement of the third harmonic at both photonic 0.049 0.138 max band gap edges. The I ω (∆n ω) dependences are oscil- 0 3 3 –0.2 –0.1 0 0.1 0.2 latory in character. Their maxima that appear, for ∆ ∆ n3ω instance, at dispersion n3ω values equal to 0.051, − λω ∆ 0.114, and Ð0.045 are reached when coincides with Fig. 6. Dispersion n3ω dependences of maximum third- the minimum of the spectrum of the reflection coeffi- harmonic peak intensity at the right (thin line) and left cient for the fundamental wave Rω and one of the min- (thick line) edges of the photonic band gap. Arrows show the dispersion values that correspond to the characteristic ima of the R3ω spectrum is observed at the third har- third-harmonic intensity maxima. The region with positive ∆ monic wavelength (Figs. 5c, 5d). The farther from the n3ω values is shown in the inset on an enlarged scale. photonic band gap minimum of the R3ω spectra that coincides with the photonic band gap edge in the Rω spectrum corresponds to the weaker the third harmonic crystal (Fig. 3) and the amplitude of inhomogeneous resonance. It follows that a key factor of third-harmonic third harmonic waves at the boundary of each layer enhancement is the coincidence of the pumping wave- [Eq. (16)]. The multiple-reflection interference effects length with the photonic band gap edge and of the third- at the third-harmonic wavelength are determined by harmonic wavelength with the minimum of the reflec- comparing the amplitude and phase of the output tion coefficient R3ω. Ð() homogeneous third-harmonic waves E1 j generated Third-harmonic resonances are observed not only at separately by each jth layer [see Eq. (8)] and the ampli- the photonic band gap edge but also at the fundamental tude and phase of inhomogeneous third harmonic wavelengths at which the Rω spectrum has the second, waves at the boundary of each layer. third, etc., minima. These resonances are enhanced to a lesser degree than the peaks at the edge of the photonic The spatial distribution of the amplitude of the inho- band gap. This is clearly seen from Fig. 7, where the mogeneous third-harmonic wave inside the photonic whole set of all third harmonic frequency spectra crystal when λω corresponds to the edge of the photonic ∆ (s) obtained as n3ω changed from Ð0.21 to 0.19 in steps of band gap is shown in Fig. 8a. The shape of the E ( j) 0.002 in wavelength steps of 1 nm is shown. Peaks at dependence reproduces that of the distribution of the the edges of the photonic band gap are quite pro- fundamental wave amplitude. When the dispersion ∆ nounced. The enhancement at the short-wavelength n3ω is introduced into calculations, the shape of the edge is more intense, and the enhancement at the other E(s)( j) dependence does not change and enhancement Rω spectrum minima is much weaker. in the middle of the photonic crystal caused by funda- The resonance enhancement of third-harmonic gen- mental field localization is retained, while the ampli- ∆ eration at the edge of the photonic band gap is deter- tude of the inhomogeneous wave decreases as n3ω mined by multiple-reflection interference of both fun- increases. The phase distribution of the inhomogeneous damental and third-harmonic waves, because the stron- third-harmonic wave at the boundary of each layer is gest enhancement of the third harmonic is observed determined by the fundamental field phase at the same when the third-harmonic wavelength occurs at an R3ω boundary. The phase difference between third-har- spectrum minimum. To elucidate the role played by monic fields in neighboring layers with equal refractive interference effects at the fundamental wavelength we indexes is close to π. For this reason, each point of the must use (4) to calculate the spatial distribution of the polar diagram in which inhomogeneous third harmon- amplitude of the fundamental wave within the photonic ics are shown with their phases (Fig. 8b) has a corre-

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λ3ω, nm 250 275 300 325 350 375 10 1.0

8 0.8

6 0.6 s ω 3 R I 4 0.4

2 0.2

0 0 750 800 850 900 950 1000 1050 1100

λω, nm ∆ Fig. 7. Solid circles are the whole set of the third-harmonic intensity spectra obtained with n3ω varied from Ð0.21 to 0.19, and the solid line is the linear reﬂection coefﬁcient spectrum.

E(s) 10 120 60 (‡) (b)

6 180 0 4

240 300 0 10203040 j

Fig. 8. (a) Dependence of the amplitude of the inhomogeneous third-harmonic wave determined by (16) on the number of the photonic crystal layer and (b) amplitudes and phases of inhomogeneous third-harmonic waves. The calculations were performed for ∆ n3ω = 0.

max ∆ sponding point with a comparable amplitude and an ima of the I3ω ( n3ω) dependence (Figs. 9aÐ9d) and at almost opposite phase. The phase of inhomogeneous its minimum (Figs. 9e, 9f). The amplitudes of inhomo- ∆ ≠ waves does not change at n3ω 0. geneous third-harmonic waves at the minimum and max ∆ The dependences of the amplitudes and phases of maximum of the I3ω ( n3ω) dependence are of the Ð same order of magnitude (Fig. 8), and the amplitude of the third-harmonic partial waves E1 emerging from the photonic crystal on the layer number j are shown in emerging free third harmonic waves at the minimum of max ∆ Fig. 9. These dependences were calculated at the max- the I3ω ( n3ω) dependence is by a factor no less than 3

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(‡) (b) E– 1 120 60 280

240

200

160 180 0 120

40 240 300 0 120 60 160 (c) (d)

120

180 0 80

0 240 300 120 60 (e) (f)

40 180 0

240 300 0 10 20 30 40 j

Ð Fig. 9. (a, c, e) Dependences of the partial third-harmonic E1 amplitude on the layer number j calculated when the fundamental ∆ wavelength is tuned to the edge of the photonic band gap for dispersion n3ω values of (a) Ð0.114, (c) Ð0.131, and (e) Ð0.085 and max ∆ (b, d, f) partial third-harmonic waves with their phases. The insets show schematically the I3ω ( n3ω) dependence shown in Fig. 6. The points at which the calculations were made are marked by arrows. smaller than at its maximum. This implies additional crystal. This is accompanied by phase matching of par- amplitude enhancement of the emerging third harmonic tial third-harmonic ﬁelds outside the photonic crystal, by constructive multiple-reﬂection interference of Ð third-harmonic waves; namely, the partial third har- and the phases of the E1 waves become localized in a monic-wave generated in the jth nonlinear layer narrow angular interval. The weakest phase matching reaches an interference maximum outside the photonic of partial third-harmonic waves is observed at the min-

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I3ω the R3ω spectra closest to the photonic band gap coin- ∆ 10 cide; namely, the larger the n3ω value, the larger phase Rs changes of partial third-harmonic waves and the stron- 1.0 ger their destructive interference. As a result, the highest intensity of the total emerging third harmonic is 1 0.8 attained because, first, the amplitude of the emerging 0.6 partial third-harmonic waves increases and, second, their phases are matched (Figs. 9a, 9b). The smallest 0.4 max I ω value is observed when the photonic band gap in 0.1 3 0.2 the Rω spectrum corresponds to the edge of the photo- 0 nic band gap in the R3ω spectrum (Figs. 9e, 9f). The 800 900 1000 1100 Ð() λω, nm maximum of the E1 j dependence for the selected 0.01 photonic crystal parameters is then shifted by six layers toward the photonic crystal surface with respect to the maximum of the spatial distribution of the inhomogeneous third-harmonic wave amplitude. This is 250 275 300 325 350 375 explained by the third-harmonic wavelength falling within the photonic band gap in the R3ω spectrum. The λ3ω, nm contributions of deeper photonic crystal layers to the Fig. 10. The intensity spectrum of the third harmonic gen- emerging third harmonic then exponentially decrease. erated in a microcavity and calculated for the SS geometry. ∆ Shown in the inset are the reflection coefficient spectra of an The plots in Fig. 9 were constructed for n3ω < 0, s-polarized wave calculated in the pumping (solid line) and when the third harmonic resonance is observed at the third harmonic (dashed line) wavelength ranges. The R3ω short-wavelength edge of the photonic band gap. The spectrum is plotted on a triply enlarged wavelength scale. ∆ dependences for n3ω > 0 are similar.

I3ω 0.8 4. THIRD HARMONIC GENERATION Rs IN MICROCAVITIES 0.05 The dependence of the intensity of the third har- 0.04 monic generated in a microcavity on its wavelength is 0.6 0.03 shown in Fig. 10. The third-harmonic spectrum has a peak corresponding to the resonance between the fun- 0.02 damental radiation and the microcavity mode. Its inten- 0.01 sity is more than three orders of magnitude higher than 0.4 the intensity of the third harmonic in other spectral 0 0.05 0.10 0.15 regions. No enhancement is observed at the edge of the λω, nm ∆ ≠ photonic band gap. If n3ω 0, the resonance peak amplitude substantially changes and THG enhance- 0.2 ment at the edge of the photonic band gap arises. The max ∆ I3ω ( n3ω) dependences of the maxima of third-harmonic intensity peaks at the resonance between the fun- 0 damental radiation and the mode and between the fun- –0.2 –0.1 0 0.1 0.2 damental radiation and the short- and long-wavelength ∆ n3ω edges of the photonic band gap are shown in Fig. 11. If ∆ Fig. 11. Maximum intensity of third-harmonic peaks gener- n3ω < 0, the resonance amplitudes in the microcavity ated in the microcavity mode (thin line) and at the long- mode and at the left edge of the photonic band gap ∆ wavelength (thick line) and short-wavelength (dot-and-dash increase. If n3ω > 0, we observe enhancement in the line) edges of the photonic band gap as a function of disper- ∆ ∆ mode and at the right edge of the photonic band gap. sion n3ω; the dependences in the n3ω > 0 region are shown in the inset on an enlarged scale. THG enhancement in the microcavity mode is maximum at zero dispersion. If dispersion is nonzero, its amplitude decreases by approximately an order of mag- max ∆ ∆ nitude and oscillates as a function of n3ω. imum of the I3ω ( n3ω) dependence. The phase jump through 2π/3 in Fig. 9d explains the reason of a maxi- The localization of the fundamental ﬁeld resonant to mum third-harmonic enhancement when the photonic the mode results in a very substantial increase in the band gap edges in the Rω spectrum and the minima in amplitude of partial third-harmonic waves, which

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– – E1 E1 18 (‡)10 (b) 120 60 120 60

12 5 6

0 180 0 0 180 0

–6 –5 –12

240 300 240 300 –18 –10

Ð() max ∆ Fig. 12. Polar diagrams of complex partial third harmonic waves E1 j corresponding to (a) the maximum of the I3ω ( n3ω) ∆ max ∆ ∆ dependence, n3ω = 0 and (b) one of the minima of the I3ω ( n3ω) dependence, n3ω = Ð0.04. depend on the third power of the fundamental field the dispersion value, whereas the peaks at the short- amplitude. Irrespective of the phase difference between wavelength edge of the photonic band gap appear at a these waves, the minimum intensity of the emerging negative dispersion, and those at the long-wavelength third-harmonic wave is then no less than the maximum edge, when dispersion is positive. amplitude of the third-harmonic wave generated at the edge of the photonic band gap. The partial third-harmonic wave amplitudes in the microcavity mode 5. AN EXPERIMENTAL STUDY OF THIRD HARMONIC GENERATION remain almost invariant as ∆n ω is varied, whereas their 3 IN MICROCAVITIES phases change substantially. The EÐ()j phases at the 1 We experimentally studied THG in microcavities. max ∆ I3ω ( n3ω) peaks are nearly equal (see Fig. 12a). At the The sample was a one-dimensional microcavity made minimum reached at ∆n ω = Ð0.04 (Fig. 12b), the partial from mesoporous silicon by electrochemical etching of 3 a heavily doped p-type silicon plate with a (100) crys- Ð() waves E1 j are out of phase, and their interference is tallographic orientation in a solution of hydrofluoric destructive. acid following the procedure described in [11]. The sample was two one-dimensional photonic crystals λ It follows that, when the fundamental wavelength is consisting of five pairs of quarter-wave ( 0 = 1300 nm) tuned to the photonic band gap edges and the microcav- porous silicon layers separated by a half-wave cavity λ ity mode, the fundamental wave is localized in the layer. The 0 value is the spectral position of the micro- neighborhood of the cavity layer. The fundamental field cavity mode at the normal fundamental wave incidence. enhancement is stronger when the fundamental wave- H The refractive index and thickness were nω ≈ 1.93 and length is in resonance with the microcavity mode. As a ≈ L ≈ result, the amplitude of inhomogeneous partial third- dH 170 nm for optically denser layers and nω 1.61 harmonic waves increases. At certain dispersion ∆n ω ≈ 3 and dL 200 nm for less dense ones. The porous silicon values, the amplitudes of partial homogeneous third- microcavity layer has the refractive index nL = 1.61 and harmonic waves become maximum because of multi- thickness dres = 400 nm. The spectral dependences of ple-reflection interference and their phases are close to the third harmonic intensity were measured using a tun- each other. This results in a resonant increase in the able optical parametric generator. Its output linearly third-harmonic intensity. The third-harmonic intensity polarized radiation had the following characteristics: peak in the microcavity mode is less sensitive to phase pulse width of 4 ns, pulse energy of about 10 mJ at a matching of partial waves because of strong fundamen- 800 nm wavelength, and tuning range of 410Ð690 nm tal field amplitude enhancement. THG enhancement in for the signal wave and 735Ð2200 nm for the idle wave. the microcavity mode is observed almost irrespective of We used the idler tuning range because the photonic

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1.0 6. CONCLUSIONS 0.8 We developed a phenomenological model of optical THG in one-dimensional photonic crystals and micro- s 0.6 (‡) R cavities. The model is based on the formalism of trans- 0.4 fer matrices. It describes the generation and propagation of third-harmonic waves through a photonic crystal 0.2 by taking into account multiple reﬂections of fundamental and third-harmonic wave at interfaces and inter- 0 ference between the homogeneous and inhomogeneous 12 I3ω (b) third-harmonic waves. The spectra of third-harmonic intensity in photonic crystals and microcavities were 10 studied for the example of nonlinear optical media with 0.04 the limiting point symmetry groups ∞2, ∞m, and ∞/mm of layers constituting photonic crystals. The spectra 8 were studied in the spectral range of wavelengths and incidence angles of fundamental waves containing the

ω 6

3 photonic band gap and regions near its edges. At the I 0 resonance between the fundamental wave and the 800 900 1000 microcavity mode, the third-harmonic intensity 4 λω, nm increases by more than three orders of magnitude. When the fundamental wavelength coincides with pho- 2 tonic band gap edges, we observe resonance enhancement of third-harmonic intensity, which depends on the 0 magnitude and sign of dispersion. If the refractive 750 800 850 900 950 1000 1050 1100 1150 indexes of photonic crystal layers at the third-harmonic λ ω, nm wavelength n3ω are smaller than the refractive indexes at Fig. 13. (a) Spectrum of the reflection coefficient of the the fundamental wavelength, we observe resonant third- s-polarized pumping wave and (b) dependence of the inten- harmonic enhancement by a factor exceeding 100 at the sity of the s-polarized third harmonic wave on the wave- short-wavelength edge of the photonic band gap. If length of s-polarized pumping. The sample is a microcavity λ n3ω > nω, THG is enhanced at the long-wavelength edge made from porous silicon with 0 = 1300 nm. of the photonic band gap. The main mechanism of resonant THG enhance- band gap and the allowed mode of the microcavity were ment in the microcavity mode is fundamental field within it. Third-harmonic radiation reflected from the localization, which manifests itself by a four- to tenfold microcavity sample was separated by UV filters and increase in the third-harmonic amplitude. An additional detected by a photoelectron multiplier. The frequency enhancement factor is phase matching of partial third- spectra of third-harmonic intensity were measured for harmonic waves from each photonic crystal layer when fundamental and third-harmonic waves polarized in the plane of the sample (SS geometry) at a 60° angle of fun- they leave the microcavity. THG enhancement at the damental wave incidence. edges of the photonic band gap is caused to a greater extent by an increase in the amplitude of emerging par- The third-harmonic intensity spectrum is shown in tial third-harmonic waves and their phase matching as Fig. 13b, and the spectrum of the linear reflection coef- a result of multiple-reflection interference of third-har- ficient for the fundamental wave is shown in Fig. 13a. monic waves and, to a lesser extent, by fundamental λ The I3ω( ω) spectral dependence has a resonance at a wave localization within the photonic crystal. fundamental wavelength of λω ≈ 1075 nm, which coincides with the microcavity mode wavelength shifted to In agreement with the calculated third-harmonic λ shorter waves with respect to 0 at an oblique angle of spectra, the experimental spectrum of the intensity of incidence. We also observe resonant third-harmonic the third harmonic generated in a one-dimensional features when λω is tuned to the spectral regions of the microcavity fabricated from porous silicon shows that photonic band gap edge and outside the gap (inset to the intensity of the third harmonic increases approxi- Fig. 13). The third-harmonic intensity in these reso- mately 1000 times in the microcavity mode. Additional nance peaks is three orders of magnitude lower than in third-harmonic peaks are observed at the left edge of the microcavity mode. The positions of third-harmonic the photonic band gap and outside the gap. The spectral intensity resonances correlate with the minima of the positions of the resonances coincide with the minima of spectrum of the linear reflection coefficient for the fun- the linear fundamental wave reflection coefficient to damental wave. within several nanometers.

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ACKNOWLEDGMENTS 13. V. V. Konotop and V. Kuzmiak, J. Opt. Soc. Am. B 17, The authors are deeply indebted to O.A. Aktsipetrov 1874 (2000). for the statement of the problem and numerous useful 14. M. Centini, G. D’Aguanno, M. Scalora, et al., Phys. Rev. discussions. E 64, 46606 (2001). 15. T. V. Dolgova, A. I. Maœdykovskiœ, M. G. Martem’yanov, et al., Pis’ma Zh. Éksp. Teor. Fiz. 73, 8 (2001) [JETP REFERENCES Lett. 73, 6 (2001)]. 1. K. Sakoda, Optical Properties of Photonic Crystals 16. T. V. Dolgova, A. I. Maidykovski, M. G. Martemyanov, (Springer, Berlin, 2001). et al., Appl. Phys. Lett. 81, 2725 (2002). 2. W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 17. T. V. Dolgova, A. I. Maœdykovskiœ, M. G. Martem’yanov, (1987). et al., Pis’ma Zh. Éksp. Teor. Fiz. 75, 17 (2002) [JETP 3. M. Scalora, J. P. Dowling, C. M. Bowden, and Lett. 75, 15 (2002)]. M. J. Bloemer, Phys. Rev. Lett. 73, 1368 (1994). 18. G. D’Aguanno, M. Centini, M. Scalora, et al., J. Opt. 4. A. V. Andreev, A. V. Balakin, A. B. Kozlov, et al., J. Opt. Soc. Am. B 19, 2111 (2002). Soc. Am. B 19, 1865 (2002). 19. A. V. Andreev, A. V. Balakin, A. B. Kozlov, et al., J. Opt. 5. I. Freund, Phys. Rev. Lett. 21, 1404 (1968). Soc. Am. B 19, 2083 (2002). 6. N. Bloembergen and J. Sievers, Appl. Phys. Lett. 17, 483 20. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Per- (1970). gamon, New York, 1964; Nauka, Moscow, 1970). 7. J. P. van der Ziel and M. Ilegems, Appl. Phys. Lett. 28, 21. N. Hashizume, M. Ohashi, T. Kondo, and R. Ito, J. Opt. 437 (1976). Soc. Am. B 12, 1894 (1995). 8. J. Martorell, R. Vilaseca, and R. Corbalan, Appl. Phys. 22. J. E. Sipe, J. Opt. Soc. Am. B 4, 481 (1987). Lett. 70, 702 (1997). 23. D. S. Bethune, J. Opt. Soc. Am. B 6, 910 (1989). 9. A. V. Balakin, V. A. Bushuev, N. I. Koroteev, et al., Opt. Lett. 24, 793 (1999). 24. N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962). 10. Y. Dumeige, P. Vidakovic, S. Sauvage, et al., Appl. Phys. Lett. 78, 3021 (2001). 25. Yu. I. Sirotin and M. P. Shaskolskaya, Fundamentals of Crystal Physics, 2nd ed. (Nauka, Moscow, 1979; Mir, 11. T. V. Dolgova, A. I. Maidykovski, M. G. Martemyanov, Moscow, 1982). et al., J. Opt. Soc. Am. B 19, 2129 (2002). 12. V. V. Konotop and V. Kuzmiak, J. Opt. Soc. Am. B 16, 1370 (1999). Translated by V. Sipachev

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 478Ð482. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 543Ð547. Original Russian Text Copyright © 2004 by Ryazanov. ATOMS, SPECTRA, RADIATION

Angular Distribution of Transient Radiation from an Ultrarelativistic Particle in a Magnetic Field M. I. Ryazanov Moscow Institute of Engineering Physics (Technological University), Kashirskoe sh. 31, Moscow, 115409 Russia e-mail: [email protected] Received September 5, 2003

Abstract—It is shown that a magnetic field acting on an ultrarelativistic charged particle escaping from a conductor changes the intensity of transient radiation. The angular and frequency distribution of transient radiation in the magnetic field is determined. The possibility of determining the energy of the ultrarelativistic particle from the change in the azimuthal asymmetry of transient radiation emitted by this particle in the magnetic field is discussed. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION In an external electric field, for a particle escaping from a conductor along the normal to the interface, this It is well known that the energy and momentum con- effect leads to the emergence of azimuthal asymmetry servation laws hold for the emission of a photon by a free in the distribution of transient radiation, which strongly ultrarelativistic particle for a small longitudinal (directed depends on the Lorentz factor γ [8]. The action of a along the particle velocity) momentum transfer, magnetic field on transient radiation differs from the action of an electric field; it would hence be interesting 2 2 to estimate the effect of a magnetic field on transient hωmc ∆p ∼ ------, radiation emitted by an ultrarelativistic particle. c E so that the wave process of radiation formation takes 2. TRANSIENT RADIATION EMITTED place over a particle path of length BY A NONUNIFORMLY MOVING CHARGE Let a particle bearing a charge e and having a veloc- h c E 2 ity v ≈ c escape at instant t = 0 from a conductor (z < 0) ------= ------. ∆p ω2 into vacuum (z > 0), where it is acted upon by a constant mc uniform magnetic field H parallel to the conductor surface. We choose the direction of the x axis along the For ultrarelativistic particles, the length of radiation magnetic field and consider transient radiation from the formation (coherence length) may be of a macroscopic particle escaping along the z axis perpendicular to the size. On such a long path, the competing processes may conductor surface. The law of motion of the particle can significantly change the motion of a particle in the region be written in the form of radiation formation, thus lowering the radiation intensity. An example of this effect is the influence of multiple r()t = R()t + s()t , scattering on bremsstrahlung [1Ð3] or the effect of polarization of the medium on bremsstrahlung [4]. where R(t) and s(t) are the normal and tangential com- In the case of an ultrarelativistic particle and the ponents of the radius vector of the particle relative to conductor-vacuum interface, transient radiation emerg- the surface, respectively. It is well known that, for such ing when the charged particle crosses the interface a motion of the charge, the field outside the conductor between the two media [5Ð7] is also formed in a spatial coincides with the field of two charges escaping from region having a length of (c/ω)(E/mc2)2 and a much the same point r = 0 at instant t = 0, viz., charge e mov- smaller transverse size. In view of the large coherence ing according to the law r(t) and an image charge Ðe length, the action of an external field on the particle moving according to the law may change the type of motion of the particle in the course of radiation formation. This affects the process r()t = Ð R()t + s()t of radiation formation since the angular distribution of radiation changes and its intensity decreases. (see [9]).

ANGULAR DISTRIBUTION OF TRANSIENT RADIATION 479 Ӷ For such motion of the two charges, the angular and particle. This means that the inequality uy v0 is satis- frequency distribution of the radiant energy has the ﬁed; i.e., Ωt Ӷ 1. In this case, the law of motion of the form (n = r/r, v(t) = dR/dt, u(t) = ds/dt) charge can be represented in the form

∞ 2 () ()≈≈Ω () d E 2ω2 ux t = 0, uy t v 0 t, v t v 0, ------= e dt[]nv{}()t + u()t (5) ω Ω ------∫ 2 d d π2 3 xt()==0, yt() ()v Ω t /2, zt() =v t. 4 c 0 0 0 × exp{}iωtiÐ kR⋅ ()t Ð iks⋅ ()t ∞ The law of motion of the image charge in this approxi- (1) mation has the form + ∫dt[]nv{}()t Ð u()t 0 () ()≈ Ω ()≈ ux t = 0, uy t v 0 t, v t Ðv 0, 2 (6) () () ()v Ω 2 () v ∫ × exp{}i(ω + ikR⋅ ()t Ð iks⋅ ()t . xt ==0, yt 0 t /2, zt =Ð 0t. ӷ Considering that v0 u, we can disregard quantity u in The action of the field may be significant when the the preexponential factor in expression (2); in this case, terms independent of the field are partly cancelled out. the angular and frequency distribution of the energy of ≡ For example, in the ultrarelativistic case, the exponent transient radiation can be written in the form (Q Ω of one of the exponential functions is on the order of kyv0 /2) ω Ð k á v and is smaller than the exponent of the other exponential, which is on the order of ω + k á v. The inte- 2 2 2 d E e ω 2 gral with the rapidly oscillating exponential is small ------= ------[]nv⋅ ω Ω 2 3 0 and can be omitted. The effect of the external field can d d 4π c be disregarded in the first approximation in all cases ∞ 2 (7) when the main terms are not mutually cancelled out. × dtiexp{}()ωti+ Ð k ⋅ v tiQtÐ 2 . This enables us to transform relation (1) to ∫ 0 0 ∞ 2 2 2 d E e ω []{}() () The integral contained in this formula cannot be ------ω Ω = ------∫dt nvt + u t d d 4π2c3 reduced to elementary functions and can be expressed 0 (2) in terms of the Fresnel integrals 2 ∫ × exp{}i(ω Ð ikR⋅ ()t Ð iks⋅ ()t . x Sx()= ()2/π 1/2∫sin2ttd , 0 (8) x 3. TRANSIENT RADIATION 1/2 2 IN A MAGNETIC FIELD Cx()= ()2/π ∫cos ttd . With the above choice of the coordinate axes, the 0 velocity components of an ultrarelativistic particle in a Ω ϑ ϕ magnetic field have the form (Ω ≡ eH/mcγ) For < 0, i.e., for ky = ksin sin < 0, integration gives

() () Ω () Ω ∞ ux t = 0, uy t = v 0 sint, v t = v 0 cos t. (3) ∫ditexp{}()ω Ð kv⋅ tiQt+ 2 It follows hence that the law of motion of the particle 0 can be written in the form π 1/2 i()ω Ð kv⋅ 2 () () ()Ω ()Ω = ------expÐ------(9) xt ==0, yt v 0/ 1 Ð cos t , 2 Q 4 Q (4)  () ()ΩΩ zt = v 0/ sin t. ω ⋅ ω ⋅ × 1 Ð kv 1 Ð kv --- Ð C------+ i --- Ð S------. We conﬁne our analysis to the case when the change 2 2 Q 1/2 2 2 Q 1/2 in the particle velocity due to the action of the magnetic ﬁeld in the region of radiation formation considered here is small as compared to the initial velocity of the The angular and frequency distribution of transient

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 480 RYAZANOV radiation for sinϑsinϕ < 0 has the form can be written as

d2E e2ω2 []nv× 2 d2E e2ω2[]nv⋅ 2 ------= ------= ------------dωdΩ πc3 Q dωdΩ π2c3 Q (10) (17) 2 2 1 ω Ð k ⋅ v 1 ω Ð k ⋅ v 2ω Ð kv⋅ 2ω Ð kv⋅ × ------× --- Ð C1/2 + --- Ð S1/2 . g ------+ f ------. 2 2 Q 2 2 Q 2 Q 1/2 2 Q 1/2

Ω ϑ ϕ ≤ ∞ For > 0, i.e., for ky = ksin sin > 0, integration In the region 0 x < , auxiliary functions f(x) and g(x) yields can be approximated with an error smaller than 2 × 10−3 by the expressions [10] ∞ {}()ω ⋅ 2 ) 1+ 0.926x ∫ditexp Ð kvtiQrÐ fx()≈ ------, 2++ 1.792x 3.104x2 0 (18) 1 π 1/2 i()ω Ð kv⋅ 2 gx()≈ ------. = ------exp------(11) 2++ 4.142x 3.492x2 +6.670x3 2 Q 4 Q More accurate approximations can be found in [11]. ω ⋅ ω ⋅ × 1 Ð kv 1 Ð kv --- + C------Ð i --- + S------. 2 2 Q 1/2 2 2 Q 1/2 4. EFFECT OF A MAGNETIC FIELD ON TRANSIENT RADIATION In contrast to expression (10), the angular and frequency distribution of transient radiation for As the magnetic field tends to zero, the argument ω | |1/2 sinϑsinϕ > 0 assumes the form ( Ð k á v)/2 Q of the Fresnel integrals in expressions (10) and (12) tends to infinity. Consequently, in 2 2 2 the range of frequencies and angles for which d E e ω 2 ------= ------()[]nv× / Q ω Ω 3 1/2 d d πc 1/2 k v eH ()ω Ð kv⋅ ӷ Q ∼ ------y 0 - , (12) mcγ ω ⋅ 2 ω ⋅ 2 × 1 Ð k v 1 Ð k v --- + C------Ð --- + S------. 2 2 Q 1/2 2 2 Q 1/2 the magnetic field practically does not affect transient radiation. The angular and frequency distribution of transient radiation is deformed by the magnetic field For large x, Fresnel integrals C(x) and S(x) oscillate when the opposite inequality holds, in the vicinity of 1/2 with an amplitude decreasing slowly upon an increase in x: ωϑ ϕ 1/2 v 0eH sin ------2 - () ()π 1/2() 2 mc γ Sx = 1/2Ð 1/2 1/x cosx , (19) (13) 2 1/2 2 2 21 Cx()= 1/2+ () 1/2π ()1/x sinx . ӷ ()ω Ð kv⋅ ∼ ω ϑ + ----- . γ 2 For small x, the Fresnel integrals increase rapidly from zero for x = 0 to values on the order of unity for x ~ 1. For characteristic angles of ϑ ~ 1/γ of the emission of In the vicinity of zero, we have radiation, this inequality assumes the form ω Sx()==()2/π 1/2()x3/3 , Cx() ()2/π 1/2x. (14) eHsinϕ ӷ ------4. (20) mc2 γ We introduce auxiliary functions f(x) and g(x), defined by the equations When inequalities (19) and (20) are satisfied, the argument of the Fresnel integrals is small; consequently, we 1/2 ÐSx() = gx()sin() πx2/2 + fx()cos() πx2/2 , (15) can use approximate expressions (14) for the Fresnel integrals. In region (20), we have 2 2 1/2 ÐCx() = gx()cos() πx /2 Ð fx()sin() πx /2 . (16) ω ⋅ ω ⋅ ω ⋅ Ð kv≈ Ð kv ӷ Ð kv C------1/2 ------1/2 S------1/2 , (21) For sinϑsinϕ < 0, the radiant energy distribution (10) 2 Q πQ 2 Q

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ANGULAR DISTRIBUTION OF TRANSIENT RADIATION 481 so that distribution (10) of transient radiation for magnetic field strength. The ratio of the intensity of sinϕ < 0 can be written in the form radiation emitted at angles ϕ = 0 and ϕ = π/2 is obviously close to unity if the argument of functions C(x) d2E e3ω2 []nv× 21 ω Ð kv⋅ and S(x) is much greater than unity. Considerable devi------= --------- Ð ------. (22) ω Ω 3 1/2 ations of this ratio from unity corresponds to small val- d d πc Q 2 πQ ues of quantity (24), i.e., to the region in which the following inequality holds: In the range of angles where sinϕ > 0, distribution (12) of transient radiation has the form ()ϑω/2 ()2 + γ Ð2 Ӷ []()ϑϕΩω/2 sin 1/2≈ (26) 1/2 2 3 2 2 []()ωϑ/γ ()ϕeH/mc sin . d E e ω []nv× 1 ω Ð kv⋅ ------= --------- + ------. (23) dωdΩ πc3 Q 2 πQ 1/2 For characteristic angles ϑ ~ 1/γ and ϕ ~ π/2, this inequality assumes the form It should be emphasized that, for small values of angle eH ω ϕ, the conditions for applicability of expressions (22) ------ӷ -----. (27) and (23) are violated; consequently, a direct transition mc γ 3 from relation (22) to (23) is ruled out. The intensity ratio for radiation emitted at angles of ϕ = 0 and π/2 can be obtained in the form 5. AZIMUTHAL ASYMMETRY OF RADIATION DISTRIBUTION 2 3 2 2 d E()ωϑϕ,, = π/2 πω mc{}1 + ()γϑ ------= ------. (28) If a charged particle escapes from a conductor at 2 ()ωϑϕ,, 8ceH ϑγ3 right angles to its surface, the angular distribution of d E = 0 radiation in zero magnetic field exhibits azimuthal sym- For very small angles ϑ, inequality (27) is violated, and metry. The action of a magnetic field breaks the axial relation (28) becomes inapplicable. It follows from the symmetry of the angular distribution. Defining the above arguments that azimuthal asymmetry in the direction of emission of radiation by angles ϑ and ϕ in angular distribution of transient radiation strongly a spherical system of coordinates with the axis directed depends on the particle energy. From the presence or along the initial velocity of the particle and considering absence of azimuthal symmetry, one can judge whether that the radiation emitted by the ultrarelativistic particle or not the field strength is high enough for affecting is concentrated in the region of small angles ϑ, we can transient radiation. represent the argument of Fresnel integrals C(x) and S(x) in expressions (10) and (12) in the form 6. DISCUSSION ω Ð kv⋅ ()ϑω/2 ()2 + γ Ð2 ------= ------. (24) It follows hence that an external magnetic field 2 Q 1/2 []()ϑϕΩω/2 sin 1/2 affects the process of formation of transient radiation from an ultrarelativistic particle only for a large length The limiting case in which the value given by for- of radiation formation and, hence, strongly depends on mula (24) tends to infinity corresponds to transient radi- the radiation frequency and the Lorentz factor of the ation in zero magnetic field. Then the angular and fre- particle. For example, for millimeter wavelengths and quency distribution (2) is transformed into the conven- for a Lorentz factor of 103, the coherence length is on tional distribution for transient radiation, the order of several meters over such a length. The magnetic field must noticeably change the motion of a par- d2E e2 []nv× 2 ticle for the effect in question to be manifested. In this ------= ------. (25) dωdΩ π2c{}()⋅ 2 2 case, for a particle escaping along the normal to the 1 Ð nv/c conductorÐvacuum interface in an external constant It can be seen from formula (24) that this limiting and uniform magnetic field, azimuthal asymmetry in case emerges for ϕ = 0 as well as for ϑ 0. However, transient radiation is observed if inequality (26) is sat- the region of ϑ Ӷ 1/γ makes a small contribution to the isfied. radiation intensity and can be disregarded. Thus, radia- The magnetic field strength for which noticeable tion emitted in the plane passing through the particle asymmetry takes place is determined by inequality (27) velocity and the magnetic field (i.e., for ϕ = 0) does not and strongly depends on the Lorentz factor of the parti- depend on the magnetic field strength on the whole. cle and the radiation frequency. However, the intensity of radiation propagating in the Azimuthal asymmetry of transient radiation in a plane perpendicular to the magnetic field and passing magnetic field can be measured quite easily, while its through the initial velocity of the particle (i.e., for ϕ = strong dependence on the Lorentz factor ensures suffi- π/2) can noticeably decrease depending on the external ciently high accuracy in energy measurements. Hence,

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 482 RYAZANOV it can be concluded that the measurement of azimuthal REFERENCES asymmetry in the angular distribution of transient radiation in an external field may become a convenient 1. L. D. Landau and I. Ya. Pomeranchuk, Dokl. Akad. Nauk method for measuring the energy of ultrarelativistic SSSR 92, 535 (1953); Dokl. Akad. Nauk SSSR 92, 735 (1953). particles. It is important to emphasize that we have considered 2. A. B. Migdal, Phys. Rev. 103, 1811 (1956). only the transient radiation formed near the surface of a 3. A. B. Migdal, Zh. Éksp. Teor. Fiz. 32, 633 (1957) [Sov. conductor over a particle path with a length on the order Phys. JETP 5, 527 (1957)]. of the coherence length. The action of the field on a par- 4. M. L. Ter-Mikaelyan, Dokl. Akad. Nauk SSSR 94, 1033 ticle also leads to emission of radiation on the subse- (1954). quent path of the particle; however, this radiation is not connected any longer with the intersection of the inter- 5. V. L. Ginzburg and I. M. Frank, Zh. Éksp. Teor. Fiz. 16, face by the particle and is conventional radiation emit- 15 (1946). ted by a particle moving in a magnetic field. The prop- 6. V. E. Pafomov, Zh. Éksp. Teor. Fiz. 33, 1074 (1957) erties of such radiation are determined by the specific [Sov. Phys. JETP 6, 829 (1957)]. nature of the subsequent motion of the particle in the field and its contribution to the total radiation can be 7. G. M. Garibyan, Zh. Éksp. Teor. Fiz. 33, 1403 (1957) different. In comparison with experiment, the contribu- [Sov. Phys. JETP 6, 1079 (1957)]. tion from such radiation should be taken into account, 8. M. I. Ryazanov, Zh. Éksp. Teor. Fiz. 122, 999 (2002) but it is inexpedient to include it in the general analysis. [JETP 95, 861 (2002)]. This is due to the fact that, first, such radiation depends 9. V. E. Pafomov, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 10, on the velocity of the particle and not on its energy; sec- 240 (1967). ond, the distribution of this radiation strongly depends on the condition of subsequent motion of the particle in 10. Handbook of Mathematical Functions, Ed. by M. Abra- the experimental setup. mowitz and I. A. Stegun, 2nd ed. (Dover, New York, 1971; Nauka, Moscow, 1979). 11. J. Boersma, Math. Comput. 14, 380 (1960). ACKNOWLEDGMENTS The author takes a pleasant opportunity to thank B.A. Dolgoshein for interesting remarks. Translated by N. Wadhwa

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 483Ð488. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 548Ð555. Original Russian Text Copyright © 2004 by Mikheev, Zonov, Obraztsov, Volkov. ATOMS, SPECTRA, RADIATION

Anisotropic Laser-Induced Evaporation of Graphite Films G. M. Mikheeva, R. G. Zonova, A. N. Obraztsovb, and A. P. Volkovb aInstitute of Applied Mechanics, Ural Division, Russian Academy of Sciences, Izhevsk, 426000 Russia bMoscow State University, Moscow, 119992 Russia e-mail: [email protected] Received April 21, 2003

Abstract—An experimental investigation of the effect of linearly polarized high-energy pulsed laser light, normally incident on a carbon thin film, is reported. The material under study consists of platelike graphite crystallites with basal crystallographic planes mostly oriented perpendicular to the substrate surface. An increase is revealed in the fraction of the graphite crystallites oriented perpendicular to the polarization plane. Laser light is found to cause significant anisotropy in diffuse scattering by the film surface. Experimental observations are explained by a model of anisotropic evaporation of graphite-like carbon material due to polarization dependence of the absorption and reflection coefficients for a rough surface. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION strates. A single-mode YAG:Nd3+ laser was used as a Currently, much attention is given to analysis of var- high-power light source. The laser cavity was designed ious allotropic forms of carbon having unique physical to produce linearly polarized light at 1064 nm with properties, such as diamond, graphite, carbon nano- pulse energy of up to 50 mJ [14]. The half-width of tubes, and fullerenes. In particular, it was shown in our Gaussian laser pulses was about 22 ns, and the beam recent publications that some graphite-like films are diameter was 2 mm. The laser beam was directed characterized by extremely low threshold electric field through a converging lens with a focal length of 10 cm strengths for electron tunneling from their surface to onto the film surface under study. The irradiance on the vacuum [1]. The field-emission properties of these carbon-film surface was varied gradually by varying the films are analogous to those of carbon nanotubes (e.g., lens-to-specimen distance between 12 and 35 cm. The see [2Ð4]). One distinctive feature of graphite-like specimens were irradiated by one or several laser light materials with low field-emission thresholds is that the pulses in air. basal crystallographic planes of the constituent well- The film structure and morphology before and after ordered platelike graphite crystallites are mostly ori- irradiation was analyzed by means of a Neophot 32 ented perpendicular to the film surface [1, 2]. Structural optical microscope with a resolution of at least 0.4 µm, properties of these graphite films can manifest them- a Solver P47 atomic-force microscope (AFM), and a selves in other phenomena as well. In this paper, we LEO 1550 scanning electron microscope (SEM). The present the results of a study of the effect of high- angular dependence of the efficiency of diffuse scatter- energy laser light on the morphology and optical prop- ing by the examined carbon films was analyzed by erties of graphite films. means of an apparatus based on an LEF-3M ellipsome- It should be noted that one well-known effect of ter (see Fig. 1), with a HeÐNe laser used as a 1 mW cir- pulsed laser light on solids is the formation of diverse cularly polarized light source. Diffuse scattered light b periodic surface structures (e.g., see [5Ð12]). Normally, was observed at a constant angle ϕ in the plane yz per- a laser-induced periodic surface structure is character- pendicular to the plane xz of incidence of beam a. The ized by a surface height varying with a period d ~ λ, angle of incidence of beam a in the xz plane was equal where λ is the wavelength of the laser light [6, 7]. to the angle of reflection of beam b in the yz plane. We Moreover, large-scale roughness (with a length scale studied the intensity IR of reflected light on the speci- varying from 10 to 300 µm) may develop on the surface men’s orientation characterized by the angle α of rota- of an opaque material in the zone irradiated by pulsed tion in the xy plane at various constant angles ϕ. In the laser light with λ ≤ 1 µm [8, 10, 12]. starting position of a specimen, the sides of the silicon substrate were aligned with the x and y axes. 2. EXPERIMENTAL We studied specimens of carbon films obtained by 3. RESULTS means of our standard technique of plasma deposition In its original surface morphology, a graphite-like from a methaneÐhydrogen mixture (e.g., see [1, 13]). film consists of platelike graphite crystallites several The films were deposited on 25 × 25 mm2 silicon sub- tens of nanometers thick, with other dimensions vary-

484 MIKHEEV et al.

z ing from one to several microns (see Fig. 2a). The predominant orientation of the platelike crystallite planes is parallel to the basal crystallographic plane of graphite (0001) and perpendicular to the substrate. There is ϕ ϕ α no particular orientation in other directions. This orien- 1 tation of the crystallites explains the changes in diffuse 3 scattering efficiency that can be observed with the 4 naked eye as the angle of incidence or observation is var- a ied. Note that the diffuse reflection efficiency is invariant b with respect to rotation of the specimen about an axis perpendicular to its surface. These observations are consistent with previously reported results [1, 2, 13, 15]. 2 x y Irradiation with high-energy laser light substantially changes the film-surface morphology, as illustrated by Fig. 1. Optical arrangement for studying diffuse light scat- images obtained by means of SEM (see Fig. 2b) and tering by carbon film surface: (1) HeÐNe laser; (2) beam AFM (Fig. 3). After irradiation with five to ten pulses chopper; (3) carbon film on silicon substrate; (4) photomul- (depending on the beam irradiance, which varied tiplier; xyz is a Cartesian coordinate system. between 10 and 30 MW/cm2), the surface had numerous elements stretched along a certain direction. Appar- ently, these were platelike graphite crystallites that had existed on the film surface and were not strongly affected by laser light. Comparative electron microscopy of the original and laser-irradiated carbon surfaces (see Figs. 2a and 2b, respectively) leads to the following conclusions. I. Irradiation by laser light results in a partial disor- dering of the graphite-like material under study, which manifests itself in changes in its surface morphology and in the efficiency of secondary electron emission. The latter effect is inferred from the observed change in SEM image contrast. Additional evidence of the disor- dering has been obtained in Raman-scattering studies. II. There is no indication of any periodic structure on the surface after repeated irradiation by laser pulses [5Ð7, 11]. The absence of periodic surface structure was (a) also noted in [7], where the effect of irradiation with 10 µm E 40 ns XeCl excimer laser pulses on rough diamond film surfaces was investigated. III. An increase is revealed in the fraction of the graphite crystallites perpendicular to the polarization plane. This conclusion about predominant orientation of crystallites on the irradiated surface segment is corroborated by atomic-force microscopy of surface morphology (Fig. 3). This last effect is most clearly manifested in the AFM image of the film surface shown in Fig. 3. More- over, the figure demonstrates that the width of an individual crystallite at a height of 0.6 µm can be as large as 2 µm in the AÐA cross section (see Fig. 3b), whereas its width in the perpendicular BÐB cross section is 0.6 µm (Fig. 3c). Analogous characteristics are exhibited by other morphological elements of the film surface irradiated by laser light. All of them are parallel to (b) the AÐA cross section. Note that the surface irradiated by five to ten pulses of laser light changes color from Fig. 2. Electron-microscopic images of carbon surface the original metallic gray to velvet black. Further (a) before and (b) after irradiation by laser light. Arrows increase in the number N of irradiating pulses leads to indicate the direction of the laser electric field vector E. decrease in the height of surface elements. When N is

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ANISOTROPIC LASER-INDUCED EVAPORATION OF GRAPHITE FILMS 485 sufficiently large, the film-surface color changes from µm black back to gray and no morphological elements with Ä predominant orientation are observed in AFM images. 1.0 An analysis of experimental results shows that the orientation of morphological elements in the SEM and Ç Ç AFM images of the graphite film surface is perpendic- 0.8 ular to the polarization plane of the incident laser light. An angular displacement of the irradiated specimen Ä about the axis of the laser beam (normal to the film 0.6 plane) results in a corresponding rotation of the predominant orientation. When the polarization plane of laser light is changed by means of two quarter-wave 0.4 plates without rotating the specimen, the structures created by laser irradiation on the carbon film surface change correspondingly. 0.2 The predominant orientation of crystallites along (a) 1 µm the direction normal to the surface is also confirmed by the aforementioned visual observations of diffuse scat- 0 tering anisotropy with respect to the angle of incidence Height, µm in the plane perpendicular to the film surface [2]. More- A–A (b) over, we found that the efficiency of diffuse scattering 1.0 by an irradiated surface depends on the angle of rotation of the specimen about an axis perpendicular to the 0.8 substrate. This effect was analyzed in detail by measur- 0.6 ing the intensity of diffuse scattered light at various angles of rotation of the specimen. 0.4

Figure 4 shows the intensity IR of diffuse scattered 0.2 HeÐNe laser light measured as a function of the angle of specimen rotation α. Whereas no orientational 0123 α a, µm dependence of IR on is observed (see Fig. 4a), the α Height, µm dependence IR( ) obtained after irradiation by high- B–B (c) energy laser light with polarization characterized by an γ 1.0 angle 1 exhibits two pronounced peaks over the interval of complete revolution about the axis (Fig. 4b). For an angle of rotation α measured relative to an arbitrary 0.8 α α ≈ ° direction, the peaks in IR( ) correspond to 11 83 and α ≈ ° α α ≈ ° 12 264 ; that is, 12 Ð 11 180 . By scrutinizing the α curve IR( ) obtained for the film irradiated by laser 0.6 γ γ light with a polarization angle 2 differing from 1 by ∆γ α , it was found that the corresponding peaks in IR( ) 0 0.4 0.8 were observed at angles differing by ∆γ. As an example, , µm α ∆γ b Fig. 4c shows the dependence IR( ) measured for = ° α ≈ ° α ≈ ° Fig. 3. AFM image of carbon film after laser irradiation (a) 45 , in which case 21 129 and 22 309 . The fol- and the mutually perpendicular cross-sectional profiles of a lowing relations were found to hold within the mea- structural element on the film surface in planes AÐA (b) and α α ≈ α α ≈ ∆γ surement error: 21 Ð 11 22 Ð 12 . BÐB (c).

4. DISCUSSION initial capillary-wave pattern formation, and other effects [12]. As noted above, our experiments give no According to [6], the formation of periodic surface indication of pattern formation on the irradiated carbon patterns is caused by diffraction of a spatially coherent film surface. However, we did observe stretched ele- incident light wave by a rough surface and interference ments mostly oriented perpendicular to the polarization of diffracted waves with the wave that penetrates the plane of laser light. It should be noted that these parallel medium and creates a periodic temperature field on the elements are chaotically distributed over the film sur- surface. Large-scale periodic pattern formation is asso- face, but their presence is clearly manifested in the ciated with instability development at the interface angular dependence of diffuse scattering. This can be between a melt and an optical-breakdown plasma, melt explained by anisotropic evaporation of platelike displacement caused by the vaporizing target material, graphite crystallites from the original film surface.

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(a) (a) ψ 40 β β

, arb. units 20 R I θ

0 (b) (b) 60

40 , arb. units R I

Fig. 5. Schematic illustration of interaction between a laser beam normally incident on a surface and a hemispherical element on the surface: (a) diametrical cross-sectional view 0 of the hemisphere; (b) top view of the hemispherical element after irradiation by a laser beam polarized in the (c) plane ψ. 60 polarization plane ψ passing through an arbitrary diameter of a hemisphere, the angle of incidence β is determined by the azimuthal angle θ: β = |90° Ð θ| (see 40 Fig. 5). The reflection efficiency and absorption coefficient

, arb. units for absorbing media depend on the angle of incidence R I and polarization of the beam. To evaluate the reflection coefficients R and R , which correspond, respectively, 20 p s to polarization in the plane of incidence (p-polarization) and in the plane perpendicular to the plane of incidence (s-polarization), we can use the following exact formulas for isotropic absorbing media [16, 17]: 0 100° 200° 300° α a2 + b2 Ð 2acosββ+ cos2 R = ------, Fig. 4. Intensity IR of scattered HeÐNe laser light versus the s 2 2 2 angle of rotation α of a specimen, measured at ϕ = 75° a ++b 2acosββ+ cos (a) before laser irradiation and after irradiation by pulsed laser light polarized at angles (b) γ and (c) γ differing 1 2 2 2 2 2 by 45°. a + b Ð 2asinββtan + sin βtan β R p = Rs------, a2 ++b2 2asinββtan + sin2 βtan2 β To elucidate this phenomenon, let us consider the effect of laser light on a rough film surface consisting 2 2 2 2 R a + b Ð 2asinββtan + sin βtan β of hemispherical elements. Suppose that a laser beam is ------p = ------, 2 2 2 2 normally incident on the film surface. For the laser Rs a ++b 2asinββtan + sin βtan β

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ANISOTROPIC LASER-INDUCED EVAPORATION OF GRAPHITE FILMS 487 where 0.8 (a) 2 1 2 2 2 2 2 2 1/2 a = ---[()()n Ð χ Ð sin β + 4n χ 0.6 2

1/2 0.4 2 2 2 , arb. units --- ()]χ β  s 1 + n Ð Ð sin , A ,

 p

A 0.2

1 2 2 2 2 2 2 1/2 b = ---[()()n Ð χ Ð sin β + 4n χ 2 0 20° 40°60° 80° 100°  β 1/2 8  (b) ---Ð ()]n2 Ð χ2 Ð sin2 β  ,  6 with n and χ denoting the real and imaginary parts of

, arb. units 4 the complex refractive index of the absorbing medium. s A

Borrowing the real and imaginary parts of the permit- / 2 p tivity of graphite at 1 µm from [18], we obtain n = 1.3 A and χ2 = 4.9 for graphite at the neodymium laser wave- 2 length. Then, we use the formulas written out above to plot the absorption coefficients Ap = 1 Ð Rp and As = 1 Ð Rs (see Fig. 6a). Figure 6b shows the dependence of Ap/As 0 20° 40° 60° 80° 100° on β plotted in a similar manner. According to Fig. 6, β β β ° Ap = As = 0.48 for = 0 and Ap = As = 0 for = 90 . As β ° β varies from 0 to 90 , the absorption coefficient As Fig. 6. Calculated dependences on the angle of incidence β for graphite at a wavelength of 1 µm: (a) absorption coeffi- monotonically decreases. The dependence of Ap on β ° cients As (curve 1) and Ap (curve 2); (b) Ap/As. exhibits different behavior. At max = 66 , the absorption coefficient Ap reaches its maximum value 0.67, whereas As = 0.23, i.e., Ap/As = 2.9. At higher values Thus, we can explain how irradiation by high- β of , the ratio Ap/As monotonically increases. energy laser light can cause a certain preferred direction to appear on an initially isotropic rough film sur- Note that laser-induced fracture and evaporation of face. The morphology of a real carbon film surface is the material under study are characterized by threshold inhomogeneous and is substantially different from the conditions depending on the absorption coefficient. model considered above. However, the anisotropic Accordingly, evaporation (or fracture) of a graphite- evaporation mechanism outlined above can obviously like material induced by laser light with a certain irra- manifest itself in a similar manner in this case as well. θ ≤ β diance at points on a spherical surface where max θ ≥ π β In experimental studies of film structure by means and Ð max (see Fig. 5a) is much more efficient in the case of p-polarization as compared to s-polariza- of a HeÐNe laser, the angle between the incident beam and the direction of observation of the scattered light tion. Even though the shapes of graphite crystallites in ° the examined films substantially differ from a hemi- was 90 (see Fig. 1). Therefore, the dependence pre- sphere, the analysis developed above explains the sented in Fig. 4a can be explained by diffuse scattering mechanism of their selective ablation with respect to by a fractal rough surface [19]. It is clear that the pat- the laser polarization direction. In the case of linearly terns developing on an irradiated surface substantially polarized laser light, both heating and subsequent frac- change the diffuse scattering function, as illustrated by ture of a hemispherical element are anisotropic. On the the experimental results shown in Figs. 4b and 4c. lateral parts of such elements, spherical half-segments of the material oriented perpendicular to the polariza- 5. CONCLUSIONS tion plane evaporate in the first place. As a result, the original hemispherical shape of the element change, It is demonstrated that irradiation of graphite-like and its top view becomes a curvilinear trapezoid whose films by linearly polarized high-energy pulsed laser parallel sides are perpendicular to the polarization light leads to the development of spatially oriented plane of the laser beam (see Fig. 5b). structures. The structure orientation determined by the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 488 MIKHEEV et al. laser polarization plane is explained by anisotropic 8. V. P. Ageev, L. L. Buœlov, and V. I. Konov, Kvantovaya evaporation of the carbon film. Élektron. (Moscow) 16, 1214 (1989). 9. D. O. Barsukov, G. M. Gusakov, and A. I. Frolov, Kvan- tovaya Élektron. (Moscow) 18, 1477 (1991). ACKNOWLEDGMENTS 10. A. B. Braœlovskiœ, I. A. Dorofeev, A. B. Ezerskiœ, et al., We are sincerely grateful to A.E. Murav’ev, Zh. Tekh. Fiz. 61, 129 (1991) [Sov. Phys. Tech. Phys. 36, A.Yu. Popov, and K. Efimovs for technical assistance 324 (1991)]. provided in the course of this study. This work was sup- 11. A. F. Banishev, V. S. Golubev, and O. D. Khramova, ported by the International Association for Cooperation Laser Phys. 3, 1198 (1993). with Scientists from the former Soviet Union (INTAS), 12. V. V. Voronov, S. I. Dolgaev, S. V. Lavrishchev, et al., grant no. 01-0254. Kvantovaya Élektron. (Moscow) 30, 710 (2000). 13. A. N. Obraztsov, A. A. Zolotukhin, A. O. Ustinov, et al., Carbon 41, 836 (2003). REFERENCES 14. G. M. Mikheev, D. I. Maleev, and T. N. Mogileva, Kvan- 1. A. N. Obraztsov, A. P. Volkov, A. I. Boronin, and tovaya Élektron. (Moscow) 19, 45 (1992) [Sov. J. Quan- S. V. Koshcheev, Zh. Éksp. Teor. Fiz. 120, 970 (2001) tum Electron. 22, 37 (1992)]. [JETP 93, 846 (2001)]. 15. A. N. Obraztsov, I. Yu. Pavlovsky, A. P. Volkov, et al., 2. A. N. Obraztsov, A. P. Volkov, I. Yu. Pavlovskiœ, et al., Diamond Relat. Mater. 8, 814 (1999). Pis’ma Zh. Éksp. Teor. Fiz. 69, 381 (1999) [JETP Lett. 16. A. P. Prishivalko, Light Reflection from Absorbing Media 69, 411 (1999)]. (Akad. Nauk BSSR, Minsk, 1963), p. 26. 3. J.-M. Bonard, H. Kind, Th. Stöckli, and L.-O. Nilson, 17. G. M. Mikheev and V. S. Idiatulin, Kvantovaya Élektron. Solid-State Electron. 45, 893 (2001). (Moscow) 24, 1007 (1997) [Quantum Electronics 27 4. A. V. Eletskiœ, Usp. Fiz. Nauk 172, 401 (2002) [Phys. (11), 978 (1997)]. Usp. 45, 369 (2002)]. 18. Optical Properties of Semiconductors: Handbook, Ed. 5. M. J. Birnbaum, Appl. Phys. 36, 3688 (1965). by V. I. Gavrilenko, A. M. Grekhov, D. V. Korbutyak, and V. G. Litovchenko (Naukova Dumka, Kiev, 1987), 6. S. A. Akhmanov, V. I. Emel’yanov, N. I. Koroteev, and p. 198. V. N. Seminogov, Usp. Fiz. Nauk 147, 675 (1985) [Sov. Phys.ÐUsp. 28, 1084 (1985)]. 19. A. A. Potapov, Fractals in Radiophysics and Radioloca- tion (Logos, Moscow, 2002), p. 194. 7. V. P. Ageev, L. L. Buœlov, V. I. Konov, et al., Dokl. Akad. Nauk SSSR 303, 598 (1988) [Sov. Phys. Dokl. 33, 840 (1988)]. Translated by A. Betev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 489Ð495. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 556Ð562. Original Russian Text Copyright © 2004 by Nasyrov. ATOMS, SPECTRA, RADIATION

Two-Scale Angular-Momentum Evolution Induced by Elliptically Polarized Resonance Radiation for a Two-Level Atom with Jg Je = Jg + 1 Optical Transition in a Constant Magnetic Field K. A. Nasyrov Institute of Automatics and Electrometry, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia e-mail: [email protected] Received August 7, 2003

Abstract—The dynamics of a two-level atom with optical transition Jg Je = Jg + 1 under the action of elliptically polarized resonance radiation in the presence of a constant field directed along the radiation wave vector is studied in the approximation of a semiclassical description of the angular momentum orientation. It is shown that the atomic distribution over the angular momentum orientation exhibits a two-scale time evolution. At the first stage, after the beginning of irradiation, the angular momenta of atoms get oriented (over comparatively short time intervals) along the magnetic field, as well as in the opposite direction, depending on their initial orientation and the ellipticity of radiation. At the second (longer) stage, the redistribution of atoms takes place, as a result of which they are oriented predominantly in one of the above directions. The duration of the second stage is an exponential function of angular momentum J. © 2004 MAIK “Nauka/Interperiodica”.

The scope of problems that involve the interaction mally manifested in the presence of narrow angular of atoms with resonance radiation in the presence of a structures with a size smaller than the quantum indeter- magnetic field has been expanded in recent years. We minacy ∆θ ~ 1/J in the angular momentum orienta- are speaking primarily of magnetooptical traps for neutral atoms [1], magnetometers based on optical pump- tion in the density matrix, can also be exhibited in such ing [2, 3], and so on. A characteristic feature of such systems [7, 8]. In the present study, we consider only problems is that the degeneracy of atomic states in the such solutions which will be referred to as semiclassi- angular momentum projection must be included in cal and in which the characteristic scale of angular view of the vector nature of the interaction of the elec- strictures exceeds the quantum-mechanical indeterminacy in the angular momentum orientation. Neverthe- tromagnetic field with the atoms. Even if we confine ӷ our analysis to the model of an atom with two energy less, the results obtained when the condition J 1 states, the problem becomes essentially a multilevel holds remain qualitatively valid for not very large val- problem in view of the above-mentioned degeneracy ues of J ~ 3Ð5 as well. This remark is especially impor- and an analytic solution can be obtained only in a few tant since atoms are mainly characterized by not very specific cases [4–6]. The dynamics of an atom can be large values of J. For example, the alkali atom Cs traced only via numerical solution of the corresponding exhibits a closed transition F F + 1 from the hyper- equations, which can provide answers to specific ques- fine state with a total angular momentum of F = 4. Only tions, although it is difficult to analyze the dependences some atoms (e.g., Fr isotopes) possess an angular on the parameters of the problem in this case. momentum sufficient for assuming that the semiclassical approximation is quite applicable for their However, semiclassical approaches to the descrip- description. tion of rotational motion and, in particular, the angular momentum orientation can be useful here [7]. In these It was shown in [8], where the classical approxima- approximations, the equations for the density matrix tion was used for describing rotational motion, that an are simplified to such an extent that analytic solutions atom with optical transition Jg Je = Jg + 1 in the can be obtained and their parametric analysis can be field of elliptically polarized resonance radiation and in carried out. A necessary condition for the validity of a the presence of a magnetic field directed along the wave semiclassical description of rotational motion is a large vector orients its angular momentum either parallel or angular momentum of the quantum system, J ӷ 1. antiparallel to the magnetic field, depending on its ini- However, essentially quantum effects, which are for- tial orientation and the ellipticity of the resonance field

490 NASYROV

t = 0 t = t0 t = 10t0 t = 100t0

Fig. 1. Evolution of the angular momentum orientation distribution for an atom with Jg = 4, Je = 5; the ellipticity of radiation is Γ 2 x0 = Ð0.3 and t0 = J /G . The z axis and the magnetic field are directed upwards. polarization. However, such dynamics of the angular about the magnetic field direction always exists). It is momentum is observed only for a certain time interval, worth noting that the second stage is much longer than the first. Γ 2 t0 = J /G , The present study aims to describe this phenomenon on the basis of a semiclassical representation of angular where J is the angular momentum, G is the Rabi fre- momentum [9]. In particular, the factors determining quency, and Γ is the line half-width. Numerical calcu- the duration of the second stage in the redistribution of lations based on the optical Bloch equations in the JM particles over the direction of angular momentum will representation for a two-level degenerate system show be clarified. that, in the case of the elliptic polarization of radiation, We consider a quantum system with two energy lev- one of the directions of the angular momentum is unstable. For long time intervals t ӷ t , the overwhelming els degenerate in the angular momentum projection, 0 which resonantly interacts with radiation so that radia- majority of particles are concentrated only in the vicin- tion induces transitions from the lower (ground) state g ity of one of the possible directions of the angular momentum (Fig. 1). The results of calculations to the upper (excited) state e and back. Passing to a obtained in the JM representation are visualized with semiclassical description of rotational motion, we will use the φθα representation introduced in [9] for a rota- the help of the method described in [7]. The figure θ φ shows that, in the case of population of the lower state tor or a spherical top. Here, angles and characterize the polar and azimuth angles of the direction of the with an initially isotropic distribution over the direction α of the angular momentum, an anisotropic distribution is angular momentum, while angle determines the orientation of the rotator axis in a plane orthogonal to the formed at the first stage over time intervals t ~ t0 (for the chosen ellipticity of polarization, the angular momenta angular momentum direction. In exact resonance with of part of the atoms are oriented along the magnetic the exciting field, the equations for the density matrix φθα field, while the angular momenta of a slightly smaller for the quantum system considered here in the rep- part of atoms are oriented in the opposite direction). For resentation have the form [9] longer periods of time, the latter part becomes smaller and smaller until the angular momenta of nearly all par- d iwˆ ticles are oriented along the magnetic field (it should be ----- + Γ ρ = Ðiexp ------Gρ e ee 2J ge stipulated that, in view of quantum indeterminacy ∆θ ~ dt Ðiwˆ 1/J in the angle of orientation of angular momentum, + iexp ------G*ρ , a certain distribution of angular momentum directions 2J eg

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TWO-SCALE ANGULAR-MOMENTUM EVOLUTION 491

d d Ðiwˆ Here, Eσ are the circular components of the polarization -----ρ + ω ------ρ = iexp ------Gρ σ ± dt gg H dφ gg 2J ge vector of the electric field of the wave ( = 0, 1); deg is the reduced matrix element of the dipole moment for iwˆ 1 ρ Γˆ ρ the eÐg transition; and Dσ, J Ð J is the Wigner rotation Ð iexp------G* eg + e ee, (1) g e 2J matrix. We will consider the case when the direction of propagation of radiation coincides with the direction of d Γ ρ iwˆ ρ ----- + eg = Ðiexp ------G gg the magnetic field. Operator wˆ acts in accordance with dt 2J the rule [9] Ðiwˆ + iexp ------Gρ , ∂ ∂ ∂ 2J ee wˆ PQ = ------Ð cosθ------P------Q ∂φ ∂α ∂θcos ρ ==ρ* ,2JJ++J 1. (3) ge eg e g ∂ ∂ ∂ Ð ------P ------Ð cosθ------Q. Γ ∂θcos ∂φ ∂α Here, e is the constant of spontaneous decay of the excited state to the ground state. We assume that there exists only one decay channel for state e. Constant Γ is We consider the situation when the radiation inten- the rate of coherence breaking between levels e and g. sity is not high; i.e., the Rabi frequency G Ӷ Γ. In this In the absence of other mechanisms of destruction of case, the population of the excited state is always much this coherence besides spontaneous breaking, we have smaller than that of the ground state and we can disre- Γ Γ ρ ρ = e/2. gard ee as compared to gg as well as the time deriva- ρ ρ We consider the case of a weak magnetic field, when tives in the equations for ge and eg. Summing the ω the Zeeman frequency H is much smaller than the equations for the diagonal elements of the density relaxation constant Γ; consequently, we disregard the matrix in this approximation for the radiation intensity, magnetic field effects in Eqs. (1) for nondiagonal ele- we arrive at the equation ments of the density matrix and for the excited state. The effect of rotation of the angular momentum of the ∂ ∂ -----ρ + ω ------ρ atom in the magnetic field is included only in the equa- ∂t gg H ∂φ gg tion for the diagonal element of the density matrix for (4) wˆ the ground state. Equations (1) are written in a coordi- = 2sin ------()ΓGρ + G*ρ + ()ρˆ Ð Γ . nate system with the z axis directed along the magnetic 2J ge eg e e ee field. Γˆ ρ Term e ee on the right-hand side of the equation In the semiclassical approximation, when it is ρ assumed that the angular scales of characteristic varia- for the diagonal element gg of the density matrix describes the arrival at the ground state due to sponta- tions of the density matrix considerably exceed the neous decay of excited state e. The specific form of this quantum indeterminacy 1/J in the angular momen- term for various types of optical transitions was deter- tum orientation, we can retain only a few first terms in mined in [8]. In the particular case of transition Jg the expansion of operators of the type exp(iwˆ /2J) into Je = Jg + 1, this term has the form a Taylor series. In particular, for our subsequent analy- ρ sis, it is sufficient to retain in Eq. (4) for gg only the 2 Γˆ e Ð Γ J Ð J J Ð J 1 terms on the order up to 1/J ; consequently, in the ------e = ------e -g 1 + ------e -g + ------∆, (2) Γ J 2J 2 expansion of the sine in Eq. (4), we retain only the e 4J ρ first nonzero term. In addition, ee and nondiagonal where ∆ is the angular Laplacian in angular variables θ elements of the density matrix can be calculated to and φ. within 1/J: In Eqs. (1), the following notation is used: ρ i ()ρ ρ wˆ ()ρ ρ ee = Ð Γ----- G ge Ð G* eg + ------Γ- G ge + G* eg , 1 e 2J e G()φθα,, = ∑Gσ Dσ, ()φθα,, Jg Ð Je σ i wˆ ˜ ()φθ, ()()α ρ = Ð --- Gρ + ------Gρ , (5) = G expiJe Ð Jg , eg Γ gg 2JΓ gg

˜ 1 Eσdeg G()φθ, = ∑Gσ Dσ, ()φθ,,0 , Gσ = ------. i wˆ Jg Ð Je ប ρ = --- G*ρ + ------G*ρ . σ 2J + 1 ge Γ gg 2JΓ gg

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 492 NASYROV

2 determining the drift velocity u of the motion of angular momentum in coordinate cosθ, while the terms proportional to 1/J2 are responsible for diffusion. 1 Thus, if we consider the case when J ӷ 1, which is necessary for the semiclassical description of the angular momentum to be valid, the diffusion term is of the 0 next order of smallness in parameter 1/J as compared to the drift term. This means that the angular momentum dynamics at the initial instant following the application –1 of laser radiation is mainly determined by the drift; at later stages, diffusion begins to play a signiﬁcant role. To simplify the subsequent analysis, we introduce –2 the notation x = cosθ. In accordance with Eqs. (7), –1.0 –0.5 0 0.5 1.0 x velocity u is alternating in the range of admissible values of x (Ð1 ≤ x ≤ 1) and assumes zero value for

Γ 2 Fig. 2. Normalized drift velocity uJ /G (solid curve), nor- 2 2 2 G+1 Ð GÐ1 malized diffusion coefﬁcient DJ2Γ/G (dashed curve), and x0 = ------2 2. (8) the ratio u/DJ of the drift velocity to the diffusion coef- GÐ1 + G+1 ﬁcient (dotted curve). The ellipticity of polarization is − x0 = 0.3. Quantity x0 also characterizes the ellipticity of the radiation polarization (in particular, x0 = 0 corresponds to ω ± We assume further that the Zeeman frequency H is the linear polarization and x0 = 1 to the right and left cir- 2 much higher than the velocity G /JΓ of rotation of the cular polarization). It follows from Eqs. (7) that u > 0 for x > x0 and, conversely, u < 0 for x < x0 (see Fig. 2). In 2 | |2 | |2 angular momentum [8] (here, G = G+1 + GÐ1 ). This other words, all particles with initial coordinate x > x0 ρ allows us to assume that population gg has a uniform move to the right and would be accumulated at point distribution over angle φ. In this approximation, we will x = 1 in the absence of diffusion, while particles with θ study the angular momentum distribution over angle , initial coordinate x < x0 would gather at point x = Ð1. If we assume that the initial distribution of particles over 2π x is uniform, the number of particles gathered at points 1 ρ = ------ρ dφ. (6) x = 1 and x = Ð1 would be 2π ∫ gg 0 1 Ð x 1 + x N+ ==------0, NÐ ------0, (9) Proceeding in the same way as described in [8] and 2 2 retaining only the terns on the order of 1/J and 1/J2, after certain calculations, we obtain respectively. We will use the following normalization for the number of particles: ∂ ∂ ∂ -----ρ +0,------uρ Ð D------ρ = (7) ∂t ∂θcos ∂θcos N+ +1.NÐ = (10)

2 However, the steady-state solution to Eq. (7) has the sin θ 2 2 u = ------[]G ()1 + cosθ Ð G ()1 Ð cosθ , form 2ΓJ Ð1 +1 x 2 θ ρ () 1 u sin [ 2()θ ()θ 0 x = --- exp ∫ ---- dx1 , D = ------2 GÐ1 1 + cos 3 Ð cos Z D 8ΓJ x 0 (11) 2()θ ()]θ 1 x + G+1 1 Ð cos 3 + cos . u Z = exp---- dx dx. ∫ ∫ D 1 It can be seen that the dynamics of the angular momen- Ð1 x0 tum distribution over its projection on the z axis obeys a diffusion-drift equation, the terms proportional to 1/J Using this equation to evaluate the total number of par-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 TWO-SCALE ANGULAR-MOMENTUM EVOLUTION 493 ticles for x > x0 and x < x0, we obtain On the other hand, owing to the quasi-stationary condition, the following equation is valid in the vicinity 1 x of point x = x0: + 1 u N0 = --- ∫ exp ∫ ---- dx1 dx, Z D d x x uρ Ð D------ρ = j. (17) 0 0 (12) dx x0 x Ð 1 u N = --- exp---- dx dx, The solution to this equation is obvious: 0 Z ∫ ∫ D 1 Ð1 x0 x ρ() u which does not coincide in the general case with the ini- x = exp ∫ ---- dx1 + Ð D tial distribution (9) for N and N . It follows hence that x 0 (18) the initial redistribution of particles, which takes the x x1 characteristic time j u ×ρ()x Ð.---- expÐ ---- dx dx 0 ∫ D ∫ D 2 1 2 Γ x0 x0 t0 ==1/uJ/G

In the vicinity of point x0, we can also use the expan- is followed by a longer process, during which a true sion steady-state distribution of particles (11), (12) sets in. The duration of this process is determined by diffusion du of particles in a region in the vicinity of point x . Inte- ux()==()xxÐ u', u' ------, (19) 0 0 dx grating Eq. (7) with respect to x from x0 to 1, we obtain xx= 0 the equation for the time variation of N+, and calculate explicitly the integral in the exponential function on the right-hand side of Eq. (18): d + -----N = j , (13) dt 0 x u ≈ u' ()2 exp Ð∫ ---- dx1 expÐ------xxÐ 0 . (20) where j0 is the particle ﬂux at point x0. D 2D0 x It follows from Eqs. (11) and (12) that function u/D 0 (the dependence of this function on x is shown in Fig. 2) Since the latter function rapidly attenuates, we can plays an important role. This function possesses the rightfully take the quantity j/D outside the integral in property Eq. (18) at point x = x0 and use in the region

u ± > ∼ ---- = 2J, (14) xxÐ20 D0/u' 1/ J D x = ±1 the following relations instead of (18): which is useful for subsequent calculations. To determine the value of j0, we will use the fact that the second x + u stage of the redistribution of particles to the stationary ρ ()x = exp---- dx value is a slow process. From the standpoint of mathe- ∫ D 1 x matics, we are using the presence of small parameter 0 (21) 1/J in the problem, which in fact is manifested in that 1 x1 j  the diffusion velocity is smaller than the drift velocity ×ρ() 0 u x0 Ð,------∫ exp Ð∫ ---- dx2 dx1 by a factor of 1/J. This circumstance suggests that, at D0 D x x large distances from point x0, in each of the regions 0 0 x > x0 and x < x0, for long time intervals, we have a x quasi-stationary distribution Ð u ρ ()x = exp---- dx ∫ D 1 +  ρ+() N ρ () > x0 x = ------0 x , xx0, (15) (22) + x0 x1 N0 j  ×ρ() 0 u x0 +.------∫ exp Ð∫ ---- dx2 dx1 D0 D Ð Ð1 x ρÐ() N ρ () < 0 x = ------Ð 0 x , xx0. (16) N0 Comparing expressions (21) and (22) with (15)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 494 NASYROV and (16), we ﬁnd that and using normalization condition (10), we obtain

()2 3/2 2 Ð + d + 3 1 Ð x0 G D0 N N -----N = ------j0 = ------------Ð ------. (23) dt 2πJ 2 Γ 1 x Ð + u N0 N0 Z expÐ ---- dx ∫ ∫ × [ ()()+ D exp ÐJ 1 + x0 Ð N (29) Ð1 x0 × ()]()() ()() exp ÐJ 1 + x0 + exp ÐJ 1 Ð x0 . + Ð We will calculate N0 and N0 by using obvious approximations, such as It follows from this equation that the duration of the second stage in the evolution of the angular momentum orientation is given by 1 x 1 + u u   J ZN0 ==∫ exp ∫ ---- dx1 dx exp ∫ ---- dx 2π 1 e D D t = t ------. (30) 1 0 2 3/2 () x0 x0 x0 3J ()cosh Jx0 1 Ð x0 1 u (24) exp∫ ---- dx Simple estimates show that the second stage dura- 1 1 D tion is t ≈ 40t for J = 5 and for ellipticity x = Ð0.3. u x 1 0 0 × expÐ ---- dx dx ≈ ------0 . ∫ ∫ D 1 2J Thus, in the description of the angular momentum x0 x based on the semiclassical approximation, the problem of the interaction between a two-level atom and ellipti- On segment x ≤ x ≤ 1, we approximately assume that cally polarized radiation in the presence of a magnetic 0 ﬁeld can be formally reduced to the diffusion-drift equation of particle transfer in the coordinate corre- u xxÐ ---- ≈ 2J------0 , (25) sponding to the angular momentum projection on the D 1 Ð x0 quantization axis. For the optical transition

Jg Je = Jg + 1, which finally gives the drift velocity is alternating; as a result, particles are ()() concentrated depending on the angular momentum ori- + ≈ exp J 1 Ð x0 ZN0 ------. (26) entation parallel or antiparallel to the magnetic field. 2J However, when the polarization differs from linear, one of these directions dominates in the sense that, after a Analogously, for NÐ we obtain long time, the angular momenta of most particles are 0 oriented in this direction. As a result of diffusion, particles are “pumped” from the less advantageous direction exp()J()1 + x of their angular momentum orientation to the opposite ZNÐ ≈ ------0 -. (27) 0 2J direction. However, the rate of such pumping is proportional to the number density of particles in the vicinity of point x , at which the drift velocity vanishes. In view Finally, using approximation (19), (20), we obtain 0 the following expression for j : of the properties of the solution to the diffusion-drift 0 equation, the number density of particles in this region is found to be exponentially small, D u' Ð j = 2J ------0 -(N exp()ÐJ()1 + x ±1 0 2π 0  expÐ ()u/D dx ∼ eÐJ , ∫ + ())() x0 Ð N exp ÐJ 1 Ð x0 as compared to the number density of particles concen- ()2 3/2 2 (28) 3 1 Ð x0 G Ð trated in the above-mentioned directions. This circum- = ------(N exp()ÐJ()1 + x 2πJ 2 Γ 0 stance determines the slow rate of the second process. After completion of the second stage, the numbers of particle oriented along and against the magnetic field + ())() Ð N exp ÐJ 1 Ð x0 . satisfy the equilibrium relation Ð ()+ Substituting this expression for flux j0 into Eq. (13) N0 = exp 2Jx0 N0 ,

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 TWO-SCALE ANGULAR-MOMENTUM EVOLUTION 495 which is quite sensitive to ellipticity x0 for large values 3. C. Andreeva, S. Cartaleva, Y. Dancheva, et al., Phys. of J. Rev. A 66, 012502 (2002). 4. E. Arimondo, Prog. Opt. 35, 257 (1995). 5. V. S. Smirnov, A. M. Tumaœkin, and V. I. Yudin, ACKNOWLEDGMENTS Zh. Éksp. Teor. Fiz. 96, 1613 (1989) [Sov. Phys. JETP This study was ﬁnanced by the Russian Foundation 69, 913 (1989)]. for Basic Research (project no. 01-02-17433). 6. A. M. Tumaœkin and V. I. Yudin, Zh. Éksp. Teor. Fiz. 98, 81 (1990) [Sov. Phys. JETP 71, 43 (1990)]. 7. K. A. Nasyrov and A. M. Shalagin, Zh. Éksp. Teor. Fiz. REFERENCES 81, 1649 (1981) [Sov. Phys. JETP 54, 877 (1981)]. 8. K. A. Nasyrov, Phys. Rev. A 63, 043406 (2001). 1. A. Aspect, E. Arimondo, R. Kaiser, et al., Phys. Rev. Lett. 61, 826 (1988). 9. K. A. Nasyrov, J. Phys. A: Math. Gen. 32, 6663 (1999). 2. R. Wynands and A. Nagel, Appl. Phys. B: Laser Opt. B68, 1 (1999). Translated by N. Wadhwa

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 496Ð502. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 563Ð569. Original Russian Text Copyright © 2004 by Zharinov. PLASMA, GASES

Vacuum Microarc A. V. Zharinov All-Russia Institute of Electrical Engineering, Moscow, 111250 Russia e-mail: [email protected] Received August 5, 2003

Abstract—The physics of stationary vacuum microarc in a wide interelectrode gap with the perveance corresponding to a geometry of the Müller electron projector type and the Langmuir–Blodgett function α2 ≥ 5 is considered on a qualitative level. Under these conditions, the electric field at the cathode can exhibit a significant (severalfold) increase due to a positive space charge of microarc, which makes field electron emission possible. The most important features of the continuity equation, Poisson equation, and thermal conductivity equation describing this system are considered. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION cathode spot on the surface of a massive cathode. This very type of discharge, referred to below as vacuum Electric microarc is a fundamental phenomenon microarc, is studied in this paper. playing the role of a living cell in the complicated “organism” of high-current arc discharge, widely used, It is commonly accepted that the typical vacuum albeit still on the basis of empirical knowledge, in both microarc consists of three parts: (i) a cathode spot; science and technology [1]. As is known, a vacuum arc (ii) an active near-cathode region called the cathode can be initiated in various ways, in particular, it appears layer, from which all ions are collected in the cathode as a result of the electric discharge between two elec- spot to provide for continuous reproduction of the nec- trodes in vacuum, thus restricting the maximum possi- essary flow of atoms from the cathode surface, the ble potential difference between electrodes to a level of working temperature, the electric field strength, and the about 105 V per millimeter of discharge gap width. electron emission current I; and (iii) a passive anode Alternatively, a vacuum arc arises upon breakage of the layer playing the role of a conductor between the outer contact between current-carrying electrodes. Depend- surface of the cathode layer and the anode surface. In ing on the circuit parameters, the current of a vacuum the case of a distant anode, the anode layer is character- arc may vary from a fraction of ampere to tens and hun- ized by predominating negative space charge and by a dreds of kiloamperes, while the voltage drop varies very low density of atoms. Ionization of atoms in the from the ionization potential to several dozens of volts. anode layer does not play any significant role in the A vacuum arc can exist between electrodes made of any mechanism of self-sustained vacuum microarc. The metals (from mercury to tungsten) possessing substan- conditions of existence and the properties of vacuum tially different evaporation rates, thermal conductivi- microarc are fully determined by interaction of the ties, and other properties. active cathode layer with the cathode spot surface. In most cases, electric arc is studied in a discharge The active cathode layer can be represented by a gap with a width below 1 cm, where an important or spherical sector of solid angle Ω with a cathode spot even decisive role is played by anode evaporation. A with an area of πR2 ≈ Ωr2 at the apex. The atomic flux high-current arc consists of numerous autonomous 0 arcs, each arising from its own cathode spot emitting from the cathode spot surface, with an angular distribution described approximately by the cosine law, propa- electrons. The discharge current via one cathode spot Ω Ω can vary from about 0.4 A for mercury up to 102 A for gates within a solid angle 0 > irrespective of the electric field structure. A part of the atomic flux within tungsten [2]. The current density measured on a cath- ∆Ω Ω Ω ode spot may reach up to 108 A/cm2. Therefore, the the angle = 0 Ð bypasses the active cathode cathode spot radius can be estimated at about 5 × layer and does not participate in discharge. 10−4 cm, the electron density is on the order of 1018 cmÐ3, Figure 1 shows a schematic diagram of such an ide- alized vacuum arc with ∆Ω = 0. The outer boundary of × Ð7 and the Debye radius is 5 10 T e cm (Te is the elec- the active cathode layer has a potential of ϕ and is tron temperature expressed in electronvolts). c spaced by r0 + a from the center (point O) of the spher- Thus, according to the results of observations, a vac- ical sector. Analysis of this model, despite its simplified uum arc with a sufficiently small current is a micro- character, allows the most important parameters to be scopic self-sustained discharge arising from a single established and their interrelation to be understood.

VACUUM MICROARC 497

In the scheme of Fig. 1, unknown parameters ϕ include the potential c, the electron emission current I, the ion current Ii to the cathode, the spot radius R, the a Ω 2 ≈ π 2 Ω solid angle (or the quantity r0 R / ), and the cathode layer thickness a. O r0 All the cathode characteristics important from the Fig. 1. Schematic diagram of the model of vacuum microarc standpoint of the microarc’s existence are considered as (see the text for explanations). known. These include the evaporation rate, the rate of cathode sputtering by ion bombardment, and the density of electron emission current as a function of the cath- approximated formula ode temperature (Tc) and the electric field strength (Ec) at the cathode, as well as thermal conductivity, electric 2.7 conductivity, etc. σ = σ ------lnW, i m W Based on the known properties of the cathode, vacuum microarc is described by jointly solving the equa- σ where m is the maximum ionization cross section, W = tion of continuity for particle fluxes in the active cath- ϕ ϕ ode layer, the Poisson equation, and the thermal bal- e/Ui, e is the kinetic energy of electron, an Ui is the ance equation for the cathode spot surface. Rigorous ionization potential. For example, i = 1.7 × 10Ð3 and σ Ð16 2 × 13 Ð2 solution of this problem is a very complicated task. For i = 10 cm correspond to Ng = 1.7 10 cm . For this reason, an analysis of the scheme in Fig. 1 will be such a low atomic density and a transport cross section σ 15 2 performed using justified simplifying assumptions of T < 10 cm , electrons are barely scattered on implying that a stationary vacuum microarc exists in a atoms in the cathode layer. It should be also noted that, certain limited region of parameters. σ × Ð16 2 for m = 5 10 cm and Ui = 8 V, the average cross σ Ð16 2 section i = 10 cm can be obtained for two values of ϕ ≈ ϕ ≈ 2. THE EQUATION OF CONTINUITY W: W1 = 1.083 ( e 8.76 V) and W2 = 54 ( e 430 V). AND THE CONDITION OF EXISTENCE σ OF VACUUM MICROARC For the known values of i and i, formula (1) gives a necessary condition for the existence of a stationary Let us restrict the consideration to a region of discharge obeying the equation of continuity for the parameters in which the ion current fraction is small: fluxes of atoms, ions, and emitted electrons. i = I /I Ӷ 1. As is known, the Langmuir collisionless i Let us consider the behavior of atoms in more detail, cathode layer is characterized by i ≥ m/M , where m assuming their radial motion at a constant velocity and M are the electron and ion masses, respectively. For of Vg. Then, atoms will travel the distance r without example, the latter ratio for tungsten is m/M ≈ 1.7 × losing electrons with a probability of Ð3 ≥ 10 . In the case of i 2m/M , the electric field Ec r r µ()r = exp Ð----0- 1 Ð ----0 , (2) accounts for greater than 0.7 of the value corresponding x r to i ӷ 1 (i.e., to a cathode layer with positive space 0 Ӷ charge). Therefore, the condition of i 1 does not where exclude field electron emission, provided that a field strength on the order of E ≈ 107Ð108 V/cm is consistent σ c x0 = eVg/ i j (3) with a joint solution of the Poisson and continuity equations. Thus, the region of parameters corresponding to is the characteristic ionization length in the flow of Ӷ i 1 admits the electron emission via the RichardsonÐ electrons with the initial current density j (at r = r0). DushmanÐSchottky and FowlerÐNordheim mechanisms. As a result of ionization, the density of atoms drops faster than according to the quadratic law, Under the condition of i Ӷ 1, we may assume that I = const in the active cathode layer. Then, the ion cur- 2 σ σ r0 rent component can be expressed as Ii = I iNg, where i () µ() ng r = n0----2 r , is the average ionization cross section and Ng is the r “atomic density” in the gas phase of the cathode layer: and the integration yields r0 + a i N ==n ()r dr ---- . (1) i ()µ g ∫ g σ Ng ==---- n0 x0 1 Ð , (4) i σ i r0 µ µ For obtaining estimates, we will use the well-known where = (r0 + a). Note that, with allowance for

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Ij of the atomic flux from the cathode spot surface by means of evaporation and cathode sputtering. If no such reproduction takes place, a vacuum arc can exist only in the case of zero integral transparency or infinite compression. It is naturally assumed that ions are reflected from the cathode in the neutral state, rather than condensed r0/a on the cathode surface, so that the atomic flux density q0 obeys the relation Fig. 2. The behavior of Ij(r0/a) for vacuum microarc with eq = j ()1 + γ + eq , χ = const. 0 i i s γ ϕ where i( c) is the cathode sputtering coefficient and q (T ) is the evaporation rate. In a stationary regime, expression (3), formula (4) is equivalent to the relation s c γ µ i ji + eqs ()µ = ------. (6) ki = 1 Ð , (5) eq0 In particular, at a sufficiently low cathode temperature where k = j/eq0 and q0 = n0Vg is the density of the atomic Tc for which cathode sputtering predominates (while flux from the cathode surface at r = r0. The probability γ Ӷ µ still i 1), we obtain (r0 + a) is naturally called the transparency of the cath- χ µÐ1 µϕ() γ()ϕ Ӷ ode layer, while the reciprocal quantity = is called c = i c 1, the compression. (7) χϕ()= µÐ1 ӷ 1. Another important process is the resonance recharge c ϕ Ӷ of atoms in the counterflow of ions. With allowance for Thus, for the given values of c = const and eqs γ this factor, the quantity k in relation (5) should be i ji, the microarc transparency and compression are replaced by constant and determined by cathode sputtering. In this case, σ ≈ n i  kk1 + -----σ --- , a a χ i 2 -----1= + ---- ln . (8) x0 r0 σ σ Ð16 2 × 4 where n is the recharge cross section. Then, relation (5) For i = 10 cm and Vg = 5 10 cm/s, we obtain an should be rewritten as estimate of x0 = 80/j, for which relation (8) can be rewritten in terms of the product of the current I and ki= h()1 Ð µ , current density j: 2 ≈ × 4r0 2 χ where the coefficient h takes into account the reverse Ij 3101 + ---- ln . σ 2σ a flux of recharged ions (1 < h < 2). Since n > 10 i, the recharge process may significantly decrease the trans- For lnχ = 5 (χ ≈ 148), we obtain parency and thickness of the cathode layer, while leav- 2 ing N = const. It should be noted that, as the ion current 5r g Ij ≈ 7.5× 10 1 + ----0 . (8') fraction i increases, the gas compression in the active a cathode layer takes place predominantly due to the res- Ӷ × onance recharge. In the limit of r0/a 1, this yields Ij 7.5 5 2 2 ӷ The model of vacuum microarc under consider- 10 A /cm = const; in the other limiting case, r0/a 1, ation, characterized by a “zero cone transparency” × 5 2 2 || ∆Ω we obtain Ij 7.5 10 r0 /a . This behavior of (Vg r; = 0), is optimum from the standpoint of a Ij(r /a) for χ = const is illustrated in Fig. 2. minimum necessary rate of reproduction of the atomic 0 flux from the cathode surface. An analogous situation Using to the results reported by Daalder for vacuum with negligibly small “edge transparency” is actually microarc on copper cathode (see review by Harris [2]), possible for R ӷ a. 2 8 A 10 A The transparency of the cathode layer can vary in I ==50 A, j ≈ 210× ------, Ij 10 ------, response to deformation of the spherical sector, which cm2 cm2 leads to a change in x0, a, and/or r0. However, for a real vacuum microarc in a stationary regime, the integral we obtain an estimate of R ≈ 2.8 × 10Ð4 cm. Taking Ij ≈ 10 2 2 ≈ transparency must exactly compensate for reproduction 10 A /cm , in relation (8'), we obtain r0/a 115. Rela-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 VACUUM MICROARC 499 ӷ tion (8) for r0/a 1 correctly reflects the dependence of ode layer is characterized by predominating positive χ space charge and the Poisson equation in the zero a/x0 on ln , since the edge transparency can be ignored. Then, applying relation (8) to the experimental approximation can be written with neglect of the space data of Daalder, we may conclude that the maximum charge of electrons. This assumption does not seem current of a field-emission microarc, I ≈ 50 A, corre- preposterous in view of a microscopic size of vacuum sponds to a “degenerate” one-dimensional geometry of microarc. ӷ the active cathode layer (r0/a 1). In order to check this, let us obtain estimates for a The scheme of vacuum microarc under consider- one-dimensional cathode layer in which ation implies that the superimposed flows of atoms, () () ions, and electrons inside the spherical segment pro- ng x = n0 exp Ðx/x0 , ceed in the radial direction and do not intersect the cone surface. This idealization leads to a severalfold and the ion current density at a low transparency can be decrease in the lower limiting value of the microarc cur- expressed as rent (called threshold current). The threshold current j ()x = en V exp()Ðx/x . corresponds to a certain minimum ratio (r0/a)min for i 0 g 0 which the atomic density Ng decreases below a mini- σ Moving toward the cathode, ions are retarded as a result mum possible level (Ng < i/ ), leading to the appearance of “vacuum” in the cathode layer volume and to of the Coulomb interaction with the intense flow of the quenching of arc discharge. In the vicinity of this electrons and the resonance recharge caused by the threshold, the discharge may be conventionally called counterflow of atoms. Therefore, it can be reasonably ≤ assumed that the average ion velocity is comparable point microarc. It is obvious that, for (r0/a)min 1, the threshold current coincides in order of magnitude with to that of the neutral atoms, Vi ~ Vg. Then, eni = that for r /a Ӷ 1. en0exp(Ðx/x0), the electric field at the cathode obeys the 0 condition In the field emission regime at j ~ 108 A/cm2, the threshold current of a point microarc according to for- E ≤ 4πen x , (9) mula (8') is approximately 7.5 × 10Ð3 A. It is interesting c 0 0 to note that a current of the same order of magnitude is and the voltage drop is observed for field emission from a microscopic point [3]. Therefore, the question naturally arises as to how can ϕ ≤ π 2 the field electron emission from a microscopic point be c 4 en0 x0 . (10) distinguished from that in the case of microarc. Formulas (9) and (10) reflect equality of the densi- In this context, it is necessary to emphasize the need ≈ for taking into account the ionization and recharge pro- ties of atoms and ions in the cathode layer, Ng Ni, and cesses in the description of stationary electron emission can be rewritten as 8 2 for j ~ 10 A/cm , when the ionization length is x0 ~ Ð6 − ≤ × [] 10 6 cm, that is, much smaller than the point radius. It Ec 1.8 10 Ng V/cm , (9') is not excluded that stationary field emission in the Müller projector can be observed for a flat cathode. N ϕ ≤ 1.44× 10Ð4------g [] V . (10') Finally, note that the above question has a physical c j meaning only provided that both degenerate and point Ð2 σ Ð16 2 microarcs involve an electric field strength of E ~ For i ~ 10 and i = 10 cm , we obtain Ng = 108 V/cm. −14 2 | | ≤ × 8 ϕ Ӷ 10 cm , Ec 1.8 10 V/cm, and c 144 V (for 8 ӷ Ec > 10 V/cm, the field emission current density is j 3. ELECTRIC POTENTIAL 108 A/cm2). These estimates seem to be quite realistic. AND FIELD DISTRIBUTION It should be also noted that, for j ~ 108 A/cm2, the IN VACUUM MICROARC frequency of the Coulomb electronÐion collisions is In many investigations, it is assumed that the active much greater than that of the electronÐatom collisions. cathode layer is filled with a quasineutral plasma and Therefore, the electric conductivity in the cathode layer the current density obeys the “3/2 law,” which is valid must correspond, in order of magnitude, to the Spitzer for the Langmuir collisionless cathode layer. However, formula there is some doubt concerning the validity of these ϕ assumptions in the case of vacuum microarc. At a low ≈ 3/2()d 14 Ð2 j 13T e x ------. density of atoms (Ng < 10 cm ) and a monotonic dx potential distribution in the cathode layer, all electrons travel the distance to the potential virtually without col- For a negligibly small ion current fraction, inelastic liding with atoms in this layer. For this reason, the cath- electron collisions are insignificant and the electron

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 500 ZHARINOV ≈ ϕ temperature Te(x) Tc + (2/3)e increases in proportion determines the characteristic scale of the Joule power to the potential ϕ. In this case, the electric field at the evolved on the cathode, cathode is |E | ≈ j/13T 3/2 . For j ≈ 2 × 108 A/cm2 and T ≤ c e c Q0 = ITc L. 0.3 eV, this yields |E | ≥ 108 V/cm. The potential ϕ(x) c × Ð8 can be determined from the formula jx ≈ 2.9ϕ5/2, which Here, L = 2.45 10 is the Lorenz number, which can σ ≈ 2 ϕ ≈ 2/5 be considered as a fundamental constant determining can be written for eVg/ i 10 as (x) 4(x/x0) . For the relationship between the thermal conductivity λ, the χ ≈ ϕ ≈ a/x0 = ln 5, this yields c 7.6 V. The electric con- electric conductivity σ, and the temperature in the ductivity according to Spitzer corresponds to electro- WiedemannÐFranz law λ = σTL. neutrality, but the order of magnitude is retained even in A heat flux from the cathode layer via the well sur- the case of a two- to threefold decompensation. face is The first special feature of vacuum microarc appar- π 2λ dT > ently consists in that the ion concentration exponen- Qλ = 2 r0 ------0. tially increases toward the cathode, rather than decreas- dr r0 ing in this direction. The second (and main) feature is Let us assume that no heat exchange with the gas dis- the positive space charge in the cathode layer. Accord- charge volume takes place on the flat cathode surface ing to estimates, the electric field strength at the cath- outside the well. Then, the temperature of the well sur- ode and the current density may reach levels typical of face is determined by the thermal conductivity equation ϕ the field emission at low values of c. Moreover, it is quite possible that the maximum field strength Ec at the j2()r cathode of a microarc may exceed the values achieved divgradT = Ðσ------()λ as a result of the field enhancement at a microscopic r point. with the boundary condition T ӷ = T . A solution of In concluding this section, it should be emphasized r r0 that the initial assumption concerning a passive role of this equation can be presented in the following form: the anode layer in vacuum microarc is valid provided ϕ αβ α that the microarc current I( c) is consistent with the t = cos Ð sin , (11) ϕ ϕ current Ia(La, c, a) limited by the perveance of the anode volume, that is, depends on the anode potential where ϕ a and the discharge gap width La. For I > Ia, a nonsta- Qλ jr L tionary regime can take place as a result of the field sag- T β α 0 t ==----- , ------, =------λ -. ging in the region of x > a or even an aperiodic instabil- T c Q0 ity of the Pierce type can develop with the formation of The current I can be expressed in terms of α as a virtual cathode and current breakage. Such processes 2 were considered in detail, with description of original 8 2 2 A experiments, by Nezlin [4]. Ij = 2.56× 10 λ α ------. (12) cm2 Thus, with exponential growth of the gas and ion density toward the cathode, field emission under sta- The angle α determines the rate of temperature tionary vacuum microarc conditions seems to be quite decrease in the bulk of cathode. In particular, for β = 0, possible even from a flat cathode surface. we obtain

Tr() αr ------= cos α Ð ------0 . 4. THERMOPHYSICS T r OF A VACUUM MICROARC CATHODE c The thermophysics of cathode spots is a very inter- In the regime of field emission from tungsten (λ ≈ 1) esting and complicated problem of independent basic for j ≈ 2.56 × 108 A/cm2, we obtain I ≈ α2. At a suffi- significance. However, judging by the available litera- ciently small cathode spot radius, the microarc current ture, this problem was never systematically nor thor- can be very small (on the order of 10Ð2 A). For α Ӷ 1, oughly studied and was not given proper attention in formula (11) yields t ≈ 1 Ð βα, which implies that the reviews. For the description of thermal processes in the spot can be “cold” for βα ≤ 1/2. cathode spot of vacuum microarc, it will be necessary Using the condition βα ≤ 1/2, we obtain to make simplifying assumptions as it was done in the preceding sections. α Q0 T c L Ð5 Consider a cathode spot in the form of a hemispher- Qλ ≤ ------= ------≈ 810× αT Ӷ 1 W. 2α 2 c ical well of radius r0 on the surface of a massive cathode π 2 β α ӷ at a temperature of Tc. The microarc current I = 2 r0 j Since ~ 1/2 1, the ion-bombardment-induced

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 VACUUM MICROARC 501 heating predominates. However, the heat flux is such dimensions. The thermophysics of the microscopic that cathode spot, including the problems of thermal instability, can also be studied in detail using a probe or a ≤ ()ϕ ≈ ()ϕ Qλ Ii c + Ui iI c + Ui . cathode filament that has been self-heated in discharge, as was done by Pustogarov [6] for plasmatrons. Model- Ð2 Ð2 ≈ α Ð1 ing seems to offer a quite justified method stimulating For i ~ 10 , I ~ 10 A, Tc 600 K, and ~ I ~ 10 , ϕ ≥ fundamental rather than purely technological approach we obtain an estimate of c 40 V. This implies that a to arc discharge and promising both new important low-current stationary microarc with I ~ 10Ð2 A is pos- basic knowledge and effective solutions of applied sible both on a big massive cathode and on a cathode problems. with dimensions on the order of several micrometers. For the above estimates, the spot diameter is smaller To summarize, the following conclusions can be than the characteristic scale of the polycrystalline struc- emphasized. ture of a massive cathode. In this case, using the values 1. At a high current density, the cathode layer fea- of λ from handbooks is physically meaningless and tures densification (described in terms of compression estimation of the minimum current of a field-emission χ) related to a higher probability of ionization and res- microarc becomes a difficult problem. onance recharge of atoms in the gas phase. The charac- β Ӷ In the opposite limiting case of = Qλ/Q0 1, teristic size of the cathode layer is on the order of the ≈ 2 7 2 expression (11) implies that, for α π/2, t 0 and ionization length x0 10 /j (for j > 10 A/cm , this the problem has no stationary solutions. For α = π/2, yields a microscopic value below 10Ð5 cm). this situation takes place when 2. On the microscopic scale, the entire cathode layer is characterized by a predominating positive space () × 8λ2 2 2 Ij cr = 6.3 10 A /cm . (13) charge. For this reason, the electric field at the cathode ϕ Ec and the potential c can be estimated by the order of For copper (λ = 3.5), this yields (Ij) = 7.74 × ≈ × Ð6 ϕ ≈ × cr magnitude as Ec 1.8 10 Ng [V/cm] and c 1.4 109 A2/cm2. This value is in satisfactory agreement Ð4 σ ≈ Ð2 ≥ 10 Ng/j [V], where Ng = i/ i n0x0 [cm ]. For Ng with the result obtained by Daalder: (Ij) =1010 A 2/cm2. −14 2 ≥ 8 ϕ 2 ≈ cr 10 cm , this yields Ec 10 V/cm, c < 10 V, and j Thus, the upper limiting microarc current is probably 108 A/cm2. Therefore, a positively charged active cath- related to a thermal instability of the cathode spot β ode layer of vacuum microarc provides conditions for developed at 0 [2]. field electron emission. Returning to the approximation of t ≈ 1 Ð αβ, note that, irrespective of the temperature, the equality αβρ = 1 3. In a stationary microarc, the compression is is valid for ρ determined by the temperature gradient on inversely proportional to the rate of reproduction of the the spot surface via the relation atomic flux from the cathode spot. In the vicinity of a cathode sputtering threshold, the compression may reach a level of χ > 104. dT T c ------= ------ρ-. 4. For the cathode layer geometry modeled by a dr r r0 0 spherical segment, the compression depends on the ion- Therefore, ρ 1 implies αβ 1 and, apparently, ization length x0, the cathode spot radius R, and the a thermal instability (t 0) in a point microarc can cathode layer thickness a. For a preset constant com- develop for α Ӷ 1. It is almost impossible to study pression, there are certain possible relations between rather complicated thermophysics of vacuum microarc the spatial characteristics x0, R, and a corresponding to in experiments on natural microscopic objects without various stationary values of the microarc current I and recourse to modeling on specially designed setups. the current density j: (i) for R ӷ a (degenerate one-dimensional regime), 5. CONCLUSIONS 2 ()≈ × 4 R ()χ 2 It is believed that a high-current low-voltage vac- Ij 1 310----- ln ; uum arc consists of numerous fragments (“group a2 spots”) representing complexes of vacuum microarcs [2, 5]. Therefore, investigation into the properties of (ii) for a ӷ R (point microarc regime), vacuum microarc is of primary significance. This task can be solved both by studying natural microscopic ()≈ × 4()χ 2 objects and by modeling separate processes and parts of Ij 2 310ln . a microarc. For example, gas compression, potential distribution, and stability of the quasispherical cathode For a point microarc with j ≈ 108 A/cm2, the layer can probably be studied in a non-self-sustained microarc current is on the same order of magnitude as discharge with an incandescent cathode of macroscopic the current of field emission from a microscopic point.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 502 ZHARINOV 5. The ascending currentÐvoltage characteristic and REFERENCES self-sustained character of vacuum microarc, together 1. G. A. Mesyats, Ectons (Nauka, Ekaterinburg, 1993). with the possibility of stationary states, admit the for- 2. L. Harris, in Vacuum Arcs: Theory and Application, Ed. mation of complexes (clusters [2]) of a large number of by J. Lafferty (Wiley, New York, 1980; Mir, Moscow, microarcs on a smooth (ﬂat) surface, arranged in a cer- 1982). tain order and satisfying the conditions of a stable ther- 3. V. N. Shrednik, in Unheated Cathodes, Ed. by M. I. Elin- mal regime and self-sustained operation. If this hypoth- son (Sovetskoe Radio, Moscow, 1974), p. 216. esis is valid, it will be possible to create stationary, 4. M. V. Nezlin, Beam Dynamics in a Plasma (Énergoat- high-current high-voltage ﬁeld emission vacuum omizdat, Moscow, 1982). diodes using smooth massive cathodes. 5. V. I. Rakhovskiœ, Physical Principles of the Commuta- tion of Electric Current in Vacuum (Nauka, Moscow, 1970). ACKNOWLEDGMENTS 6. A. V. Pustogarov, in Experimental Studies of Plasma- trons (Nauka, Novosibirsk, 1977), p. 315. The author is grateful to Yu.A. Kovalenko and A.N. Ermilov for support and promotion of this study. Translated by P. Pozdeev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 503Ð507. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 570Ð575. Original Russian Text Copyright © 2004 by Dragan. PLASMA, GASES

Electroacoustic Oscillations of Aluminum Oxide Particles in the Thermal Plasma G. S. Dragan Mechnikov National University, Odessa, 65026 Ukraine e-mail: [email protected] Received August 5, 2003

Abstract—The frequencies of natural electroacoustic oscillations of aluminum oxide particles in a laminar disperse aluminum ﬂame are determined experimentally using the capacitive method. A computational model is proposed for estimating the natural frequency of oscillations of charged particles in the smoky plasma taking into account the Doppler effect. It is shown that, for a natural frequency of oscillations of 51 kHz, two measured maxima at frequencies of 30 and 60 kHz in the oscillation spectrum correspond to the Doppler frequencies. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION the condensed phase of a smoky plasma, which emerge as a result of electric charge fluctuations on the surface Combustion products obtained as a result of burning of grains, was studied by us for the first time in [10]. of a metal powder in a gaseous oxidizer contain fine- Later [11], analogous oscillations were studied in the dispersed particles of metal oxide, whose size may dusty plasma of a dc glow discharge and were inter- change depending on the combustion regime, from preted as a consequence of the plasmaÐdust current hundredths of a micrometer (volumetric condensation instability. The most comprehensive analysis of oscilla- in the gas-phase combustion regime) to several tion processes and instabilities in dusty plasmas is micrometers (heterogeneous combustion and conden- given in review [12]. Unfortunately, smoky plasmas sation) [1]. When the temperature of combustion prod- with their specific features were outside of the author’s ucts exceeds 2000 K, condensed particles are charged interest, although some of the results considered in this as a result of thermionic emission from the particle sur- review are applicable in both cases. Thus, electroacous- face; consequently, the gas phase contains free elec- tic oscillations in smoky plasmas have not been studied trons. If the gas phase is free of easy-ionized admixture in actual practice. In addition, the above-mentioned of alkali metal atoms, the ionization of the gas can be features of plasmas with CDP (such as the formation of disregarded. Such a variety of the combustion product ordered structures and the emergence of vibrationalÐ plasma is sometimes referred to as a plasma sol, which oscillatory processes) may be interrelated. in turn is a variety of low-temperature plasma with a condensed disperse phase (CDP) [2]. A plasma contain- This study is devoted to experimental investigation ing the condensed phase in the form of smoke grains is of the spectrum of natural low-frequency electroacous- called a smoky plasma [3]. tic oscillations of the condensed disperse phase in the front of a laminar flame of aluminum powder. An A specific feature of plasmas with CDP is the ther- attempt is made to interpret the experimental results as modynamic interaction at the interface, as a result of a consequence of the formation of an ordered structure which intrinsic electric fields associated with surface of aluminum oxide particles in the plasma flow. processes, the charge state, and the mobility of charged plasma components (condensed particles, electrons, First of all, we note the distinguishing features of and ions) are induced in the plasma [2, 4]. In all proba- two common types of plasma with a CDP: bility, these processes in plasmas lead to the formation of linear chains of smoke grains [5] as well as ordered (i) The dusty plasma is formed when light dust par- 3D structures, which were discovered in [6] and dem- ticles are introduced into a gas discharge under a low onstrated in review [7]. In standing strata of gas dis- pressure or emerges in cosmic space in the presence of charge in dusty plasma, plasmaÐdust structures have a dust particles [12]; the typical features of such plasmas more perfect shape and were hence called plasma crys- are the difference between the electron and ion temper- tals [8, 9]. atures, the insignificant role of thermionic emission from the surface of particles during their charging, and On the other hand, the interfacial thermodynamic the collisionless nature of processes in the bulk; as a interaction in the plasma with CDP gives rise to new result, the self-consistent Vlasov equation for electrons modes and instabilities [10Ð12]. Acoustic oscillation in can be applied.

504 DRAGAN

I, arb.units Spectral analysis of radiation emitted by the flame in 0.20 the visible spectral range revealed that the width L of the combustion front was approximately equal to 10Ð3 m and that the combustion products contained the gaseous and 0.16 condensed phases. The gas phase contained uncontrol- lable admixture of sodium atoms and molecules with a high ionization potential. 0.12 The condensed phase was investigated by sampling of the flame followed by the electron microscopic analysis. It was found that the condensed phase was repre- 0.08 sented by Al2O3 submicrometer spherical smoke grains with a cubic mean diameter of 0.12 µm. The number 0.04 density of particles and their mean charge were determined using the techniques described in [1]. The mean value of the number density of grains was (2 ± 1) × 17 Ð3 0 10 m and their mean charge was 30 ± 10 in units of 10 100 electron charge. f, kHz The volume-averaged temperature of grains was Fig. 1. Spectrum of low-frequency electroacoustic oscilla- determined from the continuous radiation spectrum by tions in the smoky plasma of aluminum oxide. the polychromatic method, while the temperature of the gas phase was calculated from the absolute intensity of resonance lines for Na. Experiments proved that the (ii) The smoky plasma contains smoke grains, i.e., temperature of the gas phase was equal to the tempera- particles formed in the combustion products in the ture of the condensed phase to within the experimental course of volumetric condensation or as residues of the error and amounted to T = (3150 ± 100) K. burning-out fuel [2]; free electrons are generated in the The processing of the experimental data on oscilla- gas phase due to the thermionic emission from grains or tion processes in a smoky plasma in the electrode gap as a result of ionization of gas atoms; the distribution of proved that the signal recorded from resistor R was the self-consistent electrostatic potential in the vicinity alternating and bipolar in spite of the fact that a con- of free charges and charged grains can be described by stant voltage was supplied to it. The averaged ampli- the PoissonÐBoltzmann equation. tude value of the signal was U = 0.8 ± 0.2 V and the Thus, the laminar flame of aluminum particles under duration of oscillations was τ = 70 ± 30 ms. The oscil- investigation is a typical example of a smoky plasma. lation process was induced periodically and at random. The spectral composition of signal I is shown in Fig. 1. It can be seen that the oscillation process occurs 2. EXPERIMENTAL TECHNIQUE in the frequency range from 15 to 200 kHz with two AND RESULTS characteristic frequencies in the vicinity of 30 and The experiments were performed on the setup 60 kHz with clearly manifested peaks. The first peak at described in [1]. Fine-disperse aluminum powder of the a frequency of 30 kHz has a larger amplitude and low grade ASD-4 with an average grain size of 4 µm was dispersion. The half-width of the distribution function carried by the air flow through a metallic pipe into the amounts approximately to 1 kHz. The second peak with combustion zone formed above the pipe cross section a smaller amplitude has a larger half-width of 6 kHz. after the ignition of the airÐfuel mixture. The experimental technique made it possible to distribute the powder uniformly in the flow and to organize a laminar dif- 3. DISCUSSION fuse flame. The flame was conical in shape with a OF EXPERIMENTAL RESULTS height of L = 0.12 m and a base diameter of d = 0.028 m In order to explain the observed oscillations of the at the burner throat. Two plane-parallel metallic plates voltage drop across standard resistor R, we consider the of 0.16 m in height and with a width of 0.11 m were processes occurring in the electrode gap. The constant spaced 0.065 m apart and mounted along the flame axis. voltage applied to the plates obviously does not induce The plates were arranged so that there was an air gap a current in the circuit since there is no contact between between the plates and the flame. One of the plates was the plasma and the plates. The presence of an air gap grounded via a resistor R = 1 kΩ. A constant potential between the plasma and the plates also ensures the of 4 kV relative to the grounded plate was applied to the absence of conduction current in the plasma and, hence, other plate. As the flame propagated in the electrode its polarization. In this case, current may appear in the gap, a voltage drop was created across resistor R and circuit only as a result of a change in the permittivity in detected by a storage oscilloscope. The same signal was the electrode gap; i.e., oscillations are capacitive by fed to a low-frequency spectrum analyzer. nature. Considering that the flame under study is sta-

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ELECTROACOUSTIC OSCILLATIONS OF ALUMINUM OXIDE PARTICLES 505 tionary and oscillations of its size with such a frequency 4. THEORETICAL DESCRIPTION are hardly probable, we can assume that oscillations take place within the flame. We assume that a smoky plasma exhibits a certain ordering resembling the crystalline structure [8, 9] or Let us determine the main plasma parameters. In the its grains form the linear chains described in [5]. Then smoky plasma under investigation, the temperature of it becomes possible to consider a one-dimensional lin- the electronic component is close to the gas tempera- ear model, following the approach developed in [13], ture in view of a high collision rate (the pressure of the where the problem of propagation of waves in crystal ambient is 105 Pa) and the absence of conduction cur- lattices is described. rent in the plasma flow. The gas phase is formed by We assume that the system is monodisperse. Then a molecules with a high ionization potential value; con- plane wave propagating along a chain of smoke grains sequently, the ionization of these molecules is insignif- can be represented in the form icant. We can assume that free electrons appear in the gaseous phase as a result of the thermionic emission []()ω u j = Aikjaexp Ð t , (2) from the surface of Al2O3 particles. Then the number density of free electrons can be determined from the quasi-neutrality condition where uj is the displacement of the jth particle relative to its equilibrium position in the chain, A is the ampli- n = Zn , (1) tude of longitudinal oscillations, k is the wave number, e p ω is the angular frequency, a is the mean distance between particles, and t is the time. where ne is the concentration of free electrons in the gas Let us consider the interaction of the jth particle phase, np is the concentration of Al2O3 particles, and Z is the mean charge of particles in units of elementary with its nearest neighbors with numbers j Ð 1 and j + 1. charge. Substituting the experimental values of the Taking into account the screening of the surface charge of a particle by the volume charge of electrons, we can charge and number density of particles, we obtain the ϕ average value of the number density of free electrons, write the expressions for the distribution of potential which is equal to 6 × 1018 mÐ3. in the vicinity of the particle and strength E of the electric field produced by the particle in the form The Debye screening length in such a plasma medium is defined as the distance from the surface of a ϕ eZ r p Ð r particle at which the value of the Debye potential is = ------exp------, smaller than the Coulomb potential by a factor of 2.7 r D and is given by

eZ r p Ð r ez r p Ð r E = ------exp------+ ------exp------, kT Ð6 2 D rD D D ==------1.6× 10 m. r π 2 4 e ne where e is the electron charge and rp is the particle This value is close to the mean distance between parti- radius. ≈ Ð3 ≈ × Ð6 This leads to the following expression for the force cles, l n p 1.7 10 m, and is much smaller than the characteristic scale of the plasma (this is a necessary of electrostatic interaction between the jth and (j + 1)th condition for the existence of plasmas). The plasma fre- particles: × 10 Ð1 quency for the electron component is 1.4 10 s . 2 2 Z e au+ j + 1 Ð u j Clearly, electron oscillations cannot induce low-fre- F , = ------exp Ð------jj+ 1 ()2 D quency oscillations; hence, we assume that the oscilla- au+ j + 1 Ð u j (3) tions are induced by smoke grains. When charged par- 2 2 ticles are displaced relative to one another, a retrieving Z e au+ j + 1 Ð u j + ()------expÐ------. force appearing as a result of electrostatic interaction au+ j + 1 Ð u j D D leads to oscillations propagating in the condensed phase of the smoky plasma. It can easily be veriﬁed that An analogous expression can be derived for force the potential energy of interaction between particles is Fj, j Ð 1. comparable to the energy of thermal motion. Taking into account the electric origin of the interaction The resultant force acting on the jth particle is between the particles and the low-frequency nature of oscillations, we will refer to these oscillations as elec- F j = F jj, + 1 Ð F jj, Ð 1. (4) troacoustic. It is interesting to note that oscillations of particles may in turn lead to the propagation of ultra- We expand expression (3) and a similar expression sound in the gas phase. for Fj, j Ð 1 into a Taylor series; assuming that uj Ð 1, uj,

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 506 DRAGAN Ӷ uj + 1 a, we confine our analysis to the linear terms of Taking into account expression (8), we obtain the expansion. Then formula (4) is transformed to 1 ω 2 Ð1/2 µ() g()ω = ------1 Ð ------. (12) F j = 2u j Ð u j + 1 Ð u j Ð 1 , (5) π ω v 0 0 where It can be seen that the density of states exhibits an ω ω ω 2 2 explicit dependence on and turns to infinity for = 0. µ Z e r p Ð a ()2 2 = ------3 -2 exp------a ++2D 2aD . (6) If we take into account the polydisperse nature of the a D D system, the charge distribution of particles, and energy dissipation, the density of states will probably not turn The equation of motion of a particle of mass m can to infinity. However, a maximum of the frequency spec- be written in the form ω trum must be in the vicinity of 0. It should be recalled 2 that f0 = 51 kHz. Experiments show the presence of two d u j µ()peaks near 30 and 60 kHz. It should be noted in this m------= Ð 2u j Ð u j + 1 Ð u j Ð 1 . (7) dt2 connection that, in spite of quantitative agreement between the experimental results and the predictions of Substituting expression (2) into (7), we obtain the such an approximate model, a serious drawback exists dispersion equation for longitudinal waves propagating since the model predicts only one peak. along a linear chain in the condensed phase of a smoky To explain the results obtained here, we consider the plasma: possibility of formation of two peaks in the oscillation spectrum as a result of the Doppler effect. In the com- ka ωω= ± sin------, (8) bustion front, the smoky plasma moves at a velocity 0 2 ≈ close to vpl 0.6 m/s, which exceeds the wave velocity v0. For a monodisperse linear chain of particles, the µ v v ω = 2 ----. (9) phase ( ph) and the group ( g) velocities of longitudinal 0 m waves are given by

The plus and minus signs correspond to waves prop- ω sin()ka/2 v ==---- v ------, (13) agating in opposite directions. ph k 0 ka/2 In the range of long waves, when ka Ӷ 1, expression (8) assumes the form dω ka v g ==------v 0 cos------. (14) µ 1/2 dk 2 ω  ==v 0kak---- . (10) m ω ω Taking into account Eq. (8), for = 0, we obtain sin(ka/2) = 1 and cos(ka/2) = 0. Then the group velocity To estimate characteristic quantities, we will use the v plasma parameters obtained from the experiment: n = is g = 0. Consequently, a standing wave formed at a p frequency of ω = ω can be represented as the result of 2.0 × 1017 mÐ3, r = 0.06 µm, T = 3150 K, Z = 30, and 0 p summation of two counterpropagating traveling waves. ρ 3 the particle density = 3570 kg/m . In accordance with expression (13), the velocity of π Expressions (6), (9) and (10) then give these waves is vph = 2v0/ . One of these waves propa- ω gates along the plasma flow, while the other runs in the µ ≈ × Ð8 0 ≈ 8.4 10 N/m, f 0 = ------51 kHz, opposite direction. In a reference frame attached to the 2π setup, we determine the values of frequencies of per- ≈ ceived signals, v 0 0.33 m/s. v Ð1 Pay attention to the rather low velocity of wave pl α propagation (0.33 m/s), which is due to a large mass of f 1 ==f 01 + ------cos 41 kHz, v ph particles (m ≈ 10Ð18 kg) as compared to the atomic mass. v Ð1 pl α To analyze the spectral composition of oscillations, f 2 ==f 01 Ð ------cos 68 kHz, we consider the frequency (density of states) distribu- v ph tion of modes. According to [13], the density of states for a linear one-particle chain can be written in the form where α is the angle between the direction of motion of the plasma and the wave vector. In our case, this angle 1 dk is determined by the flame geometry: α = g()ω = ------. (11) πdω arctan()2L/d , where L = 0.12 m is the flame height

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ELECTROACOUSTIC OSCILLATIONS OF ALUMINUM OXIDE PARTICLES 507 f, kHz ticles can be formed in the front of a laminar disperse 100 flame. On the other hand, taking into account the assumption concerning the arrangement of particles in the flow in the form of chains, we can assume that the 80 spatial ordering of particles and their charge state are responsible for the observed effects. 2 It is interesting to note that oscillations of smoke 60 grains are macroscopic by nature; consequently, these 1 oscillations may induce acoustic oscillations of the gas 3 phase of the flame. The latter oscillations should appar- 40 ently be detected by acoustic methods.

20 ACKNOWLEDGMENTS 1017 1.5 × 1017 2.0 × 1017 2.5 × 1017 3.0 × 1017 The author expresses his sincere gratitude to –3 np, m A.V. Florko and N.I. Poletaev for fruitful discussions. Fig. 2. Dependence of the natural frequency of oscillations (1) and Doppler frequencies f1 (2) and f2 (3) on the number density of smoke grains. REFERENCES 1. A. N. Zolotko, Ya. I. Vovchuk, N. I. Poletaev, and A. V. Florko, Fiz. Goreniya Vzryva 32, 24 (1996). and d = 0.028 m is the flame diameter at the burner 2. V. I. Vishnyakov, G. S. Dragan, and S. V. Margashchuk, throat. Plasma Chemistry, Ed. by B. M. Smirnov (Énergo- Thus, the values of frequencies obtained for oscilla- atomizdat, Moscow, 1990), No. 16, p. 98. tions of the condensed phase in a smoky plasma match 3. G. S. Dragan and V. I. Vishnyakov, in Proceedings of the experimental values, which leads to the conclusion 30th EPS Conference on Controlled Fusion and that the Doppler effect can be observed in the laminar Plasma Physics (St. Petersburg, Russia, 2003); flame of a metallic powder. http://eps2003.ioffe.ru/publics/pdfs/O-1.3B-pre.pdf. Let us consider the dependence of the Doppler fre- 4. G. S. Dragan, Vestn. Odessk. Gos. Univ., Fiz.ÐMat. quencies of oscillations on the number density of Nauki 8, 163 (2003). smoke grains of aluminum oxide, shown in Fig. 2. 5. M. P. Kir’yakova and M. N. Chesnokov, Fiz. Aérodisper- Curve 1 describes the theoretical dependence of the nat- snykh Sist. 1, 126 (1969). ural frequency of oscillations in the condensed phase on 6. G. S. Dragan, A. A. Mal’gota, et al., in Proceedings of the number density of particles, while curves 2 and 3 International Scientific and Technical Meeting on Coal MGDÉS (Alma-Ata, 1982), p. 77. correspond to Doppler frequencies f1 and f2, respectively. Note that the oscillation frequency for charged 7. I. T. Yakubov and A. G. Khrapak, Sov. Technol. Rev. B smoke grains depends on their number density. As the 2, 269 (1989). value of np increases by a factor of three, the oscillation 8. J. H. Chu and Lin I, Phys. Rev. Lett. 72, 4009 (1994). frequency of particles is almost doubled. It can be seen 9. A. P. Nefedov, O. F. Petrov, V. I. Molotkov, and V. E. For- that the natural frequency of oscillations of particles tov, Pis’ma Zh. Éksp. Teor. Fiz. 72, 313 (2000) [JETP and the frequencies of detected Doppler waves are Lett. 72, 218 (2000)]. determined not only by the parameters of the con- 10. V. I. Vishnyakov, G. S. Dragan, and S. V. Margashchuk, densed phase, but also by the properties of particles, in Proceedings of III All-Union Meeting on Physics of which determine their charge state (e.g., the work func- Low-Temperature Plasma with CDP (Odessa, 1988), tion for electrons escaping from the surface of particles p. 17. to the plasma). 11. V. I. Molotkov, A. P. Nefedov, V. M. Torchinskiœ, et al., The horizontal dashed lines in the figure show for Zh. Éksp. Teor. Fiz. 116, 902 (1999) [JETP 89, 477 comparison the experimental frequency values of 30 (1999)]. and 60 kHz. It can be seen from the graphs that the 12. P. K. Shukla, Phys. Plasmas 8, 1791 (2001). experimental values of oscillation frequencies virtually 13. J. S. Blakemore, Solid State Physics, 2nd ed. (Cam- coincide to within the error in the measurements of con- bridge Univ. Press, Cambridge, 1985; Mir, Moscow, centration of smoke grains. 1988). Thus, the above analysis leads to the conclusion that an ordered structure of condensed submicrometer par- Translated by N. Wadhwa

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 508Ð514. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 576Ð583. Original Russian Text Copyright © 2004 by Kraœnov, Rastunkov. PLASMA, GASES

Generation of Even Harmonics in a Relativistic Laser Plasma of Atomic Clusters V. P. Kraœnov and V. S. Rastunkov Moscow Institute of Physics and Technology, Dolgoprudnyœ, Moscow oblast, 141700 Russia e-mail: [email protected] Received August 25, 2003

Abstract—It is shown that the irradiation of atomic clusters by a superintense femtosecond laser pulse gives rise to various harmonics of the laser field. They arise as a result of elastic collisions of free electrons with atomic ions inside the clusters in the presence of the laser filed. The yield of even harmonics whose electromagnetic field is transverse is attributed to the relativism of the motion of electrons and the consideration of their drift velocity associated with the internal ionization of atoms and atomic ions of a cluster. These harmonics are emitted in the same direction as odd harmonics. The conductivities and electromagnetic fields of the harmonics are calculated. The generation efficiency of the harmonics slowly decreases as the harmonic number increases. The generation of even harmonics ceases when the drift velocity of electrons becomes equal to zero and only the oscillation velocity of electrons is nonzero. The results can also be applied to the irradiation of solid-state targets inside a skin layer. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION polarization of the laser field in the nonrelativistic case are given by [4Ð6] Interaction between superintense femtosecond laser pulses and large clusters [1, 2] or solid-state targets (in 3ω F p|| ==------, p⊥ ------, p|| ӷ p⊥. (2) a skin layer) generates a plasma that consists of relativ- γ 3 istic electrons and multicharged atomic ions. The pro- 2 22EZ cess of multiple field-induced ionization is of tunneling or above-barrier character [3] because the Keldysh Of course, electrons may be ejected even with greater parameter γ in a superatomic field is very small: drift velocities (see formula (4) below), but with lesser probability. In the field of a titanium–sapphire laser of intensity 1019 W/cm2 and for the ionization potential of ω a multicharged atomic ion of 500 eV, we have p|| ≈ 2EZ γ = ------Ӷ 1. (1) 100 au (c = 137 au); i.e., a typical longitudinal drift F momentum is relativistic. The oscillatory motion of electrons in the filed of a Here, F and ω are the amplitude of the electric field superintense laser pulse is still more relativistic. The strength and the frequency of laser radiation, respec- relativistic momentum of the oscillatory motion is on tively, and EZ is the ionization potential of an atomic ion the order of with the charge multiplicity Z. Throughout this paper, we use the atomic system of units e = m = ប = 1. The F e pF = ----. (3) collision ionization of atomic ions is essential only in ω weak electromagnetic fields, when the electron velocity In the field of a titanium–sapphire laser of intensity is small. Cluster beams have definite advantages over 19 2 solid-state targets owing to the absence of a thin skin 10 W/cm , pF ~ 300 au; i.e., the oscillatory motion of layer and a low reflection of an electromagnetic field electrons is essentially relativistic. from the surface. Collisions between electrons and multicharged atomic ions in the presence of a laser field give rise to In the case of a linearly polarized laser field, elec- an induced emission of field harmonics due to the non- trons leave atomic ions during multiple ionization and monochromatic motion of a free electron in the laser have an essentially nonuniform angular distribution field. The nonrelativistic case for a linearly polarized with respect to the drift velocities (i.e., with respect to laser field (F/ω Ӷ c) has already been considered in the initial velocities of electrons at the moment of ion- detail by Silin [7Ð9]. In this limit, only odd harmonics ization). Indeed, the characteristic values of the initial (along the polarization vector of the laser field) are momenta of electrons along and perpendicular to the emitted. Silin also considered the case of weak relativ-

GENERATION OF EVEN HARMONICS IN A RELATIVISTIC LASER PLASMA 509 ism [10], when even harmonics are also emitted. How- the skin layer; the electron trajectory becomes similar ever, the longitudinal field of this radiation is polarized to a one-dimensional trajectory along the polarization along the wave vector of the external laser field; there- vector and closer to the surface of a solid. fore, this radiation exists only inside the plasma and is not emitted outside. A similar case in the general rela- One does not face such a problem when irradiating tivistic statement has recently been considered in [11]. atomic clusters (of course, one also does not face such a problem when irradiating atomic gases; however, In [11], the drift momenta p|| and p⊥ were small and because of the low density of gases, the harmonic yield therefore neglected compared to the oscillation is small in the case of gas targets). The cluster radius R momentum p . However, it was shown for the first time F (of about tens of angstroms) is less than the skin depth in [11] that the consideration of both oscillation and (hundreds of angstroms), so that the external electro- drift momenta leads to the generation of even harmon- magnetic filed easily passes through a whole cluster. ics, which can be experimentally observed. Indeed, the However, the oscillation amplitude of a relativistic electric-field vector of these harmonics contains a com- electron is c/ω ӷ R. Therefore, the generation of har- ponent directed along the electric field of the external monics occurs only at the moments when a relativistic laser field; i.e., the field of even harmonics is transverse and is different from zero in the wave zone outside the electron passes through a cluster during its oscillations. plasma region. According to the results of [10, 11], this In the case of large clusters, the external ionization of clusters is insignificant, so that there is not enough time component vanishes at p|| = p⊥ = 0. for a cluster to substantially expand due to the Coulomb In view of inequality (2), we assume that only the explosion during a femtosecond laser pulse. longitudinal drift momentum p|| is different from zero. To simplify the problem mathematically, we will not Accordingly, the intensity of harmonics, calculated average over the distribution of this momentum at for the motion of an electron in a cluster medium, must the moment of ionization, as it was done in Silin’s be multiplied by a small factor ωR/c Ӷ 1, which repre- works [7–9]; we just fix its value. Indeed, there is not sents a fraction of the time that a relativistic electron much difference between the dependence of the har- remains inside a cluster. Bearing this in mind, we con- monic yield on a current value of the longitudinal sider the motion of a free electron in the field of a super- momentum and on the longitudinal temperature defined intense laser wave neglecting the effects of laser pulse by formula (2). Under the tunneling ionization, the dis- focusing. The effect of the plasma medium reduces to tribution over longitudinal drift momenta formally the fact that the wave vector coincides with the Maxwell distribution [4, 12]:

2 3 ω2 ω 2 γ Ð p wp∝ exp Ð || ------. (4) k = ------3ω c

of the laser field in the medium is different from the 2. MOTION OF A RELATIVISTIC ELECTRON wave vector of a free electron in vacuum. IN A SUPERINTENSE LASER FIELD When solids are irradiated by a superintense laser The Newton equations for the motion of a relativistic field, the problem is complicated due to the fact that a electron in the field of a linearly polarized wave can be larger part of a pulse is reflected by the surface of the solved analytically (although in the implicit form) [13]. skin layer. The electric field inside the thin skin layer is Choose axis x along the propagation direction of the very small compared to the electric field of the incident wave, axis y along its polarization, and axis z along the electromagnetic wave and compared to the magnetic direction of the magnetic field. The kinematic momen- field inside the skin layer. The motion of a free relativ- tum of an electron along axis y is defined by the relation istic electron inside the skin layer is essentially different from its motion in vacuum (in the latter case, the () F ϕ electron trajectory looks like figure 8 in the case of lin- py t = p|| + ω---- cos . (5) ear polarization). In particular, in the case of vacuum, the amplitude of two-dimensional oscillations of a relativistic electron in the plane passing through the polar- Here, p|| is the drift momentum along the polarization ization vector and the wave vector of the field is on the axis and ϕ = ωt Ð kx is the phase of the electromagnetic order of c/ω. This quantity is much greater than the skin wave. The kinematic momentum of an electron along ω depth c/ p (under the standard condition of a dense axis x (when the transverse drift momentum is ω ӷ ω ω π neglected) is defined by plasma, p ), where p = 4 Ne is the plasma frequency (N is the concentration of free electrons). Thus, e 2 2 2 in the case of a solid-state target, the oscillations of an 1 F c Ð κ p ()t = ------cosϕ + p|| + ------. (6) electron are substantially distorted and damped due to x 2κω 2κ

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 510 KRAŒNOV, RASTUNKOV

C C 2 1 0

2.0 –0.1 –0.2 1.5 –0.3

1.0 (a)–0.4 (b) –0.5 0.5 –0.6 0 –0.7

C3 C0 0.5 0 0.4

–0.2 0.3 (c) (d) 0.2 –0.4 0.1

–0.6 0

C4 C5

0.25 0.20

0.20 0.15

0.15 0.10 (e) (f) 0.05 0.10 0 0.05 –0.05 0 0 0.5 1.0 1.5 2.0 2.5 3.0 0 0.5 1.0 1.5 2.0 2.5 3.0 u u

Fig. 1. Coefﬁcients (a) C1, (b) C2, (c) C3, (d) C0, (e) C4, and (f) C5 as functions of the dimensionless drift momentum u.

Here, the constant κ is given by The components of the kinematic velocities of electrons along axes y and x are equal to

2 2cκ p ()t 2 2 F v () y κ = c ++p|| ------. (7) y t = ------, 2 2 κ2 2() 2ω c ++py t (8) 2cκ p ()t v ()t = ------x , x 2 2 2 Finally, we set pz(t) = 0: there is no motion along mag- c ++κ p ()t netic ﬁeld (again, when one neglects the transverse drift y momentum). respectively. Finally, the time differential dt can be

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 GENERATION OF EVEN HARMONICS IN A RELATIVISTIC LASER PLASMA 511 expressed in terms of the differential of the phase of the The dimensionless constants u and s are deﬁned by the ﬁeld, dϕ, by the relation relations

2 2 2 2 c ++κ p ()t F p||ω ------y ϕ s ==------, u ------. (16) dt = 2 d . (9) ωκ 2ωκ F The transport cross section of relativistic elastic Equation (13) implies the following expression for the scattering of an electron by an atomic ion of charge Z at y component of the tensor of electric conductivity: small angles is defined by the Mott formula [14] (in atomic units): ϕ σ 1 ()ϕϕ y ==--- ∫djy ÐAf∫ d . (17) 4πZ2Λ F σ 0 M = ------. (10) p2()t v 2()t σ The conductivity y is a nonlinear function of the elec- Here, Λ is the Coulomb logarithm and p(t) and v(t) are tric field intensity F. the total momentum and velocity of the electron, Expanding the integrand in (17) in terms of Fourier respectively. In the limit of large velocities, the Cou- series, we obtain the following set of harmonics: lomb logarithm represents a quantum logarithm [7]. The frequency of elastic electronÐion scattering is ∞ given by σ ϕ ϕ y = ÐAC∑ n sinn Ð AC0 . (18) π 2 Λ n = 1 ν σ 4 Z Ni ei ==M Niv ------. (11) p2()t v ()t Here, the coefficients Cn of the Fourier series are defined by the following integral:

Here, Ni is the concentration of atomic ions. Multiply- 2π ing (11) by the velocity vector v of the electron, by the 1 electron concentration N , and by the time interval dt, C = ------f ()ϕ cosnϕdϕ, (19) e n πn ∫ we obtain the density of the electric current of elec- 0 trons: 2π ν 1 dj = ÐNev eidt. (12) C = ------f ()ϕϕ d . (20) 0 2π ∫ This density has components along axes x and y. Note 0 that this relation is also valid in the relativistic case (the so-called Pauli formula [13]). One can see that both odd and even harmonics of conductivity are different from zero. They are coherent to The component of current (12) along axis x gives the field of the incident electromagnetic wave. There rise to the longitudinal electric field, which, as we men- also exists a zero-order harmonic, which corresponds to tioned above, does not exist outside the plasma region. a constant electric field. Therefore, below we will concentrate only on the component of the electric current density that is directed along axis y. Substituting the expressions for the total 3. CALCULATION OF CONDUCTIVITY velocity and momentum of the electron that were AT HARMONIC FREQUENCIES obtained above into (12), we obtain The coefficients Cn in (19) and (20), which deter- ()ϕϕ djy = ÐAFf d . (13) mine the conductivity for the harmonics of the external electromagnetic field, were calculated numerically as Here, we denote functions of the dimensionless drift momentum of an electron (see (16)) π 2 Λω 4 Z Ne Ni A = ------3 - (14) ω F up= ||----. (21) F and define the function We fixed a value of the dimensionless oscillation f ()ϕ momentum of the electron ()u + cosϕ []1 + su()cosϕ + () 1/4 cos2ϕ (15) = ------. F 3/2 w = ------. (22) []()u + cosϕ 2 + su()cosϕ + ()1/4 cos2ϕ 2 ωc

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 512 KRAŒNOV, RASTUNKOV Then, the constant s, defined by relation (16), can be ductivity at the fundamental frequency, it is still suffi- expressed in terms of u and w by the relation ciently high. It slowly decreases as the drift momentum u increases. 1 s = ------. Figure 1e represents the coefficient C4 for the fourth u2 ++1/2 1/w2 harmonic. In the large, the intensity of this harmonic decreases as the harmonic number increases. As Figure 1 represents the calculated values of the coef- expected for an even harmonic, we have C4(0) = 0. ficients C with n = 0Ð5 for a typical relativistic case of n Finally, Fig. 1f represents the coefficient C5 for the w = F/ωc = 2, which corresponds to a peak intensity of conductivity at the fifth harmonic. The value of C5(0) 8 × 1018 W/cm2 of a titaniumÐsapphire laser. coincides with that obtained in [11] for the case of Figure 1a corresponds to a field generated at the fun- w = 2. The conductivity at the fifth harmonic is positive damental harmonic. For drift momenta u < 1, we have for u < 0.3 and negative for u > 0.3. C1 > 0, which corresponds to the normal (positive) con- The analysis of the results obtained allows us to ductivity of the electron current (electrons move oppo- draw the following conclusion. In a relativistic laser site to the direction of the electric field). When u > 1, plasma, not only odd but also even harmonics, as well the conductivity becomes negative (electrons move as a constant electric field along the polarization axis of along the field). As expected, the field of the fundamen- the external linearly polarized electromagnetic field, tal harmonic is the greatest among the fields of all the are efficiently generated. other harmonics. This makes it possible to determine the Joule absorption of electromagnetic energy by an atomic medium [15]. According to Fig. 1a, this absorp- 4. INTENSITY tion is determined by electrons with small drift veloci- OF RELATIVISTIC HARMONICS ties, which dominate in the expression for the absorp- The expressions for the currents obtained above can tion integrated over all drift velocities. The value of be used for determining the electromagnetic fields of the C1(0) coincides with that obtained in [11] for the case generated harmonics according to Silin’s approach [16]. of w = 2, as expected. According to (18), the Maxwell equation for the pro- In principle, an electron generated during tunneling jection of the vector potential onto the polarization axis or above-barrier ionization by an ac field may have any y of the external electromagnetic field (at the frequency value of its drift momentum. However, the probability of the nth harmonic) is expressed as of large values of the drift momentum is suppressed due ∂2 ()n 2 ()n to the exponentially small probability (4) of generation 1 A ∂ A 4π ()n 4π ()n Ð ------y ++------y ------j' = Ð------dj of such electrons and due to the small value of the coef- 2 2 2 y ∫ y c ∂t ∂x c c ficient C1 for large values of p|| (see Fig. 1a). (23) π π Figure 1b shows C as a function of the dimension- 4 σ()n 4 ϕ 2 = Ð------y F = ------ACnFnsin . less drift momentum u. According to the results of [11], c c the second harmonic along the polarization axis of the ()n field is not generated for u = 0. The probability of its Here, j' denotes the density of the electron current ≈ y generation is maximal when u 0.5 and decreases as u that is not associated with the collisions between elec- increases. The inequality C2 < 0 implies that the con- trons and atomic ions but is attributed to the electro- ductivity of the second harmonic is negative (electrons magnetic field of the generated harmonic (see below). move along the electric field vector of the electromag- The corresponding equation for the electric-field inten- netic field). Comparing Figs. 1a and 1b, we can con- sity at the harmonic frequency, clude that the intensity of the second harmonic is not much smaller than that of the fundamental harmonic. ∂ ()n However, substantial generation of the second har- ()n 1 Ay Fy = Ð------, monic occurs only for relativistic values of the drift c ∂t momentum of an electron. is obtained from (23) by differentiating with respect to The coefficient C3, which represents the amplitude time: of the third harmonic, is shown in Fig. 1c. The value of C (0) also coincides with that obtained in [11] for the () () ()n 3 ∂2F n ∂2F n ∂ j' case of w = 2, as expected. When u < 0.5, the conduc- Ð ------y - + c2------y - Ð 4π------y - tivity of the third harmonic is negative, whereas, for ∂t2 ∂x2 ∂t (24) u > 0.5, it becomes positive. = Ð4πAnωC Fncos[]()ωtkxÐ . Figure 1d corresponds to the static part of conduc- n tivity. It vanishes when u = 0 in accordance with the 2 ω2 ω 2 2 results of [11]. The static conductivity is mainly posi- Here, k = ( Ð p )/c is the square of the wave num- tive; even though its magnitude is smaller than the con- ber for the incident electromagnetic wave.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 GENERATION OF EVEN HARMONICS IN A RELATIVISTIC LASER PLASMA 513

From Newton’s second law of motion (in the nonrel- tor of 5. Moreover, in the case of clusters, one observed ativistic case, from the fact that the current generated by the generation of higher order harmonics than in the the field of a harmonic is small), we have case of a gaseous medium. There was no generation of even harmonics because the intensity of laser radiation ∂ ()n was less than 1018 W/cm2 in the experiment. The gener- j'y ()n ------= NeFy . ation of harmonics associated with the nonlinearity of ∂t the Mie oscillations (surface plasma oscillations of the Substituting this equation into (24), we find its solution electron cloud in a cluster) was insignificant in view of the small anharmonicity of the Mie oscillations. This conclusion was confirmed by numerical calculations () 4πAωnC F F n = ------n -cos[]n()ωtkxÐ . (25) for small metal clusters [18]. y ()ω2 2 n Ð 1 p The results of this work can also be applied to the ω ω irradiation of solid-state targets by superintense laser This solution is also valid in the relativistic case p > pulses, where the aforementioned phenomena occur because the field does not decay at a distance equal to a inside the skin layer. Even and odd harmonics of the cluster size. Substituting the value of the constant A laser field (from the second to the tenth) were observed from (14) into (25), we finally obtain by the authors of [19] for intensities higher than 1019 W/cm2. The generation region of the harmonics ω2ω2 Λ ()n Z p nCn []()ω corresponded to the electron concentration ranging Fy = ------cos n tkxÐ . (26) 21 23 Ð3 ()n2 Ð 1 F2 from 10 to 10 cm . The results obtained in the present study show that From (26), we obtain the following expression for the the generation of even harmonics is determined by the ratio of the harmonic intensity to the intensity of the drift velocity of electrons. During above-barrier ioniza- external electromagnetic field: tion, an electron may acquire a sufficiently high drift velocity. Of course, in a superintense laser field, this ()2 2 n 2 2 () F ω ω Λ velocity is not given by relation (2) but must be deter- η n y ZCn p n ==------2 ------. (27) mined from relativistic theory. Preliminary estimates Fcosϕ ()n2 Ð 1 F3 show that, even at intensities on the order of 1020 W/cm2, this velocity is nonrelativistic in contrast This ratio decreases as the intensity of the incident to the oscillation velocity of electrons. However, an wave increases and as the harmonic number n electron may acquire relativistic energy during a laser increases. pulse when heating a plasma. This heating is attributed Evaluating F ~ ωc for the general relativistic case, to the induced inverse bremsstrahlung of laser energy we obtain the following estimate for the generation effi- during collisions between electrons and multicharged ciency of harmonics: atomic ions, reflections from the inner surface of a cluster, elastic scattering by charged clusters, excitation of 2 2 2 surface plasma oscillations (Mie oscillations), etc. () Ze ω C nΛ η n ∝ ------p n - . (28) However, electron heating in a plasma always ()2 3ω me n Ð 1 c decreases as the kinetic energy of electrons increases because the collision rate of electrons with other Here, we recovered the charge and mass of an electron, objects decreases. The experimental results of [20] on which we set equal to unity above. The efficiency of a the irradiation of argon clusters by a superintense fem- harmonic increases as the density of the atomic tosecond laser pulse have shown that the typical elec- medium increases (whereby clusters are more efficient tron temperature is several keV. The energy spectra of than a gaseous medium) and as the laser-field frequency electrons were measured in [21, 22] under the irradia- ω decreases. The estimates obtained are also valid tion of xenon clusters by a 150-fs laser pulse with a when the plasma frequency is greater than the laser fre- peak intensity of 2 × 1016 W/cm2. The mean energy of quency. electrons was no greater than 2 keV. In spite of the fact that the drift velocity of electrons is small, it is this velocity that is responsible for the generation of even 5. CONCLUSIONS harmonics in a relativistic laser plasma. The generation of harmonics was experimentally observed by the authors of [17] on argon clusters (see ACKNOWLEDGMENTS also the survey [3]). It was shown that odd harmonics from the third to the ninth are generated on clusters This work was supported by the Russian Foundation consisting of several thousands of argon atoms; the for Basic Research (project nos. 02-02-16678 and generation efficiency is greater than that obtained with 04-02-16499), by the BRHE (project no. MO-011-0), a gaseous medium of the same average density by a fac- and by the ISTC (project no. 2155).

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REFERENCES 13. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 7th ed. (Nauka, Moscow, 1988; Pergamon 1. T. Ditmire, T. Donnelly, A. M. Rubenchik, et al., Phys. Press, Oxford, 1975). Rev. A 53, 3379 (1996). 14. V. B. Berestetskiœ, E. M. Lifshitz, and L. P. Pitaevskiœ, 2. G. Grillon, Ph. Balcou, J.-P. Chamberlet, et al., Phys. Quantum Electrodynamics, 3rd ed. (Nauka, Moscow, Rev. Lett. 89, 065005 (2002). 1989; Pergamon Press, Oxford, 1982). 3. V. P. Krainov and M. B. Smirnov, Phys. Rep. 370, 237 15. G. Ferrante, M. Zarcone, and S. A. Uryupin, Phys. Plas- (2002). mas 8, 4745 (2001). 4. P. B. Corkum, N. H. Burnett, and F. Brunel, Phys. Rev. 16. V. P. Silin, Zh. Éksp. Teor. Fiz. 47, 2254 (1964) [Sov. Lett. 62, 1259 (1989). Phys. JETP 20, 1510 (1964)]. 5. N. B. Delone and V. P. Krainov, J. Opt. Soc. Am. B 8, 17. T. D. Donnelly, T. Ditmire, K. Neumann, et al., Phys. 1207 (1991). Rev. Lett. 76, 2472 (1996). 6. V. P. Krainov, J. Phys. B: At. Mol. Opt. Phys. 36, L169 18. F. Calvayrac, P.-G. Reinhard, and E. Suraud, J. Phys. B: (2003). At. Mol. Opt. Phys. 31, 1367 (1998). 7. V. P. Silin, Kvantovaya Élektron. (Moscow) 27, 283 19. M. Tatarakis, A. Gopal, I. Watts, et al., Phys. Plasmas 9, (1999). 2244 (2002). 20. T. Auguste, P. D’Oliveœra, S. Hulin, et al., Pis’ma Zh. 8. V. P. Silin, Zh. Éksp. Teor. Fiz. 114, 864 (1998) [JETP Éksp. Teor. Fiz. 72, 54 (2000) [JETP Lett. 72, 38 87, 468 (1998)]. (2000)]. 9. V. P. Silin, Zh. Éksp. Teor. Fiz. 117, 926 (2000) [JETP 21. T. Ditmire, E. Springate, J. W. G. Tisch, et al., Phys. Rev. 90, 805 (2000)]. A 57, 369 (1998). 10. V. P. Silin, Kratk. Soobshch. Fiz. 8, 32 (1998). 22. R. A. Smith, J. W. G. Tisch, T. Ditmire, et al., Phys. Scr. 11. V. P. Krainov, Phys. Rev. E 68, 027401 (2003). 80, 35 (1999). 12. N. B. Delone and V. P. Krainov, Multiphoton Processes in Atoms, 2nd ed. (Springer, Berlin, 2000). Translated by I. Nikitin

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 515Ð526. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 584Ð597. Original Russian Text Copyright © 2004 by Vaulina, Petrov, Fortov. PLASMA, GASES

Analysis of Pair Correlation Functions for Macroscopic Particles in Dusty Plasmas: Numerical Simulation and Experiment O. S. Vaulina*, O. F. Petrov, and V. E. Fortov Institute for High Energy Densities, Joint Institute for High Temperatures, Russian Academy of Sciences, Moscow, 127412 Russia *e-mail: [email protected] Received August 29, 2003

Abstract—Pair correlation is analyzed for systems of macroscopic particles with various isotropic interaction potentials. Under certain conditions, the behavior of the pair correlation function is determined by an effective order parameter and its decrease toward inﬁnity follows an asymptotic power law. When the effective parameter is smaller than a certain critical value, the decay of pair correlation is much steeper. Experimental results concerning the form of the pair correlation function are presented for liquid-like dust structures localized in the near-electrode plasma sheath of a high-frequency capacitive discharge. An analysis of numerical and experimental results shows that melting dynamics in these systems are analogous to those characteristic of a topological phase transition. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION g(r), and B(r) represents the effects due to higher order correlations and has the form of a complicated integral Laboratory dusty plasmas are partially ionized gases of g (r , r , …, r ) if s > 2. In the popular hypernetted containing micrometer-sized dust particles, which can s 1 2 s 3 5 chain approximation, B(r) = 0, and N(r) is determined have large negative or positive charges (10 Ð10 e) and using the OrnsteinÐZernike relation [9, 10]. However, combine into quasi-steady liquid- or crystal-like struc- numerical studies have shown that the use of the hyper- tures [1Ð5]. These plasma-dust structures provide a netted chain approximation leads to unsatisfactory unique tool both for analyzing the properties of essen- results even for weakly nonideal systems [9Ð13]. Only tially nonideal plasmas and for gaining deeper under- allowance for higher order correlations (calculation of standing of self-organization of matter. Studies of the B(r) ≠ 0) ensures agreement with numerical simula- properties of nonideal dusty plasmas play an important tions [10]. Thus, determination of g(r) generally role in developing new phenomenological models of requires not only information about the binary interac- liquid-like systems. These studies are particularly tion potential, but also knowledge of the behavior of important because strong interaction between particles g (r , r , …, r ) for s > 2 or use of some approximations in liquids makes it impossible to develop an analytical s 1 2 s of these correlation functions. description of their thermodynamic characteristics based on the use of a small parameter, as done in the Unlike real ﬂuids, laboratory dusty plasmas provide theory of gases [6Ð10]. a good model for examining physical properties of non- The equilibrium properties of a liquid are compre- ideal systems, because dust particles are large enough hensively described by a set of probability density func- to be imaged, which facilitates application of direct nonintrusive diagnostic methods. Interaction between tions gs(r1, r2, …, rs) for particles at points r1, r2, …, rs. In the case of an isotropic binary interaction, the phys- dust particles in plasmas is commonly described by the ical properties of a liquid (such as pressure, density, Yukawa-type screened Coulomb potential energy density, and compressibility) are determined by | | ϕϕ ()λ a pair correlation function g(r) = g2( r1 Ð r2 ) [6Ð9]. This = c exp Ðr/ , (2) function can be expressed as follows [10]: λ ϕ where r is distance, is the screening radius, c = eZp/r gr()= exp()Ð []Ur()/T ++Nr() Br(), (1) is the Coulomb potential, and Zp is the dust-particle charge. This assumption is consistent both with measurements of forces acting between two dust particles [14] where U(r) is the potential energy of binary interaction, and with a computed structure of a screening cloud [15] T is the kinetic energy of chaotic (thermal) motion of only at relatively short distances from the particle λ particles, N(r) is determined by the functions g1(r1) and (r < 5 ). The screening weakens with increasing r, and

516 VAULINA et al. ϕ ӷ λ ϕ the asymptotic behavior of at r D is governed by the binary interaction potential (r) at rp to the particle the power law [16] temperature T. Moreover, both melting and crystallization processes (at Γ* ≈ 102Ð106) and formation of well- 2 Γ ≈ ϕ ≈ eZ a /r . ordered clusters of dust grains (at * 22Ð25) occur at p p nearly constant values of Γ* [17, 18]. One may reason- The results reported in [14Ð16] were obtained for iso- ably assume that this property holds under certain con- lated dust particles in plasmas. However, it remains ditions for potentials of more general form describing unclear how the potential of interaction between two binary interactions in many-particle systems. In the particles is modified by influence of other particles in a present study, we examine this assumption by analyz- dense dust cloud, ionization of gas both in the cloud and ing pair correlation functions and conditions for phase outside it, collisions of electrons or ions with neutrals transitions in systems characterized by various repul- in the ambient gas, and other factors. Thus, the real sive potentials. potentials of interaction between particles are not Certain physical systems exhibit topological phase known for dust grains in plasmas, and neither are they transitions between low- and high-temperature phases for many other physical systems in which interparticle [22Ð30]. Topological phase transitions are more com- interaction forces play an essential role. monly observed in low-dimensional systems. Such a Determination of the parameters responsible for the transition can be interpreted as a kind of “melting” that state of a system of interacting particles is an important eliminates the positional ordering of the low-tempera- task in the physics of nonideal dusty plasmas, as well as ture phase at T > Tm and preserves its orientational in other natural sciences. In particular, two dimension- ordering (which breaks down at T > T0 > Tm). Physi- less parameters responsible for mass transfer and phase cally, this phase transition is explained by the formation κ λ state in Yukawa dissipative systems (with = rp/ < 6) of topological defects (dislocations and disclinations) were found in [17, 18]: the effective “nonideality” in crystalline lattices. In a theory of this phenomenon parameter developed for two-dimensional systems, melting is interpreted as transformation of a crystal into an isotro- 1/2 Γ* = Γ{}()κ1 ++κκ2/2 exp()Ð pic liquid via an intermediate hexatic phase [22]. The theory was corroborated both by experimental studies of quasi-two-dimensional nonideal systems and by and the scaling parameter recent numerical simulations of extended two-dimensional many-particle systems with various binary inter- ξνÐ1 {}()κκκ2 ()π 1/2 = fr eZp 1 ++ /2 exp Ð np/ mp , action potentials [23Ð30]. In real monolayers of macroscopic particles, the topological mechanism of melting Γ 2 frequently manifests itself as topological excitations where np is the particle concentration, = (Zpe) /Trp is ν (vortices and antivortices) characteristic of finite two- the Coulomb coupling parameter, fr is the friction dimensional systems [22]. Experimental investigation Ð1/3 of topological phase transitions is a difficult task, coefficient for dust particles, and rp = np is the mean distance between particles. A numerical model was because essential qualitative changes in real systems tested against the laboratory experimental conditions in may be obscured by quasi-two-dimensional effects various dusty gas-discharge plasmas in [19Ð21]. Experi- induced by small perturbations [23Ð26]. Nevertheless, mental studies showed that dust-particle dynamics in short-range orientational ordering is observed not only these plasmas can be described in terms of the parame- in dust subsystems consisting of several (four to ten) ters Γ* and ξ. However, the parameters of the potential dust layers, but also in simulated three-dimensional liq- of interaction between particles can be determined only uid-like structures. An analysis of three-particle corre- if additional information about its form is available. lation in systems of macroscopic particles with screened interaction potential (2) has revealed the for- The behavior of the pair correlation function reflects mation of well-ordered clusters at Γ* > 25 [31]. This the phase state of a system. For example, liquid-like observation is in good agreement with numerical nonideal systems are characterized by short-range results [17]. The qualitative changes can be attributed to ordering of particles, whereas the functions g(r) used topological defects of a three-dimensional lattice, and for crystalline lattices describe long-range ordering. the transition can be associated with the existence of Numerical simulations showed that the effective Γ κ two distinct liquid phases by analogy with the topolog- parameter * of a Yukawa system (with < 6) com- ical phase transitions in two-dimensional systems. pletely determines the pair correlation function g(r) (describing both long- and short-range ordering) in the Orientational ordering is commonly analyzed in interval from Γ* < 1 to the point of crystallization into terms of boundary angular autocorrelation functions a body-centered cubic (BCC) lattice at Γ* 106. g6(t) and static correlation functions g6(r) and g(r) Thus, it was noted that the spatial correlation of parti- (see [23Ð30]). The correlation functions of two-dimen- cles in three-dimensional Yukawa systems with κ < 6 sional systems were found to exhibit universal behavior depends only on the ratio of the second derivative ϕ'' of in topological phase transitions [22]: their decay with

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ANALYSIS OF PAIR CORRELATION FUNCTIONS FOR MACROSCOPIC PARTICLES 517 κ κ increasing r follows exponential and power laws in the where a, b, 1, and 2, and n are parameters. Both high- and low-temperature phases, respectively. Thus, potential (2) and models (5) and (6) (with n = 1 in the the existence of two liquid phases must be manifested latter) are of special interest in the physics of dusty as a different variation of the correlation with distance plasmas. They can be used to allow for weaker screen- between particles. In this paper, we present numerical ing at relatively large distances between dust grains results concerning the spatial asymptotics of pair corre- [14Ð16]. These models have also been applied to lation of interacting dust grains over a wide range of an describe repulsion between atoms in covalent metals “order” parameter and examine the experimental pair [38, 39] or polymers [34, 38]. correlation functions obtained for liquid-like dust structures in the near-electrode plasma sheath of a high-fre- To analyze pair correlations in systems of particles quency capacitive discharge. with isotropic interaction potentials (4)Ð(6), three- dimensional equations of motion (3) were solved for specific values of parameters defined by analogy with 2. NUMERICAL SIMULATION those characterizing Yukawa systems: the effective order parameter 2.1. Parameters of the Numerical Analysis Correct simulation of plasma-dust particle transport Γ ()* 2 must rely on a molecular-dynamics method, in which a * = Zp e /Trp, (7) system of ordinary differential equations containing a Langevin force Fbr is solved. This force represents ran- and the scaling parameter dom impacts by molecules of the ambient gas or other random processes that underlie the relaxation of the ξω= */v , (8) kinetic temperature T of dust grains to the equilibrium fr value characterizing the energy of their stochastic motion [32Ð34]. Microscopic processes in homoge- where the frequency of collisions between macroscopic neous extended clouds of interacting macroscopic par- particles is calculated as ticles are simulated by setting periodic boundary conditions and taking into account not only the random force ω ()π 1/2 * = eZp* np/ mp (9) Fbr responsible for thermal motion, but also the forces Fint acting between pairs of particles [17Ð21]: and the effective particle charge is 2 d lk lk Ð l j ν dlk mp------= ∑Fint()l ------Ð mp fr------+ Fbr, 2 ll= k Ð l j dt lk Ð l j dt {}ϕ 1/2 j (3) Zp*eZ= pe ''/2np . (10) ∂ϕ F ()l = ÐeZ ------, int p ∂l Note that the effective particle charge does not have any particular physical meaning here. However, the use | | Γ ξ ω where l = lk Ð lj is the separation between particles, mp of (10) makes it possible to retain *, , and * as uni- ν is the particle mass, fr is the friction coefficient associ- versally applicable parameters in models with interac- ated with collisions between dust particles and ambi- tion potentials of any type. ϕ ent-gas neutrals [35, 36], and is the potential of inter- The computations were performed for 125 indepen- action between particles (the interaction energy is ϕ dent particles in the central cell, while the number of U(r) ~ eZp (r)). Computations were performed for the particles taken into account in computing binary inter- Yukawa potential with κ = 2.4 and 4.8. Correct simula- actions reached approximately 3000. The binary inter- tion of molecular dynamics was ensured by using dis- action potential was cut off at the distance L = 4l . To ӷ λ cut p cretization cells of size R [37]. In our computa- ensure that numerical results are independent of the ≈ Ð1/3 λ tions, R 5np > (12Ð24) . Additional computations number of particles and cut-off distance, we performed were performed for the following combinations of power additional computations for 512 actual particles for Γ and exponential laws frequently used to model repulsion Lcut = 7lp with * = 1.5, 17.5, 25, 49, and 92. A detailed in kinetics of interacting particles [34, 38, 39]: description of the numerical procedure can be found in [18, 21]. The value of ξ was varied between 0.04 and ϕϕ()n 3.6, i.e., within the limits characteristic of the experi- = cbrp/r , (4) mental conditions in gas-discharge plasmas. The value of Γ* was varied between 1 and 110. ϕϕ= {}aexp()Ðκ r/r + bexp()Ðκ r/r , (5) c 1 p 2 p Our computations showed that the effective parameter Γ* completely characterizes the ordering and ϕϕ{}()κ ()n = c aexp Ð 1r/rp + brp/r , (6) phase states of the simulated particle systems if the

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g 3.0 g Γ* = 107 (a)Γ* (b) 1.5 = 21 2.5 101 17 77 10 49 2.0 6 28 1.0 1.5

1.0 0.5 1.5

0.5

0123 012 r/rp r/rp

κ Γ Fig. 1. Pair correlation functions g(r/rp) for Yukawa systems ( < 6) for several values of * indicated at the curves. following empirical condition for long-range interac- also shows the maximum values of g(r) and S(q) found tion is satisfied: in [10, 40]. Our numerical study shows that spatial correlation 2πϕ'()r > ϕ''()r r . (11) of dust grains in the simulated systems is independent p p p Γ of friction (vfr) and is determined by the value of * under conditions ranging from a gaseous state (Γ* ~ 1) In the first (linear) approximation, this criterion to the point of crystallization into a BCC lattice (Γ* ≈ specifies conditions under which the force acting between two particles separated by the mean interparticle distance is greater than the force typically arising in g collisions of macroscopic particles.

2 Γ* = 77 2.2. Ordering in Dissipative Dust-Particle Systems with Various Isotropic Repulsive Potentials

Ordering in the simulated systems was analyzed by Γ* = 17.5 using the pair correlation function g(r) and the structure 1 factor S(q). Figure 1 shows the pair correlation functions obtained for Yukawa systems in a wide range of Γ*. Figure 2 compares these functions with the functions g(r) computed using various potentials subject to empirical condition (11) for two values of Γ* and two 0 12r/rp values of ξ. Figures 3 and 4 illustrate the dependence of the ﬁrst maxima g and S of the functions g(r) and S(q) Fig. 2. Comparison of g(r/rp) for several model potentials 1 1 and several values of ξ and Γ*. For Γ* = 77, the solid curve, and the corresponding distances r = d and q = d on ξ ϕ ϕ g1 S1 triangles, and circles correspond to = 0.14 and / c = Γ ξ ϕ ϕ *. Here, vertical bars represent the absolute deviations exp(Ð4.8r/rp), = 0.14 and / c = 0.1exp(Ð2.4r/rp) + ξ − ξ ϕ ϕ of these quantities for = 0.04Ð3.6 and for various exp( 4.8r/rp), and = 1.22 and / c = exp(Ð4.8r/rp) + Γ potentials satisfying (11). To compare the pair correla- 0.05rp/r, respectively. For * = 17.5, the solid curve, trian- ≠ ξ ϕ ϕ tions computed for dissipative systems (vfr 0) with gles, and circles correspond to = 1.22 and / c = − ξ ϕ ϕ solutions to reversible equations of motion for nondis- exp( 2.4r/rp), = 1.22 and / c = 0.1exp(Ð2.4r/rp) + − ξ ϕ ϕ 3 sipative Yukawa systems (vfr = 0) and with results exp( 4.8r/rp), and = 0.14 and / c = 0.05(rp/r) , respec- obtained for a one-component plasma model, Fig. 3 tively.

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102–104). Indeed, the first maximum points of the S1, g1 functions g and S corresponding to crystalline structure are characteristic of the BCC lattice: d ≈ r = q1 1 3 1/3 ≈ π 1/3 π 1/3 (/433 np) , dS1 q1 = 2 ( 2 np) , and kp = 2 (np) (see Fig. 4). Jumps in the values of the first maxima of g(r) and S(q) from 2.65 to 3.1 are observed as the normalized order parameter Γ* varies from the crystallization point Γ* ≈ 102Ð104 to the melting point Γ* ≈ jump m 2 Γ ≈ ± 106Ð107 (see Fig. 3). Thus, jumpm* 104.5( 2) can be interpreted as the point of phase transition between a liquid-like state and a BCC lattice. Γ ≈ ± Since jumpm* 104.5 ( 2%) is independent of the 1 ambient viscosity, this result is consistent with molecu- 020406080100120 lar-dynamics simulations of crystallization in Yukawa Γ* systems with zero friction [40Ð42]. The deviations of their results from Γ* ≈ 104.5 vary within ±5% and Fig. 3. First maxima of structure factor S1 (thin curve) and jumpm Γ can be attributed to difference in numerical procedures pair correlation function g1 (thick curve) vs. *: closed triangles represent g1 in the nondissipative Yukawa model (number of particles, integration step, etc.) and to (v = 0) [40]; open triangles, g in the BCC model [10]; choice of Γ* associated with either melting or crystalli- fr 1 open circles, S1 in the BCC model [10]. Vertical bars are the zation point of the system. It should be noted that absolute deviations for ξ = 0.04Ð3.6 and for various poten- Γ ≈ ± jumpm* 104.5( 2%) agrees with the theoretical results tials satisfying (11). obtained in [43], where the value of the order parameter on the phase-transition line in the BCC model was 105(±3%). (The latter value is consistent with numeri- dS1/kp, dg1/rp cal results based on various criteria for crystallization [44] and melting [45].) Note also that the form of a correlation function g(r) 1.15 satisfying condition (11) is determined by the value of Γ*. Therefore, the methods for determining the potential of interaction between particles from measurements of the structure factor based on the hypernetted chain approximation (using direct relations between g(r), 1.05 S(q), and ϕ(r) [8Ð10]) cannot be applied to the systems in question. Furthermore, the result obtained here can explain the widespread use of various phenomenological melting and crystallization criteria specifying the maximum values of correlation functions or the ratios 0.95 of their maximum and minimum values on phase-tran- 0 20406080100120 sition lines (when r ≠ 0) irrespective of the interparticle Γ* interaction potential. In particular, one can use the similarity of pair correlation functions (see Section 2.3) Fig. 4. Relative locations of the maximum of S1, dS1/kp and the numerical values of their maxima g (see Fig. 3) (thin curve), and the maximum of g1, dg1/rp (thick curve), 1 versus Γ*. Dashed curves represent the maximum points of to obtain the well-known ratio of g1 to the first mini- the correlation function for BCC lattice. Vertical bars are the mum of g(r), equal to 5, on the crystallization line. absolute deviations for ξ = 0.04Ð3.6 and for various potentials satisfying (11).

2.3. Pair Correlation in Liquid-Like Particle Systems − Γ and Dust Cluster Formation 3 5. For * between 28 and 102, the behavior of h(r)/h1 is determined by the value of Γ* (see Section 2.2). Its To analyze the asymptotic decay of pair correlation decay with increasing distance follows an asymptotic with increasing distance between particles, we normal- power law. At r > rp, it can be approximated by the ized the correlation function h(r) = g(r) Ð 1 to h1 = function (see Fig. 5a) max(h(r)). Figure 5 shows the results obtained for several values of Γ*. An analysis of numerical results Γ ()≈ ()β 2.75 ()πβ()β shows that h(r) has pronounced maxima when * > hr/rp /h1 rp/r sin 2 r/rp + Ð 1 , (12)

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h/h1 h/h1 1.0 1.0

(a) (b) Γ* = 101 Γ* = 21 Γ* = 77 Γ* = 18 Γ* = 49 Γ* = 10 Γ Γ 0.5 * = 28 0.5 * = 6

0 0

–0.5 –0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1.0 0.2

Γ* = 101 0 –0.2 Γ* = 28 Γ* = 21 Γ* = 6

–0.5 –0.4 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1.3 1.9 2.5 3.1

r/rp r/rp Γ Fig. 5. Normalized pair correlation functions h(r/rp)/h1 for Yukawa systems with * indicated in the panels, approximations (12) and (13) (thick and thin curves, respectively), and (d) an enlarged fragment comparing the numerical results obtained for several values of Γ*. where β ≈ 1.07. When the effective order parameter is is a difficult task, because the high-frequency thermal smaller than Γ* ~ 21, the decrease in pair correlation is fluctuations due to stochastic motion of dust grains much steeper and can be approximated by an exponen- affect the measured value of h(r)/h1, particularly when tial (see Fig. 5b): r is large and the fluctuation-induced error is comparable to h(r)/h1 (see Fig. 5d). In the present case, one can ()≈ {}()β hardly use a more suitable function to approximate hr/rp /h1 exp 2.75 Ð r/rp (13) numerical data for practical applications, because the × ()πβ()β sin 2 r/rp + Ð 1 . dust-subsystem correlation functions measured in actual experiments exhibit stochastic variations due Note that approximation of the pair correlation function both to fluctuations in the ambient plasma and to instru-

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h/h1 g 0.2 2.0

Γ* = 29 Γ* = 38 Γ* = 22 * 0 Γ = 24 1.5

Γ* = 17 Γ* = 101 –0.2 Γ* = 28 Γ* = 21 Γ* = 6 1.0

–0.4 1.3 1.9 2.5 3.1 r/rp 0.5 Γ Fig. 6. Functions h(r/rp)/h1 for Yukawa systems with * 0.8 1.0 1.3 1.5 indicated in the panels and approximations (14) and (15) r/rp (thick and thin curves, respectively). Shift between curves corresponding to different Γ* is due to change in the loca- Fig. 7. Maxima of pair correlation functions for Γ* ≈ 17Ð40. Γ tion of the maximum of g1 (see Fig. 4). Thick curves correspond to * = 24 and 29. mentation noise. Nevertheless, substantial qualitative behavior of the correlation function predicted for a difference in asymptotic behavior of pair correlations is crystalline phase by simulating three-dimensional sys- found between weakly (Γ* ≤ 21) and strongly (Γ* ≥ 28) tems of interacting particles is illustrated by Fig. 5c. It nonideal systems. When Γ* ≥ 28, the decay of the pair is obvious that the decay of pair correlation is much correlation with increasing distance follows power steeper here, as compared to that characteristic of liq- law (12). When the effective parameter is smaller than uid-like systems. a certain critical value Γ* ~ 21, the decrease in pair correlation can be approximated by exponential law (13). Improvement of the accuracy of approximations of Note that this result disagrees with experimental obser- g(r) by numerical ﬁtting would not be physically justi- vations reported in [30], where g(r) was found to decay ﬁed. However, such approximations may facilitate exponentially for both hexatic and isotropic liquid computations of thermodynamic characteristics of liq- phases in a monolayer, and a power-law approximation uid-like systems determined by the pair correlation was obtained only for a crystalline phase. This dis- function g(r), such as pressure, energy density, and agreement can be explained by the essentially two- compressibility (see [9, 10]). A detailed analysis of ≤ ≤ dimensional structure of the monolayer (distinct from approximating functions for g(r) curves at 0 r rp was the three-dimensional system analyzed here). The presented in [10]. The following functions can be sug-

dgl

(‡) (b) (c)

Γ Fig. 8. Slices of three-particle correlation functions g3 obtained by numerical simulation for * = 37.5 (a), 17.5 (b), and 1.5 (c).

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CCD camera RF

Particles (‡)

Laser

Ring

Fig. 9. Experiment in high-frequency capacitive discharge. (b)

≥ gested as approximations of h(r) at r rp (see Fig. 6):

()≈ ()β 2.5 hr/rp /h1 rp/r × ()()π ()β (c) sin2 dg1 r/rp + Ð 1 + 0.1 /g1, (14) 28 ≤≤Γ* 102, Fig. 10. Images of dust-cloud particles in the near-electrode plasma sheath of discharge for (a) P = 3 Pa and W = 10 W, (b) P = 3 Pa and W = 2 W, and (c) P = 7 Pa and W = 10 W. ()≈ {}()β hr/rp /h1 exp 2.25 Ð r/rp × ()()π ()β sin2 dg1 r/rp + Ð 1 + 0.1 /g1, (15) with change in the location of its maximum g1 as a Γ 4 ≤≤Γ*21. function of * (see Fig. 4). The difference in the asymptotics of the pair corre- These functions allow for the shift in g(r) associated lation functions corresponding to different values of the

(‡)

(b)

Fig. 11. Trajectories of dust grains over the averaging time interval for pair correlation functions for (a) P = 3 Pa and W = 10 W and (b) P = 7 Pa and W = 10 W.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ANALYSIS OF PAIR CORRELATION FUNCTIONS FOR MACROSCOPIC PARTICLES 523 order parameter can be compared with the results of g numerical experiments reported in [17, 31]. In particu- Γ* = 101 lar, formation of groups (clusters) of macroscopic par- 2.5 77 (‡) ticles was observed in [17] at Γ* ≈ 22Ð24. This phe- 2.0 nomenon was accompanied by a sharp decrease in dif- 37.5 fusivity D and a shift in the location of the ﬁrst maximum of g(r) (see Fig. 4), and the system’s proper- 1.5 17.5 ties changed qualitatively at this point (at least, those analogous to properties of a solid). Figure 7 illustrates 1.0 the behavior of the maximum of g(r) at Γ* ≈ 22Ð24 for three-dimensional Yukawa systems. The critical value 0.5 Γ* ≈ 23.5 corresponds to the condition of one particle per sphere of WignerÐSeitz radius 0 1.5 2.5 3.0 r/rp h/h ()π Ð1/3 1 aWS = 4 np/3 1.0 (b) (g(r) = 0 at r < a ), and the mean free path WS 0.5

≈ ()ω 2 1/2 rpÐp 3T/ * mp 0 for particleÐparticle collisions is close to a . WS –0.5 Subsequently, the formation of well-ordered clusters of macroscopic particles at Γ* > 22Ð24 was revealed in numerical simulations and laboratory –1.0 0.5 1.0 1.5 2.0 2.5 3.0 r/rp experiments by analyzing three-particle correlation functions [31]. Figure 8 shows slices of three-particle h/h1 | | 1.0 correlation functions g3(r12, r23, r31) (rij = ri Ð rj ) at Γ r12 = dg1 computed for several values of *. To facilitate (c) comparison, the slices are normalized to the maximum 0.5 of g3(r12, r23, r31) (black and white areas correspond to g3 = 1 and 0, respectively). 0 The formation of well-ordered dust clusters can be interpreted as the onset of orientational (short-range) ordering in the systems in question with increasing –0.5 order parameter (see Fig. 8a). The observed qualitative changes can be attributed to the topological defects of –1.0 three-dimensional crystalline structure responsible for 0.5 1.0 1.5 2.0 3.0 r/rp the existence of two distinct (isotropic and orientation- ally ordered) liquid-like phases (by analogy with topo- Fig. 12. Measured g(r/rp) (a) and h(r/rp)/h1 (b, c) for P = logical phase transitions in two-dimensional systems). 3 Pa and W = 10 W (diamonds), P = 3 Pa and W = 2 W (cir- Therefore, with increasing Γ*, the simulated system cles), and P = 7 Pa and W = 10 W (triangles). Solid curves Γ should exhibit a solid-like dynamical behavior that can are (a) g(r/rp) computed for several * and (b, c) approxi- be described by the jump model developed for molecu- mations (12) and (13) (thick and thin curves, respectively). lar liquids [6]. In this model, a liquid-phase molecule resides in an equilibrium position (site) until its energy becomes sufﬁciently large for the molecule to break entropy, rates of variation of autocorrelation func- free from the potential bonding with adjacent mole- tions, etc.) must be determined as functions of the cules and jump into a new site surrounded by different order parameter. molecules. Agreement of the dynamics of Yukawa sys- Γ tems with predictions of this model at * > 40Ð50 is 3. EXPERIMENT supported by numerical results [17]. To elucidate the nature of the observed effects, quantitative charac- The setup used to study pair correlations of dust par- teristics of orientational ordering (mean time of spe- ticles in the near-electrode plasma sheath of a high-fre- ciﬁc orientation of macroscopic particles, topological quency capacitive discharge is schematized in Fig. 9.

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dgl

(‡) (b) (c)

Fig. 13. Slices of three-particle correlation functions g3 obtained in experiments for (a) P = 7 Pa and W = 10 W, (b) P = 3 Pa and W = 2 W, and (c) P = 3 Pa and W = 10 W.

The experiments were conducted in argon at a pressure experimental conditions, the measured three-particle of P = 2Ð10 Pa and discharge power of W ≈ 2Ð10 W. correlation functions are also consistent with those Monodisperse melamine-formaldehyde particles of obtained by numerical simulation (see Fig. 8). Figure 13 ≈ µ ρ ≈ Ð3 radius ap 1.7 m and density p 1.5 g cm were shows slices of measured three-particle correlation used as the dust component. Four to ten layers of dust functions g3(r12, r23, r31) for r12 = dg1 (most probable particles were observed. The dust cloud was sliced with interparticle distance determined as the point of maxi- a 200Ð300 µm thick HeÐNe laser sheet and its images mum for measured g(r)). A comparison of the results were recorded with a CCD camera at a framing speed presented here for several values of discharge parame- of 25 fps. Figure 10 shows fragments of dust-cloud ters reveals a short-range orientational ordering in the images obtained under different experimental condi- dust structures, which is manifested by the maxima of tions. g3(r12, r23, r31) at the vertices of the hexagonal clusters The images were processed by means of a special depicted by dashed lines in Figs. 13a and 13b. As the computer program that determined the locations and maximum of the pair correlation function increases, displacements of individual particles in dust structures. these maxima (separated by a distance r nearly equal In all cases under analysis, we observed quasi-steady to dg1) grow and additional maxima, separated by a distance r ≈ 2d , appear (see Fig. 13a). liquid-like structures. The mean distance rp between g1 dust grains in these structures varied from 260 to µ Thus, our analysis of the spatial correlation of mac- 350 m. Since the structures under analysis consisted roscopic particles suggests that the behavior of pair cor- of several layers of macroscopic particles, no large- relation functions in experimentally observed dust scale vortices analogous to those observed in [24, 25] structures should be independent of the binary interac- were observed. Figure 11 shows the trajectories of tion potential. At Γ* ≈ 22Ð24, the dust subsystem grains recorded during the averaging time interval for appears to exhibit a transformation into an isotropic liq- pair correlation functions (between 1 and 2 s). uid analogous to the topological phase transition in The images were processed to obtain correlation two-dimensional systems. This conjecture can be sup- functions g(r) and g3(r12, r23, r31) averaged over 1 to 2 s ported by experimental studies of boundary angular under constant experimental conditions. Figure 12a autocorrelation and spatial correlation functions (g6(t), shows g(r/rp) obtained for several values of P and W. g6(r)) and by measurements of topological entropy. Figure 12b compares measured functions h(r)/h1 with approximations (12) and (13) found in numerical experiments. It is clear that the measured pair correla- 4. CONCLUSIONS tion functions agree with the functions g(r) obtained for Γ ≈ Γ A numerical analysis of correlation functions is per- the simulated systems with * 17.5 and * > 37, even formed for extended three-dimensional many-particle though the actual experimental conditions differed systems with various binary interaction potentials. It is from those set in the computed homogeneous problem. shown that the form of the pair correlation function is Note that the decay of spatial correlation of macro- determined by the value of Γ* for systems character- scopic particles follows a power law when Γ* > 37 and Γ ≈ ized by various repulsive potentials. This implies that is approximately exponential when * 17.5. the form and parameters of the potentials cannot be Despite the difference between parameters of the determined by inverting g(r). Our results show that pair three-dimensional homogeneous problem and the correlation of particles with interaction potentials con-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ANALYSIS OF PAIR CORRELATION FUNCTIONS FOR MACROSCOPIC PARTICLES 525 sidered in this study can be described (and, therefore, 7. E. M. Lifshitz and L. P. Pitaevskiœ, Course of Theoretical certain thermodynamic characteristics of liquid-like Physics, Vol. 5: Statistical Physics (Nauka, Moscow, systems can be determined) without performing the 1979; Pergamon, New York, 1980). complicated calculations required to find spatial clo- 8. R. Balescu, Equilibrium and Nonequilibrium Statistical sure approximations. For a system of this kind, the pair Mechanics (Wiley, New York, 1975; Mir, Moscow, correlation function can be approximated by a function 1978). depending on two dimensionless parameters (Γ* and 9. N. K. Airawadi, Phys. Rep. 57, 241 (1980). r/rp). Furthermore, substantially different forms of this 10. S. Ichimaru, Rev. Mod. Phys. 54, 1017 (1982). function at r/rp > 1 will be obtained only for weakly 11. H. J. Raverche and R. D. Mountain, J. Chem. Phys. 57, correlated and strongly nonideal structures (when Γ* < 3987 (1972). Γ 22 and * > 28, respectively). 12. H. J. Raverche and R. D. Mountain, J. Chem. Phys. 57, The behavior of h(r)/h1 = (g(r) Ð 1)/(g1 Ð 1) for liq- 4999 (1972). uid-like systems at Γ* ≥ 28 and Γ* ≤ 22 weakly 13. S. Wang and J. A. Krumhansl, J. Chem. Phys. 56, 4287 ≥ depends on the order parameter if r rp, in which case (1972). h(r)/h1 can be approximated by the product of a har- 14. U. Konopka, G. E. Morfill, and R. Ratke, Phys. Rev. monic function with a function describing the decay of Lett. 84, 891 (2000). spatial correlation. In strongly correlated systems (at 15. J. E. Daugherty, R. K. Porteous, M. D. Kilgore, et al., Γ* ≥ 28), the asymptotic decay of pair correlation fol- J. Appl. Phys. 72, 3934 (1992). lows a power law. When the effective order parameter 16. J. E. Allen, Phys. Scr. 45, 497 (1992). Γ ≈ is smaller than a certain critical * 22, the decay of 17. O. S. Vaulina and S. V. Vladimirov, Phys. Plasmas 9, 835 the correlation function can be described by an expo- (2002). nential law. The pair correlation of macroscopic parti- 18. O. S. Vaulina, S. V. Vladimirov, O. F. Petrov, et al., Phys. cles measured in liquid-like dust structures localized in Rev. Lett. 88, 245002 (2002). the near-electrode plasma sheath of a high-frequency capacitive discharge is in good agreement with numer- 19. V. E. Fortov, O. S. Vaulina, O. F. Petrov, et al., Phys. Rev. ical results. The change in the decay of the pair correla- Lett. 90, 245005 (2003). tion function observed in both numerical and experi- 20. O. S. Vaulina, O. F. Petrov, V. E. Fortov, et al., Fiz. mental studies can be attributed to the existence of two Plazmy 29, 698 (2003) [Plasma Phys. Rep. 29, 642 distinct liquid phases: an isotropic phase and a phase (2003)]. characterized by short-range orientational ordering. 21. V. E. Fortov, O. S. Vaulina, O. F. Petrov, et al., Zh. Éksp. This conjecture is supported by observations of well- Teor. Fiz. 123, 798 (2003) [JETP 96, 704 (2003)]. ordered clusters of dust grains. 22. J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973). 23. H. M. Thomas and G. E. Morfill, Nature 379, 806 ACKNOWLEDGMENTS (1996). This work was supported, in part, by the Russian 24. G. E. Morfill, H. M. Thomas, and M. Zuzic, in Advances Foundation for Basic Research, project no. 01-02-16658, in Dusty Plasma, Ed. by P. K. Shukla, D. A. Mendis, and and by INTAS, grant no. 01-0391. We thank A.V. Cher- T. Desai (World Sci., Singapore, 1997), p. 99. nyshev, A.V. Gavrikov, and I.A. Shakhova for provid- 25. Lin I, Wen-Tan Juan, C. H. Chiang, et al., in Advances in ing experimental images. Dusty Plasma, Ed. by P. K. Shukla, D. A. Mendis, and T. Desai (World Sci., Singapore, 1997), p. 143. 26. Wen-Tan Juan, M. H. Chen, and Lin I, Phys. Rev. E 64, REFERENCES 016402 (2001). 1. J. H. Chu and Lin I, Phys. Rev. Lett. 72, 4009 (1994). 27. K. Bagchi, Y. C. Andersen, and W. Swope, Phys. Rev. Lett. 76, 255 (1996). 2. H. Thomas, G. Morfill, V. Demmel, et al., Phys. Rev. 28. A. Jaster, Europhys. Lett. 42, 277 (1998). Lett. 73, 652 (1994). 29. K. Zahn and G. Maret, Phys. Rev. Lett. 85, 3656 (2000). 3. A. Melzer, T. Trottenberg, and A. Piel, Phys. Lett. A 191, 301 (1994). 30. A. H. Marcus and S. A. Rice, Phys. Rev. Lett. 77, 2577 (1996). 4. V. E. Fortov, A. P. Nefedov, V. M. Torchinskiœ, et al., 31. O. S. Vaulina, O. F. Petrov, V. E. Fortov, et al., Phys. Rev. Pis’ma Zh. Éksp. Teor. Fiz. 64, 86 (1996) [JETP Lett. 64, Lett. (in press). 92 (1996)]. 32. V. V. Zhakhovskiœ, V. I. Molotkov, A. P. Nefedov, et al., 5. A. M. Lipaev, V. I. Molotkov, A. P. Nefedov, et al., Zh. Pis’ma Zh. Éksp. Teor. Fiz. 66, 392 (1997) [JETP Lett. Éksp. Teor. Fiz. 112, 2030 (1997) [JETP 85, 1110 66, 419 (1997)]. (1997)]. 33. O. S. Vaulina, A. P. Nefedov, O. F. Petrov, et al., Zh. 6. Ya. I. Frenkel’, Kinetic Theory of Liquid (Nauka, Lenin- Éksp. Teor. 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34. A. A. Ovchinnikov, S. F. Timashev, and A. A. Belyœ, 40. M. O. Robbins, K. Kremer, and G. S. Grest, J. Chem. Kinetics of Diffusely Controlled Processes (Khimiya, Phys. 88, 3286 (1988). Moscow, 1986). 41. E. J. Meijer and D. Frenkel, J. Chem. Phys. 94, 2269 35. E. M. Lifshitz and L. P. Pitaevskiœ, Physical Kinetics (1991). (Nauka, Moscow, 1979; Pergamon Press, Oxford, 1981). 42. S. Hamaguchi, R. T. Farouki, and D. H. E. Dubin, Phys. 36. N. A. Fuchs, The Mechanics of Aerosols (Dover, New Rev. E 56, 4671 (1997). York, 1964). 43. W. L. Slattery, G. D. Doollen, and H. E. DeWitt, Phys. 37. R. T. Farouki and S. Hamaguchi, Phys. Lett. 61, 2973 Rev. A 21, 2087 (1980). (1992). 44. E. L. Plollock and J. P. Hansen, Phys. Rev. A 8, 3110 (1973). 38. Molecular Interactions from Diatomic Molecules to Biopolymers, Ed. by B. Pulman (Mir, Moscow, 1981). 45. H. M. van Horn, Phys. Lett. A 28A, 707 (1969). 39. I. M. Torrens, Interatomic Potentials (Academic, New York, 1972). Translated by A. Betev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 527Ð537. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 598Ð608. Original Russian Text Copyright © 2004 by Gorbunov, Frolov. PLASMA, GASES

Electromagnetic Radiation at Twice the Plasma Frequency Emitted from the Region of Interaction of Two Short Laser Pulses in a Rarefied Plasma L. M. Gorbunova and A. A. Frolovb aLebedev Physical Institute, Russian Academy of Sciences, Leninskiœ pr. 53, Moscow, 119991 Russia bInstitute for High Energy Densities of Associated Institute for High Temperatures, Russian Academy of Sciences, Izhorskaya ul. 13/19, Moscow, 125412 Russia e-mail: [email protected] Received September 12, 2003

Abstract—We study the electromagnetic radiation at twice the plasma frequency, which emerges because of the interaction of two identical counterpropagating short laser pulses in a rareﬁed plasma and caused by excitation of small-scale standing plasma waves in the pulse overlap region. The energy, spectral, and angular characteristics of radiation are investigated, and the dependence of these characteristics on the parameters of the laser pulses is analyzed. The possibility of applying this effect for diagnostics of localized plasma oscillations is discussed. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION tons) originating from a large number of elementary In our previous publication [1], we considered the processes and depend on the properties of plasma possibility of “impact” excitation of localized coherent waves (their spatial structure, damping, degree of small-scale plasma oscillations in a rarefied plasma by coherence, and method of excitation). counterpropagating short laser pulses. These oscilla- In this paper, we consider the radiation at twice the tions could become an attractive object for studying plasma frequency resulting from localized small-scale both laser pulses and the processes determining the standing plasma waves excited in a plasma by the inter- evolution of plasma oscillations (destruction of oscilla- action of two short laser pulses. It was shown in [1] that tion coherence, wave breaking, and the development of the collision of such pulses in the plasma gives rise to a Langmuir turbulence). Here, we consider a possible short-lived standing electromagnetic wave in the region method for diagnostics of localized plasma waves from where these pulses overlap, producing ponderomotive radiation produced by these waves in the vicinity of a forces, which vary periodically in space with a wave ω twofold plasma frequency 2 p. number of 2k0, where k0 is the wave number of laser ω Radiation at a frequency of 2 p has been widely dis- radiation. Under the action of these forces, small-scale cussed starting from 1950s in connection with solar coherent plasma perturbations are formed. The time flares [2]. A possible physical mechanism responsible evolution of these perturbations depends on the dura- for this radiation, which is now commonly referred to tion of the pulses. If the pulse duration τ is longer than as coalescence of two plasmons, was mentioned for the ωÐ1 the reciprocal plasma frequency p , these perturba- first time in [3]. Since the end of the 1960s, this effect tions exist only during the time of interaction of the has attracted the attention of researchers in connection pulses and disappear as the pulses move apart. If the with general problems of strong Langmuir turbulence pulse duration is on the order of or smaller than the (see, for example, [4]). Some of the observed singular- plasma period, such plasma oscillations persist even ities in the spectrum of radiation reflected from an inho- after termination of the interaction between the pulses mogeneous laser plasma in the region of the twofold ω in the form of a localized coherent plasma wave. It will laser frequency 2 0 were also attributed to two-plas- be shown below that, as a result of such an impact exci- mon fusion [5]. tation, the electromagnetic radiation at twice the The effect considered here is also based on elemen- plasma frequency can be emitted from the region of tary nonlinear coalescence of two plasmons accompa- localization of plasma oscillations. The intensity of this nied by the generation of photons. The actually radiation and its directivity diagram are essentially observed characteristics of the radiation at twice the determined by the parameters of interacting pulses and plasma frequency (such as intensity, directivity dia- carry information both on the pulses propagating in the gram, polarization, and linewidth) are determined by plasma and on the Langmuir waves remaining in the the superposition of the fields of transverse waves (pho- region of their interaction.

528 GORBUNOV, FROLOV It should be noted that the effect of radiation at twice definition (A.23) of the high-density potential and sep- ± the plasma frequency from a localized standing Lang- arating the terms proportional to exp( 2ik0z), we obtain muir wave considered here is analogous in a certain sense to the effect of generation of second harmonic e2 radiation from counterpropagating surface polaritons, φ()r, t = ------{exp()2ik z ()E ⋅ E* which is known in solid-state physics [6]. ω2 0 + Ð 4m 0 (2.3) ()()}⋅ +2exp Ð ik0z EÐ EÐ* . 2. EXCITATION OF PLASMA OSCILLATIONS DURING THE INTERACTION OF LASER PULSES We assume that the electric field amplitude of the In our previous publication [1], we studied the exci- laser pulses in the region of their interaction has an axi- tation of small-scale plasma oscillations during the ally symmetric Gaussian form, interaction of two identical short laser pulses in the hydrodynamic nondissipative approximation. The infi- ξ2 ρ2 (),  nitely long lifetime of oscillations emerging in this E+ r t = e+E0L expÐ ------Ð ------, 2L2 2R2 model leads to a singularity in the spectral density of (2.4) the energy radiated. To avoid this difficulty, we must η2 ρ2 (),  take into account the damping of plasma oscillations, EÐ r t = eÐE0L expÐ ------Ð ------, which is possible only in the framework of the kinetic 2L2 2R2 theory. For this reason, in this section, as well as in the Appendix, we will develop a kinetic approach to ξ η where = z Ð Vgt and = z + Vgt are the longitudinal describing the interaction of short laser pulses. coordinates in the systems comoving with the pulses, 2 ω We consider two identical laser pulses propagating Vg = c k0/ 0 is the group velocity of the pulses, L is the in a plasma towards each other along the z axis. We length of a pulse connected with its duration via the write the electric field EL of the pulses in the form τ ρ 2 2 relation = L/Vg, = x + y is the transverse coordinate, R is the pulse width, E0L is the maximal value of (), 1 EL r t = --- the electric field amplitude, and e+ and eÐ are the vectors 2 (2.1) determining the polarization of the laser pulses. × {}E ()r, t exp()Ðiω t + E*()r, t exp()iω t . 0 0 0 0 The Fourier component of the electric field of plasma perturbations can be expressed in terms of the ω Here, 0 is the laser frequency and E0 is the complex corresponding component of the ponderomotive poten- amplitude varying slowly with time on the ωÐ1 scale tial with the help of formula (A.27) (see Appendix). In 0 accordance with formula (2.4), the latter component and exhibiting the coordinate dependence has the form (), (), () E0 r t = E+ r t exp ik0z (2.2) mV2 π2 R2 Lτ()e ⋅ e + E ()r, t exp()Ðik z , φω(), k = ------E + Ð Ð 0 4 where k0 is the longitudinal wave number connected with frequency ω via the dispersion relation k c = ω2τ2 2 2 0 0 × k⊥ R expÐ ------Ð ------(2.5) ω2 ω2 ω π 2 4 4 0 Ð p , p = 4 e N0e/m being the plasma fre- ω quency, which is assumed to be smaller than 0; e, m, 2 2 2 2 and N0e are the charge, mass, and concentration of elec- ()k Ð 2k L ()k + 2k L × exp Ð------z 0 +,exp Ð------z 0 trons in the plasma; and E±(r, t) are the amplitudes of 4 4 laser pulses propagating from left to right (plus sign) and from right to left (minus sign), which change in ω Ð1 where VE = eE0L/m 0. space insignificantly over a scale of k0 . Bearing in mind that we are dealing with short laser In the course of interaction, the pulses generate a τ ωÐ1 standing electromagnetic wave associated with quasi- pulses of duration on the order of p , we will study static small-scale ponderomotive forces. These forces plasma oscillations remaining in the overlap region of induce periodic electron density perturbations (with a the pulses after termination of their interaction. Since wave number of 2k0) and small-scale quasi-static elec- ponderomotive forces in this case are equal to zero, tric fields in the plasma. Substituting formula (2.2) into only free plasma oscillations can exist in the plasma;

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ELECTROMAGNETIC RADIATION AT TWICE THE PLASMA FREQUENCY 529 the frequency ω and the wave vector k of these oscilla- small as compared to the pulse sizes, we will hence- tions are connected via the dispersion relation [7] forth assume that k = 2k0 in formulas (2.9).

εl()ω, k ==1 +0.δεl()ω, k (2.6) 3. ELECTRIC FIELD OF LOW-FREQUENCY RADIATION If the thermal velocity of electrons, VT = T/m (T is the electron temperature), is smaller than the phase veloc- The interaction between plasma waves induces a ity (ω/k) of plasma waves for the Maxwellian distribu- nonlinear current. In the approximation quadratic in the tion function F, expression (A.28) leads to the follow- wave fields, the Fourier component of current density j2 ing relation [7]: in the general form can be written as [8] iω dω'dk' ω 2 2 2 j ()ω, k = Ð------l p k V T i π∫ 4 δε ()ε, k = Ð------13+ ------8 ()2π (3.1) ω2 ω2 × ()ω,,ω , ()ω , ()ω , (2.7) Sijs k ' k' E j '' k'' Es ' k' , π ωω2 ω2 p ω ω ω ω 〈 ω 〉 + i ------expÐ------. where '' = Ð ', k'' = k Ð k', and E( , k) = E2( , k) . 2 3 3 2 2 k V T 2k V T In the case we are interested in, when the phase velocities of interacting waves are larger than the thermal ω ω ω Bearing in mind that the ratio of the thermal velocity of velocity of electrons ( > kVT, ' > k'VT, '' > k''VT), ω , ω electrons to the phase velocity of a plasma wave is tensor Sijs( , k ', k') has the form [8] small (in our case, this corresponds to the inequality 2V /c < ω /ω ), we can represent the Fourier compo- π 3 T p 0 ()ω,,ω , 4 ie N0e nent (A.27) of plasma waves in the form Sijs k ' k' = Ð------m2ωω'ω'' (3.2) 〈〉()ω, ik φω()ω, k k'' k' E2 k = Ð------L k L × iδ j δ s δ 2e ω---- js ++.ω------is ω----- ij (2.8) '' ' 1 1 × ------Ð ------, ωωÐ + iγ ωω++iγ Using formula (2.8), we obtain the nonlinear current (3.1) L L L L of a standing plasma wave: where 2 2 ω dω'dk'ω φω()φω, k' (), k'' j()ω, k = ------p -∫------L L L 3 2 2 32πem 4 ω'ω'' ω ()k = ω 1 + ---k r , ()2π L p2 D (2.9) ω k k' 2 k'' 2 π P 3 1 × ()⋅ γ ()k = ------exp Ð --- Ð ------, ω---- k' k'' ++ω------k'' -----ω k' L 8 3 3 2 2 2 '' ' k rD 2k rD (3.3) × 1 1 and r = V /ω is the Debye radius for electrons. ω------ω γ - Ð ------ω ω γ D T p ' Ð L + i L ' ++L i L In contrast to the hydrodynamic result [1], expression (2.8) takes into account the thermal correction in × 1 1 the dispersion relation for Langmuir waves as well as ω------ω γ - Ð ------ω ω γ . Landau damping. Although both these effects are weak '' Ð L + i L '' ++L i L in the given approximation, consistent inclusion of damping plays a fundamental role, limiting the lifetime Current (3.3) is proportional to the squared intensity of of excited plasma waves and removing the singularity the plasma wave field or the fourth power of the laser in the frequency spectrum of the radiation at twice the field. plasma frequency. The radiation field is generated only by the vortex Formula (2.8) combined with relation (2.5) defines part of current (3.3). Separating this part, we obtain the electric field of a small-scale standing plasma wave from the Maxwell equations the spectral density of the excited by the pulses in their overlap region. The life- vortex electric field Etr(ω, r), time of such a wave is characterized by decrement γ tr L(2k0), and the range of wave vectors in the transverse E ()ω, r and longitudinal directions is characterized, respec- dk exp()ikr⋅ (3.4) tively, by the transverse and longitudinal sizes R and L = Ð4πiωcurlcurl∫------j()ω, k , of the pulses. Considering that the laser wavelength is ()2π 3 k2T()ω, k

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ω ω2ε ω 2 2 ε ω ω2 ω2 where T( , k) = ( ) Ð c k and ( ) = 1 Ð p / is the spectral density of the electric field of low-fre- the transverse permittivity of the plasma. Relations (3.3) quency electromagnetic radiation: and (3.4) determine the low-frequency electromagnetic field induced in a rarefied plasma as a result of collision π3/2ω2τ2 ω V 3 Etr()ω, r = ------p ------E-E of two short laser pulses. ω 3 0L 82 0 8c Substituting relation (3.3) into (3.4), integrating 2 2 with respect to frequencies ω' and wave vectors k', and k0 R L 2 × ------()e ⋅ e εω()sin θcos θeθ discarding the small terms proportional to r + Ð ()2 2 exp Ðk0 L , we obtain ω ω2τ2 ×  εω() expi---- r Ð ------iωω π5/2m2V 4 k2 R2 Lτ2 c 8 Etr()ω, r = Ð------p ------E 0 -()e ⋅ e 2 (3.6) em + Ð ω2 2 2 θ 2 2 θ 16 2 εω()L cos + R sin  Ð ------ 2 2 2 8 ω τ ∂ c × exp Ð------curlcurle ----- 8 z∂z i()τω Ð 2ω 1+ erf ------L  ()ω ω 2τ2 2 2 2 2 8 Ð 2 L dk 1 kz L k⊥ R × ------exp Ð------× ------exp ikr⋅ Ð ------Ð ------(3.5) ω Ð22ω + iγ  ∫()π 3 2 ()ω, 8 8  L L 8 2 k T k  ()τω ω ()τω ω i + 2 L i Ð 2 L   1+ erf------2 2 1+ erf------2 2 ()ω + 2ω τ   8 ()ω Ð 2ω τ 8 L × ------exp Ð------L - Ð ------ω ω γ -expÐ------, ω ω γ ++2 L 2i L 8   Ð22 L + i L 8   where θ is the angle between the z axis and the direction i()τω + 2ω of radius vector r and eθ is the unit vector in the merid- 1+ erf ------L -  ()ω ω 2τ2  ional direction. 8 + 2 L Ð ------ω ω γ -exp Ð------, In order to find the space–time dependence of the ++2 L 2i L 8   low-frequency electric field in the wave zone, we carry out the inverse Fourier transformation in time. Taking ω ± ω into account the contributions from poles = 2 L Ð where γ 2i L, we obtain from relation (3.6)

z 3/2 2 2 3 2 2 2 2 3π ω τ ω V k R L erf()z = ------dttexp()Ð Etr()r, t = Ð------p ------p ------E-E ------0 -()e ⋅ e 2 ∫ ω 3 0L + Ð π 0 r 0 22 8c is the probability integral of the complex argument, ez  ω2 τ2 ω ×θθ γ 2r L is the unit vector in the direction of the z axis, L = sincos eθ expÐ 2 Lt Ð ----- Ð ------ 3c 2 ω 2 2 p(1 + 6k0rD ) is the plasma oscillation frequency tak- (3.7) ing into account spatial dispersion, and 3 2 2 2 2 2  Ð ---k ()L cos θ + R sin θ  8 p π ω  γ p 3 1 L = ------expÐ --- Ð ------. 88k3r3 2 8k2r2 0 D 0 D  × ω 3r()2 2 sin2 L t Ð ------12+ k0 rD . We are naturally interested in the radiation ﬁeld in 2 c the wave zone at large distances from the region of ӷ localization of plasma oscillations (r R, L). In this This relation describes a diverging spherical electro- case, the integral over wave vectors in expression (3.5) magnetic wave having a doubled plasma frequency and can be evaluated by the steepest descent method. Tak- polarized in the meridional direction. It should be noted ing into account the contribution from the pole, k = that expression (3.7) holds only at large distances from εω()ω/c , we arrive at the following expression for the region of localization of small-scale plasma oscilla-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ELECTROMAGNETIC RADIATION AT TWICE THE PLASMA FREQUENCY 531 tions (r ӷ R, L). In addition, it becomes valid only after I some time following the passage of the leading front of 7 ӷ ω τ2 2 radiation through a given point (for t Ð r/c p , ω 2 2 6 pR /c ). 5 4. ANGULAR, SPECTRAL, AND ENERGY CHARACTERISTICS 4 OF RADIATION 3 3 The energy emitted in unit interval of frequencies dω into unit solid angle do is connected with spectral 2 density (3.6) of the electric ﬁeld through the relation [9] 1 1

dW c εω()2 tr 2 ------= ------r E ()ω, r , (4.1) dωdo π2 00.5 1.0 1.5 2.0 4 x where frequency ω is regarded as positive. Fig. 1. Dependence of the dimensionless energy spectrum I of radiation emitted at angles of 45° and 135° on the dimen- ω ω γ ω Using relation (3.6), we can write the expression for sionless frequency x = /2 L for L/ L = 0.1 and for ﬁxed low-frequency radiation energy (4.1) in the form intensity and radius of laser pulses for three values of β ω τ 2 parameter = ( L ) /2, characterizing the pulse duration: β = 0.25 (1), 1 (2), and 2 (3). πω2 τ2ω4τ3 dW p ()⋅ 4 2 2ε3/2()ω ------ω - = ------e+ eÐ k0 R d do 64 ω ω ( = 2 L), formula (4.2) can be noticeably simpliﬁed and assumes the form 3 V 2 × ------E- W sin2 θθcos2 2 L πω6 τ5 2 2  dW 33 p k0 R 4 4c ------= ------()e ⋅ e dωdo 32 + Ð 2 2 ω2 2 2 θ 2 2 θ ω τ L cos + R sin 3 × exp Ð ------Ð ------εω()------(4.2) 2 2 2 2 V sin θθcos 4 c 4 × ------E- W ------2 L()ω ω 2 γ 2 4c Ð 2 L + 4 L (4.3) i()τω Ð 2ω 1+ erf ------L   2 2 ×ω2 τ2 3 2 ()2 2 θ 2 2 θ 8 ()ω Ð 2ω τ expÐ L Ð ---k p L cos + R sin × L 4 ------ω ω γ expÐ------ Ð22 L + i L 8

()τω ω 2 2 2 i + 2 L ()ω ω τ  1+ erf ------Ð 2 L ()ω ω 2τ2 Ð ------. 8 + 2 L 8  Ð ------ω ω γ -expÐ------, ++2 L 2i L 8 It follows from this formula that, for a fixed angle, both where the radiation intensity and the linewidth depend on the damping rate of plasma oscillations as well as on the τ E2 π3/2R2 L pulse duration . The reason for such dependence is that 0L the pulse duration determines not only the efficiency of W L = ------π 8 excitation of plasma waves, but also the longitudinal size of the region of their interaction. This size is asso- is the energy of a laser pulse. For weakly damped ciated with the spread in the longitudinal components ω ӷ γ plasma oscillations ( p L), expression (4.2) has a of the wave vectors of excited plasma oscillations and ω ω sharp peak in the vicinity of frequency = 2 L, corre- with the possibility that the condition for the merging of sponding to the emission of electromagnetic waves at a two plasma waves with the formation of a transverse doubled plasma frequency. In the vicinity of resonance electromagnetic wave will be fulfilled.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 532 GORBUNOV, FROLOV Figure 1 shows the dependence of the dimensionless If the pulses have identical or nearly identical longi- radiation energy spectrum, 2 2 2 Ӷ tudinal and transverse sizes (such that k p (R Ð L ) 1), 2 Eq. (4.5) leads to sin2θ = 0.5, which corresponds to dW 512ω max I = ------L the following values of the angles: ω 2 2 4 2 2 3 2 2 d do33πk R ()e ⋅ e ()V /4c cE R 0 + Ð E 0L π 3π θ = ---, ------. (4.6) max 4 4 × 3 2 2 1 exp---k p R = ------2 8 ω 2 γ Radiation is concentrated in narrow cones with angles ------Ð 1 + ------L of 45° and 135° relative to the axis along which the 2ω ω2 L L pulses propagate. This result coincides with that obtained in [10, 11] from analysis of radiation emitted 2 2 3 2 2 from solar flares in the approximation such that the ω τ ω τ ω 2 × L L 11 spectrum of Langmuir noise in the wave vector space is ------expÐ------------ω Ð 1 + ------, 2 2 2 L 4 concentrated in a small neighborhood of two opposite and rather long wave vectors. at angles of θ = 45° and 135° for three values of param- When pulses with transverse sizes much larger than ӷ eter ω2 τ2/2 characterizing the duration of pulses for their longitudinal sizes (R L) collide, the angular L directivity of low-frequency radiation essentially their fixed intensity and radius. The pulse duration depends on the parameter kpR. For narrow pulses mainly affects the efficiency of plasma wave excitation Ӷ (kpR 1), the radiation intensity has the maximal value and, hence, the radiation intensity. In addition, a rela- for angles defined by relation (4.6). If, however, pulses tively weak influence of the pulse duration on the radi- have large transverse sizes, Eq. (4.5) leads to ation linewidth is observed. The shorter the pulses, the wider the spread in the longitudinal components of the θ 4 1 π 4 1 plasma waves excited by these pulses and the larger the max = ------, Ð ------. (4.7) possible number of elementary processes of plasmon 3k p R 3k p R coalescence. It should be recalled that the coalescence In this case, radiation is emitted at small angles relative of two plasmons with opposite wave vectors into one is to the direction of pulse propagation. However, the possible only if these vectors differ in magnitude by a radiation intensity is small in accordance with (4.4). value on the order of ω /c. L If the longitudinal size of colliding pulses exceeds Integrating expression (4.3) with respect to fre- their transverse size (L ӷ R), the radiant energy attains quency, we obtain the angular distribution of the radi- ω τ Ӷ its maximal value for angles (4.6) provided that p ated energy: ω τ 1. If the opposite inequality ( p > 1) holds, Eq. (4.5) leads to ()π 3/2ω5 τ5 2 2 dW 3 p k0 R ()⋅ 4 ------= ------e+ eÐ π do 32 θ ± 4 1 max = ------ω τ. (4.8) 3 2 3 p V 2 ω × ------E- W ------p sin2 θθcos2 2 L γ (4.4) In this case, radiation is emitted in the direction perpen- 4c 2 L dicular to the direction of pulse propagation, but the value of radiant energy is exponentially small.  ×ω2 τ2 3 2 ()2 2 θ 2 2 θ Figure 2 shows the angular dependences of dimen- expÐ L Ð ---k p L cos + R sin . 4 sionless energy of the radiation at twice the plasma frequency for three values of parameter (R/L)2, character- Let us analyze the dependence of the directivity dia- izing the radii of laser pulses, for their fixed energy and gram of the low-frequency electromagnetic radiation duration: on the relation between the longitudinal and transverse 2 2 2 dW L exp()ω τ sizes of laser pulses. The equation for the optimal angle g = ------L - θ do 5 2()⋅ 4 4 max corresponding to the maximum radiation energy 24 3k pk0 e+ eÐ W L follows from relation (4.4) and has the form 3 2 2ω 2 γ 4 4m c 0 L L 2 2 4 2 8 × ------= --- sin θθcos sin θ Ð sin θ 1 + ------2 ω R max max 2 ()2 2 e L 3k p R Ð L (4.5) 2 4 3 2 22 R 2 + ------= 0. × expÐ---ω τ cos θ + ----- sin θ . 2 ()2 2 4 L 2 3k p R Ð L L

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It can be seen that the radiated energy decreases with increasing pulse radius and the directivity diagram 0 becomes narrower. These effects are due to the fact that 0.25 15 an increase in the pulse radius narrows the angular 30 spread in the wave vectors of the excited plasma waves 0.20 45 3 and reduces the number of elementary events of their 0.15 coalescence (cf. [10, 11]). In addition, in accordance 60 with results (4.7), optimal angle θ decreases with 0.10 2 max 75 increasing transverse size of a laser pulse. 0.05 1 The total energy of low-frequency electromagnetic 0 90 radiation can be determined from relation (4.4) by inte- 0.05 grating over solid angle do. For pulses with identical 105 spatial sizes L = R, integration is carried out analyti- 0.10 cally and the total energy has the form 0.15 120 0.20 135 5/2 5 5 2 2 2 3 3()π ω τ k R V 150 p 0 ()⋅ 4 E 0.25 W = ------e+ eÐ ------W L 165 20 4c2 180 (4.9) ω × p 7ω2 τ2 ------γ expÐ--- p . 2 L 4 Fig. 2. Angular dependence of the dimensionless energy g of the radiation at twice the plasma frequency for short laser ω τ α The low-frequency radiant energy (4.9) attains its max- pulses ( p = 4/3 ) for various values of parameter = imal value (R/L)2 characterizing the pulse width: α = 2 (1), 1 (2), and

3 1/2 (3). 2 2 5/22 ω 6k R 45π V W = ------0 ()e ⋅ e ------------E------p W , (4.10) max + Ð  2 γ L 5 7e 4c 2 L For example, for k0rD = 0.2, the damping decrement γ ω amounts to L = 0.1 L and the plasma oscillation fre- in the case of collision of two laser pulses with a dura- ω ω quency L is equal to 1.24 p on account of spatial dis- tion equal to persion. Let us estimate the value of energy emitted upon the ω τ 10 collision of two short laser pulses of intensity I = p = ------. (4.11) L 7 16 2 λ µ ω × 15 Ð1 10 W/cm , wavelength 0 = 0.8 m ( 0 = 2.4 10 s ), duration τ = 27 fs, and focal spot diameter 2R = 16 µm In spite of the fact that expression (4.10) for the × 19 Ð3 energy contains the product of two large parameters in a rareﬁed plasma of density N0e = 1.7 10 cm and temperature T = 20 eV. For such parameters, the energy 2 2 ӷ ω γ ӷ e k0 R 1 and p/ L 1, the energy value must be × Ð3 of the laser pulses amounts to WL = 0.96 10 J, while smaller than the energy of a small-scale standing the energy of a small-scale standing plasma wave is plasma wave [1], × Ð3 Wp = 1.4 10 WL. In this case, small-scale plasma γ 2 2 2 2 2 oscillations attenuate weakly with a decrement of L = πω τ V ω τ ω ω W = ------p ------E-W exp Ð------p . (4.12) 0.1 L, and the energy emitted at a frequency of = p 2 L  ω × 14 Ð1 2 4c 2 2 L = 1.7 10 s , which is approximately 1/15 of the laser carrier frequency, is directed at angles of 45° and ° × Ð6 × Comparing expression (4.12) with formula (4.10), we 135 and has a value of W = 2.8 10 WL = 2 ﬁnd that our treatment is valid provided that −3 × Ð9 10 Wp = 2.7 10 J. It should be emphasized once ω again that the frequency L of Langmuir waves, taking 2 2 ω V γ into account spatial dispersion, is equal to 1.24 p for k2 R2------E- Ӷ ------L . (4.13) 0 2 ω such values of plasma parameters. 4c p

Another limitation on the plasma parameters is con- 5. CONCLUSIONS nected with the condition of smallness of the Landau Radiation at twice the plasma frequency may carry γ Ӷ ω damping for small-scale plasma oscillations ( L L). information on the evolution of free small-scale local- γ ω For values of parameter k0rD < 0.2, ratio L/ L does not ized plasma oscillations (which are not sustained by exceed 0.1 and plasma oscillations are weakly damped. external sources) under the conditions of laser experi-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 534 GORBUNOV, FROLOV ments. The model considered here presumes the linear- function f(r, v, t) in the form of a power series in the ε ity of plasma waves and, in particular, their dissipation small parameter E, which is equal to the ratio of the due to Landau damping. We have considered the energy electron quiver velocity VE to the phase velocity of elec- characteristics of radiation, its spectrum, and directivity tromagnetic radiation and is proportional to the high- diagram precisely for such conditions. The discrepancy frequency electric field strength EL(r, t), between the results of measurements and calculations may indicate the importance and type of nonlinear pro- fFf= +++f …, cesses disregarded in our model of evolution of plasma 1 2 waves. For example, the effects associated with the where F is the electron distribution function in the mobility of ions (induced scattering from ions or decay absence of a laser pulse and fn is the nth order perturba- of a plasma wave into a plasma wave and an acoustic tion of the distribution function in parameter ε . wave) make the plasma wave spectrum and, hence, E directivity of radiation at twice the plasma frequency In the linear approximation (n = 1), the equation for more isotropic. A nonlinear dissipation mechanism f1 has the form affects the shape of the radiation spectrum and the radi- ∂ ∂ ∂ ation intensity. f 1 ⋅ f 1 e 1 × F ------∂ +0,v ------∂ + ----EL + ---vBL ------∂ = (A.1) In analyzing plasma waves, the linear approxima- t r m c v tion is limited to moderate intensities of laser pulses of where B is the high-frequency magnetic laser field, 15− 16 2 L 10 10 W/cm , while modern laser technology connected with the electric field via the equation makes it possible to obtain pulses with a much higher 18 20 2 ∂ intensity of 10 −10 W/cm ; for such intensities, 1 BL curlE = Ð------. plasma waves being excited may be nonlinear and may L c ∂t contain a large number of spatial harmonics, which are Keeping in mind another small parameter, ε , which is multiple of 2k0. It can be expected that the processes of T coalescence of higher order Langmuir waves will lead equal to the actual ratio of the electron thermal velocity to electromagnetic radiation not only at a doubled VT to the phase velocity of electromagnetic radiation, plasma frequency, but at higher frequencies, which are we can represent the solution to Eq. (A.1) in the form multiple of the Langmuir frequency. of a power series in the electron velocity: Nevertheless, using such high-intensity laser pulses, ∂ ∂C ∂2G it is also possible to realize the limiting case of excita- F ν i ν ν i … f 1 = ∂ν------Ð A j + i------∂ Ð i l∂------∂ + , (A.2) tion of linear plasma waves considered here. Indeed, it j x j xl x j follows from the results that the efficiency of plasma where wave excitation depends only on parameter e+ á eÐ. Plasma waves are excited most effectively for parallel l polarization vectors of laser pulses; conversely, the e (), A = ---- ∫ dt'EL r t effect of excitation of Langmuir waves disappears for m orthogonal polarizations. By changing the angle Ð∞ between the polarization vectors, it is possible to affect is the high-frequency electron velocity, the amplitudes of excited plasma waves and, hence, the t spectrum and intensity of low-frequency radiation. C = dt'Ar(), t' It should be noted that the radiation effects analo- ∫ gous to that considered here might also take place in Ð∞ other media [6] as well as in plasma for other types of is the electron displacement in the rf field, and waves. For example, doubled-frequency radiation emerging as a result of coalescence of two surface t waves was considered in [12]. G = ∫ dt'Cr(), t' . Ð∞ ACKNOWLEDGMENTS While deriving formula (A.2), we assumed that there ∞ This study was supported by the Russian Founda- was no laser pulse at a given point r for t Ð. An tion for Basic Research (project nos. 01-02-16723 and expression similar to (A.2) was discussed in [13] under 02-02-16110). the assumption that F is the Maxwell distribution function. In the approximation quadratic in ε , the equation APPENDIX E for f2 has the form Let us consider a quasi-static response of a plasma, ∂ f ∂ f e 1 ∂F quadratic in the rf field, by using the Vlasov kinetic ------2 + v ⋅ ------2 + ---- E + ---vB× ⋅ ------= ÐI, (A.3) equation [7]. We will seek the electron distribution ∂t ∂r m2 c 2 ∂v

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ELECTROMAGNETIC RADIATION AT TWICE THE PLASMA FREQUENCY 535 where the quantity I appearing on the right-hand side will also take into account the second-order terms in ε ω ∂ ∂ can be expressed in terms of functions defined by the small parameter t = (1/ 0)( / t) in the expression for linear approximation: 0 0 Dij . As a result, we find that coefficients Rj and Dij in e 1 ∂ f formula (A.5) contain terms varying slowly with fre- I = ---- E + ---vB× ⋅ ------1. (A.4) mL c L ∂v quency along with the terms varying at the doubled frequency. Accordingly, we represent distribution func- Using formula (A.2), we can write expression (A.4) in tion f2 as well as fields E2 and B2 in the form of the sum the form of rapidly and slowly varying terms, denoting the latter 〈 〉 ∂ ∂2 by angle brackets … . As a result, we obtain from F F Eq. (A.3) IR= j∂ν------+ Dij∂ν------∂ν-. (A.5) j i j ∂ ∂ Here, coefficients R and D are proportional to the ----- + v ⋅ ----- 〈〉f j ij ∂t ∂r 2 square of the rf field and are power functions of the (A.13) electron velocity vector component: e 1 ∂F + ---- 〈〉E + ---vB× 〈〉⋅ ------= Ð 〈〉I , 0 ν … m2 c 2 ∂v R j = R j ++lSlj , (A.6) 0 ν ν ν … where, in accordance with Eq. (A.5), we have Dij = Dij ++lQlij l mPlmij + , (A.7) ∂ ∂2 where 〈〉 〈〉F 〈〉 F I = R j ∂ν------+ Dij ∂ν------∂ν-. (A.14) ∂ ∂ j i j 0 Ai Ci R j = ------∂ ------∂ , (A.8) t x j The coefficients in this formula have the form 〈〉 〈〉ν0 〈〉… ∂A ∂A ∂C ∂A ∂ ∂G ∂G R j = R j ++l Slj , (A.15) S = ------i Ð ------l ------i Ð ------i ------i + ------l , (A.9) lj ∂x ∂x ∂x ∂t ∂x ∂x ∂x l i j j l i 2 ∂ 2 0 e E 〈〉R = Ð------0 - 0 1 ∂ j 2 2 ∂ D = Ð------()A A , (A.10) 4m ω x j ij 2∂t i j 0 (A.16) 2 2 i ∂ E* ∂ E 1∂ ∂ 1 ∂ +2------E ------0-i Ð E*------0-i , Q = --- A ------+ A ------A Ð ------()A A ω 0i∂ ∂ 0i∂ ∂ lij i∂ j∂ l ∂ i j 0 t x j t x j 2 x j xi 2 xl (A.11) ∂ ∂ 2 1Ai ∂ A j ∂ ie  ∂ ∂E* ∂E + ------+ ------C , 〈〉 ------------0l *------0l ∂ ∂ ∂ ∂ l Slj = Ð 3 E0i Ð E0i 2 t x j t xi 2ω ∂x ∂x ∂x 4m 0  j i i ∂ ∂ 1Ai ∂ A j ∂ ∂ ∂ ∂ E0*i E0i i Plmij = ------+ ------Cm ∂ ∂ ∂ ∂ + ------E0i------Ð E0*i------+ ------2 xl x j xl xi ∂ ∂ ∂ ω xl x j x j 0 ∂A ∂ ∂A ∂ ∂G 2 1i j m ∂ E ∂E* ∂E* Ð ---------∂ ------∂ + ------∂ ∂------∂ (A.12) × 2------0-i ------0i Ð ------0l (A.17) 2 t x j t xi xl ∂ ∂ ∂ ∂ t x j xl xi ∂ ∂ 1Al ∂ Al ∂ ∂ ∂ ∂ ------E0i ∂ E0*i E0*l Ð ∂ ∂ + ∂ ∂ Cm. Ð ------Ð ------2 xi x j x j xi ∂ ∂ ∂ ∂ x j t xl xi In deriving tensor Dij, we took into account its symme- ∂2 ∂E* ∂E*  try relative to permutation of indices i and j. 0i 0l  +3E0i∂------∂ -------∂ + ------∂ + c.c. , We have so far not used the explicit expression for t x j xl xi  the laser field. Consequently, formulas (A.5)Ð(A.12) are valid for an arbitrary dependence of EL on coordi- 〈〉D = 〈〉νD0 ++〈〉νQ ν 〈〉P , (A.18) nates and time. ij ij l lij l m lmij We define the time dependence of the laser field in 2 0 ie  i ∂ the form (2.1), retaining an arbitrary coordinate depen- 〈〉D = ------------()E E* + E E* ij 2ω ω ∂t 0 j 0i 0i 0 j dence of the amplitude. Substituting formula (2.1) into 8m 0 0 the definitions of quantities A, C, and G and integrating (A.19) them with respect to time, we confine our analysis to ∂ ∂  1 ∂ E0*i E0*j the first-order terms in the derivatives of the slowly + ------E0 j------+ E0i------Ð c.c. , ω 2 ∂t ∂t ∂t  varying amplitude E0 everywhere except in (A.10). We 0

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e2  ∂ Carrying out the Fourier transformation in 〈〉Q = Ð------------()E*E + c.c. Eq. (A.22) in coordinates and time, we obtain the fol- lij 2 2 ∂ 0i 0 j ω  xl 〈 ω 〉 8m 0 lowing expression for f2( , k, v) : ∂ ∂ i ∂ E0*l E0*l 〈〉()ω,, 1 (A.20) f 2 kv = ------+ ω------∂-----E0i------∂ + E0 j------∂ Ð c.c. i()ω Ð kv⋅ 0 t x j xi  i ∂ ∂E* ∂E*  × e 〈〉1 × 〈〉 1 φω(), 0i 0 j  Ð ----E2 + ---vB2 + ----ik k (A.24) + ω------∂------E0 j------∂ + E0i------∂ - Ð c.c. , m c m 0 xl t t  ∂F 1 ∂2F 2  ∂ ∂ × ------+ ----φ ()ω, k ------. ie ∂ E0m E0m ∂ ij ∂ν ∂ν 〈〉P = ------------E*------+ E* ------Ð c.c. v m i j lmij 2 3 ∂ 0i ∂ 0 j ∂ 8m ω  xl x j xi 0 Using this expression, we find the Fourier component of the electron current density, ∂3 ∂3 ∂2 ∂ i  E0*m E0*m E0i E0*m + ω------Ð33E0i------∂ ∂ ∂ - Ð E0 j∂------∂ ∂ + ∂------∂ ------∂ - 〈〉j ()ω, k = e dvν 〈〉f ()ω,,kv 0 t xl x j t xl xi t xl x j i ∫ i 2 2 2 2 (A.25) ∂ E ∂E* ∂E ∂ E* ∂E ∂ E* (A.21) iω i 0 j 0m 0l 0m 0l 0m = Ð------δε 〈〉E Ð -- k φ , + ------∂ ∂ ------∂ - ++------∂ ------∂ ∂ ------∂ ------∂ ∂ - 4π ij2 j e j t xl xi xi t x j x j t xi where we have assumed that distribution function F is ∂ ∂2 ∂ ∂  E0i E0*m E0 j E0*m  isotropic and introduced the standard notation [7] Ð2------∂ ------∂ ∂ - Ð2------∂ -∂------∂ + c.c. . xl t x j xl t xi  π 2 ν ∂ δε ()ω, 4 e i F ij k = ------ω ∫dvω------⋅ ------∂ν-. (A.26) Expressions (A.13)Ð(A.21) allow us to consider var- m Ð kv j ious effects to different degrees of accuracy in small It should be noted that the rf field is taken into account ε parameters T (smallness of the thermal velocity of par- in Eq. (A.22) via the two terms on the right-hand side. ε ticles as compared with the phase velocity of waves) In the lowest approximation in small parameters t and and ε (smallness of the amplitude variation over a ε t T, the contribution to current density (A.25) comes period of the rf field) in the approximation quadratic in only from the term proportional to the ponderomotive the rf field. In particular, using these relations, it is pos- potential. sible to analyze the problem of so-called nonstationary ponderomotive forces, which had been considered for a Confining our analysis to potential quasi-static long time mainly in the framework of the hydrodynamic fields and using expression (A.25), we find the Fourier description of a plasma (see the review article [14]). component of the electric field strength, ε ε δεl()ω, Including the largest terms in parameters t and E in 〈〉 ikφω(), k E2 = ----- k ------, (A.27) expression (A.14), we can write Eq. (A.13) in the form e 1 + δεl()ω, k ∂ ∂ where δεl is the partial contribution of electrons to the ----- + v ⋅ ----- 〈〉f ∂t ∂r 2 longitudinal permittivity of the plasma, 2 e 1 ∂F l 4πe 1 ∂F 〈〉 × 〈〉⋅ δε ()ω, k = ------dv------k ⋅ ------. (A.28) + ----E2 + ---vB2 ------(A.22) 2 ∫  m c ∂v mk ω Ð kv⋅ ∂v 1 ∂φ ∂F 1 ∂ ∂ ∂2F It should be emphasized that the approach to = ------⋅ ------+ ------+ v ⋅ ----- φ ------, m ∂r ∂v m ∂t ∂r ij ∂ν ∂ν describing the quasi-static response of the plasma to the i j rf field developed in the Appendix can be used for ana- where lyzing a wider range of problems.

e2 φ = ------()E E* + E* E , REFERENCES ij ω2 0 j 0i 0 j 0i 8m 0 (A.23) 1. L. M. Gorbunov and A. A. Frolov, Zh. Éksp. Teor. Fiz. 2 120, 583 (2001) [JETP 93, 510 (2001)]. e 2 φδ==φ ------E . 2. V. V. Zheleznyakov, Radioemission of the Sun and Plan- ij ij ω2 0 4m 0 ets (Nauka, Moscow, 1964); Radiation Processes in Plasmas, Ed. by G. Bekeﬁ (Wiley, New York, 1966; Mir, Quantity φ is usually referred to as the rf potential. Moscow, 1971); S. A. Kaplan, S. B. Pikel’ner, and

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V. N. Tsytovich, Physics of the Solar Atmospheric cow, 1961), pp. 81, 89; E. M. Lifshitz and L. P. Pitae- Plasma (Nauka, Moscow, 1977), p. 191. vskiœ, Physical Kinetics (Nauka, Moscow, 1979; Perga- 3. V. L. Ginzburg and V. V. Zheleznyakov, Zh. Éksp. Teor. mon Press, Oxford, 1981). Fiz. 35, 699 (1958). 8. V. V. Pustovalov and V. P. Silin, Tr. Fiz. Inst. im. 4. V. N. Tsytovich, Theory of Turbulent Plasma (Atomiz- P.N. Lebedeva, Akad. Nauk SSSR 61, 109 (1972). dat, Moscow, 1971; Plenum, New York, 1974); 9. L. D. Landau and E. M. Lifshitz, The Classical Theory E. N. Kruchina, R. Z. Sagdeev, and V. D. Shapiro, of Fields, 7th ed. (Nauka, Moscow, 1982), p. 229. Pis’ma Zh. Éksp. Teor. Fiz. 32, 443 (1980) [JETP Lett. 10. I. H. Cairns, J. Plasma Phys. 38, 179 (1987). 32, 419 (1980)]. 11. A. J. Willes, P. A. Robinson, and D. B. Melrose, Phys. 5. M. V. Goldman, D. L. Newman, D. Russel, et al., Phys. Plasmas 3, 149 (1996). Plasmas 2, 1947 (1995). 12. R. Dragila and S. Vukovic, Phys. Rev. A 34, 407 (1986). 6. G. Blau, J. L. Coutaz, and R. Reinisch, Opt. Lett. 18, 13. P. Mora and R. Pellat, Phys. Fluids 22, 2408 (1979). 1352 (1993); E. V. Alieva, G. N. Zhizhin, V. A. Sychu- gov, et al., Pis’ma Zh. Éksp. Teor. Fiz. 62, 794 (1995) 14. G. W. Kentwill and D. A. Jones, Phys. Rep. 145, 320 [JETP Lett. 62, 817 (1995)]. (1987). 7. V. P. Silin and A. A. Rukhadze, Electromagnetic Proper- ties of Plasma and Plasmalike Media (Atomizdat, Mos- Translated by N. Wadhwa

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 538Ð545. From Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 609Ð617. Original English Text Copyright © 2004 by Syromyatnikov, Maleyev. SOLIDS Structure

Low-Energy Singlet Dynamics of Spin-1/2 Kagomé Heisenberg Antiferromagnets¦ A. V. Syromyatnikov and S. V. Maleyev St. Petersburg Nuclear Physics Institute, St. Petersburg, 188300 Russia e-mail: [email protected] Received June 30, 2003

1 Abstract—We suggest a new approach for description of the low-energy sector of spin---- kagomé Heisenberg 2 antiferromagnets (KAFs). We show that a kagomé lattice can be represented as a set of blocks containing 12 spins, having the form of stars and arranged in a triangular lattice. Each of these stars has two degenerate singlet 1 ground states that can be considered in terms of pseudospin --- . Taking into consideration symmetry, we show 2 that the KAF lower singlet band is created from these degenerate states by the interstar interaction. We demonstrate that this band is described by the effective Hamiltonian of a magnet in the external magnetic field. The general form of this Hamiltonian is established. The Hamiltonian parameters are calculated up to the third order of perturbation theory. The ground-state energy calculated in the model considered is lower than those evaluated numerically in the previous finite clusters studies. A way of experimental verification of this picture using neutron scattering is discussed. It is shown that the approach presented cannot be directly extended to KAFs with larger spin values. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION onalization studies of clusters with N ≤ 36 [4, 8] revealed Unusual low-temperature properties of kagomé an exponential decay of the spinÐspin and dimerÐdimer antiferromagnets (KAFs) have attracted much attention correlation functions, and therefore the point of view that from both theorists and experimenters over the last KAF is a spin liquid is widely accepted [2, 4Ð12]. decade. Apparently, the most intriguing features were A quantum dimer model (QDM) is now considered observed in specific heat C measurements of SrCrGaO the best candidate for description of low-energy KAF 3 (spin---- kagomé material) [1]. A peak at T ≈ 5 K has properties [6, 9, 13Ð15]. In QDMs proposed for the 2 kagomé problem in some recent papers [6, 13–16], the been revealed that is practically independent of mag- spin Hilbert space is restricted to the states in which netic field up to 12 T, and C appeared to be proportional spins are paired into first-neighbor singlets. The main to T2 at T Շ 5 K. argument to support this restriction is the coincidence There is no appropriate theory describing the low- of the low-energy spectrum and the number of lower energy KAF sector. Qualitative understanding of low- singlet excitations in samples with up to 36 sites with 1 the exact diagonalization results [13, 15, 16]. At the temperature spin---- KAF physics is based mostly on same time, it was noted that further studies are required 2 to analyze this problem. As was recently demonstrated results of numerous finite-cluster investigations [2Ð6]. in [15], an effective Hamiltonian describing the low- They revealed a gap separating the ground state from energy KAF singlet sector can be written in this the upper triplet levels and a band of nonmagnetic sin- approach. Unfortunately, it appears to be quite cumber- glet excitations with a very small or zero gap inside the some and makes it possible to obtain the result under a spin gap. The number of states in the band increases number of crude approximations only [14, 15]. with the number of sites N as αN. For samples with up to 36 sites, it was found that α = 1.15 and 1.18 for even In our recent paper [17], we suggested another and odd N, respectively [2, 5]. It is now believed that this 1 approach for spin---- KAF that differs from the QDMs wealth of singlets is responsible for a low-T specific heat 2 peak and explains its field independence [1, 7]. discussed above. We proposed to consider a kagomé The origin of the singlet band and the nature of the lattice as a set of stars with 12 spins arranged in a trian- ground state are still under debate. Previous exact diag- gular lattice (see Fig. 1). Numerical diagonalization has shown that a single star has two degenerate singlet ¦This article was submitted by the authors in English. ground states separated from the upper triplet levels by

LOW-ENERGY SINGLET DYNAMICS 539 a gap. These states form a singlet energy band as a result of interstar interaction. It was assumed that this band determines the low-energy KAF singlet sector. We have shown that it is described by the Hamiltonian of a magnet in the external magnetic field where degenerate states of the stars are represented in terms of two pro- 1 jections of pseudospin --- . 2 This picture possibly reflects only the lowest part of the lower singlet sector because the number of states in the band within our approach is 2N/12 ≈ 1.06N [17], whereas it is now believed that it should be scaled by the 1.15N law obtained numerically for clusters with N ≤ 36 [13, 16]. Fig. 1. Kagomé lattice (KL). There is a spin at each lattice In this, more comprehensive paper, we develop this site. The KL can be considered as a set of stars arranged in a triangular lattice. Each star contains 12 spins. A unit cell star concept. Using the symmetry considerations pre- is also presented (dark region). There are four unit cells sented in Section 2, we prove that the singlet band aris- per star. ing from the ground states of stars determines the KAF lower singlet sector. This band is studied in Section 3, where the general form of the effective Hamiltonian is Because Hamiltonian (1) commutes with all of the established. The Hamiltonian parameters are calculated projections of the total spin operator, all of the star lev- up to the third order of perturbation theory. The ground- els are classified by the values of S, irreducible repre- state energy calculated in the model considered is lower sentations (IRs) of its symmetry group, and are degen- than these energies evaluated numerically in previous erate with respect to Sz. The star symmetry group C6v finite cluster studies. A comparison between our model contains six rotations and reflections with respect to six and the QDM is carried out. We demonstrate that our lines passing through the center. There are four one- approach cannot be directly extended to KAFs with dimensional and two two-dimensional IRs, which are larger spin values. A way to verify this picture experi- presented in Appendix 1. In their bases, the matrix of mentally using neutron scattering is discussed. We the Hamiltonian has a block structure. Each block has summarize our results in Section 4. been diagonalized numerically. As a result, it was found that the star has a doubly degenerate singlet ground state separated from the 2. SYMMETRY CONSIDERATION ∆ ≈ lower triplet level by a gap 0.26J1. Ground-state 1 wave functions can be represented as We start with the Hamiltonian of the spin---- kagomé 2 Ψ 1 ()φ φ Heisenberg antiferromagnet, + = ------1 + 2 , (2) 21/16+

Ψ 1 ()φ φ Ᏼ ⋅ ⋅ Ð = ------1 Ð 2 , (3) 0 = J1 ∑ Si S j + J2 ∑ Si S j, (1) 21/16Ð 〈〉ij, ()ij, φ φ where functions 1 and 2 are shown schematically in 〈 〉 Fig. 2. The bold line there represents the singlet state of where i, j and (i, j) denote nearest and next-nearest the corresponding two spins, i.e., (|↑〉 |↓〉 Ð neighbors on the kagomé lattice, respectively, shown in i j | | Ӷ |↓〉 |↑〉 Fig. 1. The case where J2 J1 is considered in this i j )/2 . φ φ paper. We discuss the possibility of both signs of the It can be shown that 1 and 2 are not orthogonal: φ φ next-nearest-neighbor interactions, ferromagnetic and their scalar product is ( 1 2) = 1/32. They contain six antiferromagnetic. As is shown below, although the singlets, each having the energy ÐS(S + 1)J1 = Ð3/4J1. It second term in Eq. (1) is small, it can be important for can be shown that the interaction between singlets does the low-energy properties. not contribute to the energy of the ground states which A kagomé lattice can be represented as a set of stars is consequently equal to Ð4.5J1. φ φ arranged in a triangular lattice (see Fig. 1). We ﬁrst Functions 1 and 2 are invariant under rotations of neglect the interaction between stars and set J2 = 0 in the star and transform into each other under reﬂections. Ψ Eq. (1). A star is a system of 12 spins. We now consider Hence, + is invariant under all symmetry group trans- Ψ its properties in detail. formations. In contrast, function Ð is invariant under

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540 SYROMYATNIKOV, MALEYEV Ψ Ψ 11 11 products containing + or Ð for each star. It is easy to show, by using the standard procedure of constructing 9 10 12 1 9 10 12 1 bases in irreducible representations ([18, 19] and Appendix 1), that there are at least two ground-state φ φ wave functions of the cluster discussed that are trans- 1 = 8 2 2 = 8 2 formed under any given IR. 7 3 7 3 It is important to mention that the operator of the 6 4 6 4 interstar interaction in the cluster has the same symme- 5 5 try as the intrastar interaction, which is the sum of the star Hamiltonians. Moreover, the interaction between Fig. 2. Schematic representation of the two singlet ground- stars commutes with the square of the total spin opera- φ φ state wave functions 1 and 2 of a star. The bold line shows tor. Therefore, if we increase the interstar interaction |↑〉 |↓〉 the singlet state of two neighboring spins, i.e., ( i j Ð from zero, all the levels move in energy, but their clas- |↓〉 |↑〉 sification cannot be changed. Levels can cross each i j)/2 . other as the interaction rises from 0 to J1, but crossing is forbidden for levels of the same symmetry. This is a consequence of a symmetry theorem proved in [18, 19]. (a) (b) Hence, we can conclude that the lower singlet sector of the cluster is formed by states that stem from the original 128 lower levels. We can obtain the same conclusions considering clusters with the symmetry C6v with a large number of stars. Therefore, we assume in what follows that the KAF low-energy singlet sector is formed by the states originating from those in which each star is in one of Ψ Ψ ≠ the states + or Ð. Because bands with S 0 can over- Fig. 3. (a) A cluster where operator of the interaction lap the singlet bands under discussion, we have to sup- between stars has the same symmetry group C6v as the pose that the KAF low-energy properties are deter- whole cluster; (b) the only configuration of three stars giv- mined by the lowest states in this singlet band. ing a nonzero contribution to the third term in the perturba- Because the interaction between stars commutes tion expansion. with the square of the total spin operator, bands of different S can be studied independently. The KAF lower rotations, changes sign under reflections, and is trans- singlet sector is considered in detail in the next section. ≠ formed under representation (A.3). Therefore, the Investigation of states with S 0 is outside the scope of ground state has accidental degeneracy. As shown in the this paper. next section, the next-nearest-neighbor interaction, which has the same symmetry as the original Hamilto- 3. SINGLET DYNAMICS nian, removes this degeneracy. In this section, we derive the general form of the The KAF containing ᏺ noninteracting stars has an effective Hamiltonian describing the lower singlet sec- energy spectrum with a large level degeneracy when tor. The interstar interaction is considered a perturba- ᏺ ᏺ ӷ 1. For example, the ground-state degeneracy is 2 tion. Although it is not small compared to the intrastar interaction, there are reasons presented below to use ᏺ (ᏺ Ð 1) and that of the lowest triplet level is 3 2 . Interac- perturbation expansion here. tion between stars gives rise to an energy band from each such group of levels, and it is a very difficult task Two-star coupling. We begin with considering the to follow their evolution. On the other hand, group the- interaction between two nearest stars, still neglecting ory makes it possible to draw certain conclusions about the second term in Eq. (1). Initially, there is a fourfold degenerate ground state with the wave functions the KAF low-energy sector. We now show that the sin- () () {}Ψ 1 Ψ 2 (where n = +, Ð, and the upper index glet band that stemming from the ground state cannot n1 n2 i be overlapped by those originating from the upper sin- denotes the stars) and the energy glet levels. () () () E 0 ==E 0 +9E 0 Ð J . We consider a cluster with seven stars, shown in n1n2 n1 n2 1 Fig. 3a, and we begin with neglecting the interaction As can be seen from Fig. 4, the interaction has the form between them. The symmetry group of the cluster is 7 ()1 ()2 ()1 ()2 also C v. The ground state has a degeneracy of 2 = 128. ()⋅ ⋅ 6 VJ= 1 S1 S1 + S3 S3 . (4) The corresponding wave functions transformed under IRs of C6v are constructed as linear combinations of According to the standard theory [18], the following

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 LOW-ENERGY SINGLET DYNAMICS 541 conditions should be satisﬁed to consider V as a per- 11 11 turbation: 9 10 12 1 1 12 10 9

n1n2 V 12 C = n1n2; m1m2 Ӷ 1, (5) 8 2 2 8 m1m2 ------() () E 0 Ð E 0 n1n2 m1m2 7 6 4 3 3 4 6 7 () () () () where V = 〈|Ψ 1 Ψ 2 V|〉Ψ 1 Ψ 2 , m m n1n2; m1m2 n1 n2 m1 m2 1 2 5 5 denotes excited singlet levels of the two stars, and ni = () () n1n2 1 2 +, Ð. We have calculated C for n = +, Ð using wave Fig. 4. Interactions between two stars: V = J1(S1 á S1 + m1m2 i ()1 ()2 ˜ ()1 ()2 ()1 ()2 ()2 functions obtained numerically and found that all of S3 á S3 ) and V = J2(S1 á S2 + S2 á S1 + S1 á these coefficients do not exceed 0.09. Conditions (5) ()2 ()1 ()2 are, therefore, satisfied. Then, the maximum value of S3 + S3 á S2 ), where the superscripts label the stars. The system is symmetric under reflection with respect to the n1n2 2 the sum ∑ Cm m is 0.28, which is also suffi- dotted line. m1m2 1 2 ciently small. Thus, the interaction between stars is considered a perturbation in what follows. the operator exponent up to the power 150.1 The results can be represented as We proceed with calculations of corrections to the initial ground-state energy of two stars. For this, because the state is fourfold degenerate, we must solve H++; ++ = Ð a1 + a2 Ð a3, (8) a secular equation [18]. The corresponding matrix elements in the third order of perturbation theory are given H+Ð; +Ð = Ð a1 + a3, (9) by [18]

HÐ+; Ð+ = Ða1 + a3, (10) H = V n1n2; k1k2 n1n2; k1k2 H = Ð a Ð a Ð a , (11) V V ÐÐ; ÐÐ 1 2 3 n1n2; m1m2 m1m2; k1k2 + ∑ ------()0 ()0 - , En n Ð Em m (6) where a = 0.256J , a = 0.015J , and a = 0.0017J . m1 m2 1 2 1 2 1 1 2 1 3 1 The terms of the second diagonal H = H = V V V ++; Ð Ð Ð Ð; ++ n1n2; m1m2 m1m2; q1q2 q1q2; k1k2 −H = ÐH = a = Ð0.0002J are much smaller + ∑ ∑ ------, +Ð; Ð+ Ð+; + Ð 4 1 ()()0 ()0 ()()0 ()0 than a , a , and a . Therefore, the interaction shifts all , , En n Ð Em m En n Ð Eq q 1 2 3 m1 m2 q1 q2 1 2 1 2 1 2 1 2 the levels by the value Ða1 and removes their degeneracy. The constants a2, a3, and a4 determine the level where ni, ki = +, Ð. Obviously, the ﬁrst term in Eq. (6) splitting. It is seen that the splitting is very small com- is zero, and the second term can be represented as pared to the shift. Therefore, the KAF appears to be a set of two-level ∞ δ ()0 interacting systems, and the low-energy singlet sector Ð t + iEn n t H = Ðited 1 2 of the Hilbert space can naturally be represented in n1n2; k1k2 ∫ |↑〉 Ψ |↓〉 Ψ terms of pseudospins: = Ð and = +. It follows 0 (7) ()1 ()2 from Eqs. (8)Ð(11) that, in these terms, the interaction () () Ðit()Ᏼ + Ᏼ () () ×Ψ〈|1 Ψ 2 Ve 0 0 V|〉Ψ 1 Ψ 2 , between stars is described by the Hamiltonian of a fer- n1 n2 k1 k2 romagnet in the external magnetic ﬁeld,

z z y y z () Ᏼ᏶[]᏶ Ꮿ Ᏼ i = ∑ zsi s j + ysi s j + hs∑ i + , (12) where 0 are Hamiltonians of the corresponding 〈〉, stars. The third term in Eq. (6) is to be considered later. ij i Ψ Ψ Using the symmetry of the functions + and Ð dis- where 〈i, j〉 now denotes nearest-neighbor pseudospins, cussed above and the invariance of the system under arranged in a triangular lattice formed by the stars; reﬂection with respect to the dotted line in Fig. 4, it can be shown that the only nonzero elements belong to the 1 The difference in these results from those obtained by the expansion of the exponent up to a power of 149 is on the order of ﬁrst and to the second diagonals (i.e., with n1 = k1, n2 = − ≠ ≠ 10 5%. So, the method gives nearly precise values. The results k2 and with n1 k1, n2 k2). We have calculated them would be the same if one were to use the more common expres- numerically with a very high precision by expansion of sion (6) and eigenfunctions of the star obtained numerically.

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Contributions to parameters of the effective Hamiltonian (13) from the terms V1, V2, and V3 of the perturbation expansion. 1 2 ᏺ Interaction J2 has been taken into account in V and V terms only. is the number of stars in the lattice V3* V1 V2 Totals two-stars three-stars ᏶ z 0 Ð0.007J1 + 0.002J2 Ð0.013J1 0.010J1 Ð0.010J1 + 0.002J2 ᏶ y 0 Ð0.001J1 + 0.007J2 Ð0.001J1 0.001J1 Ð0.001J1 + 0.007J2 ᏶ x 0 0 0.067J1 0 0.067J1

h Ð0.563J2 Ð0.092J1 Ð 0.218J2 Ð0.161J1 0.080J1 Ð0.173J1 Ð 0.781J2 ∆Ꮿ ᏺ ᏺ ᏺ ᏺ ᏺ ᏺ ᏺ ** Ð0.009J2 Ð0.768J1 + 1.530J2 Ð0.361J1 0.304J1 Ð0.825J1 + 1.521J2 * This term implies two-star coupling shown in Fig. 4 and three-star interaction in the conﬁguration presented in Fig. 3b. Ꮿ ᏺ ** Correction to the value 0 = Ð4.5J1 for noninteracting stars.

1 Ꮿ ᏺ ᏶ s is the spin---- operator; = Ð5.268J1 ; z = 4a3 = respectively, and are negligible in comparison with 2 those of the retained terms. We stress that, within our − ᏶ 0.007J1; y = 4a4 = Ð0.001J1; and h = Ð6a2 = precision, the Hamiltonian in Eq. (13) is an exact map- − ᏺ 0.092J1. Here, = N/12 is the number of stars in the ping of the original Heisenberg model to the low- ω ᏶ ᏶ lattice. The factor 6 appears in the expression for h energy sector (the excitation energy ~ max{ z, y, ᏶ Ӷ because each star interacts with six neighbors. We see x, h} J1). that the magnetic field in the effective Hamiltonian (12) is much larger than the exchange. In this approxima- As follows from the study of the V3 corrections in tion, stars, therefore, behave almost like free spins in the table, the common shift given by them remains external magnetic field and the ground state of the KAF much larger than the level splitting in both cases of the has a long-range singlet order, which settles on the tri- two- and three-star coupling. At the same time, the val- Ψ 3 angular star lattice and is formed by stars in Ð states. ues of the V perturbation terms are approximately two times smaller than those of the V2 ones. The change in V3 corrections. The field remains the largest term in the effective Hamiltonian, and the KAF ground state the effective Hamiltonian from the V3 terms is, there- has the same long-range order if we take the V3 terms in fore, significant, and analysis of the perturbation series the perturbation series into account. For the two-star cannot be completed at this point for correctly estab- coupling, the V3 corrections have the form given by lishing the effective Hamiltonian. Unfortunately, such 3 work requires large computer capacity that is not at our Eq. (6). The terms V imply analysis of the three-star disposal. We must restrict ourselves to the present pre- interaction as well. Nonzero contributions from them cision. have only been obtained for the configuration presented in Fig. 3b. The secular matrix for three stars is 8 × 8 in One can judge the applicability of the perturbation size. We have calculated the V3 corrections with a very series from the following values of the ground-state high precision using the integral representation similar energy of two interacting stars, shown in Fig. 4, to that in Eq. (7) for the second term in Eq. (6). All the obtained numerically and using the first two orders of operator exponents were expanded up to a power of perturbation theory. The ground-state energy of two 150. As a result, the low-energy properties of the KAF noninteracting stars is Ð9J1. That of two interacting are described by the effective Hamiltonian ones calculated numerically by power method [20] is − 9.62J1. On the other hand, the ground-state energy Ᏼ᏶[]z z ᏶ x x ᏶ y y = ∑ zsi s j ++xsi s j ysi s j obtained using table is Ð9.42J1 (the respective contribu- 2 3 〈〉, tions of the V and V terms are Ð0.27J and Ð0.15J ). ij (13) 1 1 Effective Hamiltonian structure. Although pertur- z Ꮿ + hs∑ i + , bation theory is unsatisfactory in the star model and i many perturbation terms are to be taken into account, we can now prove that Eq. (13) is the most general form where all parameters are given in the table. It describes of the effective Hamiltonian assuming that n-pseu- two-star coupling. We omit the three pseudospin terms dospin couplings with n > 2 are small, as it was in the z z z z y y case of n = 3 discussed above. We consider possible in Eq. (13) that have the form si s jsk and si s j sk and z + z Ð + Ð describe the three-star interaction. The corresponding terms of the form si s j , si s j , si , and si . In these cases, Ð3 Ð4 Ψ Ψ coefficients are on the order of 10 J1 and 10 J1, the numbers of functions + and Ð to the right of the

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corresponding matrix elements differ from those to the clear that the contribution of the next-nearest interac- | | left by unity. As has been pointed out above, a kagomé tions to the magnetic field becomes significant if J2 ~ lattice contains lines of symmetry reflections, and the 0.1J1. If J2 < 0 (ferromagnetic interaction), they can star Hamiltonian and interstar interaction are invariant even change the sign of h. Ψ under these reflections. Because + are invariant and Ψ The effect of the next-nearest ferromagnetic cou- Ð changes sign under these transformations, the pling for KAF properties was previously studied in [2] matrix elements are equal to themselves with the oppo- numerically on finite clusters with N ≤ 27 in a wide site sign and must, therefore, be zero. Another possible range of the values of J . It was shown there that, at term 2 | | Ӷ × J2 /J1 1, the ground state has the 3 3 magnetic x y i + + Ð Ð Ð + + Ð structure. At |J |/J Ӷ 1, the ground state is disordered s s = Ð---()s s Ð s s + s s Ð s s 2 1 i j 4 i j i j i j i j and there is a band of singlet excitations inside the triplet gap. As we demonstrated above, this band is a result cannot appear in the effective Hamiltonian because the of the star ground-state degeneracy in our approach. corresponding matrix elements should be imaginary. T2 specific heat behavior. There has been much ᏶ Ground state. As is clear from table, x and h are speculation on the low-temperature dependence of the the largest parameters of Hamiltonian (13) in our KAF specific heat C ∝ T2 observed experimentally for approximation. Therefore, the KAF behaves as the S = 3/2 (see [7, 21] and references therein). As we found Ising antiferromagnet in the perpendicular magnetic above, low-T properties are described by effective field. In this case, the classical value of the field at Hamiltonian (13) of a magnet, which has a spectrum of which spin-flip occurs is h = ᏶ , which is approxi- 2 2 s Ð f x the form ⑀ = ()cq + ∆' at q Ӷ 1 and can be in mately 2.6 times smaller than h. The ground state must, q therefore, remain ordered with all the stars in the ordered or disordered phases depending on particular values of the parameters. Small ∆' here implies the Ψ− state. proximity to the quantum critical point at which C ∝ The ground-state energy and that of the upper edge T 2. Such a situation arises in the singlet dynamics of of the singlet band calculated using table are the model of interactive plaquets [21]. We do not ()∆Ꮿ ᏶ ᏺ present the corresponding analysis here because the Ð4.5J1 +++h/2 3 z/4 = Ð0.452J1 N parameters of the effective Hamiltonian could be changed in subsequent orders of perturbation theory. and Experimental verification. In both cases of the ()∆Ꮿ ᏶ ᏺ Ð4.5J1 ++Ð3h/2 z/4 = Ð0.437J1 N, ordered and disordered ground state, the approach pre- ᏶ sented in this paper can be checked by inelastic neutron respectively. Corrections from x to these values in the scattering: the corresponding intensity for the singletÐ first nonzero order of perturbation theory are given by triplet transitions should have a periodicity in the recip- ᏺ᏶2 rocal space corresponding to the star lattice. This pic- (3/16)x /h and are negligible. At the same time, the ground-state energy of the largest cluster with N = 36 ture is similar to that observed in the dimerized spin- that has previously been considered numerically is Pairls compound CuGeO3 [22]. In this case, inelastic − magnetic scattering has a periodicity that corresponds 0.438J1N [5]. Hence, we believe that clusters used in the previous studies were too small to reflect the to the dimerized lattice. Heisenberg KAF low-energy sector at J = 0 properly. Comparison with QDMs. We point out that states 2 Ψ Ψ in which all stars are in the Ð or + state can be repre- Interaction J2. We now show that, in spite of its smallness, the next-nearest neighbor interaction can sented as linear combinations of some first-neighbor play an important role for low-energy properties. We dimer states proposed in [15, 16] for QDM. However, have calculated J corrections to the parameters of our approach to the kagomé problem is not equivalent 2 to the QDM. In particular, we take all intermediate effective Hamiltonian (13) for the first and the second states into account in considering the star interaction terms in Eq. (6) only. There are 12 intrinsic J2 interac- via perturbation theory in Eq. (6), whereas the QDM is tions in each star, which splits the doubly degenerate restricted to the first-neighbor dimer subspace as ground state and, as is seen from table, makes a contri- regards the dimer dynamics. bution to magnetic field h and constant Ꮿ. Unfortunately, we cannot carry out a complete com- As is clear from Fig. 4, the two-star coupling is now parison between QDM and the star approach at the given by the operator present stage. The effective Hamiltonian derived in [15] was analyzed under crude approximations only. At the ˜ ()()1 ⋅ ()2 ()1 ⋅ ()2 ()1 ⋅ ()2 ()1 ⋅ ()2 V = J2 S1 S2 ++S2 S1 S2 S3 + S3 S2 . same time, the model presented here also requires further studies of the applicability of perturbation theory Corrections proportional to J2 were calculated in the for description of the interstar interaction. We also note same way as above and are also presented in table. It is that some present-day results obtained within these two

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 544 SYROMYATNIKOV, MALEYEV approaches contradict each other. For example, our APPENDIX 1 model gives the ordered ground state, whereas the authors of [15] suggest that it is not ordered. Irreducible Representations of the Group C6v Cases of S > 1/2. We finally note that our consider- k The symmetry group C6v contains six rotations C ation of the S = 1/2 KAF cannot be extended directly to by the angles 2πk/6 (k = 0, 1, ..., 5) and six reflections, the cases of larger spins. Although functions presented k which can be written as C u1, where u1 is the operator in Fig. 2, where the bold line shows the singlet state of of a reflection. One-dimensional irreducible representa- the corresponding two spins, remain eigenfunctions of tions can be represented as follows [18, 19]: the Hamiltonian for S > 1/2, we have found numerically k ∼ ∼ that they are not ground states of the star with S = 1 and C 1, u 1 1, (A.1) S = 3/2. All details of calculations are presented in k ∼ ()k ∼ Appendix 2. Another approach for KAFs with S > 1/2 C Ð1 , u1 1, (A.2) should therefore be proposed. k ∼ ∼ C 1, u 1 Ð,1 (A.3) k ∼ ()k ∼ 4. CONCLUSIONS C Ð1 , u1 Ð.1 (A.4) In this paper, we present a model of the low-energy For two-dimensional representations, we have [18, 19]

1 2πl physics of spin---- KAFs. The spin lattice can be repre- i------k 2 6 Ck ∼ e 0 , sented as a set of stars that are arranged in a triangular 2πl Ði------k lattice and contain 12 spins (see Fig. 1). Each star has 6 0 e (A.5) two degenerate singlet ground states with different symmetry, which can be described in terms of pseu-  dospin. It is shown that interaction between the stars ∼ 01 u1 , leads to the band of singlet excitations that determines 10 the low-energy KAF properties. The low-energy 1 where two inequivalent representations are given by l = dynamics is described by the Hamiltonian of a spin---- 1 and l = 2. 2 magnet in the external magnetic field given by Eq. (13). The Hamiltonian parameters are calculated in the first APPENDIX 2 three orders of perturbation theory and are summarized in the table. Within our precision, the KAF has an Star with S = 1 and S = 3/2 ordered singlet ground state with all the stars in the state given by Eq. (3). The ground-state energy is lower In this appendix, we present the details of numerical than that calculated in the previous finite cluster stud- calculations showing that functions presented in Fig. 2, ies. We show that our model cannot be extended where the bold line depicts the singlet state of the corresponding two spins, are not ground states of the star directly to KAFs with S > 1/2. with S = 1 and S = 3/2, as this was for S = 1/2. The approach discussed in this paper can be verified A simple numerical method for determining the experimentally under inelastic neutron scattering: the eigenvalue of a Hermitian operator H of the maximum corresponding intensities for singletÐtriplet transitions modulus (power method [20]) was used. It is based on should have periodicity in the reciprocal space corre- the following statement. We consider a state of the sys- sponding to the star lattice. ψ tem f = ∑i ci i , where the sum may not include all the H eigenfunctions. For a given f, the eigenvalue Eextr of the maximum modulus is determined by ACKNOWLEDGMENTS 〈|f Hn + 1|〉f We are grateful to D.N. Aristov and A.G. Yashenkin lim ------= Eextr. (B.1) → ∞ n for interesting discussions. This work was supported by n 〈|f H |〉f the Russian Foundation for Basic Research (project This becomes evident by noting that 〈f |Hn|f 〉 = nos. SS-1671.2003.2, 03-02-17340, and 00-15-96814), c 2En . Goskontrakt (grant no. 40.012.1.1.1149), and Russian ∑i i i State Programs “Quantum Macrophysics,” “Collective Equation (B.1) can be used in numerical calcula- and Quantum Effects in Condensed Matter,” and “Neu- tions in the following way. The corresponding expres- tron Research of Solids.” sion is calculated for n = 1, 2, …, nmax. Convergence

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 LOW-ENERGY SINGLET DYNAMICS 545 can therefore be controlled by comparing results with 5. C. Waldtmann, H.-U. Everts, B. Bernu, et al., Eur. Phys. different n. Studying a full set of vectors f and taking J. B 2, 501 (1998). nmax large enough to match the necessary precision, one 6. C. Zeng and V. Elser, Phys. Rev. B 51, 8318 (1995). can ﬁnd the eigenvalue of H with the largest modulus. 7. P. Sindzingre, G. Misguich, C. Lhuillier, et al., Phys. In the case of the star, the maximum eigenvalue of Rev. Lett. 84, 2953 (2000). 2 8. J. T. Chalker and J. F. G. Eastmond, Phys. Rev. B 46, 14 the Hamiltonian is Emax = 18S J1 (this energy has the state in which all the spins are along the same direction) 201 (1992). 9. S. Sachdev, Phys. Rev. B 45, 12377 (1992). and the energy of singlet states shown in Fig. 2 is Ess = | | 10. K. Yang, L. K. Warman, and S. M. Girvin, Phys. Rev. Ð6S(S + 1)J1. Because Emax > Ess for S > 1/2, we have Ᏼ Ᏼ Lett. 70, 2641 (1993). to take H = 0 Ð WI to investigate the lower 0 levels, Ᏼ 11. V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. 59, where 0 is the star Hamiltonian given by Eq. (1), I is 2095 (1987). the unit matrix, and W = (Emax + Ess)/2 + J1. Eigenvalues Ᏼ 12. J. B. Marston and C. Zeng, J. Appl. Phys. 69, 5962 of H are therefore shifted down relative to those of 0 (1991). by the same value W such that the H eigenvalue with the 13. F. Mila, Phys. Rev. Lett. 81, 2356 (1998). Ᏼ largest modulus becomes equal to the 0 ground-state 14. G. Misguich, D. Serban, and V. Pasquier, Phys. Rev. Lett. energy minus W. 89, 137202 (2002). We have not found the ground-state energy for the 15. G. Misguich, D. Serban, and V. Pasquier, Phys. Rev. B star with S = 1 and S = 3/2 by this method because the 67, 214413 (2003). full set of vectors f should be examined for that. This 16. M. Mambrini and F. Mila, Eur. Phys. J. B 17, 651 (2000). operation requires much computer time. However, 17. A. V. Syromyatnikov and S. V. Maleyev, Phys. Rev. B 66, studying a number of vectors f, we have obtained that 132408 (2002). there are states lower than those discussed above, at 18. L. D. Landau and E. M. Livshitz, Course of Theoretical least, by the energy 1.8J1. The method yielded Eextr with Physics, Vol. 3: Quantum Mechanics: Non-Relativistic the prescribed precision to the second decimal position Theory, 3rd ed. (Nauka, Moscow, 1974; Pergamon Press, Oxford, 1977). at nmax = 100Ð300 depending on f and S. 19. M. I. Petrashen’ and E. D. Trifonov, Applications of Group Theory in Quantum Mechanics (Nauka, Moscow, REFERENCES 1967; Iliffe, London, 1969). 1. A. P. Ramirez, B. Hessen, and M. Winklemann, Phys. 20. B. N. Parlett, The Symmetric Eigenvalue Problem (Pren- Rev. Lett. 84, 2957 (2000). tice Hall, Englewood Cliffs, N.J., 1980; Mir, Moscow, 1983). 2. P. Lecheminant, B. Bernu, C. Lhuillier, et al., Phys. Rev. 21. V. N. Kotov, M. E. Zhitomirsky, and O. P. Sushkov, Phys. B 56, 2521 (1997). Rev. B 63, 064 412 (2001). 3. C. Zeng and V. Elser, Phys. Rev. B 42, 8436 (1990). 22. L. P. Regnault, M. Ain, B. Hennion, et al., Phys. Rev. B 4. P. W. Leung and V. Elser, Phys. Rev. B 47, 5459 (1993). 53, 5579 (1996).

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 546Ð555. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 618Ð629. Original Russian Text Copyright © 2004 by Ovchinnikov, Ovchinnikova. SOLIDS Electronic Properties

Effect of Spin Structures and Nesting on the Shape of the Fermi Surface and Pseudogap Anisotropy in the tÐt'ÐU Hubbard Models A. A. Ovchinnikov† and M. Ya. Ovchinnikova Institute of Chemical Physics, Russian Academy of Science, Moscow, 119977 Russia e-mail: [email protected] Received June 2, 2003

Abstract—The effect of two types of spin structures on the shape of the Fermi surface and on the map of photoemission intensities for the tÐt'ÐU Hubbard model is investigated. The stripe phase with a period of 8a and the spiral spin structure are calculated in the mean ﬁeld approximation. It is shown that, in contrast to electron- type doping, hole-doped models are unstable to the formation of such structures. Pseudogap anisotropy is different for h- and e-doping and is determined by the spin structure. In accordance with ARPES data for La2 − xSrxCuO4, the stripe phase is characterized by quasi-one-dimensional FS segments in the vicinity of points ±π M( , 0) and by suppression of the spectral density for kx = ky. It is shown that spiral structures exhibit polarization anisotropy: different segments of the FS correspond to electrons with different spin polarizations. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION information on the FS from the ARPES data. The prob- Angle-resolved photoemission spectroscopy (ARPES) lem of matrix elements was discussed in detail is an effective method for studying the electron bands in [13, 14]. However, in strongly correlated systems, and Fermi surfaces (FS) of cuprates [1, 2]. This method the topology, shape, and intensities of individual (main and shadow) segments of the FS also depend on spin provides an image of the FS projection onto the CuO2 plane. The results obtained in early studies (see [1, 2] and charge structures, which ensure the lowest energy. and the literature cited therein) corresponded to hole- This study is devoted to model analysis of the effect type FSs centered at point Y(π, π) of the two-dimen- of various periodic spin structures on the FS shape and sional Brillouin zone. Other versions of the FS topo- their manifestations in photoemission intensities. The logy were also discussed later. In particular, the existence analysis is based on the tÐt'ÐU Hubbard model. In con- of electron-type FSs with the center at point Γ(0, 0) was trast to static structures in magnesium-based com- predicted for Bi2Sr2CaCu2O8 + δ (BSCCO) [3]. The pounds, cuprates are instead characterized by dynamic revision of the problem [4, 5] apparently conﬁrmed the structures. The lifetimes of such structures are longer initial assumption. At the same time, a transition from than the corresponding times t ~ 10Ð6Ð10Ð9 s in µ-SR the h-type FS to the e-type FS was discovered for experiments. Over short time intervals and for pro- δ ប La2 − xSrxCuO2 (LSCO) upon a transition from the cesses with an energy resolution of E > /t, local spin underdoped to overdoped region of the phase diagram structures can be treated in the static approximation. In [6, 7]. On the basis of ARPES data, the existence of a this case, it is natural to ﬁnd the structures that corre- d-wave superconducting (SC) gap was proved and a spond to the FS features observed in the ARPES exper- pseudogap was discovered in antinodal directions in the iments. underdoped compound BSCCO. Recently, band split- Experimental indications of the existence of spin ting and the FS were observed in bilayer cuprates, and and charge structures in cuprates are incommensurate the phase diagram boundary was discovered beyond peaks in the spin susceptibility χ''(q, ω 0) for q = which such splitting disappears [8Ð11]. Analysis of (π ± δq, π), (π, π ± δq) in LSCO, which are obtained photoemission induced by circularly polarized light from inelastic neutron scattering [15]; a linear structure revealed a state with time-reversal symmetry breaking along the CuO bonds with a period of 4a (a is the lattice (TRSB) in the underdoped compound BSCCO [12]. constant), which was observed in BSCCO from the Wide application of photoemission intensity maps Fourier analysis of tunnel spectra [16]; and a periodic ω in the kx, ky, space involves the determination of staggered 4a × 4a structure around vortices in the matrix elements and requires methods for obtaining mixed state of BSCCO [17]. In this study, we analyze the FS topology and the † Deceased. intensity maps of photoemission spectra for a number

EFFECT OF SPIN STRUCTURES AND NESTING ON THE SHAPE OF THE FERMI SURFACE 547 of possible spin structures in the tÐt'ÐU Hubbard mod- insufficient for describing the SC order in the MF els. We will use the extremely simple language of the approximation. In accordance with some approaches, mean field (MF) method for interpreting the pseudogap the SC state arises due to attraction between electrons of (PG) to give examples of structures with different types neighboring sites, which is of the correlation type. The of PG anisotropy and to consider some properties of interaction of the type of correlated jumps in the effective photoemission of cuprates in light of the results Hamiltonian was derived, for example, in [19, 20]. In the obtained for periodic structures. Earlier [18], current empirical version, this can be, for example, Hamito- states of the type of an orbital ferromagnet were pro- nian (1) supplemented with the interaction between poses as latent order parameters (OP) responsible for nearest neighbors of the form the emergence of the pseudogap. In this work, the range κ † † of possible latent OPs is extended to stripe structures V = ∑ cnσcm, Ðσcm, Ðσcn, σ (3) and spiral spin structures. Such structures are stabilized 〈〉σnm , due to the removal of degeneracy in the states of “hot κ spots,” i.e., Van Hove singularities (VHS) of spectra or with < 0. In this study, however, we will analyze only the removal of degeneracy in bands located on parallel the normal state and confine the analysis to the MF FS segments in the case of nesting. treatment of the initial Hamiltonian (1). The article has the following structure. In Section 2, Let us consider a periodic structure with the 2D the tÐt'ÐU Hubbard model is considered and a simpli- translation vectors (), (), fies mean-field approximation is used for describing the E1 ==E1x E1y , E2 E2x E2y . (4) normal sate. The MF equations are formulated for an arbitrary periodic structure with spin density waves Suppose that a unit cell contains nc centers with coordi- (SDW) and charge density waves (CDW). The methods nates j = (jx, jy) so that an arbitrary site of the 2D lattice, for visualizing the FS and the intensity map of ARPES (), (), (), spectra are considered in Section 3. Typical examples nnLj==n1 n2 =E1 L1 ++E2 L2 jx jy (5) of FS and ARPES intensity maps are given in Section 4 for homogeneous MF solutions with the AF spin order. is described by integers L1, L2 (unit cell coordinates) On such solutions, the emergence of the pseudogap and by numbers j = (jx, jy) fixing a lattice site. Two- with various types of anisotropy in h- and e-doped sys- dimensional vectors B1, B2 of the reciprocal lattice sat- πδ tems is traced. The results obtained for inhomogeneous isfy the equations EiBj = 2 ij. (Components Ei and Bj spin and charge structures are considered. These struc- are given in units of constants of the direct and recipro- tures include stripe phases of antiphase AF domains cal lattices.) along the y bonds with a domain width of 4a, spiral spin ˜ states, and periodic 1D and 2D structures with charge We denote by k the quasimomentum within the modulation. In Section 5, various degrees of stability of principal Brillouin zone G˜ of the periodic structure in homogeneous AF solutions relative to the formation of contrast to quasimomentum k varying within the Bril- stripe and spiral spin structures for e- and h-doped mod- ˜ els are demonstrated. Features of photoemission for louin zone G of the initial lattice. The areas of G and G π2 π2 these structures, the methods for their testing, and the amount to 4 /nc and 4 and are limited by the condi- correspondence of these structures to some ARPES tions data for cuprates are discussed in Sections 6 and 7. ˜ ∈π˜ ˜ ≤ ∈π≤ k G : kBi ; kG : kxy() . (6) 2. PERIODIC SOLUTIONS The order parameters for the periodic MF solutions are OF THE HUBBARD MODEL the electron densities and the vectors of average spin for IN THE FRAMEWORK each site of the unit cell, OF THE MEAN FIELD METHOD 1 〈〉 1 〈〉 r j ==------∑ rnLj(), , Sµj ------∑ Sµ, nLj(), . (7) We will study the effect of nesting and the formation N L N L of periodic structures on the FS and the energy bands in L L the normal state of cuprates by applying the MF method Here, index µ enumerates the spin vector components to the initial Hamiltonian of the tÐt'ÐU Hubbard model, and NL = N/nc is the number of unit cells, nc being the number of centers in a cell. ⑀ † HT==+ ∑ Unn↑nn↓, T ∑ kckσckσ, (1) In the MF approximation, the mean energy in n σ, k model (1) is given by  ⑀ = 2tk()cos + cosk + 4t'cosk cosk . (2) 2 2 k x y x y HT= 〈〉+ N Ur∑Ð ∑S µ (8) L j j j µ We will assume that t = 1 and will measure all energies and parameters U and t' in units of t. Hamiltonian (1) is and the wave function is determined by the population

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 548 OVCHINNIKOV, OVCHINNIKOVA χ† of the one-electron eigenstates kλ of the linearized siderably reduces the number of independent order Hamiltonian parameters and often reduces matrix (12) to the real matrix when j0 is chosen at the symmetry center of the unit cell of the structure. {}ˆ ˆ Hlin ==TN+ L∑ 2Urjrˆ j Ð 2USµjSµj ∑ hk˜ . (9) j ˜ ∈ ˜ k G 3. METHODS FOR VISUALIZING The latter can be split into independent contributions THE FERMI SURFACE for each value of the reduced quasimomentum k˜ . Here, AND THE GENERALIZED FERMI SURFACE ˆ The Fermi boundaries and the quantity that can be rˆ j and Sµj are the operators corresponding to one-electron mean values (7). In the momentum representation, called the generalized Fermi boundary (GFS) can be the eigenstates of Hamiltonian (9) can be expanded in determined from the ARPES data using several meth- the set of 2n Fermi operators, ods. These methods are considered in detail in [13], and c each method can be put in correspondence to a method of constructing FS and GFS maps in model calcula- † † ˜ χ = c W σλ, ()k , tions. k˜ λ ∑ k˜ + Bm, σ m , σ One of these methods employs the photoemission m (10) (), intensity map I(k, ω) for electrons with momentum com- mm==1 m2 ; Bm B1m1 + B2m2; ν ω ponent k in the ab plane and with energy Ee = h Ð : λ = 1,,… 2n . c (), ω ()2 ()ω ()ω ⊗ Ik = Mk Ak f Rωk, (15) Here, k˜ varies within G˜ and the set of pairs of integers ˜ 1 2 (m , m ) is such that vectors k + Bm cover the entire Ak()ω = --- ∑ 〈|γ c σ|〉α 1 2 Z k phase space G. Matrix Wmσ, λ of eigenvectors and vector αγ, (16) β()µ E˜, λ of eigenvalues are determined by diagonalozation Eα Ð N k × e δ()Eγ Ð Eα Ð µ Ð hω . of matrix h in the basis {}c† : k˜ k˜ + Bm, σ Intensity (15) is determined by the product of the ω () squared matrix element M(k), spectral density A(k ), hk˜ mσ,,m' σ'Wm'σ', λ = Wmσλ, Ek˜, λ. (11) and Fermi function f. In order to compare the intensity with the observed ARPES signal, a convolution of the Here, we have product with the Gaussian function Rωk [13] with parameters characterizing the energy and momentum ()h = δ δσσ ⑀ k˜ mσ,,m' σ' mm' ' k˜ + Bm resolution is usually carried out in Eq. (15). In Eq. (16), (12) α and γ characterize the states of the entire system, β = ϕ(), []δ ()σ µ + U∑ jm' Ð m r j σσ' Ð Sµj µ σσ' , 1/kT, and is the chemical potential. The dependence j of matrix element M on k and its role were studied in [13, 14]. Here, we assume for simplicity that the ϕ(), []() jm = exp ijÐ j0 Bm , matrix element is constant since we are interested in the (13) Bm= B m + B m . effect of periodic structures and transition processes on 1 1 2 2 spectral density A(kω) and the photoemission intensity. The choice of j0 (the origin from which the sites in a cell In the one-electron MF approximation, we have are counted) is arbitrary and affects only the phases. In turn, order parameters (7) themselves can be calculated 1 ˜ 2 Ak()ω = ---- ∑ ∑ W σλ, ()k in terms of the eigenvector matrix W and Fermi func- N m tions f: k˜ ∈ G˜ m,,σλ (17) ×δ()δµ ω 1 E˜ λ Ð Ð , ˜ . {}r , Sµ = ------∑ ∑ {}σ , σµ ϕ()jm, ' Ð m k kk+ Bm j j 2N 0 ss' k˜ ∈ G˜ msm,,' ,s' (14) Two-dimensional index m = (m1, m2) passes through all λ × * ()˜ ()˜ ()µ independent transition vectors Bm = B1m1 + B2m2; = Wms, λ k Wm's', λ k fEk˜ λ Ð . 1, …, 2nc enumerates proper Fermi operators of linear- σ σ The Pauli matrices µ and 0 in this expression corre- ˜ ized Hamiltonian hk˜ with reduces momentum k . We spond to components Sµj and rj , respectively. Equa- − calculate function A(kω) using the standard substitution tions (11) (14) determine the self-consistent solutions δ in the MF approximation. of the function in Eq. (17) by a function of a ﬁnite For a periodic structure with a deﬁnite symmetry, width Ω, e.g., of the form δω() = coshÐ2(ω/Ω). the combination of equivalent atoms into groups con- Another method for introducing broadening is to use

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(‡) (b)

Fig. 1. Fermi surfaces for homogeneous AF solutions in hole- (a) and electron-doped (b) models with parameters U = 4, t' = 0.3, δ | | π π ±π ±π = 1 Ð n = 0.2 in the underdoped region on plane Ð < kx , ky < . Hole and electron pockets are located around points ( /2, /2) or (±π, ±π). the self-energy part chosen empirically in [13]. The but smoother variation of n(k). The connection between ω construction of map I(kx, ky, = 0) makes it possible to such segments and the emergence of a dielectric gap visualize both the main and shadow segments of the and anisotropic pseudogaps will be illustrated below. Fermi boundary, which are manifested with a lower or higher intensity. 4. TOPOLOGY OF THE FERMI SURFACE It should be noted that band energies are periodic in AND GENERALIZED FERMI SURFACE the k space: Ek˜ + Bm, λ = Ek˜, λ for any m = (m1, m2). How- FOR HOMOGENEOUS ANTIFERROMAGNETIC ()˜ 2 MEAN-FIELD SOLUTIONS ever, quantity Wmσλ, k in spectral function (17) and, accordingly, photoemission intensity (15) do not In this section, we partly repeat the well-known possess such a periodicity. For this reason, different FS results obtained in [21Ð28]. Homogeneous AF solu- segments are manifested with different intensities even tions in the MF approximation are characterized by the n + n for the matrix element in Eq. (15) independent of k in average alternating spin d = ()Ð1 x y 〈S 〉. The corre- view of the compound nature of band operators (10) in 0 nz ˜ the presence of SDW and CDW structures (i.e., transi- sponding magnetic Brillouin zone GAF is confined by | ± | ≤ π tion processes). In calculating I(k, ω), the correspon- the limits kx ky . The known energies of the upper dence of a given vector k to quantities k˜ and m = and lower Hubbard bands are given by (m , m ) is determined by the equation k = k˜ (k) + ⑀ 1 2 k˜ λ = 4t'coskx cosky m1B1 + m2B2. (19) ± 2 2 2()2 Other methods, which are adequate to the process- U d0 +4t coskx + cosky + const. ing of ARPES data, were also proposed for visualizing the FS in model calculations. One of such methods The Van Hove singularity in the density of states (DOS) employs the construction of the map of gradient gk = of the lower Hubbard band corresponds to hot spots ∇ ⊗ M = (±π, 0), (0, ±π). The shape of the FS critically knk of smoothed occupancy nk = nk Rk. It is also possible to construct the maps of intensity averaged depends of parameter t' and on the doping level. For over a definite frequency window 2∆ω: t' = 0, the band energy is constant for all values of k along the boundary of the magnetic Brillouin zone. For ⑀ ωωÐ ' t' > 0, energy k, λ = 1 at points M is lower than at diago- (), ω ω() ω I∆ω k = ∫d 'Ik, ' R  ------∆ω - . (18) nal points S ( ±π /2, ±π /2). For this reason, in accordance with the well-known pattern, the FS bounds hole pock- Here, R is the corresponding Gaussian function of ets centered at point S for a low doping level. Figure 1a width ∆ω, which imitates a finite resolution in ω. The is a map of intensity I(k, ω = 0) calculated by formu- ⑀ construction of such maps involves the normalization las (15) and (17). Since band energy M at point M lies of a function to its maximum value. Consequently, the below the chemical potential, photoemission of elec- brightness and width of the Fermi boundaries on such trons from such segments with k = kM requires an maps is determined by the width of smoothing function energy of ∆ = (µ Ð ⑀ ) > 0, which is equal to the work ∆ω PG M Rk or frequency window in Eq. (18). In particular, function. Accordingly, the curve describing the energy ∆ω for a large width , a map of functions I∆ω or gk shows distribution of photoelectrons (EDC) will be shifted by ∆ not only the actual Fermi boundaries with a sharp occu- Ð PG. Such a shift indicates the emergence of a pancy step, but also the boundaries with a substantial pseudogap in the normal state in directions k ~ kM in the

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My Y underdoped models (Fig. 1b). The main (nonshadow) segments of the FS in this case face point Y(π, π). Figure 2 shows the map of averaged intensity (18) with a frequency window width of ∆ω = 0.05t. As ∆ω 0, the maximum maxI(k, ω) ∞ on the true Fermi boundaries, and segment L1ÐL2 on the normal- L ized intensity map becomes invisible. In addition to the 1 FS around the electron pockets, we can see boundary − L1 L2 corresponding to the maximum of the occupancy gradient in the diagonal direction. On this segment of ⑀ µ the boundary, we have k, λ = 2 Ð < 0. This indicates the L − 2 emergence of a PG on segment L1 L2 around points ±π ±π kS = ( /2, /2). The neighborhoods of these points are responsible for the formation of VHS in the DOS of the upper Hubbard band in a model with t' > 0. Further e-doping transforms the FS for homogeneous MF solu- Mx tions into a single large surface around point Y(π, π). As a result of inclusion of a pairing interaction of type (3) and d superconductivity combined with nonstandard Fig. 2. Map of averaged intensity (18) with ∆ω = 0.05t on plane 0 < k , k < π for the same electron-doped model as in PG anisotropy for the underdoped region, the minimal x y energy ∆ of Fermi excitations does not vanish even Fig. 1b. Line L1ÐL2 of the generalized Fermi boundary cor- min responds to energies below the Fermi level, i.e., to the emer- in the nodal d directions of the SC gap: gence of a PG with anisotropy differing from anisotropy of the PG in BSCCO. ∆ ()∆2 ∆2 ≠ min = min PG + SC 0. (21) given homogeneous AF solutions. For small values of As regards the behavior of such quantities as heat λÐ2 t'/t ~ 0.1, the pseudogap disappears when the doping capacity and (T) for T 0, the ﬁnite value of the δ minimal gap for Fermi excitations is perceived as the level increases to a certain value of opt. At this instant, the chemical potential passes through a VHS in the generalized s symmetry of the SC order. In [32], the DOS. The peak of the DOS at the Fermi level for δ = crossover from some properties characteristic of d sym- δ ensures the maximal value of T for such a doping metry to those typical of s symmetry of the SC order opt c was considered for n-type cuprates X Ce CuO , X = level. Simultaneously, the FS changes its topology at 2 Ð x x 4 δ = δ , being transformed into a single large FS with Nd and Pr. Homogeneous model solutions predict opt reverse evolution of these properties from those typical electron-type segments in the vicinity of hot spots. In of s-type superconductors to the properties characteris- this case, optimal doping level δ increases with t'. opt tic of d-type superconductors upon an increase in the In [21−23, 25, 27], such a behavior of phase curves δ doping level. In spite of this contradiction, the solution Tc( ) is described and the geometrical interpretation of is interesting as an example of the fact that d supercon- the PG is given for the tÐt'ÐU and tÐt'ÐJ models on the ductivity combined with nonstandard PG anisotropy basis of more rigorous approaches. In these δ may imitate in some properties the SC order with the approaches, the value of opt is smaller than in the sim- generalized s symmetry. ple MF approximation. The electron pockets around points M in n-type When the effective attraction between electrons underdoped cuprates, which were predicted for the ﬁrst from neighboring sites is included, homogeneous MF time in [22, 24], were indeed observed on the ARPES solutions lead to an SC order with the d symmetry, intensity maps for underdoped compounds NCCO [33]. which is compatible with a local AF order [22, 23]. For The evolution of FSs from nonconnected small FSs to underdoped cuprates, the total gap (the shift of the EDC a large FS of the hole type around point Y(π, π) was edge in ARPES), which depends on the PG and SC gaps traced. The parameters of the tÐt'Ðt''ÐU model repro- in accordance with the relation ducing the observed evolution were selected in [28]. Among other things, the assumption that effective δ ∆∆= 2 +,∆2 (20) potential U = U( ) decreases with increasing doping PG SC level had to be made. According to [28], such a screen- explains [26, 27] the nonlinear dependence of gap ∆(z) ing of U combined with higher harmonics (~t'') in the on z = cosk Ð cosk and other features of the observed band energy leads to the simultaneous formation of x y hole pockets around point k = (π/2, π/2) and electron gap [29−31]. pockets around kM from the lower and upper Hubbard In the case of electron doping, a similar analysis bands, respectively. For δ ~ 0.2, the Mott gap in com- shows that electron pockets appear around points M in pounds NCCO is closed [28]. The emergence of a PG in

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 EFFECT OF SPIN STRUCTURES AND NESTING ON THE SHAPE OF THE FERMI SURFACE 551 diagonal directions in underdoped NCCO compounds is 〈H〉 also confirmed by Raman scattering data [34]. –0.65 Proceeding from the FS shape in underdoped mod- –0.70 2 els, general considerations concerning a possible sym- PM metry of the SC order in these compounds can be for- –0.75 1 mulated. Suppose that the SC transition is caused by 3 attraction of type (3) for electrons at the neighboring –0.80 AF sites of bonds with the x and y orientations. This inter- –0.85 AF sd() † † action could lead to states ∑ ϕ ()k c ↑c ↓ of cou- k k k –0.90 ϕs(d) ± pled pairs with (k) = coskx cosky of generalized s 〈H〉 – F(δ) or d symmetry. However, the state of a coupled pair –0.65 must be orthogonal (or nearly orthogonal) to one-site † † state c ↑c ↓ of the pair, which is suppressed by one- n n –0.70 PM center repulsion U. In underdoped model with h or e doping, in which pockets of only one (electron or hole) 1 type are present, the d symmetry is the only possibility –0.75 2 to achieve orthogonality of the function of a pair to the one-center function in view of orthogonality of the 3 AF angular parts of these functions. Indeed, the main (non- –0.80 shadow) segments of the FS in this case are either com- 0 0.1 0.2 0.3 0.4 pletely inside the magnetic Brillouin zone boundary δ = |n – 1| (see Fig. 1a) or completely outside this boundary (see Fig. 3. (top) Dependence of the mean energy (per site) in the ϕs Fig. 1b). Consequently, the s function (k) = coskx + hole-doped tÐt'ÐU model with parameters U = 4.0 and t' = cosky of a pair does not change its sign on the main 0.3 for different spin structures: for a PM state, for a homo- regions of the FS and, hence, cannot be orthogonal to geneous AF state, for a stripe phase with a period of 8a of the structure along the x axis (dashed curve), and for spiral the one-center function of the pair. states with Q = π(η, 1) (curve 2) and Q = π(η, η) (curve 3) The situation changes for δ > 0.15 in the electron- for η = 0.8. (bottom) The energies of the same structures in doped models with the parameters selected in [28]. In the electron-doped model. For convenience of representation, common function F(δ) = U(n2 Ð 1) is subtracted from this case, electron pockets around point M due to filling all energies. of the upper Hubbard band coexist with hole pockets around point S(±π/2, ±π/2) due to partial depletion of the lower Hubbard band. As a result, the main segments 〈 〉 δ | | of the FS lie partly outside and partly inside the mag- energy H = H on doping level = 1 Ð n for the nor- netic Brillouin zone boundary. Such segments corre- mal state in hole- and electron-doped models for a num- spond to different signs of pair function ϕs(k) = cosk + ber of structures in comparison with the energy of x homogeneous paramagnetic (PM) and AF solutions. In cosky; i.e., the orthogonality of the pair s-wave function addition to the latter solutions, MF solutions were to the one-center function can be ensured due to orthog- obtained (or sought) for the following structures. onality of the “radial” parts of the functions if we con- 1. A stripe structure consisting of antiphase AF ditionally refer to quantity z = coskx + cosky as a radial variable. It was very important to verify experimentally stripes parallel to the y axis with domain walls at the bonds. The structure is characterized by a unit cell with whether such an SC order of s symmetry is realized in ± NCCO and PCCO compounds for δ > 0.15. Analogous eight centers and vectors E1,2 = (4a, a). For analogous situations and problems may also arise during the for- structures with domain walls passing through lattice mation of periodic SDW and CDW structures since the sites, the mean energy is close to but slightly higher energy profile along the magnetic Brillouin zone than the energy of the first structure. The Fermi surfaces boundary changes not only upon a change in t' and t'', of these structures are also similar; for this reason, we but also as a result of the formation of these structures. will consider only the former structure. 2. Spiral spin structures with 5. INSTABILITY TO THE FORMATION 〈〉 [] OF SPIN STRUCTURES Sn = d0 ex cosQn + ey sinQn , (22) IN n- AND p-TYPE CUPRATES π η π η η η The degree of instability of homogeneous AF states Q = Qx = ( , 1) or Q = Qxy = ( , ) with ~ 0.75Ð0.8. to the formation of periodic spin and charge structures in doped models can be estimated on the basis of MF 3. Staggered structures with antiphase square 4a × calculations. Figure 3 shows the dependences of mean 4a domains. We do not consider the data on these struc-

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(‡) (b)

Fig. 4. (a) Photoemission intensity map I(k, ω = 0) for a stripe structure with AF stripes of width 4a, parallel to the y axis, and with domain walls centered at bonds. (b) The same averaged over two stripe structures with the x and y orientations. The model parameters are U = 6 and t' = 0.1. tures since their energies are higher than the energies of stripe phases or spiral spin structures in these com- the stripe structures and their FSs have a nonrealistic pounds. form.

4. The current states of an orbital antiferromagnet, 6. FERMI SURFACES FOR STRIPE STRUCTURES: which were proposed in [18], cannot be realized in ANALOGY WITH THE DATA model (1) in the MF approximation without using addi- ON COMPOUND LSCO tional interactions with U/t = 4Ð6. In the case of hole- doping, analogous solutions with alternating spin cur- Figure 4a shows the intensity map I(k, ω = 0) calcu- rents in plaquets and with the rotation of the local spin lated by formulas (15) and (17) for a stripe structure. by π/2 upon a transition between neighboring sites The structure consists of AF stripes parallel to the y axis along the perimeter of a plaquet exist only for large val- for a model with U/t = 6, t'/t = 0.1 for n = 0.8. In the case ues of U (U/t ≥ 5). The energies of such structures are of the homogeneous AF solution, this doping is close to higher than the energy for stripe and spiral structures that for which the Fermi level passes through a VHS in and the shape of their FSs is nonrealistic. For this rea- the DOS of the lower Hubbard band. The stripe struc- son, the data on these structures are not given here. ture parallel to the y axis splits the VHS, makes hot π π spots Mx = ( , 0) and My = (0, ) nonequivalent, and The calculations were made mainly for structures forms the main and shadow horizontal segments of the with a ﬁxed (commensurate) period of 8a, although the FS. Figure 4b shows an intensity map symmetrized in optimal size of domains or vector Q for the spiral state δ structures with domains of the x and y orientations. The depends on the doping level. Dependences Q( ) for the FS shape is analogous to that obtained in other calcula- spiral state were calculated by many authors (see, for tions of stripe phases [36]. It differs considerably from example, [35]). However, we are interested here in the the FS typical of homogeneous AF solutions in a model typical features of FSs and anisotropy of PGs for deﬁ- with t' > 0. The main difference, i.e., the absence of the nite structures. Fermi boundary in the diagonal direction, indicates the ± It can be seen from Fig. 3a that, in the case of h dop- emergence of a PG at kx = ky. As a consequence, we ing, homogeneous AF states are unstable to the forma- can expect that, even in the case of the d symmetry of tion of stripe structures and spiral spin states. The for- the SC state, the minimal energy of Fermi excitations mation of such structures extends the doping region in differs from zero in accordance with an equation simi- which nonzero values of local spin are preserved lar to Eq. (20) for homogeneous AF solutions with e 〈 〉 ≠ doping. As the value of t' increases to 0.3, small hole ( Sn 0); vanishing of these values corresponds to π π pockets are formed additionally at points kS = ( /2, /2). merging of energy H()δ for a given structure with the The major part of the previous zero Fermi boundary, in π energy of the PM state. At the same time, in the case of particular, at point My = (0, ), becomes a PG and electron doping, the MF energies of all the structures dielectrized Fermi boundary; only quasi-one-dimen- listed above were found to be higher than the energy of sional segments of the FS, which are normal to the the homogeneous AF state. The higher stability of the directions of stripes (y axis), are preserved. Accord- AF state with e doping corresponds to a broader doping ingly, a quasi-one-dimensional conductivity of such a region of the long-range AF order in n-type cuprates. structure can also be expected. Invisible PG segments The peak in the spin susceptibility for Q ~ (π, π) in of the generalized Fermi boundary can be visualized these cuprates (in contrast to incommensurate values of while constructing smoothed intensity (18) with a large Q in p-type cuprates) also indicates the absence of frequency window ∆ω.

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(‡) ω

YMy YYMx

(b) E

π π π Fig. 5. (a) Self-energies Eλ(k) in the MF problem as functions of quasimomentum varying along the contour Y(Ð , )ÐMy(0, )Ð π π π π π ω Y( , )ÐMx( , 0)ÐY( , Ð ) for a stripe structure with a period of 8a along the x axis. (b) Map of intensities I(k, ) for k varying along the same contour. The map reveals the same energy levels Eλ(k), but with different weights determined by the structure of band states. The parameters of the model are the same as in Fig. 4. The Fermi level is marked by the horizontal line.

Figure 5a shows one-electron energies Eλ(k) (eigen- tions), systematic suppression of the spectral weight in values of the MF problem) as functions of the quasimo- the vicinity of point (π/2, π/2) as compared to BSCCO mentum varying along the contour compounds or with overdoped LSCO samples, and straight FS segments near point (π, 0) with a width ~π/2 ()π, π (), π ()ππ, ()π, ()ππ, Y Ð Ð My 0 Ð Y Ð Mx 0 Ð Y Ð . were also interpreted as the proof of the existence of inhomogeneous structures in underdoped compound As usual, band energies Eλ(k) are periodic functions LSCO [6, 7, 37] (in particular, the combination of with a period of π/4 on the first (horizontal) segment of orderÐdisorder with stripe structures [38]). Another contour Y Ð My Ð Y. However, the intensity map reveals proof of the existence of stripe structures is the obser- only nonshadow energy levels for a given k. Figure 5b vation of neutron scattering peaks for incommensu- represents such a map on plane k, ω for k varying along rate momenta (π ± δ, π) and (π, π ± δ) in compound the same contour. The intersection of energy levels with LSCO [15]. the Fermi level in the vicinity of Mx corresponds to the quasi-one-dimensional FS shown in Fig. 4a. Anisotropy of the FS (see Fig. 4a) suggests a revision of the admissible symmetry of SC order compati- The conservation of the FS in the vicinity of Mx and ble with a stripe structure. For structures symmetric rel- the formation of a pseudogap in the vicinity of My and ative to transposition of axes x y, the d-wave SC for k ~ k are due to the action of the spin-dependent † † x y order 〈〉c , ↑c , ↓ ∝ (cosk Ð cosk ) is expected. It mean field; the principal harmonic of this field is k Ðk x y ensures the orthogonality of the pair function to one- F(n) ∝ cosQηn with Qη = (ηπ, π) (here, η = 0.75). This ⑀ π † † field elevates the zero level (0, ) at point My, repelling center pair function cn↑cn↓ suppressed by one-center ⑀ ±ηπ it from lower zero levels k at point k = ( , 0) in the repulsion U. However, in the case of the quasi-one- vicinity of Mx. The same field lowers (below the chem- dimension FS depicted in Fig. 4a, the generalized ⑀ π † † ical potential) zero level ( , 0) at point Mx, repelling it s-wave pair function 〈〉c , ↑c , ↓ ∝ (cosk + cosk ) is ⑀ ±0.25π π k Ðk x y from higher zero levels ( , ) near My. more probable. The latter function may be orthogonal ± The ARPES data for underdoped LSCO compound to the one-center pair function due to nodal lines kx ±π [6, 7, 37] are in qualitative agreement with the features ky = . The role of nodal lines in the angular depen- of the intensity maps represented in Fig. 4. The pres- dence of the SC gap will now be played by the nodal ence of two segments of the FS with different proper- lines of the “radial” part of the pair function if quantity ties (in the vicinity of points M and in diagonal direc- z = coskx + cosky can be treated as the radial variable.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 554 OVCHINNIKOV, OVCHINNIKOVA

(‡) (b)

Fig. 6. (a) Photoemission intensity map I↑(k, ω = 0) for electrons with spin polarization σ = ↑ for a spiral structure with Q = π(0.8, 1). (b) The same averaged over two spin polarizations σ = ↑, ↓ and two structures with Q = (0.8π, π) and Q = (π, 0.8π) in the model with parameters U = 6 and t' = 0.1.

Verification of this hypothesis requires appropriate cal- effect of time-reversal symmetry breaking (TRSB) in culations. the dichroism of the ARPES signal observed for the underdoped compound BSCCO [12]. (In the alternative hypothesis [12, 41], the TRSB effect is explained by a 7. SPIRAL SPIN STRUCTURES peculiar aligning of circular microcurrents.) A direct AND SPIN POLARIZATION OF VARIOUS observation of polarization anisotropy of the FS SEGMENTS OF THE FERMI SURFACE requires selective (in spin polarization) measurement of It was found earlier [39] that spiral spin structures (22) the photoemission intensity. In the measurements of the exhibit polarization anisotropy of the FS. Different seg- total photoemission intensity, such a selectivity was ments of the FS correspond to electrons with different achieved in the so-called spin-orbit photoemission [42]. predominant spin polarizations. Indeed, the mean field Such measurements are also possible in principle in from a spiral spin structure mixes one-electrons states ARPES. It is important to continue the study of TRSB {}† , † in the underdoped BSCCO compound; in particular, it ck, ↑ ckQ+ , ↓ and leads to splitting of VHS as in the is expedient to find out whether this effect and the cur- case of stripe structures. However, in contrast to stripe rent associated with it are of volume or surface nature. 〈〉† For the time being, it remains unclear whether the structures, occupancies nk, σ = ckσckσ and intensities ground state in BSCCO can be presented as a set of Iσ(kω) of photoemission of electrons with a fixed polar- quasi-static domains with a spiral structure and with a ization σ depend on the spin polarization σ. Quantities system of spin currents associated with this set. Iσ(kω) are defined by formulas (15) and (17), but do not contain summation over σ on the right-hand side of Eq. (17). 8. CONCLUSIONS ω Figure 6 shows the intensity map Iσ = ↑(k, = 0) for The MF analysis of the normal state in the tÐt'ÐU a spiral state with Q = (0.8π, π) for spin polarization Hubbard model revealed that the FS topology and PG σ = ↑ on the z axis perpendicular to the plane of rota- anisotropy in the underdoped region depend on the sign tion of mean spins with a spiral configuration. Thus, of t', the type of (e- or h-) doping, and the spin structure. FSs revealed in photoemission of electrons with polar- In hole-doped models, the homogeneous AF mean-field ization σ = ↑ exhibit anisotropy. For the opposite spin solution is found to be unstable to the formation of two polarization, the FS is the reflection of the FS depicted types of spin structures, viz., the stripe phase and the in Fig. 6a in the plane x = 0. Figure 6b shows the Fermi spiral spin structure. However, in models with electron surfaces symmetrized in spins and in two types of struc- doping, the homogeneous AF solution remains the low- tures with Q = (0.8π, π) and Q = (π, 0.8π). In the vicin- est among the solutions considered. This corresponds ity of point M, the symmetrized FSs have the intersec- to a large (in doping level) region of existence of the AF tion with MÐY line (typical of hole-type FS) as well as order in n-type cuprates (such as NCCO and PCCO) as with M Ð Γ line (typical of the electron-type FS). Such well as to the commensurate peak observed in neutron double intersections were apparently observed in the scattering for these compounds at Q = (π, π). For homo- ARPES spectra for BSCCO [1, 2, 40]. Direct compari- geneous AF solutions in underdoped models, a PG son with experiment is impossible since the two-layer arises in the antinodal direction k ~ (π, 0), (0, π) for hole splitting is disregarded in the model. doping and in diagonal (nodal) directions kx = ky for Polarization anisotropy of the FS directly reflects electron doping. The latter solution gives example of a the presence of spin currents J↑ = ÐJ↓. According system for which d-type superconductivity is combined to [39], this anisotropy could be responsible for the with a finite minimal gap of Fermi excitations. In accor-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 EFFECT OF SPIN STRUCTURES AND NESTING ON THE SHAPE OF THE FERMI SURFACE 555 dance with ARPES data for the LSCO compound, the 16. C. Howald, H. Eisaki, N. Kaneko, et al., cond- calculation of the stripe phase showed that it is charac- mat/0201546; Phys. Rev. B 67, 014533 (2003). terized by quasi-one-dimensional segments of the FS, 17. J. E. Hoffman, E. A. Hudson, K. M. Lang, et al., Science the emergence of a PG, and the suppression of spectral 295, 466 (2002). density in the diagonal direction and in the direction 18. S. Chakravarty, R. B. Laughlin, D. K. Morr, and parallel to stripe domains. Such anisotropy of the FS Ch. Nayak, cond-mat/0005443; Phys. Rev. B 63, and PG is incompatible with the d symmetry of the SC 094503 (2001). order. For a spiral spin structure, polarization anisot- 19. J. E. Hirsch, Phys. Rev. Lett. 54, 1317 (1985). ropy of the FS is detected when different FS segments 20. A. A. Ovchinnikov and M. Ya. Ovchinnikova, Zh. Éksp. correspond to different spin polarizations of electrons. Teor. Fiz. 110, 342 (1996) [JETP 83, 184 (1996)]; Zh. This property can be used for testing spiral spin struc- Éksp. Teor. Fiz. 112, 1409 (1997) [JETP 85, 767 (1997)]. tures. Among unsolved problems, we can mention the 21. N. M. Plakida, V. S. Oudovenko, R. Horsch, and study of the structures of valence bonds and current A. J. Liechtenstein, Phys. Rev. B 55, 11997 (1997). states and the transition from analysis of quasi-static 22. A. A. Ovchinnikov and M. Ya. Ovchinnikova, Phys. Lett. structures to dynamic ﬂuctuations. A 249, 531 (1998). 23. A. A. Ovchinnikov, M. Ya. Ovchinnikova, and E. A. Ple- khanov, Pis’ma Zh. Éksp. Teor. Fiz. 67, 350 (1998) ACKNOWLEDGMENTS [JETP Lett. 67, 369 (1998)]; Zh. Éksp. Teor. Fiz. 114, The author is grateful sincerely to V.Ya. Krivnov for 985 (1998) [JETP 87, 534 (1998)]; Zh. Éksp. Teor. Fiz. helpful remarks and for his assistance in the research. 115, 649 (1999) [JETP 88, 356 (1999)]. 24. R. O. Kuzian, R. Hayn, A. F. Barabanov, and L. A. Mak- This study was supported ﬁnancially by the simov, Phys. Rev. B 58, 6194 (1998). Russian Foundation for Basic Research (project 25. F. Onufrieva, P. Pfeuty, and M. Kisilev, Phys. Rev. Lett. no. 03-03-32141). 82, 2370 (1999). 26. A. A. Ovchinnikov and M. Ya. Ovchinnikova, Zh. Éksp. REFERENCES Teor. Fiz. 118, 1434 (2000) [JETP 91, 1242 (2000)]. 27. C. Kusko and R. S. Markiewicz, Phys. Rev. Lett. 84, 963 1. A. Damascelli, Z.-X. Shen, and Z. Hussain, Rev. Mod. (2000). Phys. 75, 473 (2003). 28. C. Kusko, R. S. Markiewicz, M. Lindroos, and A. Bansil, 2. Z. M. Shen and D. S. Dessau, Phys. Rep. 253, 1 (1995). cond-mat/0201117; Phys. Rev. B 66, 140513 (2002). 3. Y.-D. Chuang, A. D. Gromko, D. S. Dessau, et al., Phys. 29. M. R. Norman, H. Ding, M. Randeria, et al., Nature 392, Rev. Lett. 83, 3717 (1999). 157 (1998). 4. H. M. Fretwell, A. Kaminski, J. Mesot, et al., Phys. Rev. 30. T. Timusk and B. Statt, Rep. Prog. Phys. 62, 61 (1999). Lett. 84, 4449 (2000). 31. M. R. Norman and C. Pepin, cond-mat/0302336. 5. S. V. Borisenko, M. S. Golden, S. Legner, et al., Phys. Rev. Lett. 84, 4453 (2000). 32. H. Balci, V. N. Smolyaninova, P. Fournier, et al., Phys. Rev. B 66, 174510 (2002). 6. A. Ino, C. Kim, T. Mizokawa, et al., J. Phys. Soc. Jpn. 68, 1496 (1999). 33. N. P. Armitage, D. H. Lu, D. L. Feng, et al., Phys. Rev. Lett. 86, 1126 (2001). 7. T. Yoshida, X. J. Zhou, M. Nakamura, et al., cond- mat/0011172. 34. A. Koitzsch, G. Blumberg, A. Gozar, et al., cond- mat/0304175. 8. D. L. Feng, N. P. Armitage, D. H. Lu, et al., Phys. Rev. Lett. 86, 5550 (2001). 35. A. A. Ovchinnikov and M. Ya. Ovchinnikova, Zh. Éksp. Teor. Fiz. 116, 1058 (1999) [JETP 89, 564 (1999)]. 9. Y. D. Chuang, A. D. Gromko, A. Fedorov, et al., Phys. Rev. Lett. 87, 117002 (2001). 36. X. J. Zhou, P. Bogdanov, S. A. Kellar, et al., Science 286, 268 (1999). 10. A. A. Kordyuk, S. V. Borisenko, T. K. Kim, et al., cond- mat/0110379; Phys. Rev. Lett. 89, 077003 (2002). 37. A. Ino, C. Kim, M. Nakamura, et al., Phys. Rev. B 62, 4137 (2000). 11. A. Kaminski, S. Rosenkranz, H. M. Fretwell, et al., cond-mat/0210531. 38. X. J. Zhou, T. Yoshida, S. A. Kellar, et al., Phys. Rev. Lett. 86, 5578 (2001). 12. A. Kaminsky, S. Rosenkranz, H. M. Fretwell, et al., cond-mat/0203133. 39. M. Ya. Ovchinnikova, Zh. Éksp. Teor. Fiz. 123, 1082 (2003). 13. S. V. Borisenko, A. A. Kordyuk, S. Legner, et al., Phys. Rev. B 64, 094513 (2001). 40. D. S. Dessau, Z. X. Shen, D. M. King, et al., Phys. Rev. Lett. 71, 2781 (1993). 14. M. Lindroos, S. Sahrakorpi, and A. Bansil, Phys. Rev. B 65, 054514 (2002). 41. C. M. Varma, Phys. Rev. Lett. 83, 3538 (1999). 15. R. J. Birgeneau and G. Shirane, in Physical Properties of 42. G. Ghiringhelli, L. H. Tjeng, A. Tanaka, et al., Phys. Rev. High Temperature Superconductors, Ed. by D. M. Gin- B 66, 75101 (2002). zberg (World Sci., Singapore, 1989; Mir, Moscow, 1991), Vol. 1. Translated by N. Wadhwa

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 556Ð564. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 630Ð639. Original Russian Text Copyright © 2004 by Gavrichkov, Ovchinnikov. SOLIDS Electronic Properties

The Band Structure of n-Type Cuprate Superconductors with the T'(T) Structure Taking into Account Strong Electron Correlation V. A. Gavrichkov and S. G. Ovchinnikov Kirenskiœ Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia e-mail: [email protected]; [email protected] Received June 6, 2003

Abstract—The spectral density, dispersion relations, and the position of the Fermi level for n-doped compositions based on NCO and LCO were calculated within the framework of the generalized tight binding method. As distinguished from LCO, the dielectric gap in NCO is nonlinear in character. We observe a virtual level both at the bottom of the conduction band and at the top of the valence band in both compounds. However, its position corresponds to the extreme bottom of the conduction band in LCO and is 0.1Ð0.2 eV above the bottom in NCO. This explains why we observe Fermi level pinning in n-LCO as the concentration of the doping component grows and reproduce its absence in NCCO at low doping values. We also found both compositions to be unstable in a narrow concentration range with respect to a nonuniform charge density distribution. The relation between the phase diagram for NCCO and the calculated electronic structure is discussed. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION however, remain virtually identical to those in LSCO. Angle-resolved photoelectron spectra also revealed the Superconducting n-type cuprates Nd2 Ð xCexCuO4 appearance of intragap states when either NCCO was (NCCO) and Pr2 Ð xCexCuO4 (PCCO) have certain spe- doped with electrons or LSCO was doped with holes [5]. cial features that distinguish them from p-type cuprates. As distinguished from LSCO, where chemical potential Rare-earth metal and oxygen layers are shifted in their pinning occurs at low x, NCCO shows a more complex crystal structure with respect to CuO2 in such a way that concentration dependence of the chemical potential [6]. there is no nearest neighbor (the apical oxygen) for cop- This is one more important difference between per along the c axis; that is, the structural element of the NCCO and LSCO, which is related to the possible role CuO2 layer is a plane CuO4 cluster, whereas it is a CuO6 that neodymium f electrons can play in the formation of octahedron in La2 Ð xSrxCuO4 (LSCO). The phase dia- the state of heavy fermions at low temperatures [7]. gram of NCCO is substantially different from the phase However, ﬁrst, no unambiguous experimental substan- diagram of LSCO (Fig. 1). The initial undoped Nd2CuO4 (NCO) composition is an antiferromagnet and dielectric. Doping with electrons rather weakly T, K suppresses the antiferromagnetic state [1] because of the diamagnetic dilution mechanism [2]. The supercon- 300 Nd2 – xCexCuO4 – y La2 – xSrxCuO4 – y ducting state borders on the antiferromagnetic phase T'-structure T-structure and exists in a narrow concentration range x < x < min 200 xmax, where xmin = 0.14 and xmax = 0.17. In the normal state, the electrical conduction of NCCO is described by AFM PS a Fermi-liquid quadratic temperature dependence [3], as 100 distinguished from linear dependences for hole high-TC superconductors [4]. It was shown for NCCO by angle- SC resolved photoelectron spectroscopy [5] that the dielec- 0 SC tric gap in this compound was not rectilinear. The min- 0.3 0.2 0.1 0 0.1 0.2 0.3 imum of the conduction band and the maximum of the x valence band belong to different Brillouin zone points, Fig. 1. Phase diagram of LSCO and NCCO. Composition k = (π, 0) and k = (π/2, π/2), respectively. The disper- regions: SC, superconducting phase; PS, pseudogap state; sion relations in NCCO for the top of the valence band, and AFM, dielectric phase in the antiferromagnetic state.

THE BAND STRUCTURE OF n-TYPE CUPRATE SUPERCONDUCTORS 557 tiation [8] of the existence of such a state has been Fermi level in NCCO and n-LCO. The character of obtained and no explanation of the large coefficient γ valence band dispersion remains virtually identical in value in the linear temperature dependence of heat compounds of both types. All results obtained for capacity, c = γT, where γ ≈ 4J/k2 for x > 0.15 [9], has n-LCO have no experimental analogs and are predictive been suggested. Secondly, at a low doping level (0.05 ≤ in character, because no n-LCO materials with the T x < 0.14), the γ value is an order of magnitude smaller, structure have been prepared as yet. In particular, 2 γ = 0.3J/k [9], which leads us to conclude that, even if n-type superconducting compositions La2 Ð xCexCuO4 the state of heavy fermions of the new type does exist obtained in [16] had the T ' structure. Nevertheless, in NCCO, these effects manifest themselves at higher there is a possibility of inverting p-type LCO by the concentrations because of the low NdÐCu spinÐspin field effect, as with field-effect transistors, where coupling constant. In this work, we consider low dop- applying a positive voltage to the gate results in the for- ing levels, and, at x < 0.15, the beautiful physics of mation of an inversion layer in a p-type semiconductor heavy fermions [7] remains outside the scope of our at the boundary with the gate. A theoretical study of the analysis. electronic structure of n-LCO with the T structure is The purpose of this work was to study the electronic therefore of interest. structure of NCO and La2CuO4 (LCO) undoped and In Section 2, we discuss the most important changes weakly doped with electrons. We use the same calcula- in the generalized tight binding method for n-type tion methods taking into account strong electron corre- cuprates and give the initial equations for dispersion lation as were earlier used by us to study hole cuprates. relations and spectral density. The dispersion depen- Note that strong electron correlation is of funda- dences for the conduction band in NCCO in compari- mental importance and should be correctly taken into son with similar dependences for n-LCO and experi- account. Indeed, one-electron band calculations give mental dependences are studied in Section 3. In Sec- the ground state of LCO and NCO in the form of a tion 4, we calculate the partial contributions of various metal with a half-filled band, which is at variance with orbitals to the spectral density of the conduction band experiment. At the same time, the simplest strong elec- and study the density of states at the bottom of the con- tron correlation models like the Hubbard model are duction band in both compounds. The positions of the excessively simplified, and the degree of their applica- Fermi level in NCCO at various doping component bility to describing the band structure of a particular concentrations are determined in Section 5, where the substance is not known a priori. Our experience in instability of the state with a uniform charge density distribution is also studied. The results of our calcula- studying the electronic structure of high-TC cuprates taking into account strong electron correlation shows tions are briefly summarized in Section 6. [10Ð12] that the multiband pÐd model [13] is the most suitable at excitation energies up to 3 eV inside the 2. DISPERSION RELATIONS valence and conduction bands. This model takes into AND SPECTRAL DENSITY account two d orbitals of copper, d 2 2 ≡ d and x Ð y x OF QUASI-PARTICLE STATES IN NCCO AND n-LCO d 2 2 ≡ d . We used the generalized tight binding 3z Ð r z method to calculate the band structure of quasi-particles In the generalized tight binding method, the Hamil- taking into account strong electron correlation [14]. In tonian of the CuO2 layer can be written in the form this method, the many-electron Hamiltonian within the cell is exactly diagonalized, many-electron molecular λ λ λ † 1 1 2 † orbitals are found, and Hubbard X-operators are con- H = ∑ε a λσa λσ + ---∑ ∑ V a λ σ a λ σ i i i 2 ij i 1 1 i 1 3 structed at the first stage. At the second stage, intercell iλσ ij, λ λ σ σ σ σ jumps are included and the band structure of the crystal 1 2 1 2 3 4 (1) λ λ is calculated. Particular examples of calculations of † 1 2 † × a λ σ a λ σ + ∑ ∑ t a λ σa λ σ. hole cuprates by this method are given in [11, 12, 15]. i 2 2 i 2 4 ij i 1 j 2 〈〉, λ λ σ In this work, we calculated the spectral density and ij 2 2 dispersion for the conduction band in compounds undoped and weakly doped with electrons. The calcu- Here, aiλσ is the hole annihilation operator in the Wan- lations by the generalized tight binding method were nier state on node i (copper or oxygen) for orbital λ and performed for NCCO with the T ' structure and n-type spin σ. Two copper orbitals (d 2 2 and d 2 ) and two x Ð y z La2CuO4 with the T structure. We show that there is a p Ð and p orbitals on each oxygen node that form σ virtual level typical of systems with strong electron cor- x/y z relation both at the bottom of the conduction band and bonds with the specified copper orbitals are included. at the top of the valence band [11] in LCO and NCO. Among the Coulomb matrix elements, we can identify The positions of this level in the two compounds are, intraatomic Hubbard elements Ud(Up) for repulsion in however, different. The observed asymmetry also one copper (oxygen) orbital between electrons with results in different concentration dependences of the opposite spins, interorbital Coulomb elements Vd(Vp),

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Jd(Jp) exchange matrix elements, and interatomic Cou- persion relations and spectral density can be written in lomb repulsion parameters Vpd, which we, for simplic- the form [11] ity, consider identical for all orbitals. The last term ()δΩG in (1) describes interatomic copperÐoxygen jumps with E Ð m mn γ ()PG()γ () ------Ð2∑ λσ* m T λλ' k λ'σ n = 0, (2) x2 Ð y2, x/y z2, x G ≡ ≡ Fσ ()m the parameters t pd tpd and t pd tpd /3 and λλ' xy, ≡ oxygenÐoxygen jumps with the parameter t pp tpp. The charge transfer energy will be denoted by ∆ = ε Ð (), 1 ()λλ 1 pd p Aσ k E ==Ðπ--- ∑Im Gkσ Ðπ--- ε , and the energy of splitting of the d level in the λ d 2 2 x Ð y (3) uniaxial crystal field component, by ∆ = ε 2 Ð ε . + mn mn d dz d 2 2 ×γ()γ() ()() () x Ð y ∑ λσ m λσ n Im Dkσ AA + Dkσ BB , 2Ð Of six O ions, two apical ones are situated along the λmn c axis in the T structure of the LCO composition. Their effects are controlled by two calculation parameters, where t' and t' , which are the integrals of electron jumps pd pp λλ' 〈〉〈〉† γ ()γ† () mn from copper and in-pane oxygen to apical oxygen. Gkσ ==akλσ akλ'σ E ∑ λσ m λ'σ n Dkσ , (4) In the generalized tight binding method, the band mn structure of quasi-particles including strong electron ˆ ()ˆ () correlation effects is calculated in two stages. At the ˆ Dkσ AA Dkσ AB first stage, the CuO layer lattice is partitioned into Dkσ = , 2 ˆ ()ˆ () (5) many unit cells, and the Hamiltonian within one cell is Dkσ BA Dkσ BB exactly diagonalized. In addition to selecting the CuO6 mn m n D σ ()AB = 〈〉〈〉X σ Y σ . cluster as the unit cell, the problem of constructing the k k k E Wannier functions of b1g and a1g symmetry on the ini- Here, indices P and G run over the A and B antiferro- tial oxygen orbitals is solved [11, 12]. The many-elec- | 〉 magnetic state sublattices. Equations (2) and (3) were tron molecular orbitals n, p (where n = 0, 1, 2, … is the obtained for the antiferromagnetic phase [11, 12] using number of holes in the cell and p denotes the set of the the equations of motion for Green function (5) in the other orbital and spin indices) obtained by diagonaliz- Hubbard I approximation for intercell jumps. The ele- ing the cell Hamiltonian H0 are used to construct the ments of the tight binding matrix Hubbard operators of this cell, Xm = |n + 1, p〉〈n, q|, and m γ ikR one-electron operators, afλσ = ∑m λσ (m)X fσ . Here, AA BB 2 AA 1 T λλ ()k ==T λλ ()k ----∑T λλ ()R e , the band index of quasi-particles m numbers one-elec- ' ' N ' 1 tron excitations from the initial state |n, q〉 to the final R1 state |n + 1, p〉 [17].

AB BA 2 AB ikR2 As distinguished from LCO, oxygen and rare-earth T λλ ()k ==T λλ ()k ----∑T λλ ()R e metal elements in NCO are known to form their own ' ' N ' 2 R2 separate planes in the environment of the CuO2 plane, and the plane of the rare-earth metal is closest to the in the ﬁve-orbital dx, dz, b, a, pz basis take the form CuO2 plane. In this situation, both parameters should be ' ' () close to zero, t pd = 0 and t pp = 0. Additional changes T λλ' R in the other parameters were not introduced beforehand  and were taken from the calculations of the electronic 002Ð00t µ structure of p-type cuprates [11]. The initial parameters pd ij 2t ξ 2t λ of our Hamiltonian were: 00------pd ij ------pd ij- 0 3 3 ε ==0, ε 2 eV,  dx dz ξ (6) µ 2t pd ij ν χ ξ = Ð2t pd ij ------Ð22t pp ij t pp ij Ð2t'pp ij ; ε ==1.6 eV, ε 0.5 eV,  p pz 3 λ 2t pd ij χ ν λ t ====0.46 eV, t' 0, t' 0, U 9 eV, 0 ------2t pp ij 2t pp ij Ð2t'pp ij pp pp pd d 3  ' ξ ' λ U p ==4 eV, V pd 1.5 eV, Jd =1 eV.  002 Ð2t pp ij Ð0t pp ij µ ξ λ ν χ In the generalized tight binding method, the dis- the ij, ij, ij, ij, and ij coefﬁcients were given

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Energy, eV 3. DISPERSION OF BANDS 2.0 The band dispersion for n-LCO and NCCO at a concentration of the doping n-component of x = 0.03 in the 1.5 antiferromagnetic phase is shown in Fig. 2. Calcula- tions only reproduce dispersion in the immediate vicin- 1.0 ity of the dielectric gap. This is sufficient for analyzing the spectrum of quasi-particles involved in the super- 0.5 conducting state. The bottom of the conduction band is formed as a 0 result of the dispersion of the local state with an energy Ω 2 of c = E(1, b1g) Ð E(0, a1g), and the top of the valence –0.5 band is formed by excitations with the participation of Ω 1 2 the two-hole singlet ν = E(2, A1g) Ð E(1, b1g) and –1.0 Ω1 3 2 triplet ν = E(2, B1g) Ð E(1, b1g) (Fig. 3). Both com- π π π π π pounds have a virtual level at the bottom of the conduc- (0, 0) ( , )(, 0) (0, 0) ( , 0) (0, ) tion band. This level is similar in nature to that at the top kx, ky of the valence band (Fig. 3) [11]. Namely, there are two types of quasi-particles that correspond to possible Fig. 2. Dispersion dependences for n-LCO (solid line) and Ω Ω NCCO (dotted line). Doping level x = 0.03. transitions c and ν. One of the quasi-particles in the undoped compound corresponds to the transition between empty states, which gives zero contributions to in [11]. Equation (2) is an analog of the dispersion dispersion and spectral density. In the one-hole sector equation in the tight binding method and differs from it of the configuration space, the empty state is one of the in two respects. First, quasi-particle energies are calcu- components of the spin doublet in each of the sublattices of the Néel antiferromagnetic state of the CuO2 lated in the form ΩG = ε Ð ε , that is, in the form m 2qG 1pG layer. The vacuum sector corresponds to the a1g singlet of resonances between many-particle states from dif- state of the fully occupied p6d10 shell. The existence of ferent configuration space sectors. Secondly, the occu- two singlet states in the vacuum and two-hole sectors G pp qq (Fig. 3) is the main reason for the existence of the dis- pation number Fσ (m) = 〈〉X σ + 〈〉X σ leads to con- f G f G persionless virtual level not only at the top of the centration dependences of both dispersion and spectral valence band but also at the bottom of the conduction density amplitude (3). Quasi-particle states with differ- band in n-LCO and NCCO. ent m can overlap and interact, like singlet and triplet Conduction band dispersion in n-LCO was calcu- two-hole states of p-type cuprates do [11, 12]. lated with the parameter values obtained in studying the

3 3 (a) LSCO B1g (b) NCCO B1g

∆ε ∆ε 2 2 2 2 a1g A1g a1g A1g x

∆ε ∆ε 1 1 2 2 b1g b1g 1 – x 1 – x a1g a1g x n = 0 n = 1 n = 2 n = 0 n = 1 n = 2

Fig. 3. Conﬁguration space scheme for LSCO and NCCO. The solid lines correspond to quasi-particle transitions that form rigid bands, and the dashed lines, to impurity bands.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 560 GAVRICHKOV, OVCHINNIKOV electronic structure of p-type cuprates. Mere compari- tem of CuO6 octahedra. As a consequence, a small son of the spectra for n-LCO and NCCO in Fig. 2 shows increase in tpp in NCCO is responsible for the conduc- that the degeneracy of the spectrum of LCO at point X = tion band minimum at point X at low electron concen- (π, 0) is accidental. Conversely, the intersection of two trations. In combination, changes in precisely these conduction bands, the broad band and the band of vir- parameters qualitatively modify the dielectric gap and tual level states, at point M = (π/2, π/2) is caused by make it nonlinear in NCO. CuO2 layer symmetry and does not depend on the Among the known materials based on n-LCO and parameter values used in the calculations. having the T structure, the calculated dispersion might The broad band and the band of virtual level states be observed in La2Cu1 Ð xZnxO4 compositions [1], behave differently as the doping level increases [11, 15]. because Zn2+ has a completely filled d10 shell, which The broad bands remain virtually unchanged; they will formally corresponds to filling the vacuum sector, as in further be called rigid by analogy with the rigid band n-type materials such as NCCO. However in reality, the model. The spectral density and dispersion for the vir- Zn impurity leads to a strongly bound carrier and the tual level state bands increase as the degree of doping violation of translational invariance over the spin lat- grows, and they will be called “impurity” bands. Quo- tice. It appears that, because of strong impurity effects, tation marks (further omitted) are not meaningless, photoemission measurements for La2Cu1 Ð xZnxO4, sim- because these states have no bearing on the true local ilar to those performed for NCCO [5], cannot be made. impurity potential. The transport of quasi-particles in the valence and conduction bands occurs with different effective trans- 4. PARTIAL CONTRIBUTIONS port integrals. We therefore observe different disper- TO SPECTRAL DENSITY sion dependences for different bands. Indeed, calcula- We calculated the spectral density A(k, E) for the tions give tc.b.' /tc.b. = Ð0.05 and t'ν.b. /tν.b. = Ð0.14 for rigid and impurity bands in NCCO (Fig. 4) at a low con- ' ' centration of the doping component x = 0.03. The spec- NCO and tc.b. /tc.b. = 0.05 and tν.b. /tν.b. = Ð0.085 for tral density is characterized by two peaks correspond- LCO, where tc.b. (tc.b.' ) and tν.b. (t'ν.b. ) are the effective ing to the rigid and impurity bands. The dependences of transport integrals between the nearest (next-nearest) the peak amplitudes for (a) the rigid and (b) impurity neighbors for the conduction and valence bands, bands along the symmetrical Brillouin zone directions respectively. The most significant change in passing are plotted in Fig. 4. Figure 5b shows how the virtual from LCO to NCO is the formation of a nonlinear level with zero spectral weight at x = 0 transforms into dielectric gap because of the formation of a new mini- an impurity band with spectral weight x. The over- mum at the X point of the conduction band. Calcula- shooting of triplet states into the conduction band is tions show that the appearance of a rectilinear gap is insignificant, and this is one more source of asymmetry ' of the states of p- and n-type carriers. Similar depen- also accompanied by a change in the sign of the tc.b. /tc.b. dences for the conduction band in n-LCO are shown in ratio. There are moderate changes in the valence band, Fig. 5. These results cannot however be compared with but they do not lead to qualitative differences in the dis- experimental data because of the absence of n-type persion dependences for the n and p materials. compounds based on LCO with the T structure. The reproduction of dispersion at the bottom of the As follows from calculations of the density of states, conduction band in NCCO requires the initial parame- there is a region with a reduced density between the ters to be changed as follows: impurity and rigid bands (Fig. 6). This pseudogap vanishes at point M = (π/2, π/2) (Fig. 2). For this reason, ε ==0.2 eV, ε 2 eV, dx dz the passage of the Fermi level from the rigid to the ε ε p ==1.6 eV, p 0.5 eV, impurity band may be accompanied by a decrease in the z density of states on this level. The pseudogap itself is magnetic in nature, as follows from its absence in the t ==0.56 eV, t' 0.1 eV, pp pp paramagnetic phase. Because of the special features t'pd ==0, Ud 9 eV, inherent in the spectrum, the pseudogap is more pronounced for the density of states of NCCO.

U p ==4 eV, V pd 1.5 eV, Jd =1 eV. 5. THE CONCENTRATION DEPENDENCE It follows that dispersion calculations in NCCO OF THE FERMI LEVEL ' ' largely result in changes in the t pd and t pp values and, Calculations of the dependence of the Fermi level ∆ ∆ to a lesser extent, in pd and tpp. A smaller pd value cor- position on doping in NCCO for the antiferromagnetic responds to a smaller dielectric gap in NCO, Eg = phase exhibit considerable differences from the depen- 1.6 eV. A smaller tpp value in LCO in turn corresponds dence obtained for n-LCO. Indeed, at low concentra- to the presence of orthorhombic distortions in the sys- tions, the Fermi level in NCCO goes deep into rigid

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4000 (a) (b) 3500

3000

2500 x = 0 x = 0.03 x = 0 2000 x = 0.03 1500

1000 Spectral density, arb. units arb. Spectral density,

500

0 (0, 0) (π, π) (π, 0) (0, 0) (π, π) (π, 0) (0, 0)

Fig. 4. Spectral density of quasi-particle states along symmetrical Brillouin zone directions in NCO (a) for the rigid conduction band and (b) for the band of virtual level states.

4000 (a) (b) 3500

3000

2500 x = 0 x = 0.03 2000

1500 x = 0.03 1000 Spectral density, arb. units arb. Spectral density,

500 x = 0

0 (0, 0) (π, π) (π, 0) (0, 0) (π, π) (π, 0) (0, 0)

Fig. 5. Spectral density of quasi-particle states along symmetrical Brillouin zone directions in n-LCO (a) for the rigid conduction band and (b) for the impurity band.

conduction band states and only then, at x = x1, into The calculated concentration dependence of the ∆ impurity band states. This corresponds to x1 = 0.08Ð0.1 Fermi level contains a x interval where the rate of in Fig. 7. The Fermi level reenters rigid band states at growth of the number of states in the impurity band x2 = 0.18Ð0.2. A similar dependence of the Fermi level exceeds the rate of increasing the number of carriers ∂µ ∂ ∂2Φ ∂ 2 in n-LCO shows pinning only at low concentrations. / x = ( / x )T, P < 0, which is evidence of possible Indeed, already at very low concentrations, the Fermi phase stratiﬁcation at the given doping level. Such com- level occurs in the zone of impurity band states that are positions cannot be stably homogeneous, because the being formed. Because the number of states on the sought distribution corresponds to the thermodynamic Fermi level begins to grow more slowly than x potential Φ maximum. For instance, the dependence of (Fig. 8b), pinning ends, and the Fermi level enters the ∂µ/∂x on x for NCCO (Fig. 9) shows that Φ(x) has an − ∆ ≈ rigid conduction band. This occurs at x2 = 0.11 0.12. instability region x 0.03 wide. This region separates

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12 instability of the homogeneous normal state is related NCCO to the presence of antiferromagnetic order, because the 10 impurity band disappears in the paramagnetic phase. It follows that we observe an unusual relation between the nonuniform charge density distribution and the pres- 8 ence of antiferromagnetic ordering. The inclusion of zero spin fluctuations causes instability region ∆x nar- 6 rowing, but does not negate its existence [15]. n-LCO 4 We also observe a qualitative correlation between Fermi level movement to the antiferromagnetic state and the phase diagram of NCCO. Indeed, the concen- Density of states, arb. units Density of states, arb. 2 tration region of Fermi level residence in the impurity band, or, which is the same, in the pseudogap region 0 1.90 1.45 1.00 correlates with the superconducting region in the phase diagram. In NCCO and n-LCO, the Fermi level reaches Energy, eV the pseudogap at different dopant concentrations Fig. 6. Density of states at the bottom of the conduction x1(NCCO) > x1(n-LO) = 0. In NCCO, the Fermi level band in NCCO and n-LCO. The vertical line indicates the enters impurity band states with an already well-devel- region with a reduced density of states (pseudogap). oped spectral density. This is seen from Fig. 8a, where the spectral density in the impurity band is nonzero already at x = x1. It follows that the presence of a finite two regions with thermodynamic potential minima. spectral density of impurity band states at the Fermi Even at small dopant concentrations, n-LCO occurs in level corresponds to the superconducting region in the the ∆x ≈ 0.05 region with a maximum thermodynamic phase diagram of NCCO. The immediate consequence potential. Although the materials under consideration of a correlation of this type would be the beginning of are systems with fixed numbers of carriers, both sys- the superconducting region in the phase diagram of tems can either be divided into macroscopic regions NCCO at TC higher than TC for n-LCO, this region capable of exchanging particles or experience the tran- being narrower on the concentration scale. Accord- ≈ sition to the superconducting state, where the number ingly, we also have xmax x2. Such an equality was of particles is no longer conserved. The origin of the observed for PCCO in [18], whose authors were able to

0.25 ∆µ, eV (a) (b) 1.0 0.6 0.20 0.8 n-LCO 0.4 0.6 0.15 0.2 0.4 0.10 0 NCCO 0.2 Concentration of carriers ~~ LSCO Number of states in the band –0.2 0.05 NCCO 0 n-LCO x1 x2 –0.4 x2 –0.2 0.25 0 0.2 0.4 0 0.2 0.4 –0.6 Concentration of carriers 0.3 0.2 0.1 0 0.1 0.2 0.3 x Fig. 8. Dependence of the total number of states on the concentration of the doping component for the impurity bands in NCCO and n-LCO; x1 and x2 correspond to the entrance Fig. 7. Dependence of the chemical potential shift ∆µ(x) on to and exit from the band of virtual level states, respectively. the concentration of the doping n-component in NCCO and The solid line shows the number of states, and the dashed n-LCO. The experimental data on NCCO and LSCO were line, the concentration of electrons x. The shaded region in taken from [6]. (a) corresponds to the contribution of the rigid band.

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dµ/dx 3 (a) (b) 2 ∆x ∆x

–1 NCCO n-LCO –2

–3 x2 x1 x2 –4 0.20 0.15 0.10 0.05 0 0.05 0.10 0.15 0.20 x Fig. 9. Dependence of the second thermodynamic potential derivative ∂2Φ/∂x2 on the concentration of the doping component in NCCO and n-LCO; x2 corresponds to the exit of the Fermi level from the impurity band. study the T*(x) dependence of the characteristic tem- level with respect to the bottom of the rigid conduction perature for the pseudogap state in the superconducting band in these compositions. As a consequence, we phase of PCCO in magnetic fields higher than Hc2. In observe pinning of the Fermi level by states developed our calculations, the pseudogap state is an attribute of on the virtual level in n-type LCO at low dopant con- the impurity band rather than the precursor of the super- centrations. In NCCO, the Fermi level is immediately conducting state. It can be identified as a special feature immersed into rigid conduction band states and only of the electronic structure of materials with strong elec- enters impurity band states when the degree of doping tron correlation in the antiferromagnetic phase and with increases further. The probability of pinning the Fermi a singlet ground state in one of the configuration space level by them, however, actually decreases as the dop- sectors of the unit cell of the material under study. ing level grows. (3) We observed that, in our calculations, the regions of Fermi level pinning by the impurity band were virtu- 6. CONCLUSIONS ally ∆x regions with an instability of the form ∂2Φ ∂ 2 The results of our calculations can be summarized ( / x )T, P < 0, which could be the reason for a non- as follows: uniform charge density distribution in the compositions (1) Common to the dispersion dependences for under consideration. NCCO and n-LCO is the presence of a virtual level at (4) A qualitative correspondence exists between the the bottom of the conduction band and at the top of the phase diagram of NCCO and the concentration depen- valence band in the antiferromagnetic phase. The rea- dence of the Fermi level: namely, the concentration son for its existence is the presence of singlet states in region of Fermi level residence in the impurity band the vacuum (a1g is a closed shell) and two-particle (A1g) correlates with the concentration region of the super- configuration space sectors of both compounds. The conducting state in these compounds. There is no such rigid conduction band in NCCO has a minimum at correspondence for n-LCO because of the absence of point X of the Brillouin zone at low doping levels. the corresponding materials with the T structure. Our Because of accidental degeneracy of the rigid band and results, however, show that the hypothetical phase dia- virtual level at point X in n-LCO, its dielectric gap is gram for LCO of the p/n type with the T structure rectilinear, whereas the gap in NCCO is not. The last should be more symmetrical than the diagram of conclusion is in agreement with the angle-resolved NCCO/LSCO. photoelectron spectroscopy data on weakly doped NCCO compounds [5]. (2) The concentration dependences of the Fermi ACKNOWLEDGMENTS level for NCCO with the T ' structure and n-type LCO This work was financially supported by the Russian with the T structure are not symmetrical. We explain Foundation for Basic Research (project no. 03-02-16124), this asymmetry by the different positions of the virtual RFFI-KKFN “Eniseœ” (project no. 02-02-97705),

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INTAS (project no. 01-0654), integration program of 10. S. G. Ovchinnikov, Usp. Fiz. Nauk 167, 1043 (1997) URO and Siberian Division, Russian Academy of Sci- [Phys. Usp. 40, 993 (1997)]. ences (project no. 9), and the “Quantum Macrophysics” 11. V. A. Gavrichkov, S. G. Ovchinnikov, A. A. Borisov, and program of the Russian Academy of Sciences. E. G. Goryachev, Zh. Éksp. Teor. Fiz. 118, 422 (2000) [JETP 91, 369 (2000)]. REFERENCES 12. V. A. Gavrichkov, A. A. Borisov, and S. G. Ovchinnikov, Phys. Rev. B 64, 235124 (2001). 1. B. Keimer, N. Belk, R. J. Birgeneau, et al., Phys. Rev. B 46, 14034 (1992). 13. Ya. B. Gaididei and V. M. Loktev, Phys. Status Solidi B 147, 307 (1988). 2. S. G. Ovchinnikov, Physica C (Amsterdam) 228, 81 (1994). 14. S. G. Ovchinnikov and I. S. Sandalov, Physica C 3. S. J. Hagen, J. I. Peng, Z. Y. Li, and R. I. Greene, Phys. (Amsterdam) 161, 607 (1989). Rev. B 43, 13606 (1991). 15. A. A. Borisov, V. A. Gavrichkov, and S. G. Ovchinnikov, 4. H. Takagi, B. Batlog, H. L. Kao, et al., Phys. Rev. Lett. Zh. Éksp. Teor. Fiz. 124, 862 (2003) [JETP 97, 773 69, 2975 (1992). (2003)]. 5. N. P. Armitage, F. Ronning, D. H. Lu, et al., Phys. Rev. 16. M. Naito and M. Hepp, Jpn. J. Appl. Phys. 39, L485 Lett. 88, 257001 (2002). (2000). 6. N. Harima, J. Matsuno, A. Fujimori, et al., Phys. Rev. B 17. R. O. Zaœtsev, Zh. Éksp. Teor. Fiz. 68, 207 (1975) [JETP 64, 220507(R) (2001). 41, 100 (1975)]. 7. P. Fulde, Physica B (Amsterdam) 230Ð232, 1 (1997). 18. L. Alff, Y. Krockenberger, B. Welter, et al., Nature 422, 8. W. Henggeler, B. Roessli, A. Furrer, et al., Phys. Rev. 698 (2003). Lett. 80, 1300 (1998). 9. E. Maiser, W. Mexner, R. Schafer, et al., Phys. Rev. B 56, 12961 (1997). Translated by V. Sipachev

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 565Ð571. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 640Ð647. Original Russian Text Copyright © 2004 by Velikokhatnyœ, Eremeev, Naumov, Potekaev. SOLIDS Electronic Properties

On the Nature of Different Temperature Dependences of the Size of Antiphase Domains in Commensurate Long-Period Structures O. I. Velikokhatnyœa, S. V. Eremeeva,b, I. I. Naumova, and A. I. Potekaevb,* aInstitute of Strength Physics and Materials Science, Siberian Division, Russian Academy of Sciences, Akademicheskiœ pr. 2/1, Tomsk, 634021 Russia bKuznetsov Physicotechnical Institute, pl. Novosobornaya 1, Tomsk, 634021 Russia *e-mail: [email protected], [email protected] Received August 5, 2003

Abstract—First-principle calculations of the electronic structure and electronic susceptibility were performed to study the relation between the nesting properties of the Fermi surface and the character of the temperature dependence of long-period structures of two types exempliﬁed by Ag3Mg and Al3Ti alloys. The observed temperature dependence of the long period length 2M in the Al3Ti alloy was analyzed. It was shown that the temperature dependence of the size of the antiphase domain in long-period commensurate structures was determined by the special features of the electronic structure of the system, in particular, by the geometry of Fermi surface nesting regions. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION ensemble changes continuously and takes on various Ordered alloys with long-period structures are one values, including irrational. of the interesting and promising classes of metallic In our preceding works [8, 9] we studied the special alloys. They differ from the usual ordered systems with features of the electronic structure of the CuÐAu, CuÐ simple superstructures by the presence of antiphase Pd, and CuÐPt alloys and answered the question of why boundaries, which periodically or quasi-periodically incommensurate long-period structures characterized disturb the ordered arrangement of atoms. Antiphase by irrational periods and smeared antiphase boundaries boundaries, which are usually energetically unfavor- are formed in them. The reasons for the existence of able in ordered alloys, are equilibrium structure ele- such long-period structures only in a narrow tempera- ments in systems with long-period structures. Ordered ture range were elucidated. We also explained the alloys with long-period structures have quite deﬁnite observed dependence of the period length on the degree stability regions in temperatureÐcomposition phase of long-range ordering η in the CuÐAu alloy [8] and the diagrams [1]. reasons for the formation of two-dimensional long- Studies of the mechanical properties of such alloys period structures in the Au3Cu and Cu3Pd alloys [9]. showed [2Ð4] that strengthening against the decay of Consider the special features of alloys with com- the supersaturated solid solution could effectively be mensurate long-period structures, such as Ag Mg, combined with strengthening against atomic ordering. 3 Cu Al, Au Cd, Al Ti, Pt V, etc. This allows unusual disperse stable decay structures to 3 3 3 3 be formed. Their alloys possess high mechanical prop- It was found in studying Ag3Mg alloys with various erties, which are stable over the whole temperature magnesium contents that the antiphase half-period was range in which the ordered state of the matrix is constant, M = 2, in a certain concentration range that retained [5, 6]. corresponded to the D023 superstructure [10, 11] or M According to their character, long-period structures slightly decreased as the concentration of magnesium can be divided into two groups, namely, incommensu- increased [12]. Importantly, the antiphase boundaries rate and commensurate structures. Incommensurate structures are, for instance, formed in the CuAu, were sharply deﬁned, and M remained virtually constant as temperature varied. This type of alloys also Cu3Au, Au3Cu, Cu3Pd, and Cu3Pt alloys [7]. They are characterized by random distances M between includes Au3Zn, Au3Cd, and Au3Mn [13]. antiphase boundaries (antiphase domains of different Somewhat different results were obtained for Al Ti 〈 〉 3 lengths are stochastically distributed along the 001 alloys [14Ð17], for which high- and low-temperature direction). When the composition and temperature are phases with long-period structures were observed [14]. varied, the half-period M averaged over the chaotic Several comparatively simple commensurate structures

566 VELIKOKHATNYŒ et al. were formed at low temperatures, whereas a series of of commensurate long-period structures exemplified by fairly complex configurations were identified at high the Ag3Mg and Al3Ti alloys. temperatures. For instance, the 〈211〉 configuration (this notation corresponds to the 2Ð1Ð1 sequence of domains, where 2 and 1 stand for domains with M = 2 2. CALCULATION DETAILS and M = 1, respectively) was observed at T = 700°C, The principal value calculated in our approach is the which corresponded to M ≈ 1.33. At T = 900°C, the generalized susceptibility of noninteracting electrons χ(q), configuration was 〈21〉, and, accordingly, M ≈ 1.5; at ° 〈 〉 ≈ T = 1150 C, it was 221 with M 1.68, and, at T = 2Ω χ()q = ------° 〈 〉 ≈ 3 1200 C, the 22221 configuration with M 1.76 ()2π formed. Note that M obviously tends to increase as ()ε ()[]()ε () temperature grows, whereas the low-temperature phase × 3 f λ k 1 Ð f λ' kq+ ∫d k∑------ε ()ε() -. is formed as a mixture of antiphase domains with M = 2 λ' kq+ Ð λ k λλ, ' and M = 1 with obvious predominance of D022 superstructure elements (M = 1). The simplest long-period structure D0 was observed in the alloy of the stoichi- This value is calculated from the electron energy spec- 22 trum of the alloy. If the system contains plane Fermi ometric composition [15]. At a 60Ð73 at. % Al, struc- surface regions or Fermi surface regions whose shapes tures with 4/3 ≤ M ≤ 2 were observed and the M value coincide and these regions are separated by the nesting χ was temperature-dependent. The dependence of M on vector 2kF, the (q) function has some singularity at the composition and temperature of annealing was also the same vector. Depending on the “quality” of nesting reported in [16, 17]. In [18], PtÐV alloys with long- (that is, on the degree of similarity of the coinciding regions), the electronic susceptibility singularity may period structures at compositions close to Pt3V were studied. During annealing at 930°C, the D0 ordered be a kink, a step, or even a peak. The more pronounced 22 the singularity of χ(q), the larger the energy gain of the structure (of the Al3Ti type) transformed into a structure ° formation of a long-period structure. with L12-type ordering, and, at 1036 C, a disordered state was formed. It is directly stated in [19] that the The electron energy spectrum was calculated by the ° full-potential linear muffin-tin orbital (LMTO) method high-temperature (above 1000 C) phase has the L12 structure, and the low-temperature one (below 900°C), in the local electron density approximation [20]. The exchange-correlation potential was taken according to the D022 structure. An increase in M with temperature Barth and Hedin [21], the integration over occupied was observed. states was performed by the tetrahedron method [22], To summarize, the most characteristic features of 165 (Ag3Mg) and 126 (Al3Ti) reference points were commensurate structures are as follows: used in self-consistent calculations of the ελ(k) spectrum, and 1771 (Ag Mg) and 4851 (Al Ti) points in the (1) the low-temperature phase is formed as a com- 3 3 mensurate superstructure; irreducible part of the Brillouin zone were used in electronic susceptibility χ(q) calculations. The generalized (2) the high-temperature state is a mixture of com- susceptibility of noninteracting electrons χ(q) was cal- mensurate elements; culated only taking into account the energy bands that (3) the antiphase boundaries are sharply defined intersected the Fermi level and determined the behavior planes near which there is no essential structural pecu- of this value. The lattice parameter a was set equal to liarities; 7.766 au for Ag3Mg and 7.264 au for Al3Ti with (4) the “mean” antiphase domain size runs over c/a = 2.23. rational values as the composition of alloys changes. Nevertheless, we can distinguish between two 3. RESULTS AND DISCUSSION groups of commensurate long-period structures. The The electron energy spectrum ε(k) calculated for a first group includes such alloys as Ag3Mg, Cu3Al, hypothetical Ag3Mg alloy with L12-type ordering is Au3Cd, Au3Zn, and Au3Mn. The mean half-period shown in Fig. 1. Its characteristic feature, like that of all remains virtually constant in these alloys as tempera- noble metal-based alloys, is a bright Ag d-band local- ture varies. The second group includes the Al3Ti, ized below the Fermi level in such a way that the geom- Au4Zn, and Pt3V alloys. These alloys sometimes show etry of the Fermi surface is determined by the 17, 18, a very substantial temperature dependence of the mean and 19 bands, which are virtually fully s and p states of domain size. Ag and Mg. In this work, we study the differences in the temper- The susceptibility χ(q) calculated along the ΓÐX ature dependences of domain dimensions in two types Brillouin zone direction for the Ag3Mg alloy with

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ON THE NATURE OF DIFFERENT TEMPERATURE DEPENDENCES 567

ε, Ry

0.8

ε ε 0.6 F F

0.4

0.2

300 200 100 0Γ X M Γ R M RX n, el/Ry á cell ε ε Fig. 1. Electron energy spectrum (k) and the density of electronic states n( ) of the Ag3Mg alloy with the L12 structure.

ε L12-type ordering is shown in Fig. 2. It has a sharp max- The electron energy spectrum (k) of the Al3Ti alloy π χ imum at the wave vector qn = 2 /a [0.284, 0, 0], which with the L12 structure is shown in Fig. 4. The (q) sus- is evidence of the instability of the hypothetical L12 ceptibility along the 〈100〉 Brillouin zone direction cal- phase with respect to the formation of a long-period χ structure with half-period M = 1.76. This corresponds 30 with experimental observations. Indeed, the Ag3Mg system has never been observed to have the L12 struc- Ag3Mg ture, it undergoes the transition immediately from the 29 disordered to the long-period state characterized by a mixture of domains with the D022 (M = 1) and D023 28 (M = 2) ordering types. The elements of the D023 superstructures are more numerous, and the mean half- 27 period of the antiphase domain M is 1.75 [10], which ~ ~ is in excellent agreement with the calculated value specified above (1.76). An analysis of the partial contributions to the total 18–19 χ susceptibility (q) showed that the susceptibility maxi- 5 mum appeared because of the interband electron transitions 18Ð19 and 19Ð18 (Fig. 2). Ultimately, this maximum is caused by the geometric features of the Fermi surface shown in Fig. 3. In the vicinity of the Brillouin zone point M, there are two vast electronic regions of the 18th and 19th Fermi surface sheets which virtually π coincide in shape and are separated by the qn = 2 /a 19–18 [0.284, 0, 0] vector mentioned above. This high degree of similarity is responsible for the well-defined χ(q) maximum. It can therefore be suggested that the specified special features of the geometry of the Fermi sur- 0 0.1 0.2 0.3 0.4 0.5 face and, ultimately, the electronic structure inherent in q〈100〉 the hypothetical Ag3Mg phase with L12-type ordering Fig. 2. Electronic susceptibility χ(q) (the upper curve) and 〈 〉 contribute to the experimentally observed mixture of its partial contributions calculated for Ag3Mg in the 100 domains with the D022 and D023 superstructures. direction.

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(a) (b) Γ X Γ MMX

q qn 19e n XXRR 19 e M M 18e

17h Γ X Γ M X M π Fig. 3. Fragments of Fermi surface sections for Ag3Mg: (a) in the z = 0 plane and (b) in the z = 0.5(2 )/a plane. The nesting vector is shown by the arrow.

ε, Ry 1.0

0.8 ε F ε F 0.6

0.4

0.2

–0.2 160 120 80 40 0Γ X M Γ R M RX n, el/Ry á cell ε ε Fig. 4. Electron energy spectrum (k) and the density of electronic states n( ) of the Al3Ti alloy with the L12 structure.

culated from the electron energy spectrum of Al3Ti is of coinciding Fermi surface regions separated by the π 〈 〉 shown in Fig. 5. Of interest is the local susceptibility vector qn = 0.35(2 /a) 001 (Fig. 6). maximum at q = 0.35; it characterizes the instability of n Note that the observed low-temperature D0 structure this system with respect to long period formation with 22 has a fairly high degree of tetragonality c/a = 2.23 [23]. It M = 1.47. An analysis of the partial contributions to the would therefore be reasonable to calculate the electron χ total (q) susceptibility shows that the contribution of energy spectrum and susceptibility for the L12 structure 7Ð7 intraband transitions shown in Fig. 5 is fully with a tetragonal distortion equal to that in the D022 responsible for this local susceptibility maximum. One structure, that is, c/a = 1.115. One point should be men- can see that, at qn = 0.35, this contribution has a pro- tioned. It can be suggested from general considerations nounced maximum, which is evidence of the presence that the larger the long period length, the smaller should

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ON THE NATURE OF DIFFERENT TEMPERATURE DEPENDENCES 569 χ the usual cubic L12 structure. On the other hand, the 70 greatest degree of tetragonality should be characteristic of the simplest long-period structure of Al3Ti, that is, Al3Ti(L12) c/a = 1.0 68 D022. It follows that, for completeness, we must also study structures with intermediate tetragonal distortion 66 degrees. We analyzed the dependence of χ(q) on the c/a 64 parameter in the range of parameter values from 1 to 1.115. The singularity observed in the cubic L12 struc- π 〈 〉 62 ture at qn = 0.35(2 /a) 001 shifts to the right as c/a increases and gradually vanishes (see Fig. 7). A new singularity at q = 0.42(2π/a), however, appears at c/a = 60 n ~ ~ 1.10. This singularity not only shifts to the right but also fairly sharply increases as the degree of tetragonality 18 becomes still larger. At the experimental degree of tetragonality, this singularity corresponds to the vector q = 0.46(2π/a), which in turn corresponds to the half- 16 n 7–7 period of length M = 1.1. There exist coinciding Fermi 14 M = 1.47 surface regions for all the vectors speciﬁed above. The above results can be represented in the form of 12 the dependence of the mean half-period length M on the tetragonality parameter c/a of the base L12 cell. At 10 0 0.1 0.2 0.3 0.4 0.5 the experimentally observed degree of tetragonality, q〈100〉 calculations give M = 1.1, which is very close to the true low-temperature D0 structure (M = 1). The χ 22 Fig. 5. Electronic susceptibility (q) of the Al3Ti alloy cal- 〈 〉 antiphase domain size increases as the degree of tetrag- culated for the L12 structure in the 001 direction. The lower curve is the partial contribution of intraband 7Ð7 elec- onality decreases, and M approximates 1.5 in the cubic tronic transitions. The arrow indicates the local susceptibil- structure. Here, (c/a)Ð1 plays the role of something like ity maximum. temperature. The lower the degree of tetragonality, the higher the temperature and the longer the long period of be the degree of tetragonality of the base cells that con- the long-period structure. These results can be put in stitute the given structure. In the limit of an inﬁnitely qualitative correspondence with the experimental tem- long period, this long-period structure transforms into perature dependence of M [14].

(a) (b) åï å 6h 7e 7e 6e 8e

ï 6e ï Γ 7e

å ïå π Fig. 6. Fragments of Fermi surface sections for the Al3Ti alloy in two mutually orthogonal planes: (a) z = 0 and (b) y = 0.23(2 /a). The nesting vector is shown by the arrow.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 570 VELIKOKHATNYŒ et al. χ χ 70 72 Al3Ti c/a = 1.0 Al3Ti c/a = 1.08 68 70 66 68 64 66 62

60 64 ~ ~ ~ ~ 18 7–7 18 16 17 7–7 14 16 12

10 15 00.1 0.2 0.3 0.4 0.5 00.1 0.2 0.3 0.4 q〈001〉 q〈001〉

72 72

Al3Ti c/a = 1.10 Al3Ti c/a = 1.115 (exp.)

70 70

68 68

66~ ~ ~ ~ ~ ~ 19 18 7–7 18 17 7–7 17 16 16

15 15 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 q〈001〉 q〈001〉 χ Fig. 7. Electronic susceptibility (q) of the Al3Ti alloy in the cubic L12 structure at various degrees of tetragonality (the upper curves) and the partial contributions of the 7Ð7 intraband transitions (the lower curves). Local susceptibility maxima are indicated by the arrows.

To summarize, the temperature dependence of the face sheets is high in the Ag3Mg alloy, and a fairly sharp size of antiphase domains in commensurate long- χ(q) maximum is formed. The nesting quality is lower period structures is determined by the special features in Al3Ti, and the susceptibility singularity does not have of the electronic structure of the system, in particular, a pronounced peak. The shape of the Fermi surface and by the geometry of the nesting Fermi surface regions. the nesting vector length change as temperature The degree of coincidence of the ﬂattened Fermi sur- decreases (the degree of tetragonality increases). In

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 ON THE NATURE OF DIFFERENT TEMPERATURE DEPENDENCES 571 contrast, no nesting changes occur in Ag3Mg, and long- 9. O. I. Velikokhatnyœ, S. V. Eremeev, I. I. Naumov, and A. period structures remain stable even at low tempera- I. Potekaev, Pis’ma Zh. Éksp. Teor. Fiz. 69, 548 (1999) tures. In our view, this difference explains the differ- [JETP Lett. 69, 589 (1999)]. ence in stability of commensurate long-period struc- 10. M. Guymont and D. Gratias, Acta Crystallogr. A 35, 181 tures under temperature variations. (1979). 11. A. Gangulee and B. Bever, Trans. Metall. Soc. AIME 242i, 278 (1968). ACKNOWLEDGMENTS 12. K. Hanhi, J. Maki, and P. Paalassalo, Acta Metall. 19, 15 (1971). This work was supported by CRDF and by the Min- istry of Education of the Russian Federation under the 13. A. I. Potekaev, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 6, 3 (1995). program “Basic Research and Higher Education,” grant V1-P-16-09. 14. A. Loiseau, G. van Tendeloo, R. Portier, et al., J. Phys. (Paris) 46, 595 (1985). 15. A. Loiseau, J. Electron Microsc. 35 (Suppl. 1), 843 REFERENCES (1986). 16. R. Milida, Jpn. J. Appl. Phys. 25, 1815 (1986). 1. A. I. Potekaev, I. I. Naumov, V. V. Kulagina, V. N. Udo- dov, O. I. Velikokhatnyœ, and S. V. Eremeev, Natural 17. A. Loiseau, J. Planes, and F. Ducastelle, in Alloy Phase Long-Period Nanostructures (NTL, Tomsk, 2002). Stability: Proceedings of NATO Advanced Study Insti- tute, Maleme, 1987, Ed. by G. M. Stocks and A. Gonis 2. O. D. Shashkov, L. N. Buinova, V. I. Syutkina, et al., Fiz. (Kluwer Academic, Dordrecht, 1989), p. 101. Met. Metalloved. 28, 1029 (1969). 18. J. Planes, A. Loiseau, F. Ducastelle, and G. van Tende- 3. L. N. Buœnova, V. I. Syutkina, O. D. Shashkov, and loo, in EMAG 87: Analytical Electron Microscopy, Ed. É. S. Yakovleva, Fiz. Met. Metalloved. 29, 1221 (1970). by C. W. Lorimer (Inst. of Metals, London, 1988), 4. L. N. Buœnova, V. I. Syutkina, O. D. Shashkov, and p. 261. É. S. Yakovleva, Fiz. Met. Metalloved. 33, 1195 (1972). 19. D. Schryvers and S. Amelinckx, MRS Bull. 20, 367 5. L. N. Buœnova, V. I. Syutkina, O. D. Shashkov, and (1983). É. S. Yakovleva, Fiz. Met. Metalloved. 34, 561 (1972). 20. S. Savrasov and D. Savrasov, Phys. Rev. B 46, 12181 (1992). 6. V. I. Syutkina and É. S. Yakovleva, Fiz. Tverd. Tela (Len- ingrad) 6, 2688 (1966) [Sov. Phys. Solid State 6, 2141 21. U. Barth and L. Hedin, J. Phys. C 5, 1629 (1972). (1966)]. 22. J. Rath and A. J. Freeman, Phys. Rev. B 11, 2109 (1975). 7. M. J. Marsinkowski and L. Zwell, Acta Metall. 11, 373 23. W. B. Pearson, A Handbook of Lattice Spacings and (1963). Structures of Metals and Alloys (Pergamon, New York, 1958). 8. O. I. Velikokhatnyœ, S. V. Eremeev, I. I. Naumov, and A. I. Potekaev, Zh. Éksp. Teor. Fiz. 117, 548 (2000) [JETP 90, 479 (2000)]. Translated by V. Sipachev

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Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 572Ð581. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 648Ð658. Original Russian Text Copyright © 2004 by Kugel, Rakhmanov, Sboychakov, Kagan, Brodsky, Klaptsov. SOLIDS Electronic Properties

Characteristics of the Phase-Separated State in Manganites: Relationship with Transport and Magnetic Properties K. I. Kugela, A. L. Rakhmanova, A. O. Sboychakova, M. Yu. Kaganb, I. V. Brodskyb, and A. V. Klaptsovb aInstitute of Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Izhorskaya ul. 13/19, Moscow, 125412 Russia bKapitza Institute for Physical Problems, Russian Academy of Sciences, ul. Kosygina 2, Moscow, 119334 Russia e-mail: [email protected] Received August 19, 2003

Abstract—The temperature and magnetic ﬁeld dependence of the resistivity, magnetoresistance, and magnetic susceptibility of phase-separated manganites in the temperature range corresponding to nonmetallic behavior are considered within the framework of a model of inhomogeneous state with allowance for the existence of ferromagnetically correlated regions even in the absence of long-range magnetic order. The main attention is given to the interval of high temperatures and weak ﬁelds. The main characteristics of the phase-separated state of manganites are evaluated from a comparison of the theoretical results with available experimental data. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION magnetic dipole interactions) can result in merging of the ferrons and the formation of intermediate- to large- Unusual properties and a rich phase diagram of scale inhomogeneities [7]. There are clear experimental manganites inspired a large number of papers devoted indications suggesting that the phase separation is to various aspects of the physics of these compounds. inherent in both magnetically ordered phases and the The special interest in manganites in recent years is paramagnetic (PM) state [1Ð3, 8]. Therefore, the for- related to the possible existence of various inhomoge- mation of inhomogeneous states has proved to be a phe- neous charge and spin states such as lattice and mag- nomenon typical of manganites in various parts of their netic polarons, droplet and stripe structures, etc. [1Ð3]. phase diagrams. Moreover, the phase separation must Analogous phenomena are inherent in many strongly strongly influence the magnetic and transport proper- correlated systems where the electronÐelectron interac- ties of manganites. tion energy is greater than the kinetic energy. One of the most spectacular manifestations of such a behavior is The role of phase separation is most often consid- the formation of ferromagnetic (FM) droplets (ferrons) ered in the region of the AFM state and especially in the in slightly doped antiferromagnetic (AFM) semicon- vicinity of the transition between AFM and FM states. ductors [4]. Another example is the formation of a However, as mentioned above, manganites can be inho- string (linear trace of frustrated spins) during the mogeneous even in the PM state, at temperatures motion of a hole in an AFM insulator [5]. exceeding the corresponding phase transition temperature. An analysis of the available experimental data The above examples refer to the so-called electron reveals a substantial similarity in the high-temperature phase separation, whereby a single charge carrier behavior of the resistivity, magnetoresistance, and mag- changes its local electronic environment. In addition to netic susceptibility of various manganites with different such a small-scale phase separation, manganites can low-temperature states [9Ð12]. In addition, the magne- also feature a large-scale phase separation correspond- toresistance turns out to be rather large far from the ing to the coexistence of different phases characteristic FMÐAFM transition and even in the PM region. Fur- of first-order phase transitions (e.g., of the transition thermore, the magnetic susceptibility of manganites is between AFM and FM states). An example of large- significantly higher than that for typical antiferromag- scale phase separation is the formation of relatively nets. These experimental data clearly indicate the exist- large FM droplets in an AFM matrix. Such droplets ence of significant FM correlations in the high-temper- with linear dimensions on the order of 100–1000 Å ature range. were observed, in particular, by the method of neutron diffraction [6]. Note also that the attraction between We will proceed from the assumption that ferromag- one-electron FM droplets (mediated by either elastic or netically correlated regions exist in manganites above

CHARACTERISTICS OF THE PHASE-SEPARATED STATE IN MANGANITES 573 the temperatures characterizing the onset of long-range ρ, Ω cm magnetic (FM or AFM) ordering. This assumption 106 allows us to describe the characteristic features of the resistivity, magnetoresistance, and magnetic susceptibility of manganites in a nonmetallic state within the framework of a common model. The consideration 104 below is based on the model of conductivity in phase- separated manganites developed in [9, 13Ð15] and makes use of the experimental data for manganites of 102 various compositions reported in [9Ð12]. However, this paper is not restricted to considering only one-electron magnetic droplets (ferrons). A part of the previously obtained results will be generalized to the case of an 1 arbitrary number of electrons in ferromagnetically correlated regions. In Section 2, the temperature dependence of the 10–2 resistivity is analyzed for an inhomogeneous state with 50 100 150 200 250 300 the density of the ferromagnetically correlated regions T, K far from the percolation threshold. In Sections 3 and 4, we discuss within the same assumptions the magne- Fig. 1. Temperature dependence of the resistivity of (La1 − yPry)0.7Ca0.3MnO3 [9]: (squares) y = 1, with toresistance of manganites and their magnetic suscepti- 16 O 18O isotope substitution; (triangles) y = 0.75, with bility, respectively. It will be shown that the proposed 16 O 18O isotope substitution; (circles) y = 0.75, with model of the inhomogeneous state provides for an ade- 16 quate description of the high-temperature behavior of O isotope. The solid curve shows the results of calcula- manganites. A comparison of the theoretical results and tions using formula (1). experimental data allows us to determine the main characteristics of ferromagnetically correlated regions in droplet, l is the characteristic tunneling length, and n is manganites. the concentration of FM droplets. Formula (1) can be readily derived as described in [13] (see Appendix). 2. RESISTIVITY This expression is a straightforward generalization of The temperature dependence of the resistivity of the formula obtained in [13] for the conductivity of manganites will be analyzed based on the notions one-electron droplets. The resistivity (1) exhibits a ther- developed previously [13]. moactivation behavior, whereby the activation energy is equal to a half of the Coulomb repulsion energy (for This physical picture is essentially as follows. We consider a system comprising small ferromagnetic more detail, see [13]). droplets dispersed in a nonferromagnetic insulating Expression (1) provides for a fairly good description matrix, in which charge is transferred via tunneling of of the temperature dependence of the resistivity for var- the charge carriers from one droplet to another. In the ious manganites. This is illustrated in Figs. 1Ð4, show- general case, the tunneling probability depends on an ing the ρ(T) curves for six manganites, constructed external magnetic ﬁeld. We assume that the droplets do using the experimental results reported in [9Ð12] not overlap and the whole system is far from the perco- (detailed numerical data were kindly provided by the lation threshold. Each droplet may contain k charge carriers. Every new charge carrier tunneling to a given authors of these papers). As can be seen from these droplet experiences Coulomb repulsion from the carri- data, the samples differ in chemical composition, the ers already occurring in this droplet. The repulsion type of crystal structure, the magnitude of the resistivity (at a ﬁxed temperature, the latter varies for different energy A is assumed to be relatively large (A > kBT), so that the main contribution to the conductivity is related samples by more than two orders of magnitude), and to the processes involving droplets containing k, k + 1, the low-temperature behavior (some of the samples and k Ð 1 carriers. behave as metals and the other, as insulators). In the high-temperature range (above the point of the FM Under these conditions, an expression for the resis- ρ tivity ρ(T) has the following form: phase transition), (T) exhibits identical behavior for all samples and is well described by the universal k TAexp()/2k T relation ρ = ------B B -, (1) π 2ω 5 2 128 e 0l kn ρ()∝ () T TAexp /2kBT ω where e is the electron charge, 0 is a frequency corresponding to the characteristic energy of electrons in a represented by solid curves in Figs. 1Ð4.

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574 KUGEL et al.

ρ, Ω cm ρ, Ω cm 104

102

10–1 10–2 200 220 240 260 280 300 50 100 150 200 250 300 T, K T, K Fig. 3. Temperature dependence of the resistivity of a lay- Fig. 2. Temperature dependence of the resistivity of ered manganite with the composition Pr0.71Ca0.29MnO3: (circles) experimental data [10]; (solid (La0.4Pr0.6)1.2Sr1.8Mn2O7: (circles) experimental data [11]; curve) calculation using formula (1). (solid curve) calculation using formula (1).

Using Eq. (1) and experimental data, it is possible to the ferron size and the height of the barrier virtually determine some quantitative characteristics of the coincides with the depth of the potential well. The latter phase-separated state. In particular, an analysis carried naturally follows from the adopted model of ferron for- out by Zhao et al. [12, 16] demonstrated that interpre- mation [2]. Therefore, it is reasonable to suggest that tation of the experimental data in terms of this relation the tunneling length is on the order of the ferron size gives an accurate estimate of the Coulomb energy A. (several nanometers), although in the general case it For example, the data presented in Figs. 1Ð4 allow the can substantially differ from a. Coulomb barrier A to be determined with an accuracy of 2Ð3%; for the compounds under consideration, this Another nontrivial task is to estimate the concentra- value falls within a rather narrow range from 3500 to tion n of ferrons. On the one hand, following Zhao et al. 3700 K (see Table 1). As for the characteristic fre- [12, 16], the concentration n could be determined via ω the dopant concentration x as n ≈ x/d3. However, this quency 0 entering into formula (1), it was pointed out in [12, 13, 16] that this quantity varies within a very 13 14 restricted interval of 10 Ð10 Hz. This estimate can be ρ, Ω cm obtained, for example, from the uncertainty principle: 7 បω ប2 2 10 0 ~ /2ma , where a is the characteristic size of the droplet and m is the electron mass. Indeed, assuming that a ≈ 1Ð2 nm, we arrive at the latter interval. These values of the droplet size provide for a correct (to within an order of magnitude) estimate of the Coulomb 105 barrier energy A: taking into account that this energy is on the order of e2/εa and substituting permittivity ε ~ 10 we obtain the value of A consistent with the experimental data. 103 It is rather difﬁcult to estimate the tunneling length l. However, we can ascertain that, in the domain of applicability of relation (1), this length cannot be much smaller than the distance between droplets [13]. Other- 10 wise, the behavior of the resistivity would be different. 100 150 200 250 300 In the quasiclassical approximation, the tunneling T, K length is on the order of the characteristic size of the wave function, provided that the barrier height is com- Fig. 4. Temperature dependence of the resistivity of parable to the depth of the potential well. In our case, La0.8Mg0.2MnO3: (circles) experimental data [12]; (solid the size of the electron wave function is on the order of curve) calculation using formula (1).

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Table 1. The Coulomb energy A, the resistivity ρ (at 200 K), and the product l5n2k for some manganites, estimated using formula (1) and the experimental data presented in Figs. 1Ð4 Manganite composition A, K ρ(200 K), Ω cm l5n2k, cmÐ1 Ref. × 5 (La1 Ð yPry)0.7Ca0.3MnO3 3650 1.25 2 10 [9] (see Fig. 1) × 5 Pr0.71Ca0.29MnO3 3500 0.57 3 10 [10] (see Fig. 2) * × 5 (La0.4Pr0.6)1.2Sr1.8Mn2 O7 3600 1.5 1.5 10 [11] (see Fig. 3) × 3 La0.8Mg0.2MnO3 3700 283 1 10 [12] (see Fig. 4)

* The general chemical formula of this system is (La0.4Pr0.6)2 Ð 2xSr1 + 2xMn2O7. approach leads to at least two discrepancies. First, even fields, the magnetoresistance is described by the for- for a moderate concentration of a divalent element, x = mula (see Appendix) 0.1Ð0.2, the droplets would overlap to form a continuous FM metallic cluster. However, the material can be insulating even at higher dopant concentrations of x = µ3 S5 N3 Z2g3 J2 H MR510≈ × Ð3------B eff a H2. (2) 0.5Ð0.6, at least in a high-temperature range. Second, ()5 as can be seen from the experimental data, a relation kBT between the dopant concentration and the conductivity of manganites is rather complicated. For some com- µ Here, B is the Bohr magneton, S is the average spin of pounds, a twofold variation of x can change the resistiv- a manganese ion, N is the number of manganese ity by two orders of magnitude [12, 16], while in some eff ρ atoms in a droplet, Z is the number of nearest neighbors other cases, (x) exhibits a nonmonotonic behavior of a manganese ion, g is the Landé factor, J is the within a certain concentration range. exchange integral of the FM interaction, and Ha is the It should be noted that these difficulties are inherent effective magnetic anisotropy field of a droplet. The not only in our model of phase separation, but also arise law MR ∝ H2/T5 was experimentally observed for a in other models attempting to describe the properties of number of manganites in the region of their nonmetal- manganites (e.g., in polaronic models [17, 18]). Unfor- lic behavior [9, 10]. The same high-temperature behav- tunately, this circumstance was not given proper atten- ior of the magnetoresistance can be obtained by pro- tion in [12, 16] in the interpretation of experimental cessing the experimental data reported in [11, 12] (see data in terms of the existing models of the conductivity Figs. 5Ð8]. in manganites. The natural conclusion is that the num- The value of S depends on the relative content of the ber of carriers involved in the charge transfer process trivalent and tetravalent manganese ions and varies does not coincide with the concentration x of a divalent from 3/2 to 2. For purposes of estimation, we assume dopant. This is especially obvious in the case of charge that S = 2. The value of Z is, in fact, the number of man- ordering, when part of the charge carriers introduced by ganese ions interacting with a conduction electron doping are localized and form a periodic structure. occurring in a droplet. It is reasonable to assume that Z is of the same order of magnitude as the number of Therefore, using expression (1) and experimental ≈ data, it is possible to evaluate the combination l5n2k. nearest neighbors for a manganese ion, that is, Z 6. Table 1 summarizes the values of the Coulomb energy A, The Landé factor g is determined from experimental ρ 5 2 data. For manganese, g is usually assumed to be close the resistivity (at 200 K), and the product l n k esti- to that in the pure spin case (g = 2). The exchange inte- mated using formula (1) and the experimental data pre- gral J characterizes the magnetic interaction between a sented in Figs. 1Ð4. Note that, while the estimate for A conduction electron and a molecular field generated by ± 5 2 is accurate to within 50 K, the combination l n k can ferromagnetically correlated spins in a droplet. This be estimated only by the order of magnitude (at least, ω molecular field accounts for the low-temperature ferro- such is the uncertainty of the value of 0). magnetism. Therefore, we can evaluate J using the well-known relationship of the molecular field theory, S(S + 1)ZJ/3 ~ kBTC , where TC is the Curie temperature. 3. MAGNETORESISTANCE The latter parameter is determined from experiment (e.g., neutron diffraction or magnetization measure- Previously [9, 14, 15], it was demonstrated that the ments). For example, T of manganites of the LaÐPrÐCa model of phase separation adopted here leads to a rather C unusual dependence of the magnetoresistance system is about 100Ð120 K [6]. MR(T, H) on the temperature and magnetic field. At rel- The magnetic anisotropy of manganites related to atively high temperatures and not very strong magnetic the crystal structure of these compounds is usually not

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MR/H2, í–2 assume that the droplet is rather elongated (N˜ Ӷ l) and 10 µ π 3 ≈ Ms = Sg B /d , so that Ha 2 kOe.

The value of Neff is naturally determined by the size of a droplet and, in principle, it could be found from the neutron diffraction experiments. However, to the best of our knowledge, no such measurements were performed in a wide temperature range for the systems 10–1 under consideration. For this reason, Neff is treated here as a fitting parameter. Using Eq. (2) and the above estimates, we can determine the Neff value from experimental data on the magnetoresistance (in the range of parameters corresponding to MR ~ H2/T5). The results of such data processing are summarized in Table 2. In Figs. 5Ð8, solid curves show the data used in the fitting –3 10 procedure. The value of TC was taken to be equal to 50 100 150 200 250 300 120 K. T, K Thus, the size of the ferromagnetically correlated 2 regions for all manganite compositions under consider- Fig. 5. Temperature dependence of the MR/H ratio for ation turns out to be nearly the same at temperatures of (La1 Ð yPry)0.7Ca0.3MnO3 [9]: (circles) y = 0.75; (squares) 16 200Ð300 K. The volume of these regions is approxi- y = 0.75, with 30% of O 18O isotope substitution; 16 mately equal to that of a ball with a diameter of 7Ð8 lat- (triangles) y = 0.75, with O 18O isotope substitution; 16 tice periods. It can naturally be assumed that, within a (diamonds) y = 1; (asterisks) y = 1, with O 18O iso- droplet volume, the number of charge carriers involved tope substitution. The solid curve shows the results of cal- in the tunneling process equals the number of dopant culations using relation (2): MR ∝ 1/T5. atoms. Hence, we can write that k = Neffx, where x is the atomic fraction of the dopant. The values of x and k are presented in Table 2. too high. This implies that the main contribution to the effective magnetic anisotropy field H is related to the a 4. MAGNETIC SUSCEPTIBILITY droplet shape anisotropy and can be evaluated as H = a The concentration of droplets can be evaluated π ˜ ˜ (1 Ð 3N )Ms, where N is the demagnetization factor based on the magnetic susceptibility data, by assuming of the droplet (along the main axis) and Ms is the mag- that the main contribution to the susceptibility comes netic moment per unit volume of the droplet. Below, we from the ferromagnetically correlated regions. At high

MR MR 1 10–1

10–1 10–2

10–2 10–3 120 140 160 180 200 120 140 160 180 200 220 240 260 280 300 T, K T, K

Fig. 6. Temperature dependence of the magnetoresistance Fig. 7. Temperature dependence of the magnetoresistance of Pr0.71Ca0.29MnO3 for H = 2 T (triangles) experimental of (La0.4Pr0.6)1.2Sr1.8Mn2O7 for H = 1 T (triangles) experi- data [10]; (solid curve) calculation using formula (2). mental data [11]; (solid curve) calculation using formula (2).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 CHARACTERISTICS OF THE PHASE-SEPARATED STATE IN MANGANITES 577 ӷ µ µ temperatures (kBT BgSNeffH, BgSNeffHa), the sus- MR ceptibility χ(T) can be written as 0.10 0.09 n()µ gSN 2 χ()T = ------B eff -, (3) 0.08 3k ()T Ð Θ B 0.07 where Θ is the CurieÐWeiss constant. 0.06 The results of the experimental data processing in terms of relation (3) are presented in Table 3. The 0.05 experimental temperature dependences of the magnetic susceptibility of manganites are shown in Figs. 9Ð12, where solid curves illustrate the ﬁtting procedure. 0.04 Using these results, we can also estimate the concentration of the FM phase as p = nN d3. For all samples, the eff 0.03 value of lattice parameter d was taken to be equal to 170 180 190 200 3.9 Å. Based on the data in Tables 1Ð3, it is also possible to estimate the carrier tunneling length l. T, K Fig. 8. Temperature dependence of the magnetoresistance of La0.8Mg0.2MnO3 for H = 1.5 T (triangles) experimental 5. DISCUSSION data [12]; (solid curve) calculation using formula (2). The analysis performed in the previous sections demonstrates that a simple model of electron tunneling between the ferromagnetically correlated regions (FM neling length of the charge carriers. The obtained val- droplets) provides a description of the conductivity and ues of parameters seem to be quite reasonable. Indeed, the magnetoresistance data for a wide class of mangan- the characteristic tunneling length is on the order of the ites. A comparison of the theoretical predictions with FM droplet size, while the concentration of the FM the experimental data on the temperature dependence phase in the high-temperature range is substantially of the resistivity, magnetoresistance, and magnetic sus- smaller than the percolation threshold and varies from ceptibility allowed us to determine various characteris- about 1 to 7%. Note also that the droplets contain 50Ð tics of the phase-separated state, such as the size of FM 100 charge carriers and the parameter A evaluated from droplets, their concentration, and the number of elec- the experimental data agrees in order of magnitude with trons in a droplet, and to estimate the characteristic tun- the Coulomb energy of an extra electron in a metal ball

Table 2. The effective number Neff of manganese atoms, the number of electrons k in a droplet, and the dopant fraction x for some manganites, estimated using formula (2) and the experimental data presented in Figs. 5Ð8

Manganite composition Neff xk Ref.

(La1 Ð yPry)0.7Ca0.3MnO3 250 0.3 75 [9] (see Fig. 5)

Pr0.71Ca0.29MnO3 200 0.29 58 [10] (see Fig. 6)

(La0.4Pr0.6)1.2Sr1.8Mn2O7 250 0.4 100 [11] (see Fig. 7)

La0.8Mg0.2MnO3 265 0.2 53 [12] (see Fig. 8)

Table 3. The CurieÐWeiss constant Θ, the density of ferrons n, the FM phase fraction p, and the effective tunneling length l for some manganites, estimated using formula (3) and the experimental data presented in Figs. 9Ð12

Manganite composition Θ, K n, cmÐ3 pl, Å Ref.

× 18 (La1 Ð yPry)0.7Ca0.3MnO3 55 1.8 10 0.03 24 [9] (see Fig. 9) × 18 Pr0.71Ca0.29MnO3 105 6 10 0.07 17 [10] (see Fig. 10) 18 (La0.4Pr0.6)1.2Sr1.8Mn2O7 255 2.5 × 10 0.04 19 [11] (see Fig. 11) × 18 La0.8Mg0.2MnO3 150 0.6 10 0.01 14 [12] (see Fig. 12)

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χ–1, mol/emu model adopted here, the following expression was 25 derived for the Hooge constant [13, 14],

2 〈〉δU ωV ω 2 3 2 ω˜ α ==------s 2 π l ln ------0 , (4) 20 H 2 ω Udc

2 15 where 〈δU 〉ω is the spectral density of the voltage ﬂuc- tuations, Vs is the sample volume, Udc is the applied ω˜ ω 10 voltage, and 0 = 0exp(A/2kBT). Substituting estimated values of the parameters presented in the tables α ≈ Ð16 3 and in the text, we obtain H 10 cm at a tempera- 5 ture of 100Ð200 K and frequencies within 1Ð1000 sÐ1. α This value of H is 3Ð5 orders of magnitude higher than 0 the corresponding values for semiconductors. 50 100 150 200 250 300 Thus, we have a rather consistent scheme describing T, K the transport properties of manganites under conditions when the ferromagnetically correlated regions do not Fig. 9. Temperature dependence of the inverse magnetic form a percolation cluster. Moreover, the proposed susceptibility of (La1 Ð yPry)0.7Ca0.3MnO3 with y = 1: (tri- approach proves to be valid for a fairly wide range of angles) experimental data [9]; (solid curve) calculations using relation (3). The other manganites of this group dopant concentrations. However, as mentioned above, a exhibit analogous behavior of χ(T) in the high-temperature relationship between the concentration of FM droplets range (see [9]). and the doping level is still incompletely clear. If the above picture of the phase separation is valid, it becomes evident that not all electrons or holes intro- (7Ð8)d in diameter. The obtained estimates of the drop- duced by doping participate in the transport processes. let parameters (characteristic tunneling barrier, size, Now we will try to ﬁnd some qualitative arguments and tunneling length) are close for manganites with illustrating the possible differences in the effective con- strongly different transport properties. centrations of charge carriers below and above the tran- Another characteristic feature of the phase-sepa- sition from PM to magnetically ordered state. rated manganites is a large magnitude of the 1/f noise in The phase diagram of a typical manganite contains the temperature range corresponding to the dielectric a region of the AFM state with FM phase inclusions in state [19, 20]. In the framework of the phase separation the range of low temperatures and low doping levels.

χ–1, mol/emu χ–1, kg/cm3 300 0.5

250 0.4 200 0.3 150 0.2 100

50 0.1

0 0 100 150 200 250 300 240 260 280 300 T, K T, K Fig. 10. Temperature dependence of the inverse magnetic susceptibility of Pr0.71Ca0.29MnO3: (triangles) expe- Fig. 11. Temperature dependence of the inverse magnetic rimental data [10]; (solid curve) calculation using for- susceptibility of (La0.4Pr0.6)1.2Sr1.8Mn2O7: (triangles) mula (3). The density of a porous sample was assumed to experimental data [11]; (solid curve) calculation using for- be 0.8 of the theoretical value. mula (3).

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χ–1, kg/cm3 experimental data showing that δ ~ 1Ð7%. Thus, 8 although formula (5) correctly reﬂects the observed tendency, the nature of the charge disbalance in the PM region is incompletely clear. Probably, for x > xc (in real manganites, the concentration x can reach a level of 6 about 50%), the residual charge is localized in the PM matrix outside the temperature ferrons. A detailed study of this problem will be presented elsewhere. 4

ACKNOWLEDGMENTS 2 The authors are grateful to V.A. Aksenov, N.A. Babushkina, S.W. Cheong, I. Gordon, L.M. Fisher, D.I. Khomskiœ, F.V. Kusmartsev, V.V. Moshchalkov, 0 150 200 250 300 A.N. Taldenkov, I.F. Voloshin, G. Williams and X.Z. Zhou for fruitful discussions and kindly provided T, K experimental data. Fig. 12. Temperature dependence of the magnetoresistance of La0.8Mg0.2MnO3: (triangles) experimental data [12]; This study was supported by the Russian Founda- (solid curve) calculation using formula (3). tion for Basic Research (project nos. 02-02-16708, 03-02-06320, and NSh-1694.2003.2), INTAS (grant no. 01-2008), and CRDF (grant no. RP2-2355-MO-02). When the doping level is increased, the transition from AFM to FM phase is observed. At high temperatures, manganites are in the PM state. As the temperature decreases, the transition from PM to AFM or FM state APPENDIX takes place depending on the doping level. Let us con- We present a short derivation of formulas for the sider the behavior of such a system in the vicinity of a triple point. In the AFM phase, the radius R of a region resistivity (1) and the magnetoresistance (2). Consider converted by one electron into the FM state can be esti- a system of small ferromagnetic droplets dispersed in a mated as [3] dielectric matrix exposed to an electric ﬁeld E, in which charge is transferred via tunneling of the charge carriers ()π 2 1/5 Rd= t/4J ff S Z , from one droplet to another. In the ground state, each droplet contains k charge carriers. Every new charge where Jff is the AFM interaction constant. For the high- carrier tunneling to a given droplet experiences the temperature PM phase, the radius RT of a region con- Coulomb repulsion from the carriers already occurring verted by one electron into the FM state corresponds to in this droplet. The repulsion energy A is assumed to be the size of the so-called temperature ferron and equals relatively large (A > k T), so that the main contribution π 1/5 B to RT = d( t/4kBTln(2S + 1)) [3]. to the conductivity is related to the processes involving ≈ The critical concentration xc 0.15 corresponding to droplets containing k, k + 1, and k Ð 1 carriers. overlap of the low-temperature ferrons can be esti- π 3 Let N be the total number of droplets in the system mated as xc ~ (3/4 )(d/R) . For the high-temperature δ and N1, N2, and N3 be the numbers of droplets contain- ferrons, the corresponding estimate is c ~ π 3 ing k, k + 1, and k Ð 1 carriers, respectively. Assuming (3/4 )(d/RT) . Substituting the expressions for the radii of high- and low-temperature ferrons, we can estimate the total number of droplets in the system to be con- δ stant, so that N2 = N3 and N1 + 2N2 = N, we can write the the ratio xc/ c in the vicinity of the triple point corresponding to the coexistence of FM, AFM, and PM partition function of the system [13], phases:

N/2 3/5 ()3/5 m m A xc T ln()2S + 1 T C ln 2S + 1 ()β β ---- ∼∼------, (5) ZC==∑ NCNmÐ exp Ðm , ------, (6) δ 2 T kBT c zJff S N m = 0 where TC and TN are the Curie and the Néel tempera- m tures, respectively. For the manganites under consider- where CN are the binomial coefﬁcients. Using the ation, we have TC ~ TN ~ 120Ð150 K and ln(2S + 1) ~ Stirling formula, replacing summation by integration, δ ≤ 1.6 for S = 2, hence c xc. This is consistent with and calculating the corresponding integral by the steep-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 580 KUGEL et al. 〈 〉 est descent method, we obtain the following expres- where … V denotes the average over the sample vol- sions for the average vales of N1, N2, and N3: ume. Substituting Eq. (10) into relation (8) and integrating over space, we determine the corresponding current component. Taking into account the obvious A ρ N2 ==N3 N expÐ------, relation = j3/4E, we arrive at formula (1). As can be 2kBT (7) seen, the resistivity decreases with the temperature A according to the law N ==N Ð 2N N 12Ð.exp Ð------1 2 2k T B A ρ()T ∝ T exp ------, 2k T According to our model, electron tunneling implies B one of the four possible events: (i) two droplets with k electrons are converted into droplets with k + 1 and which is characteristic of the systems with tunneling k − 1 electrons; (ii) process inverse to that in the previ- conductivity [21]. ous case; (iii) droplets with k and k + 1 electrons Now let us proceed to the calculation of magnetore- exchange sites with each other; (iv) the same for drop- sistance MR. The probability of tunneling depends, in lets with k and k − 1 electrons. The total current density particular, on the mutual orientation of the electron spin is given by the sum of the contributions from all these and the magnetic moment of a droplet. Orientation of processes: j = j1 + j2 + j3 + j4. As was demonstrated the ferromagnetically correlated regions in the mag- in [13], all the four processes equally contribute to the netic field H leads to an increase in the transition prob- total current. ability and, hence, to a decrease in the resistance with increasing field strength—in agreement with experi- For example, consider the third process. The density ment. The conductivity of the system can be repre- of electrons involved in this exchange is (k + 1)n , σ σ 〈Σ 〉 Σ 2 sented as (H) = 0 (H) , where (H) is the “spin” where n2 is the concentration of droplets with k + 1 contribution to the probability of electron tunneling. electrons. Assuming that k ӷ 1, we obtain For this definition, MR = 〈Σ(H)〉 Ð 1. Denoting the effective magnetic moment of a drop- i i µ θ let by M = BgNeffS and assuming the interaction ≈ i r cos j3 ekn2 ∑v = ekn2 ∑------, (8) between droplets to be negligibly small, we write the τ()i, θi i i r free energy of a droplet in the magnetic field in the following form: where 〈…〉 denotes the statistical average with the sum 〈vi 〉 () () ()θ 2 ψ taken over all droplets with k + 1 electrons, is the UH = U 0 Ð MHcos + Ha cos , (11) average velocity of electrons along the electric field E, ri is the distance between droplets involved in this pro- where θ is the angle between the applied field H and the θi ψ cess, is the angle between vector E and the direction magnetic moment M, Ha is the anisotropy field, and of electron motion, and τ(ri, θi) is the characteristic is the angle between the anisotropy axis and the direc- tunneling time. The standard expression for τ is as fol- tion of the magnetic moment (for the sake of simplicity, lows [21]: we consider the case of uniaxial anisotropy). Let H be parallel to the z axis, and let the anisotropy axis lie in θ the (x, z) plane and make the angle β with vector H. In τ()ω, θ Ð1 r eErcos r = 0 exp- Ð ------, (9) this configuration, l kBT cosψ = sinθβsin cos ϕ+ cos θcos β, ω where l is the tunneling length and 0 is the characteristic frequency. Far from the percolation threshold, the where ϕ is the angle between the x axis and the projec- average in relation (8) is the spatial average of the tion of M onto the (x, y) plane. i velocity v multiplied by the number N1 of droplets In the classical limit, a given orientation of vector M accessible for the jump. corresponds to the probability Assuming the electric field strength to be small Ӷ PH(),,θϕ (eEl/kBT 1) and the repulsion energy A to be large (A/k T ӷ 1), we obtain in the first order with respect to ()θ 2 ψθϕ(), (12) B MHcos + Ha cos the field E: = AH()exp ------, kBT

eEω ÐA/2k T where A(H) is the normalization factor. The eigenstates i 0 B 〈〉2 2 θ Ðr/l ∑v = ------Ne r cos e V , (10) of an electron correspond to conservation of the spin kBT i projection s = ±1/2 onto the effective ﬁeld direction in

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 CHARACTERISTICS OF THE PHASE-SEPARATED STATE IN MANGANITES 581 a ferromagnetically correlated region. Let an electron REFERENCES interact with Z magnetic moments in the droplet. The 1. E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344, 1 energy of this interaction is Es = ÐJSZs. Since the prod- (2001). uct JSZ is on the order of the Curie temperature, Es is 2. E. L. Nagaev, Phys. Rep. 346, 387 (2001). much greater than the energy of interaction between the 3. M. Yu. Kagan and K. I. Kugel’, Usp. Fiz. Nauk 171, 577 electron spin and the magnetic field, provided that H Ӷ (2001) [Phys. Usp. 44, 553 (2001)]. 100 T. In this case, the effective field direction coin- 4. É. L. Nagaev, Pis’ma Zh. Éksp. Teor. Fiz. 6, 484 (1967) cides with the direction of vector M and the probability [JETP Lett. 6, 18 (1967)]. for the electron spin projection to be s can be written as 5. L. N. Bulaevskiœ, É. L. Nagaev, and D. I. Khomskiœ, Zh. Éksp. Teor. Fiz. 54, 1562 (1968) [Sov. Phys. JETP 27, () 836 (1968)]. exp ÐEs/kBT Ps = ------()-. (13) 6. A. M. Balagurov, V. Yu. Pomjakushin, D. V. Sheptyakov, 2cosh Es/kBT et al., Phys. Rev. B 64, 024420 (2001). 7. J. Lorenzana, C. Castellani, and C. Di Castro, Phys. Rev. Upon transfer from droplet 1 to droplet 2, an elec- B 64, 235127 (2001); Phys. Rev. B 64, 235128 (2001). tron occurs in an effective field making an angle ν with 8. N. I. Solin, V. V. Mashkautsan, A. V. Korolev, et al., that in the initial state, for which Pis’ma Zh. Éksp. Teor. Fiz. 77, 275 (2003) [JETP Lett. 77, 230 (2003)]. cosν = cosθ cosθ + sinθ sinθ cos()ϕ Ð ϕ 9. N. A. Babushkina, E. A. Chistotina, K. I. Kugel’, et al., 1 2 1 2 1 2 Fiz. Tverd. Tela (St. Petersburg) 45, 480 (2003) [Phys. Solid State 45, 508 (2003)]. (indices 1 and 2 refer to the droplet number). Then, the 10. L. M. Fisher, A. V. Kalinov, I. F. Voloshin, et al., Phys. work performed for the electron transfer from droplet 1 Rev. B 68, 174403 (2003). ∆ ν to droplet 2 is Es = Es(1 Ð cos ). Accordingly, the 11. P. Wagner, I. Gordon, V. V. Moshchalkov, et al., Euro- probability of this transfer is proportional to phys. Lett. 58, 285 (2002). −∆ exp( Es/kBT). Taking into account all the probability 12. J. H. Zhao, H. P. Kunkel, X. Z. Zhou, and G. Williams, factors introduced above, the final expression can be J. Phys.: Condens. Matter 13, 9349 (2001). written as 13. A. L. Rakhmanov, K. I. Kugel, Ya. M. Blanter, and M. Yu. Kagan, Phys. Rev. B 63, 174424 (2001). 2π 2π π 14. A. O. Sboychakov, A. L. Rakhmanov, K. I. Kugel’, et al., 〈〉Σ() ϕϕ θ θ Zh. Éksp. Teor. Fiz. 122, 869 (2002) [JETP 95, 753 H = ∫ d 1 ∫ d 2∫ sin 1d 1 (2002)]. 0 0 0 15. A. O. Sboychakov, A. L. Rakhmanov, K. I. Kugel, et al., π J. Phys.: Condens. Matter 15, 1705 (2003). ×θsin dθ P()θ , ϕ P()θ , ϕ (14) 16. J. H. Zhao, H. P. Kunkel, X. Z. Zhou, and G. Williams, ∫ 2 2 1 1 2 2 Phys. Rev. B 66, 184428 (2002). 0 17. M. Ziese and C. Srinitiwarawong, Phys. Rev. B 58, ∆ 11519 (1998). × Es ∑ Ps expÐ------. 18. G. Jakob, W. Westerburg, F. Martin, and H. Adrian, Phys. kBT s = ±1/2 Rev. B 58, 14966 (1998). 19. V. Podzorov, M. Uehara, M. E. Gershenson, et al., Phys. Rev. B 61, R3784 (2000). In the high-temperature range, where kBT is much greater compared to the Zeeman energy µ gSN H 20. V. Podzorov, M. E. Gershenson, M. Uehara, and B eff S.-W. Cheong, Phys. Rev. B 64, 115113 (2001). and the magnetic anisotropy energy µ gSN H , rela- B eff a 21. N. F. Mott and E. A. Davis, Electronic Processes in Non- tions (12)Ð(14) yield formula (2). Crystalline Materials, 2nd ed. (Clarendon Press, The limits of applicability of the above expressions Oxford, 1979; Mir, Moscow, 1982). for the resistivity and magnetoresistance of manganites are considered in more detail elsewhere [13Ð15]. Translated by P. Pozdeev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 582Ð593. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 659Ð672. Original Russian Text Copyright © 2004 by Lozovik, Ovchinnikov, Sharapov. SOLIDS Electronic Properties

Optical Properties of a Coherent Phase in ElectronÐHole Systems: Stimulated Light Scattering and Multibeam Processes Yu. E. Lozovik, I. V. Ovchinnikov, and V. A. Sharapov Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow oblast, 142190 Russia e-mail: [email protected] Received September 5, 2003

Abstract—A theoretical study is reported of stimulated light scattering, including wave-vector reversal and anomalous transmission, by a coherent phase in electronÐhole (eÐh) systems of low and high charge-carrier density. For these two cases the coherent phase is taken to be a BoseÐEinstein condensate of excitons or a BCS- like state of eÐh pairs, respectively. The scattering mechanism is laser-induced coherent recombination of two excitons or two coherent eÐh pairs, respectively. The eÐh system is assumed to exist within a GaAs/AlGaAs double quantum well or bulk GaAs. The emission rate of two photons with antiparallel momenta is estimated. Multiphoton emission due to multiexciton coherent recombination is covered. Methods for detecting the effects predicted are proposed. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION less phase, which shows quasi-long-range order (see [15] and references therein). Theory tells us that macroscopic quantum coherence will arise in the electronÐhole (eÐh) system of a In the high-density case, coherent pairing should semiconductor or semimetal if its temperature is arise between an electron and a hole in a manner resembling the BardeenÐCooperÐSchrieffer (BCS) mecha- decreased to below a certain point [1Ð8]. Considerable nism,1 the size of the eÐh pairs being much greater than interest has been shown in the detection of the phenom- the mean distance between them [1, 2, 6]. The BCS-like enon in both three-dimensional (3D) and two-dimen- state of quasi-2D eÐh systems was studied in [13, 15]. sional (2D) systems. Recent years have seen significant In particular, the case where electrons are spatially sep- experimental advances in this field, especially for the arated from holes was covered [13]. Later on, detailed quasi-2D system of crossed excitons in a double quan- research was conducted into phenomena possible in tum well (DQW) [9Ð12]. The predictions that had been dense e–h systems, such as superfluidity, phase equi- made about their behavior include anomalous transport, librium, tunneling, quasi-Josephson phenomena, and superfluidity, drag, and quasi-Josephson phenomena drag [13Ð30]. [13Ð31]. For intermediate charge-carrier densities, a liquid Let n be the charge-carrier density. Two extreme exciton phase was investigated in quasi-2D [17] and 3D cases are generally considered. The high-density case is [35] systems. The phase diagram and properties of a 3 ӷ 2 ӷ quasi-2D system below the transition point were exam- naB 1 [1] or naB 1 [13] for 3D or 2D systems, ined in [13, 15]. 3 Ӷ respectively. The low-density case is naB 1 [3, 4] or The formation of a coherent phase should confer 2 Ӷ new optical properties on the eÐh system [36Ð40]. One naB 1 [14] for 3D or 2D systems, respectively. Here, aB is the Bohr radius. 1 Consider a quasi-equilibrium dense eÐh system, which may be found in a semimetal or created by laser irradiation. It can display In the low-density case, eÐh systems can contain a coherent phase only if the respective Fermi surfaces of electrons δ ∆ excitons that should pass to a coherent state if cooled to and holes fulfill the nesting condition vF p < , where vF is the a sufficiently low temperature. For 3D systems a coher- Fermi velocity, δp is the Fermi-surface separation for the pairing particles, and ∆ is the energy gap due to pairing. The transition ent phase may be formed by BoseÐEinstein (BE) con- point falls fairly slowly with decreasing size of the nesting Fermi- densation [3, 4]. A 2D system first displays a localized surface parts [7]. Over a narrow range of δp a state with a nonuni- BE condensate, which can be characterized by an order form gap may also exist [32, 33]. A competing type of ground state is a liquid metallic phase in the form of eÐh droplets, which parameter of fluctuating phase; this state then changes are more stable in multivalley semiconductors such as Ge and into the low-temperature BerezinskiÐKosterlitzÐThou- Si [34].

OPTICAL PROPERTIES OF A COHERENT PHASE IN ELECTRONÐHOLE SYSTEMS 583 of them is a strong, narrow luminescence line related to 2. COHERENT OPTICAL PROCESSES exciton recombination in the BE condensate; its inten- AND THE ANOMALOUS GREEN FUNCTIONS sity is proportional to the density n0 of the BE condensate. However, with strong excitonÐexciton interaction, It is well known that laser irradiation of a semicon- the ground state is characterized by a fairly large den- ductor can generate eÐh pairs with a density dependent sity of uncondensed excitons, whose radiative recombi- on the radiation intensity. If the lifetime of such excita- nation makes it very difﬁcult to detect the condensate tions is far longer than their energy relaxation time, the line. It is therefore desirable that a better indicator be eÐh system can be in a quasi-equilibrium state and dis- found of a coherent phase in eÐh systems. play a number of equilibrium phases. The recombination of an eÐh pair in a direct-gap semiconductor, such This paper theoretically treats two-photon emission as GaAs, involves emitting a photon. If the temperature due to coherent recombination of two excitons or two is low enough, a coherent phase should arise in the eÐh coherent eÐh pairs in the coherent phase of an eÐh sys- system, which would be manifested in nonzero means tem in the low- and the high-density case, respectively. that relate one-particle states having antiparallel The emission should provide a clear indication of a momenta. It is this anomaly that makes it possible to coherent phase being present in the eÐh system because produce two photons with antiparallel momenta (in 3D the recombination can be represented in terms of the systems) by annihilation of two excitons or two coher- anomalous Green functions. ent eÐh pairs. This process can also be induced by resonant laser irradiation. In the low-density case the coherent phase is taken to be a BE condensate of excitons [40]. Although the The presence of a coherent phase is indicated by a two-exciton recombination is a second-order process nonvanishing anomalous Green function. In the low- with respect to excitonÐphoton interaction, its weak- density case this Green function is 2 ω ness is compensated for by its rate varying as n0 rather ˆ ()ω, i t 〈〉() () G p = Ðite∫d Tap t aÐp 0 than n0 as with one-exciton recombination. β = Ð ------, In 3D systems the total momentum of the emitted []ωεωεÐ ()()p Ð iγ []+ ()()p Ð iγ photons must be zero, because both excitons have zero momentum. The photon momenta are therefore antipar- where ap is the annihilation operator for an exciton of allel. With 2D systems, the argument applies only to momentum p; T is the Wick time-ordering operator; photon momentum components parallel to the plane of ε(p) ≡ ε and γ are the dispersion relation and the the system. This feature could be detected from the p inverse lifetime of elementary excitations in the eÐh time-dependent angular correlation of photocounts system, respectively; and β = ρ V , with ρ and V measured with appropriately arranged photon counters cond 0 cond 0 [37, 41], as in the BrownÐTwiss experiment. denoting the spatial density of the exciton BE condensate and the zeroth Fourier coefﬁcient of excitonÐexci- If the two-exciton coherent recombination in a 3D ton interaction, respectively. From here on, we set ប = system is induced by laser photons of momentum k, it 1 by taking suitable units. For simplicity, we assume must produce two photons with the respective momenta that γ does not depend on the energy or momentum of k and Ðk. The same is true of in-plane photon momen- an elementary excitation. tum components in 2D systems. This phenomenon For GaAs the dispersion relation is might be seen as the reversal of the wave vector by pho- toinduced coherent recombination of excitons. In 2D 2 systems, in-plane reversal is responsible for two anom- v 2k2 k2 ε = ------+,------(1) alous beams: reversed and transmitted (Fig. 5). Further- p M 2M more, BE-condensed excitons should exhibit multiphoton effects linked to multiexciton coherent recombina- v β tion [40]. They are considered in what follows. where = /M is the speed of sound and M is the exciton mass. Three main topics are addressed in this study. First, we examine stimulated light scattering by a 2D coherent phase in a GaAs/AlGaAs DQW (Fig. 1) for the two + extreme cases. Second, we investigate optical wave- A vector reversal for bulk GaAs in the high-density case. Third, we explore three- and four-exciton coherent B – recombination in a BE condensate of excitons in Cu2O. This line of research is taken up in light of experimental efforts to bring about the BE condensation of excitons Fig. 1. GaAs/AlGaAs double quantum well, with layers A in Cu2O [7, 8, 39, 42Ð49]. and B separated by a tunnel-thin barrier.

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584 LOZOVIK et al. ∧ and ω is the ith photon energy as reckoned from the –k|| G k|| i chemical potential µ of the eÐh system:

2 2 ω ()k||, k⊥, = c k|| + k⊥, Ð µ . Fig. 2. Two-exciton recombination in GaAs. i i i µ µ µ µ In the two extreme cases, = a = Eg Ð Eb and = b = µ µ In the high-density case the anomalous Green func- Eg + e + h, where Eg is the semiconductor energy gap; µ µ tion is Eb is the exciton binding energy; e and h are the chemical potentials of charge carriers relative to the iωt F()ω, p = Ðited 〈〉Te ()t h ()0 conduction and the valence band, respectively. Since Eg ∫ p Ðp µ µ exceeds Eb, e, and h by several orders of magnitude, Ð1 (2) µ ≈ µ ≈ 1 we have a b Eg. = ∆ω∏ Ð ---[]ε ()p + ε()p ± ()ξ()p + iη , 2 1 2 ± 3. STIMULATED LIGHT SCATTERING where ep and hp are the respective annihilation opera- BY A 2D COHERENT PHASE: ξ ≡ ξ tors for an electron and a hole and (p) p is the dis- THE LOW-DENSITY CASE persion relation of an elementary excitation in the eÐh system. As with the BCS theory, we take Let us investigate two-exciton coherent recombination for a GaAs/AlGaAs DQW in the low-density case. The Hamiltonian of excitonÐphoton interaction is ξ ∆2 ()ε () ε()2 p = +,1 p Ð 2 p where ∆ is the gap and ε (p) and ε (p) are the respective ()a gk † 1 2 Hint = ∑------akck + H.c., (4) electron dispersion relations for the electron band of the L layer A and the hole band of the layer B, assuming zero k interaction. From here on the electron and the hole band will be referred to as band 1 and band 2, respectively. where gk is the coupling constant and L is the thickness The electron dispersion relations are taken as of a constituent quantum well. Since L will not appear in the ﬁnal formula, we conveniently set L = 1.

2 2 αp Consider a system of excitons that interact with each ε () () p F 〉 α p = Ð1 Ð ------+ ------, other, Nexc being the total number of excitons before 2mα 2mα the recombination. This process (Fig. 2) is the transition from the state where m1 and m2 are the effective electron masses for the bands indicated and pF is the Fermi momentum; this |〉Φ |〉|〉 is determined by the density of excited electrons. 0 = Nexc 0 In quasi-2D systems, only an in-plane momentum component is conserved, so that the normal component to the state of photon momentum is subject to energy conservation |〉Φ = |〉()N Ð 2 |〉1 , 1 , only. f exc k1 k2 Let us consider the emission of two photons with | 〉 | 〉 respective in-plane momentum components k1 and k2 where Nexc is the exciton ground state; 0 is the photon such that k = Ðk = k||. The indices a and b will mean † † † 1 2 ground state; and |〉1 , 1 = c c |0〉, with c denot- that the corresponding quantity refers to the low- or the k1 k2 k1 k2 k1 high-density case, respectively. ing the creation operator for a photon of momentum (k , k⊥ ). To the lowest order the matrix element of the The emission rate is given by 1 , 1 transition is ᐃ πδ() ω ω ᏹ ()ω , 2 ()ab, , k|| = ∫2 1 + 2 ()ab, 1 k|| ()a 1 〈|Φ ()a ()()a ()Φ|〉 S0 → f = ---∫ f Hint t1 Hint t2 0 dt1dt2 (3) 2 (5) × dk⊥, i ∏ ------π , ᏹ ()δωω, ()ω 2 ˆ ()δωω, ()ω 2 = a k|| 1 + 2 = g G k|| 1 + 2 . i = 12, ᏹ ω where (a, b)( , k||) is the transition matrix element, k⊥, i Notice that the matrix element is nonzero only if k1 = − is the vertical coordinate of the ith photon momentum, k2 = k|| for the in-plane photon momentum compo-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 OPTICAL PROPERTIES OF A COHERENT PHASE IN ELECTRONÐHOLE SYSTEMS 585 nents. If γ Ӷ ε , we can use the approximation F k|| –pp –k|| k|| 4 2 2 πg β ᏹ ()ω, ≈ δω()± ε p–k|| k||–p a k|| ------∑ . (6) γ ()ε2 γ 2 k|| F 4 k|| + ± Fig. 3. Coherent recombination of two eÐh pairs in GaAs. As a result, Eq. (3) becomes

2 β 2 ӷ ᐃ , = ------density case, naB 1. Consider a system of coherent a k|| γ ()ε2 γ 2 4 k|| + eÐh pairs with the volume V, which initially consists of (7) N such pairs. The Hamiltonian of electronÐholeÐpho- 2 dk⊥ ton interaction is × ∏ g 2πδ() ω± ε ------, ∫ k|| 2π ±

()b f qk, † † where ω must be regarded as a function of k|| and k⊥. Hint = ∑------eqhkqÐ ck + H.c. (8) The quantity in brackets is the inverse lifetime of an qk, V exciton with the momentum k. Approximation (6) implies that the two photons differ in energy by an amount on the order of ε , which is small compared Since fq, k is almost independent of q, we shall neglect k|| the dependence and simply write f . Further, we set with µ. It follows that the magnitudes of the photon k ≈ V = 1. momenta are very close to k0 Eg/c. Thus, the photon momenta make almost the same angles with the plane The recombination is the transition from the state of the DQW. The emission rate for one photon with the in-plane |〉Φ = |〉 ψ()|〉0 τÐ1 0 N momentum component k is given by Wa, k = nk k , β ε where nk = /2 k is the total number of excitons with to the state the momentum k. The above formulas enable us to quantitatively com- |〉Φ = |〉 ψ()|〉1 |〉1 , pare the respective rates of the one- and the two-photon f N Ð 2 k1 k2 emission: where |〉ψ () is the ground-state wave function, an ana- ᐃ ε β N α a, k 2 k log of the one in the BCS theory [6]. This process a ==------2 2 -. Wa, k γ ()τ4ε + γ involves emission of two photons (Fig. 3). The corre- k k sponding matrix element is written as β × 13 Ð1 ≡ × Ð3 τ ≈ Let us take = 0.8 10 s 0.5 10 eV and k 10Ð8 s. Also, the effective exciton mass in GaAs is ()b 1 〈|Φ ()b ()()b ()Φ|〉 0.22 times the free-electron mass. The speed of sound v S0 → f = --- f Hint t1 Hint t2 0 dt1dt2 2∫ in BE-condensed excitons is then close to 2 × 105 cm/s. (9) 8 9 Ð1 ᏹ ()δωω , ()ω Assuming that γ ~ v/l ≈ 10 Ð10 s , where l is the exci- = b 1 k1 1 + 2 , ton mean free path, we ﬁnd that γ is between 108 and 9 −1 α 10 s . Thus, a varies from 0 to 10, depending on k. ᏹ ()ω, b k

4. STIMULATED LIGHT SCATTERING 2 dp (10) = f F () ω , p F ()ωωÐ , kpÐ ------dω . BY A 2D COHERENT PHASE: k∫ p p ()π 3 p THE HIGH-DENSITY CASE 2 We now proceed to the coherent recombination of ω two eÐh pairs for the same heterostructure in the high- Integration with respect to p yields

[]α ()ξξ ξ ()ξξ η ()ξ ω 2 2 , + + ++2i Ð Ð dp ᏹ ()ω, k = iπ∆ f ------kp p kpÐ p kpÐ p kpÐ ------, (11) b k∫()ξξ η ()ωαη []()2 ()ξ ξ η 2 ()π 3 p + i kpÐ + i Ð kp, Ð p ++kpÐ 2i 2

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2 |ᏹ (ω)| + 2 Ð 2 b ± ω A A α = ∆ ± ---- Ð ------+,η + ------7 2 2 2 2 6 + 2 Ð 2 ± ω A A β = ∆ ± ---- + ------+,η Ð ------5 2 2 2 4 ∆ω+ Ð A+/2 ∆ωÐ Ð A+/2 3 D = arcsin------+ Ð arcsin------Ð - 1 α α 2 ∆ωÐ + A+/2 ∆ω++A+/2 + arccos------Ð arccos------. 1 βÐ β+

0 |ᏹ ω |2 –3 –2 –1 0 1 2 3 Let us examine the function b( ) . Figure 4 shows η ∆ |ᏹ ω |2 x its graphs for different / . Notice that b( ) is non- ω ± ∆ |ᏹ ω |2 ≡ ω ∆ η ∆ zero only near the points = 2 . Owing to the fact Fig. 4. Graphs of b( ) vs. x / for / equal to that these regions disappear as η/∆ 0, we take the (1) 0.1 and (2) 0.01. approximation ᏹ ()ω 2 where b 2 (12) ε () ε() ε ()ε() M1 ∆ 4 []δω()δω∆ ()∆ 1 p + 2 p 1 kpÐ + 2 kpÐ = ------f 0Sη Ð 2 + + 2 , α , = ------Ð ------. 2π kp 2 2

where Sη ≈ 0.3ln()η/∆ . Approximation (12) enables As Eq. (11) includes a cumbersome integrand, we focus on the case k = 0, in which the emitted photon momenta us to estimate the rate of two-photon emission in the nor- are perpendicular to the plane of the DQW. In qualita- mal direction. As with Eq. (7), we calculate integral (3) tive terms the result should also work for reasonable |k|. to obtain 2∆ With k = 0, Eq. (11) becomes M Sη 2 ᐃ = ------1 ------W , (13) b 2π 2 b ᏹ ()ω, ≡ ᏹ ()ω n b 0 b where W is the rate of one-photon emission. Let us set π∆2 2 1 dp b = i f 0∫------. 12 Ð2 ∆ η ∆ 8 Ð1 ()ωξ η ()2 ()ξ η 2 ()π 3 n ~ 10 cm , ~ 0.001 eV, / ~ 0.001, Wb ~ 10 s , p + i ' Ð 4 p + i 2 and M1 = 0.1me. Then | | ≡ The integrand being actually a function of p p only, α ᐃ ∼ Ð6 Ð5 ε 2 b = b/Wb 10 Ð10 . (14) we change to the variable = p /2M1, where M1 = 2m1m2/(m1 + m2). The resultant single integral can be If the recombination is induced by resonant laser radia- ∆ Ӷ 2 calculated in analytical form. For pF /2M1, we thus tion with N0 photons per mode, the rate of emission in obtain the opposite direction to the incident beam is given by ()ᐃ Ð Ð + + Ð + Wopp = N0 + 1 b. (15) M 2 2 A ln()α β /α β + A D ᏹ ()ω = ------1∆ f Ð------b π 0 ω 5 6 2 2 r If N0 ~ 10 Ð10 , the emission rate Wopp will be compa- + ()αÐβ+ α+βÐ Ð rable with the rate of one-photon emission, so that the A ln / Ð A D former effect is, in principle, detectable. + i------ω - , 2 r Also note that in both extreme cases the 2D nature of the eÐh system implies that a laser mode will induce where the emission of photons with the wave vectors ± ± ≈ (k||, k⊥, 1) and (Ðk||, k⊥, 2), where k⊥, 1 k⊥, 2. It follows 2 r = ()ω'2 Ð44∆2 Ð η2 +,16ω'2η2 that aside from backscattering the laser-induced recombination can produce photons with a reversed in-plane momentum component that will cross the DQW (see ± 1 2 2 2 A = ------r ± ()ω' Ð44∆ Ð η , beam 4 of Fig. 5). This phenomenon might be seen as 2 anomalous transmission.

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5. OPTICAL WAVE-VECTOR REVERSAL 2 BY A 3D COHERENT PHASE 4 1 IN THE HIGH-DENSITY CASE θ θ 1 2 Let us consider stimulated light scattering by bulk 3 ӷ GaAs in the high-density case (naB 1). The Hamil- tonian of electronÐholeÐphoton interaction and the transition matrix element are again given by Eqs. (8) DQW and (9), respectively; however, k and p are now 3D vectors in (10). Note that both the magnitudes and the directions of the emitted photon momenta are subject to momentum and energy conservation, so that the two θ θ 3 3 4 photon momenta must be antiparallel and equal in mag- 5 nitude. ᏹ ω ≡ ᏹ Let us calculate b( , k) b(k). As in the 2D ω Fig. 5. Stimulated light scattering from a 2D system involv- case, integration with respect to p yields Eq. (11). The ing two-photon emission: (1) incident beam; (2) ordinary z axis being aligned with k, we recast Eq. (11) to transmitted beam, (k||, k⊥); (3) reversed beam, Ð(k||, k⊥); (4) anomalous transmitted beam, (Ðk||, k⊥); and (5) ordinary θ θ θ θ ᏹ () ()3/2 ∆ 2 () ,,… reﬂected beam, (k||, Ðk⊥). The angles are 1 = 2 = 3 = 4. b k ==2M1 f k Ipi , i 1 6. (16)

Here, I(pi) is an appropriate integral with the dimen- where bq is the annihilation operator for a phonon of sionless parameters momentum q and λ is the effective coupling constant.

2 If the BE condensate experiences N-exciton recom- k2 k2 M k p = ------, p = ------, p = ------1 , bination creating N photons, it makes the transition 1 2M ∆ 2 2M ∆ 3 2∆ 1 2 2M2 (1) 〈|i V c …c aN〈|i = 〈|f . k1 kN 0 2 ω η pF ≈ Eg p4 ==------∆, p5 ---∆------∆ , p6 =∆--- , As with two-exciton recombination, the processes 2M1 could be detected by BrownÐTwiss measurements ≠ using a coincidence circuit with N photon counters. where M2 = 2m1m2/(m2 Ð m1) with m1 m2. It can be cal- However, we here consider an alternative approach culated numerically. based on multibeam laser-induced recombination. ≈ 21 Ð3 With bulk GaAs, we have Eg 1.5 eV, n = 10 cm , Since BE-condensed excitons have zero momen- 2 tum, so must be the total momentum of the N photons p = 3 × 107 cmÐ1, M ≈ 0.2m , ∆ = 0.2 p /2M , and F 2 e F 1 created: | | ≈ | |2 ≈ k Eg/c. Calculating pi, we obtain I(pi) 250. The rate of two-photon emission is thus estimated as N ᐃ ≈ × 21 Ð1 b, k 5 10 s . ∑ ki = 0. (18) i = 1 6. MULTIEXCITON COHERENT Recall that we reckon photon energy from the exciton RECOMBINATION chemical potential µ, so that the photon dispersion rela- IN A BOSEÐEINSTEIN CONDENSATE: tion is THE CASE OF Cu2O ω = ck Ð µ ≡ ck()Ð k . This section is concerned with the coherent recom- k 0 bination of three or four BE-condensed excitons in bulk BE-condensed excitons are thus assigned zero energy Cu2O. The processes will be investigated by generaliz- (µ = 0). Due to energy conservation the photon energies ing the above results concerning two-exciton recombi- ω must obey the constraint nation. The rate of direct eÐh recombination being very i low in Cu2O, an exciton decays mostly by emission of N a photon and an optical phonon. The Hamiltonian of ∑ ω = 0, ω ≡ ω . (19) excitonÐphotonÐphonon interaction is i i ki i = 1 λ If N-exciton recombination is induced by N Ð 1 laser ()1 kq, V = ∑ ------cpq+ Qpbq + H.c., (17) beams with respective wave vectors k (i = 1, …, N Ð 1), V i pkqÐ0Ð = it should produce a plane outgoing wave whose wave

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k2 Three-exciton recombination is also connected with phononÐexciton interaction (see below). The corresponding Hamiltonian is

k3 k1 ()2 gkq, V = ∑ ------apq+ apbq + H.c. (25) pkqÐ0Ð = V

2 Fig. 6. Three-exciton recombination in a BE condensate. Assume that the coupling constant is g = 10 λ. This implies that the rate of phonon exchange between excitons is higher than that of photon emission by four ω vector kN and photon energy N are subject to con- orders of magnitude (so that the characteristic time is straints (18) and (19), respectively: about 10Ð10 s). Using the Fermi golden rule, we ﬁnd that photons N Ð 1 are created at the rate

kN = Ð ∑ ki, (20) N i = 1  W = N 2πδ∑ ω N ∫ i N Ð 1 i = 1 (26) ω ω N Ð 1 N = Ð ∑ i. (21) 2 N dki i = 1 × ᏹ ()k …k V ∏ ------, N 1 N ()π 3 i = 1 2 As compared with concurrent spontaneous emission, ᏹ the intensity of the stimulated emission is higher by a where N is the matrix element of the process and the factor of prefactor N is due to N photons being created in every ᏹ elementary process. Note that N can be expanded in Ð1/2 N Ð 1 terms of N factors of the form V , so that WN is actu- () ally independent of V. This allows us to set V = 1. ∏ Ni + 1 , (22) i = 1 6.1. Three-Exciton Recombination where N is the mean number of photons per mode for i The matrix element ᏹ (k , k , k ) of three-exciton the ith incident beam. 3 1 2 3 recombination is the sum of matrix elements obtained Let us determine conditions for an outgoing wave to from the one from Fig. 6 by all possible permutations be detectable. As with two-exciton recombination, one- of the photon-vertex arguments: exciton stimulated recombination will not contribute to ᏹ (),, the photoluminescence in the direction of the outgoing 3 k1 k2 k3 wave if the incident wave vectors are oriented appropriately. Accordingly, the background emission in the out- 3 3/2 = gλ ρ ∑Ᏻ()ω Ᏻ()ω G ()ω , (27) going-wave direction is due to one-exciton spontaneous cond i j Ðk j j ij≠ recombination only. The rate WN of N-exciton stimulated recombination should therefore be compared with ij, = 123,,, the rate W of one-exciton spontaneous recombination. 1 ω Assume that each incident beam has 103 photons per where G( , k) is the exciton Green function, mode. The detection is possible if ωε+ 2 + β2 G()ω, k = ------k , (28) 3()N Ð 1 > []ωεωε()γ []()γ 10 W N W1, (23) Ð k Ð i + k Ð i Ᏻ ω where and ( ) is the phonon Green function, 2Ω ρ τ Ᏻ()ω = ------. (29) W1 = cond/ , (24) []ωΩωΩÐ ()Ð iδ []+ ()Ð iδ The latter is assumed to be independent of k because τ π λ22 ≈ Ð5 with = / k0 10 s being the exciton lifetime in the magnitude of any wave vector involved is of the λ ≈ × 2 Ð1 3/2 Cu2O; hence 2.5 10 s cm . order of k0 (see Section 3), which is small in terms of

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 OPTICAL PROPERTIES OF A COHERENT PHASE IN ELECTRONÐHOLE SYSTEMS 589 the Brillouin zone; with such wave vectors, the optical- k3 k2 phonon energy can be regarded as a constant. The rate of the process is mainly determined by the photon-energy regions in which virtual particles are k k closest to their mass surfaces, at which resonance 4 1 occurs. The photon energies (with the dispersion relation selected) are of the order of the elementary-excita- ε tion energy k in the BE condensate or the optical- phonon energy Ω (see (31)). The orders of magnitude Fig. 7. Four-exciton recombination in a BE condensate. ε Ω of k and are much less than that of the exciton chem- µ ≈ ical potential Eg as reckoned from the valence-band sate with two laser beams making an angle of 2π/3 with maximum. We therefore take k0 as an approximation to the magnitudes of photon wave vectors (see Appendix one another (see Appendix); the wave vector of the out- ᏹ going beam will be coplanar with the incoming wave for details). Accordingly, 3(k1, k2, k3) in (27) becomes a function of photon energies only. Assuming vectors and will make the stated angle with each of them. that δ Ӷ Ω and γ Ӷ ε , where ε is the elementary- k0 k0 As already noted, the recombination rate is mainly |ᏹ |2 excitation energy, we apply the pole expansion to 3 determined by the photon-energy regions in which vir- and obtain the sum of six equal resonant terms: tual particles are closest to their mass surfaces. Specif- ically, the corresponding energies of the three emitted ᏹ ()ω ,,ω ω 2 ≈ 6g2λ6ρ3 Ᏻ()ω Ᏻ()ω G ()ω 2. photons are close to any of the following sets 3 1 2 3 cond 2 1 k0 1 ()µΩµΩµ, , Owing to the above approximation the rate W3 of three- + Ð , exciton stimulated recombination is expressed as ()µΩµε+ ,,±µΩεÐ +− , k0 k0 (31) 2 6 3 3 ()µΩ,,µε− µΩε± g λ ρ k δω Ð + k + k , cond 0 Ᏻ()ω ()ω 2 1 0 0 W3 = 18------∫ 1 Gk 1 ------πc3 0 2π where µ is the exciton chemical potential. The first set 2dω corresponds to the first term in brackets appearing × Ᏻ()ω ------2. ∫ 2 2π in (30), so that the resonant condition is fulfilled by a virtual optical phonon. The other sets correspond to the Since Ω ≈ 10Ð2 eV ӷ ε ≈ 10Ð4 eV, we arrive at second term, implying that an elementary excitation of k0 the BE condensate will be at resonance. 2λ6ρ3 3 2 Thus, the recombination rate can be increased by g condk0 1 1 1 β W3 = 18------+ ---1 + ------setting the respective photon energies of the laser πc3 Ω2δ δ γ 2ε2 beams to any two members of any energy set in (31), k0 (30) the outgoing photon energy being equal to the remain- 2λ4ρ2 2 ing member. g condk0 1 1 β = 18------+ ---1 + ------W . 2Ω2δ δ γ ε2 1 c 2 k 0 6.2. Four-Exciton Recombination

To estimate W3, we take the exciton mass as 2.7me and To the lowest order with respect to excitonÐphononÐ the other parameters as β ≈ 0.5 meV (0.5 × 1012 sÐ1), photon interaction, the matrix element of four-exciton ρ ≈ 19 Ð3 γ ≈ β δ ≈ 9 Ð1 cond 10 cm , 0.1 , and 10 s . The relative stimulated recombination is given by ε ≈ permittivity and energy gap of Cu2O are 9 and ᏹ ()ρ… λ4 ≈ ε ≈ 10 × 4 k1 k4 = cond Eg 2 eV, so that c = c0/ 10 cm/s and k0 = 3 105 cmÐ1. Thus, × ()()ω ω Ᏻ()ω Ᏻ()ω ∑ FÐ()k + k Ð l + m Ð l Ð n , ≈ Ð2 1 m W3 10 W1; lm≠ mn≠ nl≠ that is, one out of a hundred excitons decays by three- exciton stimulated recombination. Noting that the rate lmn,, = 1,,… 4. is high enough to meet requirement (23), we conclude that the process is in principle detectable. It corresponds to 12 diagrams constructed from that of Three-exciton stimulated recombination could be Fig. 7 by photon-vertex permutation. The two exciton produced and detected by illuminating the BE conden- lines that are not shown in Fig. 7 are implicitly included

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k1 + k2 that is, one out of two hundred excitons decays by four- exciton stimulated recombination. Noting that the rate k 1 is high enough to meet requirement (23), we conclude k that the process is, in principle, detectable. 1 k 2 The wave vectors of the photons created are oriented q2 q2 q3 as in Fig. 8b of Appendix. Due to condition (18) the k total momentum of any two photons is in the opposite 2 direction to that of the others. Also, the angle between k3 any two wave vectors is the same as that between the other two. However, the wave vectors are not generally coplanar. k 3 k Four-exciton stimulated recombination could be (a)4 (b) produced and detected by illuminating the BE condensate with three laser beams. The wave vectors of the Fig. 8. The magnitude of an outgoing wave vector kN relative to the mutual orientation of the incoming ones with a incoming beams should be oriented as described above, ﬁxed magnitude, for (a) N = 3 and (b) N = 4. the outgoing wave vector being given by Eq. (20). As with three-exciton recombination, the rate of four-exciton recombination is highest if the photon in the anomalous Green function of BE-condensed energies are equal to the members of the set excitons. ()µΩµΩε,,± µΩµΩε, − |ᏹ |2 + Ð k Ð + + k . Applying the pole expansion to 4 , we obtain the 0 0 sum of 12 equal resonant terms: Accordingly, the recombination rate can be increased by setting the respective photon energies of the laser ᏹ 2 4 beams to any three members of the set, the outgoing (32) photon energy being equal to the remaining member. = 12ρ2 λ8 F ()ω + ω Ᏻ()ω Ᏻ()ω 2. cond k1 + k2 1 2 1 3

As with three-exciton recombination, we take the mag- 7. CONCLUSIONS nitudes of photon wave vectors to be k0 (see Appendix We investigated stimulated light scattering, includ- for details), so that the recombination rate W4 is ing wave-vector reversal, from an eÐh coherent phase in expressed as a GaAs/AlGaAs DQW and bulk GaAs in the low- and the high-density case, the scattering mechanism being 2k 4 0 the coherent recombination of two excitons or two eÐh ρ2 λ8 k0 du ()2 pairs, respectively. The estimated rates of two-photon W4 = 48 cond ------∫------∫ dkFk u 2π2c4 2π emission indicate that the scattering is detectable in 0 both of the extreme cases. If the incident laser radiation dω dω provides 102Ð105 photons per mode, the rate of two- × ------1 Ᏻ()ω 2 ------3 Ᏻ()ω 2, ∫ 2π 1 ∫ 2π 3 photon emission will be comparable to that of one-photon emission. We also considered multiexciton coherent recombination, for bulk Cu O. Similarly, the coher- where u = ω + ω and k = |k + k |. Assuming that γ is 2 1 2 1 2 ent recombination of multiple correlated eÐh pairs is independent of k and is much less than ε , we see that k0 possible in the high-density case. Four-wave mixing is another candidate way of 2k0 studying BE-condensed excitons [50Ð52]. However, it πβ2 du ()2 is important to note that this method creates a coherent ∫------∫ dkFk u = ------. 2π 8γ 2v exciton system (in the form of exciton density waves) 0 before it is examined, whereas our approach deals with an existing one. With v ≈ 0.5 × 106 cm/s, we arrive at It should be emphasized that stimulated light scattering by an eÐh system is possible only if the system ρ2 λ8β2 4 ρ λ6β2 2 3 cond k0 3 cond k0 contains a coherent phase; therefore, this optical phe- W4 ==------W1. (33) πc c4δ2γ 2 c c2δ2γ 2 nomenon could serve as an indicator of the presence of v v such phase. Another potential test for a coherent phase is the detection of a pair of correlated photons with anti- Calculation yields parallel momenta that have been produced by coherent recombination of two excitons or two eÐh pairs in the × Ð3 W4 = 510W1; coherent phase. This could be done by BrownÐTwiss

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 OPTICAL PROPERTIES OF A COHERENT PHASE IN ELECTRONÐHOLE SYSTEMS 591 measurements with two appropriately positioned detec- Integration with respect to the angles brings expres- tors. A similar method should work for multiexciton sion (34) to a simpler form: coherent recombination. 2 k 3 dω ------0- …∏------i. (37) ACKNOWLEDGMENTS π 3∫ 2π c i = 1 Yu.E.L. is grateful to Yu.M. Kagan, L.V. Keldysh, θ L.A. Maksimov, and V.G. Lysenko for helpful discus- The fact that cos 2 = Ð1/2 implies that the angle sions of the results. between the wave vectors is 2π/3. This work was supported by grants from INTAS and the Russian Foundation for Basic Research. Four-Photon Case In the four-photon case the matrix element depends | | APPENDIX on the photon energies and k1 + k2 . To reduce the num- In general, the integral in Eq. (26) is to be taken with ber of integrations in (26), we represent the orientation respect to N Ð 1 wave vectors in a 3D space. It is possi- of k3 in terms of its angle with k1 + k2 (Fig. 8b), so that ble to reduce the total number of variables of integration if we can set k = k for i = 1, …, N Ð 1, where k is N Ð 1 i 0 i 2 2 θ the magnitude of the ith wave vector. Speciﬁcally, we ∑ ki = k1 + k2 ++k0 2k0 k1 + k2 cos 3 . introduce spherical coordinates in the 3D space and so i = 1 bring the integral to the form Expression (36) thus becomes N N Ð 1 2 k dω dcosθ dφ 2 … πδ()i i i i Ð1δ()2 θ ∫ 2 ∑ cki Ð k0 ∏ ------, (34) c k1 + k2 ++k0 2k0 k1 + k2 cos 3 Ð k0 c()2π 3 i = 1 i = 1 ()Ð1δθ() = c k1 + k2 cos 3 Ð X , the wave vectors being represented as Ð1| | ∈ where X = Ð(2k0) k1 + k2 [Ð1, 0]. Integration with ()θ θ φ θ φ θ ki = ki cos i; sin i cos i; sin i sin i . (35) respect to 3 transforms (34) into

2 2 2 With ki = k0 for each i, the product terms become k dω k dω …------0 ------3------0 i dcosθ dφ . ∫ 2 ()π ∏ 3 i i 2 2 ()π ω θ φ c k1 + k2 i = 1 c 2 k0d idcos id i ------3 -, c()2π Next, we represent the orientation of k2 in terms of its angle with k1 (Fig. 8b), so that and the delta-function term is recast to ()θ N Ð 1 k1 + k2 = k0 21+ cos 2 . 1  ---δ∑ k Ð k . (36) θ | | c i 0 Changing from cos 2 to k1 + k2 as a variable of i = 1 integration, With this approximation, energy conservation dictates θ () d cos d k1 + k2 that the incoming wave vectors be oriented so that the ------2- = ------, k +02k ∈ (), k , k + k 2 1 2 0 outgoing wave vectors as given by Eq. (20) have the 1 2 k0 magnitude k . 0 brings (34) to the form

3 Three-Photon Case k4 dω ------0 …d k + k ------i. 2 4∫ 1 2 ∏ π Without loss of generality, we introduce spherical 2π c 2 coordinates according to Fig. 8a. Consequently, i = 1

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JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 594Ð604. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 673Ð683. Original Russian Text Copyright © 2004 by Malishevskiœ, Silin, Uryupin. SOLIDS Electronic Properties

Properties of a Fast Josephson Vortex A. S. Malishevskiœ, V. P. Silin, and S. A. Uryupin Lebedev Institute of Physics, Russian Academy of Sciences, Leninskiœ pr. 53, Moscow, 119991 Russia e-mail: [email protected] Received May 22, 2003

Abstract—A model is formulated for the analytic description of vortices in a system consisting of a long Josephson junction (JJ) and a waveguide that is magnetically coupled to this junction. The application of this model made it possible to determine the allowed range of velocities of a vortex. It is established that a free vortex can move with a velocity much greater than the Swihart velocity of a Josephson junction. Such a vortex is called fast. The effect of the waveguide on the forced motion of vortices is studied. It is shown that fast vortices can be generated at relatively small values of current. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION remarkable property of the SakaiÐTatenoÐPedersen model is the possibility of describing the role of the Model equations for the difference between the Cherenkov effect in a JJ. phases of a condensate wave function on different sides of a Josephson junction (JJ) (see, for example, [1Ð6]) In the present paper, we set up a problem of the are widely used for the description of the properties of velocity of a Josephson vortex in a JJ to which a magnetically coupled waveguide is connected. An analytic Josephson vortices. The main simplification of the the- examination of such a problem is possible owing to the ory in model approaches lies in replacing a sine nonlin- model consideration of the coupling between the JJ and earity by relatively simple piecewise linear functions. the waveguide. We show that the coupling to a The efficiency of the application of model approaches waveguide characterized by the large Swihart velocity has been demonstrated both while describing vortices allows one to obtain a Josephson vortex moving with a that carry a single quantum of a magnetic flux [7] and velocity that may be much greater than that of an iso- while describing more complicated multivortex struc- lated JJ. This opens up the possibility of implementing tures [8, 9]. Up to now, the model description has been fast Josephson vortices. used for studying vortices either in an isolated JJ or in two magnetically coupled JJs [10]. Since the applica- The results of the model approach are described as tion of model approaches in the theory of simple follows. In Section 2, relying on the equations that Josephson structures proved to be efficient, it seems describe a JJ coupled to a waveguide, we describe a free expedient to use a model approach in the theory of vor- motion of an elementary vortex in such a system. We tices in a structure consisting of a JJ coupled to a indicate the range of velocities in which a fast Joseph- waveguide. The statement of such a problem was given son vortex may exist. The existence of this range of in [11]. According to this statement, in the present velocities is associated with the effect of the waveguide on the JJ. In Section 3, we investigate the forced motion study, we consider vortices in a JJ coupled to a planar of this vortex due to a transport current. For sufficiently waveguide. To obtain analytic characteristics, we apply small losses, we determine the phase differences on the the SakaiÐTatenoÐPedersen model in which the sine of JJ and on the waveguide walls within the SakaiÐ the phase difference is simulated by a sawtooth func- TatenoÐPedersen model. The use of the model tion of form (2.12) (see below). The application of this approach has allowed us to determine the contributions, model, on the one hand, allows one to correctly to the phase differences, of the terms that vary over a describe the allowed and forbidden velocity bands for a relatively large scale determined by dissipation. We freely moving vortex (these bands were determined show how this scale manifests itself in the structure of in [12] in a theory that does not take into account dissi- a vortex and determine the contributions of dissipation pation and does not use an approximate representation in the JJ and the waveguide to the relation between the for the sine of the phase difference) and, on the other vortex velocity and current. We show that the forced hand, allows one to construct an analytic solution to a motion of a fast Josephson vortex may occur under rela- system of coupled equations for the phase differences tively low densities of the transport current. In Section 4, in the JJ and in the waveguide even when dissipation is we describe the effect of Cherenkov losses in the JJ significant. The latter fact is of interest for the problem coupled to a waveguide on the transport current. In Sec- of the motion of a vortex under a transport current. A tion 5, we discuss the final results.

PROPERTIES OF A FAST JOSEPHSON VORTEX 595

2. A FREE VORTEX respectively, where

Consider a double-sandwich-type layered system L ∆λ≡ ++2d λ coth----- that consists of three superconducting layers S1, S2, and 1 2 λ 2 S3 and two nonsuperconducting layers I and W. The superconducting layers occupy the domains x < Ðd, d < ×λλ L λ2 2 L > 3 ++2dw 2 coth----- Ð 2 cosech ----- 0, x < d + L, and x > d + L = 2dw and have the London λ λ λ λ λ 2 2 depths 1, 2, and 3, respectively. The substance in the layer I of thickness 2d, which is sandwiched between ⑀ the coupling constants between the JJ and the the layers S1 and S2, has dielectric permittivity and waveguide that appear on the right-hand sides of (2.1) σ conductivity . The layer W of thickness 2dw, which and (2.2) are given by separates the superconductors S2 and S3, has dielectric ⑀ σ permittivity w and conductivity w. Assume that the λ cosech()L/λ S ≡ ------2 2 , (2.5) layer I is thin enough that Cooper pairs may tunnel λ ++2d λ coth()L/λ through it, thus creating a Josephson current of critical 3 w 2 2 density jc. The thickness of the layer W is assumed to and be so large that one can neglect the Josephson current through W compared to the displacement and conduc- λ cosech()L/λ tivity currents. This fact allows us to treat the system ≡ 2 2 Sw λ------λ ()λ , (2.6) considered as a JJ magnetically coupled to a 1 ++2d 2 coth L/ 2 waveguide. ω respectively; j is the Josephson frequency of the JJ; Let ϕ be the phase difference of the condensate and F[ϕ] is the density of the Josephson current nor- wave functions of the superconductors S and S on 1 2 malized by jc. We emphasize that the expressions for Vs their boundaries x = Ðd and x = d with the layer I and ϕ w and Vsw contain the thickness L of the superconducting be the phase difference of the condensate wave func- layer S2 through which the JJ is coupled to the tions of the superconductors S and S on their bound- 2 3 waveguide. This means that Vs and Vsw determine the aries x = d + L and x = d + L + 2dw with the waveguide Swihart velocities with regard to the above coupling. In W. Assume that the characteristic space scales of varia- the limit as L ∞, (2.3) and (2.4) reduce to ϕ ϕ tions of and w are large compared to the London lengths. Then, following [13, 14], we obtain the follow- 1 2d ϕ ϕ v ≡ ing system of equations for and w (cf. [11, 12]): V s s c --⑀------λ λ , 1 ++2 2d ∂2ϕ(), ω2 []ϕ(), zt and j F zt + ------∂t2 (2.1) 1 2dw 2 ∂2ϕ (), V v ≡ c ------, 2∂ ϕ()zt, 2 w zt sw sw ⑀ λ λ w 2 ++3 2dw = V s ------+ SVs ------, ∂z2 ∂z2 respectively. The quantities vs and vsw are the Swihart velocities of noninteracting isolated JJ and waveguide, ∂2ϕ (), ∂2ϕ (), 2 w zt 2 w zt 2 ∂ ϕ()zt, respectively. ------= V sw------+ SwV sw------. (2.2) ∂t2 ∂z2 ∂z2 For vortex structures propagating with constant ϕ ψ ζ ϕ ψ ζ ζ ≡ velocity v when (z, t) = ( ) and w(z, t) = w( ), Here, z Ð vt, Eqs. (2.1) and (2.2) yield ω2 []ψζ() ()ψ2 2 ()ζ 2ψ ()ζ λ λ ()λ j F Ð V s Ð v '' = SVs w'' , (2.7) 2 ≡ 22d 3 ++2dw 2 coth L/ 2 V s c -----⑀------∆ (2.3) ()ψ2 2 ()ζ 2 ψ ()ζ Ð V sw Ð v w'' = SwV sw '' . (2.8) and ψ ∞ ψ ∞ ψ ∞ ψ ∞ Setting (Ð ) = w(Ð ) = 0 and '(Ð ) = w' (Ð ) = 0, λ λ ()λ from (2.8) we obtain 2 ≡ 22dw 1 ++2d 2 coth L/ 2 V sw c ------⑀ ------∆ (2.4) w 2 ψ ()ζ V sw ψζ() w = ÐSw------2 2 . (2.9) are the Swihart velocities in the JJ and the waveguide, V sw Ð v

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 596 MALISHEVSKIŒ et al. ψ ζ Substituting (2.9) into (2.7), we obtain the following Using kj(v), we rewrite Eq. (2.10) for ( ) as equation for the phase difference ψ on the JJ:

2 Ð2 2 2 2 2 []ψζ() ()ψ()ζ ()v v ()v v FSTP = ---k j v '' . (2.16) ω2 []ψζ() 1 Ð 2 Ð ψ ()ζ π j F = ------2 2 - '' , (2.10) V sw Ð v The formal similarity between this equation and the where the velocities v1 and v2 are deﬁned by equation corresponding to the isolated JJ [7] makes it possible to write out the following solution to Eq. (2.16) that describes an elementary vortex (a 2π kink): v m

2 2 ()2 2 2 π π π ≡ V s + V sw ()m V s Ð V sw 2 2 > ψζ()= --- exp k ()ζv +,--- ζ < Ð------, (2.17) ------1++0,Ð ------SSwV s V sw j () 2 4 2 4 4k j v (2.11) m = 12., π ψζ() π []()ζ = + ------sin k j v , Below, in the main body of the paper, we will apply the 2 SakaiÐTatenoÐPedersen model [1Ð3] with the follow- (2.18) π π ing sawtooth function F[ψ]: <<ζ Ð------()------()-, 4k j v 4k j v []ψ []ψ F = FSTP  π π ()ψ2/π , Ð π /2 << ψπ /2, ψζ()= 2π Ð ---exp Ðk ()ζv +,---  (2.12) 2 j 4 = ()πψ2/π ()Ð , π/2 <<ψ 3π/2, (2.19)  π ()ψπ ()π π <<ψ π ζ >  2/ Ð 2 ,3/2 5 /2. ------()-. 4k j v

According to (2.12) and Eqs. (2.7) and (2.8), the wave- ψ numbers of small perturbations of the phase difference The phase difference w on the waveguide walls that that are characterized by the coordinate dependence corresponds to this vortex is determined from (2.9) and exp(ikζ) are determined by the equation is expressed as

()2 2 ()2 2 2 2 2 v Ð v v Ð v 2 π SwV sw π ± ---ω +0,------1 2 -k = (2.13) ψ ()ζ = Ð------exp k ()ζv +,--- π j 2 2 w 2 2 2 j 4 V sw Ð v V sw Ð v π ψ ζ < Ð------, where the “+” corresponds to the perturbations of 4k ()v near ψ = 0 or 2π and the “–” corresponds to the pertur- j bations of ψ near ψ = π. Equation (2.13) can be rewritten in the following equivalent form: 2 π 2 ψ ()ζ πSwV sw SwV sw []()ζ w = Ð ------Ð ------sink j v , V 2 Ð v 2 2V 2 Ð v 2 2 2 2 2 2 2 2 sw sw ± ---ω ()v ()v π j + V s Ð k V sw Ð π π (2.14) <<ζ Ð------()------()-, 2 2 2 4k j v 4k j v = SSwV s V swk .

This equation corresponds to the coupling between the 2 π 2 π ψ ()ζ π SwV sw SwV sw ()ζ ordinary (for the “+”) and extraordinary (for “–”) Swi- w = Ð 2 ------+ ------exp Ð,k j v + --- 2 v 2 2 2 v 2 4 hart waves of the JJ and an electromagnetic wave in the V sw Ð V sw Ð waveguide. π ζ > ± ------()-. In the case of “+,” we obtain k = ikj(v) from (2.13), 4k j v ± and, in the case of “–,” we obtain k = kj(v), where For the solution (2.17)Ð(2.19) to describe a solitary 2 2 ψ ∞ ψ ∞ π 2 V Ð v vortex, i.e., for the equalities (Ð ) = 0 and ( ) = 2 k ()v ≡ ---ω ------sw . (2.15) to hold, it is necessary that k (v) be real and given j π j ()2 2 ()2 2 j v 1 Ð v v 2 Ð v by (2.15). This requirement implies that the values of

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 PROPERTIES OF A FAST JOSEPHSON VORTEX 597 ≈ the vortex velocity may only belong to the two allowed v2 Ð Vsw SSw Vs/2. This means that, just as in the pre- bands vious case, the effect of the waveguide manifests itself, ﬁrst, in a slight increase in the limit velocity of a vortex 0 <

Finally, when Vsw >Vs and the coupling between the On the contrary, in the interval (v1, Vsw) and for v > v2, v JJ and the waveguide is still weak, from (2.11) we kj( ) is pure imaginary, which does not correspond to obtain the motion of a solitary vortex. Recall that, in an isolated JJ, a vortex may move with arbitrary velocities 2 v 1 V sw below the Swihart velocity s. The splitting of the v ≈ 1 Ð ---SS ------V , (2.24) domain of admissible velocities of a vortex and the 1 2 w 2 2 s V sw Ð V s appearance of the forbidden band [v1, Vsw] is associated with the effect of the waveguide on the JJ. V 2 The expressions for the velocities v1 and v2 that ≈ 1 s v 2 1 + ---SSw------V sw. (2.25) determine the right boundaries of the allowed bands 2 V 2 Ð V 2 have an especially simple form in the case of weak cou- sw s pling between the JJ and the waveguide, when the cou- ӷ In particular, in the most interesting case when Vsw pling constants S and Sw are small. When Vsw < Vs, we have Vs, the ﬁrst allowed band extends from zero to nearly the Swihart velocity of the JJ, which is followed by a ≤ ≤ 2 relatively broad forbidden band (1 Ð SSw/2)Vs v 1 V v ≈ 1 Ð ---SS ------s V , (2.22) Vsw, which, in turn, is followed by the second allowed 1 2 w 2 2 sw V s Ð V sw band 2 2 1 V 1 V V <

Ӷ The width of this band is small compared with both Vsw In particular, when Vsw Vs, this implies that the and Vs. Since this region lies near the Swihart velocity allowed band (2.20) extends from zero to nearly the of the waveguide, which was assumed to be large com- Swihart velocity of the waveguide; this band is fol- v ≈ pared with the Swihart velocity of the JJ, it can be lowed by a relatively narrow forbidden band (Vsw Ð 1 called the existence region of a fast Josephson vortex. SSwVsw/2), which, in turn, is followed by the broad second allowed band (2.21) extending from the Swihart As already mentioned, in an isolated JJ, a vortex may move with a velocity less than the Swihart velocity velocity of the waveguide to a velocity slightly greater v than the Swihart velocity of the JJ. In other words, s in the JJ. When the JJ is connected to a waveguide, a when the Swihart velocity of the waveguide is small new phenomenon arises, a fast Josephson vortex. This compared with the Swihart velocity of the JJ, the effect vortex moves with velocity much greater than the Swi- of the waveguide manifests itself in a slight increase in hart velocity of the JJ. Such an increase in the vortex velocity is attractive since the Swihart velocity of the the limit velocity of a vortex compared with Vs and in ε the emergence of a narrow (compared with both Vs and waveguide may be close to the velocity of light c/ w Vsw) forbidden band for velocities. in the substance that fills the waveguide. Thus, formula (2.26) points to the following interesting fact: a When Vsw = Vs and the coupling between the JJ and the waveguide is weak, we have Josephson vortex may move with very high velocity that is close to the velocity of light in the dielectric that 1 fills the waveguide. v ≈ 1 Ð --- SS V , 1 2 w s Thus, when the Swihart velocity of the waveguide is large compared with the Swihart velocity of the JJ, the effect of the waveguide is manifested in the fact that, ≈ 1 first, the right boundary of the first allowed band v 2 1 + --- SSw V s. 2 slightly decreases compared with Vs, second, the forbidden band, which extends nearly from Vs to Vsw, is These formulas show that the width of the first allowed broader than both allowed bands, and, third, there band is close to Vs, while the forbidden band and the appears a region of a fast vortex that moves with a ≈ second allowed band are relatively narrow: Vsw Ð v1 velocity close to the Swihart velocity of the waveguide.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 598 MALISHEVSKIŒ et al. We emphasize that the splitting of the range of case, Eq. (3.5) has three solutions. Let us write out velocities in which a Josephson 2π kink may exist and these solutions on the assumption that dissipation is the emergence of a forbidden velocity band are not small. related to the choice of the SakaiÐTatenoÐPedersen When the left-hand side of Eq. (3.5) has a “+,” two ± α nonlinearity but are attributed to the effect of the roots k = ikj(v) + i kj(v) of this equation, where waveguide on the JJ and also occur, for example, in the ()β() conventional model with sine nonlinearity (see Appen- πv k j v v dix 1). α ≡ ------Ӷ 1, (3.6) 4 ω2 j

3. FORCED MOTION OF A VORTEX 2 2 V s V sw In this section, we consider a forced motion of a vor- β()v ≡ β+ SS ------β , (3.7) w()2 2 2 w tex under the action of a dc transport current of density V sw Ð v j that flows through the JJ. As in the previous section, ± we consider a uniformly moving vortex when the accel- differ from the roots k = ikj (v) of the dissipationless α erating effect of the transport current is compensated by equation (2.14) by a small dissipative term i kj(v). The dissipative losses. third root of Eq. (3.5) with a “+” on the left-hand side, β ≡ πσ ⑀ β ≡ πσ ⑀ α Using 4 / and w 4 w/ w instead of (2.1) which is associated with dissipation, is k = i(1 Ð w)kw, and (2.2) to describe the losses in the JJ and the where waveguide, respectively, we obtain (cf. [11]) v β ≡ w 2 kw ------2 2, (3.8) 2 ∂ ϕ()zt, 2 j ∂ϕ()zt, ω F[]ϕ()zt, +++------ω ---- β------V sw Ð v j 2 j ∂ ∂t jc t (3.1) π 2 2 2 2 ∂2ϕ (), V s V sw kw 2∂ ϕ()zt, 2 zt α ≡ ---SS ------Ӷ 1. (3.9) ------w - w w 2 2 2 = V s 2 + SVs 2 , 2 V Ð v ω ∂z ∂z sw j When the left-hand side of Eq. (3.5) has a “–,” in the ∂2ϕ (), ∂ϕ (), w zt w zt low-loss limit, this equation has the following solu------+ β ------2 w ∂ ± v α v α ∂t t tions: k = kj( ) + i kj( ), i(1 + w)kw. (3.2) Conditions (3.6) and (3.9) for the smallness of α and ∂2ϕ (), ∂2ϕ(), α 2 w zt 2 zt w agree with our assumption that dissipation is small. = V sw------+ SwV sw------. ∂z2 ∂z2 Let us write a solution to the system of Eqs. (3.3) and (3.4), which describes a Josephson 2π kink and the In the case of a steady motion at a constant velocity v, associated field in the waveguide in the first allowed ψ ζ from (3.1) and (3.2), we obtain velocity band (2.20), for kw > 0. Assume that ( ) takes π π ζ ζ ζ ζ the values /2 and 3 /2 at points = Ð 0 and = 0, ω2 []ψζ() ()ψ2 2 ()ζ ω2 j β ψ ()ζ respectively. j F Ð V s Ð v '' + j ---- Ð v ' jc (3.3) We will seek a solution to systems (3.3), (3.4) as a 2 ()ζ superposition of constants and terms of the form = SVs f ' , exp(ikζ), where k is a solution to Eq. (3.5). Then, requiring that the solution do not contain terms that ()2 2 ()ζ β ()ζ 2 ψ ()ζ Ð V sw Ð v f ' Ð wvf = SwV sw '' , (3.4) grow exponentially as |ζ| ∞, we obtain ζ ≡ ψ ()ζ π j π where f( ) w' . ψζ() j = Ð------+ ---1 + ---- When F[ψ] is used in the form (2.12) corresponding 2 jc 2 jc (3.10) to the SakaiÐTatenoÐPedersen model, the wave vectors × exp[]()1 Ð α k ()ζζv ()+ , of small perturbations of ψ and f are determined from j 0 the equation ψ ()ζ []()α ()ζζ() w' = Aw exp 1 Ð k j v + 0 (3.11)

2 2 2 2 2 ± ω ()v βv in the tail of the vortex (ζ < Ðζ ), where Ðπj/2j < π--- j + V s Ð k Ð i k 0 c (3.5) ψ < π/2; × []()2 2 β 2 2 3 V sw Ð v kiÐ wv = SSwV s V swk , π j ψζ() = π ++------{}Bksin[]()ζv + Ckcos[]()ζv 2 j j j in which the “+” and “–” have the same meaning as in c (3.12) ×α[]()ζ []()α ζ Eqs. (2.13) and (2.14). In contrast to the dissipationless exp Ð k j v + Dexp Ð 1 + w kw ,

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ψ' ()ζ = {}B sin[]k ()ζv + C cos[]k ()ζv the associated ﬁeld in the waveguide in the allowed w w j w j (3.13) velocity band (2.20) to the lowest order in small dissi- ×α[]()ζ []()α ζ exp Ð k j v + Dw exp Ð 1 + w kw pation: in the middle part of the vortex (Ðζ < ζ < ζ ), where π π 0 0 ψζ()≈ j j −π/2 < ψ < 3π/2; and Ð------+ ---1 + ---- 2 jc 2 jc π j j (3.18) ψζ()= 2π Ð1------Ð Ð ---- π  × ()α ()ζv 2 jc jc exp 1 Ð k j + ------()- , 4k j v j × Eexp[]Ð()1 + α k ()ζζv ()Ð + 1 Ð ---- (3.14) j 0 j c π V 2 πα k ψ' ()ζ ≈ Ð---S ------sw k ()v 1 + ------Ð ------w π w w 2 2 j () × []()α ()ζζ 2 V Ð v 4 k j v E Ð --- exp Ð 1 Ð w kw Ð 0 , sw 2 (3.19) π × ()α ()ζ ψ ()ζ []()α ()ζζ() exp1 Ð k j v + ------w' = Ew exp Ð 1 + k j v Ð 0 4k ()v (3.15) j []()α ()ζζ + Fw exp Ð 1 Ð w kw Ð 0 in the tail of the vortex (ζ < Ðπ/4k (v)), where Ðπj/2j < ζ ζ π ψ π j c in the head of the vortex ( > 0), where 3 /2 < < 2 Ð ψ < π/2; π ( j/2jc). When writing Eqs. (3.10) and (3.14), we took into π j ψ ζ π ψ ζ π ψζ()≈ π+ ------account that (Ð 0 Ð 0) = /2 and ( 0 + 0) = 3 /2. 2 jc From the continuity condition for the functions π {}α[]ζ α []ζ []ζ ψ(ζ), ψ'(ζ), and ψ' ()ζ at ζ = ±ζ and from the relations + ------sin k j()v Ð cos k j()v exp Ð k j()v w 0 2 between the quantities B, C, D, E, Aw, Bw, Cw, Dw, Ew, (3.20) 2 2 2 and Fw that arise under the substitution of (3.10)Ð(3.15) π SS V V + ------w s sw k 2 exp[]Ð()1 + α k ζ , into any of the equations of the system (3.3), (3.4), we 2 ω 2()2 2 w w w obtain a system of 12 equations from which we can j V sw Ð v ζ determine the size of the middle region 2 0, the relation between the current density j and the velocity of a vortex π V 2 ψ' ()ζ ≈ sw structure, and the coefﬁcients of the exponential and trig- w ------Sw------2 2 onometric functions in (3.10)Ð(3.15). From this system 2 V sw Ð v of equations, in the linear approximation in dissipation, × {}[]()ζ () []()ζ kw sin k j v Ð k j v cos k j v we obtain (3.21) V 2 π ()β() ×αexp[]Ð k ()ζv + πS ------sw -k j()v π v k j v v j w 2 2 w ------= ---1 + ------2 - V sw Ð v jc 4 4 ω j × []()α ζ exp Ð 1 + w kw 2 2 π π v V Ð v = ------1 + ------sw - π ζ ω 2 2 2 2 (3.16) in the middle part of the vortex (Ð /4kj(v) < < 4 j ()() 22 v 1 Ð v v 2 Ð v π π ψ π /4kj(v)), where Ð /2 < < 3 /2; and V 2V 2 ×β+,SS ------s sw -β π j πj w 2 w ψζ()≈ 2π Ð ------Ð --- 1 Ð ---- ()2 2  V sw Ð v 2 jc 2 jc π π ζ ()β2,,β 2 ββ × ()α ()ζv 0 = ------()- + O w w . (3.17) exp Ð 1 + k j Ð ------()- 4k j v 4k j v

α α 2 2 (3.22) Again, under the assumption that and w are small, V V we obtain the expressions, given in Appendix 2, for the Ð π2SS ------s sw k2 wω 2()2 2 w ψ ζ ψ ()ζ j V sw Ð v coefﬁcients that deﬁne the functions ( ) and w' . Substituting (3.17) and the corresponding expres- π sions from Appendix 2 into (3.10)Ð(3.15), we obtain × expÐ()1 Ð α k ζ Ð ------, w w ()v the following expressions for a Josephson 2π kink and 4k j

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π V 2 πα k In addition, when considering the Cherenkov losses, we ψ' ()ζ ≈ Ð---S ------sw k ()v 1 Ð ------+ ------w will neglect dissipation in the JJ and the waveguide. w 2 w 2 2 j 4 k ()v V sw Ð v j This approach is justiﬁed for low dissipation and small Cherenkov losses, when their effect on the motion of a π vortex is additive. Under these conditions, the forced × expÐ()1 + α k ()ζv Ð ------j 4k ()v (3.23) motion of a slow vortex is described by Eqs. (3.1) and j β β (3.2) with = w = 0 when a term of the form 2  V sw π ∂4ϕ(), + 2 π S ------k exp Ð()1 Ð α k ζ Ð ------1 2 2 2 zt w 2 2 w w w () ---ω λ λ ------, (4.2) 4k j v j j J 4 V sw Ð v 2 ∂z

ζ π π 2 3 3 in the head of the vortex ( > /4kj(v)), where 3 /2 < where λ ≡ v /ω and λ ≡ (λ + λ )(λ + λ + 2d)Ð1, is ψ π π j s j J 1 2 1 2 < 2 Ð ( j/2jc). added to the right-hand side of (3.1) to take into account In the second allowed velocity band (2.21) of a vor- the weak spatial dispersion of the JJ. In writing the tex, when kw < 0, a solution to the system of Eqs. (3.3), small term (4.2), we neglect the interaction between the (3.4) is sought in a form similar to (3.10)Ð(3.15) with JJ and the waveguide. Taking into account the above the difference that the contributions (associated with the changes in Eqs. (3.1) and (3.2), we use the following ± α dissipation) of the third roots i(1 w)kw of Eqs. (3.5) equation to describe the forced motion of a slow vortex: arise in the middle part and the tail of a vortex rather 2 Ð2 1 2 2 IV j than in the middle part and the head. This solution has a []ψζ() ()ψv ()ζ λ λ ψ ()ζ F = π---k j '' + --- j J Ð ----, (4.3) form similar to (3.18)Ð(3.23) (see Appendix 3). Here, the 2 jc relation between the transport current and the vortex β β 2 2 2 velocity in a linear approximation in and w is given where k (v) ≈ (2/π)(ω v Ð v2)Ð1. Equation (4.3) for- by (3.16) as before. j j 1 mally differs from the equation considered in [7] in that To conclude this section, we note that, to obtain for- 2 λ2 λ v 2 v2 ωÐ2 mula (3.16), we can apply a method that involves the the coefficients are replaced by J and (s Ð ) j approximate solution of Eqs. (3.3) and (3.4) (see is replaced by (2/π)(kÐ2 v). This allows one to make use Appendix 4). This method allows one to write the func- j v of the mathematical result of [7]. For instance, follow- tion j( ), which differs from (3.16) only by a numerical ing [7], for the velocities of a slow vortex that satisfy factor, in the model with sine nonlinearity as well (see the condition Appendix 4): λ λ 2 2 2 2 2 2 2 ()β() v ӷ v Ð v ӷ ------J v ≈ ------J v , (4.4) j()v 42v k j v v 1 1 λ s λ 1 ------= ------. (3.24) π j π j j π3/2 ω2 c j we obtain the following relation between the current and the vortex velocity: 4. THE EFFECT OF CHERENKOV LOSSES ON THE FORCED MOTION OF A VORTEX j ε4π ε π 2 ---- = ---- sin-----ε Ð --- cos-----ε , (4.5) A remarkable property of the SakaiÐTatenoÐPeder- jc 8 2 2 2 sen approach is the fact that it makes it possible to consider the effect of Cherenkov losses on the motion of a where ε ≡ π λ λ k2 (v) Ӷ 1. Formula (4.5) gives an vortex. The results of this analysis are presented in this j J j Ӷ oscillatory relation between the current and the vortex section as applied to the conditions under which Vs Ӷ velocity, which is established when there is balance Vsw and there exists a fast vortex. When Vs Vsw, there between the effects of current and Cherenkov losses are two velocity ranges in each of which vortices exist. due to the irradiation of waves by the vortex. In this First, consider a range of small velocities when one case, the minima of the function (4.5) correspond to a can speak of the motion of a relatively slow vortex with discrete set of eigenvelocities vn of a free motion of a ≈ 1/2 a velocity of v < v1 (1 Ð SSw) Vs. To describe the Josephson vortex; this set is associated with the internal Cherenkov losses of a slow vortex, it is sufficient to take structure, of the vortex, created by extraordinary Swi- into account the spatial dispersion of the JJ. Let us hart waves Cherenkov-trapped by the vortex (see [7]). restrict the analysis to the limit of relatively weak spa- In the case of a slow vortex, the discrete set of velocities tial dispersion of the JJ, which is possible when the vor- vn is given by tex velocity satisfies the condition 2 λ v ≈ 1 Ð ------J n v , (4.6) ()2 Ӷ n λ 1 1 Ð v /v 1 1. (4.1) π j

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 PROPERTIES OF A FAST JOSEPHSON VORTEX 601 where, according to inequalities (4.4), positive integers the current and the velocity of a fast vortex is also given Ӷ Ӷ λ λ n lie in the interval 1 n j/ J. by (4.5), which now contains a new small parameter λ Let us consider the range of velocities of a fast vor- 2 2 v επλ≡ λ ()v sw w ≈ 2 j effk j = ------λ- tex when Vsw < v < v2 Vsw + (SSw/2)(V s /Vsw). The πV sw j properties of a fast vortex are primarily determined by (4.11) v the waveguide. This means that, when considering the × sV s Ӷ SSw------2 -2 1. Cherenkov losses of a fast vortex, it sufﬁces to take into v 2 Ð v account the spatial dispersion of the waveguide. As in the case of a slow vortex, we will neglect the small dis- ≈ ≈ Setting vsw Vsw and vs Vs, we can see that condi- sipation in the JJ and the waveguide. Then, to describe tions (4.8) and (4.11) are consistent if the Cherenkov losses of a fast vortex, we have β β Eqs. (3.1) and (3.2) with = w = 0 when a term of the λ 2 ӷ 4w form SSw π--------λ- . (4.12) j ∂4ϕ (), 1 2 2 zt In the case of a fast vortex, the minima of the oscillation ---v λ ------w -, (4.7) 2 sw w ∂ 4 function (4.5) are attained at a velocity close to the z eigenvelocities of freely moving fast vortices, which are approximately given by λ2 ≡ λ 3 λ 3 λ λ Ð1 where w (2 + 3 )( 2 + 3 + 2dw) is added to the λ 2 right-hand side of Eq. (3.2). The small term (4.7) does 2 w V s v ≈ 1 Ð ------SS ------n v . (4.13) not take into account weak interaction between the JJ n w λ 2 2 π j v and the waveguide. When the velocity of a fast vortex 2 satisﬁes the condition Here, according to inequalities (4.8) and (4.11), posi- Ӷ Ӷ λ λ 2 2 Ӷ 2 tive integers n lie in the interval 1 n SSw (j/ w). v 2 Ð v SSwV s , (4.8)

ψ modiﬁed equation (3.2) enables one to express w in 5. DISCUSSION terms of ψ and write the following equation for the Note that Eq. (3.16) differs from the function j(v) phase difference of the fast vortex: for an isolated JJ [7],

2 Ð2 1 2 2 IV j ()v π π v β F[]ψζ() = ---k ()ψv ''()ζ + ---λ λ ψ ()ζ Ð ----, (4.9) j  π j j eff ------= ------1 + ------ω , 2 jc jc 22 4 2 2 j v s Ð v λ where the effective length eff depends on the vortex by the velocity dependence of the term associated with velocity: losses in the JJ and by the presence of a term determined by the losses in the waveguide. We emphasize that the terms in (3.16) that contain β and β , which SS V w λ ≡ λ ------w v V ------s - characterize the dissipation in the JJ and the waveguide, eff w 2 2 sw swλ ω v Ð V sw j j respectively, depend differently on velocity. (4.10) For the most interesting case of a fast Josephson vor- V 2 ≈ SS ------sw -λ . tex, whose velocity is given by (2.26), the function (3.16) w 2 2 w can approximately be represented as v Ð V sw j()v π π 1 2 2 ≈ Since v ≈ V + SS V 2 , inequality (4.8) implies that ------1 + --- ω----- 2 sw w s jc 22 4 j 2 2 ≈ 2 v Ð v sw SSw V s . For such velocities of a vortex, the 2 λ 1 V s v Ð V sw effective length eff remains ﬁnite. However, owing to ×β β + ---SSw------2 w ------, 2 2 4 ()v Ð v ӷ λ ≈ λ v Ð V sw 2 the condition Vsw Vs, eff wV sw /V s SSw may sub- λ stantially exceed J; this makes ir possible to neglect the dispersion of the JJ when considering the Cheren- 2 1 V s kov losses of a fast vortex. A formal similarity between V <

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 602 MALISHEVSKIŒ et al. Under the condition Since a vortex moves uniformly, its energy is conserved. Therefore, taking into account (3.3) and (3.4), 2 calculating the derivative dH/dt, and equating it to zero, β < V sw β ------2 w , we obtain the following relation: SSwV s φ () φ 2 2 the dissipation is mainly determined by losses in the j v v 0 v ------0 = ------waveguide. In this case, we have c 32π3c2 ∞ (5.1) () π π β V 2 ⑀ ⑀ j v ≈ w s 2 w 2 ------1 + ------SSw------. ×ζd ---βψ[]'()ζ + ------β []ψ' ()ζ . ω 3 ∫ w w jc 82 4 j ()() d dw v 2 Ð v v Ð V sw Ð∞

≈ 2 2 For v = (Vsw + 3v2)/4 [1 + (3/8)SSw(/V s V sw )]Vsw, the The left-hand side of Eq. (5.1) represents the power of function on the right-hand side attains its minimum, the Lorenz force associated with the effect of the trans- equal to port current on the vortex. The right-hand side of (5.1) contains the power of friction forces associated with the 2 β bulk ohmic losses in the JJ and in the waveguide. Sub- jmin 42ππ 1 V sw w ------≈ ------1 + ------. stituting the terms of ψ(ζ) and ψ' ()ζ that contain j 4 SS 2 ω w c 33 w V s j ± α ζ exp[(1 w)kw ] into (5.1), we obtain Thus, the motion of a fast Josephson vortex may be 2 2 4 realized when the transport current is greater than jmin. φ ⑀ S V ------0 -----w------w sw v 3β 2 . This means that the forced motion of a fast vortex, π 2 d ()2 2 w which occurs due to the coupling between the JJ and the 16 c w V sw Ð v waveguide, does not require large values of current. The expression for ψ' ()ζ and the terms in ψ(ζ) that This expression shows that the energy losses of a vortex w per unit time that are associated with the excitation of do not contain the small quantity k in the arguments of w Ð1 ∝ βÐ1 the exponential functions are expressed to the first large-scale perturbations (whose scale ~kw w is β β α ζ 2 order in and w. The coefficients of exp[(1 + w)kw ] β determined by dissipation) are proportional to w . In ψ ζ 2 in the expressions for ( ) are proportional to kw . Tak- other words, in a linear approximation in dissipation, ing into account these small contributions to ψ(ζ) does dissipative contributions, to the phase difference, local- overlap the accuracy limit because precisely these con- ized on large scales are inessential for the dependence tributions make it possible to write correct expressions j(v). The possibility of drawing such a conclusion is for ψ' ()ζ . In (3.22), the last small dissipative term one of advantages of our analysis, which is due to the w use of the model description. (∝ k2 ) is localized within the scale ~kÐ1 , which is w w If we ignore the effect of dissipation on the coordi- Ð1 much greater than the scale ~k j (v), in which the third nate dependence of the phase differences, we can term is localized. We emphasize that, despite the fact obtain formula (3.16) by substituting dissipationless that the terms in the phase differences that correspond expressions (2.9) and (2.17)Ð(2.19) into (5.1). The to the small third roots of Eqs. (3.5) are localized in a dependence j(v) in a linear approximation in the case large spatial region, they give small corrections to the of sine nonlinearity can be obtained similarly: substi- law (3.16). To illustrate this fact, we write the energy of tuting (2.9) and (A1.2) into (5.1), which is independent the system (per unit length of axis y) as of the form of the nonlinearity F[ψ], we obtain expression (3.24). ∞ ψζ() 2 φ ⑀ω2 Finally, we established the influence of the Cheren- H = ------0 dζ------j dψF []ψ 3 2 ∫ ∫ STP kov losses on the transport current in the SakaiÐTatenoÐ 32π c  d Ð∞ 0 Pedersen model, which manifests itself both for vortex velocities less than the Swihart velocity of the JJ and for ⑀()v 2 2 ⑀ ()v 2 2 velocities of a fast vortex. In either case, the function + V s []ψ ()ζ 2 w + V sw []ψ ()ζ 2 + ------' + ------w' j(v) exhibits oscillatory behavior, which was earlier 2d 2dw established for a simple case of an isolated JJ. The minima of j(v) correspond to the velocities of a free motion ⑀ 2 ⑀ 2  of both slow (cf. [7]) and fast Josephson vortices. Under 1V s wV sw ψζ()ψ()ζ + ---S------+ Sw------w' . the conditions of small dissipation and low Cherenkov 2 d dw  losses, the oscillating part of j(v) is added to the mono-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 PROPERTIES OF A FAST JOSEPHSON VORTEX 603 tonic part of j(v) associated with dissipation in the JJ APPENDIX 2 and the waveguide. The coefﬁcients in the expressions (3.11)Ð(3.15) for the phase differences are given by 6. CONCLUSIONS π πα k A ≈ Ð---S ()v 1 + ------Ð ------w k ()v , w w ()v j Thus, we have formulated a theory for the effect of 2 4 k j a magnetically coupled waveguide on the vortex in a π Josephson junction. We have established the properties ()β2,,β2 ββ B = ------+ O w w , in a fast Josephson vortex with a velocity much greater 2 than the Swihart velocity of the JJ. π π2 SS ()v V 2 C ≈ Ð------α, D ≈ ------w s-k 2 , 2 ω 2 w ACKNOWLEDGMENTS 2 j This work was supported in part by a grant from the π ≈ () President of the Russian Federation for supporting young Bw ------Sw v kw, Russian scientists (project no. MK-1809-2003.02), a 2 grant for supporting leading scientiﬁc schools of the π Russian Federation (project no. NSh-1385-2003.2), ()() ()β2,,β 2 ββ Cw = Ð ------Sw v k j v + O w w , and by the Ministry of Science and Industry of the Rus- 2 sian Federation (project no. 40.012.11.1357). π SS ()v V 2 D ≈ πS ()v k , E ≈ --- Ð π2------w -s k 2 , w w w 2 ω2 w APPENDIX 1 j The analysis presented in Section 2 can be extended π πα k E ≈ Ð---S ()v k ()v 1 Ð ------+,------w to the case of a standard sine nonlinearity when the den- w 2 w j 4 k ()v sity of the Josephson current depends on ϕ by the law j ϕ jcsin (z, t); then, instead of Eq. (2.16), we have ≈ π () Fw 2 Sw v kw,

2 2 2 Ð2 ≡ 2 ψζ() ()ψv ()ζ where Sw(v) SwV sw /(V sw Ð v ). sin = π---k j '' . (A.1.1)

APPENDIX 3 This equation differs from the well-known sine-Gordon For low dissipation, a solution to Eqs. (3.3) and (3.4) equation in that the coefﬁcient of the second derivative corresponding to the motion of an elementary vortex in depends on velocity. A solution to (A1.1) correspond- a JJ with velocities V < v < v has the form ing to a 2π kink is given by ws 2 π π ψζ()≈ j j Ð------+ ---1 + ---- π 2 jc 2 jc ψζ() ()ζ = 4arctanexp ---k j v . (A.1.2) 2 π × exp()1 Ð α k ()ζv + ------j 4k ()v For solution (A1.2) to describe a vortex, the quantity j kj(v) should be real, or, which is equivalent, the vortex velocity should take values only within two allowed 2 2 2 V V 2 bands (2.20) and (2.21). The latter means that, in the + π SS ------s sw k wω 2()2 2 w model with sine nonlinearity, the range of allowed j V sw Ð v velocities of a 2π kink in the JJ coupled to a waveguide also splits into two regions separated by a forbidden π × ()α ζ band. Thus, the emergence of the forbidden velocity exp Ð 1 Ð w kw + ------()- , band is not associated with the SakaiÐTatenoÐPedersen 4k j v nonlinearity that we chose in the main body of the text but is determined by the coupling between an eigen 2 π V sw πα k electromagnetic mode of the waveguide and the Swi- ψ' ()ζ ≈ Ð---S ------k ()v 1 + ------Ð ------w w 2 w 2 2 j 4 k ()v hart wave of the JJ. V sw Ð v j

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 604 MALISHEVSKIŒ et al. ψ π When F[ ] corresponds to the nonlinearity in the × exp()1 Ð α k ()ζv + ------SakaiÐTatenoÐPedersen model, Eq. (A4.2) differs from j 4k ()v j that considered in [7] only in that the coefﬁcients kj(v) and β(v) depend on velocity. The similarity between 2 Eq. (A4.2) and Eq. (2.1) from [7] allows one to use the V π Ð2πS ------sw -k expÐ()1 Ð α k ζ + ------results of [7] and write the dependence (3.16) j(v) to a w 2 2 w w w 4k ()v linear approximation in dissipation. V sw Ð v j If F[ψ] = sinψ, then the formal similarity between ζ π in the tail of the vortex ( < Ð /4kj(v)) and Eq. (A4.2) and Eq. (2.5) from [15] allows one to use the results of [15] and write out the dependence (3.24). π π ψζ()≈ π j j 2 Ð ------Ð ---1 Ð ---- 2 jc 2 jc REFERENCES 1. S. Sakai and H. Tateno, Jpn. J. Appl. Phys. 22, 1374 π × ()α ()ζv (1983). exp Ð 1 + k j Ð ------()- , 4k j v 2. S. Sakai and N. F. Pedersen, Phys. Rev. B 34, 3506 (1986).

2 3. S. Sakai, Phys. Rev. B 36, 812 (1987). π V πα k ψ' ()ζ ≈ Ð---S ------sw -k ()v 1 Ð ------+ ------w 4. A. F. Volkov, Physica C (Amsterdam) 183, 177 (1991). w 2 w 2 2 j 4 k ()v V sw Ð v j 5. A. F. Volkov, Physica C (Amsterdam) 192, 306 (1992). 6. V. M. Eleonskiœ and N. E. Kulagin, Zh. Éksp. Teor. Fiz. π 119, 971 (2001) [JETP 92, 844 (2001)]. × ()α ()ζv exp Ð 1 + k j Ð ------()- 4k j v 7. V. P. Silin and A. V. Studenov, Zh. Éksp. Teor. Fiz. 117, 1230 (2000) [JETP 90, 1071 (2000)]. in the head of the vortex (ζ > π/4k (v)); in the middle 8. A. S. Malishevskii, V. P. Silin, and S. A. Uryupin, Phys. j Lett. A 253, 333 (1999). part of the vortex, a solution in the second allowed velocity band is obtained from (3.20) and (3.21) by 9. A. S. Malishevskiœ, V. P. Silin, and S. A. Uryupin, Zh. changing the signs of the last terms. Éksp. Teor. Fiz. 117, 771 (2000) [JETP 90, 671 (2000)]. 10. A. S. Malishevskiœ, V. P. Silin, and A. V. Studenov, Kratk. Soobshch. Fiz., No. 12, 3 (2000). APPENDIX 4 11. V. V. Kurin and A. V. Yulin, Phys. Rev. B 55, 11659 If dissipation is sufﬁciently small, one can obtain the (1997). ζ following approximate expression for f( ) from (3.4): 12. A. S. Malishevskii, V. P. Silin, and S. A. Uryupin, Phys. Lett. A 306, 153 (2002). 2 ()ζ ≈ V sw []ψ ()ζ ψζ() 13. A. A. Abrikosov, Fundamentals of the Theory of Metals f ÐSw------' Ð kw . (A.4.1) V 2 Ð v 2 (Nauka, Moscow, 1987; North-Holland, Amsterdam, sw 1988). Substituting (A4.1) into (3.3), by analogy with the dis- 14. V. P. Silin and S. A. Uryupin, Zh. Éksp. Teor. Fiz. 108, sipationless case, we obtain the following equation 2163 (1995) [JETP 81, 1179 (1995)]. for ψ(ζ): 15. D. W. McLaughlin and A. C. Scott, Phys. Rev. B 18, 1652 (1978). 2 Ð2 j vβ()v F[]ψ Ð ---k ()ψv ''()ζ = ---- + ------ψ'()ζ . (A.4.2) π j j ω 2 c j Translated by I. Nikitin

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 605Ð611. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 684Ð692. Original Russian Text Copyright © 2004 by Burdov, Solenov. SOLIDS Electronic Properties

Dynamic Control of Electronic States in a Double Quantum Dot under Weak Dissipation Conditions V. A. Burdov* and D. S. Solenov Lobachevskiœ State University, Nizhni Novgorod, 603950 Russia *e-mail: [email protected] Received August 13, 2003

Abstract—The problem of the time evolution of an electron wave packet in a symmetric double quantum dot under the action of a strong alternating electric ﬁeld and a slowly varying bias voltage is solved theoretically under the conditions when the electron subsystem can transfer its energy to a single resonator mode. It is shown that the possibility of energy exchange between the electron subsystem and the resonator does not hamper the formation of stable electronic states localized in the left or right quantum dot (i.e., polarized states possessing a positive or negative dipole moment). An adiabatic change in the bias voltage may alter the direction of the dipole moment of the given state (which corresponds to an electron transition from one quantum dot to the other). © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION It should be emphasized that the results obtained in the above-mentioned publications corresponded to a Considerable advances in nanotechnologies neces- completely coherent mode of evolution of a wave sitate the development of new devices operating in the packet. Dissipative processes associated with the inter- single-electron mode. In such devices, it is essential to action of the electron subsystem with radiation fields control the wave function of an individual electron were completely disregarded. This study aims at theo- using, for example, alternating external electromag- retical analysis of the effect of weak dissipation on the netic fields. The control of charge or spin dynamics of possibility of a dynamic control of electronic state in a individual electrons by external alternating fields can double quantum dot. For this purpose, we will consider be referred to as the dynamic control of electronic a model in which an electron located in a double quan- states. tum dot interacts with the resonator and can emit energy into one of the resonator modes (in actual prac- Suitable objects for investigations in this field tice, this can correspond, for example, to the phonon include structures consisting of several tunnel-coupled mode of an impurity center in a quantum dot). quantum dots. In particular, the possibility of creating electronic states localized within a quantum dot and subsequent controllable transfer of such a localized 2. BASIC EQUATIONS OF THE PROBLEM state to a neighboring quantum dot or well was con- Let us consider a symmetric double quantum dot in sidered earlier as applied to double quantum dots [1] a slowly and monotonically varying electric field E(t) and wells [2Ð4] as well as lattices of quantum wells or and in an alternating electric field Fcosωt with a con- dots [5]. stant amplitude. We assume that, in zero field, the split- Generally speaking, the studies in this field began with ting energy ប∆ for the ground level of size quantization the works on the dynamic localization effect in lattices of in the double quantum dot (splitting is due to weak tun- quantum wells and in double quantum wells [6−9]. It was nel coupling between the dots) is much lower than the χ shown in these publications that in order to “lock” the size quantization energy. The wave functions 0, 1(r) of electron density in a quantum well, a strong alternating the stationary states of two lower levels with energies field with a definite relation between the amplitude and ±ប∆/2 are symmetric and antisymmetric, respectively, frequency is required as a rule. For this reason, a strong to the sign reversal of coordinate z (we assume that the alternating field with a slowly varying amplitude [2, 3] centers of the quantum dots lie on the z axis at a dis- was proposed in [1Ð5] for controlling an electron wave tance of ±L from the origin). packet for its transfer from one site to a neighboring We assume that an electron in the quantum dot can site; alternatively, in addition to a strong harmonic field exchange energy with the resonator. For simplicity, we with a constant amplitude, it was proposed that an adi- will consider energy radiation to a single resonator abatically varying bias voltage be applied [1, 4, 5]. mode, which will be simulated here (following [10]) by

606 BURDOV, SOLENOV

Φ ( ) χ (r) បΩ 1 q R β

χ (r) L Φ បΩ 0(q) E1 បω

E0 2eLE(t)

Fig. 1. Energy diagram of a two-level double quantum dot interacting with an oscillator of frequency Ω. a harmonic oscillator of mass m and frequency Ω (see of the system in the form of the following expansion: Fig. 1). The operator of the interaction is linear in displacement q of the oscillator from the equilibrium posi- Ψ()q,,r t tion and is proportional to the electron dipole moment D(z) in the double quantum dot (i.e., this operator is an  χ () EL () eLF ω antisymmetric function of coordinate z). = L r expi------∫Etdti+ ------sin t ប បω The total Hamiltonian of the system has the form ∞  ˆ ˆ () ˆ () ()() ω () × ()Φ() Ω1 H = He r ++Hv q ez E t + Fcos t +CqD z , (1) ∑ ln t n q expÐi n + --- t (3) 2 n = 0 where Hˆ e()r is the electron Hamiltonian, Hˆ v ()q is the Ω Hamiltonian of the harmonic oscillator of frequency , eL eLF χ () () ω C is the constant of interaction of the electron with the + R r exp Ði-----ប-∫EtdtiÐ ------បω- sin t oscillator field, and –e is the electron charge. We  assume that fields E(t) and F are quite strong in the ∞ sense that the characteristic value of the electron poten-  × ()Φ() Ω1 tial energy in the electric field is much higher than the ∑ rn t n q expÐi n + --- t . 2 splitting energy (eLF, eLE(t) ӷ ប∆). At the same time, n = 0 we assume that the values of eLF and eLE(t) are much smaller than the characteristic energy of size quantiza- Here, ln(t) and rn(t) are unknown expansion coeffi- Φ tion in the quantum dot, which is of the same order of cients, n(q) are the stationary eigenfunctions of the magnitude as the potential barrier height. The latter χ oscillator Hamiltonian, and L, R(r) are orthonormal condition enables us to disregard the probability of sys- functions connected with χ (r) via the relations tem excitation to the upper levels. As a result, the elec- 0, 1 tron quantum dynamics essentially affects only two χ () χ() χ () χ() lower levels with energies ±ប∆/2. These assumptions χ () 0 r Ð 1 r χ () 0 r + 1 r L r ==------, R r ------. enable us to use the two-level approximation for the 2 2 electron Hamiltonian. χ Ψ The expansion in basis L, R(r) is more convenient for In order to determine the wave function (q, r, t) of χ the system, we must solve the Schrödinger equation us than the expansion in basis 0, 1(r) of stationary χ states since functions L, R(r) are almost completely ∂Ψ localized in the left and right quantum dots, respec- iប------= Hˆ Ψ (2) tively. Consequently, the squares of the moduli of coef- ∂t ficients ln and rn are the joint probabilities of an electron being in the left or right quantum dot and of the oscilla- with Hamiltonian (1). We will seek the wave function tor being in the nth state.

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The substitution of wave function (3) into WL Schrödinger equation (2) leads to equations of the form 0.8 ∆  dln 2eL () 2eLF ω i------= Ð--- rn expÐi------∫EtdtiÐ ------sin t dt 2 ប បω 0.4

βω( {}Ω {}Ω ) 0 Ð nln Ð 1 exp i t + n + 1ln + 1 exp Ði t , –5000 0 5000 10000 15000 τ dr ∆ 2eL 2eLF (4) n () ω Fig. 2. Probability of population of the left quantum dot for i------= Ð--- ln exp i------ប ∫Etdti+ ------បω sin t dt 2 β = 0, λ = 7.7, µ = 10Ð4, and ∆/ω = 0.2.

βω( {}Ω {}Ω ) + nrn Ð 1 exp i t + n + 1rn + 1 exp Ði t , ing form, where the resonance terms with different values of s are explicitly separated: where the dimensionless coupling parameter ∞ µτ2 dln i δ {}()τ ប 〈|χ Dz()χ|〉 i------= Ðrn expÐ------∑ s exp Ðis+ 1 β = C ------R R- dτ 2 2mΩ បω s = Ð∞ Ð β( nl exp{}i()τ1 + η is introduced. In all subsequent calculations, we will n Ð 1 assume that β Ӷ 1, which indicates a weak coupling of {}()τη ) the electron subsystem with the resonator. + n + 1ln + 1 exp Ði 1 + , Let us suppose that field E(t) varies with time ∞ µτ2 (7) according to the law drn i δ {}()τ i------= Ðln exp------∑ s exp is+ 1 dτ 2 () ()µω s = Ð∞ Et = E0 1 + t , (5) β( {}()τη where the dimensionless parameter µ characterizes the + nrn Ð 1 exp i 1 + rate of variation of field E(t) and is small. The E(t) {}()τη ) dependence in form (5) is naturally not the single pos- + n + 1rn + 1 exp Ði 1 + . sible dependence. We choose this dependence as the τ ω simplest from an infinite set of various monotonic Here, we introduced the dimensionless time = t; parameter η = Ω/ω Ð 1 characterizes the closeness of dependences. δ the oscillator frequency to the resonance value, s = Note that in the case when an electron experiences λ ∆ ω Ӷ λ បω the action of a constant field alone (i.e., for µ = F = 0), Js( ) /2 1, and = 2eLF/ . the two-well potential becomes asymmetric, the energy In the subsequent numerical calculations (whose levels in the double quantum dot diverge by results are presented in Figs. 2Ð6), we assume every- 2 λ µ Ð4 ∆ ω ប2∆2 + 4e2 L2E , and the wave functions of station- where that = 7.7, = 10 , and / = 0.2. Choosing 0 energy បΩ = 10Ð2 eV (which corresponds to the charac- ary states are localized in each potential well separately. teristic values of phonon frequencies) and distance L = In the limit of a strong field, the quantum transition 10 nm and taking into account the fact that frequency ω energy approaches a value of 2eLE0 and the wave func- is close to oscillator frequency Ω, we obtain a value of tions corresponding to each of these energy levels χ the alternating field amplitude F approximately equal become equal to L, R(r) as shown in Fig. 1. If we now to 40 kV/cm. The resonance value of field E in this ω 0 “switch on” an alternating field of frequency , the case is close to 5 kV/cm, which corresponds to bias quantum system can attain resonance when voltages in the structure equal to 0.1Ð0.01 V. The split- បω ប∆ 2eLE0 = s, (6) ting energy is found to be 2 meV; i.e., it is the smallest energy scale of the problem, which was presumed where s is an arbitrary integer. earlier. It should be noted that the chosen value of For µ ≠ 0, with an appropriate choice of the time ori- parameter µ corresponds to the increase in field E(t) gin, we can always make the constant component of from the first resonance value (s = 1) to the second one field E(t) exactly satisfy the resonance condition, e.g., (s = 2) over a time approximately equal to 10Ð9 s. As with s = 1. Assuming that instant t = 0 is chosen pre- regards the coupling parameter β, its value is deter- cisely in this way, we will consider, without loss of gen- mined by a specific type of interaction and may vary erality, the fact that E0 satisfies the resonance condition over a wide interval (from nearly indefinitely small val- with s = 1. Then, Eqs. (4) can be written in the follow- ues to several units).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 608 BURDOV, SOLENOV For µ = 0, the two-level electron system can be in Thus, for τ < 0, the separation between two lower exact resonance with the alternating electric field for an levels of the double quantum dot (the instantaneous infinitely long time. For µ ≠ 0, for each instant, we can value of this quantity is 2eLE(t)) is found to be smaller introduce the concept of instantaneous energy of level than energy បω of an alternating field quantum. The splitting, which is approximately equal to 2eLE(t) in a larger the value of |τ|, the farther the two-level system strong field. In this case, the quantum system is effec- from resonance and the higher the probability (10) of tively close to resonance (when the action of the exter- occupation of the left quantum dot. With increasing nal field is the strongest) for a certain finite time interval time, the separation between the energy levels of the determined by expression (5) and the relation between double quantum dot increases and, for τ = 0, this sepa- parameters µ and δ. After this time, the two-level sys- ration coincides with បω, which means that the two- tem deviates from resonance and the type of the action level system is in strict resonance (6). In this case, prob- τ of the alternating electric field on this system changes. ability WL is equal to 1/2. A further increase in leads We will now consider the solution of system (7). We to further divergence of the energy levels and to a grad- assume that, at the initial instant (here, this instant ual withdrawal of the system from resonance. This pro- τ τ cess is accompanied by a decrease in probability W to should be a certain time = Ð 0 exceeding all character- L istic time scales of the problem, but not large enough zero and virtually complete relocation of the electron for the field E(Ðτ ) to be negative), the oscillator is not density to the right quantum dot. The characteristic 0 transition time is determined by ratio δ/µ, which excited and the left quantum dot is populated, while the Ð10 right dot is absolutely free; i.e., we assume that amounts to a value on the order of 10 s in dimensional units. ()τ ()τ ≥ l0 Ð 0 ==1, ln Ð 0 0, n 1, Note that the exact solution to Eq. (9) represented in (8) r ()Ðτ = 0, n ≥ 0. Fig. 2 slightly differs from the approximate solution n 0 described by expression (10). This is due to the follow- Obviously, initial condition (8) corresponds to the min- ing two circumstances. First, the numerical calculations imum energy of the electron + oscillator system. were made for a wider range of time τ embracing two neighboring resonances with s = 1 and s = 2, while solution (10) is valid only in the vicinity of one resonance. τ 3. DYNAMICS OF POPULATION Second, as can be seen from Fig. 2, probability WL( ) OF QUANTUM STATES OF THE SYSTEM rapidly oscillates, although the amplitude of these In the simplest case, when there is no coupling oscillations is small. Obviously, expression (10) between the double quantum dot and the resonator describing a smooth decreasing function is the result of (β = 0), system (7) splits into independent pairs of averaging of the exact solution over these rapid oscilla- τ τ tions in the region of the first resonance. equations for amplitudes ln( ) and rn( ) at each level of the oscillator. To find the solution in the vicinity of res- The oscillations observed in the exact solution are in onance, it is sufficient to retain only one term with fact analogous to Rabi oscillations, which now occur in s = −1 in each sum over s in Eqs. (7). All these pairs of a strong alternating field. If field E(t) were strictly con- τ equations are completely identical and can be reduced to stant, the solution to Eq. (9) would have the form l0( ) = τ cosδτ. In accordance with this solution, the electron a single second-order equation, say, for coefficient ln( ), wave packet would oscillate between the two quantum d2l dl dots with constant frequency δ, which has the meaning ------n ++0iµτ------n δ2l = (9) 2 τ n of the Rabi frequency in a strong field (since the field is dτ d strong, frequency δ is not linear in field amplitude F). (we have omitted index Ð1 on δ). However, in view of variability of field E(t), the cosinu- soidal dependence is observed only for times |τ| Շ δ2/µ; An approximate solution to Eq. (9) was obtained for chosen values of δ and µ (see the caption to Fig. 2), earlier in [4] by using the WKB method; here, we give the value of this time is on the order of unity. For τ > only the final expression for the total probability of fill- δ2/µ, the second term that “quenches” the amplitude of ing for the left quantum dot: oscillations dominates in Eq. (9). Since the value of δ2/µ if found to be much smaller than the period 2π/δ of δ2 µ2τ2 µτ ()τ 4 + Ð Rabi oscillations, not a single oscillation with complete W L = ------. (10) δ2 µ2τ2 transfer of charge from the left to the right quantum dot 24 + is observed. τ While deriving this relation, we chose the value of 0 Let us now consider a more interesting situation equal to infinity since Eq. (9) was derived in the single- when the electron subsystem interacts with the resona- resonance approximation. Expression (10) describes tor field; i.e., β ≠ 0. The interaction with the resonator the transfer of the electron density from the left to the can obviously break the dynamic control in the system right quantum dot during an adiabatic transition of the since it becomes possible for the electron located, say, in two-level electron system through a resonance. the right quantum dot in a state disadvantageous from the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 DYNAMIC CONTROL OF ELECTRONIC STATES IN A DOUBLE QUANTUM DOT 609 energy point of view to transfer its excess energy to the 0.004 2 2 resonator. As a result, the resonator is excited and leaves |l | |r | its ground state, while the electronic state that initially 1 1 0.002 possessed a certain polarization can be destroyed. In order to find out whether it is possible to preserve the dynamic control of the electron distribution in a –5000 5000 10000 15000 double quantum dot under the conditions of energy τ exchange with the resonator, we must solve system (7) for β ≠ 0. In this case, system (7) obviously cannot be | |2 | |2 l1 r1 represented in a closed form and can be solved only 0.00007 numerically. At a certain step, it is necessary to truncate Eqs. (7), which corresponds to the replacement of the oscillator by an N-level system with an equidistant spectrum. –2000 –1000 0 1000 2000 Calculations show that the number of retained energy levels of the oscillator and the solutions to Fig. 3. Probability of excitation of the oscillator to the first Eqs. (7) themselves differ significantly for different level for η = 0.1, β = 0.01, λ = 7.7, µ = 10Ð4, and ∆/ω = 0.2. values of parameters η and β. Further, we consider two basically different cases, i.e., the nonresonance case, when the frequency of the oscillator significantly dif- to corrections on the order of 10Ð10 to the values of prob- fers from the frequency of the external alternating field abilities, which is beyond the required accuracy limits. (i.e., η ≠ 0), and the resonance case, when η = 0. We also calculated the total probability (11) of pop- The main problem is to evaluate the total (i.e., ulating the left quantum dot with an electron; in the summed over all states of the oscillator) probability of nonresonance case, this probability coincides with | τ |2 population of any of quantum dots (e.g., left) as a func- l0( ) to within fractions of percent (see Fig. 3). Calcu- tion of time, which is now defined as lations proved that the difference of the probabilities ∞ ∆W ()τ = W ()τβ, = 0.01 Ð W ()τβ, = 0 ()τ ()τ 2 L L L W L = ∑ ln . (11) Ð4 n = 0 remains small everywhere and amounts to ~10 . It can be proved analytically that this difference in the nonres- Further, we calculate and discuss the behavior of indi- onance case must be determined by the second power | τ |2 | τ |2 vidual probabilities ln( ) and rn( ) , which are essen- of coupling parameter β, which is confirmed by the tial for determining the sum in Eq. (11). results of calculations. Let us begin with the nonresonance case. Figure 3 Consequently, the dynamics of the electron density shows the probability of exciting the oscillator to the remains the same as in the absence of coupling with the first level for η = 0.1. It can be seen from the figure that resonator. Upon each passage of the two-level system | τ |2 | τ |2 through the resonance, the double quantum dot reverses it is as though probabilities l1( ) and r1( ) alternate on the τ axis: when one probability decreases, the other the direction of polarization since the electron density starts to increase, and vice versa. As regards the oscilla- performs transition from one quantum dot to the other. tor, it is excited most strongly when the two-level elec- Let us now consider the resonance situation, when tron system is in the vicinity of the second resonance η = 0. Figure 4 shows the joint probabilities of populat- for τ ~ 104. Outside this neighborhood, probabilities ing one of the quantum dots by an electron and of the | τ |2 | τ |2 oscillator being in one of its states for β = 0.01 and λ = ln( ) and rn( ) decrease by one or two orders of magnitude. However, the excitation probability does not 7.7. It can be seen from Fig. 4 that, after the transition | τ |2 exceed a few thousandths in the vicinity of the reso- from the first resonance s = 1, probabilities ln( ) nance as well. Calculations show that the probabilities decrease to values close to zero until the two-level sys- of excitation of the oscillator to higher levels have val- tem finds itself in the vicinity of the next resonance | τ |2 ues rapidly decreasing with increasing level number. with s = 2. Conversely, probabilities rn( ) increase in For example, the probabilities of excitation of the oscil- this range of τ from almost zero values to a few tenths lator to the second level turn out to be lower than the for several first energy levels of the oscillator. probabilities represented in Fig. 3 by 3Ð4 orders of After the passage through the second resonance, the magnitude. Consequently, the oscillator is almost probabilities of populating the right quantum dot always unexcited and, hence, hardly affects the electron decrease to zero for all levels of the oscillator, while distribution in quantum dots. | τ |2 probabilities ln( ) increase. Upon the passage through Calculating the probabilities in the nonresonance each next resonance, an increase or a decrease in the case (η = 0.1), we replaced the oscillator by a four-level probabilities obviously changes to a decrease or an system; i.e., 0 ≤ n ≤ 3. The addition of the fifth level leads increase, respectively. The following tendency can be

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1.0 | |2 2 r0 0.15 |l | 0.8 0 | |2 0.6 0.10 l2 0.4 0.05 | |2 0.2 r2 –50000 5000 10000 15000 –50000 5000 10000 15000

2 0.05 0.3 |l | 2 | |2 1 |l | 0.2 r1 0.03 3 0.1 | |2 0.01 r3 –50000 5000 10000 15000 –50000 5000 10000 15000 τ τ

Fig. 4. Dynamics of excitation of the oscillator in the resonance case (η = 0) for β = 0.01, λ = 7.7, µ = 10Ð4, and ∆/ω = 0.2.

1.0 | |2 | |2 –5 l0 r0 4 × 10 0.8 2 –5 | | 0.6 3 × 10 l2 0.4 2 × 10–5 0.2 × –5 | |2 1 10 r2 –5000 0 5000 10000 15000 –5000 0 5000 10000 15000

0.008 2 | | –7 l1 1.2 × 10 2 0.006 |l | | |2 × –7 3 0.004 r1 0.8 10 0.002 × –7 | |2 0.4 10 r3 –5000 0 5000 10000 15000 –5000 0 5000 10000 15000 τ τ Fig. 5. Dynamics of excitation of the oscillator for β = 0.001. The values of the remaining parameters are the same as in Fig. 3. traced: the longer the time of interaction between the appreciably. In the region Ð5000 < τ < 15000, the values | τ |2 | τ |2 β electron subsystem and the resonator, the higher the of probabilities ln( ) and rn( ) for = 0.001 differ degree of resonator excitation; i.e., higher and higher insignificantly from their values in the case of complete energy levels of the oscillator become involved in the absence of coupling. However, with increasing τ, these process of energy exchange between the resonator and differences gradually accumulate and the behavior of the external alternating field. For this reason, the num- probabilities for β = 0.01 qualitatively repeats the ber N of oscillator levels that should be retained in the dependences shown in Fig. 4 for β = 0.01. calculations obviously depends on the time interval As in the nonresonance case, we calculate the total during which we are going to observe the system. For τ τ probability WL( ) by using formula (11). The results of instance, for the time interval Ð5000 < < 15000 over calculation for β = 0.01 are shown in Fig. 6, where the which our calculations were made (see Fig. 4), the probability difference ∆W (τ) is depicted. It can be seen number of energy levels of the oscillator was N = 8. L that the difference is extremely small in region τ < 0 (up Figure 5 illustrates the case of a weaker coupling to resonance region), constituting a small fraction of a τ (β = 10Ð3 for λ = 7.7). It can be seen that the general percent of the probability WL( ) proper, which is shown | |2 | |2 in Fig. 2. In the vicinity of the resonance, the difference form of the dependence of probabilities ln and rn on time τ does not change as compared to the case when noticeably increases and reaches several percent. Fur- β ther, in region τ > 0, as we move away from the reso- = 0.01. As before, the oscillator is gradually excited τ nance, the value of WL( ) itself is insignificant, while with time to its higher levels, but the process of its exci- ∆ τ tation occurs much more slowly in view of a weaker difference WL( ), whose absolute value slightly coupling with the electron subsystem. For this reason, decreases in this region, nevertheless becomes of the τ in the interval of time variation shown in Fig. 5, coeffi- order of probability WL( ) itself. τ τ cients l0( ) and r0( ) play the major role, while all In the region of the second resonance, a noticeable ∆ τ remaining amplitudes still have no time to increase “spike” of WL( ) is observed (although the probability

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∆ WL

0.10

0.05

–0.05

–0.10

–5000 0 5000 10000 15000 τ Fig. 6. Variation of population probability for the left quantum dot, associated with the interaction between the electron subsystem and the resonator, in the resonance case for λ = 7.7, µ = 10Ð4, and ∆/ω = 0.2.

τ WL( ) itself sharply increases in this region), after not change in this case since any energy losses in it are which the difference decreases again. On the whole, as compensated by the strong alternating field. τ follows from Fig. 6, the deviation of probability WL( ) from its value calculated for β = 0 does not exceed 0.1 ACKNOWLEDGMENTS in absolute value. It should be noted that, in contrast to the nonresonance case, the difference is two orders of The authors thank Yu.A. Romanov and A.M. Sata- magnitude larger and is apparently determined by the nin for their interest in this research. first power of coupling parameter β. This study was financed by the Ministry of Education It should be noted that the value of difference of the Russian Federation (grant nos. E02-3.1.-336 and ∆W (τ) for β = 0.001 turns out to be one or two orders UR.01.01.057) and the Russian Foundation for Basic L Research (project no. 02-02-17495). of magnitude smaller than for β = 0.01. For this reason, τ dependence WL( ) repeats the dependence shown in Fig. 2 even to a higher degree of accuracy. REFERENCES τ Thus, probability WL( ) of populating the left quan- 1. V. A. Burdov, Pis’ma Zh. Éksp. Teor. Fiz. 76, 25 (2002) tum dot for various values of parameter β remains vir- [JETP Lett. 76, 21 (2002)]. tually the same as in the absence of coupling both in the 2. R. Bavli and H. Metiu, Phys. Rev. A 47, 3299 (1993). nonresonance (η = 0) and in the resonance (η = 0.1) 3. A. A. Gorbatsevich, V. V. Kapaev, and Yu. V. Kopaev, Zh. cases. Such a behavior indicates that the electron den- Éksp. Teor. Fiz. 107, 1320 (1995) [JETP 80, 734 (1995)]. sity, as before, performs transitions (as in the case when 4. V. A. Burdov, Zh. Éksp. Teor. Fiz. 116, 217 (1999) [JETP β = 0) from one quantum dot to the other upon an adia- 89, 119 (1999)]. batic passage of the two-level electron system through 5. V. A. Burdov and D. S. Solenov, Phys. Lett. A 305, 427 resonance. Even in the case when an electron wave (2002). packet is localized in the right quantum dot, such an 6. D. H. Dunlap and V. M. Kenkre, Phys. Rev. B 34, 3625 “energetically disadvantageous” electronic state polar- (1986). ized against the external field E(t) is found to be stable. Its 7. S. Raghavan, V. M. Kenkre, D. H. Dunlap, et al., Phys. stability is due to the “locking” effect of the strong alter- Rev. A 54, R1781 (1996). nating field, which confines the electron density in the 8. F. Grossmann, T. Dittrich, P. Jung, and P. Hanggi, Phys. right quantum dot despite the possible energy exchange Rev. Lett. 67, 516 (1991); F. Grossmann and P. Hanggi, between the electron subsystem and the resonator. Europhys. Lett. 18, 571 (1992). On the whole, it can be concluded that the interaction 9. Y. Dakhnovskii and R. Bavli, Phys. Rev. B 48, 11010 of the electron subsystem with the resonator does not dis- (1993). turb the dynamic control over electron states. This is 10. A. B. Pippard, Physics of Vibration (Cambridge Univ. mainly due to the fact that the external alternating field in Press, Cambridge, 1978; Vysshaya Shkola, Moscow, our model possesses an infinitely large energy, whose 1989). finite part is transferred via the two-level electron system to the resonator. The state of the “transmitter” itself does Translated by N. Wadhwa

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004

Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 612Ð618. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 125, No. 3, 2004, pp. 693Ð700. Original Russian Text Copyright © 2004 by Gorbenko, Demin, Kaul’, Koroleva, R. Szymczak, H. Szymczak, Baran. SOLIDS Electronic Properties

Experimental Evidence of a Two-Phase Magnetic State in Thin Epitaxial Films of Re0.6Ba0.4MnO3 (Re = La, Pr, Nd, Gd) O. Yu. Gorbenkoa, R. V. Demina, A. R. Kaul’a, L. I. Korolevaa, R. Szymczakb, H. Szymczakb, and M. Baranb aMoscow State University, Moscow, 119992 Russia bInstitute of Physics, Polish Academy of Sciences, 02668 Warsaw, Poland e-mail: [email protected] Received October 9, 2003

Abstract—Thin epitaxial films of Re0.6Ba0.4MnO3 (Re = La, Pr, Nd, Gd) on (001)-oriented single crystal SrTiO3 and ZrO2(Y2O3) substrates have been prepared and studied. All films possess a cubic perovskite structure, except for the film with Re = La, which exhibited a rhombohedral distortion of the perovskite lattice. The results show evidence for the presence of two magnetic phases, ferromagnetic (FM) and antiferromagnetic (AFM), in the films studied: (i) the magnetization isotherm M(H) appears as a superposition of a linear component (characteristic of antiferromagnets) and a small spontaneous magnetization component; (ii) the magnetic moment per formula unit is significantly reduced as compared to the value expected for the complete FM or ferrimagnetic ordering; (iii) there is a difference between magnetizations of the samples cooled with and without an applied magnetic field, which is preserved in the entire range of magnetic fields studied (50 kOe); (iv) the temperature dependence of the magnetization M(T) in strong magnetic fields is close to linear (for the composition with Re = Gd, M(T) is described by a Langevin function for superparamagnets with a cluster moment µ of 22 B); and (v) the magnetization hysteresis loops of the field-cooled samples are shifted along the field axis. The exchange integral (characterizing the MnÐOÐMn coupling via the FMÐAFM phase boundary) estimated from the latter shift is |J| = 10Ð6 eV. This value is two orders of magnitude lower than the negative exchange integral between the FM layers in ReMnO3, which makes the presence of a transition layer at the FMÐAFM phase boundary unlikely. The temperature dependences of electrical resistance and magnetoresistance exhibit maxima at the Curie temperature (TC), where the magnetoresistance reaches a colossal value. This behavior indicates that the two-phase magnetic state is caused by a strong sÐd exchange. © 2004 MAIK “Nauka/Inter- periodica”.

〈 〉 1. INTRODUCTION radius rA of A cations, since it is known that the Curie temperature of manganites of the ABO3 type increases In recent years, much attention has been devoted to 〈 〉 with rA value [8, 9]. At the same time, there is an manganites of the Re1 Ð xAxMnO3 system, where Re is a opposite trend (so-called misfit effect) related to a dif- rare earth ion and A = Ca, Sr, or Ba. This interest is ference between the radii of different A cations (Re3+ related to the phenomenon of colossal magnetoresis- 2+ tance observed in some of these compounds at room and Ba ), which decreases the Curie temperature [10]. temperature. The compounds with A = Sr and Ca were At present, there is no commonly accepted opinion studied most thoroughly, while Re1 Ð xBaxMnO3 compo- about the crystal structure of Re1 Ð xBaxMnO3 com- ≤ ≤ sitions were characterized to a much lower extent, espe- pounds. In La1 Ð xBaxMnO3 compositions with 0.2 x cially in the case of thin films. Only thin films of the 0.4, Radaelli et al. observed a hexagonal structure La1 Ð xBaxMnO3 system with x = 0.2 and 0.33 were stud- (space group, R3c ), while Barnabe et al. [7] reported ied [1Ð3], in which the room-temperature magnetore- on a more complicated crystal structure (in agreement sistance R0/RH reached about 50% at H = 0.8 and 5 T for with the neutron diffraction data [11]) in stoichiometric the former and latter composition, respectively. It was compositions of the Re1 Ð xBaxMnO3 systems (Re = La, also reported [4] that the magnetoresistance of the com- Pr) with x = 0.4. At the same time, Raman spectroscopy pound with x = 0.33 strongly depends on the degree of and X-ray data [12] for polycrystalline samples of the nonstoichiometry with respect to oxygen. ≥ La1 Ð xBaxMnO3 system with x 0.35 showed evidence Compounds of the La1 Ð xBaxMnO3 system are of of a phase separation into cubic La0.65Ba0.35MnO3 and considerable interest because of very high Curie tem- hexagonal BaMnO3 phases. It was suggested [12] that ≥ peratures: TC = 362 was observed for x = 0.3 [5Ð7]. the structure of compounds with x 0.35 cannot 2+ Such a high TC value is due to a relatively large average accommodate Ba ions because of their large size.

EXPERIMENTAL EVIDENCE OF A TWO-PHASE MAGNETIC STATE 613

We have studied the crystal structure and the mag- type can be explained by an increase in the parameter netic and electrical properties of thin epitaxial films of of disorder with decreasing radius of the rare earth ion Re1 Ð xBaxMnO3 compounds (Re = La, Pr, Nd, Gd). The (and, accordingly, by an increase in the difference of observed features have been interpreted within the ion radii of the rare earth element and barium statisti- framework of the modern theory of magnetic semicon- cally occupying the A sublattice sites in the perovskite ductors. As was noted above, only films of the structure). This effect should be most pronounced in La1 − xBaxMnO3 system were reported previously [1Ð3]. compositions with the level of barium doping The films of Re1 Ð xBaxMnO3 with other rare earth ele- approaching 0.5. ments have been prepared and characterized for the first The films grown on (001)-oriented ZrO2(Y2O3) sub- time. strates exhibited simultaneous growth in two directions, (001) and (110). In the films of perovskite manganites studied previously, we usually observed only 2. EXPERIMENTAL METHODS one of these orientations, namely, (110) [13, 14]. The All films were grown by metalorganic chemical appearance of another type we explain by an increase in vapor deposition (MOCVD) using aerosols of volatile the lattice parameter of barium-doped perovskite, organometallic compounds. The aerosols of diglyme which leads to a change in the lattice mismatch between solutions of the initial compounds with a total concen- film and substrate. tration of 0.02 mol/l were produced by an ultrasonic source. The initial compounds for MOCVD were 3.2. Magnetic Properties R(thd)3 (R = La, Pr, Nd, Gd), Mn(thd)3, and Ba(thd)2(Phen)2, where thd = 2,2,6,6-tetramethylhep- Figure 1 shows the temperature dependence of the tane-3,5-dionate and Phen = o-phenanthroline. The magnetization M(T) measured in various magnetic samples were prepared in a reactor with an induction fields for films of the PrBaMnO and GdBaMnO sys- heated substrate holder at a substrate temperature of tems with x = 0.4 on SrTiO3 substrates (below, this sub- 800°C, an oxygen partial pressure of 3 mbar, and a total script will be omitted). The M(T) curves of the pressure of 6 mbar. The deposition rate was 1 µm/h and LaBaMnO and NdBaMnO films on the SrTiO3 sub- the film thicknesses ranged within 300–400 nm. The strates, as well as of the NdBaMnO film on the films were deposited onto (001)-oriented single crystal ZrO2(Y2O3) substrate, were very much like those SrTiO3 and ZrO2(Y2O3) substrates. depicted in Fig. 1a for PrBaMnO films on the SrTiO3 The films were studied by scanning electron micros- substrates. These magnetization measurements were copy (in combination with electron probe microanaly- performed in two modes. The upper curves for each sis) and by X-ray diffraction. The magnetization of thin field strength H were obtained by initially cooling a films was measured with a SQUID magnetometer and sample from T = 300 to 5 K in the given field, after the electric resistance was determined by the conven- which the temperature variation of the magnetization tional four-point-probe technique. was measured in the course of heating of this field- cooled (FC) sample. The lower curves were obtained for a sample cooled in the absence of a magnetic field, 3. EXPERIMENTAL RESULTS after which the field was applied and the M(T) curve of AND DISCUSSION the zero-field-cooled (ZFC) sample was measured in 3.1. Structural Characteristics of Thin Films the heating mode. The data in Fig. 1 reveal differences between the According to the X-ray diffraction data, the films temperature dependences of the magnetization for the obtained on (001)-oriented SrTiO3 substrates repre- FC and ZFC samples. The magnetization of FC sam- sented single-phase perovskites epitaxially grown in ples is higher than that of the otherwise identical ZFC the “cube over cube” mode. The pseudocubic lattice samples. This difference increases with decreasing parameter of the perovskite phase monotonically temperature and is more pronounced in lower magnetic decreased with the ion radius of Re3+. Only the X-ray fields. The M(T) curves of the ZFC samples studied in diffraction pattern of the film of La Ba MnO exhib- 0.6 0.4 3 small fields exhibit maxima at a certain temperature Tf; ited weak superstructural reflections and splitting of the at T > Tf, the curves of the FC and ZFC samples coin- pseudocubic reflections expected for a rhombohedral cide. In sufficiently large fields, the maximum in the distortion of the perovskite lattice (space group, R3c ), M(T) curves of the ZFC samples is not observed, but a in accordance with the behavior of this material in the difference between the cures of FC and ZFC samples is ceramic state [7]. In all other Re0.6Ba0.4MnO3 films, we retained in the entire range of magnetic fields studied observed neither peak splitting nor superstructural (up to 50 kOe). The only exception was the GdBaMnO reflections characteristic of the rhombohedral, tetrago- film, for which the difference disappeared at H < 6 kOe nal, or orthorhombic distortions encountered in the per- (see the inset to Fig. 1b). ovskite structures of rare earth manganites described in Figure 2 shows the magnetization isotherms mea- the literature. The increase in symmetry up to the cubic sured at various temperatures for the FC samples of

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 614 GORBENKO et al.

M, emu/g NdBaMnO ﬁlms on SrTiO3 substrates. The magnetiza- 12 tion isotherms of the other ﬁlms studied in our experi- (a) ments were analogous to those presented in Fig. 2. As can be seen, the M(H) curves appear as superpositions 10 H = 6 kOe 20 of a small spontaneous magnetization component and a linear component characteristic of antiferromagnets. 8 emu/g 10 H = 50 kOe By extrapolating the linear parts of the M(H) curves to ,

M intersection with the M axis, we determined the sponta- 6 1 kOe neous magnetization component and calculated the µ 0 100 200 300 experimental magnetic moments exp (expressed in T, K µ 4 Bohr magnetons per formula unit, B/FU). The values µ of exp at T = 5 K for the films of all studied composi- 2 0.2 kOe tions are given in the table. For comparison, we also µ present the values of theoretical magnetic moments th µ 0 50 100 150 200 250 300 ( B/FU) calculated for three variants of spin ordering for Mn3+, Mn4+, and Re3+ ions: the first value refers to T, K FM ordering only between Mn3+ and Mn4+; the second, M, emu/g to FM ordering of Mn3+, Mn4+, and Re3+; and the third, 3.0 3+ 4+ (b) 80 to FM ordering of Mn and Mn , and AFM ordering 3+ 70 of Re . These values were calculated using the follow- H = 500 Oe µ 3+ 2.5 60 ing pure spin moments of the rare earth ions: 2 B (Pr ), µ 3+ µ 3+ 50 3 B (Nd ), and 7 B (Gd ). As can be seen, the values kOe µ µ 2.0 40 H = 50 of exp are much lower than th in all cases. This exper- emu/g , 30 imental fact also provides evidence for the presence of M 20 two magnetic phases in the films under consideration. 1.5 6 kOe 10 In the films with Re = La, Pr, and Nd, the behavior 100 Oe of M(T) in strong magnetic fields is close to linear (see 0.5 0 50 100 150 200 250 300 Fig. 1a for the PrBaMnO films), which is untypical of T, K ferromagnets. As is known, the temperature dependence of the magnetization in ferromagnets is 0 50 100 150 200 250 300 described by the Brillouin function [15]. Obviously, the T, K exact value of the Curie temperature can be determined only in experiments performed in the absence of exter- Fig. 1. Temperature dependence of the magnetization M(T) nal magnetic fields, because such a field suppresses and of thin films of Re0.6Ba0.4MnO3 with Re = Pr (a) and Gd (b) smears the phase transition. In practice, however, the on SrTiO3 substrates measured in various magnetic fields. Curie temperature is frequently determined by extrapo- The upper and lower curves for each field strength H refer lating the steepest (upper) part of the M(T) curve to the to the FC and ZFC samples, respectively (see the text). temperature axis. In the general case, this yields a cer-

tain characteristic temperature T C' that is close to the M, emu/g Curie temperature. The characteristic temperatures 5 K determined in this way for the films studied in our 30 25 experiments strongly depend on the field applied during 50 the M(T) measurements. The values of T C' obtained by 75 this method are listed in the table. As can be seen, the 20 ' 100 T C value significantly increases with increasing H. For 140 example, the T C' values for the LaBaMnO film in a field of 100 Oe is 283 K and that in a field of 50 kOe is 90 K 10 higher, amounting to 373 K. In magnetically homogeneous materials possessing spontaneous magnetization (for example, in ferromagnets) this difference does not 0 10 20 30 40 50 exceed 10 K. H, kOe As can be seen from Fig. 1b, the film of GdBaMnO exhibits a difference between magnetizations of the FC Fig. 2. Magnetization isotherms of Nd0.6Ba0.4MnO3 films and ZFC samples only in sufficiently small fields: no on SrTiO3 substrates measured at various temperatures. such difference is observed for H = 6 kOe. The shape of

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Magnetic properties of thin epitaxial ﬁlms of Re0.6Ba0.4MnO3 (Re = La, Pr, Nd, Gd) on (001)-oriented SrTiO3 and ZrO2(Y2O3) substrates Re La Pr Nd Nd Gd

Substrate SrTiO3 SrTiO3 SrTiO3 ZrO2(Y2O3) SrTiO3 µ µ exp, B/FU 1.93 1.15 1.52 1.35 3.28 µ µ th, B/FU 3.6 3.6, 4.8, 2.4 3.6, 5.4, 1.8 3.6, 5.4, 1.8 3.6, 7.8, 0.6 ∆H, Oe 57 400 230 300 380 3 4 × 4 × 4 × 4 × 4 Ku, erg/cm 10 1.9 10 1.2 10 3.2 10 2.7 10 ' TC , K (H = 100 Oe) 283 160 142 145 75 ' TC , K (H = 6 kOe) 308 213 170 210 26 ' TC , K (H = 50 kOe) 373 369 260 50

Tf, K (H = 100 Oe) 100 95 73 75

Tf, K (H = 6 kOe) 50 50 41 38

Tf, K (H = 50 kOe) 30 33 33

Tρ(max), K 284 116 the M(T) curves for this film also differs from that of the was noted above, by the magnetization curves. For H ≥ curves in Fig. 1a (the latter are also typical of the films 6 kOe, the compensation point in the M(T) curves is no of other compounds studied). In weak fields, the M(T) longer observed and the magnetization strongly curves of the ZFC samples of GdBaMnO (Fig. 1b) pass increases, so that the magnetic moment reaches µ through a maximum and then exhibit a minimum fol- 3.28 B/FU at 5 K in a field of 50 kOe. This value is sig- lowed by a sharp increase. The curves of the FC sam- nificantly greater than that in a sample with complete µ ples exhibit only an inflection point (instead of the max- ferrimagnetic ordering (0.6 B/FU), but still markedly imum and minimum observed for the ZFC samples) smaller that the magnetic moment in the case of µ followed by a sharp growth with decreasing tempera- complete FM ordering (7.8 B/FU. Apparently, the ture. In a field of H > 6 kOe, the difference between moments of Gd3+ and manganese ions exhibit FM M(T) curves of the FC and ZFC samples disappears and ordering, but the FM phase occupies only a part of the the magnetization in both cases monotonically sample volume. decreases with increasing temperature, showing no singularities observed in lower fields. The magnetic The above considerations suggest that the phase moment of the film measured at 5 K in a field of 50 kOe with spontaneous magnetization in a GdBaMnO film is 3.28µ /FU. occurring in a two-phase magnetic state exhibits a mag- B netic field-induced transition from ferrimagnetic to FM At the same time, the M(T) curves measured in the ordering. The experimental M(T) curve observed in a fields H < 6 kOe resemble those of a ferrimagnet with field of 50 kOe is well described by the Langevin func- the compensation point. In this case, the theoretical tion for an ensemble of superparamagnetic clusters µ with a ferromagnetic moment of µ = 22µ and a true low-temperature magnetic moment th per formula unit B at low temperatures must be equal to the difference magnetization of M0 = 73.6 emu/g, between the magnetic moments of Gd3+ and manganese ()µ µ µ M/M0 = coth H/kT Ð kT/ H, (1) ions, that is, to 0.6 B/FU. The experimentally observed magnetic moment at H = 500 Oe, where the compensa- where M is the magnetization at a given temperature µ tion point is still observed, does not exceed 0.08 B/FU, and M is the true magnetization. This is illustrated in µ 0 which is lower than th by a factor of 7.5. This suggests Fig. 3, showing a good fit of experimental data (sym- that only a part of the sample (about 13%) is ferrimag- bols) to a theoretical curve constructed according to netic, while the remaining part is antiferromagnetic. It relation (1). Assuming that the spins of Gd3+ and man- should be noted that the ferrimagnetic state with a com- ganese ions in a cluster are ferromagnetically ordered, pensation point was previously observed in the related the cluster includes approximately three formula units. compound Gd0.67Ca0.33MnO3 [16]. The presence of an These results also provide evidence that a two-phase AFM phase in the GdBaMnO film is also evidenced, as (FMÐAFM) magnetic state exists in the GdBaMnO

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 616 GORBENKO et al.

M, emu/g loop was originally observed in partly oxidized cobalt 80 [17], where it was attributed to the exchange interaction between FM Co particles and their AFM shells of CoO. This phenomenon was called exchange anisotropy. Later, the shift of hysteresis loops in the samples cooled 60 in a weak magnetic field was considered evidence for a spin glass state. Nevertheless, this phenomenon has been explained only for cluster spin glasses and must be 40 absent in a true spin glass containing only randomly oriented spins. Kouvel [18] explained the shift of the magnetization hysteresis loops observed in CuMn and 20 AgMn spin glasses by an inhomogeneous distribution of manganese ions: the regions depleted of Mn were assumed to be ferromagnetic, while the Mn-rich regions featuring exchange interaction were considered antiferromagnetic. 0 50 100 150 200 The shift of the magnetization hysteresis loops T, K observed in our samples unambiguously point to the Fig. 3. Temperature dependence of the magnetization M(T) existence of a two-phase (FMÐAFM) magnetic state of a thin film of Gd0.6Ba0.4MnO3 on a SrTiO3 substrates with exchange interact ion between the FM and AFM measured in a magnetic field of H = 50 kOe (black squares). regions of the films. This shift can be expressed as Solid curve shows the Langevin function calculated for an ensemble of superparamagnetic clusters with the magnetic ∆HK= /M , (2) µ µ u s moment = 22 B and the true magnetization M0 = 73.6 emu/g. where Ku is the exchange anisotropy constant and Ms is the saturation magnetization. The exchange anisotropy constants Ku calculated for all films have proved to be M, emu/g on the order of 104 erg/cm3 (see table). Using these val- 10 ues, it is possible to determine exchange integral J characterizing the MnÐOÐMn coupling via the FMÐAFM phase boundary, provided that the area of this interface 5 is known. Unfortunately, no such data are available at present.

0 M, emu/g 3.3. Electrical Properties 10 The resistivity ρ and magnetoresistance ∆ρ/ρ = ρ ρ ρ –5 0 ( H Ð H = 0)/ H = 0 of LaBaMnO, PrBaMnO, and –10 NdBaMnO films on SrTiO3 substrates were studied at ≤ –4 –2 0 2 4 T > 78 K and H 8.2 kOe. The resistances of –10 GdBaMnO films on SrTiO and NdBaMnO films on –4 –2 0 2 4 3 ZrO2(Y2O3) were so high (because of very small film H, kOe thicknesses) that we failed to measure the ρ values Fig. 4. Magnetization hysteresis loop of a thin film of below TC by the four-point-probe technique. Figure 5 Pr0.6Ba0.4MnO3 on a SrTiO3 substrates measured at 5 K shows the ρ(T) curves of the films studied. Figure 6 pre- upon cooling the sample in a magnetic field of 4 kOe. The sents data on the behavior of {∆ρ/ρ}(T) for PrBaMnO inset shows the same for a thin film of Gd0.6Ba0.4MnO3 on and LaBaMnO. The magnetoresistance is negative and a SrTiO3 substrate. the ρ(T) and |{∆ρ/ρ}|(T) curves exhibit maxima. The temperatures of maxima for the latter curves are lower than those for the former ones, which is typical of mag- film, with the magnetic moment of the FM cluster netic semiconductors. The resistivities at maxima were amounting to approximately 22µ in a magnetic field of B as follows: ~10Ð3 Ω cm for PrBaMnO and NdBaMnO, 50 kOe. and ~10Ð5 Ω cm for LaBaMnO. The magnetoresistance The existence of a two-phase magnetic state in the at maximum is very large, reaching 43% for PrBaMnO. films under consideration is also confirmed by the fact The presence of peaks in the temperature depen- that the magnetization hysteresis loops of FC samples dences of ρ and ∆ρ/ρ, together with the colossal mag- are shifted along the field axis. This is illustrated in netoresistance, are indicative of the existence of a two- Fig. 4, showing such shifted loops for PrBaMnO and phase magnetic state related to a strong sÐd exchange in GdBaMnO films. An analogous shift of the hysteresis the films studied. Evidently, these manganites are

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ρ, Ω m ∆ρ/ρ, %

2 40 100 1 Pr0.6Ba0.4MnO3 30 10–1 3 20 10–2

–3 10 10 4 La0.6Ba0.4MnO3

10–4 0 50 100 150 200 250 300 350 50 100 150 200 250 300 350 T, K T, K Fig. 5. Temperature dependence of the resistivity ρ of thin Fig. 6. Temperature dependence of the magnetoresistance films of Re0.6Ba0.4MnO3 with Re = Gd (1), Nd (2), Pr (3), of thin Pr0.6Ba0.4MnO3 and La0.6Ba0.4MnO3 films on and La (4) on SrTiO3 substrates. SrTiO3 substrates measured in a magnetic field of 8.5 kOe.

essentially the AFM semiconductors LaMnO3, and ZFC samples (Fig. 1), the magnetic moment per 2+ PrMnO3, NdMnO3, and GdMnO3 doped with Ba formula unit at 5 K is strongly reduced (see table), and ions. Judging by the resistivity ρ, the films of the M(T) curve shape differs from that of the Brillouin function. On the other hand, there are significant dis- LaBaMnO, PrBaMnO, and NdBaMnO on SrTiO3 substrates occur in a conducting two-phase magnetic state tinctions as well. First, the M values of the FC samples with AFM clusters deprived of charge carriers (holes) are of spin glasses are independent of the temperature at dispersed in a conducting FM matrix. This conducting T < Tf, provided that the cluster size does not vary with two-phase magnetic state related to a strong sÐd the temperature (this condition is usually valid for spin exchange has been described in review [19]. This state is glasses). In contrast, the magnetizations of our FC sam- characterized by a sharp increase in resistivity in the ples increase with decreasing temperature. Second, in vicinity of the Curie point. There are two mechanisms by spin glasses, the behavior of the magnetization in FC which the impurity magnetism influences the resistance: and ZFC samples differs only in small fields not the scattering of charge carriers (decreasing their mobil- exceeding several kOe, whereas our films (except ity) and the formation of a tail (representing localized GdBaMnO) exhibit this difference in the entire range of states) in the conduction band. In the vicinity of the Curie magnetic fields studied (up to 50 kOe). These facts can point, charge carriers sharply lose the mobility and be explained assuming that a decrease in the tempera- exhibit partial localization in the band tail, which ture leads to an increase in the volume of the FM phase ρ in the two-phase magnetic state. The same factor can explains the appearance of a maximum in (T) near TC. Under the action of a magnetic field, these charge carri- account for the difference between the M(T) curve and ers are delocalized from the band tail and their mobility the Brillouin function. Third, the magnetization iso- increases that leads to a colossal magnetoresistance. therms of spin glasses are substantially nonlinear (see Fig. 2), while those of our films are superpositions of a As can be seen from Fig. 5, the film of GdBaMnO small spontaneous magnetization component and the possesses the maximum resistivity among the samples linear component characteristic of antiferromagnets. studied, exceeding 1 Ω m at T = 150 K (i.e., significantly above TC). This suggests the presence of an insu- The shift ∆H of the magnetization hysteresis loops lating two-phase magnetic state comprising FM drops, of the FC samples along the H axis is unambiguous evi- where charge carriers (holes) are concentrated due to an dence in favor of the two-phase magnetic state, sÐd exchange energy gain [19], dispersed in an insulat- although this state is also observed in spin glasses ing AFM matrix. (where it is entirely due to the presence of FM and AFM regions and the exchange interaction between them, as was pointed out by Kouvel [18]). Using the field shift 4. CONCLUSIONS ∆H, we have estimated the exchange integral J charac- The magnetic properties of the films described terizing the MnÐOÐMn coupling via the FMÐAFM above resemble those of cluster spin glasses. Indeed, phase boundary in some manganites occurring in an there are differences between magnetizations of the FC insulating two-phase (FMÐAFM) magnetic state [20].

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 618 GORBENKO et al.

It was found that |J| ~ 10Ð6 eV. This value is two orders 3. T. Kanki, H. Tanaka, and T. Kawai, Phys. Rev. B 64, of magnitude lower than a negative exchange integral 224418 (2001). | | × Ð4 4. H. L. J. Ju, J. Gopalakrishnan, J. L. Peng, et al., Phys. between the FM layers in LaMnO3 ( J1 = 5.8 10 eV) determined from the neutron scattering data [21]. This Rev. B 51, 6143 (1995). result indicates that the presence of a transition layer 5. G. H. Jonker and J. H. van Santen, Physica (Amsterdam) with tilted spins at the FMÐAFM phase boundary is 16, 337 (1950); G. H. Jonker, Physica (Amsterdam) 22, unlikely. 702 (1956). 6. P. G. Radaelli, M. Marezio, H. Y. Hwang, and In the material occurring in a two-phase magnetic C. W. J. Cheong, Solid State Chem. 122, 444 (1996). state, the charge carriers are concentrated in the FM 7. A. Barnabe, F. Millange, A. Maignan, et al., Chem. phase and are absent from the AFM phase. For this rea- Mater. 10, 252 (1998). son, the topology of the two-phase magnetic state is 8. F. Millange, A. Maignan, V. Caignaert, et al., Z. Phys. B determined by the Coulomb forces and the interfacial 101, 169 (1996). energy. As can be seen from data presented in the table, 9. A. Maignan, Ch. Simon, V. Caignaert, et al., Z. Phys. B the FM phase volume in LaBaMnO, PrBaMnO, and 99, 305 (1996). NdBaMnO on SrTiO substrates occur in a conducting 3 10. L. M. Rodriguez-Martinez and J. P. Attfied, Phys. Rev. B two-phase magnetic state with the FM phase filling the 54, R15622 (1996). space between insulating AFM spheres. Since the Ku 11. Z. Jirak, E. Pollet, A. F. Andersen, et al., Eur. J. Solid values are on the same order of magnitude for all the State Inorg. Chem. 27, 421 (1990). films studied (a part of which occur in the conducting 12. C. Roy and R. C. Budhani, J. Appl. Phys. 85, 3124 two-phase magnetic state) and for the manganites (1999). reported in [20] (occurring in an insulating two-phase 13. O. Yu. Gorbenko, R. V. Demin, A. R. Kaul’, et al., Fiz. magnetic state), we may suggest that the area of the Tverd. Tela (St. Petersburg) 40, 290 (1998) [Phys. Solid FMÐAFM phase boundary in the two cases is also com- State 40, 263 (1998)]. parable. Therefore, the conclusions made in [20] can be 14. A. I. Abramovich, L. I. Koroleva, A. V. Michurin, et al., expanded to include the films considered above, so that Zh. Éksp. Teor. Fiz. 118, 455 (2000) [JETP 91, 399 the presence of a transition layer with tilted spins at the (2000)]. FMÐAFM phase boundary is unlikely. The GdBaMnO 15. S. V. Vonsovskiœ, Magnetism: Magnetic Properties of films apparently occur in an insulating two-phase mag- Dia-, Para-, Ferro-, Antiferro-, and Ferrimagnets netic state. For this material, the magnetic moment of the (Nauka, Moscow, 1971; Wiley, New York, 1974). FM clusters estimated using the Langevin function (1) 16. G. J. Snyder, C. H. Booth, F. Bridges, et al., Phys. Rev. µ for H = 50 kOe is 22 B, which corresponds to three for- B 55, 6453 (1997). mula units. 17. W. H. Meiklejohn and C. P. Bean, Phys. Rev. 105, 904 (1957). 18. J. S. Kouvel, J. Phys. Chem. Solids 21, 57 (1961); ACKNOWLEDGMENTS J. Phys. Chem. Solids 24, 795 (1963). This study was supported by the Russian Founda- 19. É. L. Nagaev, Usp. Fiz. Nauk 166, 833 (1996) [Phys. tion for Basic Research, project no. 03-02-16100. Usp. 39, 781 (1996)]; Phys. Rep. 346, 381 (2001). 20. R. V. Demin, L. I. Koroleva, R. Szymszak, and H. Szym- szak, Pis’ma Zh. Éksp. Teor. Fiz. 75, 402 (2002) [JETP REFERENCES Lett. 75, 331 (2002)]. 1. R. von Helmolt, J. Wecker, B. Holzapfel, et al., Phys. 21. F. Moussa, M. Hennion, and J. Rodriguez-Carvajal, Rev. Lett. 71, 2331 (1993). Phys. Rev. B 54, 15149 (1996). 2. G. C. Xiong, Q. Li, H. L. Ju, et al., Appl. Phys. Lett. 66, 1427 (1995). Translated by P. Pozdeev

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