Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1006Ð1018. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1145Ð1159. Original Russian Text Copyright © 2003 by Bolgova, Ovsyannikov, Pal’chikov, Magunov, von Oppen. , SPECTRA, RADIATION

Higher Orders of for the Stark Effect on an Atomic Multiplet I. L. Bolgovaa, V. D. Ovsyannikova, V. G. Pal’chikovb,*, A. I. Magunovc, and G. von Oppend aPhysics Department, Voronezh State University, Voronezh, 394006 Russia bNational Research Institute for PhysicalÐTechnical and Radiotechnical Measurements, Mendeleevo, Moscow oblast, 141570 Russia *e-mail: [email protected] cInstitute of General Physics, Russian Academy of Sciences, Moscow, 119991 Russia dTechnical University, Berlin, D-10623, Germany Received November 28, 2002

Abstract—The contribution of higher order corrections to the Stark energy is calculated in the anticrossing region of atomic multiplet sublevels. Perturbation theory for close-lying levels is presented that is based on the Schrödinger integral equation with a completely reduced Green’s function. Analytic formulas are obtained for the splitting of two interacting fine-structure sublevels as a function of the field strength. These formulas take into account fourth-order and nonresonance corrections to both the diagonal and the off-diagonal matrix elements of the moment operator. By the method of the Fues model potential, a numerical anal- ysis of radial matrix elements of the second, third, and fourth orders is carried out that determine a variation in 3 3 the transition energy between n P0 and n P2 sublevels of a helium for n = 2, 3, 4, 5 in a uniform electric field. It is shown that the contribution of the fourth-order corrections in the vicinity of anticrossing of levels for n = 2, 3, 4, 5 amounts to 0.1, 5, 10, and 15% of the total variation of energy, respectively. A comparative anal- ysis is carried out with the results of calculations obtained by the method of diagonalization of the energy matrix, which, together with resonance terms, takes into account other states of the discrete spectrum with n ≤ 6. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION A comparison of the numerical data obtained by these expressions with the results of measuring the field The Stark effect in a constant electric field was cal- dependence of the probability of radiative transitions culated for a atom and described in detail in between highly excited Rydberg states [11] has shown textbooks and monographs almost immediately after that the first three terms in powers of the field strength the origination of quantum (see, for exam- are quite sufficient for calculating the transition proba- ple, [1, 2]). Nevertheless, this phenomenon has not yet bilities in virtually all situations of interest up to the been completely studied and still attracts attention in field value when the above-barrier ionization of the our time. The experimental studies of the Stark effect upper level becomes possible. have been stimulated by the development of precision By the end of 1970s, methods of laser spectroscopy methods of laser spectroscopy [3Ð6]. Theoretical calcu- of Rydberg levels had been developed that allowed one lations of the shift and broadening of the Stark states in to accumulate a large volume of experimental data on hydrogen atoms carried out by the beginning of the the Stark effect in highly excited multielectron atoms 1980s were based on the iterative solution of the [12Ð15]. These data stimulated the development of sim- Schrödinger equation involving the separation of vari- ple semiempirical calculation methods for the polariz- ables in a parabolic system of coordinates [2]. The abilities of atomic levels [16, 17] (including Rydberg development of computer programs for analytic com- levels [18]), as well as the development of exact ab ini- putations that allowed one to derive general expressions tio methods that enable one to consistently take into for the coefficients of power series in the field strength consideration relativistic and quantum-electrodynamic for both the shift and the broadening of atomic levels effects, which play a significant role in the spectra of [7, 8] has served as a powerful incentive for the calcu- of high degree of ionization [19Ð22]. In [13Ð15, lation of higher order corrections to energy. Analytic 23], it was observed that the Stark effect on multiplet programming has also enabled one to obtain general sublevels deviates from the quadratic law even in weak expressions for perturbation theory series for the wave fields. This phenomenon is accounted for by the Stark functions, matrix elements, and radiative transition prob- interaction between sublevels that leads to their repul- abilities between the Stark states of hydrogen [9, 10]. sion in a field and is determined by the hyperpolariz-

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HIGHER ORDERS OF PERTURBATION THEORY FOR THE STARK EFFECT 1007 ability of an atom. It was demonstrated that the main hyperpolarizabilities are rapidly increasing functions, (resonance) contribution to the hyperpolarizability of a α ~ ν7 and γ ~ ν17 [32], the anticrossing for higher levels multiplet sublevel can be expressed in terms of the ten- occurs in weaker fields, and the contribution of higher sor of this multiplet [24]. Therefore, pre- order corrections becomes more significant. The first cision calculation [25] and measurement [3, 5, 26] of calculations of the nonresonance hyperpolarizability irreducible components of the polarizability of for the triplet state 3 3P in helium atoms [30] has shown multiplet states of atoms becomes topical. that the contribution of the fourth-order field correc- Precision information about the Stark effect in tions to the diagonal elements may be several times higher order terms of perturbation theory plays a key greater than experimental errors. Therefore, the deter- role in modern optical frequency standards based on the mination of corrections to the off-diagonal elements application of magnetooptical traps in combination also becomes topical. with the methods of laser cooling of atoms up to several The main goal of the present paper is to determine nanokelvins [27], as well as in the problems of fre- the contribution of the fourth-order corrections in the quency stabilization in atomic standards of a new gen- field strength to the energy of multiplet states of an eration—atomic fountains [28]. atom near the anticrossing of the fine-structure sublev- The interaction between sublevels of a multiplet in a els. In Section 2, we give a generalization of higher field may lead to an important phenomenon, the so- order perturbation theory for degenerate states [33] to called anticrossing [29]. This phenomenon manifests the case of close-lying levels that have nonzero splitting itself when the polarizability of a higher energy sub- in zero field. We obtain expansions of the matrix ele- level is greater than the polarizability of a lower energy ments of the interaction Hamiltonian of an atom and a state. Then, in a weak field, the sublevels move closer field in powers of the field strength F up to the fourth to each other; i.e., the fine-structure splitting decreases. order. In Section 3, the coefficients of these expansions In a stronger field, the sublevels are repulsed from each are expressed in terms of irreducible parts—scalar and other; this repulsion is determined by the resonance tensor and hyperpolarizabilities— part of the hyperpolarizability. The field strength at which, in turn, are represented by linear combinations which attraction turns into repulsion and the minimal of radial matrix elements of the second, third, and value of splitting are uniquely determined by the fourth orders. In Section 4, we derive general expres- atomic susceptibilities—the components of the polariz- sions for the field-dependent splitting of two interacting fine-structure sublevels. The fine-structure splitting of ability and hyperpolarizability—and can be calculated 3 theoretically. Usually, the field value corresponding to n P states of helium (n = 2, 3, 4, 5) is calculated numer- the anticrossing of levels is sufficiently high; therefore, ically as a function of F; in zero field, these states have an appreciable contribution to the energies of states total momenta of J = 0, 2 and a projection of M = 0. The may be made not only by the polarizability and the res- contribution of the fourth-order corrections in the field onance part of the hyperpolarizability but also by non- region of level anticrossing is determined. resonance additions to the hyperpolarizability. In [30], fourth-order nonresonance corrections have 2. HIGHER ORDER PERTURBATION THEORY been calculated to the energy of separate sublevels of a FOR CLOSE-LYING LEVELS helium atom multiplet. In this case, the resonance The calculation of higher order corrections to the γ(res) hyperpolarizability , which is determined by the Stark effect for atomic multiplet sublevels requires a off-diagonal matrix elements of the interaction operator consistent development of perturbation theory for between atom and field through the fine-structure states ˆ and is expressed in terms of the tensor polarizability αt, close-lying levels. Suppose that H0 (r) is a one-electron is calculated only in the second order of perturbation Hamiltonian of a valence electron with eigenfunctions φ theory. This is associated with the fact that, formally, nLSJ(r) and eigenvalues EnLSJ that correspond to the the fourth-order corrections to an off-diagonal element sublevels of an atomic multiplet with the principal make a correction of order F6/δ, where δ is a fine-struc- n and the quantum numbers of orbital ture splitting value at F = 0, to the expression for the L, spin S, and total J momenta (the Hamiltonian Hˆ 0 (r) fine-structure splitting δ(F) depending on the field takes into account the spinÐorbit interaction). Then, strength F. However, in the anticrossing region, the t 2 ˆ ()φ () φ () field amplitude may reach such values that α F ~ δ; H0 r nLSJ r = EnLSJ nLSJ r . (1) therefore, the fourth-order corrections to diagonal and off-diagonal matrix elements may make identical con- Let us project the Schrödinger equation tributions to the energy of the multiplet sublevels. Since ()Ψˆ () ˆ () () Ψ () the fine-structure splitting is a rapidly decreasing func- H0 r + V 0 r nLSJM r = E nLSJM r (2) tion of the effective principal quantum number of an for an atom in a constant electric field, whose interac- ν atomic level = 1/Ð2En (En is the binding energy of tion is described by the operator Vˆ (r) = Fz, onto a wave the level), δ ~ νÐ3 [31], and the polarizabilities and function of a certain state of degenerate basis, assuming

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

1008 BOLGOVA et al. Ψ that all projections of the vector of state nLSJM onto the Using an iterative procedure, we represent the for- basis vectors are of the same (zero) order in the field, mal expression for a as the BrillouinÐ Wigner series 〈〉Ψ ()φ () ()λ nLSJM r nLSJ r = aJ . λ Ψ() Here, J = J1, …, Jk; = 1, 2, …, k; and k is the number r = ∑aJ' of interacting sublevels (which may not coincide with J' the multiplicity of the multiplet). Taking into account (1), ∞ (12) q we obtain the following expression as a result of the × []G' ()∆rr, ' ()EVÐ ˆ ()r' |〉φ ()r' . projection: ∑ E nLSJ' q = 0 ()〈〉φ ˆ Ψ EnLSJ Ð E aJ +0.nLSJ V = (3) Applying the formula for the sum of a geometrical pro- Next, we project Eq. (2) onto the Green’s function gression, we rewrite this expression as GE (r, r') of the unperturbed atom that satisfies the Ψ() [](), ()ˆ () ∆ Ð1 inhomogeneous equation r = ∑aJ' 1 + G' rr' V r' Ð E E (13) ()ˆ ()δ, () J' H0 Ð E GE rr' = rrÐ ' , (4) ×φ|〉() nLSJ' r' . whose right-hand side contains the Dirac δ function, and Substituting (13) into (3), we obtain the following sys- tem of equations for coefficients aJ: Jk 1 E = --- ∑ EnLSJ (5) []()δε ∆ k ∑aJ' W JJ' + J' Ð E JJ' = 0, (14) JJ= 1 J' represents the mean value of the energy of interacting sublevels. For convenience, we introduce the following where notation: 〈|φ ˆ []' ()ˆ ∆ Ð1|〉φ ε W JJ' = nLSJ V 1 + GE V Ð E nLSJ' (15) J = EnLSJ Ð E (6) is the energy of a sublevel with a given J with respect to is a matrix element of the operator of interaction Vˆ the mean value (5), and between the atom and the field, which formally includes all orders of perturbation theory. ∆EEE= Ð (7) ˆ is the relative energy of the atom in the field. As a result In the absence of an external field (V = 0), system (14) ∆ ε of projection, we obtain is solvable if E = J. Then, aJ = 1 and aJ' ≠ J = 0. When Vˆ ≠ 0, the system has a nonzero solution if ∆E satisfies Ψ()r +0.G ()rr, ' ()ΨVˆ ()r' Ð ∆E |〉()r' = (8) E the secular equation Let us select the states of the multiplet under consider- ()δε ∆ ation from the Green’s function expressed as a spectral det W JJ' +0.J' Ð E JJ' = (16) expansion [16, 17], To solve this equation, one has to know the matrix ele- φ ()φr * ()r' G ()rr, ' = ∑ ------n'L'S'J' n'L'S'J' -, (9) ment WJJ', which depends on the strength of the exter- E E Ð E nal field F. Numerical calculations show that, when F < n'L'S'J' n'L'S'J' F0 (F0 is the field value at which the upper sublevel of by representing the latter function as the multiplet |nLSJ 〉 occurs above the top of the poten- G ()rr, ' = G' ()rr, ' tial barrier), one can use the expansion of WJJ' in powers E E of F2. In spite of the fact that such a series is asymp- Jk φ ()φ () (10) totic, its first several terms form a decreasing sequence nLSJ' r nLSJ* ' r' + ∑ ------. such that their sum determines WJJ' to a high degree of EJ' accuracy up to F = F . Practically, it suffices to take into J' = J1 0 consideration the first two terms of the series, in which Substitute (10) into (8). Taking into account (3) and (6), 2 4 we obtain the coefficients of F and F are determined by the polarizability and the hyperpolarizability of the atomic ∆ ε 2 ӷ ∆ 2 4 ӷ 6 Ψ()r = ∑a φ ()r state. Since E ~ J' ~ F EF ~ F F …, such an J' nLSJ' expansion of the matrix element W is given by J' (11) JJ' ' (), ()Ψˆ () ∆ |〉() 2 ()2 2∆ 4 ()4 Ð GE rr' V r' Ð E r' . W JJ' = F wJJ' ++F EuJJ' F wJJ', (17)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 HIGHER ORDERS OF PERTURBATION THEORY FOR THE STARK EFFECT 1009 where  × ()L'0 SL J SLJ' 11 () ∑ CL010 RL'J'' S ()2 〈|' |〉  wJJ' = Ð nLSJ zGEz nLSJ' , L' = L ± 1 1 J'' L' 1 J'' L'

2 is the irreducible part of the matrix element. We will use u = Ð〈|nLSJ zG()' z|〉 nLSJ' , (18) JJ' E the standard notation for the ClebschÐGordan coeffi- cients and the 6j symbols [34]. The radical factor ()4 〈|' ' ' |〉 wJJ' = Ð nLSJ zGEzGEzGEz nLSJ' in (19) is introduced in order that the coefficients of this expansion, as well as the coefficients of the expansion are matrix elements that contain one, two, and three of the irreducible part (20) in terms of radial matrix ele- Green’s functions. The second and third terms on the ments R, should be rational numbers that do not contain right-hand side of (17) are on the order of F4; moreover, radicals. For a matrix element of the (N + 1)th order it suffices to use a quadratic approximation for the fac- with N Green’s functions, we use the notation ∆ tor of E in the second term. Applying the apparatus of … n1n2 nN + 1 () angular momentum theory [34], we can reduce these RL J ,,,L J … L J S matrix elements to a sum of irreducible parts that con- 1 1 2 2 N N tain scalar and tensor polarizabilities and hyperpolariz- n n ≡ 〈|nLSJ r 1g ()r , r r 2 (21) abilities. 1 L1 J1 1 2 2

The general formulas (16)Ð(18) of degenerate per- n n …r N g ()r , r r N + 1 |〉nLSJ' , turbation theory, which contain the second-, third-, and N LN J N N N + 1 N + 1 fourth-order field corrections, can be used for calculat- ing the Stark effect in a multielectron atom. In this case, where gL'J'(r, r') is the radial part of the Green’s func- the state vector |nLSJ 〉 is constructed within a given tion (9) that determines its series expansion in spherical scheme of coupling between angular momenta (for harmonics. The notation for the components of the example, in the LS or JJ representations); by the opera- polarizability tensor that is conventionally used in the tor z in (18), we mean the z component of the dipole literature is determined by the irreducible parts of the moment αs ()0 αt diagonal matrix element (19): nLSJ = aJJ and nLSJ = () N a 2 for scalar and tensor polarizabilities, respectively. θ JJ Dr= ∑ i cos i Thus, for the diagonal matrix element (19), we have i 2 () ()2 1 αs 3M Ð JJ+ 1 αt ' wJJ = Ð--- nLSJ +,------nLSJ (22) of the atom, and GE corresponds to the definition of a 2 J()2J Ð 1 reduced Green’s function of the N-electron atom. where

s 2 L'0 2 3. DECOMPOSITION OF MATRIX ELEMENTS α = ---()2L + 1 ∑()2J''+ 1 ()C INTO IRREDUCIBLE PARTS nLSJ 3 L010 L'J'' (23) Using the WignerÐEckart theorem and the proper- 2 ties of 6j symbols and the ClebschÐGordan coefficients × SL J 11 () RL'J'' S , [34], we obtain the following expression for the first 1 J'' L' term in the matrix element (17): 1/2 ()2 10() 2J Ð 1 w αt 3 () JJ' nLSJ = ------()2L + 1 32J + 2 2 ()()1/2 (19) 1 JM 2J + 2 2J'1+ ()j = Ð--- C ------j a , ∑ J'Mj0 ()JJ'  2 2J + 1 Ð j j + 1 JJ+ '' 11 2 j = 02, × ∑()Ð1 ()2J''+ 1 (24)  J'' JJJ'' where (b)n = b(b + 1)…(b + n Ð 1) is the Pochhammer symbol and 2 L'0 211 1/2 × ()SL J () () ()2J + 1 Ð j ()2 j + 1 ∑ CL010 RL'J'' S . a j = 22()L + 1 C j0 ------j + 1  JJ' 1010 () L' = L ± 1 1 J'' L' 2J + 2 j

The decomposition of the matrix element uJJ' into  irreducible parts is described by expressions that for- × ()JJ+ ''()11 j ∑ Ð1 2J''+ 1 (20) mally coincide with (19), (20), and (22)Ð(24) when the JJ' J'' J'' radial matrix elements with a single Green’s function

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1010 BOLGOVA et al. are replaced by a matrix element with two Green’s tiplet. Substituting the numerical values for the coeffi- functions, cients of the vector summation into (26), we obtain the following expressions for three independent compo- 11 () 〈|(), |〉 RL'J'' S = nLSJ rgL'J'' rr' r' nLSJ' nents of the hyperpolarizability: 101 () () RL'J'', L'J'' S γ ()0 8 2LLÐ 1 1111 () nLS = ------()------RL Ð 1,,L Ð 2 L Ð 1 S 〈|(), (), |〉 52L + 1 2L Ð 1 = nLSJ rgL'J'' rr' gL'J'' r' r'' r'' nLSJ' . 2 Similarly, a fourth-order diagonal matrix element L()4L + 1 1111 4LL()+ 1 1111 + ------R , ()S + ------R , () S can be expressed in terms of the hyperpolarizability 2 L Ð 1 L, L Ð 1 L Ð 1 L, L + 1 4L Ð 1 2L + 1 components of an atomic level that generally has three (27) ()j 2 γ ()L + 1 ()4L ++8L 5 1111 independent irreducible parts nLSJ , j = 0, 2, 4 [17, 30, () + ------RL + 1, L, L + 1 S 32]. The irreducible parts of the hyperpolarizability ()2L + 1 ()2L + 3 represent a linear combination of fourth-order radial ()() 1111 2 L + 1 L + 2 1111 () matrix elements R ,, (S). The explicit form of + ------RL + 1,,L + 2 L + 1 S , L1 J1 L2 J2 L3 J3 2L + 3 these expressions is rather awkward, but they can be substantially simplified by neglecting the effect of the () γ ()2 8L 4 L Ð 1 1111 () fine structure on the radial matrix element. In this nLS = Ð------()------RL Ð 1,,L Ð 2 L Ð 1 S approximation, the matrix elements (21) do not depend 72L + 1 2L Ð 1 on the indices of the total momenta J1, J2, …, JN. As a 2 result, the dependence of the matrix elements on the 8L Ð56L + 1111 + ------RL Ð 1, L, L Ð 1()S total angular momentum of the initial, intermediate, 4L2 Ð 1 and final states is expressed only in terms of the coef- 2 ficients of vector summation, which can be performed 82()L ++2L 3 1111 + ------R , ()S (28) analytically, similar to the summation over J'' in (20) ()2L + 1 ()2L + 3 L Ð 1 L, L + 1 [34]. Then, a fourth-order matrix element can be repre- sented as 2 ()2L Ð 1 ()8L ++22L 19 1111 + ------R , ()S ()()2 L + 1 L, L + 1 ()4 1 JM ()J' ++LS 2L + 1 2L + 3 wJJ' = Ð------∑ CJ'Mj0 Ð1 24 ()() j = 024,, 4 L + 2 2L Ð 1 1111 () (25) + ------2 RL + 1,,L + 2 L + 1 S , 1/2 ()2L + 3 ()2L + 1 ()2J'1+ () × J + 1 LLj γ j ------()- nLS, 2L + 1 Ð j j J' JS ()4 48LL()Ð 1 1111 γ = Ð------R ,,()S - nLS ()2 L Ð 1 L Ð 2 L Ð 1 where 35 4L Ð 1 ()()1/2 ()j 2L + 1 Ð j J + 1 2 j + 1 2L Ð 3 1111 γ + ------R , ()S NLS = 24 ------() 2L + 1 L Ð 1 L, L Ð 1 2L + 2 j

j 0 j 0 22()L Ð 3 ()2L Ð 1 1111 1 2 j0 () × ∑ ()2 j + 1 ()2 j + 1 C C C + ------RL Ð 1, L, L + 1 S (29) 1 2 1010 1010 j10 j20 ()2L + 1 ()2L + 3 j1 j2 2 ()2L Ð 1 ()2L Ð 3 1111  + ------R , ()S × ()LL j ()()2 L + 1 L, L + 1 ∑ 2L2 + 1  (26) 2L + 1 2L + 3 j1 j2 L2 L2 2 ()2L Ð 1 ()2L Ð 3 1111 + ------R ,,()S .  2 L + 1 L + 2 L + 1 L10 L10 L30 L30 LL j ()2L + 3 ()2L + 5 × ∑ C C C C 2 1 L010 L2010 L2010 L010 11L1 L1 L3 Thus, in the approximation used, all fourth-order matrix elements in the basis of the multiplet states are  expressed in terms of three different components of the LL2 j2 1111 × R ,,()S L1 L2 L3 hyperpolarizability tensor (27)Ð(29) that depend only 11L3 on the principal, spin, and orbital quantum numbers. ()4 is the irreducible part of the hyperpolarizability of the The dependence of wJJ' on J and J' is described by the |nLS 〉 level that is independent of the total momentum coefficients of vector summation in (25). As a rule, this J and is identical for all components of the atomic mul- approximation is fairly sufficient because the fourth-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 HIGHER ORDERS OF PERTURBATION THEORY FOR THE STARK EFFECT 1011 order matrix element determines only the quadratic (in Table 1. Coefficients of singlet–triplet mixing for a helium- field) correction to the main quantity. like atom If we also neglect the effect of the fine structure on the second- and third-order matrix elements, we can Term AL BL also obtain expressions analogous to (25) for these 1L cosϕ sinϕ quantities: L nL nL 3 ϕ ϕ LL Ðsin nL cos nL ()2 1 JM J' ++LS 3 () LL Ð 1 01 wJJ' = Ð--- ∑ CJ'Mj0 Ð1 2 3 j = 02, LL + 1 01 (30) 1/2 ()2L + 1 ()2J'1+ () × J + 1 LLj α j ------()- nLS, 2L + 1 Ð j j J' JS the scalar and tensor components of the polarizability of a helium-like atom in these two representations are α()0 ≡ αs α()2 ≡ αt related by the following simple formulas: where nLS nLS and nLS nLS are the scalar and αs()α13, s()13, tensor polarizabilities of the level, respectively. Both of n LJ n L , these quantities can be represented as a linear combina- (33) αt()α1 t()1 tion of second-order radial matrix elements: n LL n L , αt()α3 t()3 αs 2 []11 ()11 n LL + 1 n L , (34) nLS = ------LRL Ð 1()S + L + 1 RL + 1()S , (31) 32()L + 1 2 t 3 L + L Ð 3 t 3 α ()n L ------α ()n L , (35) L LL()+ 1 αt 2L 11 2L Ð 1 11 nLS = Ð------RL Ð 1()S +.------RL + 1()S (32) 32()L + 1 2L + 3 αt()3 n LL Ð 1

()L Ð 1 ()L + 1 ()2L Ð 3 ()2L + 3 t 3 (36) In this case, the scalar polarizability appears only in the ------α ()n L . expressions for the diagonal matrix elements and deter- L2()2L Ð 1 ()2L + 1 mines the energy shift that is identical for all sublevels of the multiplet, while the tensor polarizability deter- Similar formulas for the scalar and tensor components mines the splitting into magnetic components, different of hyperpolarizabilities are more cumbersome and are for different sublevels, and appears in both the diagonal not presented here to save space. and the off-diagonal matrix elements. An expression for The above formulas do not take into account the sin- the third-order matrix element uJJ' is obtained from (30) gletÐtriplet mixing of excited states of a helium atom. by replacing α(j) β(j), where β(j) is a superposition The contribution of these phenomena is quite signifi- of third-order radial matrix elements, which is given by cant for the states with L ≥ 3; its correct consideration relations (31) and (32) in which one should make the is required first for the comparative analysis of theoret- 11 101 ical data with the results of precision measurements. following substitution: RL' (S) RL'L' (S). For the quantitative and qualitative analysis of these Using the definition of the Green’s function (9), one phenomena, one should use an intermediate scheme for can easily verify that the maximal contribution to the the coupling of angular momenta; in this case, the wave radial matrix elements is made by the levels n' = n. For functions for the atomic multiplet components with J = 3 example, in [35], it was shown that, for 1s3p PJ levels L are determined by the superposition of states in a of helium, the contribution of resonance terms amounts “pure” LS coupling scheme of angular momenta to about 97%. For circular states with n = L + 1, the 1 3 11 |〉 |〉 |〉 nLSJ = AL n L + BL n L . (37) dominant correction with n' = n is absent in RL + 1 , and, 11 Ӷ 11 hence, the inequality RL + 1 RL Ð 1 holds for the radial The coefficients AL and BL in (37) are calculated by the matrix elements in (31) and (32). Under this condition, method of diagonalization of the energy matrix with we obtain the approximate equality αs ≈ Ðαt for the cir- regard to relativistic interactions (first of all, a spin– cular states. orbit interaction) in the total Hamiltonian of an atom Note also that relation (30) for an off-diagonal (Table 1). matrix element actually determines a transition from For a helium atom, the numerical values of the mix- ϕ ° ° the LS scheme of coupling of angular momenta to the J ing angles nL for L = 1, 2, 3, 4 are equal to 0.02 , 0.5 , representation for the second-order scalar and tensor 30°, and 44°, respectively, and virtually do not depend components of electric susceptibilities of a multielec- on the principal quantum number of a valence electron tron atom. This transition was first analyzed in [36]. [37]. Taking into account the singletÐtriplet mixing The comparison of (22)Ð(24) with (30)Ð(32) shows that alters formulas (23) and (24); then, according to the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1012 BOLGOVA et al. definition (37), the components of the polarizability In other words, the singletÐtriplet mixing is determined tensor are expressed as by

2 3 t 3 t 1 s 2 ϕ α ()α() α = ---∑Φ()JJ,,,' LL' , (38) sin nF --- F Ð.F (43) 3 4 L'J' For instance, for n = 4, this phenomenon is abnormally 1/2 t 5J()2J Ð 1 high (20%); for n = 5, its contribution is no higher than α = 4 ------62()J + 3 ()J + 1 1%; and, again, it amounts to several percent for n > 5. (39) Thus, up to the fourth order in the field strength, the  JJ+ ' matrix element (17) is determined by relations (25)Ð × ∑()Ð1 112 Φ()JJ,,,' LL' . JJJ' (32), which are sufficiently universal for all multiplet L'J' levels and can be used for calculating the variation of Here, energies of specific atoms in a field with regard to the fine-structure effects in an atomic multiplet. Φ()JJ,,,' LL' 4. THE STARK EFFECT ON TRIPLET STATES () AL AL' 11 () OF HELIUM = 2J'1+ L> ------RL'J' 0 (40) ()2L + 1 ()2L'1+ 3 In this section, we calculate the energy of n P0, 2 states of helium up to the fourth-order corrections with 2 respect to field F neglecting the singletÐtriplet mixing LJ1 11 () Ð BL BL'RL'J' 1 , in the spectrum of intermediate states. The fourth-order J' L'1 corrections should be taken into consideration for the field corresponding to the anticrossing region of the δ and L> is the maximum of L and L'. triplet sublevels where the energy difference 02 = E0 Ð As a result, the use of the intermediate coupling E2 is minimal. The attraction of sublevels to each other scheme of angular momenta requires that one should in a weak field and their subsequent repulsion are take into account the singletÐtriplet mixing not only in observed when the polarizability of the upper sublevel the wave functions of the considered states with L ≥ 3 is greater than that of the lower one. This situation is 3 but also in the spectrum of intermediate states. Note characteristic of n PJ states of helium (n = 2, 3, ...) with that the weak dependence of the coefficients AL and BL J = 0, 2 and the projection of the total momentum M = on the principal quantum number n allows one to apply 0. The state with J = 1 and M = 0 remains isolated since the method of Green’s functions for summing over the there is no nonzero matrix element for the dipole elec- whole set of intermediate states. tric transition to a state with a different value of the total 3 | | For example, for a tensor component of a singlet 1D momentum J. The sublevels n PJ with M = J = 2 also level of helium, one can easily obtain an approximate remain isolated. The “repulsion” of interacting levels is formula that takes into account the mixing phenomena attributed to the off-diagonal matrix element of the for the F component of the intermediate spectrum operator Vˆ (r) that makes a positive contribution to the energy of the upper level and a negative contribution to αt()α1 ≈ t()1 nd D2 nd D the energy of the lower level. Note that the states with |M| = 1 and J = 1, 2 of the multiplets considered have E 1 Ð E 1 (41) 4 11()2 ϕ n F n D identical polarizabilities and hyperpolarizabilities; Ð ------R3 0 sin nF 1 Ð.------35 E 3 Ð E 1 therefore, they immediately start to diverge, as the field n F n D strength increases, due to the interaction between sub- A numerical calculation has shown that the contribu- levels, which is determined by the off-diagonal matrix tion of the mixing phenomena is no greater than 0.1% element. for arbitrary n. The case of a tensor component of the 1 F3 level is more interesting since, on the one hand, 4.1. Field Dependence for Energies there are experimental data [38] for this case and, on the of Two Interacting Sublevels other hand, this case exhibits strong dependence of the mixing amplitude on the principal quantum number n: To calculate the Stark splitting of two close-lying ε sublevels with relative energy shifts = E1 Ð E = αt()α1 ≈ t()1 J1 nf F3 nf F −δ ε δ /2 and = E2 Ð E = /2, where E = (E1 + E2)/2, (42) J2 2 3 t 3 t 1 ∆ + sin ϕ ---α ()αnf F Ð.()nf F one has to solve the secular equation (16) for E. To nF 4 determine the corrections to the energy up to the fourth

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 HIGHER ORDERS OF PERTURBATION THEORY FOR THE STARK EFFECT 1013 order in F, we substitute expression (17) for the matrix order of magnitude in the neighborhood of the anti- element WJJ' into Eq. (16), which can be rewritten as crossing of levels as those taken into account in the diagonal elements. c c det 11 12 = 0, (44) c21 c22 4.2. Polarizabilities and Hyperpolarizabilities of Triplet Levels of Helium where Using (46), we determine the field dependence of δ ∆ 2 ()2 2∆ 4 ()4 the fine-structure splitting for the sublevels of triplet nP c11 = Ð--- Ð E ++F wJ J F EuJ J +F wJ J , 2 1 1 1 1 1 1 states with the total momenta J1 = 2 (the lower sublevel) () () and J2 = 0 (the upper sublevel) and the projections M = c ==c F2w 2 ++F2∆Eu F4w 4 , (45) 12 21 J1 J2 J1 J2 J1 J2 0 onto the field direction. According to (25) and (30), the matrix elements are expressed in terms of irreduc- δ () () c = --- Ð ∆E ++F2w 2 F2∆Eu +F4w 4 . ible parts as 22 2 J2 J2 J2 J2 J2 J2 ()2 1αs 1βs To determine ∆E from (44) up to the fourth-order w00 ==Ð--- 3 , u00 Ð--- 3 , n P n P terms in the field, one should retain the terms up to the 2 2 sixth order in the equation: ()4 1 γ ()0 w00 = Ð------3 , ~(∆E)2F2 ~ ∆EF2δ ~ (∆E)2δ ~ ∆Eδ2 ~ F2δ2 ~ F4δ ~ F6. 24 n P Solving quadratic equation (44) for the field-depen- ()2 1()αs αt w22 = Ð--- 3 Ð 3 , dent splitting of two interacting sublevels of the multip- 2 n P n P let up to the fourth-order terms, we can obtain an expression of the form 1()βs βt u22 = Ð--- 3 Ð 3 , 2 n P n P 2 + Ð  F (48) δ() ∆ ∆ δ () () () () F ==E Ð E  1 + ----- uJ J + uJ J 4 1 ()γ 0 γ 2 2 1 1 2 2 w22 = Ð------3 Ð 3 ,  24 n P n P

2 2 () () F ()2 ()2 2 2 2 1 t ()() w ==w Ð------α 3 , + ----- wJ J Ð wJ J + F gJ Ð gJ 20 02 n P δ 2 2 1 1 2 1 2 (46) 2 () () ()u + u 1 t 4F 2 2 4 2 J1 J1 J2 J2 β u20 ==u02 Ð------3 , + ------wJ J + F wJ J + wJ J ------n P δ2 1 2  1 2 1 2 2 2

1/2 ()2 ()2 2 ()4 ()4 1 γ ()2 w + w  w20 ==w02 Ð------3 . J J J J  n P + u ------1 1 2 -2  , 12 2 J1 J2 2   The same irreducible parts of polarizabilities and where hyperpolarizabilities enter into similar expressions for the matrix elements that determine the field-depen- () () g ≡ w 4 + w 2 u (47) dence of the splitting of states n 3P Ð n 3P with the pro- Ji Ji Ji Ji Ji Ji Ji 1 2 jection |M| = 1 of the total momentum: is the hyperpolarizability of a sublevel with the total momentum Ji. Note that the states with the total ()2 ()2 1s 1 t w ==w Ð--- α 3 Ð ---α 3 , 11 22  momenta J1 and J2 are mixed due to the field, so that the 2 n P 2 n P total momentum is no longer an integral of . In the expression analogous to (46) that was 1βs 1βt u11 ==u22 Ð--- 3 Ð --- 3 , obtained in [30] and in which the fourth-order phenom- 2n P 2 n P ena were taken into account only in the diagonal matrix elements, all corrections to the off-diagonal matrix ele- ()4 ()4 1 γ ()0 1γ ()2 w11 ==w22 Ð------3 Ð --- 3 , ment (the term proportional to F2 in the second set of 24n P 2 n P square brackets under the sign of the radical) were (49) absent. The interaction of levels was not taken into ()2 ()2 3αt w21 ==w12 Ð--- 3 , account when calculating the hyperpolarizabilities of 4 n P individual sublevels in diagonal elements; this resulted 3βt in other combinations of products of matrix elements u21 ==u12 Ð--- 3 , n P (2) 4 w and u entering into gJ' and led to the absence of the second term in the first set of square brackets in (46). ()4 ()4 1 3 w ==w Ð------γ ()n P . Calculations show that the above terms have the same 21 12 16 2

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Table 2. Scalar and tensor polarizabilities αs, αt, βs, and βt and hyperpolarizabilities γ(0) and γ(2) for the n 3P state of a helium atom (in au)

State αs αt βs βt γ(0) γ(2)

2 3P 47.95 74.86 4.556(3) Ð3.086(3) 7.797(6) Ð3.427(6) 3 3P 17 255.6 374.0 7.935(6) Ð1.031(6) 1.142(11) Ð1.911(10) 4 3P 1.697(5) 1.664(3) 1.863(8) Ð2.409(7) 1.314(14) Ð1.921(13) 5 3P 8.976(5) 7.264(3) 1.936(9) Ð2.477(8) 8.895(15) Ð1.292(15) Note: The number in parentheses indicates a power of ten: a(k) ≡ a × 10k.

δ Using (27)Ð(29), (31), and (32), we represent the irre- well as the fine-structure splitting (Fa) in this field, ducible parts of polarizabilities and hyperpolarizabili- may serve as quantitative characteristics for determin- ties entering into these expressions in terms of radial ing the components of the polarizability and hyperpo- matrix elements as larizability of the states considered. αs 2[]11() 11() To calculate the radial matrix elements in a one- 3 = --- R 0 1 + 2R2 1 , n P 9 electron approximation, we applied the Green’s func- (50) tion method to the Fues model potential [16, 17]. The αt 2 11() 1 11() numerical values of susceptibilities α, β, and γ for the 3 = Ð --- R0 1 +,---R2 1 n P 9 5 triplet nP states of helium atoms (n = 2, 3, 4, 5), obtained for the parameters of the model potential cho- γ ()0 8 ( 1111() 1111() sen in [25], are shown in Table 2. 3 = ------25 R 010 1 + 40R012 1 n P 225 (51) These parameters were used for calculating the field 1111() 1111()) δ 3 +34R212 1 + 36R232 1 , dependence of the splitting 02(F) between the n, P2 3 and n, P0 sublevels of the fine structure with M = 0. The γ ()2 8 ( 1111() 1111() results are shown in Fig. 1 for the states with (a) n = 3 3 = Ð ------175 R010 1 + 280R012 1 n P 1575 (52) and (b) n = 5. This figure also represents the following () 1111() 1111()) field dependences calculated by (46): δ 2 (F), which +49R212 1 + 36R232 1 . 02 takes into account only the quadratic (in the field) cor- Formulas (51) and (52) show that, when calculating the δ()4 scalar and tensor components of hyperpolarizabilities, rections to the matrix elements, and 02 (F), which one has to take into account the F levels in the spectrum takes into account the fourth-order corrections together of intermediate states (in the radial matrix element with the quadratic ones. For all values of the field 1111 strength that are presented in the figures, the inequality R (1)) that are characterized by significant singlet– 232 δ()4 δ()2 ϕ ≈ ° 02 (F) < 02 (F) holds. triplet mixing, nF 30 . However, just as in the esti- mates of the polarizabilities of the D levels given above, 3 γ(0) For 2 P states, the contribution of the fourth-order the numerical contribution of these phenomena to corrections is very small; it amounts to less than 0.1% γ(2) ≤ and is small ( 0.1%). of the second-order corrections even near the anticross- Formulas (46) and (48) show that the sublevels of ing. For 3 3P states, the fourth-order corrections near the state n 3P with J = 0, 2 and M = 0 become closer to the anticrossing amount to 5% of the second-order cor- each other in a weak electric field if the tensor polariz- rections; for 4 3P states, 10%; and for 5 3P, more than αt 15%. Here, the fourth-order corrections to the off-diag- ability 3 is positive. Then, the leading (first) term in n P onal elements constitute more than half of the total the expression under the radical sign in (46) decreases fourth-order corrections. as F increases. It continues decreasing until it becomes equal to the increasing second term, which is propor- In these calculations, we used the data available in 4 tional to F for F = Fa. When F > Fa, the levels start to the literature (see, for example, [39–42]) for the fine- δ 3 3 diverge, which corresponds to the anticrossing of lev- structure splitting of triplet states n P0Ðn P2 of a free els. As one can see from relations (49), the sublevels atom. The numerical values of these quantities are with J = 1, 2 and |M| = 1 are repulsed starting from shown in Table 3. This table also presents the numerical δ F = 0 because the upper and lower levels have identical values of the splitting 02(Fa) at the anticrossing point polarizabilities (the hyperpolarizabilities of these levels of levels, as well as the corresponding values of the are also identical). Thus, the anticrossing field Fa, as field strength Fa.

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3 3 Exact estimates for the contribution of continuum E(3 P0) – E(3 P2), MHz were obtained in [35] when calculating the polarizabil- 8800 3 ities of helium. For instance, for 1snp P0, 2 (n = 2) lev- els, this contribution to the scalar and tensor polariz- (a) abilities amounted to 23% and 3%, respectively, while, 3 8600 for 1snp P0, 2 (n > 2) levels, it was no greater than 1%. To verify the results presented above, we also applied the method of diagonalization of the energy 8400 matrix that contains dipole matrix elements of the first order with regard to all possible fine-structure sublevels for states with n ≤ 6, n ≤ 5, and n ≤ 4. For the matrix ele- 8200 ments between states with n ≤ 4 and L = 0Ð3, we used 0 50 100 150 200 250 3 3 the data of precision relativistic calculations based on E(5 P0) – E(5 P2), MHz the method of configuration interaction, which involves 1800 the Coulomb and the Breit interaction operators in the (b) total atomic Hamiltonian [25, 30]. In this case, one-par- ticle basis orbitals used in the method of configuration 1750 interaction included s, p, d, f, and g partial waves with the use of a spline approximation for each wave. The relative accuracy of calculating the matrix elements was 10Ð4Ð10Ð5 [25, 30]. For other states, the data of non- 1700 relativistic variational calculations given in [43] were used. 1650 Figure 2 represents the F dependence of the splitting 0 5 10 15 20 δ 3 3 02 between n P0 and n P2 sublevels of the fine struc- F, kV/cm ture with M = 0 as a function of the dimension of the δ Fig. 1. Fine-structure splitting 02 of sublevels with the basis set in the original energy matrix. The total number 3 of matrix elements for n ≤ 6 with regard to all fine-struc- momenta J = 0 and J = 2 of the triplet state 1snp PJ of a 3 helium atom as a function of the electric-field strength; (a) ture components is greater than 10 . Figures 2a and 2b n = 3 and (b) n = 5. The dashed curve corresponds to calcu- 3 3 show that, for the energy differences E(2 P0) Ð E(2 P2) lations with regard to the quadratic (in field) corrections to 3 3 the matrix elements, and the solid curve corresponds to cal- and E(3 P0) Ð E(3 P2), the result is virtually indepen- culations with regard to the second- and fourth-order terms. dent of the dimension of the basis and agrees with the numerical data presented in Table 3 to within a few per- cent. This fact may provide a basis for the applicability the energy matrix with a substantially greater number of the resonance approximation discussed in Section 3. of basis elements. However, for highly excited states of helium-like atoms with n ≥ 4, the resonance approximation is not suffi- Thus, the application of the Green’s function for- cient (Fig. 2c). This fact requires the diagonalization of malism to calculating the Stark effect allows one to

3 Table 3. Numerical values of the splitting between the fine-structure sublevels of the n PJ state with the total momenta J = 0, 2 for a free atom (δ) and at the anticrossing point of levels with the projection M = 0, as well as the corresponding values of the field strength Fa at the anticrossing point

δ δ()2 ()()2 ()2 δ()4 ()()4 ()4 State , MHz 20 Fa , MHzFa , kV/cm20 Fa , MHzFa , kV/cm

2 3P 31908 30081.6 617.05 30080.3 617.2 3 3P 8772.5 8243.8 146.31 8216.2 146.91 4 3P 3576.8 3348.7 44.82 3323.8 47.47 5 3P 1797.4 1687.6 15.07 1672.2 15.53

δ()2 ()()2 Note: The quantity 20 Fa is calculated with regard to the quadratic corrections alone to the matrix elements and the relativistic cor- δ()4 ()()2 rections to the difference between the scalar components of polarizabilities [25], while 20 Fa is calculated with regard to the fourth-order corrections.

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3 3 E(2 P0) – E(2 P2), MHz data (obtained without taking into account field correc- 32000 tions ~F4) is attributed to the contribution of relativistic corrections to the difference between scalar compo- (a) nents of the polarizabilities of the atomic multiplet. 31500 However, the results of exact relativistic calculations of 3 the polarizabilities of helium levels 1s3p P0, 2 (M = 0) 31000 [25, 30] did not confirm this assumption; moreover, the contribution of relativistic corrections proved to be of 30500 opposite sign to the expected one; i.e., the consideration of these corrections led to a greater discrepancy δth 30000 between theory and experiment: 02 = 8244 MHz [25, 0 200 400 600 800 1000 30]. This fact stimulated calculations for the Stark 3 3 effect with regard to higher orders of perturbation the- E(3 P0) – E(3 P2), MHz δ ory (hyperpolarizabilities) whose contributions to 02 8800 are rather substantial for strong fields (on the order of (b) 100Ð200 kV/cm) at the points of anticrossing of levels (Fig. 1). In [25, 30], together with relativistic correc- 8600 tions to the polarizability and the resonance hyperpo- larizability, the authors also calculated nonresonance corrections to the hyperpolarizability of states; in this 8400 case, the field corrections ~F4 only to the diagonal matrix elements of the secular equation (16) were taken into account. To control the accuracy and reliability of 8200 calculations, two different methods were used in these 50 100 150 200 250 works, a method of summation of relativistic forces of E(4 3P ) – E(4 3P ), MHz oscillators for a finite set of discrete states of the inter- 0 2 mediate spectrum and a method using a Green’s func- 3700 tion for the model potential of Fues. These calculations (c) yielded 8231 MHz and 8234 MHz, respectively [30]. 3600 The results of calculations carried out with regard to the full set of corrections ~F4 both to the diagonal and off- diagonal matrix elements of Eq. (16) (see Table 3) also 3500 do not agree with the results of measurements pre- sented above. 3400 Thus, we should admit that there is no satisfactory δ agreement between theoretical calculations of 02 and 3300 the corresponding experimental data for the levels with 0 1020 30 40 50 60 70 n = 3 [26]. This fact makes topical new measurements F, kV/cm of the anticrossing parameters for the triplet levels of δ helium in a static field. The details of the future experi- Fig. 2. Fine-structure splitting 02 of sublevels with the 3 mental investigations of anticrossing for singlet and momenta J = 0 and J = 2 of the triplet state 1snp PJ of a triplet levels of helium are described in [44, 45]. helium atom as a function of the electric-field strength and the dimension of the energy matrix. The dashed curve cor- responds to calculations with regard to the basis set of dis- 5. CONCLUSIONS crete states with n ≤ 4, the dotted curve, with n ≤ 5, and the solid curve, with n ≤ 6. This paper has been initiated first of all by the neces- sity to give a theoretical interpretation to the results of precision measurements of the fine-structure intervals obtain exact values of electric susceptibilities for both in helium atoms (the method of anticrossing of levels in the ground and the highly excited states by a simple and an external field with the use of high-resolution laser rational method and to analyze the F dependence in the spectroscopy [26]). The measurement error is ±5 MHz; spectra of helium-like atomic systems on the basis of however, the application of microwave techniques the results obtained. combined with the methods of laser cooling of atomic For triplet states of helium with n = 3, experimental beams will allow one to reduce this error significantly measurements of δ at the anticrossing point of levels in the nearest future [44]. An adequate interpretation of 02 such measurements requires that one should take into δexp ± yield 02 = 8257 5 MHz [26]. According to [26], the consideration not only the quadratic Stark effect but discrepancy between the experimental and theoretical also the contributions of higher orders of perturbation

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 HIGHER ORDERS OF PERTURBATION THEORY FOR THE STARK EFFECT 1017 theory associated with nonlinear corrections to the 7. N. Hoe, B. d’Elat, and G. Couland, Phys. Lett. A 85, 327 dipole moment of an atom induced by an external field. (1981). In this paper, we have formulated an approach to the 8. R. D. Damburg and V. V. Kolosov, in Rydberg States of calculation of the Stark effect of higher orders in the Atoms and Molecules, Ed. by R. F. Stebbings and F. B. Dunning (Cambridge Univ. Press, Cambridge, spectra of helium-like atomic systems. For the first 1983; Mir, Moscow, 1985). time, we have investigated analytically and numerically 9. A. A. Kamenski and V. D. Ovsiannikov, J. Phys. B 33, a full set of fourth-order corrections with respect to an 491 (2000); J. Phys. B 33, 5543 (2000). external field and with regard to the fine-structure phe- 10. A. A. Kamenskiœ and V. D. Ovsyannikov, Zh. Éksp. Teor. nomena. We used the Fues model potential method Fiz. 120, 52 (2001) [JETP 93, 43 (2001)]. [16, 17] as a basis for the numerical calculations; the 3 11. M. Bellermann, T. Bergeman, A. Haffmans, et al., Phys. parameters of this method for metastable S states were Rev. A 46, 5836 (1992). chosen according to [25]. The summation of intermedi- 12. R. T. Hawkins, W. T. Hill, F. V. Kovalski, et al., Phys. ate states over the whole spectrum was based on the Rev. A 15, 967 (1977). Green’s function method for a model potential. This 13. M. G. Littman, M. L. Zimmerman, T. W. Ducas, et al., fact has allowed us to take into account the contribution Phys. Rev. Lett. 36, 788 (1976). of discrete states and continuum and to simplify and 14. T. F. Gallagher, L. M. Humphrey, R. M. Hill, et al., Phys. unify the calculations, guaranteeing a reliable control Rev. A 15, 1937 (1977). of accuracy in all stages of theoretical analysis. We 15. C. Fabre, S. Haroche, and P. Goy, Phys. Rev. A 18, 229 have presented theoretical results for the scalar and ten- (1978). sor components of the second- and fourth-order suscep- 3 16. L. P. Rapoport, B. A. Zon, and N. L. Manakov, Theory of tibilities for helium levels 1snp P0, 2 (n = 2Ð5), deter- Multiphoton Processes in Atoms (Atomizdat, Moscow, mined the values of the electric-field strengths and the 1978). differences between energy levels at the anticrossing 17. N. L. Manakov, V. D. Ovsiannikov, and L. P. Rapoport, point, and analyzed the relative contributions of the Phys. Rep. 141, 319 (1986). field corrections. 18. V. A. Davydkin and B. A. Zon, Opt. Spektrosk. 52, 600 The theoretical approach presented in this work is (1982) [Opt. Spectrosc. 52, 359 (1982)]. sufficiently universal and allows one not only to obtain 19. W. R. Johnson and K. T. Cheng, Phys. Rev. A 53, 1375 a full set of fourth-order field corrections in the spec- (1996). trum of a helium-like atom but also to outline a method 20. V. G. Pal’chikov and V. P. Shevelko, Reference Data on for systematic calculations of the Stark effect in higher Multicharged Ions (Springer, Berlin, 1995). orders of perturbation theory that are of interest from 21. K. Pachucki and J. Sapirstein, Phys. Rev. A 63, 012504 the viewpoint of modern experiment. (2001). 22. A. K. Bhatia and R. J. Drachman, Phys. Rev. A 58, 4470 (1998). ACKNOWLEDGMENTS 23. D. Normand, G. Petite, and J. Morellec, Phys. Lett. A 65, This work was supported by the Russian Foundation 290 (1978). for Basic Research (project no. 01-02-97-013P), by the 24. V. A. Davydkin and V. D. Ovsiannikov, J. Phys. B 17, German Research Society (grant no. 96-02-00257, 436 L207 (1984). RUS 113/164/O(R,S)), by the US Civil Research and 25. A. Derevyanko, W. R. Johnson, V. D. Ovsyannikov, Development Foundation (US CRDF), and by the Min- et al., Zh. Éksp. Teor. Fiz. 115, 494 (1999) [JETP 88, istry of Education of the Russian Federation (grant 272 (1999)]. no. VZ-010-0). 26. R. Schumann, M. Dammasch, U. 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34. D. A. Varshalovich, A. N. Moskalev, and V. K. Kherson- 41. P. B. Kramer and F. M. Pipkin, Phys. Rev. A 18, 212 skiœ, Quantum Theory of Angular Momentum (Nauka, (1978). Leningrad, 1975; World Sci., Singapore, 1988). 42. D. H. Yang, P. McNicholl, and H. Metcalf, Phys. Rev. A 35. A. I. Magunov, V. D. Ovsiannikov, V. G. Pal’chikov, 33, 1725 (1986). et al., in The Hydrogen Atom, Ed. by S. G. Karshenboim, F. S. Pavone, G. F. Bassani, et al. (Springer, Berlin, 2001), 43. G. W. F. Drake, in Handbook of Atomic, Molecular, and p. 753. Optical Physics, Ed. by G. W. F. Drake (AIP Press, New 36. J. P. P. Angel and P. G. H. Sandars, Proc. R. Soc. London, York, 1996), Chap. 11. Ser. A 305, 125 (1968). 44. R. Schumann, C. Schubert, U. Eichmann, et al., Phys. 37. E. S. Chang, Phys. Rev. A 35, 2777 (1987). Rev. A 59, 2120 (1999). 38. A. S. Aynacioglu, G. von Oppen, W. D. Perschmann, and 45. O. Reusch, C. Dieste, S. Garnica, and G. von Oppen, D. Szostak, Z. Phys. A 303, 97 (1981). J. Phys. B 34, 2145 (2001). 39. T. A. Miller and R. S. Freund, Phys. Rev. A 4, 81 (1971). 40. T. A. Miller and R. S. Freund, Phys. Rev. A 5, 588 (1972). Translated by I. Nikitin

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1019Ð1036. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1160Ð1178. Original Russian Text Copyright © 2003 by Sazonov, Sobolevskiœ. ATOMS, SPECTRA, RADIATION

On Nonlinear Propagation of Extremely Short Pulses in Optically Uniaxial Media S. V. Sazonov* and A. F. Sobolevskiœ Kaliningrad State University, Kaliningrad, 236041 Russia *e-mail: [email protected] Received February 13, 2002; in final form, February 4, 2003

Abstract—Nonlinear wave equations describing the propagation of optical pulses of duration up to a period of electromagnetic oscillations in transparent media with uniaxial optical anisotropy are derived on the basis of a quantum-mechanical model of material response. The electron and electron-vibrational nonlinearities, electron and dispersion, and are taken into account. It is shown that the inclusion of the electron response alone leads to a system of two constitutive equations for the ordinary and extraordinary polarization compo- nents. When a pulse propagates across the optical axis, this system is reduced to an inhomogeneous model of the HenonÐHeiles type and, hence, generalizes the Lorentz classical electron model. In order to take into account stimulated Raman scattering (SRS) processes, an anisotropic analog of the BloembergenÐShen quan- tum-mechanical model taking into account the population dynamics of SRS sublevels is obtained. The genera- tion of an extraordinary wave video pulse with the help of the high-frequency ordinary component in the ZakharovÐBenney resonance mode is investigated. Some analytic soliton-like solutions in the form of propa- gating bound states of ordinary and extraordinary video pulses corresponding to different birefringence modes are considered and their stability to self-focusing is analyzed. © 2003 MAIK “Nauka/Interperiodica”.

ω ωτ ӷ 1. INTRODUCTION ably smaller than its carrier frequency ; i.e., p 1. The effects of nonlinear propagation of ESP in isotropic In recent years, studies devoted to the interaction of media have been investigated quite thoroughly [8Ð26]. In light pulses of duration up to a period of electromag- view of the existence of a symmetry center in such netic oscillations (video pulses or extremely short media, the expansion of polarization P into a power pulses (ESP)) with matter have become very popular. series in the electric field E contains only odd powers This interest stems mainly from the generation of ESP of the field. Such an expansion can be carried out if the in laboratory conditions [1Ð6]. The absolute duration τ p spectrum of a pulse belongs to the region of optical of such pulses varies from hundreds of [6] to several [7] transparency of the medium, i.e., under the conditions femtoseconds. The term “extremely short pulses” (or [10, 13, 22Ð26] “ultimately short pulses”) is being used more and more widely (see reviews [4, 5, 8] and article [9]), although ()ω ω Ð1 Ӷ ()ω δω Ð1 Ӷ the terminology has not settled as yet; some authors 0/ 1, 0/ 1, (1) also use the terms “ultrashort” or “supershort” pulses in ω their communications and papers. The latter terms where 0 is the characteristic resonance frequency of appear to be slightly confusing since they are some- the medium and ω is the frequency corresponding to times applied to quasimonochromatic pulses to empha- the center of the pulse spectrum. size their short duration in absolute meaning. The term “few-cycle pulses” is also used quite often. The latter The meaning of conditions (1) is that the frequen- term is employed especially frequently in foreign liter- cies of the Fourier components of the pulse lie much ature [3]. The term “video pulses” is also encountered lower than the frequencies of resonance electron-opti- sometimes. cal absorption. In the case of quasimonochromatic ω ≈ ω δω Ӷ ω Since an ESP contains approximately one period of pulses, we have , . In this case, the first ω ω Ð1 Ӷ oscillations, its spectrum is so broad that the concept of condition in (1) has the form ( 0/ ) 1, while the carrier frequency loses its meaning. For obvious rea- second is satisfied automatically. In the other limiting ω τ Ӷ sons, the standard approximation of slowly varying case p 1 corresponding to video pulses amplitudes and phases (SVAP) from the optics of ω Ӷ δω ∼ τ quasimonochromatic pulses is inapplicable for theoret- (1/ p ), it is sufficient to require that the ical investigations of the interaction between ESP and second condition, which can be written in the form ω τ Ð1 Ӷ matter. The quasimonochromaticity condition implies ( 0 p) 1, is satisfied. In the general case, the fulfill- δω τ that the spectral width ~ 1/ p of a pulse is consider- ment of both conditions in (1) is required.

1063-7761/03/9606-1019 $24.00 © 2003 MAIK “Nauka/Interperiodica”

1020 SAZONOV, SOBOLEVSKIŒ η y uniaxial single crystals, organic molecular structures, z and so on [36]. In the case of single crystals, we must generally specify a symmetry class [36, 37]. In such cases, it is difficult to carry out appropriate quantum- ϕ mechanical calculations to take into account the above- ζ mentioned types of material response. Uniaxial crystals x include crystals of the tetragonal, trigonal, and hexa- gonal systems with the same structure of linear suscep- χ χ ≠ tibility tensor reduced to the principal axes: xx = yy χ zz [38]. In the optical transparency spectral range, the main contribution to the polarization response of the Fig. 1. Geometry of propagation of an ESP in a uniaxial medium is determined by linear effects; in this case, birefringent medium; ζ is the optical axis; the pulse propa- nonlinearity plays the role of a perturbing factor. Since gates along the z axis at an angle ϕ to the ζ axis. The ordi- the structure of the linear susceptibility tensor is insen- nary component is polarized in the xz plane normal to the sitive to a change in the symmetry class of uniaxial plane of the figure, while the extraordinary wave is polar- crystals and nonlinearity is weak under conditions (1), ized in the plane of the figure along the y axis. we can disregard the dependence of nonlinear suscepti- bility on the symmetry class. In this connection, the By virtue of this condition, the ESP spectrum con- quantum-mechanical model proposed by us combines tains no resonance Fourier components; for this reason, the optical properties of all media with natural uniaxial the interaction with the medium is relatively weak [12]. anisotropy in their spectral transparency region. On the basis of this material model and Maxwell equations, we Anisotropy of the medium is a necessary condition can arrive at nonlinear wave equations describing the for the presence of even powers of electric field in the ESP dynamics in the spectral transparency range of above-mentioned expansion. In the general case, the uniaxial media and carry out their analysis; this forms electromagnetic wave in such a medium is not com- the subject of the present paper. pletely transverse. In a uniaxial medium, there exist two preferred directions (along and across the optical axis) along which the wave field is strictly transverse [27]. In 2. ELECTRON AND ION RESPONSES the former case, nonlinearities of only odd orders are left [28, 29], while, in the latter case, even powers Let an electromagnetic pulse propagate in a uniaxial appear, and quadratic nonlinearity plays the major role medium along the z axis at an angle ϕ to the optical axis [30]. When a pulse propagates at right angles to the (Fig. 1). The ordinary component Eo of the electric field optical axis, we must generally take into account two of the ESP is perpendicular to the plane of the figure components of its electric field, ordinary Eo and and is parallel to the x axis, while the extraordinary extraordinary E , and assume that nonlinear suscepti- e component Ee lies in this plane and is normal to the z bilities are tensor quantities. Some authors employ a axis (parallel to the y axis). scalar model, taking into account only one electric field component of the electric field of the ESP [31–33]. It The evolution of the state of the medium is should be noted that quadratic nonlinearity does not described by the equations for the elements of the den- appear in media with induced anisotropy [34, 35] in sity matrix ρˆ : view of the weakness of applied fields (electric, mag- netic, and deformational) as compared to intra-atomic ∂ρµν i fields. ------= Ð iωµνρµν + ---V µν()ρµµ Ð ρνν ∂t ប The theoretical models proposed in [28, 30, 31, 33, (2) 35] are phenomenological by nature. These models are i ()ρ ρ based on the expansion of P(E) in the presence of non- Ð ប--- ∑ V µj jν Ð µjV jν , linear susceptibility tensor of rank two and/or three j ≠ µν, [30, 35], or an anisotropic oscillator with a cubic or quadratic nonlinearity is proposed for the material where ប is Planck’s constant, Vµν is the matrix element model [28]. Under conditions (1), both approaches lead of the Hamiltonian of the electric-dipole interaction to identical systems of nonlinear wave equations for between the pulse and the field, and ωµν is the fre- ordinary Eo and extraordinary Ee pulse components. quency of the quantum transition µ ν . The sub- Here, we propose a simple quantum-mechanical scripts in Eq. (2) run through the values µ, ν = 1, 2, 3, model of an optically uniaxial medium, including the …, K, where K is the total number of electron quantum electron, electron-vibrational, and ion responses. We levels formed by the strong intrinsic field and partici- assume that anisotropy is natural, i.e., formed by a pating in the interaction with the ESP field (the value of strong intrinsic electric field in which both electrons K ≥ 3 is regarded as arbitrary). Equations for diagonal and ions are located. Such properties are typical of elements can be obtained from Eq. (2) for µ = ν.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 ON NONLINEAR PROPAGATION OF EXTREMELY SHORT PULSES 1021 η ζ x y z T x T In a strong internal electric field of the medium, where d' = (,dµν dµν , dµν ) and d = (dµν , dµν , dµν ) , electron levels are not degenerate in the modulus of the we obtain total angular momentum component M [39]. As a result, π (∆M = 0) and σ (∆M = ±1) transitions, which are allowed in accordance with the electric-dipole x dµν dµν = ------∆Mµν , selection rules, are formed in the electron subsystem. 2 2 In view of the uniaxial anisotropy, []Hˆ , Mˆ differs y dµν 0 dµν = i------∆Mµνcosϕ Ð Dµν()ϕ1 Ð ∆Mµν sin , (4) from zero (Hˆ 0 is the intrinsic Hamiltonian of electrons 2 in the internal electric field of the medium and Mˆ is the z dµν dµν = i------∆Mµνsinϕ + Dµν()ϕ1 Ð ∆Mµν cos . angular momentum operator). At the same time, 2 []Hˆ 0, Mˆ ζ = 0, where Mˆ ζ is the operator of the angular momentum component along the ζ axis, which has Accordingly, for the matrix elements of the interaction eigenvalues equal to M. For this reason, the wave func- ˆ tion of an optical electron in an axisymmetric field can Hamiltonian V , we have be written in the form V µν = Ðdµν ⋅ Er(), t ψ ζ, ()ϕ µM = Rµ()r exp iM , (3) ∆ ϕ dµν∆ ϕ = Dµν()1 Ð Mµν sin Ð i------Mµνcos Ee 2 where r, ϕ, and ζ are the components of the cylindrical ζ (5) system of coordinates (the axis of axial symmetry dµν ∆ coincides with the optical axis) and µ is the set of quan- Ð ------Mµν Eo tum numbers corresponding to the cylindrical symme- 2 try. dµν ()ϕ ∆ ∆ ϕ Ð D µν 1 Ð Mµν cos + i------Mµνsin Ez. In addition, we choose the Cartesian system of coor- 2 dinates x, η, ζ turned through angle ϕ relative to the x, y, z system in the (yz) plane (see Fig. 1). Using the ψ In the axisymmetric field, quantum levels do not above expression for µM, we obtain the Cartesian possess any definite . This circumstance deter- components of vector dµν of the dipole moment of the mines the selection rules for electric-dipole transitions µ ν η ζ transition in the coordinate system x, , : in the strong field of the medium. In contrast to an iso- tropic medium, a much larger number of transitions are allowed in such a medium due to mixing of states with x dµν ∆ η dµν∆ dµν ==------Mµν , dµν i------Mµν, various parities with certain quantum levels. This is the 2 2 reason for the emergence of quadratic and all other even nonlinearities. The internal field of the medium is ζ dµν = Dµν()1 Ð ∆Mµν , strong and is able to form initially the electron quantum levels. For this reason, we assume that conditions (1) are satisfied for all (including forbidden) electron-opti- ∆ ≡ ± where Mµν Mµ Ð Mν = 0, 1. The maximal values for cal transitions, except those between vibrational sub- dipole moments for σ (ϕ = 0) and π (ϕ = π/2) transitions levels of the ground state (see below). The electric field can be written in the form E of the pulse depends on coordinates r and time t. This dependence presumes the fulfillment of conditions (1). ∞ ∞ In this case, system (2) can be solved by the method of π 2 , , successive approximations in small parameters (in the dµν = Ð 2 er∫ drR∫ µ()zrRν()zrdz, sense of relations (1)), which are proportional to the 0 Ð∞ time derivatives in relation (2), as well as the summands of the sum appearing on the right-hand side of this ∞ ∞ equation. In this case, in the zero-order approximation, Dµν = Ð2πerrd zRµ()zr, Rν()zr, dz, we obtain from Eq. (2) ∫ ∫ 0 Ð∞ V µν()ρµν Ð ρνν ρµν = ------. where e is the elementary charge. បωµν Carrying out the above-mentioned transformation Substituting this expression into the terms omitted ear- of rotation through angle ϕ around the x axis, d' = Lˆ d, lier, we obtain ρµν in the first approximation, and so on.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1022 SAZONOV, SOBOLEVSKIŒ χ χ χ As a result, in the second approximation, we have where e, z, and ez can be expressed in terms of the principal diagonal components χ⊥ and χ|| of the linear ρνν Ð ρµµ ρµν = Ð------V µν instantaneous susceptibility tensor, បωµν χ χ 2 ϕ χ 2 ϕ e = ⊥ cos + || sin , 1 ()α α χ χ 2 ϕ χ 2 ϕ (11) + ------∑ νλ Ð λµ V µλV λν∫ z = ⊥ sin + || cos , បωµν λµν≠ , χ ()ϕϕχ χ ez = ⊥ Ð || sin cos ,

2 V λν dµν ∆Mµν + ------()αηµV µλV ηλ Ð αληV µηV ηλ (6) χ () បω ∑ ⊥ = N ∑ ------W ν Ð Wµ , µλ បωµν ηµν≠ , µν≠ (12) ∆ V µλ 2 1 Ð Mµν χ|| = 2NDµν------()W ν Ð Wµ , + ------∑ ()αηλV ληV ην Ð ανηV ληV ην ∑ បω បωνλ µν ηλν≠ , µν≠ and nonzero components of the nonlinear susceptibility 2 ανµ∂V µν ανµ∂ V µν tensor of rank two are given by Ð i------+ ------, ω ∂ 2 2 ()2 ()2 ()2 ()2 µν t ωµν ∂t χ χ χ χ eo ===xxy xyx yxx where αµν ≡ (Wν Ð Wµ)/បωµν, Wµ being the initial occu- 4N Dµν (13) pancy of the µth level. In expression (6), we included = ------sinϕ ∑ ------∑ dµλdνλ()αµλ Ð ανλ , ប ωµν the nonlinearity of not higher than third order and took µν≠ λµν≠ , into account the fact that, under conditions (1), the ()2 ()2 2N 3 nonlinearity and dispersion (the last two terms in rela- χ ==χ ------sin ϕ tion (6)) behave as additive quantities [12]. e yyy ប

Substituting relation (6) into the right-hand side of (14) Dµν Eq. (2), taking into account the Hermitian nature of × ∑ ------∑ DµλDνλ()αµλ Ð ανλ . ωµν operator Vˆ , and integrating, we obtain the following µν≠ λµν≠ , expressions for the diagonal elements of ρˆ : Cumbersome expressions for the third-order nonlin- χ()3 χ()3 χ()3 ear susceptibilities o , e , and eo are given in the Wµ Ð Wλ 2 ρµµ = Wµ Ð ∑ ------V µλ . (7) Appendix. ប2ω2 λµ≠ λµ It was noted above that, owing to condition (1), the frequencies of the Fourier components of the spectrum We obtain the values of the polarization components are far from the resonance frequencies of the medium. corresponding to the electric field components Eo, Ee, Consequently, for χ(2) and χ(3), the zero-dispersion and Ez of the pulse with the help of relations approximation is realized, for which the Kleiman rule [36] holds. According to this rule, the components of ()xyz,, nonlinear susceptibility tensors are invariant to any P ,, = Ndµν ρµν* + c.c., oez ∑ transposition of their indices. µν, While deriving Eqs. (8) and (9), we took into where N is the concentration of valence electrons, and account the above-mentioned degeneracy of electron using relations (4)Ð(7). After cumbersome but simple states in the modulus of M. The real parts of the terms transformations, we obtain in Eq. (6), containing the first derivatives with respect to time, correspond to σ transitions (see also Eq. (5)). χ χ()2 χ()3 3 Po = ⊥Eo ++2 eo EeEo o Eo In the course of summation in the expressions for polar- 2 ization components, terms with ∆M = ±1 and with iden- () ∂ E (8) χ 3 2 κ o tical values of ωµν mutually cancel out on account of + eo Ee Eo Ð ⊥------, ∂t2 degeneracy. The imaginary parts of the terms in ques- tion, containing ∆Mµν in π and σ transitions (see χ χ χ()2 2 χ()2 2 Pe = eEe ++zEz e Ee +eo Eo Eqs. (5) and (6)), also cancel out during the summation ∂2 (9) with complex conjugate quantities in the expressions χ()3 3 χ()3 2 κ Ee for components of P. + e Ee + eo EoEe Ð e------, ∂t2 It should be noted in connection with the latter remarks that the application of a constant external mag- χ χ Pz = zEx + ezEe, (10) netic field removes the degeneracy in modulus of M.

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In this case, doublets are formed from each degenerate Expansions (8) and (9) can be reversed and written ∆ ± ω level, for which M = 1 and the frequencies µν of the in the form of a system of differential equations for Po corresponding transitions are not identical any longer. and Pe with the right-hand sides depending on the elec- As a result, the right-hand sides of expansions of type (8) tric field of the pulse. In the zero-order approximation, χ and (9) acquire the first time derivatives of E [29], it follows from relations (8) and (9) that Eo = Po/ ⊥ and χ which leads to Faraday’s rotation of the polarization Ee = Pe/ e. Substituting these relations into the next plane of the pulse. However, in our case, there is no terms of expansions (8) and (9), we obtain external magnetic field, and the internal crystal electric 2 field removes degeneracy only in the modulus of M. ∂ () () Po ω2 β 2 β 3 3 ------+ oPo Ð 2 oe PePo Ð o Po The origin of the second time derivatives in Eqs. (8) ∂t2 (16) and (9) can be explained most easily by using the β()3 2 ω2χ dependence of linear electron susceptibilities on fre- Ð oe Pe Po = o ⊥Eo, quency ω: 2 ∂ P 2 ()2 2 ()2 2 2 e ω β β ω ------+ e Pe Ð e Pe Ð eo Po χ ω χ eo, ∂t2 (17) e, ⊥()= e, ⊥------2 -2. ω , Ð ω eo β()3 3 β()3 2 ω2χ Ð e Pe Ð eo PoPe = e eEe. For ω/ω Ӷ 1 (see relations (1)), we have the expan- e, o Here, sion ()2 2 χ⊥ 2 χ ()2 χ 2 ω ω e β eo ω o ==κ------, e κ-----, oe =χ------, χ , ⊥()ω ≈ χ , ⊥1 + ------. ⊥ e ek⊥ e e ω2 eo, ()2 ()2 ()3 () χ χ () χ () χ β 2 ===------eo -e, β 2 ------e , β 3 ------o -, ω ∂ ∂ eo 2 e χ κ o 2 Using the substitution i / t, we arrive at the sec- χ κ e e χ κ ond time derivatives in relations (8) and (9). Dispersion ⊥ e ⊥ ⊥ was disregarded in expression (10) since P Ӷ P , P , ()3 ()3 ()3 z e o () χ () χ () χ while dispersion in approximation (1) is an effect with β 3 ==------e -, β 3 ------eo , β 3 =------eo . e χ2κ oe χ2κ eo χ2 κ a higher order of smallness than the linear instanta- e e e ⊥ ⊥ e neous response. This system generalizes the Lorentz classical aniso- κ κ The dispersion parameters ⊥ and e in Eqs. (8) and tropic model [41] to the case when the nonlinearity of (9), which take into account the weak (in the sense of the electron response is taken into account. It should be conditions (1)) inertia of the electron response, can be noted that this system has been derived here on the χ χ written in terms of ⊥ and e, respectively, in the form basis of quantum-mechanical concepts concerning the medium by using the low-frequency resonance approx- 2 2 ω ω ∂ χ ∂ χ imation (1). In this case, eigenfrequencies o and e are κ 1 ⊥ κ 1 e ⊥ ==---------, e ---------determined by the entire set of electron quantum levels 2∂ω2 2∂ω2 ω ω = 0 ω = 0 (see relations (11), (12), (15) and expressions for o ω and e). In addition, it should be noted that frequency and express the extent of dependence of linear suscep- ω ϕ e generally depends (although weakly) on angle . tibilities on frequency ω. In explicit form, we have ω Ӷ ω ω Obviously, in the nonresonant case, when o, e 2 2 (conditions (1) are satisfied), it is practically impossible κ = κ⊥ cos ϕ + κ|| sin ϕ, e to single out a quantum transition which interacts with 2 the field most strongly. This also explains the collective dµν ∆Mµν κ⊥ = N ∑ ------αµν, nature of frequencies ω and ω . Usually, ω , ω ~ ω2 o e o e µν≠ µν (15) 1016 sÐ1 [24]. Consequently, frequencies ω of the visible ω Ӷ ω ω range can easily satisfy the condition o, e (see 2 1 Ð ∆Mµν κ|| = 2ND∑ µν------αµν. also conditions (1)). ω2 µν≠ µν The system of equations (16), (17), as well as expan- sions (8)Ð(10), is invariant to substitutions Po ÐPo Usually, |χ⊥ − χ||| Ӷ χ⊥, χ|| [40]. In this case, it can be and Eo ÐEo, but is not invariant to transformations seen from relations (11) that χ Ӷ χ⊥, χ . This circum- ez e Pe ÐPe and Ee ÐEe. The validity of this state- stance is responsible for the relative smallness of the ment can easily be explained by the axial symmetry of Ӷ longitudinal electric field component of the ESP: Ez the medium: reflections in planes normal to the optical Eo, Ee (see Section 4). For this reason, only the linear axis are symmetry transformations, while reflections in local response to component Ez is taken into account in planes parallel to this axis are not (see Fig. 1). The only expansions (8)Ð(10). exception are reflections perpendicular to the optical

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1024 SAZONOV, SOBOLEVSKIŒ

j Eqs. (16), (17) a system of constitutive equations of the type of the generalized HenonÐHeiles model, ∂2 Po ω2 β()2 ω2χ ------+ oPo Ð 2 oe PePo = o ⊥Eo, (18) ∂t2 ∂2 Pe ω2 β2 ()2 β()2 2 ω2χ ------+ || Pe Ð e Pe Ð eo Po = || ||Ee, (19) ∂t2

where ω|| = χ||/κ||. It should be noted that the HenonÐHeiles system permits both random and regular dynamics depending on the relation between the coefficients [42]. 1'' Ionic (vibrational) degrees of freedom, correspond- ω ⊥ v 1' ing to the optical branch of vibrations of atomic nuclei ωv|| at crystal lattice sites, may considerably affect the type 1 of dispersion [24]. The characteristic frequencies of 13 Ð1 these vibrations are wi ~ 10 s . In this case, for the Fig. 2. Diagrams of transitions taking into account the SRS τ Ð14 Ð15 time scale p ~ 10 Ð10 s of the ESP for ordinary and processes; 1' and 1'' are the SRS sublevels in the vicinity of ()i ()i the ground electron state in an optically uniaxial medium; extraordinary ionic polarizations P and P , we have ω ω o e v⊥ and v|| are the frequencies of normal optical modes of ω τ Ӷ 1. ion vibrations across and along the optical axis, respec- i p tively; j are upper-lying electron levels. In this approximation, we can neglect ionic anhar- monism as well as elastic retrieving forces in the oscil- lator equations of motion [24]. This gives axis and corresponding to ϕ = 0. This circumstance 2 ()i 2 2 2 ∂ ω Poe, pEoe, explains the absence of nonlinearities of type Po , Pe ------= ------, (20) 2 π 2 ∂t 4 and Po Pe in Eq. (16) and nonlinearities of type PoPe ω 2 where p is the ionic frequency [43]. and Pe Po in Eq. (17). Let us consider two particular cases. 1. Let us assume that ϕ = 0. It follows from rela- 3. DESCRIPTION OF SRS PROCESSES tions (11)Ð(15), formulas of the Appendix, and expres- WITH THE HELP OF THE GENERALIZED BLOEMBERGENÐSHEN sions for ω , ω , and nonlinear constants that χ = χ⊥, o e e MODEL ω ω β()2 β()2 β()2 β()3 β()3 β()3 o = e, e = oe = eo = 0, and o = e = oe = Let us now consider the stimulated Raman scatter- () β 3 β(3) eo = . In this case, Eqs. (16) and (17) can be writ- ing (SRS) processes corresponding to electron-vibra- ten in the form of the Duffing equation for complex tional nonlinearity. In view of the axial symmetry, SRS- ≡ active centers have at least two normal vibrational polarization P Po + iPe: modes with frequencies ωv|| and ωv⊥ along and across 2 the optical axis, respectively. We will first derive a ∂ P 2 ()3 2 2 ω β ω χ⊥ closed system of equations for the matrix elements of ------2- + oP Ð P P = o E, ∂t ρˆ , corresponding to the vibrational sublevels of the ground electron state (Fig. 2), eliminating the remain- where E = Eo + Ee. ing elements in the framework of the adiabatic approx- Here, the division of polarization into ordinary and imation (1). The subscripts of the corresponding matrix extraordinary components Po and Pe is completely arbi- elements will be labeled by Latin letters. In this case, in trary since both components behave as ordinary com- accordance with Eq. (2) (j, k = 1, 1', 1''), we write ponents. Using the operation of rotation around the ∂ρ optical axis, we can equate to zero one of the compo- jk i ------= Ðiω ρ Ð --- ∑ ()V λρλκ Ð ρ λV λ . (21) nents. In this case, we obtain for polarization the con- ∂t jk jk ប j j k ventional Duffing model, which is often used as the λ > 1'' constitutive equation for isotropic dielectrics [15]. Here, we have taken into account the fact that electron 2. Let us now suppose that ϕ = π/2. In this case, the transitions between the sublevels in question are forbid- quadratic nonlinearity plays a dominating role. Disre- den. Assuming in Eq. (21) that j = k, we obtain equa- garding the cubic nonlinearity, we obtain from tions for diagonal elements of ρˆ .

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 ON NONLINEAR PROPAGATION OF EXTREMELY SHORT PULSES 1025 ρ λ We can derive expressions for λj ( > 1'', j = 1, 1', to the description of SRS processes in an optically ∂ρ ∂ 1'') from Eq. (2) for λj/ t = 0 (see relations (1)): uniaxial medium. It should be noted in this connection that system (23) is completely equivalent to the equa- 1'' tions for the for a three-level atom in ρ 1 ρ λj = Ð------បω ∑ V λk kj which all three transitions excited by the square of the λj k = 1 field are allowed. It is well known that the Bloember- (22) genÐShen model [44, 45] has the corresponding anal- 1 ()ρ ρ ogy with a two-level atom. Ð ------∑ V λµ µj Ð λµV µj . បωλ jµ > ω ω 13 Ð1 ω τ Ӷ 1'' Since v⊥, v|| ~ 10 s , we have v⊥(||) p 1. Con- The terms appearing in parentheses in this equation sequently, we can disregard the first term on the right- have been taken into account in the above analysis. hand side of Eq. (23) in the zeroth approximation and These terms contribute to the electron polarization and, write this equation in the following symbolic form: hence, can be omitted here. Substituting Eq. (22) into ∂Rˆ Eq. (21) and taking into account the above remark, we ------= iG[]ˆ , Rˆ . obtain ∂t ∂ R jk ω []ˆ , ˆ Σˆ ------∂ = Ð i jkR jk + iGR jk, (23) Matrix commutes with itself at various instants. t In this case, the solution to this operator equation can be where elements of matrix Gˆ have the form written in the form + Rˆ ()t ==Uˆ Rˆ ()Ð∞ Uˆ , Uˆ exp()iθˆ /3 . (25) ∑ V jλV λl បÐ2λ > 1'' G jl = ------ω , Here, λ1 θˆ = θΣ()ˆ Ð ˆI , and matrix Rˆ contains only those elements of ρˆ which correspond to the three lower quantum levels 1, 1', and t 3 2 2 (26) 1'' (see Fig. 2): θ = ------()χ E + χ⊥E dt'. 4បN ∫ e e o Ð∞ ρ ρ ρ 1''1'' 1''1' 1''1 Rˆ = ρ ρ ρ . Obviously operators Σˆ and ˆI commute with each 1'1'' 1'1' 1'1 other; consequently, we have ρ ρ ρ 11'' 11' 11 iθ iΣˆ θ Here, we have disregarded the difference in frequencies Uˆ = expÐ----- exp------. ω λ 3 3 λj of electron-optical transitions ( > 1'', j = 1, 1', 1'') ω ≈ ω for different values of j, setting λj λ1. It can be seen ω ω ω ω Σˆ 2 Σˆ Σˆ 3 2 Σˆ Σˆ k from the figure that 1''1 = v⊥ and 1'1 = v||. Since = (K + 1) , = (K + 1) , …, = (K + In our case, sublevels 1' and 1'' have exclusively 1)k Ð 1 Σˆ , …, the series corresponding to the exponential vibrational origin; for this reason, they have identical in the last expression can be easily summed. As a result, angular momentum components Mj (j = 1, 1', 1'') along neglecting an insignificant C-number factor exp(Ðiθ/3), with the first level. For the sake of simplicity, we we obtain assume that Mj = 0. Following the approach developed ≈ by Bloembergen and Shen [44], we assume that Dλj Σˆ 2 θ ≈ Uˆ = ˆI Ð --- 2sin --- Ð isinθ . (27) Dλ1 and dλj dλ1 in view of the small spacing between 32 the vibrational sublevels. Taking into account the above arguments, using relation (5), and neglecting compo- Matrix Rˆ ()Ð∞ is diagonal. The corresponding occu- ˆ nent Ez, we obtain for matrix G pancies of the electron state and of the SRS sublevels are W , W , and W . Using expressions (25)Ð(27), we 1 2 2 1 1' 1'' Gˆ = ------()Σχ E + χ⊥E ()ˆ Ð ˆI , (24) obtain the following expression for j ≠ k: 4បN e e o θ ˆ Σˆ 1 ()2 ()θ where I is a unit operator and is matrix all of whose Rkj = --- 23Wl Ð 1 sin --- Ð,3iWk Ð W j sin elements are equal to unity. 9 2 (28) We can assume that the system of equations (23) ljk≠ , . derived by us together with relation (24) generalizes the quantum-mechanical BloembergenÐShen model [45] The expression for the electron contribution P(ev) to

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1026 SAZONOV, SOBOLEVSKIŒ polarization has the form ering that E2 ≈ (4π/c)I, where I is the ESP intensity, we obtain 1'' ()ev * ρ P = Nv ∑ ∑ dλj λj + c.c. χ Iτ θ ∼ 3π------p. λ > 1'' j = 1 N បc 1'' ≈ * ρ Nv ∑ dλ1 ∑ λj + c.c., Taking polarizability χ/N ~ 10Ð26 cm3, I ~ 1013 W/cm2, λ > 1'' j = 1 τ Ð14 θ Ð2 Ð3 Ӷ and p ~ 10 s, we have ~ 10 Ð10 1. In these cases, we can set sinθ ≈ θ in Eq. (30). Then, the expres- where Nv is the concentration of molecules contributing sion for P for ϕ = 0 assumes the form derived in [24] for to SRS. an isotropic dielectric disregarding the variation of Substituting Eq. (22) into this relation and subtract- occupancies in SRS sublevels. This estimate shows that ing the second sum taking into account the electron the dynamics of populations of SRS sublevels can polarization considered above, we obtain indeed be neglected for intensities of 1013 W/cm2 and for ESP durations of 10 fs. However, for I ~ 1015 W/cm2 and τ ~ 100 fs, we must take into account the change ()ev dλ*1V λ1 p P = ÐNv ∑ ------∑ Rkj + c.c. (29) in occupancies since θ ~ 1 in this case. បωλ λ > 1 > 1'' kj Let us estimate the relative contribution to the polar- It can easily be seen from relation (5) that the factors ization from the electron cubic nonlinearity and the in front of the second sum in Eq. (29) are real valued for SRS nonlinearity. The above estimate, relations (8), (9), χ(3) both the ordinary and the extraordinary components. (12), (30), and the expressions for (see the Appen- Using relation (28), we obtain dix) imply that

2 4 4 2 3 () () Nd ω τ E ∑ Rkj +0.c.c. = χ ∼ Nd χ 3 ∼ Nd ev ∼ v p ------, ------3, P ------3 2 kj> បω ()បω ប ω 0 0 0 ω τ In the first approximation in parameter v⊥(||) p, we ≈ obtain from Eq. (21) (here, Nv N and d is the characteristic value of the dipole moments of the electron-optical transitions). ∂ Then, the sought ratio is ∑ R + c.c. = i∑ω ()R* Ð R . ∂t kj kj kj kj > > ()ev ω kj kj P v 2 ------∼ ------()ω τ , ()3 ω 0 p This relation together with Eq. (29) gives P 0

∂ 2 where P(3) is the third-order nonlinear electron polar- ∑ R + c.c. = ---∑ω ()θW Ð W sin . ω 13 Ð1 ω 16 Ð1 τ ∂t kj 3 kj j k ization. Setting v ~ 10 s , 0 ~ 10 s , and p ~ kj> kj> 10−15 s, we obtain P(ev)/P(3) ~ 0.1; i.e., the contribution of SRS can be disregarded as compared to the cubic Integrating this expression with respect to t, substitut- electron nonlinearity for an ESP duration of several ing the result into Eq. (29), and using relation (5), we femtoseconds. This estimate is in accordance with the obtain data given in [24].

t A similar estimate shows that ()ev Nv ω χ θ Peo, = ------v eo, Eeo, ∫ sin dt', (30) 3N ()ev ()τ 2 Ð∞ P wv p I ------∼ d------. ()2 បω P v c where Ð19 ω τ 2 ω 13 Ð1 ω ω ()ω() For d ~ 10 CGSE units, ( v p) ~ 0.1, v ~ 10 s , v = v || W1 Ð W1' + v⊥ W1 Ð W1'' and I ~ 1013 W/cm2, we obtain P(ev)/P(2) ~ 0.1. Consid- ()ω ω () + v⊥ Ð v || W1' Ð W1'' . ering that the quadratic nonlinearity plays the major role in the electron response for propagation across the Let us estimate the value of the quantity θ appearing optical axis, we arrive at the conclusion that this nonlin- in the argument of the sine in expression (30). Consid- earity in such geometry also dominates over the SRS

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 ON NONLINEAR PROPAGATION OF EXTREMELY SHORT PULSES 1027 mechanism for pulse durations on the order of a few In this case, we obtain the following equation in the tens of femtoseconds. x and y components from Eq. (31): Let us now estimate the relative dispersion contribu- 2 2 2 2 ∂ E , n , ∂ E , 4π ∂ (i) oe ------oe------oe ------tion of the ionic P and electronic P degrees of free------2 Ð 2 2 = 2 2 dom. From relations (8), (9), (15), and (20), we have ∂z c ∂t c ∂t 2 () ∂ E , () () ()i ω2 × nl κ oe i ev ∆ P 4 Poe, Ð ⊥, e------++Poe, Poe, + ⊥Eoe, , ∼ p ()ω τ ∂ 2 ------ω------ω 0 p , t P 0 c πχ where no, e = 14+ ⊥, e are the refractive indices of ω π 2 ប where c = 4 d N/ is the cooperation frequency. the ordinary and extraordinary waves and ∆⊥ is the ω ω 12 13 Ð1 ω 16 Ð1 transverse Laplacian. Setting p ~ c ~ 10 Ð10 s , 0 ~ 10 s , and τ Ð15 (i) The right-hand sides of these equations describe p ~ 10 s, we obtain the following estimate: P /P ~ 1Ð10. It can be seen that this ratio strongly depends on nonlinearity, dispersion, and diffraction (which are parameter ω τ . effects with a higher order of smallness as compared to 0 p the instantaneous linear response appearing on the left-hand sides) through renormalization of the rates 4. NONLINEAR WAVE EQUATIONS of field components with the help of refractive indices no and ne . In order to analyze the self-consistent dynamics of This circumstance allows us to use the slowly vary- pulses and the medium, the relations derived above for ing profile approximation in the comoving frame of ref- material responses should be supplemented with the erence [46]. In accordance with this method, we can Maxwell equations assume that the right-hand sides are equal to zero in the zeroth approximation in nonlinearity and dispersion Σ 1 ∂2E 4π∂2P and assume that the two components propagate in only ∆ ∇∇()⋅ ------τ E Ð E Ð 2 2 = 2 2 , (31) one direction (along the z axis). Then, Eo, e = Eo, e( o, e), c ∂t c ∂t τ where o, e = t Ð no, ez/c. In the first approximation, the Σ effect of the right-hand sides will be taken into account where the total polarization P = P + P(i) + P(ev). by introducing the “slow” coordinate z' = εz in addition τ Henceforth, we will assume that diffraction effects to o, e in the arguments of Eo and Ee [46]: Eo, e = τ ε Ӷ are weak, considering that the pulse field components Eo, e( o, e; z'), where 1 is a small dimensionless depend mainly on z and t. It was mentioned above that parameter taking into account the effect of the right- the longitudinal field component is much smaller than hand sides of the latter equations. Passing from vari- τ the transverse component. For this reason, we can dis- ables t and z to variables o, e and z' in the equations for regard the derivatives with respect to transverse compo- Eo and Ee, we can write nents in the equation for E , taking them into account z ∂ ∂ ∂ ∂ ∂ for components E and E only. Integrating twice the noe, ε 0 e ∂ ==∂τ , ∂ Ð------∂τ + ∂ , ()i t oe, z c oe, z' z component of Eq. (31) and considering that Pz = ()ev 2 2 2 2 π ∂ n , ∂ 2εn , ∂ Pz = 0, we obtain Ez = Ð4 Pz. This expression ≈ oe oe ------2 ------2 ------2 Ð ------, together with Eq. (10) gives ∂ ∂τ c ∂τ , ∂z' z c oe, oe πχ where we have neglected the term of the order of ε2 in 4 ezEe Ez = Ð------πχ-. the last relation. This allows us to integrate the wave 14+ z τ τ equations for Eo and Ee with respect to o and e, respec- tively, in view of the fact that the field of the pulse and In order to write equations for Eo and Ee, we first all its derivatives tend to zero at infinity: represent components Po and Pe of the electronic part of 2n , ε∂E , 4π ∂ polarization in the form Ð------oe------oe = ------∂ 2 ∂τ c z' c oe, ∂2 τ Eoe, ()nl 2 oe, P , = χ⊥, E , Ð κ⊥, ------+ P , , () ∂ E , () () oe e oe e ∂ 2 oe × nl κ oe i ev ∆ τ' t Poe, Ð ⊥, e------2 ++Poe, Poe, + ⊥ ∫ Eoe, d oe, . ∂τ , oe Ð∞ ()nl where Poe, are the parts of the electron polarization Returning to the initial variables t and z and using containing quadratic and cubic nonlinearities. formulas (8), (9), (20), and (29), we arrive at the follow-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1028 SAZONOV, SOBOLEVSKIŒ ing system of nonlinear wave equations for the ordinary system (32), (33) is transformed into the following and extraordinary components of the pulse: equation in E = Eo + iEe: ∂ ∂ ∂()∂()2 τ Eo no Eo EeEo Ee Eo 3 ∂E ∂ 2 ∂ E ------∂ - ++------∂ - a2------∂ - +a3------∂ ------++a ()δE E Ð ------σ Edτ' z c t t t ∂z 3∂τ o ∂τ3 ∫ t Ð∞ ∂E ∂  τ τ 2 o Λ θ  + b3oEo------∂ - + o∂ Eo ∫ sin dt' (32) ∂ c t t + Λ E sinθτd ' = ------∆⊥ Edτ'; Ð∞ o∂τ ∫ ∫ 2no Ð∞ Ð∞ ∂3 t t δ Eo σ c ∆ τ Ð o------3- + ∫ Eodt' = ------⊥ ∫ Eodt', here, = t Ð noz/c. ∂t 2no Ð∞ Ð∞ Disregarding the dynamics of population of SRS ∂E n ∂E ∂E ∂E sublevels, we obtain ------e ++----e ------e a E ------o +b E ------e ∂ ∂ 2 o ∂ 2e e ∂ τ' z c t t t 3χ sinθθ≈ = ------o- E 2dτ''. ∂()2 ∂ t 4បN ∫ EoEe 2 Ee ∂ + a ------++b E ------Λ E sinθdt' Ð∞ 3 ∂t 3e e ∂t e∂te ∫ (33) Ð∞ In this case, the last equation exactly coincides with the t t one derived in [24] for the ESP propagation in isotropic ∂3 δ Ee σ c ∆ dielectrics. Ð e------3- + ∫ Eedt' = ------⊥ ∫ Eedt'. ∂ 2no ϕ π t ∞ ∞ Let us now suppose that = /2. It was mentioned Ð Ð in the previous section that we can neglect the elec- Here, tronic cubic and electron-vibrational nonlinearities for 13 2 τ πχ()2 πχ()2 πχ()3 pulses of intensity I ~ 10 W/cm and of duration p ~ 4 eo 4 e 2 eo χ a2 ===------, b2e ------, a3 ------, 1Ð100 fs. In addition, ez = 0 (see formula (11)). In this noc noc noc case, Eqs. (32), (33) assume the form () () πχ 3 πχ 3 π ω χ 3 6 e 6 o 2 Nv v o ∂ n ∂E ∂()E E ∂ E b ===------, b ------, Λ ------, Eo o o e o δ o 3e 3o o ------++------a2------Ð o------noc noc 3Nnoc ∂z c ∂t ∂t ∂t3 π ω χ πκ t t (34) Λ 2 Nv v e δ 2 ⊥ e ==------, o ------, σ c ∆ 3Nnoc noc + ∫ Eodt' = ------⊥ ∫ Eodt', 2no Ð∞ Ð∞ 2πκ ω2 δ ==------e, σ ------p , ∂ ∂ ∂ ∂ e Ee ne Ee Eo Ee noc 2noc ------++------a E ------+b E ------∂z c ∂t 2 o ∂t 2e e ∂t and the dynamic parameter θ is defined by formula (26). t t (35) ∂3 The system of nonlinear wave equations (32), (33) Ee c Ð δ ------+ σ E dt' = ------∆⊥ E dt'. describes the propagation of an ESP in a uniaxial e 3 ∫ e ∫ e ∂t 2no medium at an arbitrary angle to the optical axis. These Ð∞ Ð∞ equations take into account the electron and electron- It should be noted that, if a pulse polarized in the vibrational nonlinearities as well as the electron and ion plane of the principle cross section (i.e., in the plane dispersions. The terms on the right-hand sides describe formed by the optical axis and the direction of pulse the pulse diffraction. propagation) enters the medium, it can be seen from It should be noted that the SVP approximation is not Eqs. (34) and (35) that Eo = 0. In an anisotropic connected in any way with the approximation of slowly medium, only the extraordinary component of the ESP varying envelopes [47], which is traditional for quasi- can propagate. If, in addition, we disregard ionic dis- monochromatic pulse optics, where the reduction in persion, the one-dimensional dynamics of this compo- derivatives is attained due to the fact that the field enve- nent will be described by the KortewegÐde Vries equa- lope covers a large number of electromagnetic oscilla- tion, which has soliton video-pulse solutions among tions. System (32), (33) is written not for envelopes other solutions. However, the input signals required for (which cannot be introduced for ESP), but directly for obtaining such solutions must also be in the form of the electric field components Eo and Ee. video pulses, which can subsequently split into several If the signal propagates along the optical axis (ϕ = solitons. A more interesting case when ESP (or video χ 0), coefficients a2, b2e, and ez vanish; in this case, ne = pulses) can be generated by the envelope pulses will be χ χ δ δ Λ Λ no, e = o, o = e, a3 = 3b3o = 3b3e, and o = e, and considered in the next section.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 ON NONLINEAR PROPAGATION OF EXTREMELY SHORT PULSES 1029

5. GENERATION OF EXTRAORDINARY tions [51]. The YadjimaÐOikawa system is integrable by COMPONENT VIDEO PULSE the method of the inverse scattering problem [50]. The IN THE ZAKHAROVÐBENNEY RESONANCE corresponding single-soliton two-parametric solution in MODE WITH ORDINARY WAVE the laboratory system of coordinates has the form Let us write the ordinary component of a pulse in the v ω ξξ []()Ω tzÐ / form of the envelope pulse with a carrier frequency = m exp Ði tqzÐ sech------τ - , p and wave number k: (39) ξ(),, []()ω 2tzÐ /v Eo = tzr⊥ expi tkzÐ + c.c., (36) Ee = ÐEmsech ------τ - . where ξ is the slowly varying complex envelope: p |∂ξ ∂τ| Ӷ ω|ξ| |∂ξ ∂ | Ӷ |ξ| / , / z k . In this case, Ee has no Here, carrier frequency. 6δ 6δ Substituting Eq. (36) into Eqs. (34) and (35), ξ ==------o ωΩ, E ------o , m a τ m τ2 neglecting rapidly oscillating terms, and using the 2 p a2 p asymptotic expansion [24, 47] Ω 2 1 t t q = ---- + g Ω + ----- , v τ2 ξ []()ω p ∫ Eodt' = ∫ exp i tkzÐ dt' + c.c. Ð∞ Ð∞ the velocity of the pulse satisfies the relation ξ 1 ∂ξ i ∂2ξ 1 1 = Ði---- ++------+… exp[]i()ωtkzÐ + c.c., ---- = ------2Ð gΩ, (40) ω 2 ∂ 3 2 ω t ω ∂t v v g we obtain, after simple transformations, a system of parameter Ω determines the nonlinear frequency shift equations for ξ and E interacting in the ZakharovÐBen- of the short-wave component to the red region since e Ω ξ ney resonance mode: > 0 (see expression for m and Eq. (36)), while the τ other free parameter p, which has the meaning of soli- ∂ξ ∂2ξ ∆ω τ c ton duration, determines the spectral width: ~ 1/ p. i------+ g------= a ωξE + ------∆⊥ξ, (37) ∂ 2 2 e ω z ∂τ 2no The ordinary component of the pulse is an envelope τ soliton, while the extraordinary component is a video 3 ∂E ∂E ∂ E soliton. It follows from Eqs. (37) and 38) that, if Ee = 0 ------e + b E ------e Ð δ ------e + σ E dτ' ∂z 2e e ∂τ e ∂τ3 ∫ e at the input, a pulse of the envelope of the ordinary Ð∞ wave in the medium can generate an ESP of the extraor- τ (38) dinary wave. In this case, each photon of the ordinary ∂ ()ξ 2 c ∆ τ component increases the wavelength by transferring its = Ð a2∂τ + ------⊥ ∫ Eed '. 2no energy to the extraordinary wave, which explains the Ð∞ nonlinear shift of the spectral peak of the pulse: ω δ ω σ ω3 τ ω Ð Ω. In view of the positive value of electron disper- Here, g = 3 o Ð / , = t Ð z/vg, the group velocity v of the ordinary component is defined as sion, the group velocity of the ordinary component g (and, accordingly, of the extraordinary component) 1 dk n 2 σ acquires a positive shift: ------==------3o ++δ ω ------, v dω c o ω2 g 1 no δ ω2 no δ ()ωΩ2 ------= -----3++o ----- 3 o Ð and the dispersion equation has the form v g c c

ω σ no 2 1 no δ ω3 ≈ ----- + 3δ ω Ð 6ωδ Ω = ------2Ð gΩ, k = ------+ o Ð ----. o o c ω c v g In Eq. (38), the ZakharovÐBenney resonance condi- which coincides with formula (40) tion [48] is taken into account, according to which the group velocity of the short-wave (ordinary) component It should be noted that the mechanism of ESP gen- is equal to the phase velocity of the long-wave (extraor- eration associated with the ZakharovÐBenney reso- dinary) component: v = c/n . nance is very close to the corresponding Cherenkov g e mechanism analyzed for the first time in [52]. The only If we disregard diffraction (∆⊥ = 0) and ionic disper- difference is that the generation mode in the latter case sion (σ = 0), we arrive at the system analyzed in [49]. is noncollinear: angle γ between the directions of prop- δ If, in addition, b2e = e = 0, system (37), (38) is trans- agation of ESP and the pulse generating it in a quadrat- formed into the YadjimaÐOikawa equations [50], ically nonlinear medium, whose spectrum contains ω ω which is a unidirectional version of the Zakharov equa- two close extreme frequencies 1 and 2 correspond-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1030 SAZONOV, SOBOLEVSKIŒ ing to wave numbers k1 and k2 , is determined by the Nonlinearity and dispersion of the extraordinary formula [52, 47] component in Eq. (38) are effects of the same order of ω ω smallness. Consequently, the condition under which these γ k2()2 Ð k1()1 effects can be neglected and for which solutions (39) cos = ------()ω ω -. k 2 Ð 1 2 Ӷ were written can be represented in the form b2eEm The velocity of the nonlinear polarization wave at fre- ξ2 2 ω ω a2 m . Substituting the above expressions for Em and quency 2 Ð 1 must be larger than the phase velocity ξ2 ωτ Ωτ ӷ of the wave in this medium at the same frequency: m into this condition, we obtain p p b2e/a2. ω Ð ω ω Ð ω Considering that b and a are quantities of the ------2 1 - > ------2 1 -. 2e 2 k ()ω Ð k ()ω k()ω Ð ω same order of magnitude and taking into account the 2 2 1 1 2 1 quasimonochromaticity condition ωτ ӷ 1 of the ordi- ω ω p Proceeding to the limit 2 1 in this inequality and nary component, we conclude that the condition in assuming that γ = 0 in the preceding equality, we arrive question can easily be satisfied in a wide range of free ω Ω τ at the condition d /dk = vph(0), where vph(0) is the parameters and p. phase velocity in the low-frequency dispersion-free limit. This case corresponds to the ZakharovÐBenney The generation of the extraordinary component with resonance condition. This synchronism condition is the help of the ordinary component in the second har- usually difficult to satisfy in the collinear propagation monic generation mode was considered in [30]. The sit- mode [47]. In our case, the ZakharovÐBenney reso- uation of long-waveÐshort-wave resonance considered nance condition can be written in the form here corresponds to energy pumping from the high-fre- quency ordinary wave to the zeroth harmonic; as a n Ð n result, ESPs of the extraordinary wave are generated. It ------e o = 3 δ ω2. (41) c o can easily be proved that, using a representation of form (36) for Ee and taking into account the substitu- Since the electron dispersion is positive in the transpar- tions ω 2ω and ξ ξ , Eqs. (34) and (35) in the δ e ency region ( o > 0), the ZakharovÐBenney resonance SVAP approximation lead to the well-known system condition can be satisfied for ne > no; i.e., the medium [47] describing the nonstationary process of second must possess positive birefringence. This conclusion harmonic generation. It can easily be found, using the remains unchanged if we take into account ionic disper- dispersion relations that takes into account the electron sion (it can easily be seen from the expression for vg dispersion alone, that the phase synchronism condition δ ω2 δ ω2 σ ω2 ω ω that the substitution 3 o 3o + / has been 2ko( ) = ke(2 ) for the second harmonic generation in carried out on the right-hand side of the last relation). our case has the form Taking into account the closeness of the values of ne n Ð n 2 2 and no, we can write ------o e = ()ω 4 δ Ð δ ≈ 3δ ω , 2 e o o 2 2 πχ()χ ≈ ne Ð no 2 e Ð o ne Ð no ------= ------. which has the sign opposite to that in condition (41). 2no no This means that the collinear mode of the second har- In addition, we have monic generation can be realized in media with nega- tive birefringence. This must enable us to distinguish 2πκ⊥ 2πχ δ = ------≈ ------o . between the two effects under the experimental condi- o n c ω2 tions. o noc o The stability of solutions (39) is an equally impor- In this case, condition (41) can be written in the form tant factor. Let us analyze the stability using the RitzÐ χ Ð χ ω 2 Whitham method of averaged Lagrangian [53]. Since ------e o ≈ ------. δ σ 3χ ω solution (39) corresponds to b2e = e = = 0, we will o 0 consider the stability under the same conditions. It

For crystalline quartz, ne = 1.55 and no = 1.54 [40]. should be noted that this problem can also be solved for χ χ χ ≈ ω ω ≈ b , δ , σ ≠ 0, but the expressions will be cumbersome. Then, ( e Ð o)/3 0 0.01. Consequently, / 0 0.1. 2e e ω 16 Ð1 ω 15 Ð1 In view of the above remarks, system (37), (38) can be Setting 0 ~ 10 s , we obtain ~ 10 s . For the ωτ ӷ τ obtained from the Lagrangian density input pulse, we have p 1; hence, its duration p ~ 10Ð100 fs. In accordance with Eq. (39), the generated ∂ξ∗ ∂ξ ESP (or video soliton) will have a duration of approxi- i ξ ξ∗ δ ∂ξ 2 c ∇ ξ 2 L = ------------Ð ------+ 3 o ------Ð ------⊥ mately the same order of magnitude. Selecting appro- 2ω ∂z ∂z ∂τ 2n ω2 o (42) priately the carrier frequency of the input signal, we can ∂ ∂ ∂ satisfy the ZakharovÐBenney resonance condition and, 1 U U c ()∇ 2 ξ 2 U + ------∂ ------∂τ Ð ------⊥U + a2 ------∂τ . hence, realize the effective generation of ESP. 2 z 2no

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Here, the electric field of the extraordinary wave is Ee = the ordinary component through the ZakharovÐBenney ∂U/∂τ. mechanism under the experimental conditions. In accordance with Eq. (39), we choose the trial Let us estimate the intensity of the input pulse for solution in the form which a video soliton can be formed in the medium. Equation (38) for b = δ = σ = ∆⊥ = 0 leads to Φ 2e e ξ Ω ω z = Aiexp Ðt + no---- sech RtÐ ---- , Ð1 ξ 2 c v 1 1 2 a2 E ∼ ---- Ð ------a ξ ∼ Ð------. (43) e v v 2 2gΩ z g UB= tanh RtÐ ---- , v Substituting this estimate into Eq. (37), we arrive at the nonlinear Schrödinger equation (NSE) where A, B, and R are slowly varying functions of z and 2 r⊥; Φ is a rapidly varying function of the same vari- ∂ξ ∂ ξ i------++g------q ξ 2ξ = 0, ables; and Ω and v are constant parameters connected ∂z ∂τ2 through relation (40). Substituting relations (43) into Eq. (42), taking into where q = ωa2/2gΩ (naturally, we can speak about an account the derivatives of “fast” variables only [53], equation only conditionally since the expression for and integrating with respect to t, we obtain the “aver- coefficient q contains parameter Ω of the soliton solu- age” Lagrangian tion of system (37), (38)). It is well known [48] that the ∞ formation of a soliton of the NSE requires the fulfill- δ Ω2 2 ment of the threshold condition 1 n 2∂Φ 2 3 A 〈〉L ≡ --- Ltd = ------o A ------Ð δ A R Ð ------o ∫ ∂ o χ 2 cR z R o ωΩ Ð∞ ξ > ------. 0 χ 2 eo ω no 2 2 2 2 2 0 + ------A ()∇⊥Φ Ð 2δ ωΩB R Ð ---a A B, 2cR o 3 2 Since whose variation over dynamic parameters A, B, R, and χ χ()1 បω Φ ------o- ∼∼------0 leads to the following system of equations in planar χ ()2 fluid dynamics of an ideal liquid (continuity equations eo χ d and Cauchy integral): (see [3]), where d is the characteristic value of the ∂ρ dipole moment of quantum transitions participating in ------+0,∇⊥()ρv⊥ = ∂z the interaction with the pulse, we have 2 (44) ប ωΩ ∂Φ v⊥ dP c 2 ξ > ξ ∼ ------++------= 3-----δ Ω . 0 th ------ω τ . ∂ ∫ ρ o d 0 p z 2 no Ω τ 14 Ð1 ω 15 Ð1 ω 16 Ð1 Here, the z coordinate plays the role of time, v⊥ = ∇⊥Φ, Setting ~ 1/ p ~ 10 s , ~ 10 s , 0 ~ 10 s , Ð20 and “pressure” P is connected with “density” ρ = A2/R and d ~ 10 CGSE units, we obtain the following esti- ξ2 π 13 through the equation mate for the threshold intensity: Ith ~ c th /4 ~ 10 Ð 4 1014 W/cm2. Then, the intensity of the video soliton dP c a2 2 ------6= -----δ ------ρ . (45) being generated is dρ n oδ ωΩ o 6 o I ∼∼cE2 /4π cξ2 /4πωτ Parameters A and B are given by e m m p ∼ 0.1I ∼ 1012Ð1013 W/cm2. 6δ R ωΩ 6δ R th A ==------o -, B Ð.------o a a Such an estimation procedure cannot be regarded as 2 2 rigorous, but nevertheless provides reasonable values τ In the one-dimensional case, R = 1/ p, the solutions of threshold intensities attainable in modern lasers. obtained here are transformed into solution (39) on ∂ ∂τ account of the fact that Ee = U/ . Obviously, the sta- bility of the solutions in question is equivalent to the 6. SOLITON-LIKE SOLUTIONS stability of the ideal liquid flow of type (44), (45). In OF THE TYPE OF VIDEO PULSES this case, soliton (39) is stable for dP/dρ > 0, which fol- In this section, we consider two solutions of sys- lows from Eq. (45). Thus, the above analysis leads to tem (34), (35) in the form of bound states of the ordi- the conclusion that soliton solutions (39) exhibit trans- nary and extraordinary components of an ESP, propa- verse stability. Consequently, a video pulse of the gating across the optical axis. As before, we will disre- extraordinary wave can be generated with the help of gard ionic dispersion.

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Eom, 6a2, which corresponds to one of the cases of integra- Eem bility of the HenonÐHeiles model [42]. Analysis shows that, in the case under investigation the duration of an ESP (and, hence, the velocity of its 1 propagation) is bounded from below. This question is closely related to the stability of solution (46) and will 2 therefore be considered in greater detail. As in the pre- vious section, we will use the method of averaged Lagrangian. The density of the Lagrangian correspond- ing to system (34), (35) in the absence of ionic disper- τ–2 τ –2 sion has the form 0 c p

2 Fig. 3. Amplitudes of the ordinary (1) and extraordinary (2) ∂ ∂ ∂ 2 ∂2 1 Ui Ui niUi δ Ui components of a soliton-like ESP of type (46) as functions L = --- ∑ ------++------i------of the square of reciprocal duration. Dashed segments of the 2 ∂z ∂t c ∂t ∂ 2 ioe= , t curves correspond to the instability region. (48) ∂ 3 ∂ 2∂ c 2 b2eUe a2Uo Ue () ∇ Ð ------⊥ U i ++ ------ ------∂ -----------∂ ------∂ . Direct substitution readily shows that, for b2e/a2 = 2n 6 t 2 t t δ δ o 6 e/ o, system (34), (35) has one-dimensional solutions tzÐ /v Here, the ESP field components can be expressed in E = ±E sech ------, ∂ ∂ o om τ terms of “potentials” Uo and Ue as Eo = Uo/ t and p ∂ ∂ (46) Ee = Ue/ t. tzÐ /v In accordance with Eqs. (46), we choose trial solu- Ee = ÐEemsech------τ - , p tions in the form where A U = ------o arctan{}sinh[]Rt()Ð Φ , o R 2 n Ð n 4δ Ð δ E = ------δ ------e o Ð ------e -o , om a τ oc τ2 2 p p A U = Ð-----e tanh[]Rt()Ð Φ , e R 12δ E = ------e , em τ2 where A , A , and R are slowly varying functions of b2e p o e variables z and r⊥, while Φ is a rapidly varying function τ and velocity v is connected with pulse duration p of the same variables. through the relation Substituting these expressions into Eq. (48) and integrating with respect to t, we arrive at the “averaged” δ 1 no o Lagrangian which leads to equations of motion of ---- = ----- Ð. ---- - (47) δ v c τ2 type (44) except for the substitution 3c o/no no/c in p the Cauchy integral. In this case, v⊥ = ∇⊥Φ and ρ The spectral widths of the ordinary and extraordi- “density” is connected with parameter R through the ∆ω τ relation nary components (46) can be estimated as o ~ 1/ p ∆ω τ ∆ω and e ~ 2/ p ~ 2 o. Thus, the spectrum of the 4δ ()n Ð n R 28δ2 R3 extraordinary component is twice as wide as that of the ρ = ------o e o Ð ------o . ordinary component. Consequently, solutions (46) can a2c 3a2 be regarded as an analog of the second harmonic gener- 2 2 ation for quasimonochromatic pulses. Quantities Ao and Ae coincide with Eom and Eem upon It follows from the expressions for Eom and Eem that τ δ δ the substitution 1/ p R and for e = o, b2e = 6a2. solution (46) can be realized, for example, in a medium The relation between the “pressure” and “density” is with positive birefringence (n > n ); in the transpar- e o ρ δ 2 ency region of this medium, we can disregard the dif- given by the equation ∫dP/ = oR , whence ference in the dispersion of the ordinary and extraordi- δ ≈ δ nary refractive indices (i.e., we assume that e o). In Ð1 ()()δ 2 δ δ dP dPdρ 2 ne Ð no /c Ð 7 o/3 R the case of strict equality e = o, the condition for the ------==------2δ R ------. dρ dRdR o ()δ 2 existence of the exact solution (46) has the form b2e = ne Ð no /c Ð 7 o R

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ρ ω In this case, the stability criterion dP/d > 0 taking no( ), τ ω into account the fact that R = 1/ p can be written in the ne( ) form 1 δ τ > τ ≡ 7 oc p c ------. 2 ne Ð no It follows hence that, in accordance with Eq. (47), the value of velocity v lies in the interval ω ω ω c c n Ð n 0 o || ----- < c, the amplitudes of both compo- τ nents increase with decreasing p (Fig. 3). Conse- For testing relations (49) for stability, we choose the quently, the intense central part of the ESP in the cross corresponding test solutions in the form section leads the peripheral parts, and the pulse is stable on the whole. −Ao []()Φ Uo = +------sech RtÐ , Using the estimates obtained in the previous section, R πχ()χ πχ A 2 e Ð o 2 o e []()Φ n Ð n ≈ ------, δ ∼ ------, Ue = Ð----- tanh RtÐ . e o n o ω2 R o noc o The application of averaged variational principle τ we can write the expression for c in the form using these expressions and Lagrangian (48) also leads in this case to a system of equations of type (44) taking χ into account the substitution 3cδ /n n /c in the τ ωÐ1 7 o o o o c = 0 χ------χ , e Ð o Cauchy integral. As in the previous case, ∫dP/ρ = δ 2 ρ which at least does not contradict condition (1). oR , and the relation between R and “density” is given by the relation As in the case of solutions (46), we can verify by direct substitution that, for ne = no and 3b2e/a2 = 1 + 2 3 δ δ 12δ ()3a Ð 2b R 2 e/ o, system (34), (35) has the solutions ρ o 2 2e = ------3 . a2 ± tzÐ /v tzÐ /v Eo = Eomtanh------τ - sech------τ - , p p Consequently, dP/dρ = 2ρ2/3 > 0 and an ESP of type (49) (49) is stable to self-focusing. This conclusion is also con- 2tzÐ /v Ee = ÐEemsech ------τ - , firmed by qualitative considerations formulated above p for solution (46) taking into account the fact that the amplitudes of both components and the velocity where increase with decreasing duration. It can be seen from the expression for Eom that solution (49) can be realized δ δ 2 6δ for o > 4 e. At the same time, ne = no. Thus, solution (49) E ==------3δ ()δ Ð 4δ , E ------o , om a τ o 0 e em τ2 corresponds to the situation when birefringence 2 p a2 p emerges exclusively due to dispersion and is absent in the dispersion-free (low-frequency) region. From and the velocity and duration are connected through expressions (18) and (19) for the ordinary (χ⊥(ω)) and relation (47) as before. extraordinary (χ||(ω)) electron susceptibilities, we The profile of the ordinary component in (49) has a obtain the following dispersion relations: bipolar form; consequently, its spectrum is centered at frequency ω ~ 1/τ . In view of the unipolarity of E , ω2χ ω2χ c p e o ⊥ || || the spectrum of the extraordinary component of the χ⊥()ω ==------, χ||()ω ------. ω2 ω2 ω2 ω2 ESP is centered at zero frequency. o Ð || Ð

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1034 SAZONOV, SOBOLEVSKIŒ At the same time, we have electron-vibrational response taking into account the dynamics of population of the SRS sublevels. 2 4 ∂ χ⊥||, π χ⊥||, The obtained system of wave equations (32), (33) is δ , ==------------. oe 2 2 written directly for the ordinary and extraordinary com- cno∂ω cno ω ω = 0 o, || ponents of the electric field of the pulse, and not for their envelopes as in earlier publications. This circum- χ χ χ χ Since ne = no, ⊥ = ⊥(0) = || = ||(0). From these rela- stance makes it possible to use this system for analyz- δ δ ω tions and from the condition o > 4 e, we obtain || > ing the propagation of quasimonochromatic pulses as ω 2 o. It can be seen from relation (12) that the suscepti- well as optical pulses of duration of only a few periods bilities of the ordinary and extraordinary waves are (including video pulses). The mechanism of generation formed by σ and π transitions, respectively. Conse- of a video pulse of the extraordinary wave in the quently, the characteristic frequency of π transitions ZakharovÐBenney resonance mode with a quasimono- must be more than twice the frequency of σ transitions chromatic ordinary component (see Section 5), which for the realization of solution (49). can be realized in media with positive birefringence, may serve as an illustration of the previous statement. Thus, solution (49) can be realized in media with The role of SRS processes in the generation of video dispersion origin of birefringence (Fig. 4). In this case, ω ω pulses due to continuous energy pumping from high- no( ) > ne( ); i.e., the medium must possess negative frequency Fourier components to the Stokes compo- birefringence in the dispersion frequency range. nents of the spectrum is well known [45]. In this con- nection, the investigation of the combined effect of the 7. CONCLUSIONS electron quadratic nonlinearity and SRS on the process of ESP generation with the help of pulses initially pos- The analysis carried out by us here reveals differ- sessing a clearly manifested carrier frequency is of con- ences in the ESP dynamics in isotropic and optically siderable interest. uniaxial media. In the latter media, the quadratic non- linearity of the electron response, which is absent in an The soliton-like solutions (46) and (49) in the form isotropic dielectric, plays a significant role. The system of coupled states of the ordinary and extraordinary of constitutive equations (16), (17) as an analog of the ESP components presented here are, in addition to Lorentz classical model, which receives its quantum- relations (39), only a minor illustration of possible mechanical substantiation here, can be used in subse- solutions contained in Eqs. (32), (33). In all probability, quent investigations for describing the electron other solutions will mainly be obtained with the help of response in the low-frequency transparency range. In numerical experiments, which does not preclude fur- the Voigt geometry (when ESP propagates at right ther analytic investigations. angles to the optical axis), these equations are trans- formed into the HenonÐHeiles system (18), (19). It is remarkable that the HenonÐHeiles system (its homoge- ACKNOWLEDGMENTS neous version) permits both regular and chaotic motion This study was supported by the Russian Founda- depending on the relation between the coefficients [42]. tion for Basic Research (project no. 02-02-17710a) Consequently, it cannot be ruled out that chaotic modes of ESP propagation can be observed in some anisotro- pic media. APPENDIX The HenonÐHeiles system is an analog of the Duffing equation describing a nonresonant nonlinear The expressions for third-order nonlinear suscepti- response of an isotropic medium in the low-frequency bilities in terms of the microscopic parameters of the region. It was shown in [21] that the Duffing equation medium have the form fails to provide an adequate response of an isotropic dielectric to an intense external action in the high- ()3 ()3 ()3 2N frequency region. In the same publication, a nonlinear χ ===χ χ ------2 ∑ ∆µν()ϕ eo xyyx yxxy ប2 model is proposed in the form of two parametrically  µν≠ coupled oscillators; it was found that this model holds in the low- and high-frequency regions. Proceeding from this remark, we can say that an analogous α α α × 2 µj µν µν modification of the Henon–Heiles model is forth- ∑ dµjω------ω - ++ω------ω - ω------ω - jµ µν jν µν µj jν coming. j ≠ µν, The generalization of the BloembergenÐShen quan- tum-mechanical system proposed by us here for α 2 ϕ 4 µν 2 ϕ Dµν describing SRS processes in media with uniaxial Ð cos ∑ dµν------2 + sin ∑ ------ω ωµν anisotropy enabled us to derive expressions (30) for the µν≠ µν µν≠

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× ()11. É. M. Belenov, P. G. Kryukov, A. V. Nazarkin, et al., ∑ ∑ dµjdkν D jk ++d jkdkν Dµj dµjdkjDkν Pis’ma Zh. Éksp. Teor. Fiz. 47, 442 (1988) [JETP Lett. j ≠ µν, k ≠ µν,,j 47, 523 (1988)]. 12. É. M. Belenov, A. V. Nazarkin, and V. A. Ushchapovskiœ, α α α α  Zh. Éksp. Teor. Fiz. 100, 762 (1991) [Sov. Phys. JETP × µj + kj kν + kj ------ω + ------ω - , 73, 422 (1991)]. µk jν  13. A. I. Maœmistov and S. O. Elyutin, Opt. Spektrosk. 70, 101 (1991) [Opt. Spectrosc. 70, 57 (1991)].  14. A. I. Maimistov and S. O. Elyutin, J. Mod. Opt. 39, 2201 χ()3 χ()3 2N ∆ ϕ ∆ ϕ (1992). e ==yyyy ------ ∑ µν()∑ µj() ប2  µν≠ j ≠ µν, 15. A. V. Vederko, O. B. Dubrovskaya, F. M. Marchenko, and A. P. Sukhorukov, Vestn. Mosk. Univ., Ser. 3: Fiz., α α α Astron. 33, 4 (1992). × µj µν µν 16. A. I. Maœmistov, Opt. Spektrosk. 76, 636 (1994) [Opt. ω------ω - ++ω------ω - ω------ω - jµ µν jν µν µj jν Spectrosc. 76, 569 (1994)]. 17. S. V. Sazonov and E. V. Trifonov, J. Phys. B 27, 369 (1994). 4 4 4 4 αµν Ð ∑ ()dµνcos ϕ + Dµνsin ϕ ------18. A. I. Maœmistov, Opt. Spektrosk. 78, 483 (1995) [Opt. ω2 µν≠ µν Spectrosc. 78, 435 (1995)]. 19. A. V. Andreev, Zh. Éksp. Teor. Fiz. 108, 796 (1995) [JETP 81, 434 (1995)]. 1 2 ϕ Dµν + --- sin ∑ ------∑ ∑ Dµj Dkν D jk 20. A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. Lett. 75, 2 ωµν µν≠ j ≠ µν, k ≠ µν,,j 2316 (1995). 21. S. A. Kozlov, Opt. Spektrosk. 79, 290 (1995) [Opt. Spec- trosc. 79, 267 (1995)]. αµ + α α ν + α  × j kj k kj 22. S. V. Sazonov, Fiz. Tverd. Tela (St. Petersburg) 37, 1612 ------ω + ------ω - , µk jν  (1995) [Phys. Solid State 37, 875 (1995)]. 23. S. V. Sazonov, Opt. Spektrosk. 79, 282 (1995) [Opt. where Spectrosc. 79, 260 (1995)]. 24. S. A. Kozlov and S. V. Sazonov, Zh. Éksp. Teor. Fiz. 111, 2 404 (1997) [JETP 84, 221 (1997)]. dµν 2 2 2 ∆µν()ϕ = ------cos ϕ + Dµνsin ϕ. 25. I. V. Mel’nikov and D. Mihalache, Phys. Rev. A 56, 1569 2 (1997). 26. S. V. Sazonov, Zh. Éksp. Teor. Fiz. 119, 419 (2001) χ()3 χ()3 The expression for o = xxxx can be obtained from [JETP 92, 361 (2001)]. () the formula for χ 3 for ϕ = 0. 27. M. Born and E. Wolf, Principles of Optics, 4th ed. (Per- e gamon Press, Oxford, 1969; Nauka, Moscow, 1973). 28. A. I. Maœmistov, Opt. Spektrosk. 87, 104 (1999) [Opt. REFERENCES Spectrosc. 87, 96 (1999)]. 1. P. C. Becker, H. L. Fragnito, J. Y. Bigot, et al., Phys. Rev. 29. S. V. Sazonov, Zh. Éksp. Teor. Fiz. 107, 20 (1995) [JETP Lett. 63, 505 (1989). 80, 10 (1995)]. 2. K. Tamura and M. Nakazawa, Opt. Lett. 21, 68 (1996). 30. O. B. Dubrovskaya and A. P. Sukhorukov, Izv. Akad. Nauk, Ser. Fiz. 56, 184 (1992). 3. T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 (2000). 31. E. V. Kazantseva and A. I. Maœmistov, Opt. Spektrosk. 4. A. M. Zheltikov, Vestn. Mosk. Univ., Ser. 3: Fiz., 89, 838 (2000) [Opt. Spectrosc. 89, 772 (2000)]. Astron., No. 4, 3 (2001). 32. S. V. Sazonov and A. F. Sobolevskiœ, Opt. Spektrosk. 90, 5. A. M. Zheltikov, Usp. Fiz. Nauk 172, 743 (2002) [Phys. 449 (2001) [Opt. Spectrosc. 90, 390 (2001)]. Usp. 45, 687 (2002)]. 33. E. V. Kazantseva, A. I. Maimistov, and B. A. Malomed, 6. D. H. Auston, K. P. Cheung, J. A. Valdmanis, and Opt. Commun. 188, 195 (2001). D. A. Kleinman, Phys. Rev. Lett. 53, 1555 (1984). 34. G. Agrawal, Nonlinear Fiber Optics (Academic, San 7. A. V. Kim and M. Yu. Ryabikin, Usp. Fiz. Nauk 169, 58 Diego, 1995; Mir, Moscow, 1996). (1999) [Phys. Usp. 42, 54 (1999)]. 35. S. V. Sazonov and A. F. Sobolevskiœ, Kvantovaya Élek- 8. A. I. Maœmistov, Kvantovaya Élektron. (Moscow) 30, tron. (Moscow) 30, 917 (2000). 287 (2000). 36. Nonlinear Properties of Organic Molecules and Crys- 9. A. A. Zabolotskiœ, Zh. Éksp. Teor. Fiz. 121, 1012 (2002) tals, Ed. by D. S. Chemla and I. Zyss (Academic, [JETP 94, 869 (2002)]. Orlando, 1987; Mir, Moscow, 1989). 10. É. M. Belenov and A. V. Nazarkin, Pis’ma Zh. Éksp. 37. L. N. Ovander, Usp. Fiz. Nauk 86, 3 (1965) [Sov. Phys. Teor. Fiz. 51, 252 (1990) [JETP Lett. 51, 288 (1990)]. Usp. 8, 337 (1965)].

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38. R. H. Pantell and H. E. Puthoff, Fundamentals of Quan- 46. M. B. Vinogradova, O. V. Rudenko, and A. P. Sukho- tum Electronics (Wiley, New York, 1969; Mir, Moscow, rukov, The Theory of Waves, 2nd ed. (Nauka, Moscow, 1972). 1990). 39. I. I. Sobel’man, Atomic Spectra and Radiative Transi- 47. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, The tions (Nauka, Moscow, 1977; Springer, Berlin, 1979). Optics of Femtosecond Pulses (Nauka, Moscow, 1988). 48. R. K. Dodd, J. C. Eilbeck, J. Gibbon, and H. C. Morris, 40. R. E. Stoiber and S. A. Morse, Microscopic Identifica- Solitons and Nonlinear Wave Equations (Academic, tion of Crystals (Ronald, New York, 1972; Mir, Moscow, New York, 1982; Mir, Moscow, 1988). 1974). 49. E. S. Benilov and S. P. Burtzev, Phys. Lett. A 98, 256 41. E. I. Butikov, Optics (Vysshaya Shkola, Moscow, 1986). (1983). 50. N. Yadjima and M. Oikawa, Prog. Theor. Phys. 56, 1719 42. A. J. Lichtenberg and M. A. Lieberman, Regular and (1976). Stochastic Motion (Springer, New York, 1982; Mir, Mos- cow, 1984). 51. V. E. Zakharov, Zh. Éksp. Teor. Fiz. 62, 1745 (1972) [Sov. Phys. JETP 35, 908 (1972)]. 43. D. N. Klyshko, Physical Fundamentals of Quantum 52. U. A. Abdullin, G. A. Lyakhov, O. V. Rudenko, and Electronics (Mir, Moscow, 1986). A. S. Chirkin, Zh. Éksp. Teor. Fiz. 66, 1295 (1974) [Sov. 44. Y. R. Shen and N. Bloembergen, Phys. Rev. A 137, 1738 Phys. JETP 39, 633 (1974)]. (1965). 53. S. K. Zhdanov and B. A. Trubnikov, Quasi-Gaseous Unstable Media (Nauka, Moscow, 1991). 45. É. M. Belenov, P. G. Kryukov, A. V. Nazarkin, and I. P. Prokopovich, Zh. Éksp. Teor. Fiz. 105, 28 (1994) [JETP 78, 15 (1994)]. Translated by N. Wadhwa

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Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1037Ð1044. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1179Ð1187. Original Russian Text Copyright © 2003 by Vaulina, Samarian, James, Petrov, Fortov. PLASMA, GASES

Analysis of Macroparticle Charging in the Near-Electrode Layer of a High-Frequency Capacitive Discharge O. S. Vaulinaa,*, A. A. Samarianb, B. Jamesb, O. F. Petrova,*, and V. E. Fortova aInstitute of High Energy Densities, IVTAN, Russian Academy of Sciences, Izhorskaya ul. 13/19, Moscow, 127412 Russia bSchool of Physics, University of Sydney, NSW 2006, Sydney, Australia *e-mail: [email protected] Received November 6, 2002

Abstract—Spatial variation of dust particle charges are estimated numerically for typical laboratory experi- ment conditions in a radio-frequency (rf) capacitive discharge. The surface potentials of macroparticles levitat- ing in the upper part of the near-electrode layer of the rf discharge are measured. A model is proposed for the formation of irregular dust oscillations due to stochastic motion of dust in the bulk of a spatially inhomogeneous plasma (in the presence of a dust charge gradient). This mechanism is used for analyzing the results of mea- surements of the amplitude of vertical vibrations of dust particles in the near-electrode layer of the rf discharge. It is found that the dust charge gradient may be responsible for the development of such vibrations. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION plasma. The formation of various self-induced dust vibrations in the field the gravitational force orthogonal The charge of a dust particle is an important param- to the macroparticle charge gradient was considered eter for investigating various transport processes in dusty plasmas, such as phase transitions, diffusion pro- in [12]. However, this mechanism can hardly be cesses, propagation of waves, and formation of self- responsible for the evolution of the observed vibrations excited dust vibrations. Considerable attention in the of dust particles in capacitive rf discharges in view of study of plasmas is paid to the methods and results of considerable uniformity of the parameters of the capac- measurements of dust particle charges. A large number itive rf discharge plasma in the radial direction (ortho- of methods for determining macroparticle charges are gonal to the gravitational force). Stochastic fluctuations based on the measurement of the dynamic response of of dust charges due to discreteness of plasma currents dust particles to various external perturbations [1Ð9]. charging a macroparticle may lead to “anomalous heat- The charges of macroparticles can also be determined ing” of dust particles in gas-discharge plasmas [18], but cannot be responsible for high kinetic energies (>0.1 eV) without perturbing the system in question by external µ agencies, but from an analysis of their diffusion or from acquired by light particles of radius 1Ð2 m (density the equilibrium conditions for a stationary particle in ρ ≈ 1.5Ð2 g/cm3) under gas pressures P > 0.02 Torr. the gravitational field of the Earth and in the electric field of the trap [9–11]. One of the possible mechanisms of evolution of irregular vibrations of dust particles is associated with Since the charge of dust particles is a function of the stochastic changes in their charge, which are deter- parameters of the surrounding plasma (concentrations mined by the random position of a particle in a spatially ne(i) and velocities ve(i) of electrons and ions), the varia- inhomogeneous plasma (in the presence of a dust tion of these parameters may lead to a change in the charge gradient in the direction of gravity) due to ther- macroparticle charge and to evolution of various insta- mal or other fluctuations, e.g., due to the above-men- bilities in plasma-dust systems [12, 13]. Available tioned discreteness of the charging current. This mech- experimental observations demonstrate that, under cer- anism was considered for the first time in [14]. How- tain conditions (upon a variation of pressure or an ever, in the proposed model, it was proposed that increase in the number of particles), dust particles in the particles move in the free diffusion mode, which is strata of a dc glow discharge or in the near-electrode unsuitable for describing spatially bounded trajectories layer of a capacitive rf discharge may acquire energies of macroparticles, which are observed both in the dust on the order of 1Ð100 eV and perform regular or sto- layer formed in the rf discharge and under the condi- chastic vertical vibrations (in the direction of the grav- tions of a bulk dust cloud in a dc glow discharge [13, 16, itational field) [13–17]. 17, 19]. It should also be noted that the very possibility The reason for the evolution of such vibrations may of formation of irregular dust vibrations in the frame- be related to the inhomogeneity of the surrounding work of this model was determined by the initial energy

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1038 VAULINA et al. of the system, which is not always observed in real ϕ is the ) provided that the directional experiments, including the one described below. velocity of ions ui is much higher than their thermal Here, we propose a model of formation of irregular velocity v T : dust vibrations due to stochastic motion of dust in a i spatially inhomogeneous plasma, which takes into ∆ δ ϕ() account the limitations imposed by an electric field on iZ p ≈ 2z n/n0 Ð e zsÐ /T e ------〈〉Ð------()-. (2) the displacements of macroparticles in a preferred Z p z 1 ++sz direction. This mechanism is used for analyzing verti- cal vibrations of macroparticles in the near-electrode Here, 〈Z 〉 is the equilibrium charge in a plasma layer layer of a capacitive rf discharge. The material is p δ ≈ 2 arranged in the following order. In Section 2, the esti- with n 0 and concentration n0 and s = mi ui /2Te. mates of spatial variation of the charge of a dust particle Relation (2) can be used for estimating the charge vari- in the near electrode layer of the discharge are obtained. ation in the vicinity of the upper boundary of the elec- In Section 3, basic relations are derived for estimating trode layer, where levitation of macroparticles is nor- the kinetic energy acquired by a dust particle in an inho- mally observed. In this case, s = 0.5, considering that mogeneous plasma. The last two sections are devoted to ions enter the layer at the Bohm velocity v = T /m experiments on dust particle charging and on the B e i ≈ dynamics of formation of stochastic dust vibrations in and n0 = nB n0* /2.7, where n0* is the concentration of the rf discharge plasma. the unperturbed plasma. It should be noted that such an approach is suitable only for very low pressures, when the mean free path li for ions colliding with gas neutrals 2. SPATIAL VARIATIONS λ OF THE MACROPARTICLE CHARGE is much longer than the electron Debye radius De. For IN GAS-DISCHARGE PLASMAS average pressures (0.05Ð1 Torr), which are working pressures in most experiments on dusty plasmas, li ~ In gas-discharge plasmas, where emission processes λ De. In this case, the velocity of ions ui(0) at the layer are insignificant as a rule, the charge of a dust particle boundary is smaller than the Bohm velocity v by is negative. The estimate of the macroparticle surface B πλ potential obtained in the orbital-motion-limited (OML) approximately a factor of De/2li [22]. approximation gives the following expression for its value: It should be noted that we assumed, while deriving relation (2), that ionization processes in the plasma 〈〉 ϕ eZp ≡ zTe layer can be neglected (niui = const). The analytic the- e = Ð------Ð------; ory of the near-electrode layer in an rf discharge devel- a p e oped for this case is described in [22]. In this case, it is 〈 〉 here, Zp is the equilibrium (time-averaged) charge of assumed that the layer is in contact not with the unper- a dust particle, Te is the electron temperature in elec- turbed plasma, but with a preliminary layer in which tronvolts, and z ≈ 2Ð4 for most experiments on dusty the electroneutrality of the plasma is quite high: δ Ӷ plasmas under the discharge conditions in inert gases n/n0 1. An analysis of the proposed system of equa- λ Ӷ [20, 21]. tions for low pressures ( De li) gives for the averaged Using the formulas of the OML approximation, we electric field E(y) of the near-electrode layer in the ∆ can estimate the small variation nZp of the equilibrium vicinity of its upper boundary a distribution close to a 〈 〉 charge Zp of a macroparticle due to violation of electro- linear function: δ neutrality of the surrounding plasma, n = ni Ð ne [19]: Ey()= C y. (3) ∆ Z divE 1 ------n -p ≈ Ð------. (1) 〈〉Z 4πen ()1 + z λ p 0 For average pressures ( De ~ li), the solution of the Here, E is the electric field strength and n0 is the con- equations of the analytic theory [22] leads to the fol- lowing linear approximation for the gradient E(y) of centration of the neutral plasma, where ne = ni = n0. δ Ӷ this field: Obviously, as long as condition n n0 holds, the requirement of the smallness of charge variation, 2 Ey()= C y . (4) ∆ Ӷ 〈〉 2 nZ p Z p , Let us analyze the conditions realized in some is satisfied automatically. experiments in a dusty plasma of a capacitive rf dis- A similar relation charge [16, 17]. These experiments show that levitation ∆ 〈〉 Ӷ 〈〉 of macroparticles is observed in the vicinity of the upper iZ p = Z p Ð Z p Z p boundary of the near-electrode layer whose thickness for estimating the charge variation in a plasma layer can dmax lies approximately between 0.5 and 1.5 cm under | ϕ | Ӷ be obtained under the assumption e /Te 1 (where experimental conditions (P = 0.015Ð0.2 Torr).

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We can estimate the macroparticle charge gradients Video camera

dZ d()∆ Z β = ------p ≡ ------i p y dy dy Particles arising as a result of a change in the charging conditions Laser µ ρ for light particles of radius ap = 1Ð2 m and density = 3 1.5Ð2 g/cm , assuming that the electric fields E confin- Rf ing particles in the gravitational field of the Earth are generator equal to 1Ð4 V/cm, the plasma density at the layer 8 9 Ð3 boundary is ne = 10 Ð10 cm , the electron temperature Fig. 1. Simplified diagram of experimental setup. is Te = 2 eV, and the gas used in experiments is argon (z ≈ 3Ð4 [20, 21]). Then, in the case of linear field (3) 2 for C1 = 12 V/cm , the particle in question will be sus- account the fluctuations of particle interaction forces ≈ pended at distance y = y0 0.1Ð0.33 cm from the upper determined by particle charge variations boundary of the layer, where the condition of balance ˜ between the gravitational force and the electric field is Zs = β yΣ observed: y 〈〉 due to a random variation in their positions in the layer m pgZ+0.p eE() y0 = (5) in question. Deviations (r, y) of an individual particle in such a layer from its equilibrium (r , y ) position can be The relative change in the macroparticle charge gradi- 0 0 β 〈 〉 described by the linearized system of equations ent y/ Zp in this region, which is obtained from joint solution of Eqs. (2) and (3) for the conditions of the m y'' = Ð m ν y' Ð α eZ〈〉y ++eβ E yF, (6a) problem, varies from Ð0.1 to Ð0.3 cmÐ1. Thus, in the p p fr y p y y y model described here, the charge of a dust particle ν α 〈〉 ˜ decreases as it approaches the electrode, the rate of this m pr'' = Ð m p frr' Ð reZp r ++eZsEr Fr, (6b) approach being the higher, the lower this particle is located. The same qualitative pattern is observed for where ν is the friction coefficient, which is defined in nonlinear field (4) also. In this case, the charge gradi- fr β 〈 〉 ≈ the free-molecule approximation as ents y/ Zp of macroparticles levitating at distance y0 0.25Ð0.5 cm from the upper boundary of the layer [] (C = 12 V/cm3) changes with increasing y from Ð0.15 ν []Ð1 ≈ Cv P Torr 2 0 fr s ------Ð1 []µ ρ []3 to −0.39 cm . a p m g/cm

Concluding the section, we note that, in spite of con- ≈ µ 3 β 〈 〉 Ð1 (where Cv 820 m g/s Torr cm for argon); primes siderable charge gradients y/ Zp = Ð(0.1Ð0.4) cm , 〈 〉 denote the time derivatives of coordinates; F = (Fy, Fr) the relative change in its value Zp did not exceed 7% |∆ 〈 〉| is a random force leading to stochastic motion of parti- ( iZp/ Zp < 0.07) in all analyzed cases. The perturba- α |δ | cles; and y, r = dEy, r /dy are the gradients of the external tion of plasma electroneutrality was n/n0 < 0.09, and | ϕ | electric field E = (Ey, Er). Here, e /Te < 0.35, which is a good approximation for linear- izing equations of the analytical theory of the layer [22] m g E = ------p as well as equations of the OLM approximation [20, 21] y eZ〈〉 and, accordingly, for estimating the charge variation in p the layer from relations (2)Ð(4). is determined by the balance between the vertical elec- tric force and the gravitational force of the Earth; the value of Er can be estimated by taking into account the 3. EFFECT OF CHARGE FLUCTUATIONS balance of the radial electric force and the forces of OF MACROPARTICLES interaction between particles. For a homogeneous ON THEIR KINETIC TEMPERATURE extended layer of particles, we can assume that radial IN A SPATIALLY INHOMOGENEOUS PLASMA fields are linear, i.e., ≈ α ≈ α Let us consider the 2D problem in the cylindrical ry Er rr r Nrl p, geometry, simulating a layer of macroparticles levitat- ing above an electrode of the rf oscillator (see Fig. 1 where lp is the average particle spacing and Nr is the β below) in the presence of a dust charge gradient y = number of particles located in the region between the dZp/dy in the direction of gravity (y axis), taking into axis of the cylindrical system and the particle in ques-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1040 VAULINA et al. tion, the gradient of this field for a planar dust cloud by a macroparticle is redistributed uniformly over being degrees of freedom: 〈〉 ∆sT ≈ ∆sT ≈ ∆sT. α ∝ eZp r y r ------3 . l p Simulation shows that, for parameters close to experi- mental conditions, such an assumption is justified for a large number of particles (n ), low buffer gas pressures It should be recalled that the origin of force eE Z˜ p r s (small ν ), and the formation of additional dust layers in (6b) is determined by the collective action of the fr ∆s remaining particles of the layer on an individual parti- [12, 18]. In this case, the value of the kinetic energy T cle. In the formulation considered here, the change in acquired by a dust particle due to random variation of its ˜ charge in an inhomogeneous plasma can be defined as Zs is determined by the random quantity yΣ defined by f ˜ s T + ∆ T Eq. (6a) with a different value of random force Fsy, ≠ ∆ ≈ 0 T ------θ -, (8) Fy, whose parameters can be determined using the pro- 1 Ð 1 cedure described in [18]. It should be noted that, in where order to solve the problem, it is sufficient to assume that 2 2 2 2 the action of the random forces considered here is not β 2 e 〈〉Z E ω θ = ------y ------p r r . (9) 〈〉˜ 1 〈〉 2 2 2 2 2 correlated (FsF = 0); the correlation of these forces Z p m ν ()ωω + ω with “slow” displacements l = (r, y) of particles is also p fr r y y ˜ Here, absent (〈Fl〉 = 0 and 〈〉Fsl = 0) [23, 24]. In this case, an ∆s eZ〈〉α additional kinetic energy Tr, proportional to the ω2 p r r = ------. amplitude of charge fluctuations m p 〈〉˜ 2 β2 〈〉2 If necessary, relation (8) can take into account the frac- Zs = y y , tion γ of redistributed energy: is supplied to the system in the radial direction r ()∆ f ()()θγ ∆s T 0 + T 11+ Ð 1 〈〉2 T = ------γθ . (10) (see [18]), where y is the mean square deviation in 1 Ð 1 the y direction: Thus, for γ ≈ 1, the value of kinetic energy ∆sT is deter- f s mined by the value of coefficient θ , which strongly T ++∆ T ∆ T 1 〈〉y2 = ------n y. (7) depends on the parameters of the dust system. If we ω 2 m p y take into account the fact that β a In this equation, y Ӷ y ------〈〉 ----- Z p Ey 〈〉α β ω 2 eZp y Ð yEy in the vicinity of the upper boundary of the near-elec- y = ------; m p trode layer (see Section 2) and also assume that 2 2 2 ∆f ≈ 〈〉 ω ≈ ω Tn is the temperature of the surrounding gas; T is the Er NreZp /l p , r y, stochastic energy acquired by a particle in the plasma we can obtain a simpler relation for estimating θ : due to other mechanisms, e.g., due to discreteness of 1 ∆s β 2 2 2 〈〉2 charging currents; and Ty is a part of the kinetic y Nr e Z p s θ ≈ ------, energy ∆ T transferred in the y direction through the 1 〈〉 2 r Z p 2m ν l particle interaction. The coefficient p fr p θ ≈ µ 〈 〉 which gives 1 0.25 for Nr = 10, lp = 300 m, Zp = ∆sT 5 × 103e, |β |/〈Z 〉 = 0.2 cmÐ1, ν ≈ 13 sÐ1 (argon, P ≈ γ = ------y y p fr s µ ρ 3 ∆ T 0.03 Torr), ap = 1 m, and = 2 g/cm . It can easily be r θ seen that coefficient 1 tends to unity under the same |β | 〈 〉 of energy transfer due to particle interaction in the dust conditions if Nr 20 or y / Zp 0.4. In this cloud differs from zero and is determined by the reac- case, we find that kinetic energy ∆sT increases indefi- tion of the dust system to transverse perturbations of the nitely with an increase in deviation (y, r) of particles system and by the amplitude of particle displacement from their equilibrium positions. However, the linear- [12, 18]. The derivatives corresponding to these pertur- ized system of equations (6a), (6b) in this case does not bations are excluded from system (6a), (6b) since we provide a correct analysis of the dynamics of particles will henceforth assume that the kinetic energy acquired since the amplitude of their motion can be limited due

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Table

(ϕ E) , R, µm r, g cmÐ3 h, mm s exp E, V/cm 〈Z 〉 〈〉Zc δ, % V2/cm p p

1 2 10.06 7.38 1.12 4579 4632 1 1.39 1.5 9.65 10.71 1.67 6200 6438 4 2.12 1.5 7.70 24.89 3.93 9332 9820 5 3.07 1.5 4.80 52.21 7.38 15079 14220 6 to various nonlinear effects. In addition, relation (8) particles in a dust cloud is large enough and, accord- was obtained under the assumption of smallness of ingly, can develop upon an increase in the concentra- mean square deviations of particles, tion of the dust component. 〈〉2 Ӷ 2 ≈ 2 〈〉2 y l0 l p + y , 4. DETERMINING PARTICLE CHARGE where l0 and lp are particle spacings in the perturbed IN THE NEAR-ELECTRODE LAYER and unperturbed layers of macroparticles. OF RF DISCHARGE An estimate of the maximum value ∆sTmax of kinetic The experiment was carried out in a capacitive rf energy (disregarding nonlinear effects and possible discharge in argon under pressure P = 0.1 Torr for a dis- variation of system parameters within the trajectory of charge power W = 60 W. The schematic diagram of the particles) can be obtained by additionally taking into experimental setup is shown is Fig. 1. Melamine form- 2 account deviations 〈〉y (7) in Eq. (6b) through varia- aldehyde particles with different sizes ap were sus- 2 pended above the lower electrode at different distances tion of field E ∝ l . In this case, for θ 1, we obtain ϕ r 0 1 h from its surface (see table). The surface potential s of macroparticles could be derived from balance equa- ∆s max 4 2 ω2 T = ---l p ym p, (11) tion (5) for the gravitational force mpg and the electric 3 〈 〉 force e Zp E(y); proceeding from this equation, the ϕ max 〈〉2 relation between the value of s and electric field E(y) while the maximum amplitude Ay = 2 y of par- can be defined by the relation ticle displacement in the y direction is given by m g max ≈ ϕ ()y Ey()= ------p -, (13) Ay 1.6l p. (12) s a p µ ρ For the example considered above (ap = 1 m, = which can be used for determining the dependence of 3 µ θ α ≈ ϕ 2 g/cm , lp = 300 m, and 1 1), for y 6Ð function s(y)E(y) on height h from the results of mea- 12 V/cm2 (see Section 2), the value of ∆sTmax ≈ 25Ð surements (see table and Figs. 2a and 2b). max ≈ µ The spatial potential ϕ in the near-electrode plasma 50 eV and Ay 480 m. It should be noted that the growth of amplitude and kinetic energy is limited both layer was measured by a compensated Langmuir probe at various heights relative to the electrode. The mea- in real experiments and in simulation of systems with a ∆ϕ macroparticle charge gradient [15, 16]. sured potential difference between points h = 1.1 cm and h = 0.6 cm amounted to 1.5 V. Then, the It should be noted in conclusion that the mechanism experimental data were approximated under the considered here can explain parametric buildup of 〈 〉 ϕ ≈ assumption of small variation of charge Zp ( s const, vibrations of particles with charge gradients, which is see Section 2) by linear (3) and quadratic (4) functions observed in a numerical experiment [12] upon a for E, where y = d Ð h. The near-electrode layer decrease in the frictional force (ν ) below a certain crit- max fr thickness d and coefficients C and C in these ical value. The reason behind such a buildup remains max 1 2 unclear in the framework of [12] since the effects asso- approximations were obtained through the best fitting ≈ 2 ≈ ciated with collective thermal fluctuations of particles of experimental data and were C1 12 V/cm and dmax ≈ 3 in a dust cloud were eliminated from the theoretical 1.1 cm for the linear field (3), and C2 16.2 V/cm and ≈ analysis. In the case of “anomalous heating” of macro- dmax 1.26 cm for the quadratic dependence (4). The particles due to discreteness of charging currents, exter- results of approximation are shown in Figs. 2a and 2b. nal electric forces also serve as the main source of addi- The mean-square errors of the linear and quadratic tional energy of a dust particle [18]. On the other hand, approximations are approximately equal to 4% and the proposed mechanism is ensured by collective 10%, respectively. The higher value of error in the latter effects, which are possible only when the number of case is explained by the strong mismatching between

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1042 VAULINA et al.

1.09 4 55 1.06 2 3 (a) 1 (b) 4 3 45 1.03 1 exp ) E

s 2 /cm 2 35 ϕ 1.00 /( , V ap ) E s E

25 s 0.97 ϕ ϕ ( 15 0.94

5 0.91 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 h, cm h, cm ϕ ϕ ϕ Fig. 2. Dependence of (a) sE and (b) the ratio of various approximations ( sE)ap to experimental values ( sE)exp on the height h over the electrode cut for a linear field E (3) (1), formula (15a) (2), formula (15b) (3), and quadratic approximation of E (4) (4). The dots show the results of measurements. the nonlinear approximation (4) and the results of mea- In spite of a low probability of determining the mac- β surements that is observed as we approach the electrode roparticle charge gradient y = dZp/dy correctly, we (upon a decrease in h). approximated the experimental data by the following 〈 〉 functions: The electric fields and charges Zp of macroparti- cles reconstructed in the linear electric field approxima- ϕ ()β 2 tion are given in the table. It can easily be seen that the sEC= 1 y + y y , (15a) error ϕ ()2 β 3 sEC= 2 y + y y . (15b) 〈〉Z Ð 〈〉Zc δ = ------p p 〈〉c Thus, we have taken into account possible gradients of Z p macroparticle charges in the linear (3) and in qua- in the measured charge 〈Z 〉 relative to the value given dratic (4) approximations of the field for the measured p function (ϕ E) (13). As a result, we obtained close val- by the relation s ues of fields E(y), charges Zp, and the layer thickness ϕ d in both cases. The charge gradient for the linear 〈〉c sa p ≡ max Z p = Ð------Cap, (14) β 〈 〉 ≈ Ð1 e field (15a) turned out to be positive ( y/ Zp 0.2 cm ), which readily follows from the behavior of the linear where C = 4632 µmÐ1, is completely determined by the approximation (curve 1 in Fig. 2a) of the results of error in the linear approximation of E. Since the field E measurements, but contradicts the theoretical predic- was determined by gravity both in the linear (3) and in tions (see Section 2). For (15b), the reconstructed β 〈 〉 ≈ Ð1 quadratic (4) approximation, a close coefficient C = charge gradient was y/ Zp Ð0.37 cm , which may be 4598 µmÐ1 for relation (14) was determined with an true in fact since the relative changes in the charge, δ ∆ 〈 〉 error from 7% (for h = 0.77 cm) to 30% (h = 0.48 cm) iZp/ Zp , did not exceed 20% in this case even for the in this case also (see Figs. 2a and 2b). Thus, we could closest point to the electrode with h = 0.48 cm. not detect in our experiments any appreciable changes in the charge of macroparticles due to a change in their charging conditions in the layer, which is in complete 5. ANALYSIS OF THE RESULTS accordance with the theoretical predictions described in OF EXPERIMENTAL OBSERVATIONS Section 2. OF VERTICAL VIBRATIONS OF PARTICLES ϕ IN THE NEAR-ELECTRODE LAYER Considering that s = ÐzTe/e in the OLM approxima- tion (z ≈ 3Ð4 for argon [19, 20]), we can estimate the OF AN RF DISCHARGE electron temperature from the reconstructed value of The results of experiments described in the previous µ Ð1 ≈ C = 4600 m : Te 1.7Ð2.2 eV. This value matches the section indicate that variations of the charging condi- ≈ ± values of Te 1.9 0.3 eV obtained from independent tions in the upper part of the near-electrode layer of an probe measurements of electron temperature in the rf discharge do not affect significantly the charge of near-electrode layer of the experimental setup in the light macroparticles levitating in this region. However, absence of dust. even an insignificant change in the dust charge (see

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 ANALYSIS OF MACROPARTICLE CHARGING IN THE NEAR-ELECTRODE LAYER 1043

10000 100 (a) (b) )

1000 2 0 1 P ( y m A µ 10 )/ , i y P A ( 100 y 1 A 2

10 1 10 30 50 70 90 10 30 50 70 90 Pressure, mTorr Pressure, mTorr

Fig. 3. Dependence of (a) amplitude Ay of vertical vibrations and (b) the ratio Ay(Pi)/Ay(P0) on pressure P of the rf discharge in argon µ max max for particles of radius ap = 1 (1) and 2.1 m (2). Dashed lines correspond to the values of Ay (a) and Ay /Ay(P0) (b), while solid lines in (b) correspond to approximation (16).

β Section 3) may cause gradients y sufficient for the evo- age kinetic energy of lighter particles of radius ap = lution of stochastic dust vibrations. 1 µm, which was obtained from an analysis of their Experiments were conducted for particles with dif- velocity spectrum for the minimal pressure, exceeded µ 3 eV, while the energy of particles with a = 2.1 µm ferent sizes (ap = 1 and 2.1 m) in the near-electrode p layer of a capacitive rf discharge in argon under pres- attained ~10 eV. sures from 0.1 to 0.015 Torr. Under certain conditions 2θ 2θ (upon a decrease in pressure or upon an increase in the Considering that Pi 1()Pi = P0 1()P0 (see rela- θ number of particles), dust particles acquired energies tion (10)), the value of 1(P0) can be obtained from on the order of 1Ð10 eV and performed irregular verti- relation (16) with the help of best matching of calcu- cal vibrations (in the direction of the gravitational lated and experimental data in the range where Ay(Pi) < θ ≈ field). Here, we consider only one of the possible mech- l0. This procedure gives 1 0.5 for P0 = 0.05 Torr for µ θ ≈ anisms of evolution of such vibrations due to stochastic particles of radius ap = 1 m and 1 0.28 (P0 = µ variation of their charges in a spatially inhomogeneous 0.1 Torr) for particles with ap = 2.1 m. The results of plasma on the basis of the numerical estimates calculation of Ay(Pi)/Ay(P0) (solid line) are shown in described in Section 3. Since the value of kinetic energy Fig. 3b. The dashed line shows the ratio of the maximal s ∆ T acquired by a dust particle due to macroparticle max charge gradients strongly depends on the accuracy of amplitude Ay (12) of particle displacement to its ini- determining the parameters of particles and the sur- tial value Ay(P0) measured in experiments. rounding plasma, we will analyze the relative changes Thus, the evolution of the amplitudes of vibrations in the amplitude Ay of vibrations of dust particles upon being analyzed upon a change in the discharge pressure a decrease in pressure P in the discharge. We will is in qualitative agreement with the proposed mecha- assume that such a decrease in P changes the friction nism of formation of such vibrations. The quantitative ν coefficient fr for macroparticles, but does not lead to a difference between the proposed approximations and noticeable perturbation of the surrounding plasma the results of measurements can be due to the fact that parameters. In this case, we obtain from relations (7) the possible change in the plasma parameters upon a and (8) decrease in pressure or within the trajectory of dust par- ticles was disregarded in the calculation of amplitude θ Ay()Pi 1 Ð 1()P0 max θ ------= ------θ -, (16) Ay and function 1(Pi). Ay()P0 1 Ð 1()Pi where A (P ) is the amplitude of vibrations of particles y i 6. CONCLUSIONS for various pressures Pi (i = 1, 2, ..., N). The effect of nonuniform conditions on dust particle Dependences Ay(P) measured for particles of two sizes are shown in Fig. 3a. The dashed line in the charging in the upper part of the near-electrode layer of same figure shows the boundaries at which the ampli- an rf discharge is analyzed numerically. Simple ana- tude of particle vibrations attains a value close to lytic expressions are given for determining the macro- particle charge gradients. The surface potentials of max ≈ Ay 1.6lp (12), where lp corresponds to the radial macroparticles of different sizes are measured in the particle spacing in an unperturbed dust layer. The aver- near-electrode plasma of the rf discharge. The measure-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1044 VAULINA et al. ments show that the inhomogeneity of the surrounding 7. V. Fortov, A. Nefedov, V. Molotkov, et al., Phys. Rev. plasma does not noticeably affect the charges of light Lett. 87, 205002 (2001). dust particles levitating in the upper part of the near- 8. C. Zafiu, A. Melzer, and A. Piel, Phys. Rev. E 63, 066403 |β | 〈 〉 electrode layer of the discharge in question ( y / Zp < (2001). 0.4 cmÐ1). The results of measurements are in good 9. E. B. Tomme, D. A. Low, B. M. Anaratone, and agreement with the analytic estimates of the macropar- J. E. Allen, Phys. Rev. Lett. 85, 2518 (2000). ticle charge gradients obtained in the OLM approxi- 10. E. Thomas, B. Annaratone, G. Morfill, and H. Rother- mation. mel, Phys. Rev. E 66, 016405 (2002). A possible mechanism of evolution of random 11. A. P. Nefedov, O. S. Vaulina, O. F. Petrov, et al., Zh. vibrations of macroparticles due to their stochastic Éksp. Teor. Fiz. 122, 778 (2002) [JETP 95, 673 (2002)]. motion in the bulk of a spatially inhomogeneous 12. O. S. Vaulina, A. P. Nefedov, O. F. Petrov, and V. E. For- plasma is proposed (in the presence of slight variation tov, Zh. Éksp. Teor. Fiz. 118, 1319 (2000) [JETP 91, of dust charges). Analytic expressions are derived for 1147 (2000)]. estimating the amount of kinetic energy acquired by 13. S. Nunomura, T. Misawa, N. Ohno, and S. Takamura, macroparticles due to the given mechanism. These esti- Phys. Rev. Lett. 83, 1970 (1999). mates show that the kinetic energy of light dust parti- 14. V. E. Fortov, A. G. Khrapak, S. A. Khrapak, et al., Phys. cles may attain values on the order of 1Ð10 eV, which is Plasmas 7, 1374 (2000). close to the experimentally observed energies. The pro- 15. V. V. Zhakhovskiœ, V. I. Molotkov, A. P. Nefedov, et al., posed mechanism forms the basis of analysis of the val- Pis’ma Zh. Éksp. Teor. Fiz. 66, 392 (1997) [JETP Lett. ues of amplitude of macroparticle vibrations with dif- 66, 419 (1997)]. ferent sizes measured in the near-electrode layer of the 16. A. Samarian, B. James, O. Vaulina, et al., in Proceedings rf discharge under various pressures. It is shown that of 25th International Conference on Phenomena in Ion- the given mechanism may be responsible for the evolu- ized Gases, Nagoya Univ., Nagoya, Japan (2001), Vol. 1, tion of stochastic vertical vibrations. p. 17. 17. A. Samarian, B. James, S. Vladimirov, and N. Cramer, ACKNOWLEDGMENTS Phys. Rev. E 64, 025402 (2001). 18. O. S. Vaulina, S. A. Khrapak, A. P. Nefedov, and This work was partly financed by the Russian Foun- O. F. Petrov, Phys. Rev. E 60, 5959 (1999). dation for Basic Research (project nos. 01-02-16658 19. O. S. Vaulina, A. P. Nefedov, O. F. Petrov, et al., Zh. and 00-02-17520), INTAS (grant no. 01-0391), and Éksp. Teor. Fiz. 120, 1369 (2001) [JETP 93, 1184 Australian Council of Scientific Research. The research (2001)]. work of A.A. Samarian was supported by the Univer- 20. S. A. Khrapak, A. P. Nefedov, O. F. Petrov, et al., Phys. sity of Sydney research scholarship U2000. Rev. E 59, 6017 (1999). 21. J. Goree, Plasma Sources Sci. Technol. 3, 400 (1994). REFERENCES 22. Yu. P. Raœzer, M. N. Shneœder, and N. A. Yatsenko, High- 1. T. Trottenberg, A. Melzer, and A. Piel, Plasma Sources Frequency Capacitive Discharge: Physics; Experiment Sci. Technol. 4, 450 (1995). Technology; Applications (MFTI–Nauka “Fizmatlit,” Moscow, 1995). 2. J. B. Pieper and J. Goree, Phys. Rev. Lett. 77, 3137 (1996). 23. A. A. Ovchinnikov, S. F. Timashev, and A. A. Belyœ, Kinetics of Diffusely-Controlled Chemical Processes 3. A. A. Homann, A. Melzer, S. Petrs, and A. Piel, Phys. (Khimiya, Moscow, 1986). Rev. E 56, 7138 (1997). 4. A. A. Homann, A. Melzer, and A. Piel, Phys. Rev. E 59, 24. Photon Correlation and Light Beating Spectroscopy, Ed. R3835 (1999). by H. Z. Cummins and E. R. Pike (Plenum, New York, 1974; Mir, Moscow, 1978). 5. U. Konopka, G. E. Morfill, and L. Ratke, Phys. Rev. Lett. 84, 891 (2000). 25. A. A. Samarian and B. W. James, Phys. Lett. A 287, 125 (2001). 6. A. A. Samaryan, A. V. Chernyshev, O. F. Petrov, et al., Zh. Éksp. Teor. Fiz. 119, 524 (2001) [JETP 92, 454 (2001)]. Translated by N. Wadhwa

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1045Ð1054. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1188Ð1199. Original Russian Text Copyright © 2003 by Anshukova, Golovashkin, Ivanova, Rusakov. SOLIDS Structure

The Effect of Superstructural Ordering on the Properties of High-Temperature Oxide Superconductor Systems N. V. Anshukovaa, A. I. Golovashkina,*, L. I. Ivanovab, and A. P. Rusakovb aLebedev Institute of Physics, Russian Academy of Sciences, Moscow, 119991 Russia bMoscow Institute of Steel and Alloys, Moscow, 117936 Russia *e-mail: [email protected] Received November 12, 2002

Abstract—The effect of superstructural ordering in the oxygen sublattice (in addition to the influence of the antiferromagnetic interaction of copper ions) on the electron and phonon characteristics of oxide high-temper- ature superconductor (HTSC) systems has been studied. Taking into account this ordering effect, it is possible to explain a wide range of experimental data, including doping-induced changes in shape of the Fermi surface, features of the phonon spectra, the existence of stripes, the presence of a pseudogap and its coexistence with the superconducting gap, and some peculiarities in the phase diagrams of HTSCs. © 2003 MAIK “Nauka/Inter- periodica”.

≈ 1. INTRODUCTION In a low-doped La2 Ð xSrxCuO4 Ð δ compound (x 0.07) occurring in the metallic state, ARPES measurements In recent years, there was considerable progress in revealed coexisting dielectric and metallic phases on a the development of methods for the synthesis of high- microscopic level [1, 2]. In these experiments, a sample quality single crystals of various high-temperature exhibited two branches in the plot of energy E versus superconductor (HTSC) systems—compounds charac- wavevector k, one of these branches being characteris- terized by high Tc values in a certain range of composi- tic of a purely dielectric phase and the other, of an opti- tions. By changing the level of doping, it is possible to mum doped metallic phase. Only one of these branches vary the state of such compounds from insulating to E(k)—namely, that with a higher binding energy—is metallic. The availability of high-quality samples retained in the dielectric state at x < 0.05, while only the allowed the electron, phonon, and other characteristics second branch remains in the metallic state at a high of oxide superconductors, as a function of the level of level of doping (x ≥ 015). doping, to be thoroughly studied. Such investigations have been performed for HTSC compounds belonging Analogous coexistence of two phases (dielectric and to various systems, including La Sr CuO δ, metallic) on the microscopic level was revealed by neu- 2 Ð x x 4 Ð tron diffraction in the study of the dispersion of high- YBa Cu O , Bi Sr Ca Y Cu O δ, etc. Some new 2 3 6 + x 2 2 1 Ð x x 2 8 Ð frequency longitudinal optical (LO) phonons in features in the electron and phonon properties were La − Sr CuO δ [3, 4], YBa Cu O [5, 6], and other found for HTSCs at low doping levels and in the vicin- 2 x x 4 Ð 2 3 6 + x HTSC systems. At an intermediate level of doping, two ity of the dielectricÐmetal transition. For example, the ω data of angle-resolved photoelectron spectroscopy LO phonon frequencies LO were observed, one of (ARPES) showed [1, 2] that compounds of the these being characteristic of a metallic phase (observed La − Sr CuO δ system with x ≥ 0.05 are character- in strongly doped compounds) and the other, of a purely 2 x x 4 Ð dielectric low-doped phase. When the level of doping ized, in the vicinity of the dielectricÐmetal transition, ω by a Fermi surface with wide flat regions in the plane of was varied, both LO values remained virtually unchanged and only the intensities of the correspond- (kx, ky) wavevectors, these regions being parallel to the [100] and equivalent directions. The cross section of ing neutron diffraction lines exhibited redistribution. this Fermi surface initially possesses a finite area close As the x value increased, the volume of the metallic to that characteristic of the optimum doping level, phase exhibited growth at the expense of decreasing rather than changing in proportion to the parameter x. content of the dielectric phase. As the x value increases up to x ≈ 0.15, the shape of the The whole body of these and other experimental Fermi surface remains virtually unchanged, since only observations (including data on the static and dynamic intensity of the ARPES lines increases. Only upon dop- magnetic superstructure modulation [7], negative ther- ing to a level exceeding the optimum (x ≥ 0.17) does the mal expansion at low temperatures [8], etc.) poses a Fermi surface exhibit significant changes, and, on question about coexistence and mutual ordering of the reaching x ≈ 0.25, it becomes parallel to directions of metallic and dielectric phases on the microscopic level the [110] type. in HTSCs. All these experimental data can hardly be

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1046 ANSHUKOVA et al. explained within the framework of the existing theoret- This formula unit precisely corresponds to the unit cell ical models assuming uniform electron density distri- of the dielectric compound La2CuO4. bution in the course of doping. The above mechanism of period doubling is essen- As will be demonstrated below, the new data have a tially a manifestation of the JahnÐTeller effect typical natural explanation if we take into account the presence of oxygen-containing copper compounds. This effect of a superstructural charge order in the oxygen sublat- removes degeneracy between the 2px and 2py states of tice, in addition to the well-known antiferromagnetic oxygen ions in the CuO planes, which is accompanied order in the sublattice of copper ions in cuprate HTSCs. 2 We will consider some features of the superstructural both by elastic straining and by charge redistribution order in the oxygen sublattice, which influence the between ions in the lattice. As a result, the system electron and phonon spectra of these HTSC systems. achieves an energy gain, typically of about 1 eV per Allowance for this superstructural charge order in the unit cell [11]. In the case under consideration, this is oxygen sublattice explains the nature of coexisting manifested by the appearance of a bandgap Eg ~ 2 eV. dielectric and metallic phases on the microscopic level According to the neutron diffraction data [12, 13], a characteristic displacement of oxygen ions in the CuO2 observed in doped samples of high-Tc cuprate super- conductors. plane for a period modulation comparable with the lattice period is on the order of 0.004 Å. Such changes can hardly be detected at the existing level of accuracy 2. DIELECTRIC STATE OF HTSC SYSTEMS in phonon dispersion measurements or X-ray diffrac- tion analysis. Let us consider the dielectric state of HTSC systems Ð1.5 using an example of the La2 Ð xSrxCuO4 Ð δ system, Four oxygen ions in the charged state O (per unit beginning with a compound with x = 0 and δ = 0. The- cell with a period doubled in three directions) is the oretical calculations show that this undoped compound minimum number of ions for which the degeneracy is in the initial state must occur in the metallic state [9]. removed for all oxygen octahedra. The lattice period According to these calculations, the lower and upper doubling leads to the appearance of a new reciprocal π Hubbard bands (composed mostly of the 3d states of lattice vector G2 = G/2 = ( /a)[100], where G = copper) are separated by a wide band composed pre- (2π/a)[100] is the vector of a reciprocal lattice without dominantly of the 2p states of oxygen. It is the presence period doubling and a is the CuÐCu distance in the of the latter band that imparts metallic properties to the CuO2 plane. Recently, Sachdev [14] has independently compound; such a metal is referred to as the praphase. demonstrated that the period doubling follows from a Previously [10], we have demonstrated that oxygen general theoretical analysis of HTSC systems. ions in the CuO2 planes of the praphase can exist in two charged states, OÐ2 and OÐ1.5, which correspond to the Since OÐ1.5 ions are less strongly bound to the lattice CuÐOÐ2 ionic bonds and the CuÐOÐ1.5 ionic-covalent than the OÐ2 ions, the electron states of the former ions bonds, respectively. Ordering of the latter covalent are situated most closely to the chemical potential µ of bonds, with allowance of the antiferromagnetic order- the dielectric. Therefore, these states are at the top of a ing of copper ions, leads to doubling of the lattice complex valence band including a narrow oxygen band period of the metallic praphase of the CuO2 plane in the and the lower Hubbard band. The aforementioned [100] and equivalent directions. This situation is equiv- ordering of these weakly bound OÐ1.5 ions leads, as alent to the appearance of a charge density wave noted above, to the appearance of a narrow about (CDW) in the sublattice of oxygen ions. For this reason, (0.3 eV), almost purely oxygen valence band formed a narrow oxygen band (instead of the aforementioned by 2p states of OÐ1.5 ions, lying above the wide (~3 eV) wide band) and a dielectric gap with a width of Eg ~ mixed copperÐoxygen Hubbard band formed by 2 eV appear at the top of the lower Hubbard band. As a Cu(3d)ÐOÐ2(2p) states. This energy band diagram is result, a dielectric state rather than a metallic state is schematically depicted in Fig. 1a, indicating typical observed in real undoped HTSC systems of the type experimental energy values [15]. A small width of the under consideration. upper valence band formed by the 2p states of OÐ1.5 ions In the neighboring CuO plane, the CuÐOÐ1.5 cova- is determined by a relatively small overlap of these 2 states, since the unit cell contains only four such ions lent bonds are ordered in the perpendicular direction, Ð2 whereby a dielectric gap appears in the c-axis direction (against 28 of O ions). as well. As a result, a new unit cell is formed that con- The upper valence oxygen band formed by 2p states tains four OÐ1.5 ions in addition to the OÐ2 ions. An exact of OÐ1.5 ions contains 4 × 1.5 = 6 electrons per cell. chemical formula describing the dielectric compound These electrons fill the Brillouin zones for a square La CuO with allowance for the period doubling in all 2 4 quasi-two-dimensional lattice including two CuO2 three directions can be written as planes. In the case of a dielectric, the structure of three first Brillouin zones filled with electrons is schemati- ()+3 +1.75 Ð1.5 Ð2 8La2CuO4 = La16 Cu8 O4 O28. (1) cally depicted in Fig. 2a. The third Brillouin zone con-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 THE EFFECT OF SUPERSTRUCTURAL ORDERING ON THE PROPERTIES 1047

(a) (b) E E Upper Upper Hubbard Hubbard band band

∆* ~ 2 eV 2pO–1.5 µ ~ 0.3 eV EF

~ 0.3 eV 2pO–1.5

~ 3 eV Lower Lower Hubbard Hubbard band band 3dCu + 2pO–2 3dCu + 2pO–2

N(E) N(E)

Fig. 1. Energy band diagram of cuprate HTSCs in the (a) dielectric and (b) metallic state at an optimum level of doping. The arrow in (a) indicates a narrow band formed by the 2p states of OÐ1.5 ions; blackened region in (b) is filled by holes upon doping (E is the µ ∆ energy; is the chemical potential; EF is the Fermi energy; N(E) is the density of electron states; * is the Peierls energy gap).

tains two electrons, which corresponds to x = xc = 1/8 = within the framework of the tight binding method, by 0.25 electron per copper ion in formula unit (1). the formula [16Ð18] () E()k = Ð4 2tkcos x + cosky Ð t'coskx cosky (2) 3. DOPING AND ELECTRON STRUCTURE ()()2 Ð2t''cos 2kx + cos 3ky Ð t⊥ coskx Ð cosky /4, In La Sr CuO slightly doped with strontium, 2 Ð x x 4 where k = (kx, ky) is the dimensionless wavevector of charge carriers (holes) might be expected at the top of the quasi-two-dimensional reciprocal lattice; t, t', and t'' the narrow oxygen valence band. In the case of degen- are the overlap integrals with the nearest neighbor, eracy, the Fermi surface would pass near the boundaries next-to-nearest, and third-shell ions; and t⊥ is the over- of the third Brillouin zone. The distance between the lap integral for the interaction between adjacent CuO2 boundaries of the third Brillouin zone and the Fermi planes. The maxima calculated for the valence bands surface must be proportional to the dopant concentra- coincide with the boundaries of the Brillouin zones for tion (i.e., to the x value). However, as mentioned above, the dielectric state (Fig. 2a) corresponding to t = ARPES reveals the Fermi surface far from the bound- 386 meV, t'/t = −0.272, t''/t = 0.223, and t⊥ = 150 meV. aries of the third Brillouin zone (Fig. 2b). For example, the Fermi surface passes approximately in the middle Figure 2b presents the shape of the Fermi surface between points (0, 1) and (1/2, 1), that is, close to the calculated [16] for doped La2 Ð xSrxCuO4 with x = point (1/4, 1). This boundary remains almost unshifted 3xc/4 = 0.1875, which corresponds to t = 0.5 eV, t'/t = when the level of doping increases up to the optimum −0.3, t''/t = 0.2, and t⊥ = 0.15 eV in formula (2). Using level of x = 0.15 [1]. In the case on weak doping (0.05 < the overlap integrals, it is possible to estimate the dis- x < 0.13), the system exhibits, as was also noted above, persion E(k) for the line between points (0, 1) and coexistence of the metallic and dielectric phases [2]. (1/2, 1). The corresponding dispersion curves are con- structed in Fig. 3 for both pure dielectric and a doped These and other new experimental data can be compound. In Fig. 3a, point (1/2, 1) corresponds to the explained within the framework of the model consid- boundary of the third Brillouin zone. The width of the ered below. The results presented in Fig. 2 are obtained band filled with electrons is about 0.3 eV. In the case of by direct calculations of the electron structure. The val- slight doping, holes might appear at the top of this ues of the dispersion E(k) obtained by these calcula- valence band as depicted in Fig. 3b. In the case of tions for the upper valence bands can be approximated, degeneracy, the Fermi surface would pass near the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1048 ANSHUKOVA et al.

(a) E (a) 11, 0, 1 1, 1 µ 3 3 2 3 1 3 10, 221, 0 0, 0 3 3 2 0 k 3 3 (0, 1) 1, x --- 1 11, 01, 11, 2 E G = 2π/a[100] (b) (b) (c) 1 1 E 11, ---, 1 ---, 1 1, 1 F 4 4 11, 0, 1 1, 1

1 1 1, --- 1, --- 4 4 0 10, 1, 0 1 1 0, 0 1 0, 0 (0, 1) kF ---, 1 kx 1, --- 1, --- 2 4 4 E (c) 1 1 11, ---, 1 ---, 1 11, 11, 01, 11, 4 G4 4 G2

Fig. 2. The Brillouin zones and the Fermi surface shapes in EF cuprate HTSCs at various levels of doping. Positions of the π symmetry points are indicated in /a units (a is the mini- 0 mum CuÐCu distance). Crosses indicate the experimental 1 1 (0, 1) ---, 1 ---, 1 kx ARPES data [1, 2]: (a) the first three Brillouin zones (indi- 4 2 cated by numbers) for a flat CuO2 lattice with doubled period (for the dielectric phase); dashed line shows the first E (d) Brillouin zone for the flat CuO2 lattice without period dou- bling; (b) the Fermi surface (solid curve) for La2 Ð xSrxCuO4 with 0.05 ≤ x ≤ 0.15; dashed line shows the boundary of the third Brillouin zone; cross-hatched region corresponds to µ filled electron states; G4 = G2/2 = G/4 are reciprocal lattice ∆* vectors, where G = (2π/a)[100]; (c) the Fermi surface (solid curve) in the case of strong doping (x > 0.25); dashed line shows the boundary of the second and third Brillouin zones 1 1 1 1 (for x = 0.25, the Fermi surface coincides with the bound- ---, 1 ---, 1 (0, 1) ---, 1 ---, 1 kx aries of the second Brillouin zone); cross-hatched region 2 4 4 2 corresponds to filled electron states (for x = 0.3). G4 G2 boundaries of the third Brillouin zone. However, in the Fig. 3. Schematic dispersion curves E(k) for cuprate case of a narrow band and weak screening, as in the sys- HTSCs in the direction between points (0, 1) and (1/2, 1): tem under consideration, it is energetically favorable to (a) dielectric state; (b) the case of weak doping in the model of uniform hole distribution (shaded region corresponds to fill this band in separate regions of the crystal to half of the states occupied by holes); (c) the case of a Brillouin the reciprocal lattice vector (Fig. 3c) at the expense of zone half-filled with holes with respect to the momentum, holes liberated from some intermediate regions. For i.e., up to the point (1/4, 1); (d) the formation of the Peierls such regions half-filled with holes, the lattice period gap ∆* and the new reciprocal lattice vector as a result of can exhibit another doubling in the [100] direction. nesting by the vector G4 = G/4. These regions are characterized by the reciprocal lattice vector G = G/4 (Fig. 3d) representing the nesting vec- 4 over the whole sample as depicted in Fig. 3b, the crystal tor (here, G is the reciprocal lattice vector for the initial exhibits separation into dielectric regions free of holes undoubled direct lattice). (with the dispersion such as in Fig. 3a) and half-filled As is well known, nesting gives rise to the Peierls regions with the Peierls gap and fourfold lattice period instability with the formation of a dielectric gap ∆* (with the dispersion such as in Fig. 3d). The densities of (Fig. 3d). As a result, instead of the uniform band filling states corresponding to the dielectric regions and the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 THE EFFECT OF SUPERSTRUCTURAL ORDERING ON THE PROPERTIES 1049 regions with Peierls gaps ∆* are presented in Figs. 1a E (a) and 1b, respectively. This scheme provides for a certain µ energy gain: the dielectric regions are characterized by a decrease in the Coulomb energy of repulsion between holes, while the regions with Peierls gaps exhibit a decrease in the kinetic energy as a result of the for- mation of gap ∆*. It should be emphasized that this phase separation takes place in the crystal with ordered 1 1 1 1 1 1 Ð1.5 (0, 1)0, --- (0, 0) ---, --- (0, 1) ---, 1 ---, --- (1, 0) CuÐO covalent bonds. 2 2 2 2 2 2 In the above scheme, the system separates into the E

metallic stripes with half-filled zone and the intermedi- (b)

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❙ ❙ (holes) appears upon the formation of the first metallic ❙ EF stripes. For example, in La2 Ð xSrxCuO4 the degeneracy is experimentally observed for x ≈ 0.02Ð0.05, depend- ing on the crystal quality [20]. ~0.3 eV ~0.3 eV Figure 4a shows schematic dispersion curves E(k) (0, 1), 1 (0, 0) 1, 1 (0, 1)1, 1 1, 1, 1 (1, 0) for the upper valence band along some symmetric 0 --- --------------- 1 ------directions in the Brillouin zone of cuprate HTSCs. The 2 2 2 2 2 2 2 2 curves, obtained from an analysis of relation (2), refer Fig. 4. Schematic dispersion curves E(k) for the upper to a dielectric state (Fig. 4a) and a metallic state at opti- valence band of cuprate HTSCs along some symmetric directions in the Brillouin zone: (a) dielectric state; mum doping (x = x0 = 0.1875). (b) metallic state at the optimum doping (x = x0); cross- Since doping in La2 Ð xSrxCuO4 is realized in the hatched regions correspond to the states filled by holes. form of homogeneously distributed strontium ions, the inhomogeneous distribution of holes must lead p (according to the above scheme) to an additional Cou- (a) lomb interaction between the metallic regions, result- ing in a certain ordering of these regions (Fig. 5). Owing to the charge density fluctuations, this order can p0 possess a dynamic character. Indeed, a dynamic order of this kind was observed in La2 Ð xSrxCuO4 [19]. On 0 the other hand, in the presence of defects (appearing a upon doping with atoms possessing strongly different 4a L atomic dimensions, such as La and Nd), a static order p in the metallic regions can appear as well. Such a (b) static ordering was also experimentally observed in (La,Nd)2 Ð xSrxCuO4 [12, 20] and La2CuO4 + δ [21]. p0 Experimental data [22] showed that ordering leads to the formation of a stripe structure. The stripes are 0 spatially separated and oriented either along vector [100] or along [010] [22]. In the case under consider- L = 4a ation, with the unit cell containing two CuO planes, it 2 Fig. 5. Schematic diagrams of charge distribution in one of would be natural to assume that direction [100] is char- the symmetric directions in the CuO2 plane (a) in the case acteristic of one of these planes and direction [010], of of intermediate doping (0 < x < x0) and (b) in the absence of ≈ the adjacent plane. This ordering of stripes in two mutu- dielectric spacers between metallic stripes (x x0 = 0.1875) ally perpendicular directions in the adjacent CuO2 (p is the hole density; p0 = 0.1875 is the hole density per planes in a particular case of (La,Nd)2 Ð xSrxCuO4 was copper ion in a metal stripe; L is the period of charge mod- experimentally revealed [12] by analysis of the experi- ulation). mental neutron diffraction data. Figure 5 schematically illustrates the distribution of of the (1/4, 1/4) type. Note also that the Peierls gap is charge density in one of the symmetric directions in the not formed along the stripes. CuO2 plane. An analogous pattern is observed in the perpendicular direction in the adjacent CuO2 plane. As According to the scheme under consideration, the can be seen in Fig. 2b, the conditions of nesting with the metallic conductivity of doped samples is related to the vector G4 = G/4 are satisfied on a considerable part of fact that the Peierls gap is not formed in the vicinity of the Fermi surface, so that this pattern holds for almost points of the (1/4, 1/4) type. According to this, the all these states (except regions of the vicinity of points Peierls gap ∆* in Fig. 1b is depicted as partly filled and

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1050 ANSHUKOVA et al. main Bragg diffraction reflections and their splitting upon superstructural modulation. The superstructural reflections differ in intensity by many orders of magni- tude from the main reflections (observed for the lattice without superstructural modulation). All the super- (0, 1, 0) (1, 1, 0) structural reflections were separately observed in ε experiment [12, 14, 20]. In particular, reflections corre- G sponding to the period doubling were reported in [14]. Reflections related to the antiferromagnetic ordering of G2 copper ions, as well as the 2ε reflections due to a peri- 2ε G4 odic distribution of metallic regions with the spatial period L, we reported in [12, 20] and in some other 1 ε ---,,00 papers. The intensity of, for example, 2 reflections was 2 smaller by six orders of magnitude (about one million (0, 0, 0) (1, 0, 0) times) as compared to that of the main Bragg reflections [12]. The intensity of superstructural reflections related Fig. 6. Schematic diagrams showing the pattern of Bragg to the period doubling can be even smaller. Apparently, reflections and their splitting for the reciprocal lattice of this smallness accounts for the absence of experimental oxide HTSCs in the region of coexistence of the dielectric data simultaneously revealing all types of reflections and metal phases according to the proposed model of super- depicted in Fig. 6. structural ordering: large black circles indicate reflections from the lattice without superstructural modulation; open The value of 2ε varies, depending on the level of circles indicate reflections due to the lattice period dou- doping, from 2ε = 0 for x = 0 (i.e., for L ∞) to 2ε = bling; dots indicate reflections due to fourfold lattice peri- G/4 for L = 4a (G is the absolute value of vector G). The ods; squares indicate reflections due to the distribution of metallic stripes with the period L (2ε = 2π/L); crosses indi- latter case (depicted in Fig. 5b) corresponds to the cate reflections due to the antiferromagnetic ordering of absence of dielectric spacers between metallic stripes, copper ions. Positions of the symmetry points are indicated whereby the entire crystal consists of regions with four- π π ≤ ε ≤ in 2 /a units; G4 = G2/2 = G/4 = (2 /a)/4; 0 2 G4. The fold lattice period. In this case, holes fill three-quarters intensity of superstructural reflections is lower by many of states in the third Brillouin zone (see the cross- orders of magnitude than that of the main Bragg reflections. hatched region in Fig. 7a). Complete filling of the third Brillouin zone is attained at a hole density correspond- the Fermi level (E ) is indicated rather than the chemi- ing to x = 0.25 per copper ion, while filling of the three- F quarters of states in this zone corresponds to a doping cal potential. × level of x = x0 = 0.25 3/4 = 0.1875. The charge ordering depicted in Fig. 5 must be accompanied by elastic straining with the same spatial Figure 7b shows dependence of the superstructural ε period. According to the neutron diffraction data for charge modulation 2 on the doping level x. Here, the dashed curve shows our estimate constructed as (La,Nd)2 Ð xSrxCuO4 [12, 13], the deformation (e.g., the shift of oxygen ions in the CuO2 plane) amounts to 2ε()x = () 4x/3 ()2π/a ,0≤≤xx, (4) approximately 0.004 Å for x = 0.12. The period L of 0 charge ordering in Fig. 5 is related to the doping level 2ε(x )= () 1/4 ()2π/a , x ≤≤x 0.25. (5) x as 0 L = 3a/4x, (3) Symbols in Fig. 7b represent the experimental data for La2 Ð xSrxCuO4 and (La,Nd)2 Ð xSrxCuO4 [12]. As can be ≤ ≤ where x can vary within the interval 0 x x0 = 0.1875. seen, the experimental points fit the calculated curve Theoretical calculations (reviewed in [12]) indicate that well. At x > 0.1875, the value of 2ε ceases to change ε charged metallic stripes in the strained CuO2 planes and remains equal to 2 = G4 = G/4 = 0.25. The corre- with antiferromagnetically ordered spins of copper ions sponding magnetic modulation is ε = 0.125. The value play the role of the boundaries of domains with differ- of x0 = 0.1875 corresponds to a certain critical point on ent phases of antiferromagnetic order. From this, it fol- the T(x) phase diagram. lows that the charge modulation period L corresponds For x ≥ 0.1875, holes appear at the top of the valence to the superstructural antiferromagnetic modulation band (below the Peierls gap ∆*) near the point (1/4, 1) with a period of 2L. In the reciprocal space, the charge ε as depicted in Fig. 3d. A further increase in the level of ordering corresponds to the vector 2 with the modulus doping (x > 0.1875) leads to a decrease in the Peierls 2ε = 2π/L, and the antiferromagnetic ordering corre- ∆ ε ε π π gap width * (related to an increase in the hole screen- sponds to the vector with the modulus = 2 /2L = /L. ing) and is accompanied by a decrease in intensity of The relationship between the charge and magnetic the neutron diffraction lines corresponding to super- ordering in the reciprocal space according to the pro- structural charge and spin ordering. In addition, it is posed scheme is illustrated in Fig. 6, which shows the possible to calculate a change in the Fermi surface for

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 THE EFFECT OF SUPERSTRUCTURAL ORDERING ON THE PROPERTIES 1051 x > 0.1875 by formula (2) taking into account the (a) dependence of t' and t'' on x, G4 () , t' = t0' 0.25 Ð x /0.25, (6) 11 0, 1 1, 1 () t'' = t0'' 0.25 Ð x /0.25, (7) where t0' = Ð0.15 eV and t0'' = 0.1 eV are the parameters for the state with x = 0.1875 (Figs. 2b and 3c) in which the metallic stripes are not separated by dielectric , spacers. 10 0, 0 1, 0 2ε (b) A maximum value of x, corresponding to the third 0.30 Brillouin zone completely filled with holes, is 0.25 per copper ion. For x ≥ 0.25, the Fermi surface acquires the 0.25 shape represented by the solid curve in Fig. 2c, where 0.20 crosses represent the experimental data [1] for x = 0.3. As can be seen, the calculated shape of the Fermi sur- 0.15 face qualitatively agrees with experiment. At x = 0.25, 0.10 the Fermi surface coincides with the boundaries of the 0.05 second Brillouin zone. In this case, according to the proposed model, the Fermi surface acquires an electron 0 0.05 0.10 0.15 0.20 0.25 0.30 character. This fact has been established in numerous x experiments. Fig. 7. Superstructural modulation 2ε and filling of the third Thus, the proposed scheme for x < 01875 features Brillouin zone with holes: (a) three-quarters of the zone are coexistence of two phases, dielectric and metallic, on a filled with holes for x = x0 = 0.1875; (b) the dependence of local level. This situation is reflected by the ARPES 2ε on x calculated by Eqs. (4) and (5) (dashed line) and plot- data [2]. At a higher level of doping, there exists a single ted from the experimental data for La1.6 Ð xNd0.4SrxCuO4 (metallic) phase, as experimentally confirmed in [1]. and La2 Ð xSrxCuO4 (dots and squares, respectively) [12].

4. EFFECT OF DOPING (a) (b) (c) ON THE PHONON SPECTRUM ω ω ω LO LO LO ω ω ω As noted above, doped HTSC systems exhibit an TO TO TO anomaly in the high-frequency LO phonon dispersion ω curve LO(Q) in the metallic phase. The presence of such anomaly also follows from the proposed scheme of elec- Q QQ tron ordering in HTSCs. As is known [23], the longitudi- 0 G/2 00G/4 G/2 G/4 G/2 nal optical phonons obey an approximate relation Fig. 8. The effect of doping on the dispersion of the phonon ω ω2 ≈ ω2 ω2 ε frequency LO for HTSCs (a) in the dielectric state (the dis- LO()Q TO + p/ (),Q (8) ω persion of TO is ignored), (b) in the case of intermediate doping (0.05 < x < 0.l875; the thickness of lines approxi- ω where TO is the frequency of transverse optical mately reflects the dielectric to metallic phase ratio in the phonons (for simplicity, the dispersion of these crystal), and (c) for x ≥ 0.1875. ω phonons is ignored), p is the plasma frequency of ions given by the formula Figure 8a shows a schematic diagram illustrating the ω ω dispersion of LO and TO in the dielectric state (with ω2 π ()∗ 2 Ω ε p = 4 Ne / M, (9) neglect of the dispersion of (Q)). Upon doping, the system exhibits separation into the dielectric and metal- Ω lic phases on the microscopic level. For the dielectric e* is the effective charge of oxygen ions, N/ is the ω number of oxygen ions per unit cell, ε(Q) is the macro- phase, the dependence of LO on Q remains qualita- scopic permittivity of the electron subsystem, and M is tively the same as in Fig. 8a for x = 0. In contrast, the the mass of an oxygen ion. For the dielectric phase (x = metallic phase exhibits significant changes. The ≤ ≤ 0 for La Sr CuO ) in the long-wave approximation phonons with G/4 Q G/2 connect flat congruent 2 Ð x x 4 regions of the Fermi surface as depicted in Fig. 9. (Q 0), we have ε(Q) ε∞, where ε∞ is the optical dielectric constant. As the Q value increases from 0 to As is known [24]Ð27], in the presence of congruent Q = G/2, the permittivity ε(Q) remains positive. regions of the Fermi surface and nesting for the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1052 ANSHUKOVA et al. ω Thus, the frequency LO at Q = G4 = G/4 must ω2 | | decrease in a jumplike manner by the value 2p / e(G4) as schematically depicted in Fig. 8b. Taking into account that a crystal with 0.05 < x < 0.1875 contains two phases, the real spectrum will display two lines 0, 2 ω with different frequencies LO. The intensity of these phonon lines will be proportional to the volume of the corresponding phase in the crystal. As the content of the metallic phase increases with the level of doping, the intensity of the neutron diffraction lines corresponding ω , to the LO value of this phase grows, while the intensity 11 0, 1 1, 1 of lines corresponding to the dielectric phase in the ≤ ≤ interval Q4 Q G2 drops. This is reflected by differ- ent line thicknesses in Fig. 8b. For x ≥ 0.1875, whereby there is no dielectric spac- ers between metallic stripes, the dispersion of ω 0, 0 LO acquires the shape schematically depicted in Fig. 8c. When x 0.25, the shape of the Fermi surface changes, as can be seen from the comparison of Figs. 2b and 2c. This must be accompanied by qualitative changes 11, 11, ω in the dispersion of LO, but these effects are beyond the scope of this paper. The model described above is quali- tative corroborated by experiment [3Ð6, 15]. G4 G As was noted above, the appearance of double and fourfold lattice periods is accompanied by very small Fig. 9. Schematic diagram showing reciprocal lattice vector shifts of ions (on the order of 0.004 Å). Direct observa- G4 = G/4 connecting flat congruent regions of the Fermi tion of such shifts through the measurement of phonon surface (solid arrow) and G4 + Q vector connecting states dispersion is difficult, but the effect can be detected on the Fermi surface that are symmetric relative to the point through broadening of the neutron diffraction lines [3, 4]. (0, 1) (dashed arrow). It was established [4] that the line half-width signifi- cantly increases and exhibits a maximum for the wavevectors corresponding to a doubled lattice period. wavevectors Q, whereby E(k) = E(k + Q), the static It was also demonstrated [3] that the appearance of the electron susceptibility fourfold lattice period also leads to considerable broad- ening of the lines. 1 f ()k Ð f ()kQ+ χ()Q = ---- ∑------(10) With decreasing temperature, the low-frequency Ω E()kQ+ Ð E()k transverse acoustic phonon branches ω exhibit soft- k TA ening at the boundaries of the Brillouin zone. It can be exhibits divergence when Q is equal to the nesting vec- shown that, without allowance for the interaction tor (f is the distribution function). In the case under con- between ions and the charge density wave, the HTSC sideration, this must take place for Q = G4 = G/4. Under structure would be unstable [10]. The structure stabi- ω these conditions, the macroscopic permittivity ε(Q) of lizes (i.e., TA becomes positive at the boundaries of the the electron subsystem becomes negative. This follows Brillouin zone) only in the presence of this interaction. from the relation [24, 26, 27] On heating from T = 0, the amplitude of the charge den- sity wave and, hence, the intensity of this interaction ()χ4πe2/Q2 ()Q decrease, which must lead to contraction of the crystal ε()Q = 1++------∆ε, (11) 14Ð ()πe2/Q2 L()Q χ()Q (in the temperature range where the charge density wave contribution is decisive). Thus, HTSC systems where e is the electron charge, ∆ε is the nonspecific (especially their dielectric phases) must exhibit nega- contribution to the permittivity, and L(Q) is a correction tive thermal expansion at low temperatures. This anom- for the local crystal field (0 < L(Q) < 1). In Eq. (11), alous thermal expansion was actually observed in χ(Q) denotes the modulus of the static electron suscep- experiment [34Ð37]. tibility. In the case under consideration, ω2 ≈ ω2 ω2 ε 5. SUPERCONDUCTIVITY IN HTSC SYSTEMS LO()Q TO Ð p/ ()G4 (12) Let us study the possibility of superconducting pair- ≤ ≤ for G4 Q G2. ing within the framework of the model proposed above.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 THE EFFECT OF SUPERSTRUCTURAL ORDERING ON THE PROPERTIES 1053 π In Fig. 9, the vector G2 = G/2 = ( /a)[100] is a transla- ∆, ∆* tion vector and the point (0, 1) can be considered as the center of symmetry. The pairing of carriers occurring on the Fermi surface and connected by vectors G4 + Q ∆ passing through the point (0, 1) yields zero total momentum and can be superconducting (such a vector ∆* is shown by the dashed arrow in Fig. 9). These vectors can be the wavevectors of phonons. Thus, at a suffi- ciently low temperature, the states on the Fermi surface that satisfy the condition of nesting and are symmetric with respect to point (0, 1) are involved in interactions of two types. One interaction leads to the formation of the dielectric Peierls gap ∆*, while the other leads to the ∆ appearance of a superconducting gap s. The spectrum of single-particle excitations E(k) in such cases can be described by the formula [24, 28] 0 1 1 1 1 1/2 , , , []()2 ∆∗ 2 ∆2 --- 1 ------1 --- E()k = k /2m + + s , (13) 4 4 4 4 k where k is the electron wavevector for the states near EF Fig. 10. Dependence of the superconducting gap ∆ (dashed and m is the electron mass. This leads to the appearance ∆ of a common temperature-dependent gap curve) and the Peierls gap * (solid curve) on the direction in the (kx, ky) plane of an HTSC crystal. ∆ ()∆∗ 2 ∆2 ()T = +.s (14) conclude that the experimentally observed pseudogap ∆ At low temperatures (T < Tc), the total gap width is essentially the Peierls gap ∆*. ∆ exceeds s and the material is superconducting. At T > ∆ ∆ ∆ Tc, s(T) = 0 and (T) = *. As can be seen from for- ∆ 6. CONCLUSION mula (14), the value of (and Tc) with an allowance for the Peierls pairing can be significantly greater than in New experimental data obtained using high-quality the absence of such pairing, even for a usual phonon single crystals can be explained within the framework mechanism of superconductivity. The dependence of of the model of superstructural ordering in the oxygen the ∆* value on the position of the state on the Fermi sublattice, with allowance for the antiferromagnetic surface is illustrated by Fig. 2b, showing that no Peierls ordering of copper ions. In particular, the metallic gap appears (i.e., ∆* = 0) in the vicinity of points stripes with a width of 4a and dielectric spacers pre- (±1/4, ±1/4) where the conditions of nesting are vio- dicted by the model agree well with the stripe models lated. discussed in the literature. Thus, ∆* depends on the direction in the (k , k ) According to formula (3), the distance between the x y ≈ plane as described by the solid curve in Fig. 10. This metallic stripes in the La2 Ð xSrxCuO4 system with x figure shows an approximate position of the point 0.06 is about 30 Å. This value is close to the coherence ξ where ∆* = 0 (exact calculation of the dispersion curve (correlation) length (T) in the ab plane of CuO2. In is difficult, since the regions of all three Brillouin zones these compositions, the stripes exhibit Josephson’s occur in the vicinity of this point). The dashed curve in coupling and the entire crystal occurs in a coherent Fig. 10 shows variation of the gap ∆(T) depending on state. This behavior is qualitatively consistent with the the direction at T < Tc according to formula (14). As can experimental phase diagram Tc(x) of La2 Ð xSrxCuO4. be seen, ∆* and ∆ vary in a similar manner. Analogous The same conclusion follows from an analysis of the curves are obtained even for s pairing (imitating the effect of doping on Tc in some other HTSC systems. At behavior for d pairing). Such dependences for HTSCs x ≥ 0.25, the third Brillouin zone is fully depleted of are observed experimentally, for example, by ARPES, electrons and the Fermi surface acquires the shape for the superconducting gap ∆ [29, 30] and pseudogap depicted in Fig. 2c. The conditions of nesting for vector ∆ * [31]. G4 = G/4 are violated and the Peierls gap disappears, ∆ which leads to breakage of the superconducting state. The existence of a pseudogap * at T > Tc and of a superconducting gap ∆(T) described by formula (14) at The proposed model provides for a natural explanation of the shape of the Fermi surface for La Sr CuO and T < Tc was confirmed by the tunneling study of the tem- 2 Ð x x 4 perature dependence of the gap (see, e.g., [32, 33]). other HTSC systems. Qualitative agreement of the experimental data with the The dielectric Peierls gap ∆* formed according to behavior predicted by the proposed model allows us to the proposed model is identified with the experimen-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1054 ANSHUKOVA et al. tally observed pseudogap. The dependences of the 15. Electronic Properties and Mechanisms of High-Tc superconducting gap (∆) and the dielectric gap (∆*) on Superconductors, Ed. by T. Oguchi, K. Kadowaki, and the direction in the (kx, ky) plane predicted by the model T. Sasaki (Elsevier, Amsterdam, 1992). agree well with experiment even in the case of s pairing. 16. A. I. Liechtenstein, O. Gunnarsson, O. K. Andersen, and R. M. Martin, Phys. Rev. B 54, 12505 (1996). The proposed model qualitatively explains the 17. O. P. Sushkov, G. A. Sawatzky, R. Eder, and H. Eskes, nature of the anomalous phonon dispersion, the onset Phys. Rev. B 56, 11769 (1997). and breakage of superconductivity in doped HTSC sys- 18. T. Mishonov and E. J. Penev, J. Phys.: Condens. Matter tems, and the origin of superstructural reflections in the 12, 143 (2000). neutron diffraction patterns. High critical temperatures observed for cuprate HTSCs are related to their quasi- 19. K. Yamada, C. H. Lee, K. Kurahashi, et al., Phys. Rev. B 57, 6165 (1998). two-dimensional character and the coexistence of dielectric and metallic pairing in such systems. 20. N. Ichikawa, S. Uchida, J. M. Tranquada, et al., Phys. Rev. Lett. 85, 1738 (2000). 21. Y. S. Lee, R. J. Birgeneau, M. A. Kastner, et al., Phys. ACKNOWLEDGMENTS Rev. B 60, 3643 (1999). 22. J. M. Tranquada, Physica B (Amsterdam) 241Ð243, 745 This study was supported by the Russian Founda- (1997). tion for Basic Research (project no. 01-02-16395) and 23. M. J. Rice and Y. R. Wang, Physica C (Amsterdam) 157, by the Federal Targeted Scientific-Technological Pro- 192 (1989). gram “Superconductivity.” 24. L. N. Bulaevskiœ, V. L. Ginzburg, G. F. Zharkov, et al., in Problems in High-Temperature Superconductivity, Ed. by V. L. Ginzburg and D. A. Kirzhnits (Nauka, Moscow, REFERENCES 1977). 1. A. Ino, C. Kim, T. Mizokawa, et al., J. Phys. Soc. Jpn. 25. Yu. V. Kopaev, Tr. Fiz. Inst. im. P.N. Lebedeva, Akad. 68, 1496 (1999). Nauk SSSR 86, 3 (1975). 2. A. Ino, C. Kim, M. Nakamura, et al., Phys. Rev. B 62, 26. E. G. Maksimov, Tr. Fiz. Inst. im. P.N. Lebedeva, Akad. 4137 (2000). Nauk SSSR 86, 101 (1975). 3. L. Pintschovius and M. Braden, Phys. Rev. B 60, 27. V. L. Ginzburg and E. G. Maksimov, Sverkhprovodi- R15039 (1999). most: Fiz. Khim. Tekh. 5, 1543 (1992). 28. R. S. Markiewicz, C. Kusko, and V. Kidambi, Phys. 4. R. J. McQueeney, Y. Petrov, T. Egami, et al., Phys. Rev. Rev. B 60, 627 (1999). Lett. 82, 628 (1999). 29. H. Ding, M. R. Norman, J. C. Campuzano, et al., Phys. 5. Y. Petrov, T. Egami, R. J. McQueeney, et al., cond- Rev. B 54, R9678 (1996). mat/0003414. 30. J. Mesot, M. R. Norman, H. Ding, et al., Phys. Rev. Lett. 6. R. J. McQueeney, T. Egami, J.-H. Chung, et al., cond- 83, 840 (1999). mat/0105593. 31. H. Ding, T. Yokoya, J. C. Campuzano, et al., Nature 382, 7. N. Ichikawa, S. Uchida, J. M. Tranquada, et al., Phys. 51 (1996). Rev. Lett. 85, 1738 (2000). 32. V. M. Krasnov, A. Yurgens, D. Winkler, et al., Phys. Rev. 8. A. I. Golovashkin, N. V. Anshukova, L. I. Ivanova, et al., Lett. 84, 5860 (2000). Physica C (Amsterdam) 341Ð348, 1945 (2000). 33. V. M. Krasnov, Phys. Rev. B 65, 140504 (2002). 9. W. E. Pickett, Rev. Mod. Phys. 61, 433 (1989). 34. N. V. Anshukova, A. I. Golovashkin, L. I. Ivanova, et al., Int. J. Mod. Phys. B 12, 3251 (1998). 10. A. I. Golovashkin and A. P. Rusakov, Usp. Fiz. Nauk 35. N. V. Anshukova, A. I. Golovashkin, L. I. Ivanova, and 170, 192 (2000) [Phys. Usp. 43, 184 (2000)]. A. P. Rusakov, Pis’ma Zh. Éksp. Teor. Fiz. 71, 550 11. B. Normand and A. P. Kampf, Phys. Rev. B 64, 024521 (2000) [JETP Lett. 71, 377 (2000)]. (2001). 36. H. You, U. Welp, and Y. Fang, Phys. Rev. B 43, 3660 12. J. M. Tranquada, J. D. Axe, N. Ichikawa, et al., Phys. (1991). Rev. B 54, 7489 (1996). 37. Z. J. Yang, M. Yewondwossen, D. W. Lawther, et al., 13. M. Braden, M. Meven, W. Reichardt, et al., Phys. Rev. B J. Supercond. 8, 223 (1995). 63, 140510 (2001). 14. S. Sachdev, Science 288, 475 (2000). Translated by P. Pozdeev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1055Ð1064. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1200Ð1211. Original Russian Text Copyright © 2003 by Troyanchuk, Trukhanov, Shapovalova, Khomchenko, Tovar, Szymczak. SOLIDS Structure

The Influence of Oxygen Vacancies on the Magnetic State of La0.50D0.50MnO3 Ð g (D = Ca, Sr) Manganites I. O. Troyanchuka, S. V. Trukhanova,*, E. F. Shapovalovaa, V. A. Khomchenkoa, M. Tovarb, and H. Szymczakc aInstitute of Solid-State and Semiconductor Physics, Belarussian Academy of Sciences, ul. Brovki 17, Minsk, 220072 Belarus bHahn-Meitner-Institute (BENSC), D-14109, Berlin, Germany cInstitute of Physics, Polish Academy of Sciences 02-668, Warsaw, Poland *e-mail: [email protected] Received December 25, 2002

Abstract—The crystal structure and magnetic and electric transport properties of polycrystalline La0.50D0.50MnO3 Ð γ manganites (D = Ca, Sr) were studied experimentally depending on the concentration of oxygen vacancies. The La0.50Sr0.50MnO3 Ð γ system of anion-deficient compositions was found to be stable and possess a perovskite structure only up to the γ = 0.25 concentration of oxygen vacancies, whereas, for the La0.50Ca0.50MnO3 Ð γ system, we were able to obtain samples with the concentrations of oxygen vacancies up γ to = 0.50. The stoichiometric La0.50D0.50MnO3 (D = Ca, Sr) compositions had O-orthorhombic (Ca) and tet- ragonal (Sr) unit cells. The unit cell of the anion-deficient La0.50Sr0.50MnO3 Ð γ manganites also became O-orthorhombic when the concentration of oxygen vacancies increased (γ > 0.16). Oxygen deficiency in γ La0.50Sr0.50MnO3 Ð γ first caused the transition from the antiferromagnetic to the ferromagnetic state ( ~ 0.06) and γ then to the spin glass state ( ~ 0.16). Supposedly, the oxygen vacancies in the reduced La0.50Sr0.50MnO3 Ð γ sam- γ ≥ ples with 0.16 were disordered. The special feature of the La0.50Ca0.50MnO3 Ð γ manganites was a nonuni- form distribution of oxygen vacancies in the La0.50Ca0.50MnO2.75 phase. In the La0.50Ca0.50MnO2.50 phase, the type of oxygen vacancy ordering corresponded to that in Sr2Fe2O5, which led to antiferromagnetic ordering. The specific electric resistance of the La0.50D0.50MnO3 Ð γ anion-deficient samples increased with increasing oxygen deficiency. The magnetoresistance of all samples gradually increased as a result of the transition to the magnetically ordered state. Supposedly, the La0.50Ca0.50MnO3 Ð γ manganites in the range of oxygen vacancy concentrations 0.09 ≤ γ ≤ 0.50 had a mixed state and contained microdomains with different types of magnetic ordering. The experimentally observed properties can be interpreted based on the model of phase layering and the model of superexchange magnetic ordering. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION Currently, the systems studied most thoroughly are 3+ 2+ 3+ 4+ 2+ 2+ The discovery of such collective electron phenom- La1 Ð xDx Mn1 Ð xMnx O3 (D = Ca , Sr ). The base ena as giant magnetoresistance and metalÐdielectric compound for the type of compositions under consider- or charge orderÐdisorder phase transitions induced by ation is LaMnO3. It exhibits antiferromagnetic dielec- an external magnetic field aroused interest in tric properties [5], and its magnetic structure is of the A Ln1 − xDxMnO3-type compounds, where Ln is a triva- type and represents a collection of antiferromagneti- lent rare-earth metal (La3+ or Y3+) and D is a divalent cally arranged (001) ferromagnetic planes. The small metal such as Ca2+, Sr2+, Ba2+, Cd2+, and Pb2+ or Bi3+ ferromagnetic component, which is a consequence of [1Ð3]. The magnetic and electric properties of these the noncollinearity of the magnetic moments of manga- hole-substituted manganites have been the object of nese ions, arises because of antisymmetric Dzyaloshin- many experimental and theoretical studies [4Ð7]. The skiÐMoriya exchange [11]; LaMnO3 is therefore a reason for this interest is the abundance and diversity of weak ferromagnet. properties of metals and dielectrics combined in com- pounds of one type, which include systems with crystal The replacement of La3+ by Ca2+ (Sr2+) ions structure, spin, orbital, and charge ordering and, lastly, increases the mean manganese valence, which pre- systems that experience phase layering. These proper- serves compound electroneutrality, and formally results ties are a consequence of close interactions between the 4+ 3 lattice, charge, and spin degrees of freedom, which in the formation of Mn ions with the t2g electronic result in complex phase diagrams of compounds of this configuration (total spin S = 3/2) [12]. It is assumed 3+ class [8Ð10]. that, in these systems, the eg electrons of Mn are delo-

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1056 TROYANCHUK et al. calized and play the role of charge carriers. Such sub- and Mn4+. However, the ferromagnetic properties of stituted manganite systems exhibit metalÐdielectric manganites cannot be explained solely by double concentration phase transitions at 0.12 < x < 0.50 [12]. exchange. According to Goodenough [25], the ferro- The properties of ferromagnets and metals appear in the magnetic properties are determined not only by double La1 Ð xCaxMnO3 and La1 Ð xSrxMnO3 systems almost exchange but also by the special nature of superex- simultaneously. It is, however, not completely clear change interactions in the Mn3+ÐOÐMn3+ and Mn3+ÐOÐ whether or not the arising of properties of one type Mn4+ JahnÐTeller ionic systems. The orbital configura- favors the appearance of properties of another [13]. tion of 3d electrons depends on the positions of manga- Two special concentrations x of substituent ions are nese nuclei if static JahnÐTeller distortions are removed of interest for the La1 Ð xCaxMnO3 system. Most of the (the Goodenough quasi-static hypothesis). In the super- recent works on these materials were concentrated on exchange model, the ferromagnetic contribution is deter- the compositions with x = 0.30, which exhibit giant mined by the virtual electron transitions from the Mn3+ magnetoresistance (on the order of 108% [14]) related half-filled eg orbitals to the unoccupied eg orbitals. to the first-order phase transition from the paramag- The contributions of double exchange and superex- netic dielectric to the ferromagnetic metallic state at change interactions can be controlled by varying the TC = 270 K [15]. Another critical level of doping is x = Mn3+/Mn4+ ratio. It is, therefore, of interest to study the 0.50. In contrast to x = 0.30, the magnetoresistance of properties of the La1 Ð xDxMnO3 Ð γ compounds depend- this composition is related to the first-order antiferro- ing on the mean valence of manganese ions. The magnetÐferromagnet transition and the dielectricÐmetal Mn3+/Mn4+ ratio can be varied by at least three meth- transition induced by an external magnetic field [16, 17]. ods: (1) by the replacement of lanthanum with divalent The La0.50Ca0.50MnO3 compound is a paramagnetic alkaline-earth metal ions, (2) by the replacement of ≈ semiconductor above TC 260 K and a charge-ordered manganese by magnetic and nonmagnetic ions of vari- ≈ antiferromagnet (of the CE type) below TN 180 K ous valences, and (3) by controlling oxygen nonstoichi- (during heating). The CE magnetic structure type is a ometry [26]. chessboard ordering of C- and E-type magnetic unit The dependence of magnetic and magnetoresistive cells. The C magnetic structure type is in turn a collec- properties on oxygen nonstoichiometry in hole-substi- tion of antiferromagnetically coupled (110) ferromag- tuted manganites with perovskite structures has scarcely netic planes, and the E structure type is a collection of been studied. The removal of oxygen anions from the antiferromagnetically coupled (11 0) ferromagnetic 3+ 2+ 3+ 4+ 2Ð crystal lattice of La1 Ð xDx Mn1 Ð x + 2γ Mnx Ð 2γ O3 Ð γ planes. Between TC and TN, La0.50Ca0.50MnO3 consists transforms Mn4+ ions into Mn3+ and decreases the coor- of ferromagnetic and paramagnetic phases [18, 19]. In dination number of manganese (6 5). zero field, La0.50Ca0.50MnO3 is a dielectric in the whole temperature range. Note that long-range charge and Very interesting magnetic and magnetoresistive properties have recently been reported for the antiferromagnetic orders in La0.50Ca0.50MnO3 are estab- La1 − xCaxMnO3 Ð γ [27Ð29] and La1 Ð xBaxMnO3 Ð γ [30, 31] lished simultaneously (TN = T ). Short-range charge C0 oxygen-deficient compositions. The results of these ordering, however, begins to arise at about 210 K, that studies were evidence of such unusual properties of is, at a temperature much higher than that of the ferro- these strongly reduced manganites as a large ferromag- magnetÐantiferromagnet phase transition [20]. The eg netic component, a high magnetic ordering tempera- electrons in the charge-ordered state are localized in the ture, and a large magnetoresistance in spite of the crystal lattice, which results in a time-independent peri- absence of Mn3+ÐMn4+ pairs. 3+ 4+ odic distribution of the Mn and Mn ions. In this work, we studied the magnetic and electric The properties of La Sr MnO resemble those 0.50 0.50 3 properties of the anion-deficient La0.50D0.50MnO3 Ð γ of La0.50Ca0.50MnO3 in many respects. Ferromagnetic manganites (D = Sr, Ca). In spite of equal substitution ordering occurs in the La0.50Sr0.50MnO3 manganite at levels, the reduced samples exhibited quite different TC = 320 K [12, 21]. Long-range antiferromagnetic properties determined by the special features of the order arises at TN = 180 K, and the ferromagnetic and arrangement of oxygen vacancies. antiferromagnetic phases coexist even to helium tem- peratures [22]. No ordering of manganese ions of dif- ferent valences (no charge ordering) occurs in the 2. EXPERIMENTAL La0.50Sr0.50MnO3 manganite; accordingly, an A-type Polycrystalline stoichiometric La0.50D0.50MnO3 antiferromagnetic structure is formed. (D = Sr, Ca) samples were prepared by the standard In is generally believed that double exchange ceramic technique; namely, La2O3 (99.99%), CaCO3 3+ 4+ between Mn ÐMn pairs determines the magnetic and (99.99%), and MnO2 (99.99%) were mixed in the electric properties of manganites with perovskite struc- required ratio between the cations and thoroughly tures [23, 24]. This model is based on real electron ground. Prior to weighing, La2O3 was annealed at 3+ ° exchange between two partially filled d shells of Mn 1000 C for 5 h to remove H2O and CO2. The mixtures

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 THE INFLUENCE OF OXYGEN VACANCIES ON THE MAGNETIC STATE 1057 were pressed into pellets 2 cm in diameter and 1.5 cm them to oxidation in air at 900°C for 24 h. The corre- high with a hydraulic press in a steel press mold under sponding chemical reaction can be written as an approximately 108 Pa pressure and calcined at ° γ 1100 C for 2 h in air. The pellets were then reground, La0.50Ca() Sr 0.50MnO3 Ð γ + ---O2 pressed, and sintered at 1550°C for 2 h in air, which 2 (3) was followed by slow cooling in a furnace to room tem- La0.50Ca() Sr 0.50MnO3. perature. The rate of cooling was 100 K hÐ1. During the synthesis, the samples were on the surface of platinum. An increase in the weight of the samples after their The temperature in the furnace with chromideÐlantha- oxidation corresponded to the weight loss in the reduc- num heaters was measured by a platinumÐplatinumÐ tion. This is evidence of a topotactic reduction charac- rhodium (10%) thermocouple. The cold thermocouple ter. Note that an important feature of the anion-deficient junction was in ice. The rate of heating and cooling the perovskite compounds is the possibility of their oxida- samples was controlled by an RIF-101 device. tion with restoration of the initial composition, struc- ture, and physical properties. The chemical reaction of the synthesis of The magnetization measurements were performed La0.50D0.50MnO3 manganites (D = Sr, Ca) can be writ- on an MPMS-7 quantum SQUID magnetometer and an ten as OI-3001 commercial vibrating-coil magnetometer in the temperature range 4Ð380 K. The magnetic transi- 0.25La2O3 ++0.50Ca()CO Sr 3 0.50Mn2O3 tion temperature was assumed to correspond to the (1) onset of a sharp increase in the dynamic magnetic sus- La Ca() Sr MnO + 0.50CO ↑. 0.50 0.50 z 2 ceptibility or static magnetization measured in a 100-Oe field. Dynamic magnetic susceptibilities were The content of oxygen in all synthesized samples measured with a mutual induction bridge in the temper- was determined thermogravimetrically. It was found ature range 77Ð350 K. The field amplitude was that oxygen concentrations were slightly lower than 200 A/m, and the field frequency was 1200 Hz. The stoichiometric and corresponded to the formula specific resistance of the samples was measured by the La0.50D0.50MnO2.99 ± 0.01 (D = Ca, Sr). The X-ray powder standard four-point technique in the temperature range patterns were obtained at room temperature on a 77Ð350 K. Indium contacts were formed by ultrasonic DRON-3 diffractometer (CrKα radiation) in the angle deposition. Magnetoresistance calculations were per- range 30° ≤ 2θ ≤ 100°. The neutron diffraction patterns formed by the equation of La0.50Ca0.50MnO2.50 were recorded at the Neutron Scattering Center (BENSC, HahnÐMeitner Institute, MR[] % = {}[] ρ()H Ð ρ()0 /ρ()0 × 100%, (4) Berlin) on an E9 (FIREPOD) neutron powder diffracto- λ ∆θ where MR[%] is the negative isotropic magnetoresis- meter with a = 1.79686 Å wavelength and a ~ tance, ρ(H) is the specific electric resistance in a 9-kOe 0.002 scan step. magnetic field, and ρ(0) is the specific electric resis- Oxygen vacancies in the samples synthesized as tance in zero magnetic field. The electric current was described above were formed by annealing the samples directed along the longer side of the samples. The mag- in evacuated quartz ampules at 900°C for 24 h with the netic field was applied parallel to the electric current in use of tantalum metal as an oxygen absorber. The fol- the sample. lowing anion-deficient compositions were prepared: γ La0.50Ca0.50MnO3 Ð γ ( = 0.01, 0.04, 0.10, 0.12, 0.17, 3. RESULTS AND DISCUSSION 0.20, 0.22, 0.25, 0.27, 0.30, 0.31, 0.32, 0.35, 0.37, 0.45, γ The anion-deficient samples of the 0.48, and 0.50) and La0.50Sr0.50MnO3 Ð γ ( = 0.01, 0.06, La0.50Sr0.50MnO3 Ð γ series (LaÐSr) were obtained in the 0.09, 0.12, 0.16, 0.17, 0.20, and 0.25). The reduction γ followed the reaction single-phase state up to the = 0.25 concentration of oxygen vacancies. The sample with the nominal con- γ tent of oxygen vacancies γ = 0.30 contained small 2 4- La0.50Ca() Sr 0.50MnO3 + ------Ta amounts of two impurity phases with K NiF and 5 2 (2) NaCl-type structures in addition to the major perovskite γ phase. Supposedly, these impurities were the La0.50Ca() Sr 0.50MnO3 Ð γ + ---Ta2O5. 5 La2 − xSrxMnO4 and MnO oxides. According to the X- ray data, the LaÐSr samples with 0 ≤ γ < 0.16 had tet- The final oxygen contents were calculated from ragonal unit cells (Fig. 1). An increase in the concentra- sample mass changes after the reduction. Usually, a tion of oxygen vacancies above γ = 0.16 resulted in the 2−3 g sample was loaded into the ampule to decrease formation of O-orthorhombic unit cells, although the the error of measurements. The relative error of mea- degree of symmetry distortion decreased. The unit cell surements did not exceed 0.3%. The content of oxygen parameters for the anion-deficient La0.50Sr0.50MnO3 Ð γ in the reduced samples was controlled by subjecting compositions are listed in the table.

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M, G cm3/g 150 (a) 100 4 La0.50Sr0.50MnO2.99

100 50 M, G cm3/g 3 80 0 91° 92° 93° 94° 95° 50 60 FC 2 FC 40 0 H = 15 kOe 1 20 150 (b) H = 100 Oe 40 0 40 80 120 T, K 100 20 0 50 100 150 200 250 300 350 T, K 0 50 91° 92° 93° 94° 95° Fig. 2. Temperature dependence of magnetization for the La Sr MnO anion-deficient sample in a 100-Oe Intensity, arb. units 0.50 0.50 2.99 external magnetic field. Shown in the inset is the low-tem- perature dependence of magnetization of the same sample 0 in a 15-kOe field. Measurements were performed after cool- (c) ing the sample in the corresponding magnetic field (FC). 150 150 Arrows indicate the direction of measurements.

100 75 structure reflections corresponded to those characteris- 0 tic of an Sr2Fe2O5-type structure with MnO4 tetrahedra 50 91° 92° 93° 94° 95° and MnO6 octahedra as basic structure units [32, 33]. The unit cell parameters of several anion-deficient La Ca MnO γ samples were reported in [28, 29]. 0 0.50 0.50 3 Ð 30° 40° 50° 60° 70° 80° 90° 100° According to the dynamic magnetic susceptibility 2θ data, the La0.50Sr0.50MnO2.99 anion-deficient sample has Fig. 1. X-ray powder patterns of La0.50Sr0.50MnO3 Ð γ a magnetic ordering temperature of 320 K, which is γ γ anion-deficient samples with (a) = 0.01, (b) = 0.12, and almost equal to the temperature TC of stoichiometric (c) γ = 0.17 recorded at room temperature. Shown in the La Sr MnO [22]. The temperature of the magnetic insets are the (211) X-ray reflections for the corresponding 0.50 0.50 3 samples. transition from the antiferromagnetic to the ferromag- netic state, however, sharply decreases compared with the stoichiometric sample, for which it equals 180 K [22]. The anion-deficient samples of the According to magnetization measurements, the transi- La0.50Ca0.50MnO3 Ð γ series (LaÐCa) had O-orthorhom- tion in La0.50Sr0.50MnO2.99 in a 100-Oe field during bic unit cells. Starting with γ = 0.27, X-ray crystal heating begins at 55 K and ends at 100 K (Fig. 2). A 5 K

γ ≤ Symmetry type and parameters a, b, c, and V of unit cells for La0.50Sr0.50MnO3 Ð γ (0 < 0.25) anion-deficient compositions Chemical composition Unit cell symmetry a, Å b, Å c, Å V, Å3

La0.50Sr0.50MnO2.99 Tetragonal 5.457 7.760 231.08

La0.50Sr0.50MnO2.94 Tetragonal 5.454 7.792 231.76

La0.50Sr0.50MnO2.91 Tetragonal 5.454 7.793 231.78

La0.50Sr0.50MnO2.88 Tetragonal 5.484 7.753 233.19

La0.50Sr0.50MnO2.84 O-Orthorhombic 5.462 5.484 7.757 232.37

La0.50Sr0.50MnO2.83 O-Orthorhombic 5.462 5.484 7.757 233.88

La0.50Sr0.50MnO2.80 O-Orthorhombic 5.480 5.487 7.759 233.52

La0.50Sr0.50MnO2.75 O-Orthorhombic 5.506 5.511 7.783 236.16

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M, G cm3/g M, G cm3/g γ = 0.06 80 (a) 0.12 (a) 0.12 6 60

0.01 0.09 FC 40 3 ZFC 20 T = 5 K H = 100 Oe 0.25 γ 0 = 0.16 0 (b) La0.50Sr0.50MnO3 – γ γ = 0.01 La Sr MnO γ 30 0.50 0.50 3 – 0.03 (b) FC 20 H = 100 Oe ZFC 0.02

10 T = 40 K γ = 0.25 0.01

0481216 H, kOe 0 100 200 300 Fig. 3. Field dependences of magnetization for the γ La0.50Sr0.50MnO3 Ð γ anion-deficient samples with = 0.01, T, K 0.06, 0.12, and 0.25 at (a) 5 and (b) 40 K. The samples were zero-field-cooled (ZFC). The arrows denote the directions Fig. 4. Temperature dependences of FC magnetization in a of field variations. The measurements shown in the upper 100-Oe field for the La0.50Sr0.50MnO3 Ð γ anion-deficient figure were performed in the field decreasing mode. samples with γ = 0.09, 0.12, and 0.16 (a) and 0.25 (b). temperature hysteresis is evidence of a first-order phase in the spontaneous magnetic moment, which then grad- transition. ually decreases. A maximum spontaneous magnetic moment is observed for the La0.50Sr0.50MnO2.94 anion- An increase in the field to 15 kOe weakly influences deficient sample; it equals 2.97µ /formula unit. This the temperature interval of the phase transition in the B La Sr MnO anion-deficient sample during heat- value is below that calculated for completely ferromag- 0.50 0.50 2.99 netic ordering of manganese ion spins (3.62µ /formula ing. During cooling, a large magnetization hysteresis is B observed, which is evidence that the magnetic field unit). Below 100 K, the La0.50Sr0.50MnO3 Ð γ anion-defi- effectively stabilizes the ferromagnetic phase in cient compositions exhibit metamagnetic behavior in fields higher than 6 kOe. The highest attainable field on La0.50Sr0.50MnO2.99. The magnetic moment in the ferro- magnetic phase of this La Sr MnO sample is the unit that we used, 16 kOe, however, proved to be too 0.50 0.50 2.99 low for effecting the complete transition to the ferro- close to the expected value for the parallel orientation magnetic state. A field magnetization hysteresis was of all manganese ion spins, the magnetic moments of observed for the La Sr MnO sample at 40 K 3+ 4+ µ µ 0.50 0.50 2.99 Mn and Mn being equal to 4 B and 3 B, respec- (Fig. 3). tively (Fig. 2). An increase in the concentration of oxygen vacan- γ The field dependences of magnetization at various cies to = 0.06 lowers the Curie temperature to TC = temperatures for the anion-deficient La0.50Sr0.50MnO3 Ð γ 300 K, and the low-temperature phase transition from samples are shown in Fig. 3. An increase in the concen- the antiferromagnetic to the ferromagnetic state disap- tration of oxygen vacancies initially causes an increase pears. In La0.50Sr0.50MnO2.94, the Curie temperature

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M, G cm3/g The smeared phase transition and the characteristic 1.5 temperature behavior of the ZFC magnetization show that the magnetic state of the 0.17 ≤ γ ≤ 0.25 anion-defi- FC cient compositions is drastically different. The La0.50Sr0.50MnO2.83 La0.50Sr0.50MnO2.83 composition exhibits different ZFC and FC magnetization behaviors. The temperature 1.0 dependences of magnetization obtained by heating this H = 100 Oe composition in a 100-Oe field after field cooling (FC) and zero field cooling (ZFC) are shown in Fig. 5. Both curves have maxima at about Tf = 50 K. The ZFC and 0.5 FC magnetizations are approximately equal above this Tf temperature but substantially different below it. The ZFC ZFC magnetization first increases to its maximum value at 50 K as temperature grows and then smoothly decreases to zero, whereas the FC magnetization is 0 30 60 90 120 150 180 almost constant at low temperatures. A sharp increase T, K in the ZFC magnetization in the temperature interval Fig. 5. Temperature dependences of ZFC and FC magneti- 40Ð45 K is evidence of a strong decrease in magnetic zations in a 100-Oe field for the La0.50Sr0.50MnO2.83 anion- anisotropy, which is characteristic of cooperative phe- deficient sample. nomena. Above Tf, no anomalous magnetization behav- ior is observed. The low spontaneous magnetic moment T, K value corresponding to the La0.50Sr0.50MnO2.83 anion- deficient composition is likely caused by a nonuniform magnetic state, which is a collection of antiferromagnet- La0.50Sr0.50MnO3 – γ 300 ically and ferromagnetically ordered clusters. The com- petition between antiferromagnetic and ferromagnetic P cluster interactions can lead to the spin glass state [34]. Also note that a temperature of 50 K is typical of cluster 200 spin glass states in manganites [35]. Spontaneous magne- tization measurements for La0.50Sr0.50MnO2.83 encounter F certain difficulties because of the Langevin shape of the field dependence of its magnetization, which is charac- 100 A teristic of spin glasses or superparamagnets.

SG A further decrease in the content of oxygen to γ = 0.25 sharply decreases the magnetic susceptibility of 0 0.05 0.10 0.15 0.20 0.25 the LaÐSr samples, but the field and temperature depen- γ dences of magnetization remain qualitatively unchanged (Figs. 3, 4). The temperature corresponding Fig. 6. Magnetic phase diagram of the system of to the ZFC and FC curve maxima for the La0.50Sr0.50MnO3 Ð γ anion-deficient compositions; A stands for antiferromagnet; F, for ferromagnet; P, for para- La0.50Sr0.50MnO2.75 sample (Tf = 40 K) is slightly lower magnet; and SG, for spin glass. than that for the LaÐSr sample with γ = 0.17. Magnetic property measurements allowed us to con- decreases to 212 K (Fig. 4). No magnetization satura- struct the magnetic phase diagram shown in Fig. 6. The tion is observed in fields up to 16 kOe, as is character- loss of oxygen causes a sharp decrease in the tempera- istic of magnets with weakened cooperative magnetic ture and then the disappearance of the transition to the interactions. The Curie temperatures for the antiferromagnetic state for the LaÐSr samples with La0.50Sr0.50MnO2.88 and La0.50Sr0.50MnO2.84 samples are 0.01 < γ < 0.06, whereas the Curie temperature and 248 and 209 K, respectively (Fig. 4). For the samples spontaneous magnetization decrease comparatively with 0.17 ≤ γ ≤ 0.25, the temperature of the transition to insignificantly. The transition from the long-range to the paramagnetic state is difficult to determine because short-range ferromagnetic order occurs close to the of substantial transition smearing. The temperature threshold value of the concentration of oxygen vacan- dependences of zero-field-cooled (ZFC) and field- cies, at γ ~ 0.16. Simultaneously, the critical tempera- cooled (FC) magnetizations differ insignificantly for ture decreases severalfold in a very narrow interval of the compositions with 0 ≤ γ ≤ 0.16. Such a magnetiza- vacancy concentrations. We observe a well-defined tion behavior is evidence that these compounds retain trend of magnetic susceptibility and spontaneous mag- long-range ferromagnetic order in the temperature netization lowering as the concentration of oxygen interval under consideration. vacancies increases.

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A fundamentally different trend is observed for the 4 LaÐCa series. According to [28, 29], the behavior of γ ≤ (a) La0.50Ca0.50MnO3 Ð γ at 0.15 is in many respects sim- ilar to that of the LaÐSr samples. In both systems, the T = 300 K low-temperature antiferromagnetic state is suppressed because of the development of the ferromagnetic com- 2 ponent, which is subsequently replaced by the spin glass component. However, at concentrations of vacan- cies above γ = 0.15, the behaviors of the two systems become qualitatively different. In the LaÐSr series, an increase in the concentration of oxygen vacancies causes a sharp decrease in magnetic susceptibility, whereas, in the LaÐCa series, the ferromagnetic com- 4 (b) ponent sharply increases and reaches a maximum at a T = 100 K γ ~ 0.30 concentration of vacancies [29]. A further increase in the number of oxygen vacancies (γ > 0.30) in the LaÐCa samples fairly smoothly decreases the fer- Intensity, arb. units romagnetic component and the temperature of the tran- 2 sition to the paramagnetic state. The decrease in the fer- romagnetic component coincides with the development of a perovskite-like structure of the Sr2Fe2O5 type based on the initial perovskite cell. In order to understand what happens with the mag- 4 (c) netic structure at high oxygen vacancy concentrations, T = 10 K we performed a neutron diffraction study of La0.50Sr0.50MnO2.50 at various temperatures. The neu- tron diffraction patterns show that additional reflections begin to appear below 100 K (Fig. 7). As the transition 2 from the magnetically ordered to the paramagnetic state occurs at 120 K (this follows from the results of magnetic measurements), the conclusion can be made that the additional reflections have a magnetic nature. 10° 20° 30° 40° 50° 60° Note that the intensity of different magnetic reflections 2θ increases differently as temperature lowers, which can be caused by a rearrangement of the magnetic structure Fig. 7. Neutron diffraction powder patterns of the or the presence of a weak magnetic sublattice. The La0.50Ca0.50MnO2.50 anion-deficient sample at (a) 300, spontaneous magnetization value is very small, and the (b) 100, and (c) 10 K. magnetic contribution to nuclear reflections is insignif- icant. This leads us to conclude that the magnetic struc- gested in [28], taking into account the new experimen- ture is antiferromagnetic. tal data. The evolution of the magnetic state of the The electric resistance of the La0.50Ca0.50MnO3 Ð γ La−Ca system is depicted in the phase diagram (Fig. 9). anion-deficient samples with 0.09 ≤ γ ≤ 0.50 is charac- Long-range ferromagnetic order in the stoichiometric teristic of semiconductors and continuously increases La0.50Ca0.50MnO3 sample is established at T = 250 K, as temperature lowers (Fig. 8). At low temperatures, the and long-range antiferromagnetic order, at T = 180 K specific resistance of the La–Ca samples satisfies the [18, 19]. Deviations from stoichiometry weakly influ- equation ln(ρ) ∝ TÐ1 (see inset in Fig. 8). The behavior ence the ferromagnetic critical point, whereas the anti- of the magnetoresistance of the samples with 0.09 ≤ γ ≤ ferromagnetic state becomes sharply suppressed by the 0.50 correlates with the absence of any electric resis- ferromagnetic state. In the concentration range 0.02 < tance anomalies close to the temperature of the transi- γ < 0.09, the samples behave as ferromagnets with tion to the magnetically ordered state. Below this tem- inclusions of clusters with substantially weakened perature, magnetoresistance begins to increase continu- exchange interactions. Below 40 K, the ZFC magneti- ously toward liquid nitrogen temperatures without a zation of the anion-deficient compositions increases as peak on its temperature dependence. Note that the elec- temperature grows. This is evidence of changes in tric transport properties of the La−Sr samples with anisotropy likely caused by sample inhomogeneity. At 0.06 ≤ γ ≤ 0.25 are similar to those of the LaÐCa sam- γ > 0.09, we observe a sharp decrease in spontaneous ples shown in Fig. 8. magnetization, whereas the anomalous behavior of We modernized the magnetic phase diagram of the magnetization below 40 K becomes more pronounced. La0.50Ca0.50MnO3 Ð γ system, which was initially sug- Nevertheless, the divergence of the ZFC and FC mag-

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3 8 γ = 0.50 γ 10 = 0.50 La Ca MnO γ 15 0.35 0.50 0.50 3 – (a) cm] Ω

[ 10 T = 6 K 6 0.35 ρ 0.25 2

10 ln 5 cm Ω /formula unit , 0.006 0.012 B ρ 0.25 µ –1 –1 4 , 10 T , K s

M 1

102 (a) 0 270 0 P (b) γ = 0.50

180 H = 9 kOe –20 F + P , K MR, % La0.50Ca0.50MnO3 – γ T 0.35

90 A1 + F A2 + F

–40 0.25 (b) SG 0 0 0.1 0.2 0.3 0.4 0.5 100 200 300 γ T, K

Fig. 8. Temperature dependences of (a) specific electric Fig. 9. (a) Dependence of spontaneous magnetic moment resistance and (b) magnetoresistance measured in a 9-kOe on the concentration of oxygen vacancies and (b) magnetic phase diagram of the La Ca MnO γ system of external magnetic field for the La0.50Ca0.50MnO3 Ð γ anion- 0.50 0.50 3 Ð deficient samples with γ = 0.25, 0.35, and 0.50. Shown in anion-deficient compositions; A1 stands for charge-ordered the inset are the dependences of the logarithm of the specific antiferromagnet; A2, for charge-disordered antiferromag- electric resistances of the same samples on reciprocal tem- net; F + P, for inhomogeneous ferromagnet; P, for paramag- perature. net; and SG, for spin glass. netizations begins at fairly high temperatures and lin- The behavior of magnetic properties shows that early decreases as the concentration of vacancies vacancy distributions of the same type as in the increases. Close to the γ = 0.30 concentration of oxygen La0.50Ca0.50MnO2.75 phase are observed in a fairly wide vacancies, spontaneous magnetization sharply range of oxygen vacancy concentrations, namely, ≤ γ ≤ increases (Fig. 9) and reaches a maximum of about 0.09 0.50. It is most likely that spin-glass-type 40% of the value calculated for purely ferromagnetic systems with a fairly uniform distribution of oxygen vacancies and phases of the La Ca MnO type ordering of the manganese ion spins. According to 0.50 0.50 2.75 coexist in the 0.09 < γ < 0.30 concentration interval. In Gonzáles-Calbet et al. [36Ð38], the La0.50Ca0.50MnO2.75 ≤ γ ≤ the 0.30 0.50 interval, the La0.50Ca0.50MnO2.50 sample is a pure phase with a unique temperature antiferromagnetic phase gradually replaces the dependence of the distribution of oxygen vacancies. La0.50Ca0.50MnO2.75 phase. As the Curie (Néel) temper- Judging from high-resolution electron microscopy ature depends on the concentration of vacancies almost data, oxygen vacancies predominantly concentrate in linearly, it can be suggested that either the composition the domain walls of microdomains, which have a small of the La0.50Ca0.50MnO2.75 phase is not strictly constant, thickness compared with the other two dimensions. or microdomains of this phase are exceedingly small,

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 THE INFLUENCE OF OXYGEN VACANCIES ON THE MAGNETIC STATE 1063 and the Curie point is determined by the interaction of ordered clusters resulted in the formation of the spin magnetic domains of different types. La0.50Ca0.50MnO2.75- glass state. The specific electric resistance of type phases are not formed in the LaÐSr samples, La0.50D0.50MnO3 Ð γ anion-deficient samples grew and because the perovskite phase becomes thermodynami- the metalÐsemiconductor transition disappeared as cally unstable in these systems. oxygen deficiency increased. The magnetoresistance of The Mn3+ÐOÐMn3+ superexchange interactions are all anion-deficient samples gradually increased anisotropic in orbitally ordered phases (positive in the depending on temperature in the transition to the mag- (001) plane and negative in the [001] direction) and iso- netically ordered state. Supposedly, oxygen vacancies tropic in orbitally disordered phases (positive in all were disordered in the La0.50Sr0.50MnO3 Ð γ reduced compositions with γ ≥ 0.16. The special feature of the directions). The La0.50Ca(Sr)0.50MnO3 stoichiometric compositions are orbitally ordered in the ground state. La0.50Ca0.50MnO3 Ð γ manganites was a nonuniform dis- For this reason, Mn3+ÐOÐMn3+ superexchange is nega- tribution of oxygen vacancies in the La0.50Ca0.50MnO2.75 tive and does not contribute to magnetization. The phase. This increased the ferromagnetic component. In appearance of oxygen vacancies removes orbital order- the La0.50Ca0.50MnO2.50 phase, oxygen vacancies were ing and increases double exchange (Mn3+ÐOÐMn4+) ordered as in Sr2Fe2O5, which resulted in antiferromag- and superexchange (Mn3+ÐOÐMn3+) contributions to netic ordering. The observed experimental properties the resultant magnetization. The mean valence and the can be interpreted based on the model of phase layering coordination number of manganese ions and, accord- and the superexchange magnetic ordering model. ingly, the ferromagnetic double exchange contribution decrease as the concentration of oxygen vacancies ACKNOWLEDGMENTS increases. A decrease in the coordination number of manganese changes the sign of Mn3+ÐOÐMn3+ superex- This work was financially supported by the change from positive to negative. It follows that the Belarussian Foundation for Basic Research (grant antiferromagnetic component of exchange interactions no. F02R-122), the Committee on Science of the increases as the concentration of oxygen vacancies Republic of Poland (KBN grant 5 PO3B 016 20), and grows. In the LaÐSr compositions with 0.06 ≤ γ ≤ 0.16, the Scientific Program of the European Community this results in a gradual Curie temperature and sponta- (The Improving Human Potential Program, contract neous magnetic moment lowering without radical HPRI-CT-1999-00020). changes in the magnetic state. The γ = 0.17 concentra- tion of oxygen vacancies is likely to be critical; at this REFERENCES concentration, the volumes of two phases (ferromag- netic and antiferromagnetic) become comparable. The 1. R. M. Kusters, D. A. Singleton, D. A. Keen, et al., Phys- system is divided into clusters with different types of ica B (Amsterdam) 155, 362 (1989). magnetic ordering. The competition between ferromag- 2. Y. Tokura and Y. Tomioka, J. Magn. Magn. Mater. 200, 1 netically and antiferromagnetically ordered cluster (1999). interactions results in the arising of a spin-glass-type 3. E. Dagotto, T. Hotta, A. Moreo, et al., Phys. 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17. Y. Moritomo, A. Machida, S. Mori, et al., Phys. Rev. B 29. S. V. Trukhanov, N. V. Kasper, I. O. Troyanchuk, et al., 60, 9220 (1999). J. Solid State Chem. 169, 85 (2002). 18. F. Rivadulla, M. Freita-Alvite, M. A. López-Quintela, 30. S. V. Trukhanov, I. O. Troyanchuk, I. M. Fita, et al., et al., J. Appl. Phys. 91, 785 (2002). J. Magn. Magn. Mat. 237, 276 (2001). 19. P. Levy, F. Parisi, G. Polla, et al., Phys. Rev. B 62, 6437 31. S. V. Trukhanov, I. O. Troyanchuk, M. Hervieu, et al., (2000). Phys. Rev. B 66, 184424 (2002). 20. C. H. Chen and S.-W. Cheong, Phys. Rev. Lett. 76, 4042 32. J. M. González-Calbet, M. Vallet-Regi, M. A. Alario- (1996). Franko, et al., Mater. Res. Bull. 18, 285 (1983). 21. H. Watanabe, J. Phys. Soc. Jpn. 16, 433 (1961). 33. J. B. Wiley, M. Sabat, S. J. Hwu, et al., J. Solid State 22. S. I. Patil, S. M. Bhagat, Q. Q. Shu, et al., Phys. Rev. B Chem. 87, 250 (1990). 62, 9548 (2000). 34. S. V. Trukhanov, I. O. Troyanchuk, N. V. Pushkarev, and 23. C. Zener, Phys. Rev. 82, 403 (1951). G. Szymczak, Zh. Éksp. Teor. Fiz. 122, 356 (2002) [JETP 95, 308 (2002)]. 24. P.-G. De Gennes, Phys. Rev. 118, 141 (1960). 35. S. V. Trukhanov, N. V. Kasper, I. O. Troyanchuk, et al., 25. J. B. Goodenough, A. Wold, R. J. Arnott, and N. Menyuk, Phys. Status Solidi B 233, 321 (2002). Phys. Rev. 124, 373 (1961). 36. J. M. González-Calbet, E. Herrero, N. Rangavittal, et al., 26. I. O. Troyanchuk, S. V. Trukhanov, D. D. Khalyavin, J. Solid State Chem. 148, 158 (1999). et al., Fiz. Tverd. Tela (St. Petersburg) 42, 297 (2000) [Phys. Solid State 42, 305 (2000)]. 37. J. Alonso, E. Herrero, J. M. González-Calbet, et al., Phys. Rev. B 62, 11328 (2000). 27. I. O. Troyanchuk, S. V. Trukhanov, H. Szymczak, et al., Zh. Éksp. Teor. Fiz. 120, 183 (2001) [JETP 93, 161 38. J. Alonso, A. Arroyo, J. M. González-Calbet, et al., Phys. (2001)]. Rev. B 64, 172410 (2001). 28. S. V. Trukhanov, I. O. Troyanchuk, H. Szymczak, and K. Barner, Phys. Status Solidi B 229, 1417 (2002). Translated by V. Sipachev

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Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1065Ð1077. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1212Ð1226. Original Russian Text Copyright © 2003 by Zyubin, Rudnev, Kashurnikov. SOLIDS Structure

Ordered States and Structural Transitions in a System of Abrikosov Vortices with Periodic Pinning M. V. Zyubin, I. A. Rudnev, and V. A. Kashurnikov* Moscow Institute of Engineering Physics, Kashirskoe sh. 31, Moscow, 115409 Russia *e-mail: [email protected] Received November 27, 2002

Abstract—A system of Abrikosov vortices in a quasi-two-dimensional HTSC plate is considered for various periodic lattices of pinning centers. The magnetization and equilibrium configurations of the vortex density for various values of external magnetic field and temperature are calculated using the Monte Carlo method. It is found that the interaction of the vortex system with the periodic lattice of pinning centers leads to the formation of various ordered vortex states through which the vortex system passes upon an increase or a decrease in the magnetic field. It is shown that ordered vortex states, as well as magnetic field screening processes, are respon- sible for the emergence of clearly manifested peaks on the magnetization curves. Extended pinning centers and the effect of multiple trapping of vortices on the behavior of magnetization are considered. Melting and crys- tallization of the vortex system under the periodic pinning conditions are investigated. It is found that the vortex system can crystallize upon heating in the case of periodic pinning. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION and magnetic flux distribution in a quasi-two-dimen- sional HTSC plate with a random distribution of pin- A large number of recent publications are devoted to ning centers were reported in our earlier publications experimental investigations on the interaction of an [12Ð14], where a new method based on the Monte Abrikosov vortex lattice with a periodic artificially cre- Carlo algorithm was developed for a large canonical ated structure of pinning centers in the form of micro- ensemble with a number of singularities reflecting the holes as well as submicrometer particles of a magnetic behavior of vortex systems in layered HTSC materials. or nonmagnetic material (see, for example, [1Ð3] and This method enabled us to obtain the equilibrium distri- the literature cited therein). Experiments reveal singu- bution of vortex density upon a change in the external larities on the magnetization curves as well as on the magnetic field H and to calculate the M(H) depen- curves describing the magnetic field dependence of the dences for an arbitrary arrangement of pinning centers critical current and resistivity, which are interpreted as at various temperatures. regimes of matching between the vortex lattice and the Here, we report on new results of simulation of the lattice of pinning centers. Direct observation of the Abrikosov vortex system using the Monte Carlo regimes of matching the vortex system to the periodic method in the case of periodic pinning. The magnetiza- lattice of pinning centers was carried out using tion curves and the patterns of vortex density distribu- Lorentz force microscopy [4] and scanning Hall magne- tion are calculated for various defect lattices in a wide tometry [5]. range of temperatures and fields. It is shown that, in the The complexity of such experiments and the inter- case of periodic pinning, the curves describing the mag- pretation of the observed effect necessitate the use of netic field dependence of magnetization display a num- numerical simulation, including that based on the ber of peaks associated with the interaction between the Monte Carlo method. The Monte Carlo technique was vortex lattice and the lattice of pinning centers. Differ- used earlier for obtaining new results for phase transi- ent reasons for the emergence of these singularities on tions and the dynamics of a two-dimensional Abrikosov the magnetization curve are indicated for the first time. vortex lattice in a model system imitating layered high- In addition, we discovered an ordering of the vortex temperature superconductors (HTSC). For example, it system with periodic pinning upon heating, viz., the was shown in [6Ð11] that a phase transition (melting of inverse crystallization effect. Such a peculiar behavior a triangular lattice resulting in the formation of vortex of correlated systems is extremely uncommon in liquid) is observed in the absence of defects. In the nature. Inverse crystallization was observed in some presence of defects, a phase of “rotating lattice” is magnetic materials [15] as well as in polymer systems formed between the phases of the vortex crystal and the [16]. In a recent publication [17], inverse crystallization vortex liquid. The vortex system in this phase has the was discovered in a system of vortices with random form of lattice islands rotating around pinning centers. pinning. Inverse crystallization in a system of Abriko- The results of numerical calculation of magnetization sov vortices with periodic pinning is predicted by us for

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1066 ZYUBIN et al. ξ the first time and has a fundamentally different physical superconductor surface [14]; and 0 is the vortex core origin as compared to random pinning. size for T = 0. We introduce pinning centers to study the behavior 2. MODEL of a system with defects. In this case, the energy of AND COMPUTATIONAL PROCEDURE interaction with a pinning center was chosen in the model form, Let us consider a three-dimensional bulk sample of a layered (in the xy plane) HTSC. The sample has a U0()T 1 r U ()Tr, = Ðα------exp Ð------, finite size l in the x direction and is infinitely large in the p U ()0 r/ξ + 1 2ξ y and z directions. It is placed in a magnetic field paral- 0 lel to the z axis, which rules out demagnetization where α is the parameter characterizing the depth of the effects. Assuming that the interaction between layers in ξ HTSC is weak, we will carry out calculations for a potential well of a defect and is the coherence length. quasi-two-dimensional (in the xy plane) plate of thick- Such a choice of the sizes and depth of pinning corre- ness d, which simulates a superconducting layer. In sponds to the case when only one vortex can be pinned other words, we “cut” a layer of thickness d, which will at the center. be treated below, along the z axis. The magnetic field induction in the given geometry was calculated using the following formula taking into We consider a 2D system of Abrikosov vortices in account the contribution of Meissner currents: the form of model classical particles with a long-range potential in the plate placed in an external magnetic NΦ 2λH Ðl/λ field H. The thermodynamic Gibbs potential of the vor- B = ------0 + ------()1 Ð e . tex system in such a plate has the form S l

Φ We have also taken into account the fact that the flux ε 0 H GN= dNdÐ ------π carried by each vortex depends on the distance from the 4 plate edge [14]. Φ 0 H xi lxÐ i In our computations, we used the approach devel- + d ------∑ exp Ð --- - + exp Ð------4π λ λ oped earlier and based on the Monte Carlo algorithm i for a grand canonical ensemble [13, 14]. The method enables us to obtain the equilibrium vortex density dis- 1 + ---∑Ur()+ ∑U ()r + U , tribution for given extrinsic parameters (magnetic field 2 ij p i surf H, temperature T, and the distribution and type of pin- ≠ ij i ning centers). Using this distribution, we can calculate where N is the number of vortices in the system; magnetization M and induction B. Thus, the method makes it possible to determine both integrated charac- teristics of a superconductor and visual patterns of Φ 2 λ ε 0 0 magnetic flux distribution. Such an approach differs = ------πλ- ln----ξ-+ 0.52 4 0 fundamentally in some respects from familiar computa- tional methods; namely, it ensures the most correct is the self-energy of a vortex [18]; Φ = hc/2e; λ is the inclusion of the effect of the plate boundary, operation 0 in a wide range of temperatures 0 < T < T , and inclu- magnetic field penetration depth in the superconductor; c λ λ sion of any distribution of any type of defects. 0 = (T = 0); d is the superconducting layer thickness; NdΦ H/4π is the energy of interaction of a vortex with For simulating, we use parameters of a layered 0 λ the external field H; U(r ) = U K (r /λ) is the energy of superconductor Bi2Sr2CaCu2O8: d = 0.27 nm, 0 = ij 0 0 ij 180 nm, ξ = 2 nm, and T = 84 K [19]. The temperature Φ2 π2λ2 0 c paired interaction of vortices; U0 = 0d/8 ; K0 is dependence of the magnetic field penetration depth is the Bessel function of the imaginary argument; rij is the given in the form [11] distance between vortices; λ λ ()0 Φ ()T = ------. 0 H xi lxÐ i ()3.3 d------exp Ð---- + exp Ð------1 Ð T/T c 4π λ λ Computations were made for plates having a size of 5 × is the energy of interaction between the ith vortex and 3 and 5 × 2.25 µm2. The x size in the region in question Meissner currents passing in the y direction over the was chosen so that our analysis could be confined to plate surface; Up(ri) is the energy of interaction only the first terms in the expression for the interaction between the ith vortex and pinning centers; Usurf is the of vortices with the surface and, on the other hand, con- energy of interaction of the vortex system with the siderable errors due to the application of periodic

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 ORDERED STATES AND STRUCTURAL TRANSITIONS IN A SYSTEM 1067 boundary conditions in the computation of vortex inter- –4πM, T action could be avoided. The y size varied depending on a the pinning lattice period so that the periodic boundary conditions were not violated. The maximal range of variation of the external field H is limited only by the 1 computer power and, accordingly, computer time. In 0.03 2 b the computations described here, the range of the exter- d c nal field variation was Ð0.1 T ≤ H ≤ 0.1 T. 3 e g 3. RESULTS AND DISCUSSION 0.02 The presence of the lattice of pinning centers leads to the emergence of commensurability effects between the number of vortices and the number of defects f responsible for the formation of ordered configurations 0.01 of the system of vortices, which often differ consider- ably from the conventional triangular lattice. In turn, the ordered structure of the vortex system leads to sin- gularities in the magnetization and in the magnetic field dependences of the critical current and resistivity. 0 0.05 0.10 Reichhardt et al. [20] used the molecular dynamics H, T method to analyze the behavior of a vortex system in the presence of square and triangular lattices of point Fig. 1. Magnetization curves for a triangular lattice of point defects and discovered a variety of ordered configura- pinning centers for different concentrations of defects nd, tions. They discovered that the vortex system is ordered µmÐ2: 11.18 (1), 5.7 (2), and 3.63 (3). Temperature T = 1 K. only for definite matching fields, when the number of Points aÐg correspond to vortex density distributions in vortices is multiple to the number of defects. However, Fig. 2. the calculations in [20] were made under periodic boundary conditions in the canonical ensemble, i.e., for to consider the difference between the value of magne- a given fixed number of vortices. In addition, the tization in the case of ordered arrangement of defects authors of [20] considered the behavior of the vortex and its value in the case of random pinning for the same system only at T = 0. concentration of defects as a quantitative characteristic The formation of ordered configurations in the case of singularities emerging on the magnetization curve. of matching between the number of vortices and the The results of calculations show that, in the case of ran- number of defects can also be demonstrated in the µ Ð2 dom pinning with concentration nd = 11.18 m , the framework of a canonical ensemble for periodic bound- initial segment of the magnetization curve coincides ary conditions. However, while studying the effect of with the magnetization of a defect-free superconductor. the defect structure geometry on the magnetic flux pen- The effect of random pinning on the behavior of mag- etration and distribution, we must allow for the vortex netization is manifested for concentrations nd > production/destruction, i.e., consider a grand canonical µ Ð2 ensemble and take into account surface effects. 16.7 m [14]. For this reason, the emergence of sin- gularities on the magnetization curves for concentra- We carried out computations for the following lat- tions n ≤ 11.18 µmÐ2 should be attributed just to the tices of point pinning centers: square, triangular, and d kagome lattices.1 We also considered square lattices of ordered arrangement of pinning centers. extended pinning centers, at which more than one vor- Table 1 shows the number and characteristics of sin- tex could be pinned. We will analyze in detail all the gularities emerging on the magnetization curves for the above-mentioned configurations of pinning centers. investigated defect concentrations nd = 11.8, 5.7, and 5.63 µmÐ2, corresponding to the defect spacing a = 3.1. Triangular Lattice of Point Defects 0.32, 0.45, and 0.56 µm. Line H gives the positions of Figure 1 shows the magnetization curves obtained peaks corresponding to the point of maximal ascent of for various concentrations of defects forming a triangu- magnetization (it will be shown below that ordered con- figurations are formed at the peak base), while line ∆M lar lattice. We considered the concentrations nd = 11.18, µ Ð2 gives the heights of the peaks measured from the mag- 5.7, and 3.63 m corresponding to the triangular lat- netization M of a defect-free superconductor, and δ = tice periods a = 0.32, 0.45, and 0.56 µm. The depen- 0 ∆M/M . Letters S and M in the line Reason indicate, dence exhibits a number of singularities. It is expedient 0 respectively, that a singularity appears due to screening 1 This term indicates a superlattice with an increased period super- or due to matching of the vortex system with the lattice imposed on the initial lattice and is often used for spin systems. of pinning centers (formation of an ordered configura-

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(‡) (b) (c)

(d) (e) (f)

(g) (h)

µ Ð2 µ Ð2 Fig. 2. Vortex density distribution for nd = 11.18 m , H = 0.049 (a), 0.07 (b), 0.074 (c), 0.09 (d), 0.096 T (e); for nd = 5.7 m , µ Ð2 × µ 2 H = 0.068 (f), 0.1 T (g); and for nd = 11.18 m , H = 0 (h). The size of the system is 5 2.25 m . Temperature T = 1 K. tion). We will call the singularities associated with The reasons for the emergence of singularities can screening first-type singularities, while the singularities be explained by comparing the vortex density distri- associated with matching the vortex system to the defect bution with corresponding points on the magnetiza- structure will be called second-type singularities. tion curves.

Table 1. Characteristics of singularities on the magnetization curves for a triangular lattice of point defects (see text) a, µm 0.32 0.45 0.56 µ Ð2 nd, m 11.18 5.7 3.63 Number of peaks 3 3 2 H, T 0.051 0.05 0.048 ∆M, 10Ð3 T 13.67 4.34 1.01 Peak 1 δ, % 62.7 19.9 4.5

Nv/Nd Ð23 Reason S M M H, T 0.071 0.06 0.054 ∆M, 10Ð3 T 5.05 2.06 0.77 Peak 2 δ, % 25.9 10.6 1.9

Nv/Nd 234 Reason M M M H, T 0.094 0.072 Ð ∆M, 10Ð3 T 3.26 1.73 Ð Peak 3 δ, % 17.1 8.9 Ð

Nv/Nd 34Ð Reason M M Ð

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µ Ð2 π For H = 0.049 T, nd = 11.18 m (Fig. 2a), lines –4 M, T from pinned vortices formed in the surface regions pre- vent further penetration of the magnetic flux. Free vor- 0.05 tices, which cannot enter the plate as yet, also form lines. A delay in the magnetic flux penetration leads to a considerable increase in magnetization. It can be seen from the vortex density distribution that this singularity on the magnetization curve is due not to the formation of an ordered configuration of the vortex system, but to the screening of the surface regions by pinned vortices. It should be noted that a certain number of vortices managed to penetrate to the bulk of the sample even –0.1 0.1 prior to the formation of lines of pinned vortices. This H, T indicates the low efficiency of point pinning centers as far as the probability of vortex pinning is concerned. µ Ð2 For H = 0.07 T, nd = 11.18 m (Fig. 2b), a lattice corresponding to Nv/Nd = 2 is formed, where Nv and Nd are the numbers of vortices and defects, respectively. A local ordering is observed: the number of vortices at the center is insufficient, while an excess concentration –0.05 emerges in the surface regions. However, the formation of the given configuration leads to a singularity on the magnetization curve. The point lies at the base of the Fig. 3. Magnetization loop for a triangular lattice of pinning µ Ð2 peak. centers with defect concentration nd = 11.18 m . Temper- µ Ð2 ature T = 1 K. For H = 0.074 T, nd = 11.18 m (Fig. 2c), an insig- nificant change in the external field leads to the destruc- tion of the vortex lattice. The lines of entrance of “new” the peaks and are not shown in the figures). Field H = vortices can be clearly seen. The destruction of the 0.068 T is characterized by the formation of a lattice ordered structure is accompanied by a decrease in mag- with Nv/Nd = 4 (Fig. 2f). In the strongest field H = 0.1 T netization. (Fig. 2g), even the seventh matching is realized µ Ð2 For H = 0.09 T, nd = 11.18 m (Fig. 2d), the vortex (Nv/Nd = 7). In this case, several regions with different system forms a nearly perfect triangular lattice. The orientations of the vortex lattice are observed. All the given matching is characterized by the ratio Nv/Nd = 3. ordered configurations of the vortex structure described The point lies at the base of a peak on the magnetization above are in accordance with the results obtained curve. in [20], in which the behavior of the vortex system was For H = 0.096 T, n = 11.18 µmÐ2 (Fig. 2e), destruc- also investigated for stronger fields and the existence of d a triangular lattice was detected for N /N = 9, 12, 13, tion of the ordered configuration is observed, which v d leads to a decrease in magnetization. The limited power 16, 19, 21, 25, and 28. of computer equipment does not permit the tracing of In the case of a still lower defect concentration nd = higher order matching in the case of a large number of 3.63 µmÐ2, the effects associated with screening are not pinning centers. However, by reducing the number of observed. Insignificant singularities corresponding to defects, we can obtain other configurations of the vortex the third and fourth matching are observed. An analysis system in the same range of fields. For example, for nd = of the vortex density distribution shows that ordered µ Ð2 5.7 m , the effect associated with the screening of the configurations are extremely sensitive to a change in surface by pinned vortices is also observed. However, the external magnetic field. Lattices are destroyed even in contrast to the previous case, a delay in the magnetic for small variations of the external field, which does not flux propagation does not lead to the emergence of a result in noticeable singularities on the magnetization peak, but is accompanied just by a change in the slope curves. of the curve. The magnetization curve displays three singularities associated with the formation of ordered It should be noted that a decrease in the number of configurations. As expected (Figs. 2f and 2g), a defects leads to coordinated disappearance of singular- decrease in the defect concentration leads to a displace- ities associated with screening as well as singularities ment of singularities towards lower fields and to a associated with the formation of vortex lattices. This is decrease in the magnitude of the singularities. For H = not surprising since both types of defects appear due to 0.048 and 0.057 T (these cases are not shown in the fig- the interaction of free vortices with vortices pinned at ure), the second and third matching are realized, pinning centers. Figure 3 shows a complete magnetiza- respectively (the fields correspond to the beginning of tion loop in the case of a high concentration of defects,

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–4πM, T “rigid” vortex lattice prevents the magnetic flux pene- a tration. When the field is removed, the vortex system b tends to expand upon a decrease in the surface barrier; as a result, some of the vortices are expelled from the c plate. The vortex density distribution after the removal 0.03 of the magnetic field (H = 0) is shown in Fig. 2h for µ Ð2 defect concentration nd = 11.18 m . Vortex trapping is 3 2 1 observed both due to pinning at the pinning centers and d due to collective interaction with pinned vortices. 0.02 When the overheating field of the Meissner state is attained, vortices of the opposite sign (antivortices) start entering the plate. However, antivortices are e f pinned at the surface pinning centers and prevent sub- 0.01 sequent penetration of the magnetic flux, leading to the emergence of a singularity on the magnetization curve. It can be seen from Fig. 3 that this singularity is observed in the same fields as for the singularity appearing during initial magnetization and associated with screening. 0 0.05 0.10 H, T 3.2. Square Lattice of Point Defects Fig. 4. Magnetization curves for a square lattice of point pinning centers for different concentrations of defects Figure 4 shows the magnetization curves for various µ Ð2 nd, m : 9.68 (1), 7.11 (2), and 4.94 (3). Temperature T = concentrations of pinning centers forming a square 1 K. Points aÐg correspond to vortex density distributions in lattice. Fig. 5. µ Ð2 For a high concentration of defects (nd = 9.68 m ), strong effects associated with the screening of the sur- µ Ð2 nd = 11.18 m . In the reverse direction, no singulari- face by pinned vortices are observed (Fig. 5a for H = ties are observed on the magnetization curve. This is 0.043 T and Fig. 5b for H = 0.054 T). The vortex den- apparently due to the fact that singularities appear on sity distribution shown in Fig. 5c (H = 0.068 T) corre- the magnetization curve due to repulsion between vor- sponds to the penetration of vortices into the sample. tices. As the applied magnetic field increases, the The square lattice of defects forms peculiar channels

(‡) (b) (c)

(d) (e) (f)

(g) (h)

µ Ð2 µ Ð2 Fig. 5. Vortex density distribution for nd = 9.68 m , H = 0.043 (a), 0.054 (b), 0.068 T (c); for nd = 4.94 m , H = 0.046 (d), µ Ð2 × µ 2 0.062 (e), 0.066 (f), 0.072 T (g); and for nd = 9.68 m , H = 0 (h) (magnetic field trapping). The size of the system is 5 2.25 m . Temperature T = 1 K.

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Table 2. Characteristics of singularities on the magnetization curves for a square lattice of point defects (see text) a, µm 0.32 0.375 0.45 µ Ð2 nd, m 9.68 7.11 4.94 Number of peaks 2 3 3 H, T 0.048 0.044 0.043 ∆M, 10Ð3 T 12.7 10.29 7.37 Peak 1 δ, % 55.3 41.7 29.3

Nv/Nd ÐÐÐ Reason S S S H, T 0.057 0.054 0.047 ∆M, 10Ð3 T 13.02 9.93 5.39 Peak 2 δ, % 62.4 47.1 23.1

Nv/Nd ÐÐ2 Reason S S M H, T Ð 0.079 0.064 ∆M, 10Ð3 T Ð 0.31 1.59 Peak 3 δ, % Ð 0.95 4.68

Nv/Nd Ð44 Reason Ð M M along which the magnetic flux propagates. Naturally, As in the case of a triangular lattice of pinning cen- the active entrance of vortices into the sample is accom- ters, an ordered system of trapped vortices (Fig. 5h) is panied by a decrease in magnetization. left in a superconductor with a square lattice of pinning µ Ð2 centers after the removal of the applied field. Even the defect concentration nd = 4.94 m is suf- ficient for the emergence of new singularities associ- On the whole, noticeable singularities associated ated both with screening and with matching. In a field with the screening of the surface (often even more sig- H = 0.043 T, a singularity associated with screening is nificant than in the case of a triangular lattice with the observed. In a field of 0.047 T (Fig. 5d), a square cen- same period) are observed in the case of a square lat- tered lattices is formed but only in the surface regions; tice. However, the formation of vortex lattices either this is reflected in the emergence of a peak on the mag- does not affect the magnetization at all or is accompa- netization curve (see Fig. 4). nied by extremely weak effects, which are observed The vortex density distributions shown in Figs. 5e predominantly when the vortex system forms triangular and 5g show that the vortex system forms ordered con- lattices. figurations in stronger fields also. However, the forma- Basic characteristics of singularities on the magne- tion of lattices is not accompanied by the emergence of tization curve are given in Table 2. noticeable singularities on the magnetization curve. The point with H = 0.062 T can be put in correspon- dence with the beginning of a magnetization peak, 3.3. Kagome Lattice of Point Defects while the point with H = 0.072 T corresponding to the emergence of a configuration with Nv/Nd = 5 (Fig. 5g) A kagome lattice can be obtained from a triangular is not singled out on the magnetization curve. lattice by eliminating alternate defects in alternate rows The competition between the tendency of the vortex (Fig. 6a). For the same period, the density of the kag- system to form a triangular lattice and the symmetry of ome lattice is 3/4 of the density of the triangular lattice. the square lattice of defects leads to the emergence of By virtue of symmetry, we can expect the formation of various ordered configurations. For example, the vortex triangular lattices in the vortex system, which is con- system exhibits a structural transition from the triangu- firmed by simulation (Figs. 6b and 6d). However, the lar lattice of pinning centers to the square lattice via a ratios Nv/Nd are different in the configuration formed: disordered state; this is illustrated by the sequence of Nv /Nd = 4 (Fig. 6b) and Nv /Nd = 16/3 (Fig. 6d) as Figs. 5eÐ5g. against Nv /Nd = 3 and Nv /Nd = 4 for the triangular

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(‡) (b) (c)

(d) (e) (f)

µ Ð2 Fig. 6. Configuration of pinning centers (a) and vortex density distribution for nd = 4.27 m , H = 0.057 (b), 0.06 (c), 0.068 T (d), µ Ð2 and for nd = 10.95 m , H = 0.077 (e), 0.091 T (f). Temperature T = 1 K. defect lattice. In intermediate fields, no configurations the triangular lattice); as a result, cells in which vortex with a long-range order are formed. rotation takes place are formed. Since the kagome lattice was obtained from the tri- Figures 6e and 6f show the configurations of the angular lattice, it would be interesting to compare the vortex system, illustrating a transition to the interstitial singularities on the magnetization curve in the cases of phase. For H = 0.077 T, interstitial positions are still kagome and triangular lattices. It was noted above that, vacant, and the cells between defects contain three lat- for the same lattice period, the vortex system forms tices each and no rotation is observed. As the applied identical ordered configurations in the same fields (see magnetic field increases, the second coordination Figs. 6b and 6d), which is reflected in the behavior of magnetization. Figure 7 shows the magnetization curves for triangular (curve 1) and kagome (curve 3) –4πM, T lattices of pinning centers with a period a = 0.45 µm. However, the height of the peaks on the magnetization curve for the kagome lattice is smaller since the density of the kagome lattice is 3/4 of the density of the trian- 0.03 gular lattice with the same period. 3 In the case of identical density, the period of the kag- 12 ome lattice is smaller than the period of the triangular lattice (curve 2 in Fig. 7 depicts the magnetization 0.02 curve with a density close to that of the triangular lat- tice and with period a = 0.375 µm); for this reason, a noticeable peak associated with the screening effect is b observed. Since ordered configurations are formed for d the kagome lattice for larger values of Nv /Nd , the sin- 0.01 gularities associated with the matching of the vortex system to the lattice of pinning centers are observed in stronger fields. For example, a peak observed for H = 0.091 T due to the emergence of a configuration with Nv /Nd = 16/3 has a height approximately equal to that 0 0.05 0.10 of the peak for the triangular lattice in the field H = 0.072 T. H, T The configurations realized in the structure consid- Fig. 7. Magnetization curves for different concentrations of µ Ð2 ered here were called by Laguna et al. [21] an intersti- point defects: triangular lattice, nd = 5.7 m (1); kagome µ Ð2 tial phase (Fig. 6f). Interstitial configurations are char- lattice, nd = 6.15 m (2); and kagome lattice, nd = acterized by filling of the second coordination sphere 4.27 µmÐ2 (3). Temperature T = 1 K. Marked points corre- (we are speaking of the second coordination sphere for spond to the vortex density distributions shown in Fig. 6.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 ORDERED STATES AND STRUCTURAL TRANSITIONS IN A SYSTEM 1073 sphere is filled, which is accompanied by active rota- –4πM, T tion of vortices in the “cells.” This transition is not accompanied by the formation of a singularity on the a magnetization curve. b

0.03 c 3.4. Square Lattice of Extended Defects d We have considered so far the lattices of point pin- ning centers at which only one vortex can be pinned. e f g However, artificially created defects are not point 0.02 1 2 defects as a rule. Recent experiments proved that sev- eral vortices could be trapped at an “extended” pinning center [22]. Reichhardt et al. [23] analyzed the effect of multiple trapping of vortices on the currentÐvoltage characteris- 0.01 tics in the case of a square defect lattice by the molecu- lar dynamics method. It was shown that the matching effects between the number of vortices and the number of defects must also be observed in the case of multiple trapping, the critical depinning force being larger in the 0 0.05 0.10 case of multiple trapping of vortices at a pinning center. H, T Here, we consider the effect of multiple trapping of vortices on the behavior of magnetization. Following Fig. 8. Magnetization curves for extended pinning centers µ Ð2 Reichhardt et al. [23], we choose the potential of inter- with concentration nd = 4 m , distributed chaotically action of a vortex with an extended pinning center in (curve 1) and forming a square lattice (curve 2). Tempera- the following model form: ture T = 1 K. Marked points correspond to the vortex density distributions shown in Fig. 9.  U ()T r Ð r 2 α------0 -------v p Ð 1 , r Ð r < r , 2 v p pin nated since two vortices are pinned at each defect near U p =  U0()0 r  pin the surface. The magnetization curve displays a clearly  > manifested peak. 0, rv Ð r p rpin, For H = 0.054 T (Fig. 9c), three vortices are pinned where α is a coefficient with the dimensions of energy at each pinning center in the surface rows of defects, characterizing the depth of the pinning center; rpin is the preventing the penetration of new vortices. This leads radius of the pinning center; and rv and rp are the radius to an increase in the magnetization. vectors of the vortex and the pinning center, respec- For H = 0.064 T (Fig. 9d), an interstitial center tively. between defects exhibits a tendency to the formation of Calculations were made for a square lattice of pin- a triangular lattice; two vortices are pinned at each ning centers with a period a = 0.6 µm, which corre- defect. In the surface region, three vortices are trapped µ Ð2 at a defect. sponds to density nd = 4 m . We analyzed pinning µ α centers of radius rpin = 0.15 m and depth = 0.1 eV as For H = 0.076 T (Fig. 9e), an ordered configuration µ α is observed. Three vortices are pinned at each defect. well as of radius rpin = 0.2 m and depth = 0.2 eV. A stronger pinning will be considered below. Figure 8 The trapped vortices rotate. Free vortices form a lattice shows the magnetization curves for extended pinning arranged between defects and having fourfold symme- centers arranged chaotically and forming a square lat- try. The formation of this configuration leads to an tice. In the case of a random distribution of defects, the increase in the magnetization. magnetization curve displays no effects, while, in the For H = 0.079 T (Fig. 9f), the previous configuration case of the square lattice of pinning centers, the curve is destroyed. Vortices flow along the channels formed in has a number of singularities. Let us consider the vortex the square lattice of defects. Point f in Fig. 8 corre- density distributions in greater detail and compare these sponding to this case lies in the region of decreasing distributions with the behavior of magnetization. magnetization. For H = 0.043 T (Fig. 9a), extended defects effec- For H = 0.09 T (Fig. 9g), four vortices are trapped at tively trap vortices. In the rows of surface defects, two each defect. Free vortices form a square lattice with vortices are pinned simultaneously at each defect. The interstitial vortices. rotation of pinned vortex pairs takes place. For H = 0.1 T (Fig. 9h), five vortices are trapped at For H = 0.048 T (Fig. 9b), the propagation of the a defect in the surface region and four vortices are magnetic flux to the bulk of the semiconductor is termi- pinned at each defect at the center. Free vortices form a

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(‡) (b) (c)

(d) (e) (f)

(g) (h)

µ Ð2 Fig. 9. Vortex density distribution for a square lattice of extended defects with concentration nd = 4 m , H = 0.043 (a), 0.048 (b), 0.054 (c), 0.064 (d), 0.076 (e), 0.079 (f), 0.09 (g), and 0.1 T (h). The size of the system is 5 × 3 µm2. Temperature T = 1 K. fourfold symmetry lattice with interstitial vortices. Vor- annihilation [14] cannot propagate any further to the tices flow along the channels, in which free vortices bulk of the superconductor until all vortices at a defect rotate in the regions between defects. are suppressed (Fig. 10). This leads to a partial arrest of the annihilation front at the rows of pinned vortices. Thus, the analysis of the vortex density distribution leads to the conclusion that singularities on the magne- tization curve in the case of extended pinning centers 3.5. Behavior of the Vortex System appear as a result of structural transitions as well as the upon a Change in Temperature in the Case screening of the plate surface by pinned vortices. On of a Triangular Lattice of Point Defects the whole, it can be stated that multiple trapping of vor- tices leads to a considerable enhancement of both types Ordered configurations of the vortex system are of singularities (associated both with matching and obviously observed at quite low temperatures. For this with screening) on the magnetization curve. It should reason, we analyzed in the previous sections the effect be recalled that, according to our calculations, a square of ordered configurations of the vortex system on the µ Ð2 behavior of magnetization using curves calculated for lattice of point defects with density nd = 4.94 m (see Fig. 4) does not lead to any appreciable effects in the T = 1 K. In this section, we consider the effect of tem- behavior of magnetization. However, in view of the perature of a triangular lattice of point defects on the large size of extended defects, their density cannot be behavior of the vortex system in the case of periodic increased further. At the same time, the results of com- pinning. putations proved that an increase in the density of point Figure 11 shows the magnetization curves obtained at µ Ð2 defects leads to considerable effects associated with the different temperatures for concentration nd = 5.7 m . It surface screening. was established in Section 3.1 that, for such a concen- In the case of multiple trapping of vortices at tration, all three singularities appear as a result of defects, the process of magnetization reversal of the matching of the vortex lattice to the lattice of defects. superconductor can be demonstrated more visually on As the temperature increases, the Meissner state the magnetic induction distribution. This structure is overheating field decreases for all magnetization distinguished by the fact that several vortices can be curves, and the curves become lower. The peaks associ- pinned at a defect, and the front of the magnetic flux ated with the formation of ordered configurations are

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(‡) (b) (c)

(d) (e) (f)

Fig. 10. Magnetic field distribution (dark regions correspond to the field) for a square lattice of extended defects with concentration µ Ð2 × µ 2 nd = 4 m , H = Ð0.025 (a), Ð0.03 (b), Ð0.035 (c), Ð0.04 (d), Ð0.045 (e), and Ð0.046 T (f). The size of the system is 5 3 m . Temperature T = 1 K. displaced towards lower fields, and the peak heights It was noted in Section 3.1 that ordered configura- decrease upon heating. A further increase in tempera- tions are observed at the base of the peaks on the mag- ture suppresses the singularities associated with match- netization curve, and the segments on which the mag- ing. Analysis of the vortex density distributions indi- netization decreases correspond to the disordered state cates that the peaks vanish due to melting of the vortex of the vortex system. Since the positions of the peaks lattice. change with temperature, a situation is possible where

–4πM, T S6, rel. units 0.04 0.7

0.6 3 0.03 0.5

0.4 1 2 0.02 1 2 0.3 3 0.01 0.2

0.1

0 0.05 0.10 0 10 20 30 H, T T, K Fig. 11. Magnetization curves for a triangular lattice of Fig. 12. Temperature dependence of structural factor S6 for point pinning centers with concentration n = 5.7 µmÐ2 for d a triangular lattice of defects, n = 5.7 µmÐ2 (1); random dis- different temperatures T, K: 1 (1), 5 (2), and 10 (3). The d µ Ð2 arrow corresponds to the magnetic field in which inverse tribution of defects, nd = 5.7 m (2); and pure supercon- crystallization of the vortex system is observed. ductor (3).

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(‡) (b)

(c) (d)

µ Ð2 Fig. 13. Vortex density distribution for different temperatures in field H = 0.062 T for a triangular lattice of defects, nd = 5.7 m : T = 1 (a), 5.2 (b), 10 (c), and 25 K (d). The size of the system is 5 × 2.25 µm2. the region of decrease at a low temperature corresponds temperature both in the case of a random distribution of to the base of the peak at a higher temperature (e.g., the defects and for a pure defect-free sample. region of decrease behind the peak marked by the arrow It should be observed that inverse crystallization of at T = 1 K and the base of the peak at T = 5 K in Fig. 11). the system of vortices was experimentally observed in Thus, as the temperature increases, it is possible to pass the case of chaotic pinning by Avraham et al. [17]. For from a disordered configuration to the ordered one. a random distribution of defects, the vortex system In order to verify this assumption, we calculated the ordering is due to the fact that the effect of pinning configurations of the vortex system for a fixed field H = becomes weaker upon heating and the interaction of 0.062 T at various temperatures. We analyzed the fol- vortices leads to the formation of a triangular vortex lat- lowing situations: a triangular lattice of point defects tice. It was noted above that pinning centers were µ Ð2 (nd = 5.7 m ), a random distribution of point defects assumed to be rather deep to exclude temperature- µ Ð2 induced depinning. For this reason, inverse crystalliza- (nd = 5.7 m ), and a pure sample. tion in the case considered here is of fundamentally dif- In order to characterize the extent of ordering of the ferent physical origin. The ordering of the system upon system, we calculated the standard structural factor S6 heating occurs due to matching between the number of reflecting the tendency of the system to be aligned into vortices and the number of defects [24]. a triangular lattice:

N Z j θ 4. CONCLUSIONS 1 6i ij S6 = ∑ ---- ∑ e , We have considered the effect of periodic pinning Zi i = 1 j = 1 on the behavior of magnetization and on the magnetic flux penetration, distribution, and trapping. It is shown where Zi is the number of nearest neighbors of the ith that, for a periodic pinning, there exist two types of θ vortex and ij is the angle between the nearest neigh- effects leading to the emergence of singularities on bors. magnetization curves, namely, the screening of the Figure 12 shows the temperature dependences of superconductor surface by the vortices pinned in the surface region and the formation of ordered configura- factor S6. In the case of a periodic arrangement of pin- ning centers in the range T = 1Ð5.8 K, an increase in the tions by the vortex system. Effects of the first type are observed in weak fields, while effects of the second value of S6 is observed; i.e., the system of vortices becomes ordered upon heating. At T = 1 K (Fig. 13a), type occur in fields in which the number of vortices is the vortex system is disordered despite the low temper- multiple to the number of defects. ature because the number of vortices is not a multiple It is found that ordered configurations are not of the number of defects. As the temperature increases, located at the point of a local magnetization peak, but new vortices enter the sample, and a stable ordered con- lie at the base of the segment of increasing magnetiza- figuration is formed (Fig. 13b); upon a further increase tion. On segments where magnetization decreases, the in temperature, it melts (Figs. 13c and 13d). Thus, in destruction of vortex lattices takes place. The position the case of a periodic arrangement of defects, the “crys- of singularities on the magnetization curve is deter- tallization” of the vortex system is possible upon heat- mined by the density of defects. As the density of ing. It should be noted that a gradual decrease in the defects decreases, the singularities are displaced structural parameters is observed upon an increase in towards weaker fields and their absolute magnitude

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 ORDERED STATES AND STRUCTURAL TRANSITIONS IN A SYSTEM 1077 decreases. A decrease in the density of defects leads to 4. E. Rossel, M. Van Bael, M. Baert, et al., Phys. Rev. B 53, a correlated disappearance of singularities associated R2983 (1996). both with screening and with the formation of ordered 5. A. N. Grigorenko, G. D. Howells, S. J. Bending, et al., configurations. Phys. Rev. B 63, 052504 (2001). We have analyzed the behavior of magnetization 6. M. E. Gracheva, V. A. Kashurnikov, and I. A. Rudnev, Pis’ma Zh. Éksp. Teor. Fiz. 66, 269 (1997) [JETP Lett. upon a change in temperature in the case of a triangular 66, 291 (1997)]. lattice of point defects. It is found that an increase in temperature suppresses the singularities, which are dis- 7. M. E. Gracheva, M. V. Katargin, V. A. Kashurnikov, and I. A. Rudnev, Fiz. Nizk. Temp. 23, 1151 (1997) [Low placed to the region of weak fields. After the attainment Temp. Phys. 23, 863 (1997)]. of the melting point for the vortex lattice, the singular- ities associated with the formation of ordered configu- 8. M. E. Gracheva, V. A. Kashurnikov, and I. A. Rudnev, Fiz. Nizk. Temp. 25, 148 (1999) [Low Temp. Phys. 25, rations disappear. 105 (1999)]. It is shown that inverse crystallization of the vortex 9. M. E. Gracheva, V. A. Kashurnikov, O. V. Nikitenko, and system, which is caused by the entrance of new vortices I. A. Rudnev, Fiz. Nizk. Temp. 25, 1027 (1999) [Low and the formation of a stable configuration, is possible Temp. Phys. 25, 765 (1999)]. in the case of periodic pinning. Inverse crystallization 10. V. A. Kashurnikov, I. A. Rudnev, M. E. Gracheva, and of the vortex system can be observed visually on super- O. V. Nikitenko, Zh. Éksp. Teor. Fiz. 117, 196 (2000) conductors with a periodic arrangement of artificial [JETP 90, 173 (2000)]. pinning centers with the help of magnetooptical meth- 11. I. A. Rudnev, V. A. Kashurnikov, M. E. Gracheva, and ods or using high-resolution scanning magnetometry. O. A. Nikitenko, Physica C (Amsterdam) 332, 383 (2000). Various lattices of pinning centers are considered. It is found that the most striking effects associated with 12. A. V. Eremin, O. S. Esikov, V. A. Kashurnikov, et al., the matching of the vortex system to the defect lattice Supercond. Sci. Technol. 14, 690 (2001). are observed for a triangular lattice of defects. 13. V. A. Kashurnikov, I. A. Rudnev, and M. V. Zubin, Supercond. Sci. Technol. 14, 695 (2001). The effect of multiple trapping of vortices on the 14. V. A. Kashurnikov, I. A. Rudnev, and M. V. Zyubin, Zh. behavior of magnetization is analyzed. In the case of Éksp. Teor. Fiz. 121, 442 (2002) [JETP 94, 377 (2002)]. extended pinning centers at which more than one vortex 15. Y. Yeshurun, M. B. Salamon, K. V. Rao, et al., Phys. Rev. can be pinned, the magnetization curve displays con- Lett. 45, 1366 (1980). siderable effects associated both with screening and 16. A. L. Greer, Nature 404, 34 (2000). with matching. 17. N. Avraham, B. Khaykovich, Y. Myasoedov, et al., Nature 411, 451 (2001). ACKNOWLEDGMENTS 18. V. V. Pogosov, A. L. Rakhmanov, and K. I. Kugel’, Zh. Éksp. Teor. Fiz. 118, 676 (2000) [JETP 91, 588 (2000)]. This study was supported financially by the Russian 19. S. L. Lee, P. Zimmermann, H. Keller, et al., Phys. Rev. Foundation for Basic Research (project no. 00-02-17803), Lett. 71, 3862 (1993). the Federal Special Program “Integration” (project 20. C. Reichhardt, C. J. Olson, and F. Nori, Phys. Rev. B 57, no. B0048), and the State Science and Engineering Pro- 7937 (1998). gram “Important Trends in the Physics of Condensed 21. M. F. Laguna, C. A. Balseiro, D. Domínguez, and State” (Subprogram “Superconductivity”). F. Nori, Phys. Rev. B 64, 104505 (2001). 22. S. Field, S. S. James, J. Barentine, et al., cond- mat/0003415. REFERENCES 23. C. Reichhardt, G. T. Zimanyi, R. T. Scalettar, and 1. V. V. Moshchalkov, M. Baert, V. V. Metlushko, et al., I. K. Schuller, cond-mat/0102266. Phys. Rev. B 57, 3615 (1998). 24. M. V. Zyubin, I. A. Rudnev, and V. A. Kashurnikov, 2. V. Metlushko, U. Welp, G. W. Grabtree, et al., Phys. Rev. Pis’ma Zh. Éksp. Teor. Fiz. 76, 263 (2002) [JETP Lett. B 59, 603 (1999). 76, 227 (2002)]. 3. D. J. Morgan and J. B. Ketterson, J. Low Temp. Phys. 122, 37 (2001). Translated by N. Wadhwa

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1078Ð1088. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1227Ð1238. Original Russian Text Copyright © 2003 by I. G. Kuleev, I. I. Kuleev, Taldenkov, Inyushkin, Ozhogin, Itoh, Haller. SOLIDS Structure

Normal Processes of PhononÐPhonon Scattering and the Drag Thermopower in Germanium Crystals with Isotopic Disorder I. G. Kuleeva,*, I. I. Kuleeva, A. N. Taldenkovb, A. V. Inyushkinb, V. I. Ozhoginb, K. M. Itohc, and E. E. Hallerd aInstitute of Physics of Metals, Ural Division, Russian Academy of Sciences, Yekaterinburg, 620219 Russia *e-mail: [email protected] bInstitute of Molecular Physics, Russian Research Center Kurchatov Institute, Moscow, 123182 Russia cDepartment of Applied Physics and Physico-Informatics, Keio University, Yokohama 223-8522, Japan dUniversity of California and Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA Received December 10, 2002

Abstract—A strong dependence of the thermopower of germanium crystals on the isotopic composition is experimentally found. The theory of phonon drag of electrons in semiconductors with nondegenerate statistics of current carriers is developed, which takes into account the special features of the relaxation of phonon momentum in the normal processes of phononÐphonon scattering. The effect of the drift motion of phonons on the drag thermopower in germanium crystals of different isotopic compositions is analyzed for two options of relaxation of phonon momentum in the normal processes of phonon scattering. The phonon relaxation times determined from the data on the thermal conductivity of germanium are used in calculating the thermopower. The importance of the inelasticity of electronÐphonon scattering in the drag thermopower in semiconductors is analyzed. A qualitative explanation of the isotope effect in the drag thermopower is provided. It is demonstrated that this effect is associated with the drift motion of phonons, which turns out to be very sensitive to isotopic disorder in germanium crystals. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION lifetime. Therefore, a decrease in the degree of isotopic disorder must further result in an increase in the abso- Recently, high-quality single crystals of germanium lute values of the phonon drag thermopower. However, of different chemical compositions have been success- the Herring theory [6] predicts a very weak dependence fully grown, including a uniquely pure (both chemi- of α (T) on the impurity concentration in the case of a 70 ph cally and isotopically) crystal with 99.99% Ge isoto- fairly pure semiconductor (see also [7Ð9]). Within a 70 pic enrichment [1], hereinafter referred to as Ge standard one-parameter approximation, the phonon (99.99%). Experimental investigations of the thermal relaxation rate in the normal processes (N processes) of conductivity [2, 3] of these crystals have revealed that, phononÐphonon scattering was included in the total 70 for monoisotopic samples of Ge (99.99%), the maxi- phonon relaxation rate, which was the only parameter mal values of thermal conductivity are an order of mag- defining the nonequilibrium phonon distribution func- nitude higher than those for crystals with natural isoto- tion. This approach is justified in the case of “impure” pic composition. It is evident that this effect is associ- semiconductors, when the phonon relaxation rate in the ated with the increase in the free path of thermal N processes, ν (q), is much lower than the phonon phonons because of the decrease in the scattering by phN relaxation rate in the resistive processes of scattering, “impurity” isotopes; the normal processes of phonon– ν phonon scattering play an important part in the case of phR(q), caused by the phononÐphonon scattering in the isotopically pure crystals at temperatures in the vicinity umklapp processes, and from the defects and bound- of the maximum of thermal conductivity [3Ð5]. A vari- aries of the sample. In the opposite extreme case of ation in the isotopic composition must also affect the fairly pure semiconductors, one must take into account thermoelectric phenomenon of the phonon drag ther- the phonon system drift caused by the N processes of α mopower ph(T), which explicitly depends on phonon phononÐphonon scattering [10, 11].

1063-7761/03/9606-1078 $24.00 © 2003 MAIK “Nauka/Interperiodica”

NORMAL PROCESSES OF PHONONÐPHONON SCATTERING 1079

In nondegenerate conductors, the electrons interact phononÐphonon umklapp processes) to thermal resis- only with long-wavelength phonons whose wave vector tance. The drift motion of phonons must be taken into is much less than the wave vector of thermal phonons account under conditions in which the phonon relax- ν making the main contribution to thermal conductivity. ation rate in the N processes, phN(q), is higher than or Because the probability of isotopic scattering of a comparable to the rate of relaxation in resistive pro- ν phonon is proportional to the fourth power of its wave cesses of scattering, phR(q). It is evident that, in isoto- vector q, the thermopower calculated within a one- pically pure Ge samples at low temperatures, when the parameter approximation turns out to be insensitive to phononÐphonon umklapp processes are largely frozen, the degree of isotopic disorder. Kozlov and Nagaev [12] the rate of relaxation of longitudinal phonons in the N called attention to the anomalies of thermopower aris- processes significantly exceeds the resistive rate of ν ing in such a situation as long as 30 years ago. They relaxation phR(q) which is mainly due to isotopic dis- have demonstrated that, in the case of very perfect crys- order. In this paper, we will demonstrate that the inclu- tals, the thermal phonon drag of long-wavelength sion of the phonon drift caused by the N processes phonons may cause anomalously high values of ther- enables one to qualitatively explain the significant mopower. In contrast to the Herring thermopower, this effect of isotopic disorder on the drag thermopower in thermopower (two-stage drag thermopower) is inversely Ge crystals. proportional to the impurity concentration [13] and is closely associated with the mechanism of relaxation of In describing the drag thermopower, in contrast to long-wavelength phonons from thermal phonons in the the previous investigations, we will separate the contri- normal processes of phononÐphonon scattering. butions by longitudinal and transverse phonons and take into account the redistribution of the phonon The first attempt at detecting the effect of isotopic momentum in the N processes of scattering both within phonon scattering on the thermopower was made by each vibrational branch (Simons mechanism [19]) and Oskotskii et al. [14], who investigated the thermal con- between different vibrational branches of phonons ductivity and thermopower of Te crystals with two dif- (Herring mechanism [20]). In this approximation, the ferent isotopic compositions, of which one was sub- nonequilibrium of the phonon system is described by jected to 92% 128Te isotopic enrichment. The isotopic six parameters, namely, by the rates of phonon relax- enrichment resulted in a threefold increase in the max- ation in the resistive and normal processes of scattering imal values of thermal conductivity; however, Oskot- and by the average drift velocities for each branch of skii et al. [14] observed no effect of isotopic disorder on the phonon spectrum. This description of phonon non- the phonon drag thermopower at low temperatures. equilibrium enables one to reveal new features of relax- This negative result is possibly due either to the differ- ation of the momentum of quasi-particles and their ent concentrations of charged impurities in the investi- effect on the thermopower and thermal conductivity of gated samples or to the relatively weak contribution of semiconductors. We will demonstrate below that the the N processes to the overall phonon relaxation rate. drift velocity of phonons (as well as the thermal con- ductivity) is largely defined by thermal phonons for In recent measurements of the thermopower in ger- which the scattering from defects plays a significant manium crystals of different isotopic compositions, we part. Therefore, when the drift of the phonon system is found an almost twofold increase in the thermopower at taken into account, the thermopower becomes sensitive low temperatures in a monoisotopic sample of 70Ge to the degree of isotopic disorder. We further give the (99.99%) compared to Ge of natural isotopic composi- results of measurements and quantitative analysis of the tion [15]. This result is indicative of the important part isotope effect in the drag thermopower. played by the N processes in the relaxation of the phonon system for isotopically enriched germanium crystals. The importance of these processes in the lat- 2. EXPERIMENTAL RESULTS tice thermal conductivity without the separation of the In this paper, we analyze the experimental data on contributions made by longitudinal and transverse α phonons is studied quite well [16Ð18]. In the N pro- the thermopower (T) of single crystals of germanium cesses of scattering, the phonon momentum is con- with three different isotopic compositions, namely, the served. These processes make no direct contribution to natural composition and compositions subjected to 70 the thermal resistance; they provide for the relaxation Ge isotopic enrichment of 96.3% and up to 99.99%. of the phonon subsystem to the drift locally equilibrium Ge crystals of the n and p types with the concentration | | × 13 Ð3 distribution. Therefore, the N processes redistribute the of charged impurities of Nd Ð Na < 2 10 cm were energy and momentum between different phonon used. Note that Geballe and Hull [21] found that, in the modes to form the nonequilibrium phonon distribution case of highly pure samples of Ge of the n and p types, function and prevent a strong deviation of each phonon the phonon drag thermopower very weakly depends on mode from the equilibrium distribution. This is accom- the concentration of electrically active impurities at a panied by a variation of the relative contribution by var- doping level below 1015 cmÐ3 and decreases in magni- ious resistive processes of scattering (scattering from tude at higher concentrations. Our samples were shaped defects and boundaries of the sample and in the as parallelepipeds of square cross section. The samples

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1080 KULEEV et al.

α, mV/ä doing, the thermopower at the maximum for isotopi- 20 cally pure 70Ge (99.99%) is approximately twice as 3 high as that for germanium with the natural isotopic composition (natGe). Compared to the thermal conduc- 10 4 tivity, the thermopower of germanium turned out to be approximately five times less sensitive to the variation 0 of the degree of isotopic disorder. Note that, in the case of samples with the same isotopic composition, the thermopower is independent of the degree of doping –10 5 within the experimental error. This is in good agree- ment with the well-known fact of the weak sensitivity 1 of the magnitude of the drag thermopower to the dopant –20 concentration in fairly pure germanium crystals [6, 21]. 2 These special features of thermopower call for detailed –30 theoretical treatment. 10 100 T, K Given below are the results of quantitative analysis Fig. 1. The magnitude of differential thermopower as a of the isotope effect in the thermopower of germanium. function of temperature for samples of germanium crystals Attention is focused on the investigation of the effect of with different isotopic compositions: (1) sample no. G2, the drift motion of the phonon system, due to the nor- (2) G7, (3) G70, (4) Gn21, (5) S1. mal processes of phonon scattering, and of the inelas- ticity of electronÐphonon scattering on the drag ther- had a total length of approximately 40 mm, with the mopower. The effect of the normal processes of phonon square side in cross section of approximately 2.5 mm. scattering on the mutual drag of electrons and phonons The thermopower was measured using the method of in metals and in degenerate semiconductors is treated steady longitudinal heat flux in vacuum in the tempera- in [10, 11]. In our paper, this theory is generalized to ture range from 8 to 300 K. The heat flux was directed the case of semiconductors with nondegenerate statis- along the longer axis of the sample; the temperature dif- tics of current carriers. We have treated the redistribu- ference along the sample did not exceed 1% of its aver- tion of the momentum of longitudinal and transverse age temperature. The parameters of five investigated phonons in the N processes of scattering both within samples are given in the table. each vibrational branch and between different vibra- tional branches. Previously, this approach made it pos- The experimental data on the temperature depen- sible to successfully explain the effect of the isotopic dence of the thermopower are given in Fig. 1. One can composition on the thermal conductivity of germanium see in the figure that, at temperatures above 70 K, the and silicon crystals [22, 23]. Here, this method is used thermopower is almost independent of temperature. to investigate the effect of isotopic disorder on the drag α The diffusion component of thermopower e(T) pre- thermopower. In calculating the emf, we used the times dominates in this temperature range; this component is of phonon relaxation determined from the data on the defined by the degree of doping and by the band param- thermal conductivity for the same samples of germa- eters of the semiconductor and is independent of the nium [3, 22]. It is demonstrated that, in fairly pure phonon lifetime. At low temperatures, where the semiconductors, both the thermopower and the lattice α α phonon drag thermopower ph(T) predominates, (T) thermal conductivity [22] (with the separation of the increases with decreasing isotopic disorder; in so contributions by longitudinal and transverse phonons)

Parameters of investigated samples of Ge crystals Isotopic composition, Sample no. g, 10Ð5 Axis |N Ð N |, 1012 cmÐ3 % 70Ge d a G2 99.99 0.008 [100] 2.7 G7 99.99 0.008 [111] 20 G70 96.6 7.75 [100] 2 Gn21 natural 58.9 [100] 0.5 S1 natural 58.9 [111] 4

2 Mi Ð M Note: g = ∑ f i------is the factor characterizing the isotopic disorder of the crystal [3]. i M

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 NORMAL PROCESSES OF PHONONÐPHONON SCATTERING 1081 largely depend on the mechanism of relaxation of the tion functions in the form [3Ð5] phonon momentum in the N processes of scattering. ∂ ()ε δ δ f 0 ⋅ ε f k ==f 0 k + f k, f k Ð------∂ε vk c(), k (3) 3. THE EFFECT OF THE N PROCESSES λ 0 OF PHONONÐPHONON SCATTERING Nq = Nqλ + gλ(),q ON THE RELAXATION OF MOMENTUM ε OF ELECTRONS AND PHONONS where f0( k) is the equilibrium electron distribution δ IN A NONEQUILIBRIUM ELECTRONÐPHONON function, and fk and gλ(q) are nonequilibrium addi- SYSTEM tions to the distribution functions, which are linear as regards the external effects. We will linearize the colli- For simplicity, we will treat a semiconductor with sion integrals with respect to these additions. We will the isotropic law of dispersion of current carriers. We δ represent the collision integrals Iie( fk) and Iph e(f0, gλ(q)), will calculate a charge flow caused by the effect of 0 as well as I (δf , N λ ) in the approximation of elastic electric field E = {Ex, 0, 0} and the temperature gradient e ph k q ∇ ∇ T = ( xT, 0, 0). The set of kinetic equations for the scattering, in terms of relaxation rates [27]. In calculat- λ nonequilibrium electron f(k, r) and phonon N (q, r) ing the collision integral Iph e(f0, gλ(q)), we will not distribution functions in view of the N processes of restrict ourselves to the linear approximation with scattering has the form [11] respect to the inelasticity parameter [7Ð9, 24Ð28] and will take into account the inelasticity of collisions ∂ between nonequilibrium phonons and equilibrium elec- e f k λ ---E ⋅ ------+ ()v ⋅ ∇ f = I ()f + I (), f , N trons. ប 0 ∂k k r k ei k e ph k q λ ⋅ ∇ λ ()νλ ()0 ()λ1 (1) We will substitute expressions (2) and (3) into (1) to vq r Nq = Ð Nq Ð Nqλ ph derive, similar to [11], the expression for the phonon ()νλ ()⋅ λ ()λ, distribution function gλ(q), Ð Nq Ð N quλ phN + Iph e N q f k . 0 ()0 បω λ Nqλ Nqλ + 1 qλ λ ⋅ ∇ Here, vq = sλq/q is the group velocity of acoustic gλ()q = Ð------λ ------vq T ν ()q k T 2 λ 0 ph B phonons with polarization ; Nqλ is the Planck func- (4) ប ⋅ νλ λ quλ 0 0 phN()q tion; ν ()q is the phonon relaxation rate in the N pro- + ------N λ()N λ + 1 ------. phN k T q q νλ cesses of scattering; and the rate B ph()q

ν()λ1 νλ νλ νλ Here, ph ()q = phi()q ++phB()q phU()q νλ ν()λ1 νλ νλ includes all of the nonelectron resistive rates of phonon ph()q = ph ()q ++ph e () q ph N () q relaxation, due to the phonon scattering from phonons νλ νλ = phN()q + phR()q νλ in the umklapp processes, phU()q , from defects and λ is the total rate of relaxation of phonons with the wave isotopic disorder, ν ()q , and from the boundaries of λ phi vector q and polarization λ, and ν ()q is the rate of νλ ph e the sample, phB()q . The collision integrals of electrons relaxation of momentum of phonons from electrons with impurities, Iei , and with phonons, Ie ph, and of [24Ð27]. The first term in expression (4) is defined by phonons with electrons, Iph e, were determined in [7Ð9, the diffusion motion of phonons, and the second term 24Ð27]. In Eq. (1), it is taken into account that the takes into account the drift phonon motion and is asso- N processes of scattering bring the phonon subsystem ciated with the normal processes of phononÐphonon to the locally equilibrium Planck distribution with the scattering. The phonon drift velocity uλ is found from drift velocity uλ which may be different for phonons of the balance equation for phonon momentum, which fol- different polarizations [16Ð18], lows from the law of conservation of momentum in the normal processes of phononÐphonon scattering, បω ប ⋅ Ð1 qλ Ð quλ N()qu, λ = exp ------Ð 1 k T B 1 ប νλ ()λ (), 1 ប νλ (2) --- ∑ q phN()q Nq Ð N quλ = --- ∑ q phN()q បqu⋅ λ V V ≈ 0 0 ()0 , λ , λ Nqλ + ------Nqλ Nqλ + 1 . q q (5) kBT ប ⋅ × quλ 0 ()0 gλ()q Ð0.------Nqλ Nqλ + 1 = We will represent the electron and phonon distribu- kBT

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1082 KULEEV et al.

( ) ν L ut = uH. The Simons mechanism of relaxation [19] phR uL = ut = uH L-ph R involves phonons of one polarization. In the case of this ν(L) mechanism of scattering, the law of conservation of

phN momentum in the N processes is valid for each branch ∆ T uH of the phonon spectrum, and the drift velocity of longi- tudinal phonons differs from that of transverse ν(t) phN phonons. Therefore, we will treat below two options for (a) t-ph R the relaxation of the phonon momentum in the N pro- ν(t) cesses. phR We use expressions (4) and the balance equation for phonon momentum (5) to find the phonon drift velocity uL uλ for the Herring (H) and Simons (S) mechanisms of ≠ ν(L) uL ut phN relaxation, as was done in [11]. After this, we derive, for the phonon distribution function gλ(q), L-ph R

ν(L) ∆ phR 0 0 T N λ()N λ + 1 បω λ λ ν(t) q q q ⋅ ∇ phR gλ()q = Ð------λ ------vq T, t-ph R ν˜ ()q k T 2 (b) ph B (6) ν(t) λ()SH, λ λ Ð1 phN ν ν ()ν β ˜ ph ()q = ˜ ph()1q + ˜ phN()q ()SH, , ut λ Ψ s 2 ΨL + 2S2 Ψt Fig. 2. A scheme illustrating the relaxation of momentum in β N β L N * N S ==------λ -, H ------L 5 t . (7) a phonon system for two mechanisms of phonon scattering Ψ sλ Ψ + 2S Ψ in the normal processes: (a) for the Herring mechanism, NR NR * NR (b) for the Simons mechanism. Here, S /s , and the remaining functions are defined * = sL t A scheme illustrating the redistribution and relaxation by the expressions of the momentum received by a phonon system from λ the temperature gradient is given in Fig. 2. The phonon λ ν ()q scattering in the resistive processes of scattering (R) Ψ = ------phN - N νλ (from isotopic disorder, electrons, and sample bound- ph()q zdλ aries, and the phononÐphonon scattering in the zdλ λ umklapp processes) brings about the relaxation of the λ λ ν ≡ 4 phN()q 0 ()0 momentum of the phonon system. The N processes ∫ dzq()zq ------λ -Nqλ Nqλ + 1 , (8) ν ()q redistribute the momentum between different phonon 0 ph modes (L-ph and t-ph) and bring about the phonon drift λ λ λ ν ()q ν ()q with an average velocity uλ. As in [10, 11, 16Ð18], we Ψ = ------phR phN , NR νλ assume that the drift velocity is independent of the ph()q z λ wave vector of phonons. Two mechanisms of normal d three-phonon processes of scattering are usually exam- where ined, namely, the Herring [20] and Simons [19] mech- anisms. In the Herring mechanism of N processes, បω phonons of different polarizations are involved: the rate λ qλ q kBT zq ==------, qTλ =------ប -, of relaxation of transverse phonons in the Herring kBT qTλ sλ mechanism is defined by the scattering processes (t + L L) in which one transverse and two longitudinal បω λ 2k dλ phonons are involved; in this case, the main contribu- z2k ==------, zdλ ------tion to the rate of relaxation of longitudinal phonons is qTλ kBT made either by the processes of decay of a longitudinal ω phonon into two transverse phonons belonging to dif- ( dλ is the Debye frequency of phonons with polariza- ferent branches or by the fusion of two transverse tion λ). One can see in expression (6) that the inclusion phonons to form a longitudinal phonon (L t 1 + t 2 ). of the normal processes of phononÐphonon scattering This relaxation mechanism provides for redistribution reduces to the renormalization of the rate of relaxation of the drift momentum between longitudinal and trans- of phonon momentum. The phonon drift motion caused verse phonons (see Fig. 2) and tends to establish a by the N processes leads to a decrease in the effective locally equilibrium distribution with a drift velocity phonon relaxation rate; this decrease is different for the that is the same for phonons of both polarizations, uL = Herring [20] and Simons [19] relaxation mechanisms.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 NORMAL PROCESSES OF PHONONÐPHONON SCATTERING 1083

We will now consider the electron subsystem. The electron–phonon collision rate is defined by the expres- ν electron scattering from impurities, ei, results in the sion relaxation of electron momentum, and the mechanisms of electronÐphonon relaxation characterized by the λ λ ν 〈〉ν , λ rates ν and ν bring about the redistribution of e ph () k = ∑ e ph ()kq , e ph ph e z2k momentum within an electronÐphonon system; in so λ doing, the electrons interact only with long-wavelength 2 phonons. We do not treat the renormalization of the λ me C0λ 5 0 0 ν ()kq, = ------q N λ()N λ + 1 thermopower due to the mutual drag of electrons and e ph πប2 3 q q phonons. Note that the quantities such as the ther- 2 k (11) mopower and thermal conductivity are found from the 2msλ 2msλ condition that the total current through the sample is × + ε Ð ε 1 Ð ------F ()k +  1+ ------F ()k , zero. In this case, the average velocity of ordered elec- បq បq tron motion in any physically small sample volume is zero. Therefore, the transfer of momentum of ordered where electron motion to the phonon subsystem is low, and the effect of electron nonequilibrium on the electrons via 2 ± f ()ε Ð f ()ε ± បω E λប the phonon subsystem may be ignored [11]. On the ε ± 0 k 0 k q 2 0 F ()k ==------ε ()ε -, C0λ ------ρ , other hand, a steady phonon flow from the hot end of f 0()1k Ð f 0()k sλ the sample to the cold end exists in the presence of a ρ temperature gradient, and the magnitude of ther- E0λ is the deformation potential constant, and is the mopower is largely defined by the transfer of momen- density. For semiconductors with nondegenerate statis- tum of ordered phonon motion to electrons. Note that, tics of current carriers, for longitudinal phonons in Ge crystals at low temper- εζ atures, when the electronÐphonon drag makes a marked ε ≈ Ð f 0() expÐ------, contribution to the thermopower, the relaxation rate kBT ν ӷ ν phN(q) phR(q) [22]. It follows from the foregoing that the relaxation of phonon momentum in a nonequi- the F± functions may be represented in the form librium electronÐphonon system must be taken into account more rigorously than was done in the case of F+ ≈ 1 Ð eÐz, FÐ ≈ ez Ð 1; one-parameter approximation [7Ð9, 24Ð28]. The purpose of this theoretical analysis is to investi- then, in view of the inelasticity of electronÐphonon gate the effect of the phonon drift caused by the N pro- scattering, we find cesses on the drag thermopower. In this case, one can ignore the mutual drag of electrons and phonons and ()τ ε Ð1 ()τλ Ð1 Ð3/2 obtain, as was done in [27], the following solution for e ph() = ∑ 0e ph x J1λ(),x the function c(ε): λ

1 + Ð εζ ---() ε τε() kB˜ ε Ð ∇ J1λ()x = J1λ()x + J1λ()x , c()= Ðe E +,-----Aph()+ ------T (9) 2 e kBT ± zmax, λ λ ± ± z 2 2k λ J1λ()x = ∫ J1λ()z dz, msλ λ ν ()kq , ˜ ε e ph Aph()= ∑------∫ dzq ------λ -. (10) 0 (12) kBT ν˜ ()q λ 0 ph 3 + z ()z Ð δλ J1λ()z = ------, ez Ð 1 Here, τ(ε) is the total relaxation time of electrons;

3 z ()z + δλ τÐ1 ε ν ν ν ν Ð Ðz ()k ==e()k ei()k ++e0()k e ph (); k J1λ()z = ------e , ez Ð 1 ν the rates of electron relaxation from neutral, e0(k), and 2 charged, ν (k), impurities have the known form (see, ()τλ Ð1 me C0λ 3 δÐ3/2 ei 0e ph = ------qTλ λ . for example, [7], formulas (10.29) and (10.50)); and the 2πប3

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1084 KULEEV et al. The upper limits of integration in Eqs. (12) are defined duction current j by dividing it into three parts propor- by the expressions tional to nonequilibrium additions to the electron distri- bution function c(ε), ± 2បsλk 1/2 1/2 z , λ ==------2 δλ x ± δλ, ∞ max k T ∂ 3 B e εf 0 k ε j = Ð------∫d Ð------c() 2 3π2 ∂ε m()ε (16) 2mesλ T sλ 0 δλ ==------, (13) k T T ≈ B jdr ++jdrag jdif = 0. 2 ε 2mesλ From the condition j = 0, we find T sλ ==------, x ------, kB k0T αα α = ph + dif. (17) where δλ is the inelasticity parameter. In the case of semiconductors, the effective temperature Tsλ defining We will not consider below the diffusion component of the inelasticity of electronÐphonon scattering is, as a thermopower: for germanium crystals at T < 100 K, this ≈ contribution is small. In the case of nondegenerate sta- rule, less than 1 K; for example, in the case of Ge, TsL ≈ × 5 ≈ tistics, the expression for the phonon drag thermopower 0.8 K at sL 5.21 10 cm/s, me 0.22m0. Therefore, δ Ӷ Ӷ may be represented in the form even at T > 10 K, λ 1 and zmax 1; therefore, expres- sions (12) may be expanded in powers of z. In a zero 〈〉〈〉τε ˜ ε approximation with respect to the inelasticity parame- α kB ()Aph() ph = Ð------〈〉〈〉τε -, ter δλ, we will derive from Eqs. (12) the known expres- e () sion for the electronÐphonon relaxation time, ∞ Ðx 3/2 ε ∫dxe x f () (18) τλ ε τλ δ Ð2 Ð1/2 ε e ph()= 0e ph()2 λ x . (14) 〈〉〈〉ε 0 f () ==------∞ -, x ------, k0T In the same approximation, the expression for the inverse ∫dxeÐx x3/2 time of phononÐelectron relaxation has the form 0

2 2 2 λ me C0λ kBT η Ðz Ðxm ν ()kq, = ------e ()1 Ð e e , ˜ me C0λ 2 Ð1/2 Ð3/2 + Ð ph e 5 Aph()ε = ∑------q λδλ x ()J λ()x + J λ()x , πប πប3 T 2 2 (15) λ 4 2 (19) ()z Ð δλ xm = ------. ± 4δλ zmax, λ ± ± J1λ()z dz J2λ()x = ∫ ------λ , The concentrations of electrons, ne, and of charged ν˜ ()q η ζ 0 ph donors, Nd+, and the reduced Fermi level = /kBT are 3()δ found from the condition of electroneutrality for ger- + z z Ð λ (20) ≈ 12 13 Ð3 J λ()z = ------, manium (see [7], formula (6.9)): Nd 10 Ð10 cm , 1 z ε ≈ ≈ e Ð 1 d 0.01 eV, and me 0.22m0. For these values of the 3 parameters, the criterion of nondegenerate statistics is Ð z ()z + δλ Ðz well valid. J1λ()z = ------e . ez Ð 1 In the approximation we adopted, the c(ε) function allows for the direct effect of the electric field and tem- The upper integration limits in Eqs. (20) are defined perature gradient on the electron subsystem, as well as by expressions (13). First of all, note that the drag ther- for the effect of the phonon drag of electrons. mopower includes, as does the lattice thermal conduc- tivity [11, 22], the phonon momentum relaxation rate 4. THE DRAG THERMOPOWER renormalized by the N processes. Unlike standard IN SEMICONDUCTORS one-parameter approximations for the drag ther- WITH NONDEGENERATE STATISTICS mopower [7Ð9, 24Ð28], expressions (18)Ð(20) include OF CURRENT CARRIERS the inelasticity of electronÐphonon scattering, as well as the contribution made by the phonon drift motion. We will examine the effect of the normal processes This contribution has different forms for the Herring of phononÐphonon scattering on the thermal electro- and Simons mechanisms of relaxation. Because the motive force of semiconductors with nondegenerate phonon drift velocity is defined by all thermally excited statistics of current carriers. We will calculate the con- phonons, the thermopower becomes sensitive to the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 NORMAL PROCESSES OF PHONONÐPHONON SCATTERING 1085 degree of isotopic disorder. It follows from formulas (6), Therefore, the inferences previously made with respect to (19), and (20) that the inclusion of the drift of the the temperature and field dependences of the drag ther- phonon subsystem, which is associated with the normal mopower [7–9, 24–28] need to be refined. It is evident processes of phonon scattering, brings about a decrease that, in the case of one-parameter approximation (22), the α in the effective relaxation rate of phonons and, accord- drag thermopower ph is insensitive to the degree of ingly, an increase in the fraction of momentum trans- isotopic disorder. Note that a different approach to the ferred to electrons by phonons. This result is of practi- calculation of the drag thermopower was suggested cal importance as regards the interpretation of experi- in [12, 13]. This approach is based on dividing the mental data on the thermopower of germanium crystals entire system into two subsystems, namely, the sub- with isotopic disorder. system of long-wavelength phonons (q < 2k) with ν Ӷ ν which electrons interact and the subsystem of thermal In the extreme case of phN(q) phR(q), one can phonons (q > qTλ). The authors of [12, 13] suggested a ignore the contribution of the phonon drift motion and mechanism of two-stage drag: the phonon drift motion use the expression for the drag thermopower that was is defined by the thermal phonons, which, in turn, drag previously derived with a one-parameter approximation ν ӷ ν the long-wavelength phonons. By their physical con- [7Ð9, 24Ð28]. With phN(q) phR(q), the normal pro- tent, our method and the approach developed in [12, 13] cesses of phononÐphonon scattering and the drift of the coincide, because, in our theory, it is the thermal phonon system associated with this scattering lead to a phonons that define the phonon drift motion, as well as significant increase in the absolute values of ther- the thermal conductivity. However, our method is more mopower. Note that, in interpreting the experimental general: we treat correctly the N processes of scattering data on the drag thermopower in previous studies of thermal phonons with regard for their drift and diffu- involving the use of a one-parameter approximation sion motion, identify the contributions by phonons of (see [7Ð9, 24Ð28]), the relaxation rate in the normal different polarizations, and treat both the intrabranch ν processes phN(q) was included in the total phonon and interbranch redistribution of the phonon momen- νλ tum in the N processes of scattering. relaxation rate ph()q as the resistive mechanism of ν ӷ ν phonon scattering, and, at phN(q) phR(q), it was treated as the only mechanism of relaxation of momen- 5. THE RESULTS OF CALCULATION tum of long-wavelength phonons [7, 8]. However, it OF THE DRAG THERMOPOWER follows from expressions (18)Ð(20) that, in this extreme OF GERMANIUM CRYSTALS ν case, the relaxation rate phN(q) is eliminated from the OF DIFFERENT ISOTOPIC COMPOSITIONS α drag thermopower, and ph is fully defined by the aver- aged relaxation rate of phonons in the resistive pro- Given below are the results of numerical analysis of cesses of scattering, the drag thermopower in germanium crystals of differ- ent isotopic compositions, which, in view of the assumptions made, may only pretend to be a qualitative 2 〈〉〈〉τενλ ε ()SH, kB mFsλ () e ph() explanation of the effect. The main results include the α ≈ Ð-----∑------, ph e k T λ ()SH, isotropic band approximation and the assumption that λ B 〈〉ν ()q 〈〉〈〉τε() ph R zdλ the phonon drift velocity is independent of the wave λ vector, i.e., the drift velocities of thermal and long- λ () 〈〉ν 〈〉ν S phR wavelength phonons are the same. The calculation of phR()q = ------()4 , Jλ the drag thermopower with a real band structure of ger- L 5 t (21) manium within the suggested method of inclusion of λ ()H 〈〉ν + 2S 〈〉ν 〈〉ν phR * phR the normal processes of phonon scattering, with the phR()q = ------() () -, J 4 + 2S3 J 4 long-wavelength and thermal phonons treated sepa- L * t rately, is of interest per se. In this analysis, we restrict z λ d 4 z ourselves to examining the effect of the phonon drift ()4 dzz e motion and of the inelasticity of electronÐphonon scat- Jλ = ∫ ------2. ()ez Ð 1 tering on the drag thermopower in germanium crystals. 0 The values of the parameters defining the phonon relax- A one-parameter approximation yields in this case an ation rate were borrowed from the results of analysis of entirely different result, the data on the thermal conductivity of Ge crystals of different isotopic compositions, obtained in [3, 22]. The use of these parameters made it possible to fit the α()SH, ≈ kB ph Ð----- results of calculations of thermal conductivity for the e Herring mechanism of relaxation [22] to the experi- λ z mental data of [3] in a wide temperature range in the 2 2k λ (22) λ ν , entire investigated range of isotopic enrichment. In our × mFsλ 1 τε e ph()kq ∑------〈〉〈〉τε - ()∫ dzq ------λ - . calculation of the drag thermopower, these parameters kBT () ν ()q λ 0 ph N are not varied. The fitting parameter of the theory is the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1086 KULEEV et al. ≈ and me m0. One can see in the figures that the exact 0 inclusion of the inelasticity of electronÐphonon scatter- ing brings about a marked suppression of the contribu- –5 tion of phonon drag in the thermopower. The maximal |α | 1 values of thermopower max decrease by a factor of 2 ≈ –10 1.6Ð1.7 for the value of me 0.22m0. However, the

, mV/ä importance of inelasticity increases with the effective ph mass of electrons: at m ≈ m , the value of |α | α –15 e 0 max decreases by a factor of 3 for natGe and by a factor of 2.2 1a 70 –20 for Ge (99.99%). This result came as a surprise to us. 2a The thing is that analysis of the time of relaxation of electrons from phonons [7Ð9] revealed that, for temper- –25 (a) ӷ ≈ ≈ atures T Tsλ (TsL 0.8 K for Ge at me 0.22m0), the 0 importance of inelasticity is minor and, at temperatures 1 above 5 K, it may be ignored. Therefore, in the previ- –10 ously published papers dealing with the drag ther- mopower in semiconductors [6Ð9, 24Ð30], the inelas- –20 2 ticity of electronÐphonon scattering was taken into account in a linear approximation with respect to the –30 inelasticity parameter បω λ/k T. 1a q B

, mV/ä Note that the inclusion of scattering from charged

ph –40

α and neutral donor impurities at concentrations of the order of 1012Ð1013 cmÐ3 has little effect on the magni- –50 2a tude of the thermopower (this scattering introduces a (b) contribution of less than 3%), while the magnitude of –60 the electron mobility varies more significantly in the 0 1020304050low-temperature region. T, K Figure 4 gives the contributions of the phonon drift and diffusion into the drag thermopower for Fig. 3. The temperature dependence of the drag ther- nat 70 mopower for the following values of parameters: (a) m ≈ Ge and Ge (99.99%). One can see in the figure that, e in the case of natGe, the predominant contribution to the 0.22m0, E0L = 16 eV; (b) me = m0, E0L = 4 eV. Curves 1 and 1a are for germanium of natural isotopic composition thermopower is made by the phonon diffusion motion. × 12 Ð3 The contribution by the drift motion is small and (Nd = 4 10 cm ), and curves 2 and 2a are for germa- nium with 99.99% 70Ge (N = 2 × 1013 cmÐ3). Curves 1 amounts to 21% of the diffusion contribution at the d |α| 70 and 2 allow for the inelasticity of electronÐphonon scatter- maximum of . In contrast, in the case of Ge ing, and curves 1a and 2a are plotted in a linear approxima- (99.99%), the drift contribution to the drag ther- tion with respect to the parameter of inelasticity of electronÐ mopower predominates. It is six times the diffusion phonon scattering. contribution. In view of the foregoing, indeed, the isotope effect in the thermopower for Ge is associated with the drift motion of thermal phonons. As was already deformation potential constant. Given a fixed effective observed in analyzing the thermal conductivity of Ge and mass of electrons, this constant is selected on the basis Si crystals of different isotopic compositions [22, 23], a of the condition of agreement between the calculated decrease in the degree of isotopic disorder brings about value of absolute thermopower at the point of maxi- an abrupt increase in the contribution made by the drift mum and the experimentally obtained values for ger- motion of longitudinal phonons to the thermal conduc- manium of natural isotopic composition and is then 70 tivity. The same effect shows up in the drag ther- used to calculate the thermopower of Ge (99.99%). mopower. Because the effective mass of one of four ellipsoids in ≈ Figure 5 gives the theoretically and experimentally the crystallographic direction [111] is me 1.68m0, its average magnitude was varied from the value of the obtained temperature dependence of the drag ther- ≈ mopower for natGe and 70Ge (99.99%). One can see in effective mass of the density of states me 0.22m0 to the value of m ≈ m . the figure that the theory provides a qualitative explana- e 0 tion of the isotope effect in the thermopower: the max- |α | nat We will first examine the part played by the inelas- imal values of max in the case of transition from Ge ticity during the transfer of momentum from nonequi- to 70Ge (99.99%) increase by a factor of 1.3 for the ≈ ≈ librium phonons to equilibrium electrons. Figures 3a value of me 0.22m0 and by a factor of 2.25 for me m0, and 3b give the results of calculations of the drag ther- which actually agrees with the experimentally observed nat 70 ≈ mopower for Ge and Ge (99.99%) at me 0.22m0 increase in the direction [111]. This may point to the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 NORMAL PROCESSES OF PHONONÐPHONON SCATTERING 1087

0 0 1b 1a 1 –5 2a –5

–10 1 –10 , mV/ä –15 , mV/ä –15 3 ph ph α α –20 –20 2b –25 2 –25 2

–30 –30 10 100 0 20406080100 T, K T, K Fig. 5. The temperature dependence of the drag ther- Fig. 4. The temperature dependence of (1, 2) the drag ther- mopower (E0L = 4 eV). Curve 1 is for germanium of natural mopower, as well as of the contributions by (1a, 2a) the dif- × 12 Ð3 fusion and (1b, 2b) drift of phonons for germanium of dif- isotopic composition (me = m0, Nd = 4 10 cm ), curve 2 70 × ferent isotopic compositions (me = m0, E0L = 4 eV) allow- is for germanium with 99.99% Ge (me = m0, Nd = 2 ing for the inelasticity of electronÐphonon scattering. 1013 cmÐ3) in the direction [111], and curve 3 is for germa- Curves 1, 1a, and 1b are for germanium of natural isotopic nium with 99.99% 70Ge (m ≈ 0.9m , N = 2 × 1012 cmÐ3) × 12 Ð3 e 0 d composition (Nd = 4 10 cm ), and curves 2, 2a, and 2b in the direction [100]; the symbols indicate the experimen- 70 × 13 Ð3 are for germanium with 99.99% Ge (Nd = 2 10 cm ). tal data. predominant part played by one of four ellipsoids with parameter into the theory hardly added anything to the the maximal effective mass along the direction [111]. physical content of this paper. However, the position of maxima for natGe (see Figs. 3, 4, and 5) turns out to be shifted to the low-temperature ≈ ≈ 6. CONCLUSIONS region, Tmax 6 K, while experiment gives Tmax 17 K. 70 ≈ For Ge (99.99%), calculation gives Tmax 10 K, while In this paper, we have interpreted the experimentally ≈ experiment produces Tmax 15 K. In calculating the found strong dependence of the thermopower of germa- thermopower in the direction [100] (see Fig. 5, curve nium crystals on the isotopic composition. We have 3), the deformation potential constant was not varied, developed a theory of phonon drag of electrons in semi- × and the velocities of sound were taken to be sL = 4.92 conductors with nondegenerate statistics of current car- 5 × 5 riers, which takes into account the effect of the phonon 10 cm/s and st = 3.55 10 cm/s, in accordance with [31]. In this case, the isotope effect in the thermopower drift motion associated with the normal processes of with the same constant of deformation potential turned phonon scattering. A qualitative explanation has been out to be 35% lower, which may be indicative of some given of the isotope effect in the drag thermopower. It anisotropy of the drag thermopower. has been demonstrated that the rigorous inclusion of inelastic electron scattering brings about a significant Note that the contribution by longitudinal phonons (by factor of more than two) reduction of the absolute alone was taken into account in the calculation of the values of the drag thermopower. In our opinion, the iso- drag thermopower. Analysis revealed that, within the tropic band approximation for conduction electrons, as assumptions made, the isotope effect for transverse well as the assumption of the equality of the drift veloc- phonons was low and, upon transition from natGe to ities of long-wavelength and thermal phonons, failed to highly enriched germanium, this contribution increased provide for quantitative agreement with the experimen- by approximately 10%. This is associated with the pre- tal data on the drag thermopower, in contrast to calcu- dominant part played by the diffusion motion of trans- lations of thermal conductivity [22]. verse phonons (for more detail, see [22]). Therefore, in this analysis, we ignored the contribution of transverse The inclusion of both of the above-identified factors phonons, although the position of the maximum of requires significant mathematical effort, namely, a sep- arate study of the relaxation of thermal and long-wave- αt ph is found at approximately 20Ð22 K. The inclusion length phonons and analysis of the Simons mechanism of this contribution could have markedly improved the of normal processes of scattering, which leads to the agreement between the calculated curves and the exper- redistribution of momentum between the thermal and imental data at temperatures above the maximum. long-wavelength phonons of different vibrational However, the introduction of an additional fitting branches.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1088 KULEEV et al.

ACKNOWLEDGMENTS 14. V. S. Oskotskiœ, A. M. Pogarskiœ, I. N. Timchenko, and S. S. Shalyt, Fiz. Tverd. Tela (Leningrad) 10, 3247 We are grateful to A.P. Tankeev for the discussion of (1968) [Sov. Phys. Solid State 10, 2569 (1968)]. our results and critical comments. 15. A. N. Taldenkov, A. V. Inyushkin, V. I. Ozhogin, et al., in This study was supported by the Russian Founda- Proceedings of IV Conference on Physicochemical Pro- tion for Basic Research (project nos. 00-02-16299 and cesses under Atom and Molecule Selection, Zvenigorod, 01-02-17469), by the CRDF (grant no. RP2-2274), and 1999 (TRINITI, Troitsk, 1999), p. 243. by the Dinastiya and MTsFFM Foundations. 16. J. Callaway, Phys. Rev. 113, 1046 (1959). 17. R. Berman, Thermal Conduction in Solids (Clarendon Press, Oxford, 1976; Mir, Moscow, 1979). REFERENCES 18. B. M. Mogilevskiœ and A. F. Chudnovskiœ, Thermal Con- 1. K. Itoh, W. L. Hansen, E. E. Haller, et al., J. Mater. Res. ductivity of Semiconductors (Nauka, Moscow, 1972), 8, 1341 (1993). p. 536. 2. V. I. Ozhogin, A. V. Inyushkin, A. N. Taldenkov, et al., 19. S. Simons, Proc. Phys. Soc. London 82, 401 (1963); Pis’ma Zh. Éksp. Teor. Fiz. 63, 463 (1996) [JETP Lett. Proc. Phys. Soc. London 83, 799 (1963). 63, 490 (1996)]. 20. C. Herring, Phys. Rev. 95, 954 (1954). 3. M. Asen-Palmer, K. Bartkowski, E. Gmelin, et al., Phys. 21. T. H. Geballe and G. W. Hull, Phys. Rev. 94, 1134 Rev. B 56, 9431 (1997). (1954). 4. A. P. Zhernov and D. A. Zhernov, Zh. Éksp. Teor. Fiz. 22. I. G. Kuleev and I. I. Kuleev, Zh. Éksp. Teor. Fiz. 120, 114, 1757 (1998) [JETP 87, 952 (1998)]. 1952 (2001) [JETP 93, 568 (2001)]. 23. I. G. Kuleev and I. I. Kuleev, Zh. Éksp. Teor. Fiz. 122, 5. A. P. Zhernov, Fiz. Tverd. Tela (St. Petersburg) 41, 1185 558 (2002) [JETP 95, 480 (2002)]. (1999) [Phys. Solid State 41, 1079 (1999)]. 24. L. É. Gurevich and I. Ya. Korenblit, Fiz. Tverd. Tela 6. C. Herring, Phys. Rev. 96, 1163 (1954). (Leningrad) 6, 856 (1964) [Sov. Phys. Solid State 6, 661 7. V. M. Askerov, Electronic Transport Phenomena in (1964)]. Semiconductors (Nauka, Moscow, 1985), p. 318. 25. I. G. Lang and S. T. Pavlov, Zh. Éksp. Teor. Fiz. 63, 1495 8. I. M. Tsidil’kovskiœ, Thermomagnetic Effects in Semi- (1972) [Sov. Phys. JETP 36, 793 (1973)]. conductors (Nauka, Moscow, 1960; Academic, New 26. I. G. Kuleev, Fiz. Met. Metalloved. 87, 5 (1999). York, 1962). 27. I. G. Kuleev, Fiz. Tverd. Tela (St. Petersburg) 42, 979 9. P. S. Zyryanov and M. I. Klinger, Quantum Theory of (2000) [Phys. Solid State 42, 1009 (2000)]. Electron Transport Phenomena in Crystalline Semicon- 28. E. Kaden and H.-L. Günter, Phys. Status Solidi B 126, ductors (Nauka, Moscow, 1976), p. 480. 733 (1984). 10. I. G. Kuleev, Fiz. Met. Metalloved. 90 (1), 14 (2000). 29. A. G. Samoilovich, I. S. Buda, and I. V. Dakhovski, 11. I. G. Kuleev, Fiz. Tverd. Tela (St. Petersburg) 42, 649 Phys. Status Solidi 23, 229 (1967). (2000) [Phys. Solid State 42, 423 (2000)]; Fiz. Tverd. 30. A. G. Samoœlovich and I. S. Buda, Fiz. Tekh. Polupro- Tela (St. Petersburg) 44, 215 (2002) [Phys. Solid State vodn. (Leningrad) 3, 400 (1969) [Sov. Phys. Semicond. 44, 223 (2002)]. 3, 340 (1969)]. 12. V. A. Kozlov and É. L. Nagaev, Pis’ma Zh. Éksp. Teor. 31. R. Truel, C. Elbaum, and B. B. Chick, Ultrasonic Meth- Fiz. 13, 639 (1971) [JETP Lett. 13, 455 (1971)]. ods in Solid State Physics (Academic, New York, 1969; Mir, Moscow, 1972). 13. A. A. Bel’chik and V. A. Kozlov, Fiz. Tekh. Poluprovodn. (Leningrad) 20, 53 (1986) [Sov. Phys. Semicond. 20, 31 (1986)]. Translated by H. Bronstein

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1089Ð1103. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1239Ð1255. Original Russian Text Copyright © 2003 by Zabolotskii. SOLIDS Structure

The Evolution of Longitudinal and Transverse Acoustic Waves in a Medium with Paramagnetic Impurities A. A. Zabolotskii* Institute of Automatics and Electrometry, Siberian Division, Russian Academy of Sciences, Universitetskii pr. 1, Novosibirsk, 630090 Russia *e-mail: [email protected] Received December 10, 2002

Abstract—We study the bipartial interaction of longitudinal and transverse acoustic pulses with a system of paramagnetic impurities with an effective spin S = 1/2 in a crystalline layer or on a surface in the presence of an arbitrarily directed external constant magnetic field. We derive a new system of evolution equations that describes this interaction and show that, in the absence of losses, for equal phase velocities of these acoustic components, and under the condition of their unidirectional propagation, the original system reduces to a new integrable system of equations. The derived integrable system describes the pulse dynamics outside the scope of the slow-envelope approximation. For one of the reductions of the general model that corresponds to the new integrable model, we give the corresponding equations of the inverse scattering transform method and find soli- ton solutions. We investigate the dynamics and formation conditions of the phonon avalanche that arises when the initial completely or incompletely inverted state of the spin system decays. We discuss the application of our results to describing the interaction dynamics of spins and acoustic pulses in various systems with an exter- nal magnetic field. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION verse, i.e., generally, a three-component one [7, 8]. The At present, the nonlinear coherent optical phenom- approximation of a quasi-monochromatic wave was ena associated with soliton and other self-similar solu- used to find soliton solutions of the equations that tions [1, 2] have been analytically studied in detail in described the surface acoustic waves in the papers terms of integrable models [3]. When elastic waves known to us, except [7] and our paper [15]. In general, the characteristic length of an acoustic pulse is no less propagate in paramagnetic crystals, similar soliton-like Ð4 pulses can be produced by effects related to anhar- than 10 cm, i.e., much larger than the lattice constant. monic oscillations and dispersion [4] as well as under At the same time, for a picosecond acoustic pulse of τ Ð7 Ð6 conditions of nonlinear coherent interaction between duration a ~ 10 pc, its length is 10 Ð10 cm, and pros- acoustic waves and paramagnetic impurities in the pects for the physics and engineering of such ultrashort medium and in the case of acoustic self-induced trans- acoustic pulses look appealing. The spatial extent of parency (ASIT) [5Ð8]. The nonlinear coherent phenom- such a pulse in a solid is only an order of magnitude ena associated with the acoustic paramagnetic reso- larger than the characteristic size of the lattice cell, nance and the propagation of acoustic pulses have also which is of fundamental interest in acoustic spectros- been studied for a long time. In several studies (see, copy and diagnostics. Therefore, ideally, media with e.g., [5Ð10]), the authors constructed models for the lengths of only hundreds or tens of characteristic lattice evolution of acoustic pulses in bulk crystals with impu- sizes can be used to produce such pulses. In spectral rity paramagnetic particles and found the simplest soli- language, the passage to picosecond acoustic pulses ton solutions. The coherent effects that arise during the implies mastering the hitherto inaccessible frequency evolution of Rayleigh-type surface acoustic waves were range above 100 GHz. The concentration of acoustic investigated in [6, 10] and other works. Such waves can energy for such short intervals also allows strong propagate along the interface. Similar phenomena can acoustic pressures exceeding 109 bars to be produced. be observed during the evolution of plane waves in bulk Recently [4], the generation of picosecond acoustic media [11Ð14]. solitons has been observed experimentally. The solitons As a rule, the authors of the above papers used an were formed at a distance much smaller than 1 mm due analogy between optical and acoustic effects. At the to the balance between the dispersion attributable to the same time, the evolution of an acoustic pulse in a crys- positions of atoms in the crystal lattice and the nonlin- tal with paramagnetic impurities has a number of qual- earity that arose from the anharmonicity of interatomic itative distinctions from the dynamics of light waves in forces. An acoustic resonance effect, an analogue of a medium, which stem, for example, from the fact that optical self-induced transparency, was observed in low- an acoustic wave in a crystal can be longitudinalÐtrans- temperature crystalline samples with paramagnetic

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1090 ZABOLOTSKII impurities. This effect was observed on Fe2+ impurities surface waves—bipartial Rayleigh surface waves and in MgO [11] and LiNbO3 [12] crystal matrices when shear surface waves (see, e.g., [17])—can correspond the longitudinal acoustic pulse propagated at an angle to this case. Other physical prerequisites for the realiza- to the external field. tion of this model in a bulk crystal can be associated In general, the group velocities of the longitudinal with its strong anisotropy, for example, when the spinÐ phonon coupling coefficients that correspond to the y (v||) and transverse (v⊥) components of an acoustic pulse in a solid are different. As a result, the local inter- acoustic wave component are relatively small. Other action between the pulses of these components with a conditions can be a similarity of the group velocities for characteristic length l is limited by the time the longitudinal and transverse waves and their large a deviation from the group velocity of another transverse component (see below and [17]). ∼ la tint ------. v || Ð v ⊥ In this interaction geometry, we show that, to pass from the original evolution equations that describe the The interaction of the pulses is most effective at close coherent dynamics of acoustic pulses to an integrable group velocities, v|| ≈ v⊥. This situation takes place in system of equations, it will suffice to use the approxi- elastic-isotropic crystals, in which the velocities of the mation of unidirectional wave propagation and the con- longitudinal and transverse elastic field components do dition of equal phase velocities for the longitudinal and not depend on the direction. These conditions are best transverse acoustic waves. satisfied for ion crystals of alkali metal galogenides Since our new integrable system of evolution equa- with central forces of interatomic interaction, for exam- tions is complex for analysis, we developed the appara- ple, in NaBr [16]. tus of the inverse scattering transform method (ISTM) The theory of ASIT in a medium of paramagnetic [3] for its special case. The ISTM application to this impurities with spin 1/2 for longitudinalÐtransverse model allows the various evolution regimes of picosec- acoustic waves was developed by Voronkov and ond acoustic pulses to be studied outside the scope of Sazonov [7, 8], who derived the equations describing the slow-envelope approximation. Based on the ISTM, the dynamics of acoustic pulses. These authors used we found a soliton solution of the model that explicitly several constraints on the interaction geometry and described, in particular, the dependence of the soliton dynamics to solve the general evolution equations. As a shape on the relative contribution of the longitudinal result, they obtained complex (for analysis), exactly and transverse components. nonintegrable systems of equations and then reduced Apart from the soliton solutions associated with the problem either to the standard (sine-Gordon-type) ASIT, other pulse evolution regimes can also be ana- equations or to the dispersion equations that corre- lyzed in terms of the ISTM. For example, an unstable sponded to the evolution of low-amplitude pulses. The state of the system arises in the case of initially com- latter equations cannot be used to describe the dynam- plete or incomplete spin inversion, with a weak seed ics of intense picosecond acoustic pulses. acoustic wave being sufficient to remove it from this At the same time, the rich structure of the evolution state (in the case of complete inversion). We discuss the equations that describe the dynamics of acoustic waves use of the ISTM to describe the emerging phonon ava- in paramagnetic media makes it possible to reduce lanche that was observed experimentally [13, 14] and them, for quite realistic approximations, to integrable for other physical situations related to the dynamics of models without imposing similar stringent constraints. spin–phonon systems in a magnetic field. These equations not only can correspond to a more gen- This paper has the following structure. The basic eral physical model but also allow new physical phe- system of evolution equations that describes the nomena in a similar or different interaction geometry to dynamics of a longitudinalÐtransverse wave is derived be described analytically. This paper is a continuation in the next section. The most general integrable reduc- of our paper [15], in which we derived such integrable tion of the original system of equations for this system equations and used them to describe ASIT. Here, how- is found in Section 3. The ISTM apparatus for the ever, we consider a completely different interaction reduction of the general model is developed in the next geometry of acoustic waves described by a different section, and a one-soliton solution of this model is (than the system found in [15]) new original physical found. In Section 5, we discuss our results and their system of equations and, accordingly, by qualitatively applications. In the Appendix, we give the Lax repre- new integrable reductions of this system. sentation for the general integrable model and integra- In contrast to [15], in which we studied the interac- ble reductions of the general model. tion dynamics of three acoustic pulse components, here, we consider the “bipartial” dynamics of acoustic waves; i.e., we take into account the interaction of one 2. THE DERIVATION transverse and one longitudinal acoustic wave compo- OF BASIC EQUATIONS nent in the xz plane. The contribution of the y acoustic Below, we derive the equations that describe the field component was disregarded. The simplest types of dynamics of a longitudinalÐtransverse wave in a crystal

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

THE EVOLUTION OF LONGITUDINAL AND TRANSVERSE ACOUSTIC WAVES 1091 with paramagnetic impurities following [7, 15]. The Hamiltonian (1) takes the form external constant and uniform magnetic field B is ˆ µ ˆ ()a assumed to be directed along the z axis. The Zeeman Hs = ∑∑ Bg jjB jS j . (5) ()a interaction of the magnetic moment µˆ at point a α j gives the following contribution to the total Hamilto- Ᏹ The spinÐphonon interaction in the linear (in pq) nian: approximation is described by the Hamiltonian ()a ˆ µ () Ha = Ð ˆ B. ˆ µ Ᏹ ˆ a Hint = ∑ ∑ B B jF jk, pq pqSk . (6) () The µˆ a components can be expressed in terms of the α jkpq,,, (a) ∂ ∂Ᏹ S (ra) spin components, where ra is the radius vector Here, Fjk, pq = ( gjk/ pq) are the spinÐphonon coupling of spin a, as constants [12, 18]. The dynamics of the acoustic field in a crystal with- µ()a µ ˆ ()a out anharmonicity is described by the Hamiltonian ˆ j = Ð∑ Bg jkSk . k ∂ ∂ 1 2 1 U U µ H = ------p + --- λ ------j ------l dr, (7) Here, B is the Bohr magneton and gjk are the Lande a ∫ ∑ j ∑ jklm ∂ ∂ 2n0 2 xk xm tensor components. Thus, j jklm,,,

N where n0 is the mean crystal density, pj (j = x, y, z) are z z ()a Hˆ ==∑ Hˆ a µ ∑∑B g Sˆ k , (1) the momentum density components that arise during B j jk λ a = 1 a jk, dynamic displacements, and jklm is the elastic modulus tensor of the crystal [19]. The integral in (7) is taken where N is the total number of spins. The diagonal val- over the crystal volume. We assume that the number of ues of the Lande tensor can differ. phonons is large and that the classical description of the Since the effective spin is 1/2, it can be decomposed acoustic-field dynamics is valid. At the same time, a into Pauli matrices: two-level spin system requires a quantum-mechanical description. For S = 1/2 and in the presence of a suffi-   a 1 01 a 1 0 Ði ciently strong magnetic field, the terms quadratic in Sˆ x ==---, Sˆ y ---, spin operators can be disregarded (for more detail, 2 2 10 i 0 see [20]). Here, an analogy with the interaction of a (2) classical electromagnetic field with an optical quantum  a 1 10 two-level medium holds [21]. Sˆ z = ---. 201Ð As in the case of an optical medium, we can pass from the description of the spin dynamics to the evolu- We assume that the x, y, and z coordinates along the tion equations for the density matrix elements ρˆ of the principal Lande tensor axes coincide with the crystal two-level medium: symmetry axes. The Lande tensor is then diagonal in a ∂ρˆ nondeformed unperturbed medium: iប------= []Hˆ , ρˆ , (8) ∂t ()0 δ g jk ==g jk g jj jk, ∂U ∂H ∂p ∂H δ ------==------, ------Ð------. (9) where jk is the delta function. The deformation of the ∂t ∂p ∂t ∂U crystal by an acoustic wave is described by linear cor- rections to the Lande tensor: Here,

∂ HH= + 〈〉Hˆ int , ()0 g a ------jk- Ᏹ … g jk = g jk ++∑∂Ᏹ pq , (3) , pq 0 where the interaction between the spin and the field of pq an elastic pulse is described by the classical Hamilto- where Ᏹ is the elastic strain tensor of the crystal at the nian equations for a continuous medium: spin location. The derivatives are taken at the point of zero deformation. The strain tensor components can be () 〈〉ˆ µ Ᏹ 〈〉ˆ a expressed in terms of the components of displacement Hint = ∑ ∑ B B jF jk, pq∫ pq()r Sk ()r dr. (10) α ,,, vector U = (Ux, Uy, Uz) as jkpq Here, n is the paramagnetic impurity density. The sum- 1 ∂U ∂U Ᏹ p p mation over the spin-1/2 ions uniformly distributed in pq = ---------∂ - + ------∂ - . (4) 2 xq x p the crystal is substituted with integration over the entire

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1092 ZABOLOTSKII space. The angular brackets denote an averaging over Below, when deriving the general integrable system of the quantum states: equations, we impose no additional constraints on the coefficients fk, k = 1Ð4. Given the above conditions, the ()a ()a 〈〉Sˆ ()r = TrSˆ ρˆ . k k expressions for Hˆ s and Hˆ int take the form We consider the bipartial dynamics of the acoustic fields; i.e., we assume that only the longitudinal and ˆ បω ˆ ()α Ᏹ Ᏹ Hs = ∫∑n BSz dr, (12) transverse field components ( xz and zz) contribute to the interaction. Such a situation is possible if the coef- α ficients F and F are much smaller than the coef- yx, zz yz, yz បω ficients F , F , and F . A different mechanism n B xz, zz xz, xz zz, zz Hˆ int = ∑------∫ g that leads to quasi-two-dimensional dynamics can be α (13) associated with the difference between the elasticity λ λ ≈ λ × {}()Ᏹ Ᏹ ˆ ()α ()Ᏹ Ᏹ ˆ ()α coefficient yz and the coefficients xz zz. The phase f 1 zz + f 2 xz Sz + f 3 xz + f 4 zz Sx dr. velocities of the acoustic wave components are propor- ω µ ប λ Here, B = g BB/ is the Zeeman splitting frequency of tional to yz . Since the interaction efficiency in a non- linear pulse regime is determined by the interaction the Kramers doublet, g = gˆ , and n is the ion density length, for waves with almost equal phase velocities in the medium. The coefficients fk, k = 1Ð4, describe the ≈ ≈ coupling of the longitudinal and transverse acoustic (v1 v2 v), this interaction is much more effective than the interaction with a wave whose phase velocity fields with the spin system; the first two of these coeffi- significantly differs from v. The “escape” of the acous- cients correspond to the frequency shift that depends on tic wave component from the interaction region is also the acoustic-field amplitude. The physical nature of this used to motivate the passage to the “bipartial” descrip- shift can be different. By analogy with the case of an tion of the dynamics of acoustic waves (see, e.g., [17]), optical medium with a constant dipole moment i.e., with the contribution of the y acoustic wave com- described above (see [22]), the nonzero f1 may result ponent disregarded. Since, as we show below, the group from the presence of a constant magnetic moment in the velocities of the generated acoustic pulses in the mod- two-level medium. The mechanism described in [10] els under study are close to the phase velocities, the gives a similar contribution to the frequency shift. restriction to the bipartial interaction is justified. Taking Under the above symmetry conditions, the Hamilto- into account these conditions (see also the Introduc- nian H takes the form tion), we may formally set the transverse field compo- a nent equal to zero (Ᏹ ≡ 0). yz 2 2 ∂U 2 ∂U 2 ˆ 1 px + pz λ z λ x We assume that the acoustic waves propagate along Ha = --- ------++11 ------44 ------dr. (14) 2∫ n ∂z ∂z the z axis. The direction of the magnetic field B in space 0 can be arbitrary. Since only the B projection onto the xz plane contributes to the interaction, we assume, without Here, the Vogt notation [18] is used for the subscripts: loss of generality, that xx 1, zz 3, xz 5. 2 2 BB= x +,Bz Since the anharmonic effects are disregarded, there is no direct interaction between the longitudinal and where Bj are the components of vector B. Under these transverse fields (it arises indirectly, only through their conditions, the following arbitrary real effective spinÐ interaction with the spin system). This and the condition phonon coupling coefficients remain: of equal phase velocities (see below) allow us to pass to a new effective transverse (or quasi-transverse [17]) 1 field—a linear combination of the stress tensor compo- f = --- ∑ B F , , 1 B j jz xz nents, which interacts with the x spin component (see jxz= , Hamiltonian (13)): 1 f f = --- ∑ B F , , ᐃᏱ 4Ᏹ 2 B j jz zz = xz + ----- zz. (15) , f 3 jxz= (11) 1 With this substitution, the interaction of the acoustic f = --- ∑ B F , , 3 B j jx xz field ᐃ with a two-level spin medium (see formula (8)) jxz= , is described by the Hamiltonian 1 nបω f 4 = --- ∑ B jF jx, zz. ˆ ˆ B[]σ ()Ᏹ νᐃ B Hs + Hint = ∫------3 gf+ zz + f 3 ˆ dr, (16) jxz= , g

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 THE EVOLUTION OF LONGITUDINAL AND TRANSVERSE ACOUSTIC WAVES 1093 where Ux are equal. As a result, we obtain the system of evo- lution equations for the acoustic fields with averaging f f over the quantum states ff= Ð ------1 -4. 2 f 3 2 2 បω 2 ∂ ᐃ 2∂ ᐃ n f ∂ Trνˆ ρˆ ------Ð v ------= ------B 3 ------, (20) µ ν 2 1 2 2 The effective magnetic operator moment B ˆ , where ∂t ∂z gn0 ∂z

 ∂2Ᏹ ∂2Ᏹ 2nបω f ∂2S ν a 1 zz v 2------zz------B ------3 ˆ = (17) ------2 - Ð 2 2 = 2 , (21) 1 Ða ∂t ∂z gn0 ∂z where and a = f1/f3 is the constant magnetic moment, corre- sponds to the last term in (16). λ λ 11 44 Using (8)Ð(10) and (16), we derive the equations of v 1 ===------v 2 ------. motion for the components of spin S describing the n0 n0 transitions in an effective two-level medium arising during Zeeman splitting: On the right-hand side of (20), νρ ∂ f Tr ˆ ˆ = 2aSz + 2Sx. S = -----3ᐃS , ∂t z ប y The derived new system of evolution equations (18)Ð ∂ ω f Ᏹ f 1ᐃ f 3ᐃ (21) describes the propagation of acoustic pulses in a ∂ Sy = B ++--ប- zz -----ប Sx Ð -----ប Sy, (18) two-level medium. In the Bloch equations (18), the lon- t Ᏹ gitudinal acoustic field component zz leads only to ∂ f f 1 nonlinear phase modulation. However, as we show S = Ð ω ++---Ᏹ ----- ᐃ S , ∂t x B ប zz ប y below, the longitudinal component is related to the transverse component and its allowance leads to quali- where tatively new acoustic-pulse dynamics.

()a Sγ ==TrSˆ γ ρˆ , γ xyz,,, 3. THE DERIVATION OF A GENERAL INTEGRABLE MODEL i.e., In this section, we derive the most general integrable reduction of the basic system of equations (18)Ð(21), 1 1 S ==---()ρ Ð ρ , S ---()ρ + ρ , which arises for a minimum number of physical con- z 2 11 22 x 2 12 21 straints. To be more precise, we find the reduction of the physical system of nonlinear equations (18)Ð(21) i derived above in the Lax representation and show that S = ---()ρ Ð ρ . y 2 12 21 the ISTM apparatus is applicable. Above, we have already assumed that the phase It is easy to show that velocities of the effective longitudinal and transverse acoustic waves are equal: 2 2 2 ()ρ ρ 2 ≡ Sz ++Sx Sy = 11 + 22 1 (19) v 1 ==v 2 v . (here, the spin length was normalized to unity). In deriving system (18), we assume that the classical We seek additional integrability conditions by assum- description of the acoustic field (the number of phonons ing that the reduction of the original system (18)Ð(21) is large) is valid. must describe the dynamics of acoustic pulses with a πωÐ1 To derive the evolution equations that describe the duration of the order of or shorter than B . Under dynamics of the classical fields (linear combinations of this condition, the slow-envelope approximation is ᐃ Ᏹ the stress tensor components) and zz, we first inapplicable. As was noted above, in real media, pico- obtain the wave equations for the displacements U and second acoustic pulses can correspond to this case. We then differentiate them with respect to z and use the also disregard the relaxation effects, which is valid for expressions for these fields in terms of the differentials the chosen range of pulse durations. of the displacements with respect to z that follow from The equations that describe the dynamics of such definition (4). We also assume that the phase velocities pulses are complex for analysis, but system (20) and that correspond to the displacement components Uz and (21) can be simplified at a sufficiently low density of

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1094 ZABOLOTSKII paramagnetic impurities. Such a physical situation solitons and other coherent structures can be formed in takes place in all of the known cases. This approxima- crystals less than one millimeter in thickness [4]. tion is similar to the approximation used in [23] to Therefore, we can easily create conditions under which derive the reduced MaxwellÐBloch equations for a two- the interval where level optical medium and is called the condition of uni- directional wave propagation. The latter condition cor- ω ប U ()t ≠ ------B - responds to the following approximate formal equality: 0 f ∂ ≈ Ð1∂ ⑀ z Ð v t + O(), is much larger than the formation time scales of solitons and other nonlinear structures. ⑀ where is a small parameter. Physically, this means that Using (26), we obtain a new integrable system of the acoustic pulses propagate in the medium at a veloc- evolution equations from (18), (24), and (25): ity close to the phase velocity v . The fulfillment condi- tion for the approximation is that the normalized impu- ∂χEbUS= Ð , rity density is of the same order of smallness as the y ∂ () derivative of the acoustic-field amplitudes τSy = aE+ bU Sx Ð ESz, (27) ∂ () ∂ ∂ Ð1∂ τSx = Ð aE+ bU Sy, χ˜ = z + v t. ∂ τSz = ESy. In this approximation, the derivative with respect to z on the right-hand sides of Eqs. (20) and (21) can be sub- Here, all functions are real and Ð1∂ ⑀2 stituted with v t with an accuracy O( ). Thus, when ᐃ χτ, the condition of unidirectional acoustic pulse propaga- χτ, () 2 χτ, 2 χτ, tion is satisfied, system (20) and (21) reduces to E()==------, U ()+1,E () U0()t បω ∂ᐃ n B f 3 ∂Tr()νˆ ρˆ ------= ------, (22) ω 2 t ∂χ˜ 2 ∂t n B f 3 f 3 f 2v n0g χχ==˜ ------, τ ----- U ()t' dt', b =----- . 2 ប ∫ 0 gn v f 3 0 0 ∂Ᏹ nបω f ∂S ------zz- = ------B ------3. (23) ∂χ˜ v 2n g ∂t The representations of this system as a condition for the 0 simultaneity of two linear systems with an arbitrary Here, we chose f ≠ 0. The case f 0 is discussed in (spectral) parameter are given in the Appendix. It is also the Appendix. When calculating the derivatives with pointed out in the Appendix that system (27) is complex respect to t, we take into account the Bloch equations (18) enough for the ISTM to be applied. Therefore, here, we and reduce Eqs. (22) and (23) to restrict our analysis to its physically interesting reduc- tion that arises if we set a = 0. This reduction, probably, also corresponds to the new integrable system of equa- ∂ᐃ nបω ff បω ------= Ð------B -3 ------B- + Ᏹ S , (24) tions. ∂χ˜ 2 f zz y v n0g 4. THE ISTM APPARATUS FOR a = 0 ∂Ᏹ nបω ff ------zz- = ------B -3ᐃS . (25) ∂χ˜ 2 y The condition a = 0 implies that the contribution of v n0g the effective transverse acoustic field to the frequency ρ modulation of the density matrix component 12 is dis- It is now easy to find from Eqs. (24) and (25) that the regarded. Such a physical situation arises if the x and z amplitudes of the longitudinal and effective transverse coordinates along the principal Lande tensor axes coin- fields are related by cide with the crystal symmetry axes. In this case, the

2 Lande tensor is diagonal in a nondeformed unperturbed 2 ω ប 2 ᐃ + Ᏹ + ------B - = U ().t (26) medium. The magnetic field B is assumed to be zz f 0 directed along the z axis, i.e., along the axis of propaga- tion of the acoustic waves. ≠ Here, the arbitrary real function U0(t) 0 is determined We solve the problem on the entire axis for a rapidly by the boundary conditions. The dependence of U0 on t decaying field at infinity: leads only to a renormalization of the variable t and the Ᏹ ᐃ ∀ E()τ 0, τ∞± . functions zz and . It follows from (26) that t ≠ U0(t) 0 and the ISTM apparatus developed for an infi- We assume that the spin system at the initial and final nite interval can be used. On the other hand, acoustic times is in a stable ground state corresponding to mini-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 THE EVOLUTION OF LONGITUDINAL AND TRANSVERSE ACOUSTIC WAVES 1095 mum energy, i.e., spond to a zero field, E(χ, τ) = 0. These states are dif- | | τχ, τ∞± ferent for different b . It is easy to show that the follow- Sz()= Ð 1, . ing ground states of the system are stable in the linear A pulse of acoustic field E(τ, 0) with an area large approximation: enough for the formation of solitons is assumed to be ± E ==0, S3 1, S+ =0, injected into the crystal. To describe the corresponding (32) > soliton dynamics, it is convenient to choose the follow- SÐ = 0, b 1, ing Lax representation for the system of equations (27) at a = 0: E ==0, S3 Ð1,1, S+ = (33) <  SÐ = 0, b 1, ÐiλU ()λβ+ E ∂τΦ ==Φ Lˆ 0Φ, (28) Ð()λβÐ E iλU ES3 ===Ð0,S+U, SÐ b 1. (34) For |b| = 1, the ground state (34) is indifferently stable. b ∂χΦ = ------For a potential that vanishes at infinity (τ = ±∞), we 2 λ2 b Ð 4 introduce the Jost function Ψ± in a standard way (the (29) solutions of (28) with asymptotics): ÐiλS ()λβ+ ()bS Ð 2iλS × z x y Φ ± Ψ ()λσ τ τ∞± < ()βλ()λ λ = exp Ði 3 , , b 1, (35) Ð 2i Sy + bSx i Sz  = Aˆ 0Φ. ξ 1 ------± βλ Here, Φ = Ð exp()Ðiλσ τ , ξ 3 (36) 2 2 1 2  U +1,E ==β --- b Ð,1 ------1 2 βλ+ where λ is the spectral parameter. τ∞± , b > 1. The spectral problem (28) for real fields may be The symmetry properties (30) and (31) correspond to considered to be related to the problems for complex the following matrix form of the Jost functions: fields that arise when solving the integrable LandauÐ Lifshitz equations [24] and the equations of Raman ± ±∗ scattering or four-wave mixing [25] in terms of the ± ψ ψ κ Ψ 1 Ð 2 ISTM. The ISTM apparatus for these problems has = . ψ± ψ±∗ been developed in sufficient detail. Soliton and periodic 2 1 solutions were found for these related models. Previ- ously [26], we found an expression that defined the These solutions are related by the scattering matrix Tˆ : quasi-self-similar asymptotics describing the decay of an initial unstable state. Given the specifics attributable ΨÐ = Ψ+Tˆ . (37) to the fact that the field E is real and that the problem is symmetric, these results can be used for model (28) It follows from the symmetry properties (30) and (31) after appropriate modification. that the scattering matrix can be chosen in the form The solutions of the spectral problem (28) have the involution Ꮽ∗ Ꮾκ Tˆ = . (38) Ð1 ∗ Φ = Mˆ Φλ()∗ ∗Mˆ , (30) ÐᏮ Ꮽ where The Jost functions have standard analytical properties (cf., e.g., [27]). The function a(λ) is holomorphic in the upper half-plane λ, where its zeros correspond to the λβ+ κ ------, b > 1, soliton solutions. Mˆ =,0 κ = λβÐ (31)   Let us represent the Jost functions as Ð01 1, b ≤ 1.  Ψ+ τ ()λσ τ ()= exp Ði 3 ∞ In finding soliton solutions, we should determine the λK ()τ, s ()λβ+ K ()τ, s stable ground state of the system against the back- + ∫1 2 (39) ()λβ τ, λ τ, ground of which the solitons propagate. Consider the τ Ð Ð K2*()s K1*()s regime in which the asymptotic limits τ ±∞ corre- × ()λσ spond to the stable ground states that, in turn, corre- exp Ði 3s ds.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1096 ZABOLOTSKII It follows from system (37) that Substituting expression (39) for Ψ+ into the spectral problem (28) and equating the expressions for different λ ∗ powers of yields, in particular, ∗ ψÐ Ꮾ ψ+ = ------1 - Ð -----κψ+, (40) ττ, []τ τ []ττ, 1 Ꮽ Ꮽ 2 K2()1+ U() = E()1Ð iK1(). (46) Using this relation, we can easily find a relationship Ð∗ between the potential F and the kernels K in the ∗ ψ Ꮾ 3 1, 2 ψ+ ------2 ------ψ+ form 2 = Ð Ꮽ + Ꮽ 1 . (41) 21[]Ð iK ()ττ, K*()ττ, E(τ ) = ------1 2 -.(47) We substitute the components of these functions 2 []1 + iK*()ττ, []1 Ð i K () ττ , + K ()ττ, from (39) into (40) and (41) and integrate the resulting 1 1 2 equations (40) and (41) over λ from Ð∞ to ∞ with the λ π λ β Ð1 λ πλ Ð1 weights exp(Ði y)[2 ( Ð )] and exp(Ði y)(2 ) , 4.1. A One-Soliton Solution respectively. As a result, we obtain the Marchenko of the Model for |b| < 1 equations for the right end of the axis (y ≥ τ): The soliton solutions associated with ASIT describe ∞ the propagation of acoustic pulses without any change * τ, ()τ τ, ∂ of their shape against the background of a stable ground K2 ()y = Fβ + y + iK∫ 1()s i yFβ()sy+ ds, (42) state. This state for system (28) at |b| < 1 is (33). The τ condition for the field E being real imposes constraints on the soliton parameters. In particular, the discrete ∞ eigenvalues must either be imaginary or enter into the λ λ τ, τ, ()±β ∂ form of anticonjugate pairs { , Ð *}. K1*()y = Ð∫K2()s + i y F0()sy+ ds. (43) τ Let us find the one-soliton solution of the problem associated with the only eigenvalue λ. We represent the kernel F that corresponds to this value of λ = iη as On the right-hand side of Eq. (43), the plus (+) and 2 χ ()λ minus (Ð) signs in front of β correspond to b > 1 and iC1()exp Ði y 2 Fβ()y = ------ηβ . (48) b < 1, respectively. The kernel Fβ is defined as Ð 0 Here, Ꮾ χ Ðiλy () e λ β β β Fβ()y = ∫ ------d , (44) ==i 0,Im0 0. Ꮽ()χ 2πλ()Ð β Ꮿ The dependence of the real quantity where Ꮿ is the contour that includes the real axis and Ꮾ()χ; λ C1 = ------that passes above all poles in the upper half of the com- ∂ηᏭ()χ; η β ηλ= plex plane, F0 = Fβ ( = 0). χ λ on is derived below. Given the residues at poles k in the upper half- plane, the kernel F can be represented as To solve the Marchenko equations, we introduce 0 new functions:

∞ ∞ Ðiλy Ꮾ e τ τ, Ðiλs , Fβ()y = ------dλ Qn()==∫Kn()s e ds, n 12. ∫ Ꮽ2πλ()Ð β τ Ð∞ (45) Ðiλ y Ꮾ e k Substituting these functions into the Marchenko equa------Ð i∑∂ Ꮽ λλλ β. tions (43) and (44) and integrating them over y yields λ ()= k k Ð k γ 2 ()η , Ði 1 exp 4 z K1()zz = ------, (49) The Marchenko equations for the left end of the axis 2η2[]1 + γ ()χ 2 exp()4ηz (for y y ≤ τ) can be found in a similar way. Using the 1 results obtained below, we can then easily show that the C exp()2ηz corresponding solutions are joined at y = τ (for more K ()zz, = ------1 -, (50) 2 ()ηβ[]γ χ 2 ()η detail, see [27]). Ð 0 1 + 1() exp 4 z

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 THE EVOLUTION OF LONGITUDINAL AND TRANSVERSE ACOUSTIC WAVES 1097 where we denoted E C ()χ 1.0 γ ()χ = ------1 -. 1 2η χ 0.5 Next, it is necessary to find the function C0( ), which can be determined from system (29) for ±∞χ, ≡ ±∞χ, 0 Sz()1,Ð E()= 0. τ ∞ In a more general case, for the time interval [ 0, ], this –0.5 function can be found by using the formula (see [26] and the derivation in [27]) –1.0 ∂ ˆ ˆ ˆ ˆ τ , χ ˆ Ð1 ˆ ˆ ∞χ, ˆ Ð1 ˆ χT = ÐTV 0 As()0 V 0 + V ∞ As()V ∞ T. (51) 0 5 10 15 20 25 τ Here, Vˆ 0 and Vˆ ∞ are the matrices composed of the eigenvectors, the solutions of problem (27) that were Fig. 1. The amplitude of the effective transverse acoustic found at the initial time τ and for τ = ∞, respectively. pulse component E versus τ. The solid line corresponds to 0 b = 0.9, and the dashed line corresponds to b = 0.1 and Let η = 0.5. The soliton position on the τ axis is arbitrary. τ, ττ, ∞ E()0,0 ==0 , then Sz 1.0 ˆ ˆ ()σ λτ τ τ , ∞ V 0 ==V + exp Ði 3 , =0 . 0.5 Hence, for the chosen initial and boundary conditions that correspond to soliton dynamics, we obtain 0 χη γ χ γ Ð2b 1()= 0 exp------, (52) 4η2 + b2 –0.5 γ where 0 is a constant. Using (47), (49), (50), and (52), –1.0 we find a one-soliton solution of model (28) for a = 0 in 0 5 10 15 20 25 the form τ γ ηη() β ()γ 2 2φ φ 4 0 Ð 0 1 + 0 e e Fig. 2. The amplitude of the level population difference ver- E()τχ, = ------, (53) sus τ for a soliton regime. The solid and dashed lines corre- ()ηβ2[]γ 2 2φ 2 ()γ η 2 2φ Ð 0 1 + 0 e + 2 0 e spond to b = 0.9 and 0.1, respectively. where dynamics in our model and the dynamics studied, e.g., bχ in [5]. The soliton solution (53) describes the energy φ ητ transfer between the longitudinal and transverse field = 2 Ð ------2 2 . b + 4η components caused by the interaction with the spin sys- tem. In Fig. 2, the level population difference is plotted We see from solution (53) that the soliton shape and against τ. velocity depend on the coefficient b. The soliton veloc- ity decreases with increasing b, starting from the phase velocity v in the medium. At b ~ 2η, the velocity 4.2. The Decay of an Inverted Initial State reaches a minimum and then again tends to the phase of the Spin System velocity. In Fig. 1, the soliton amplitude E(τ) is plotted A completely inverted spin system is unstable and against τ for different values of b. We see from the fig- tends to a stable ground state after the action of a weak ure that E(τ) has a dip to zero at the center. As b perturbation. This dynamics in nonlinear two-level decreases, a breakup into a soliton and an antisoliton optical media is described by the self-similar asymp- whose separation tends to infinity when b 0 takes totic solution. In [26], we showed that the asymptotic place. behavior of the fields in the case of Raman scattering It also follows from formula (50) that the normal- for general initial and boundary conditions is described ized soliton amplitude does not exceed unity. Therein by a complex first-order differential equation with the lies an important difference between the acoustic-field right-hand side proportional to a known function. This

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1098 ZABOLOTSKII function can be expressed in terms of the solution to the and reduce it to the form transcendental Painleve equation of the third type (P ) III ∞ [28]. The solution corresponds to the nonsoliton, radia- ρ 0 exp[]()νϑ/2 ()Ð 1/ν tion dynamics associated with the continuum of the Fβ()χτ, = ------dν 2π ∫ ν β τ χ corresponding spectral problem. Following the results Ð 2 0 /b Ð∞ (57) of [26], we can show that this dynamics is characteristic χ χ τ of arbitrary Sγ( , 0) slowly changing with and a suffi- ≈ iρ I ()ϑ ++, 2β ------I ()ϑ … ciently small seed acoustic field pulse |E(τ, 0)| Ӷ 1. For 0 0 0 bχ 1 such small field amplitudes and its derivatives, it is easy ϑ to determine the scattering coefficient where I0, 1 are modified Bessel functions and = χτ Ꮾ()χ; λ 4b . We assumed that ρχ(); λ = ------Ꮽ χ λ (); β τ χ Ӷ 4 0 /b 1. at χ = 0 by using the spectral problem (28): It follows from (42), (47), and (57) that the field ampli- Ꮾ()0; λ tude increases at the linear stage: ρ()0; λ ==ρ ------0 Ꮽ()0; λ χτ, ≈≈χττ,, χ, τ ∼ ρ ϑ ∞ (54) E()2K2()2Fβ()2 0e . (58) Ðλ 2iλτ ≈ ------E∗()τ, 0 e dτ. 2 ∫ Another conclusion that follows from the derived Ð∞ expressions for kernel (57) is that the solution for the ϑ ӷ τ ρ acoustic field is concentrated in the range 1 for If E( , 0) = const, then the scattering coefficient 0 does ρ |ρ | ӷ λ small 0 such that Ðln 0 1. In this range, the inte- not depend on . grals in the Marchenko equations (41) and (43) can be We solve the initial-value (for χ = 0) problem with approximately calculated by the saddle-point method. the trivial boundary conditions that correspond to an It follows from the Marchenko equations that the ker- τ τ unstable (at = 0) state (complete inversion) and a nels K1, 2 exponentially increase with at the initial small (seed) acoustic deformation of the crystal: stage; i.e., the solution for |E| can reach values of the τ, Ӷ , χ order of unity for arbitrarily small seed field amplitudes E(0 )== const 1, Sz()1.0 (55) E(0, τ). The seed field E(τ, 0) causes this state to decay and the Complete spin inversion can be achieved by using system tends to a stable state, sufficiently strong laser radiation [13]. Incomplete ini- , χ tial inversion of the medium, i.e., E ==0, Sz()0 Ð 1, <<, χ which is reached for χ ∞. Ð1 Sz()1,0 ρ χ λ The scattering coefficient 0( ) depends on . Let corresponds to a more general situation. In this case, a us calculate the kernel F (45) by taking into account mixed initial-valueÐboundary-value problem arises on the dependence of the scattering data on χ and condi- the semiaxis (τ ∈ [0, ∞)), whose solution is much more χ ± tions (55), i.e., with the discrete spectrum disregarded. complex than the cases where Sz(0, ) = 1. For some For the initial and boundary conditions (55), it will suf- of the nonlinear systems of equations, this problem was fice to take into account the contribution of only the solved in the case where E(τ, 0) = 0 (the Dirichlet prob- continuous spectrum of the problem. The dependence lem [26, 29]; see also the method for solving a more ρ(χ) can be determined by using expression (51): general problem in [30, 31]). χλ ρχ ρ 2ib Below, we use the results of [26] to find the asymp- ()= 0 exp------. (56) b2 Ð 4λ2 totic solution of the Dirichlet problem that is applicable to describing the dynamics of the phonon and spin ava- Let us now find the linear solution of the problem that lanches that arise under the following initial and bound- corresponds to the initial stage of the decay of an ary conditions: inverted state and the formation of a phonon avalanche. , χ ≡ , ()0 Since, as we show below, the range of large |λ| mainly Sz()0 Sz()00 = Sx , |λ|2 ӷ 2 contributes to the solution, we assume that 4 b . , χ ≡ , ()0 (59) To calculate the kernels, we introduce a new integration Sx()0 Sx()00 = Sx , variable on the right-hand side of (45), , χ ≡ τ, ≡ Sy()0,0 E()0.0 iν λ = ------, To this end, following [26], we will reduce the spectral 2 bχ/τ problem (28) to the ZakharovÐShabat problem [3] by a

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 THE EVOLUTION OF LONGITUDINAL AND TRANSVERSE ACOUSTIC WAVES 1099 simple gauge transformation of the form E Ð1 Ð1 Ð1 1.0 Ψ Dˆ Φ, Lˆ 0 Dˆ Lˆ 0 Dˆ Ð Dˆ ∂τ Dˆ , (60) ()σ () 0.8 Dˆ = ˆI cos v /2 + i ˆ 1 sin v /2 . 0.6 σ Here, ˆI and ˆ 1 are the unit and Pauli matrices, respec- tively. We use the following notation: 0.4 0.2 E ==sinv , U cosv . 0 Transformation (60) reduces the spectral problem (28) to –0.2 0 20406080100 λ ∂ Ψ Ði V Ψ ≡ Ψ τ z = L1 , (61) ÐV∗ iλ Fig. 3. The acoustic-pulse amplitude E versus τ for incom- ()0 π plete initial inversion of the medium Sx = sin( /2.7). The i 2 V()χτ, = ---()1 Ð b sinv + ∂τv , (62) solid line corresponds to b = 0.9, and the dashed line corre- 2 sponds to b = 0.1 and η = 0.5. where we assume that b2 < 1. The solution of the prob- lem reduced to analyzing a simpler spectral problem solution for nonsmall χ, we should determine the total and restoring E from the known solution V(τ, χ) using contribution of all these poles. For sufficiently large χ, (62). the kernels Fr in the Marchenko equations (76) and (77) The Marchenko equations for the ZakharovÐShabat can be calculated asymptotically in the same way as problem are well known (for more detail, see [3]). They was done in [26] for a different problem. Using the are given in the Appendix for reference. For an arbi- results of [26], after modification associated with the ()0 trarily inverted medium at the initial time (S ≠ 0), the values of matrix Aˆ 0' at zero, it is easy to show that the x χ scattering coefficient ρ(χ) and the kernels in the March- kernel Fr for conditions (59) and large is enko equations can be calculated by the asymptotic χ ()0 method suggested in [26]. To determine the dependence ()ττ, χ 2 bSx θ χ Fr + ' = ------I1(),' (65) of the scattering data on using formula (51), we θ []Ω ()0 2 should find the values of the matrix ' 0 + Sz where Ð1 Ð1 Ð1 Aˆ 0' ()τ = Vˆ τ()Dˆ Aˆ 0 Dˆ Ð Dˆ ∂τ Dˆ Vˆ τ (63) θ ()ττ χ Ω []()0 2()2 1/2 '2==+ ' b , 0 1 Ð Sx 1 Ð b , at the ends of the interval (τ = 0, ∞). In [26], we showed that the large λ տ λ , i.e., in the case under consid- I1 is a Bessel function. This kernel corresponds to the a condition eration,λ ӷ b, 1 Ð b2 /2, mainly contributed to the a π >>α π ()0 α radiation asymptotic solution. For this asymptotics and /2 0 /4, Sx = sin 0. ˆ ' π ≥ α conditions (59), the components of matrix A0 are For /4 0 > 0, we should substitute I1 J1.

()0 For this kernel, in [26], we found a solution to the ibS []A' ()0 = Ð[]A' ()0 ≈ ------z , Marchenko equations (76) and (77) and an explicit 0 11 0 22 4λ solution for V(τ, χ) composed of a set of oscillations ()0 that were damped with increasing χ. This solution, to bS []A' ()0 = Ð[]A' ()0 ≈ ------x , within the factor χ, is a self-similar solution that 0 21 0 12 4λ (64) depends on the variable θ = 2bτχ . Ðib []' ∞ []' ∞ ≈ ------A0()11 = Ð A0()22 λ , In the limit b 1, the asymptotic solution for the 4 acoustic-wave amplitude is described by the equation

[]A' ()0 = ()Ð[]A' ()∞ = 0. χ 0 12 0 21 ∂ 8b Ꮾ θ τv = ------θ (), (66) These boundary conditions give rise to an infinite series of ρ(χ) poles whose positions depend on χ. To find the where Ꮾ(θ) is the self-similar solution of the sine-Gor-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1100 ZABOLOTSKII don equation solution that describes the transition to a stable state is a nonsoliton one. Similar phonon dynamics was d 2 1 d observed in a series of experiments by the authors Ꮾ + --- Ꮾ = sinᏮ(),θ (67) dθ θdθ of [13, 14]. These authors investigated the stimulated emission of resonant acoustic phonons produced by the decay of the initial population inversion in a system of with the boundary conditions Kramers doublets E˜ ()2E of impurity Cr3+ ions in a ()0 Ꮾ θ Ꮾ()0 ==S , d () 0. ruby (Al2O3). An external magnetic field of approxi- x ------θ - d θ = 0 mately 3.48 T in strength led to Zeeman doublet split- ting. The crystal was at a temperature of 1.8 K. The ini- This self-similar solution can be expressed in terms tial level population inversion, i.e., the spin inversion, of the solution to P in a standard way [28]. was caused by laser pumping at a wavelength of III 693 nm. These authors found that a nonmonotonic time The results of our numerical calculations of the orig- dependence of the lower level population arose for a inal system for conditions (59) are shown in Fig. 3. sufficiently large effective length of the inverted These results and analysis of the asymptotics for the medium. This kind of field dynamics can be described original system of equations and solutions (66) indicate in terms of model (27) under the initial condition (59). that the general solution consists of a packet of damped In [13, 14], this effect, called a phonon avalanche, is nonlinear pulsations with an increasing duration. In real explained by using an analogy between the dynamics of media, the relaxation and diffraction processes lead to transverse acoustic waves and the dynamics of photons a significant relative suppression of the oscillation during superfluorescence for quasi-monochromatic amplitudes compared to the leading edge. Therefore, in waves. Our results indicate that a similar phonon ava- practice, it is often sufficient to find an expression for lanche can be observed for acoustic pulses with a dura- the first nonlinear oscillation. Our solution shows that ωÐ1 the asymptotics (at large χ) is a nonsoliton one and tion close to B and in the more general case of a lon- characterized by the self-similar variable bχτ . This gitudinalÐtransverse wave and for incomplete inver- solution for the acoustic-wave amplitude describes the sion. phonon avalanche and the avalanche transition of the The new integrable models obtained here can also spin system to a stable state. be used in other fields of physics, for example, in sys- tems with spatially localized electrons. Let us consider a phonon-induced transition with the change in electron 5. DISCUSSION OF THE RESULTS orientation between the Zeeman sublevels in a system AND THEIR PHYSICAL APPLICATIONS of quantum dots in GaAs. In [32], the rate of this tran- We studied the dynamics of acoustic pulses with a sition was estimated for various spinÐorbit interaction duration close to ωÐ1 in terms of integrable reductions mechanisms. The spinÐlattice relaxation for the elec- B trons localized at quantum dots was shown to be much of the evolution equations. These systems of equations smaller than that for free electrons. For a sufficiently describe the evolution of longitudinalÐtransverse waves large strength of the applied magnetic field, the contri- that propagate along the magnetic field in a medium of bution of the spin–phonon interaction can be signifi- impurity ions with an effective spin of 1/2. In studying cant. It was shown in [32] that, for a phonon wavelength the coherent dynamics of acoustic pulses, we used only much larger than the size of one quantum dot (g µ B Ӷ the condition of equal phase velocities for the longitu- 0 B 2បω dinal and transverse waves and the approximation of mv 0 , where v is the speed of sound, m is the unidirectional acoustic-wave propagation. As a result, បω electron mass, and 0 is the typical separation we were able to find an integrable model that corre- between the orbital levels at a quantum point), the spinÐ sponded to the most general physical situation for the phonon interaction is described by a Hamiltonian simi- interaction geometry under consideration. On the other lar to (13). Taking into account the evolution of the lon- hand, milder physical conditions than those in the mod- gitudinalÐtransverse acoustic wave in a system of such els constructed and studied by other authors, for exam- quantum dots and following the above assumptions, we ple, by Voronkov and Sazonov [7], are required to obtain a system similar to (27). Since the picosecond observe the behavior of the field described here, duration of an acoustic pulse corresponds to its limiting because our models can be applied at higher tempera- Ð7 tures and lower magnetic-field strengths. length ls ~ 10 cm and since an acoustic soliton can be formed at several lengths l , an allowance for the coher- In pure form, the soliton dynamics associated with s ASIT requires producing a sufficiently intense pulse ent phonon dynamics for such evolution scales can be with a nearly soliton shape at the boundary of the of importance in controlling the electron behavior in an medium for its observation. On the other hand, as we ensemble of quantum dots. showed above, if the spin system is initially partially or Analysis of the soliton solutions (53) indicates that completely inverted and is in an unstable state, then the the time dependence of the spin direction for a longitu-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 THE EVOLUTION OF LONGITUDINAL AND TRANSVERSE ACOUSTIC WAVES 1101 dinalÐtransverse wave depends on the relative contribu- been observed in a two-level medium suppress each tion of the longitudinal wave. The relation between E other. This picture was observed experimentally [33]. and Sz for a one-phase soliton solution can be easily Here, we disregarded the nonlinear effects related to derived from system (27). For solution (53) and the the anharmonicity of the crystal lattice. They can be conditions when τ Ð∞, we find that included in the integrable models found above in the form of additional perturbing terms. ()∞χ, U Sz = Sz Ð Ð ------, b2 + 4η2 (68) ACKNOWLEDGMENTS ∞, χ 1 I am grateful to E.V. Podivilov for fruitful discus- Sz()Ð = ------Ð 1 . b2 + 4η2 sions of the physical problems touched on in this paper. This study was partly supported by the Russian Foun- τ The dependence Sz( ) for different b is shown in Fig. 2. dation for Basic Research, project no. 03-02-16297. Below, we give another example of the physical sit- uation that can be described by using the qualitative APPENDIX results obtained above. The relaxation of magnetic spins from an inverted initial state to a final (stable) The Lax representation for the system of equa- state in the presence of a magnetic field at low temper- tions (27) is atures (1Ð5 K) can be accompanied by elastic crystal  a deformations [33]. A Hamiltonian similar to Hˆ int (13) λ Ði U G---ZU+ E can be used to describe this spinÐphonon interaction. b ∂τΦ = Φ, (69) This interaction gives rise to a spinÐphonon avalanche a [33], which was observed in experiments aimed at G˜ ---ZU+ E iλU b studying the decays of Mn12 magnetization in the pres- ence of a magnetic field and at temperatures of 1.9–5 K. In contrast to the model considered above, which leads λ () iY b ÐSz + aSx A12 ∂χΦ = Φ, (70) to a two-level medium, the situation studied in [33] cor- λ () responds to a multilevel (more precisely, 21-sublevel) A21 iY bSz Ð aSx medium. In this case, the cascade transitions (or, fol- 2 2 λ lowing the terminology of [33], the phonon-induced where U + E = 1, is the spectral parameter, tunneling) between sublevels can be described in the []()λ2 λ quasi-classical approximation in terms of the model of A12 = G aYZSz + 14+ Y Sx Ð i2Yb Sy , an adiabatically changing spin. Since we consider the time range of acoustic pulses that corresponds to the A = G˜ []aYZS ++()14+ λ2Y S i2YbλS , spectral range including all sublevel transitions, the 21 z x y effective Hamiltonian of the spinÐphonon interaction in this approximation is similar to (10). Some of the qual- Y itative results obtained above can be used to explain the Ð2 = ------, dynamics of spins in such a multilevel medium with 2 2 2 2 2 2 2 2 2 cascade tunneling, for example, the behavior of the sys- 4λ Ð 1 Ð a Ð4b + ()λ ++1 a Ð b + 4a b tem during the decay of an inverted state. 1 2 2 2 At the same time, when the cascade transition in the Z = ------[Ð4λ Ð 1 Ð a + b adiabatic approximation is replaced with a two-level 2a2 medium, the smoothing of the oscillating tail structure 2 of the phonon avalanche is disregarded. Indeed, the for- +4()λ2 ++1 a2 Ð b2 + 4a2b2 ]. mation time of the leading edge of the avalanche is determined by the initial fluctuation (noise), to which ˜ λ λ the system is insensitive. To be more precise, the delay Here, G() and G( ) are arbitrary piecewise smooth of the leading edge is determined by the logarithm of functions that are not identically equal to zero and that ρ are related by the seed area (proportional to ln 0; cf. (57)). On the other hand, a phase difference of the order of π between 1 2 2 2 the pulses generated at different transitions is accumu- GG˜ = Ð---[4λ ++1 a Ð b 8 lated in a time of the order of the duration of the first (71) pulse, because the coupling constants between phonons 2 and different tunneling transitions differ significantly +4()λ2 ++1 a2 Ð b2 + 4a2b2 ]. (by several times). As a result, the shapes of the gener- ated phonon and spin avalanches must consist of one This parametrization contains ambiguities related to intense pulse. The subsequent pulses that must have the square roots of the powers of λ in the matrix ele-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1102 ZABOLOTSKII ments on the right-hand sides of (69) and (70). Below, tion for system (27) with the parametrization of the we give a different special form of the Lax representa- coefficients via the Jacobi elliptic functions:

 B cn()dnξ, k ()ξ, k ab B sn()ξ, k Ði------+ U ------U + ------Ð -E ξ, ξ, 8sn()k 2BÐsn()k 8 ∂τΦ = Φ, (72) ab B sn()ξ, k B cn()dnξ, k ()ξ, k Ð------U Ð ------Ð -E i------+ U ξ, ξ, 2BÐsn()k 8 8sn()k

b Formally passing to the limit in (27), ∂χΦ = ------ξ, []()2 2 2 ξ, 2BÐsn(k )21+ a Ð BÐsn ()k a ≠ 0, b 0, bχχ', (74) (73) iA ()ÐS + aS A χ × 0 z x 12 Φ. where ' is a new variable, we obtain the well-known A iA ()S Ð aS integrable system [28]—the reduction of the original 21 0 z x system of equations (27): ∂ θ Here, U2 + E2 = 1; ξ is the spectral parameter; sn(ξ, k), χ'EU= Ð sin , (75) cn(ξ, k), and dn(ξ, k) are the Jacobi elliptic functions with the modulus where

τ B+ θ 2 τ τ θ k = ------; ==1 + a ∫E()' d ', Sy sin , BÐ 0

2 2 2 2 2 2 2 1/2 1/2 2 θ B± = {}1 + a Ð1b ± []()+ a Ð b + 4a b , aSx Ð1Sz = + a cos . However, we made this passage to the limit in system (27) ξ, ξ, A0 = BÐ B+cn()dnk (),k after applying the unidirectionality approximation and then used relation (26) between the longitudinal and quasi-transverse fields. If we take the original physical []2 2 2 ξ, A12 = 2abSz Ð b 2a Ð BÐ sn ()k Sx system (18), (20), and (21) with f = b = 0 as the basic + i2B B cn()dnξ, k ()ξ, k S , one, then we will obtain (after applying the same Ð + y approximations) a different integrable system of equa- tions that is formally equivalent to the integrable sys- A = Ð2abS + b[]2a2 Ð B 2sn2()ξ, k S tem recently found by Agrotis et al. [22] and used by 21 z Ð x these authors to describe the dynamics of optical soli- ξ, ξ, + i2BÐ B+cn()dnk ()k Sy. tons in a two-level medium with a constant dipole moment. The same (to within the notation) system emerges as the reduction (27) in the low-amplitude This system can be solved in terms of the ISTM on a limit, E Ӷ 1, for which the function U in this system can torus in a way similar to the cases of a biaxial chiral be formally substituted with unity. field [34] and related Landau–Lifshitz equations [35]. However, the following additional difficulty arises Below, we give the Marchenko equations for the ZakharovÐShabat problem (61) [3] on the entire axis here: the nondiagonal matrix elements on the right- τ ∈ ∞ ∞ τ hand sides of (69) and (70) do not become zero for the (Ð , ) and for the field V( , 0) and its derivatives vacuum solution E = 0 against the background of which that rapidly become zero at infinity: the solitons propagate. Therefore, in contrast to the τ main chiral field and the Landau–Lifshitz equations, K ()ττ, ' ++F()ττ+ ' F()τ' + s K ()τ, s ds = 0,(76) another ambiguity arises here. This ambiguity makes 1 ∫ 2 the analytical properties of the problem much more 0 complicated. τ ττ, τ, τ For the special case a = 0, the Lax representation (69) K2()' Ð0.∫K1()s F()' + s ds = (77) and (70) transforms into (28) and (29), respectively. 0

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 THE EVOLUTION OF LONGITUDINAL AND TRANSVERSE ACOUSTIC WAVES 1103

Here, the kernel 16. C. Kittel, Introduction to Solid State Physics, 4th ed. (Wiley, New York, 1971; Nauka, Moscow, 1974). ()λτ 17. M. K. Balakirev and I. A. Gilinskiœ, Waves in Piezoelec- τχ, ρχ λ exp Ði λ tric Crystals (Nauka, Novosibirsk, 1982). Fr ==F()∫ (); ------d (78) 2π 18. J. W. Tucker and V. W. Rampton, Microwave Ultrasonics ޒ in Solid State Physics (North-Holland, Amsterdam, for conditions (59) is completely determined by the real 1972; Mir, Moscow, 1975). spectrum of problem (61). The relation between the 19. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 7: Theory of Elasticity, 3rd ed. (Nauka, potential V and the kernel K1 has the standard form Moscow, 1987; Pergamon, New York, 1986). τχ, ττχ,, V()4= K1(). (79) 20. S. A. Al’tshuller and B. M. Kozyrev, Electron Paramag- netic Resonance in Compounds of Transition Elements, 3rd ed. (Nauka, Moscow, 1981; Halsted, New York, REFERENCES 1975). 1. A. I. Maimistov and A. M. Basharov, Nonlinear Optical 21. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, Waves (Kluwer Academic, Dordrecht, 1999). New York, 1984; Nauka, Moscow, 1989). 2. A. I. Maœmistov, Kvantovaya Élektron. (Moscow) 30, 22. M. Agrotis, N. M. Ercolani, S. A. Glasgow, and 287 (2000). J. V. Moloney, Physica D (Amsterdam) 138, 134 (2000). 3. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and 23. J. D. Gibbon, P. J. Coudrey, J. K. Eilbeck, and R. K. Bul- L. P. Pitaevskiœ, Theory of Solitons: the Inverse Scatter- lough, J. Phys. A: Math. Gen. 6, 1237 (1973). ing Method (Nauka, Moscow, 1980; Consultants Bureau, 24. A. E. Borovik and S. I. Kulinich, Pis’ma Zh. Éksp. Teor. New York, 1984). Fiz. 39, 320 (1984) [JETP Lett. 39, 384 (1984)]. 4. H.-Y. Hao and H. J. Maris, Phys. Rev. B 64, 064302 25. A. A. Zabolotskii, Physica D (Amsterdam) 40, 283 (2001). (1989). 5. G. A. Denisenko, Zh. Éksp. Teor. Fiz. 60, 2269 (1971) 26. A. A. Zabolotskii, Zh. Éksp. Teor. Fiz. 115, 1158 (1999) [Sov. Phys. JETP 33, 1220 (1971)]. [JETP 88, 642 (1999)]. 6. G. T. Adamashvili, Zh. Éksp. Teor. Fiz. 97, 235 (1990) 27. L. A. Faddeev and L. A. Takhtajan, Hamiltonian Meth- [Sov. Phys. JETP 70, 131 (1990)]. ods in the Theory of Solitons (Nauka, Moscow, 1986; 7. S. V. Voronkov and S. V. Sazonov, Fiz. Tverd. Tela (St. Springer, Berlin, 1987). Petersburg) 43, 1969 (2001) [Phys. Solid State 43, 2051 28. A. C. Newell, Solitons in Mathematics and Physics (2001)]. (SIAM, Philadelphia, PA, 1985; Mir, Moscow, 1989), 8. S. V. Voronkov and S. V. Sazonov, Zh. Éksp. Teor. Fiz. CBMS-NSF Regional Conference Series, Vol. 48. 120, 269 (2001) [JETP 93, 236 (2001)]. 29. J. Leon and A. V. Mikhailov, Phys. Lett. A 53, 33 (1999); 9. P. A. Fedders, Phys. Rev. B 12, 2046 (1975). M. Boiti, J.-G. Caputo, J. Leon, and F. Pempinelli, 10. G. T. Adamashvili, Physica B (Amsterdam) 266, 173 Inverse Probl. 16, 303 (2000). (1999). 30. A. Fokas, J. Math. Phys. 41, 4188 (2000). 11. N. S. Shiren, Phys. Rev. B 2, 2471 (1970). 31. A. Degasperis, S. V. Manakov, and P. M. Santini, 12. V. V. Samartsev, B. P. Smolyakov, and R. Z. Sharipov, nlin.SI/0210058. Pis’ma Zh. Éksp. Teor. Fiz. 20, 644 (1974) [JETP Lett. 32. A. V. Khaetskii, Physica E (Amsterdam) 10, 27 (2001). 20, 296 (1974)]. 33. E. del Barco, J. M. Hernández, M. Sales, et al., Phys. 13. H. W. de Wijn, P. A. van Walree, and A. F. M. Arts, Phys- Rev. B 60, 11898 (1999). ica B (Amsterdam) 263Ð264, 30 (1999). 34. I. V. Cherednik, Teor. Mat. Fiz. 47, 755 (1981). 14. L. G. Tilstra, A. F. M. Arts, and H. W. de Wijn, Physica B (Amsterdam) 316Ð317, 311 (2002). 35. A. M. Mikhailov, Phys. Lett. A 92, 51 (1982). 15. A. A. Zabolotskii, Pis’ma Zh. Éksp. Teor. Fiz. 76, 709 (2002) [JETP Lett. 76, 607 (2002)]. Translated by V. Astakhov

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1104Ð1112. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1256Ð1265. Original Russian Text Copyright © 2003 by Granovsky, Bykov, Gan’shina, Gushchin, Inoue, Kalinin, Kozlov, Yurasov. SOLIDS Structure

Magnetorefractive Effect in Magnetic Nanocomposites A. B. Granovskya,*, I. V. Bykova, E. A. Gan’shinaa, V. S. Gushchina, M. Inoueb,c, Yu. E. Kalinind, A. A. Kozlova, and A. N. Yurasova aMoscow State University, Vorob’evy gory, Moscow, 119992 Russia bToyohashi University of Technology, Toyohashi 441-8580, Japan cCREST, Japan Science & Technology Corporation, Kawaguchi 332-0012, Japan dVoronezh State Technical University, Moskovskiœ pr. 14, Voronezh, 394026 Russia *e-mail: [email protected] Received December 30, 2002

Abstract—The magnetorefractive effect in ferromagnetic metalÐinsulator granular nanostructures (CoFeZr)Ð SiOn, CoÐAlÐO, FeSiOn, and (CoFe)Ð(MgÐF) is investigated in the infrared spectral region in a wavelength range from 5 to 20 µm. The magnitude of the effect varies from 0.1 to 1.5% for different nanocomposites and strongly depends on the frequency of light and magnetoresistance. It is shown that the reflection coefficient changes in a magnetic field not only due to the magnetorefractive effect, but also due to the even magnetooptical effect. Simple relations describing this effect are given for the case when the reflection from the substrate is insignificant and in the case of a three-layer (insulator–film–substrate) system. The expression for the frequency dependence of the magnetorefractive effect in nanocomposites is derived and its features in the case of high- frequency spin-dependent tunneling are analyzed. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION tization do not exceed 0.01%. For this reason, the The magnetorefractive effect (MRE) is a frequency search for materials with a strong MRE is of consider- analog of giant magnetoresistance and is manifested in able practical importance. The MRE can also be used a change in light reflectance R, transmittance T, and for a contactless analysis of giant magnetoresistance absorptance A for samples with a considerable magne- [3]. Finally, the study of the MRE in metalÐinsulator toresistance upon their magnetization [1, 2]. systems is a direct method for investigating high-fre- quency spin-dependent tunneling. Traditional magnetooptical effects (MOE), even and odd in magnetization, are associated with the influence The MRE was investigated theoretically and con- of spinÐorbit interaction on intraband (in the IR spectral firmed experimentally for the first time for Fe/Cr mul- region) or interband (in the visible spectral region) opti- tilayers [1]. The MRE theory for metallic multilayers cal transitions. The MRE is not associated with the was also developed in [4Ð6]. In spite of certain incon- spinÐorbit interaction and is due to spin-dependent sistencies in the results of these publications (see, for scattering or tunneling. The conductivity σ(ω) of mag- example, discussion in [6]) concerning the relative netic materials with a giant, tunnel, or colossal magne- magnitude and frequency dependence of the MRE, this toresistance changes significantly upon magnetization; effect is reliably observed in the near-IR spectral region for this reason, the permittivity and amounts to 0.1–0.5% for reflection [1, 6]. πσ() ω εω() ε()ω 4 The MRE theory for granular metalÐmetal alloys = r Ð i------ω , was constructed in [2] and modified recently in [7]. The experimentally measured values of MRE for CoÐAg which determines the refractive indices and is a linear granular systems [3, 7] in strong magnetic fields did not function of conductivity σ(ω), is a function of the exceed 1%. The simplest relation for the MRE in metal- ε ω applied magnetic field also. Here, r( ) is the permittiv- lic systems was derived in [2] for the HagenÐRubens ity taking into account the contributions of displace- spectral range (ωτ Ӷ 1, where τ is the electron relax- ment currents. The MRE must be manifested most ation time) for the normal incidence of light: clearly in the IR spectral region, in which intraband transitions dominate. ∆R RH()= 0 Ð RH() ------= ------It should be noted that, in the case of reflection in R RH()= 0 ferromagnetic metals as well as in alloys and compos- (1) 1 ρ()ρH = 0 Ð ()H 1 ∆ρ ites based on such metals in the IR spectral region, tra- = Ð---()1 Ð R ------= Ð---()1 Ð R ------. ditional magnetooptical effects odd and even in magne- 2 ρ()H = 0 2 ρ

1063-7761/03/9606-1104 $24.00 © 2003 MAIK “Nauka/Interperiodica”

MAGNETOREFRACTIVE EFFECT IN MAGNETIC NANOCOMPOSITES 1105

Here, R(H = 0) and R(H) are the coefficients of light the MRE contribution. The nondiagonal elements of reflection from samples in zero magnetic field and in a the permittivity tensor are responsible for the magne- magnetic field H, ρ(H = 0) and ρ(H) are the correspond- tooptical Kerr and Faraday effects, which are linear in ing resistivities, and ∆ρ/ρ is the absolute value of mag- magnetization, while the orientational magnetooptical netoresistance. Expression (1) implies that high values effect, which is even in magnetization [12, 13], is asso- ε 2 of the MRE must be observed in systems with a large ciated both with the contribution dbaM to the diagonal magnetoresistance and with a low reflectance, i.e., in elements and with nondiagonal elements. Conse- nonmetallic systems. Indeed, a considerable MRE was quently, the results of MRE measurements always con- recently detected in granular metal–insulator films of tain the contribution from the even magnetooptical CoÐAlÐO [8, 9] and CoFeÐMgF [10]. effect along with the true MRE. In this study, an attempt is made at a detailed inves- tigation of the MRE in magnetic composites. Section 2 is devoted to analysis of possible contributions to the 2.1. Model of Semi-infinite Space measured reflectance from the MRE and from even Let us first consider a thick sample for which the magnetooptical effects. It was assumed earlier (see, for reflection from the substrate can be disregarded. For example, [1–10]) that the change in the reflectance in a p-polarized light incident in the xy plane from a transpar- magnetic field is precisely the magnetorefractive effect; ent insulator (medium 1 with a real refractive index n1) at however, this is justified only if the MRE exceeds 0.1%. φ an angle 0 on the sample (medium 2 with a complex In the same section, simple relations are given for cal- η refractive index 2 = n2 Ð ik2), the reflectance R can be culating the MRE in the model of a semi-infinite space written in the form [14] (insulatorÐmagnetic medium) and possible features of the MRE are analyzed in the case of high-frequency p 2 spin-dependent tunneling. It should be noted in this Rr= 12 , connection that expression (1) is obviously inapplica- g η2 Ð g n2 g η2ε (4) ble for describing the MRE in nanocomposites since it p 1 2 2 1 ------1 2 xy - r12 = ------2 -2 Ð 2, has been derived on the basis of the frequency depen- g η + g n g n2()g η2 + g n2 dence of a DrudeÐLorentz-type metallic conductivity. 1 2 2 1 2 1 1 2 2 1 The methods for preparing samples are described in Section 3. Section 4 deals with the experimental meth- g ==n2 Ð,n2 sin2 φ g η2 Ð.n2 sin2 φ (5) ods and details and, in particular, describes a setup 1 1 1 0 2 2 1 0 modified as compared to that in [8]. The results of ε η2 2 experiments for a number of nanocomposites and their Since ( 2)xx = 2 = (n2 Ð ik2) by definition, it can easily analysis are given in Section 5. Main attention is paid be seen, in view of relations (2) and (3), that the term to analysis of the frequency dependence of the MRE, quadratic in magnetization in relation (4) is determined the dependence of the signal on the polarization of radi- by the MRE, induced anisotropy, and nondiagonal ation, the correlation between the MRE and magnetore- terms. The contributions of induced anisotropy and sistance, and the influence of optical parameters on the nondiagonal terms are on the order of the square of the MRE. The results are summarized in the Conclusions. magnetooptical factor Q, i.e., are quadratic in the spinÐ orbit interaction. The magnetooptical factor Q in the visible spectral region does not exceed 0.02, and there 2. THEORY are no grounds to expect an increase in this factor in the The permittivity tensor for a medium magnetized IR spectral region.1 Consequently, the even magne- along the z axis has the form tooptical effect may lead to a change in R upon magne- tization by not more than 0.1%. The numerical calcula- ε ε tions made for specific alloys with experimentally xx xy 0 determined optical and magnetooptical parameters [15] εˆ = ε ε 0 , (2) yx yy confirmed this estimate. Thus, the influence of the even ε 00 zz magnetooptical effect on the MRE can be neglected in all cases when the MRE exceeds 0.1%. Then we can set where it follows from symmetry considerations that a = 0 and ba = 0 in relation (3), ε ε ε ε xx = yy and yx = Ð xy. The nondiagonal components are linear, while the diagonal components are quadratic n ==n0()1 + cM2 , k k0()1 + dM2 , (6) in magnetization M; i.e., 2 2 2 2 1 The nondiagonal components of the permittivity tensor depend ε ε ()2 ε xx = d 1 + bM , b = ba +bMRE, xy = aM. (3) on the ratio of the anomalous Hall coefficient to the square of resistance. The anomalous Hall effect in nanocomposites in the vicinity of the percolation threshold attains giant values which Here, ba characterizes the contribution due to induced are four orders of magnitude larger than in metals; however, the anisotropy of the magnet [11Ð13], while bMRE describes resistivity increases even more strongly in this case.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1106 GRANOVSKY et al. where coefficients c and d are determined by spin- by the resistive and capacitive components. We will dependent scattering or tunneling. prove that the former two factors are insignificant for the formation of the MRE, while the latter factor con- If medium 1 is vacuum (n1 = 1) and the incidence of φ siderably modifies the frequency dependence as com- light is close to normal ( 0 = 0), expression (4) is sim- plified considerably, pared to a dependence for metallic systems. The simplest estimate for the tunneling time τ is the ()2 2 ratio of the tunnel gap width, i.e., the spacing s between 1 Ð n2 + k2 R = ------, (7) granules, to the Fermi velocity vF. For a spacing s = ()2 2 1 + n2 + k2 1−3 nm typical of tunneling, we find that the tunneling time is on the order of 10Ð16 s; i.e., in the IR range and then ωτ Ӷ 1, when λ = 1Ð10 µm, and the tunneling proba- bility is the same as in the static case [16]. Since ωτ Ӷ 1 ∆ R ()2 and the electron tunneling probability at frequency ------= 1 Ð R M បω ωτ 2 R EF + depends on the factor [e Ð 1] and on the inci- dent radiation power [16], the tunnel resistivity ρ in the 0 2 0 2 0 2 (8) 1 Ð ()n + ()k ()k IR range is independent of frequency for a radiation × c------2 2 - Ð.2d------2 - 0 2 0 2 0 2 0 2 power in the flare spot much lower than 1 W/cm2 ()1 Ð n + ()k ()1 Ð n + ()k 2 2 2 2 (which is obviously the case). However, the tunnel gap can be regarded a resistor and a capacitor with permit- tivity ε connected in parallel. Then, the expression for 2.2. The InsulatorÐMagnetic FilmÐSubstrate ins Three-Layer System the conductance of such a system and of the granular film as a hole at finite frequencies can be written in the In the case of a three-layer system consisting of a form magnetic film (medium 2) of thickness h on a substrate (medium 3) and an insulator (medium 1), the expres- 1 + iωε ρ()H /4π σω(), H = ------ins -, (11) sion for the reflectance of p-polarized radiation incident ρ()H from medium 1 has the form which differs significantly from the Drude–Lorentz fre- p 2 p quency dependence for metallic systems. It should be 2 r + F r p p jk k kl noted that expression (11) was successfully used in [17] Rr==jkl , r jkl ------2 p p , 1 + Fk r jkrkl for interpreting the experimental data on magne- (9) 2 2 toimpedance of tunnel systems. By definition, we have p g η Ð g η 2 2 2 r ==------j k k -j , g η Ð,η sin φ 4πσ() ω jk η2 η2 j j 1 0 ε ==ε Ð i------ε' Ð iε''. (12) g j k + gk j xx r ω ε 2 2 ε ε ()π λÐ1 Here, ' = n Ð k , '' = 2nk, and r = 1 for the Drude Fk = exp Ð2 i gkhk , (10) model. It follows from relations (11) and (12) that only the imaginary part of the permittivity depends on the where the presence of factor Fk, which appears due to tunnel resistivity of nanocomposites. Assuming that the transmission of light through the film and reflection magnetoresistance from the substrate and contains the complex refractive ∆ρ()H ρ()0 Ð ρ()H index of the magnetic medium, may lead to a consider------= ------(13) able enhancement of the MRE. Some examples of the ρ ρ()0 results of calculations based on formulas (4) and (9) is small and using relations (8), (12), and (13), we will be given below together with microscopic expres- obtain sions for parameters c and d. ∆ρ 0 2 ()2 c k cd+ M ==------, --- ----- , (14) 2.3. Peculiarities of the Frequency Dependence ρ d n0 of Magnetorefractive Effect for Nanocomposites which is equivalent to the relations At low frequencies, tunneling probability is inde- 2 ∆ρ 1 pendent of frequency. However, at high frequencies, dM = ------, ρ 0 0 2 both a decrease in the tunnel transparency (when the 1 + ()k /n period of an electromagnetic wave becomes smaller (15) 0 0 2 than the characteristic time of tunneling) and an 2 ∆ρ ()k /n cM = ------, increase in the tunneling probability due to absorption ρ ()0 0 2 may be observed. In addition, a tunnel junction is 1 + k /n essentially a capacitor; for this reason, the conductance in which index “2” indicating the magnetic medium of the tunnel junction at finite frequencies is determined has been omitted. Expression (15) combined with for-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 MAGNETOREFRACTIVE EFFECT IN MAGNETIC NANOCOMPOSITES 1107 mulas (4) or (10) for the reflectance completely defines glomerates and chains, which in turn form a labyrinth the MRE in nanocomposites for p-polarized light in structure. terms of magnetoresistance and optical parameters. In The use of CoFeZrÐSiOn films for studying the the particular case of normal incidence, it follows from MRE makes it possible to minimize the influence of the Eqs. (8) and (15) that even magnetooptical effect on the MRE since the induced anisotropy effect is obviously insignificant for 2 2 amorphous metals (see discussion following formula (5) ∆R ∆ρ 2 3n Ð1k Ð ------= Ð() 1 Ð R ------k ------. (16) in Section 1). In addition, the wide range of composi- ρ 2 2 2 2 R ()n + k []()1 Ð n + k tions obtained makes it possible to study the MRE in the range of metal and insulator phases as well as near This expression represents the main result of this sec- the percolation threshold. tion. It shows that the MRE in nanocomposites with a Granular ferromagnetic metal–insulator films of tunnel magnetoresistance is a complex function of opti- Cox(Al2O3)1 Ð x with a high magnetoresistance attaining cal parameters. In particular, the MRE must be mani- 8% in a field of 12 kOe at room temperature were pre- fested most clearly in the spectral regions where weak pared by the method of tandem rf magnetron sputtering reflection takes place and both negative and positive from various targets of CoxAl1 Ð x alloys in argon and MRE are possible; this enables us to interpret the exper- oxygen atmosphere onto uncooled glass substrates. imental data obtained in [9]. The film thickness was 2 µm and the size of Co grains varied from 2 to 5 nm. A detailed description of the sample preparation procedure, as well as the methods 3. SAMPLES and results of measurements of chemical, structural, electrical, and magnetic parameters of the samples, is The objects of investigations were nanocomposite given in [18, 19]. Nanogranular samples of the films with a considerable tunnel magnetoresistance. (FeCo)−(MgF) system, whose magnetoresistance attained 13.3% at room temperature in a magnetic field The films of amorphous ferromagnetic alloy of 10 kOe, were synthesized according to basically the Co45Fe45Zr10 in the amorphous matrix of silicon diox- same technology [20]. α ide -SiO2 were obtained by ion-beam sputtering of Films of the FeÐSiOn system were prepared by the compound targets. A multicomponent target had the double ion-beam sputtering of Fe and SiO from a com- × 2 2 form of a molded base having a size of 270 80 mm pound target onto silicon substrates, which permitted and made of the corresponding ferromagnetic alloy. one to vary the proportion between the ferromagnet and Quartz plates having a width of 9 mm were placed on the insulator. The characteristic grain size was approx- the surface of the base at right angles to its longitudinal imately equal to 4 nm and the film thickness was axis. Sputtering was carried out in a vacuum of 1 × 0.2−0.8 µm. The maximal value of magnetoresistance 10−5 Torr onto fixed glass-ceramic substrates. A granu- in a field of 10 kOe at room temperature attained values lar structure with a wide and continuous set of metal- of 1–3% depending on the film composition. The struc- phase concentrations was formed in a single production tural, electrical, and magnetoresistive properties of cycle during simultaneous sputtering of the metallic films belonging to this system are described in [21]. alloy and the dielectric from the compound target with a varying spacing between the quartz plates. The values of the metal phase concentration x varied from 30 to 4. METHODS OF INVESTIGATION 65 at. %. The middle of this interval (so-called percola- AND EXPERIMENTAL DETAILS tion threshold) corresponds to samples with structures Optical reflection and magnetooptical effects in in which the highest values of magnetoresistance are magnetic nanocomposites were studied in a wide range observed. The film thickness in the region of the perco- of wavelengths from 1.43 to 20 µm (7000Ð500 cmÐ1). In lation threshold amounted to approximately 4 µm. the frequency range 500Ð7000 cmÐ1, we used a FTIR According to the results of electron microscopic inves- PU9800 commercial Fourier spectrometer with a spec- tigations carried out with the help of a high-resolution tral resolution of approximately 4 cmÐ1 and a photomet- transmission electron microscope, the synthesized ric transmission accuracy ∆T = 0.1Ð0.01%. Optical composites consisted of amorphous metallic grains reflection will be represented below by the frequency with a size from 2 to 5 nm, distributed in the amorphous dependence of energy reflectance R(ω), while the mag- matrix. Smaller grain sizes correspond to lower con- netooptical effects will be characterized by the relative centrations of the metallic phase, while larger sizes are variation of the intensity of radiation reflected from the typical of samples with concentration x exceeding ferromagnet during its magnetization. In the experi- 60 at. %. The grains formed as a result of growth are not mental geometry used for observing the magnetoopti- insulated absolutely in the dielectric matrix (even in the cal effect for a p wave of linearly polarized light ⊥ case of a high SiO2 concentration), but form small con- (E M), three intense effects can be detected simulta-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1108 GRANOVSKY et al.

λ, µm light measured for H = 0 determines the value of the 20 10 5 MRE, which is given by the formula

0.6 I()0 Ð IH() R()0 Ð RH() ξω()==------, (17) I()0 R()0 0.5 1 where the light intensities are replaced by the energy 0.4 reflectances, which are proportional to them. R 2 Such modifications and improvements enabled us to 0.3 3 measure the optical reflectance and the MRE not only in the case of oblique incidence of light, but also for 4 0.2 angles close to normal incidence and not in the state of residual magnetization, as before, but on completely demagnetized samples. Despite the complication of the optical path of the setup, the noise level did not exceed 500 1000 1500 2000 that in [8]. The noise level was lowered to 1 × 10Ð4 in ν, cm–1 the frequency range 500Ð1000 cmÐ1 and to 3 × 10Ð4 for 2000 cmÐ1 due to the application of storage operations Fig. 1. Frequency dependence of the reflectance of granular (over 1000 scans), mutually reversible cycles, and (Co45Fe45Zr10)x(SiO1.7)100 Ð x films: x = 57 (1), 47 (2), 40 (3), and 34 (4); ϕ = 10°. smoothing. Optical reflection and MRE measurements were made for angles of incidence ϕ = 10° and 50° in the frequency range 500Ð5000 cmÐ1 with a spectral res- neously, viz., the transversal Kerr effect (which is linear olution of 2 cmÐ1 at room temperature. in magnetization M) and two effects which are qua- dratic in M (the orientation effect and the MRE). Magnetoresistance was measured using a two-point potentiometer at room temperature in a magnetic field We measured the magnetorefractive effect on a of strength up to 12 kOe oriented parallel to the sample setup described in [8] with a number of modifications surface and was determined, as usual, in accordance concerning mainly the design of the magnetooptical with formula (13). attachment of the Fourier spectrometer. First, the radiation from the IR spectrometer was 5. EXPERIMENTAL RESULTS turned through 90° with the help of plane and aspheri- AND DISCUSSION cal off-axis mirrors, which made it possible to direct radiation at an angle ϕ ≈ 10° to the normal of the sam- An analysis of the frequency dependence of optical ple placed in the gap of the magnet. The MRE measure- reflection in nanocomposites belonging to the (CoFeZr)x(SiO2)100 Ð x system proved (Fig. 1) that the ments for an angle of light incidence close to the nor- ν mal allowed us to avoid the influence of the transversal reflectance R( )) is 2Ð3 times lower than for pure met- Kerr effect (which is equal to zero for the normal inci- als constituting grains for all concentrations in the fre- Ð1 dence); such a geometry is most convenient for inter- quency range 500Ð7000 cm and is practically inde- preting the experimental results. Second, the rotating pendent of frequency in the range 2500Ð7000 cmÐ1. For permanent magnet was replaced by an electromagnet, frequencies below 2500 cmÐ1, the value of R(ν) which created (in a gap of 7 × 10 mm2) either a constant decreases insignificantly for samples with concentra- tions below 47 at. %. In the frequency range 1100Ð magnetic field Hmax = 1700 Oe or a varying field whose amplitude value also attained 1700 Oe depending on 1400 cmÐ1, a sharp decrease in R(ν) followed by its whether a dc or an ac current was fed to the magnet increase associated with absorption in the silicon diox- winding. By reducing the varying field to zero, we ide matrix is observed. The minimal values of R(ν) cor- could demagnetize the sample almost completely (i.e., respond to samples with concentrations from the range obtain a state with M = 0). 34Ð47 at. %, which corresponds to the region of perco- r lation threshold. The oscillatory behavior of R(ν) for We also developed a new method for measuring the samples in this concentration range in the frequency difference in the intensities I(0) and I(H) of radiation interval 1100Ð1400 cmÐ1 is associated with the interfer- reflected from the sample in the demagnetized and ence of light reflected at the film–air and film–substrate magnetized states instead of the ratio of these intensi- interfaces. The strongest changes in R(ν) correspond to ties, as was done earlier [8]. This allowed us to reduce a narrow interval of 1300 ± 100 cmÐ1 in which the the “contribution” from noise, which is especially reflectance changes by a factor larger than 2. The reflec- important for small values of the effects being mea- tion spectra for IR radiation in Fig. 1 correspond to an sured. The division of difference ∆I by the intensity of angle of incidence ϕ = 10°. For ϕ = 50°, individual

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 MAGNETOREFRACTIVE EFFECT IN MAGNETIC NANOCOMPOSITES 1109 absorption bands are shifted by 20Ð40 cmÐ1 towards |∆ρ/ρ|, % higher frequencies. 4 Figures 2 and 3 show the results of measurements of magnetoresistance and MRE of the (CoFeZr)x(SiO2)100 Ð x system. The data on magnetoresistance and MRE are 3 given for the same value of field (1700 Oe). The percolation threshold determined from the results of resistance measurements for this system of 2 amorphous granular alloys amounts to x ≈ 43 at. %; near this threshold, the magnetoresistance attains its maximal value of 3.5%. For samples with a high con- 1 centration of the metal phase (x = 57 at. %), the magne- toresistance is close to zero (∆ρ/ρ ≈ 0.2%), which cor- related with an insignificant MRE in accordance with 0 the theory developed in Section 1. It is interesting to compare the MRE for samples with the same magne- 20 40 60 80 100 x toresistance, but with concentrations x smaller and larger than 43 at. %, i.e., on the left and right of the per- Fig. 2. Concentration dependence of the magnetoresistance colation threshold. It can be seen from Fig. 3 that the of granular (Co45Fe45Zr10)x(SiO1.7)100 Ð x composite. MRE for samples in the dielectric phase is stronger than in the metallic phase (x > 47 at. %) for virtually equal ξ, % values of magnetoresistance (see Fig. 2). In accordance with formula (16), this is due to smaller values of the 0 reflectance for the “dielectric” sample (see Fig. 1). 1 Finally, the frequency for which the maximal value of MRE is observed exactly coincides with the frequency –0.1 (1200 cmÐ1) for which the reflectance has the minimal 2 value. 0 Since the MRE is weak (less than 0.1% on the aver- age) in this system of alloys, we had to prove that the –0.1 measured dependences are associated precisely with 3 the MRE. Figure 4 shows the frequency dependences of MRE for one of the samples for natural light and for a 0 p wave of linearly polarized light. It can be seen that the polarization dependence is practically absent in the –0.1 spectral region where the signal exceeds 0.05%, which is direct proof of the fact that the signal being measured 0 is just the MRE. However, the situation is different in 4 Ð1 the frequency ranges 2500Ð1300 and 1000Ð800 cm , –0.1 where the signal is weaker than 0.05% and the influ- ence of the even orientation effect becomes noticeable. 1000 1400 1800 2200 A correlation between magnetoresistance and MRE ν, cm–1 is reliably observed for all nanocomposites. For exam- Fig. 3. Frequency dependence of the magnetorefractive ple, the magnetoresistance of FeÐSiOn composites in a effect of granular (Co45Fe45Zr10)x(SiO1.7)100 Ð x compos- field of 2.2 kOe does not exceed 1.2%; accordingly, the ites: x = 57 (1), 47 (2), 40 (3), and 34 (4); magnetic field is frequency dependence of the MRE shows that the mag- 1700 Oe; ϕ = 10°. nitude of this effect attains a value of 0.2% (Fig. 5). In the vicinity of the percolation threshold, nano- ° composites become relatively transparent (see Fig. 1); which were obtained for an angle of incidence of 10 and for which the effect of interference is pronounced consequently, the interference of light associated with ξ ν the reflection from the substrate cannot be disregarded most clearly. The theoretical ( ) dependences corre- completely even for relatively thick films. In particular, sponding to these spectra are shown in Fig. 7. interference beats of the reflectance can be clearly seen Figure 8 shows the experimental data on dispersion in Fig. 1 for films with x = 40, 34 at. %. A quantitative of the MRE (for two values of magnetic field of 1500 description of the effect must be based on the theory for and 1700 Oe) and of the reflectance of a nanocomposite the model of a three-layer system described in Section 1. (Co0.4Fe0.6)48(Mg52F) film with a tunnel conductivity Figure 6 shows the spectra describing the frequency and a high magnetoresistance (7.5% in a field of ϕ ° dependence of the MRE for a Co43Al22O35 sample, 1700 Oe) for an angle of incidence = 10 . It can be

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ξ ξ , % , % R 0.2 0 0.45 0.1 0.35 –0.05 0 0.25

–0.10 –0.1 0.15

–0.2 0.05 –0.15 800 1200 1600 2000 900 1000 1100 1200 1300 ν, cm–1 ν, cm–1 Fig. 4. Frequency dependences of the MRE for natural (dot- ted curve) and p-polarized (solid curve) light for granular Fig. 5. Frequency dependences of the reflectance (dotted (Co45Fe45Zr10)47(SiO1.7)53 film in a magnetic field of curve) of a FeÐSiO2 film on a Si substrate and of the MRE 1700 Oe; ϕ = 10°. (solid curve) for a magnetic field H = 2200 Oe.

ξ, % ξ, %

0 0.2

0 –0.2

–0.2 –0.4

–0.4 –0.6 –0.6

–0.8 –0.8 800 1000 1200 800 1000 1200 ν, cm–1 ν, cm–1

Fig. 6. Frequency dependences of the MRE of a Fig. 7. Theoretical dependences of the MRE for a ∆ ∆ρ ρ ∆ρ ρ Co43Al22O35 film for two values of field H = 2200 and Co43Al22O35 film: / = 3% (solid curve); / = 1%, 100 Oe: solid and dotted curves correspond to angles of ϕ = 45° (dotted curve); and ∆ρ/ρ = 1%, ϕ = 10° (dashed incidence ϕ = 45° and 10°, respectively. curve). seen from the figure that the MRE attains a value of expression (16) does not rule out higher values (on the 1.5%, which is a record high for all metallic and non- order of 10%). Figure 9 shows the spectra of frequency metallic systems studied and is two orders of magni- dependences of the MRE for the same sample in a tude stronger than the traditional magnetooptical wider frequency range for ϕ = 45° and for three values effects. In accordance with relation (16), such a magni- of the magnetizing field H = 0, 1500, and 1700 Oe. The tude of the MRE is due to a low reflectance and a high spectrum for H = 0 shows the noise level from which magnetoresistance. It should be emphasized that the response of the investigated signal to the action of

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λ, µm ity of λ = 10 µm does not correlate with the behavior of 20 10 5 the reflectance either. Finally, we cannot explain a con- siderable MRE “noise” in fields of 1.5 and 1.7 kOe in a 0 wavelength range of 15Ð10 µm. These features are 40 probably associated with an additional influence of the magnetic field on elementary excitations in the dielec- –0.5 tric MgF matrix. It should also be borne in mind that the 30 simple theory of high-frequency tunneling developed , % , % above disregards such factors as, for example, the effect ξ 20 R of electronÐelectron interaction [20]. –1.0 10 6. CONCLUSIONS The experiments described here prove the existence –1.5 0 of a new magnetooptical effect, viz., MRE, associated 500 750 1000 1250 1500 1750 2000 with spin-dependent tunneling in magnetic nanocom- ν, cm–1 posites in the vicinity of the percolation threshold. The MRE obviously correlates with the tunnel magnetore- Fig. 8. Spectra of MRE ξ in fields H = 1700 Oe (solid curve) sistance. Depending on the magnetoresistance and opti- with a resolution of 2 cmÐ1 and H = 1500 Oe (dotted curve) cal properties of the nanocomposite, the magnitude of Ð1 with a resolution of 4 cm and reflectance RH = 0 for a the MRE varies over a wide range from 0.1 to 1.5%, (Co0.4Fe0.6)48(Mg52F) film after averaging over 300 scans; which is one or two orders of magnitude higher than the ϕ = 10°. even magnetooptical effect. The developed theory pro- vides a qualitative and semiquantitative explanation for λ, µm a number of experimental data and promises the attain- ment of even higher magnitudes of the MRE. It would 10.0 5.0 2.5 undoubtedly be interesting to continue the investigation 0 of the angular, polarization, and temperature depen- dences of the MRE; the search for mechanisms respon- sible for the MRE in metalÐsemiconductor systems; and the advancement to the far-IR spectral region as well as to the visible region for which new effects asso- ciated with the dependence of the tunnel transparency –0.5 on the light frequency should be expected. , % ξ ACKNOWLEDGMENTS The authors are grateful to S. Ohnuma, B. Aronzon, and M. Sedova, who supplied a number of CoÐAlÐO, –1.0 FeÐSiO2, and CoFeÐMgF samples, and to A. Vedyaev 1000 2000 3000 4000 and E. Shalygina for fruitful discussions. ν, cm–1 This study was supported financially by the Russian Foundation for Basic Research (project no. 00-02-17797), Fig. 9. Spectra of MRE ξ in fields H = 1700 Oe (solid curve) the Ministry of Education, Science, Sports, and Culture and H = 1500 Oe (dotted curve) and of noise with a resolu- of Japan (grant no. 14-205045), and the program Ð1 tion of 4 cm for a (Co0.4Fe0.6)48(Mg52F) film after aver- Universities of Russia (project nos. 01.03.003 and aging over 300 scans; ϕ = 45°. 01.03.006). the magnetic field is separated. The magnitudes of the REFERENCES MRE for the above values of the field imply that the ξ 1. J. C. Jacquet and T. Valet, Mater. Res. Soc. Symp. Proc. (H) dependence is nonlinear. 384, 477 (1995). The following two facts, which cannot be described 2. A. B. Granovskiœ, M. V. Kuzmichev, and J. P. Clerc, Zh. in the framework of the theory developed in Section 1, Éksp. Teor. Fiz. 116, 1762 (1999) [JETP 89, 955 (1999)]. are worth noting. First, it can be seen from Fig. 8 that 3. M. Gester, A. Schlapka, R. A. Pickford, et al., J. Appl. the values of reflectance for λ = 20, 5 µm differ by a Phys. 85, 5045 (1999). factor of 4, while the magnitude of the MRE is practi- 4. H. F. Kubrakov, A. K. Zvezdin, K. A. Zvezdin, and cally the same for these wavelengths. Second, the pres- V. A. Kotov, Zh. Éksp. Teor. Fiz. 114, 1101 (1998) ence of two narrow and sharp MRE peaks in the vicin- [JETP 87, 600 (1998)].

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5. G. M. Genkin, Phys. Lett. A 241, 293 (1998). 15. A. B. Granovskiœ and A. N. Yurasov, Fiz. Met. Metall- 6. S. Uran, M. Grimsditch, E. Fullerton, and S. D. Bader, oved. (2003) (in press). Phys. Rev. B 57, 2705 (1998). 16. M. Buttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 7. V. G. Kravets, D. Bosec, J. A. D. Matthew, et al., Phys. (1982). Rev. B 65, 054415 (2002). 17. H. Kaiju, S. Fujita, T. Morozumi, and K. Shiiki, J. Appl. 8. I. V. Bykov, E. A. Gan’shina, A. B. Granovskiœ, and Phys. 91, 7430 (2002). V. S. Gushchin, Fiz. Tverd. Tela (St. Petersburg) 42, 487 18. S. Mitani, H. Fujimori, and S. Ohnuma, J. Magn. Magn. (2000) [Phys. Solid State 42, 498 (2000)]. Mat. 165, 141 (1997). 9. D. Bozec, V. G. Kravets, J. A. D. Matthew, and 19. S. Ohnuma, K. Hono, E. Abe, et al., J. Appl. Phys. 82, S. M. Thompson, J. Appl. Phys. 91, 8795 (2002). 5646 (1997). 10. A. B. Granovsky, E. A. Ganshina, V. S. Gushchin, et al., 20. N. Kobayashi, S. Ohnuma, T. Masumoto, and H. Fuji- in Digests of INTERMAG-2003, Boston, USA (2003). mori, J. Appl. Phys. 90, 4159 (2001). 11. G. S. Krinchik and M. V. Chetkin, Zh. Éksp. Teor. Fiz. 21. B. A. Aronzon, A. E. Varfolomeev, A. A. Likal’ter, et al., 36, 1924 (1959) [Sov. Phys. JETP 9, 1368 (1959)]. Fiz. Tverd. Tela (St. Petersburg) 41, 944 (1999) [Phys. 12. G. S. Krinchik and V. S. Gushchin, Pis’ma Zh. Éksp. Solid State 41, 857 (1999)]. Teor. Fiz. 10 (1), 35 (1969) [JETP Lett. 10, 24 (1969)]. 22. S. T. Chui and L. Hu, Appl. Phys. Lett. 80, 273 (2002). 13. G. A. Bolotin, Fiz. Met. Metalloved. 39, 731 (1975). 14. V. M. Maevskiœ, Fiz. Met. Metalloved. 59, 213 (1985). Translated by N. Wadhwa

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Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1113Ð1121. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1266Ð1275. Original Russian Text Copyright © 2003 by Nefedova, Alekseev, Lazukov, Sadikov. SOLIDS Electronic Properties

The Thermodynamic Properties and Special Features of Spectra of Elementary Excitations of Unstable Valence Sm- and Ce-Based Compounds E. V. Nefedova*, P. A. Alekseev, V. N. Lazukov, and I. P. Sadikov Kurchatov Institute, Russian Research Center, Moscow, 123182 Russia *e-mail: [email protected] Received December 5, 2002

Abstract—The experimental data on the spectra of elementary excitations measured by inelastic neutron scat- tering and on the heat capacity and the coefficient of thermal expansion are used to analyze the correlation between the spectral characteristics of the electron and phonon subsystems and the special features of the tem- perature dependence of the thermodynamic properties of a number of unstable valence Sm- and Ce-based com- pounds. The anomalous behavior of the thermodynamic properties of these compounds is defined by the special features of their phonon and electron (4f and conduction electrons) spectra. The rearrangement of the 4f-elec- tron spectrum as a result of temperature variation plays a decisive part in the formation of temperature depen- dences of the heat capacity and the coefficient of thermal expansion of unstable valence systems. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION namic properties [5Ð7]. The development of concepts of the effect of valence was marked by the need for tak- Unstable valence compounds on the basis of rare- ing into account various interactions observed in unsta- earth elements are characterized by the presence of a ble valence systems (in particular, the electronÐphonon fairly large anomalous (or additional) contribution to interaction [8]) and leading to a variation of the internal the temperature dependences of the heat capacity and energy and entropy of unstable valence systems. Note the coefficient of thermal expansion as compared to that the treatment of partial components associated their isostructural analogs with an open or filled 4f shell with different interactions made it possible to interpret [1, 2]. It is known that, in the case of rare-earth com- the thermal properties of heavy-fermion compounds in a pounds with a stable magnetic moment, the anomalous wide temperature range, in particular, of CeRu2Si2 [9]. contribution to the thermodynamic properties is due to However, in the case of quantitative calculations of the the presence of 4f multiplets split in a crystal electric anomalous thermodynamic and elastic properties of field [3]. This “traditional” interpretation of the anom- unstable valence compounds, an integral characteristic alous contribution to thermodynamics cannot be was introduced, as a rule. In particular, Takke et al. [10] employed in the case of unstable valence compounds, used a phenomenological scaling function for the elec- because no effects in a crystal electric field in the ordi- tron part of free energy, proportional to the universal nary sense are present in these systems as a result of energy scale T0 characteristic of the systems being partial delocalization of 4f electrons. treated. The volume dependence of the electron part of As a rule, the anomalous contribution in unstable free energy is defined by the volume dependence of T0. valence compounds, which is obtained as the difference This enables one to introduce the electron Grueneisen Γ between the heat capacities or the coefficients of ther- parameter el = Ðd[lnT0]/d[lnV] (where V is the vol- mal expansion of an unstable valence compound and its ume) for the description of the electron and electronÐ structural analog, is taken to be electronic by nature and phonon contributions: in terms of this parameter, the directly associated only with the valence instability, majority of thermodynamic quantities may be calcu- i.e., with a nonintegral population of the 4f shell. In one lated. The above-described approach produces adequate of the early papers [4], the electron anomaly of the agreement between theory and experiment at T < T0; coefficient of thermal expansion of unstable valence however, the problem of physical understanding of spe- compounds was interpreted as a result of variation of cial features of the anomalies of the thermal expansion the population of the 4f shell of a rare-earth ion with coefficient for different compounds (sign, amplitude, temperature. As a matter of fact, in all of the subsequent range of existence) cannot be fully solved. For exam- papers, the population of the 4f shell was one of the ple, in spite of the fact that the characteristic tempera- main parameters treated in analyzing the thermody- ture (T0 ~ 150 K) and the values of variation of valence

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1114 NEFEDOVA et al. |∆ | in the temperature range from 4 to 300 K ( vRE ~ 0.05) (4f and conduction electrons) and lattice subsystems are comparable for SmB6 [11] and CeNi [12], the and to use the resultant estimates to determine the part anomalous contributions to the coefficient of thermal played by separate contributions in the formation of expansion of SmB6 [13, 14] and CeNi [15, 16] have anomalous heat capacity and coefficient of thermal opposite signs, as well as essentially different scales expansion in a wide temperature range. Samarium and ranges of this anomaly. hexaboride (SmB6) is known as a “classical” unstable valence compound with a “strong” intermediate In analyzing the thermodynamic properties of a valence. The valence of Sm ions at room temperature is number of specific unstable valence compounds (the v ~ 2.55 and decreases with cooling to v ~ 2.50 [11]. so-called Kondo-insulators) in which, at a low temper- Sm Sm An important feature of SmB is the presence of a nar- ature, a gap is formed in the electron density of states in 6 the vicinity of the Fermi energy, it is the presence of the row gap in the electron density of states in the vicinity gap that is regarded as the main reason for the emer- of the Fermi energy, which was estimated in early gence of a significant additional contribution to the heat papers at approximately 5Ð10 meV [2, 13]. Detailed capacity and to the coefficient of thermal expansion. investigations of the kinetic characteristics of SmB6 For example, the marked anomaly revealed in the elec- were recently performed using high-quality samples tron component of heat capacity and coefficient of ther- [20Ð22]. It was experimentally shown in the latter papers and theoretically substantiated in [23] that the mal expansion of SmB6 was interpreted as the contribu- tion arising as a result of excitation of electrons via a electron transport in SmB6 is defined by at least two dif- semiconductor gap of the order of 5 meV [17]. Ade- ferent energy scales, namely, the hybridization gap of approximately 10–20 meV and the “impurity” band in quate agreement was obtained between a model calcu- ∆ lation and experiment; however, in this case, Mandrus the vicinity of the bottom of the conduction band ( ~ et al. [17] examined a fairly narrow temperature range 3 meV). The smaller energy scale is defined by the for- (T < 80 K) and restricted themselves to one binary com- mation of a bound electronÐpolaron complex as a result pound. The significant negative component of the coef- of fast valence fluctuations on each Sm ion. The carrier ficient of thermal expansion in the higher temperature concentration in the “impurity” band is estimated at range (T > 100 K) [14]) remained unexplained. approximately 1017 cmÐ3 [23]. CeNi is an intermetallic unstable valence compound with a “moderate” inter- Therefore, in analyzing the thermodynamic proper- mediate valence (vCe ~ 3.11 at room temperature and ties, the authors of the publications known to us either v ~ 3.14 at T = 10 K [12]). The spectra of lattice and introduced (within the phenomenological approach) Ce 4f-electron excitations of SmB and CeNi have been the universal integral parameter or treated a concrete 6 special feature of the system being examined. However, studied in sufficient detail [24]. In addition, the valent they have, first of all, failed to take into account the real state of a rare-earth ion in both compounds may be pur- energy spectrum of excitations of 4f electrons and its posefully varied when a Sm (or Ce) ion is replaced by rearrangement with temperature, which is known to be a La ion. A variation of the valent state leads to a marked significant. The second important point to be noted is transformation of the 4f-electron and phonon spectra. the need for taking into account the electronÐphonon Therefore, a study of dilute systems (Sm1 Ð xLaxB6 and interaction. Indeed, a number of Sm- and Ce-based Ce1 Ð xLaxNi) offers a possibility of checking the gener- unstable valence compounds exhibit significant ality of the inferences made for stoichiometric com- changes in the phonon dispersion curves compared to pounds and of more reliably determining the general structural analogs [18, 19]. A renormalization of regularities of the formation of anomalies in the ther- phonon frequencies in unstable valence compounds modynamic properties. may bring about the emergence of an additional contri- bution to the heat capacity and coefficient of thermal expansion compared to the data for an isostructural 2. CALCULATION analog. And, thirdly, in treating Kondo insulators, i.e., OF THERMODYNAMIC PROPERTIES systems with a valence instability and a gap in the elec- tron density of states in the vicinity of the Fermi energy, Anomalous contributions to the thermodynamic prop- one must take into account the special features of the erties of unstable valence Sm1 Ð xLaxB6 and Ce1 Ð xLaxNi density of states of conduction electrons in the vicinity compounds were treated within a unified approach as a of the Fermi energy along with (rather than instead of) result of summation of independent (in a first approxi- the 4f-electron contribution and the special features of mation) components associated with the actual special the lattice vibration spectrum. features of the phonon and electron (4f electrons and conduction electrons) spectra. It was the objective of our study to estimate, using the example of Sm1 Ð xLaxB6 and Ce1 Ð xLaxNi com- The following expression was used to calculate the pounds, the contribution made to the thermodynamics anomalous contribution to the heat capacity: of unstable valence compounds due to the special fea- ∆ ∆ tures of the real spectral characteristics of the electron CT()= C f ()T ++Clat()T Cg(),T (1)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

THE THERMODYNAMIC PROPERTIES AND SPECIAL FEATURES OF SPECTRA 1115

–1 where Cf is the heat capacity due to the special features ∆C, J (mol K) of the 4f-electron excitation spectrum. The value of Cf 20 was determined on the basis of the experimentally mea- 120

sured 4f-electron excitation spectrum for unstable –1 SmB6 valence compounds; the structure of this latter spec- 80 trum provides information about the density of states of

15 , meV 4f electrons. In calculating the heat capacity, the multi-

mag 40 plicity of degeneracy of the ground state of a rare-earth S ∆ SmB ion was taken into account; Clat is the additional lattice 6 contribution to the heat capacity of unstable valence 10 0 10 20 30 40 50 compounds, which arises as a result of renormalization E, meV of phonon frequencies in unstable valence compounds Cf compared to isostructural materials. This latter contri- bution was obtained as a difference between the lattice 5 contribution to the heat capacity of an unstable valence ∆ compound and the corresponding contribution to the Clat heat capacity of an isostructural analog containing no Cg 4f electrons. In the case of SmB6, the density of phonon states was obtained from the experimentally measured 0 10 20 30 40 50 dispersion curves [18] on the basis of model calcula- tions which take into account the contribution by the T, K excitonÐphonon interaction. For CeNi, we used the Fig. 1. The temperature dependence of the experimentally phonon density of states obtained from experiments in obtained anomalous contribution to the heat capacity of ᭹ SmB6 ( ) [14] and the calculated partial contributions. The inelastic neutron scattering [25]; the value of Cg is the contribution to the heat capacity, which is due to the dashed line indicates the contribution associated with the 4f-electron states (Cf). The dotted line indicates the addi- gap in the electron density of states in the vicinity of the ∆ tional lattice contribution ( Clat). The dot-and-dash line Fermi level. This contribution was obtained for indicates the contribution describing the electron gap in the Sm1 − xLaxB6 compounds as the difference between the vicinity of the Fermi energy (Cg). The solid line indicates total experimentally obtained anomalous contribution the total calculated contribution to the anomalous heat capacity of SmB . Shown in the inset is the low-energy part to the heat capacity and the calculated contributions Cf 6 and ∆C . of the magnetic response for SmB6 [24]. The bold arrow lat ≈ indicates the low-energy excitation at Eex 14 meV, and the The corresponding partial contributions to the coef- thin arrow indicates the strongly broadened spinÐorbit tran- 2+ ≈ ficient of thermal expansion of unstable valence com- sition J0 J1 for Sm (E 36 meV). α pounds ( i) were determined on the basis of the Grue- neisen relation by fitting the Grueneisen coefficients Γ × Ð4 3 Ð1 ( i). In the calculation, the possible temperature depen- CeNi (1.02 10 cm J ) were borrowed from [26] dence of the Grueneisen coefficients was ignored. The and [27], respectively. total anomalous contribution to the coefficient of ther- mal expansion (∆α) was obtained as the sum of sepa- rate partial contributions, 3. RESULTS AND DISCUSSION All of the calculations and analysis were based on ∆α α ∆α α ()T = f ()T ++lat()T g()T the experimental information obtained by us over a (2) number of years for spectra of 4f-electron and lattice Ð1 []Γ Γ ∆ () Γ = V XT f C f ()T ++lat* Clat T gCg()T , excitations [14, 18, 19, 24, 25] and detailed data on the temperature dependences of the coefficient of thermal expansion for Sm La B and Ce La Ni [14, 16] α 1 Ð x x 6 1 Ð x x where f is the contribution defined by the 4f-electron and of the heat capacity for Sm La B [14]. ∆α 1 Ð x x 6 excitation spectrum, lat is the additional lattice con- tribution associated with the transformation of phonon α spectra, g is the contribution by the gap in the density 3.1. Sm La B Compound of electron states in the vicinity of the Fermi energy, V 1 Ð x x 6 is the molar volume, and XT is the isothermal compress- We will examine the possible components (or reasons ibility. For the compounds treated by us, the molar vol- for) the anomalous heat capacity of SmB6. Figure 1 ume and isothermal compressibility depend little on gives the temperature dependences of the experimen- temperature. In the calculation, their values were taken tally observed anomalous contribution to the heat to be constant parameters. The values of isothermal capacity of SmB6 [14] in the low-temperature region, × Ð4 3 Ð1 compressibility for SmB6 (0.125 10 cm J ) and which was already analyzed in the literature [2, 17], as

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–1 Clat, J (mol K) trum of 4f-electron excitations [24] (Fig. 1, inset). The main special features of the 4f-electron spectrum of SmB include the great width of spinÐorbit excitation 6 6 SmB6 LaB6 (J0 J1) and the presence of additional narrow low- 100 4 energy excitation (marked by arrow in the inset in Fig. 1) associated with the formation, at low tempera-

, rel. units 2 ture, of a new ground, quantum-mechanically mixed D state of samarium ion. As the temperature rises, the 0 2.0 2.5 3.0 intensity of low-energy excitation decreases; however, a quasi-elastic component arises. Such a spectrum of ω, THz 4f-electron states and its transformation with tempera- 50 ture produce additionally a significant contribution Cf SmB6 LaB6 to the anomalous heat capacity of SmB6 (Fig. 1).

After the sum of two components (Clat + Cf) is sub- tracted from the experimentally determined anomalous ≈ heat capacity of SmB6, the part with a maximum at T 30 K remains in ∆C(T); this latter part is apparently 0 100 200 300 associated with the effect of excitation of electrons via T, K the gap in the spectrum of electron states in the vicinity of the Fermi energy (Cg). As was mentioned in the Fig. 2. The temperature dependence of the lattice contribu- Introduction, it has now been reliably found that two tion to the heat capacity for SmB6 (solid line) and LaB6 (dashed line), calculated from the phonon density of states energy scales are characteristic of SmB6, namely, the (see the text). Shown in the inset is the phonon density of hybridization gap and “impurity” band [21]. If a two- states of acoustic branches in SmB6 (solid line) and LaB6 level model is used to evaluate the gap from the temper- (dashed line). ature dependence Cg(T), a value on the order of 60 K is obtained. This value is close to the scale of hybridiza- tion effects (in this case, one must take into account the well as the calculated (total and partial) contributions to possible temperature dependence of the gap size the heat capacity of SmB6. proper). The absence of a marked effect of the “impu- The phonon dispersion curves of SmB significantly rity” band on the thermodynamic properties is appar- 6 ently due, first, to the low density of states in this band differ from those of its isostructural analog LaB6 [18]. and, second, to the thermal dissociation of the “impu- These differences are associated with the presence of rity” states at T ~ 15 K [23]. resonant interaction of normal lattice vibrations with dipole (fÐd) and monopole (fÐf) excitations of the Therefore, the formation of an anomalous contribu- 4f shell of Sm; i.e., it would appear that the electronÐ tion to the heat capacity of samarium hexaboride is phonon interaction is superimposed on the phonon associated with three special features of this system, spectrum of LaB6 and distorts this spectrum by virtue of namely, the strong electronÐphonon interaction, the the nonadiabaticity of the electron subsystem that specific spectrum of 4f-electron states, and the exist- arises upon formation of the unstable valence state. As ence of a gap in the density of electron states in the a result, the density of phonon states of SmB6 shift to vicinity of the Fermi energy. In accordance with the the region of lower frequencies with respect to LaB6 Grueneisen relation given by Eq. (2), the anomalous (Fig. 2, inset). This cannot but reflect on the tempera- contribution to the coefficient of thermal expansion of ture behavior of the heat capacity. Figure 2 gives the SmB6 must also have three components. Figure 3a calculated lattice components of heat capacity Clat for gives the anomalous contribution to the experimentally SmB6 and LaB6. One can see that the value of heat obtained coefficient of thermal expansion of SmB6 [14] capacity Clat for SmB6 markedly exceeds that for LaB6 and the calculated components of the coefficient of in a wide temperature range. Consequently, because of thermal expansion. One can see that the overall calcu- the strong electronÐphonon interaction and correspond- lated coefficient of thermal expansion reproduces the ing renormalization of the phonon frequencies, the main special features of the temperature dependence of unstable valence state of Sm brings about the emer- the experimentally determined anomalous contribution gence of an additional contribution to the lattice heat to the coefficient of thermal expansion. However, the ∆ capacity ( Clat) at T < 300 K in SmB6 (compared to temperature dependence curve of the calculated coeffi- cient of thermal expansion is smoother than that of the LaB6), which represents an appreciable fraction of the entire anomalous heat capacity of SmB (Fig. 1). experimentally obtained coefficient. This is possibly 6 associated with the somewhat simplified approach Along with singularities in the atomic vibration employed by us. It was assumed in the calculation that spectrum, samarium hexaboride has a specific spec- the Grueneisen coefficient was temperature-indepen-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 THE THERMODYNAMIC PROPERTIES AND SPECIAL FEATURES OF SPECTRA 1117 dent. Apparently, when treating systems with a strong ∆α, 10–6 K–1 interaction of the electron and lattice subsystems, one must take into account some temperature dependence SmB6 (a) of the Grueneisen coefficient. 0 α f The calculated negative values of partial Grueneisen α coefficients (Table 1) are experimentally validated. The g ∆α Γ lat information about the sign of may be provided, for –2 example, by experiments performed under pressure, because Γ ∝ d(lnE)/dP (where E is the characteristic energy of the subsystem and P is the pressure). Investi- gations of the electric resistance of samarium –4 hexaboride as a function of pressure revealed a linear decrease in the gap size with increasing pressure [28]. Γ Consequently, the Grueneisen coefficient g must be Sm0.78La0.22B6 (b) 0 negative. α g Neutron experiments performed under pressure α f have revealed that, at Eex = 14 meV (the inset in Fig. 1), –2 the exciton-like magnetic excitation observed under ∆α normal external pressure shifts to the region of lower –4 lat energies (Eex = 7 meV at P = 7 GPa) [29]. This indicates Γ –6 that the value of f is negative. The situation is somewhat more complicated in the –8 case of the Grueneisen coefficient for the additional lat- tice contribution. No measurements have been per- Sm La B (c) formed up to now of the phonon dispersion curves of 0.5 0.5 6 0 SmB under pressure. Therefore, we restrict ourselves α 6 g α f to qualitative considerations with respect to the possi- –2 bility of emergence of a negative value of Γ* . In nor- lat ∆α mal cubic systems with integral valence, a decrease in –4 lat the volume of a unit cell leads to an increase in the phonon frequencies. In SmB6, which is characterized –6 by partly delocalized 4f electrons, a softening of the photon spectrum has been experimentally observed as –8 compared to LaB6 [18], in spite of the significantly 0 100 200 smaller volume of a unit cell of samarium hexaboride. Because, when a minor external pressure is applied, T, K samarium ions become more “intermediately valent” [29] Fig. 3. The temperature dependence of the experimentally (the valence of samarium ions under pressure is obtained anomalous contribution to the coefficient of ther- + mal expansion for Sm La B [14] and the calculated par- vSm 3 [30]), it is logical to assume that this results 1 Ð x x 6 tial contributions: (a) x = 0, (b) x = 0.22, (c) x = 0.5. The dot- in a still further softening of the phonon frequencies in ∆α SmB . That is, the negative value of the Grueneisen ted line indicates the additional lattice contribution ( lat). 6 The dashed line indicates the contribution associated with Γ the 4f-electron states (α ). The dot-and-dash line indicates coefficient lat* does not contradict the physical patterns f the contribution describing the electron gap in the vicinity discussed above for this compound. In order to arrive at α of the Fermi energy ( g). The solid line indicates the total Γ a final conclusion with respect to the sign of lat* , one calculated contribution to the coefficient of thermal expan- sion of Sm La B . needs to perform direct measurements of phonon dis- 1 Ð x x 6 persion curves under pressure. Analysis of the anomalous coefficient of thermal perature variation of the valence of a Sm ion [2]. At T > expansion of SmB has revealed (Fig. 3a) that, at a low 6 100 K, the significant negative contribution is largely temperature (T < 100 K), coefficient ∆α(T) is largely associated with special features of the 4f-electron spec- defined by the strong electron–phonon interaction. trum and with the gap in the density of electron states Note that the reasons for the emergence of the addi- in the vicinity of the Fermi energy. Previously, the tional component of the coefficient of thermal expan- anomalous contribution to the coefficient of thermal sion at T > 100 K were not understood heretofore, expansion at T < 100 K was interpreted either as a result because the valence of samarium ions at temperatures of the presence of the gap [17] or as a result of the tem- above 110 K remains a constant quantity [11], and the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1118 NEFEDOVA et al. α Table 1. The Grueneisen coefficients of partial components and to its weakening. Finally, the g(T) component of the anomalous contribution to the coefficient of thermal remains in dilute samples, which is largely due to the expansion for Sm1 Ð xLaxB6 compounds gap in the vicinity of the Fermi energy, by analogy with SmB6. This fact must be given special attention, Sample Γ Γ* Γ f lat g because it is generally taken that, when SmB6 is doped with lanthanum, the gap in the spectrum of electron SmB6 Ð1.5 Ð0.4 Ð1.5 states disappears even at a low concentration of La [2]. Sm La B Ð1.2 Ð0.9 Ð1.0 The results of our thermodynamic measurements [14] 0.78 0.22 6 point to the existence of a gap in all dilute samples up Sm0.5La0.5B6 Ð1.0 Ð1.2 Ð1.0 to x = 0.5. The statement about the absence of a gap in dilute samples is based on the results of measurements of electric resistance alone and apparently fails to fully effect of the electron components is significantly reflect the microscopic properties, because the additional weakened. states introduced by lanthanum ions “shunt” the gap in We will treat SmB -based compounds for which the kinetic measurements. However, the form of the density 6 of states suffers no cardinal changes. Additional states valent state of samarium ion varies upon substitution of show up, for example, in the marked increase in the value Sm by La (Sm1 Ð xLaxB6). All ternary compounds are of the Sommerfeld coefficient at low temperatures [14]. likewise characterized by the presence of the additional ∆α negative contribution to the coefficient of thermal By and large, the values of cal(T) for Sm1 Ð xLaxB6 expansion (Figs. 3b and 3c); however, the main singu- are in adequate agreement with experiment. Conse- larity (a minimum of ∆α(T) at T < 100 K) shifts to the quently, the anomalous contribution to the coefficient region of higher temperatures. We will discuss the of thermal expansion for Sm1 Ð xLaxB6 has three compo- dependence ∆α(T) for dilute compounds from the nents. For dilute systems, the high-temperature anoma- standpoint of the importance of all components respon- lous negative contribution to the coefficient of thermal sible for the anomalous coefficient of thermal expan- expansion is largely defined by the electron–phonon sion in SmB6. An investigation of spectra of lattice interaction and, at T < 150 K, it is defined by the special excitations of Sm1 Ð xLaxB6 has revealed that all dilute features of the spectrum of states of 4f electrons. One compounds exhibit a general softening of acoustical can clearly see that the shift of position of the minimum ∆α phonons compared with LaB6; however, compared with of (T) at T < 100 K in dilute compounds relative to SmB is largely the result of qualitative variation of the SmB6, the variation of the phonon frequencies in those 6 dilute compounds is insignificant [14]. Therefore, the 4f-electron spectrum. exciton model suggested for the description of spectra Analysis of the temperature dependence of the coef- of lattice vibrations for SmB6 is apparently valid for ficient of thermal expansion in a wide temperature dilute systems as well. Consequently, the anomalous range for Sm La B compounds revealed the follow- coefficient of thermal expansion of dilute samples must 1 Ð x x 6 ∆α ing. For all of the treated compounds, the anomalous incorporate the component lat associated with the contribution ∆α(T) may be interpreted in view of the electronÐphonon interaction. Because the lanthanum main special features of these compounds, namely, the doping failed to bring about significant changes in the presence of a narrow gap in the spectrum of electron phonon spectrum of samarium hexaboride, the value of states in the vicinity of the Fermi energy and the unsta- ∆α lat in dilute systems remained the same as in SmB6 ble valence state of samarium ion, which affects the (Figs. 3b and 3c). parameters of spectra of magnetic and lattice excita- In Sm La B compounds, the component α (T) tions. It was found that, at low temperatures, the value 1 Ð x x 6 f ∆α due to the presence of 4f electrons significantly differs of (T) is largely defined by the contribution associ- ated with the 4f-electron spectrum and by the presence from the analogous component for SmB6; this is asso- ciated with qualitative changes in the 4f-electron exci- of a gap in the spectrum of electron states. The signifi- tation spectra. According to the data on the inelastic cant contribution to the coefficient of thermal expan- magnetic scattering of neutrons, the low-energy excita- sion and to the heat capacity at T > 80Ð100 K is associ- tion in La-substituted samples, which is associated with ated with the manifestation of electronÐphonon interac- the new ground state of samarium ions, is characterized tion, i.e., with the result of the effect of the unstable by different experimentally observed values of param- valence state of Sm ions on the lattice dynamics. eters such as the energy, intensity, and temperature dependence [29]. The substitution by La brings about 3.2. Ce La Ni Compound an increase in the energy of “low-energy excitation” 1 Ð x x observed in SmB6 to 25Ð30 meV; in so doing, the tem- Since CeNi is a metal, ∆C(T) and ∆α(T) will be perature dependence of the intensity also varies and determined only by the effect of the unstable valence becomes smoother. This leads to a marked shift of the state on the spectrum of magnetic and lattice excita- α f(T) maximum to the region of higher temperatures tions.

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We will treat the temperature dependence of the ∆C, J (mol K)–1 anomalous contribution to the heat capacity of CeNi. In CeNi, as in SmB , one can expect the emergence of a 6 CeNi 4 contribution associated with the renormalization of 10 phonon frequencies, because a strong softening of

rel. units 2 phonon vibrations in CeNi compared to LaNi was , experimentally observed recently [19]. In contrast to D SmB , the difference between the spectra of lattice 0 6 12345 vibrations of CeNi and LaNi leads to the emergence of ∆ ω, THz the heat capacity component Clat only at low tempera- 5 tures (Fig. 4). The reason for the difference between the ∆ additional lattice contributions Clat in CeNi and SmB6 is associated with the much “softer” vibrational spec- Cf trum of CeNi (Fig. 4, inset). The entire RNi spectrum ∆ fits the 0–5 THz range; the acoustic region in which the Clat softening is observed corresponds to the 0Ð2 THz range [24]. The total SmB6 spectrum extends up to 40 THz. 0 100 200 300 The inset in Fig. 2 shows only the acoustic region for T, K SmB . 6 Fig. 4. The temperature dependence of the experimentally The heat capacity component C due to the presence obtained anomalous contribution to the heat capacity of f CeNi (᭺) [15] and the calculated partial contributions. The of 4f electrons was calculated using the experimental dashed line indicates the contribution associated with the data on the 4f-electron spectrum in CeNi in view of the 4f-electron states (Cf). The dotted line indicates the addi- ∆α temperature dependence of the latter spectrum [24]. tional lattice contribution ( lat). The solid line indicates Figure 4 shows that the calculated heat capacity by and the total calculated contribution to the anomalous heat large enables one to describe the main special features capacity of CeNi. Shown in the inset is the generalized func- of the experimentally obtained heat capacity. The main tion of phonon states for CeNi (᭹) and LaNi (ᮀ) obtained from reason for the formation of an anomalous contribution experiments in inelastic neutron scattering [25]. to the heat capacity of CeNi was found to be due to the existence of an unusual spectrum of states of 4f elec- values. The positive value of Γ may be validated by the trons at a low temperature and to its rearrangement with f results of investigation of the spectrum of magnetic increasing temperature. However, the experimentally excitations with applied “chemical” pressure. Indeed, observed anomalous contribution to the heat capacity the substitution of Ce by La (the introduction of lantha- differs in magnitude from that calculated at T < 200 K. num ions into the CeNi lattice causes the latter to The value of entropy for the 4f-electron contribution Ð1 ≈ expand, which is equivalent to the application of “neg- (about 9 J (mol K) Rln3, where R is the gas con- ative” external pressure) brings about the shift of the 4f- stant) obtained upon integration of the calculated electron spectrum to the region of lower energies [25]. anomalous contribution to the heat capacity is much Γ* lower than the value of entropy determined by integra- In order to validate the positive sign of lat , additional tion of the experimentally obtained value in the same experiments are required to investigate phonon curves temperature range (about 13.5 J (mol K)Ð1 ≈ Rln5). This under pressure or with chemical substitution. fact indicates that, along with the treated contributions, One can see in Fig. 5a that, for CeNi, the great pos- this system is characterized by additional degrees of itive anomaly in ∆α(T) is largely defined by the spec- freedom which were disregarded in our calculations but trum of states of 4f electrons. This inference was could have resulted in changes of the internal energy of strongly supported by the results of calculation of the system. In particular, the presence of coherence in ∆α (T) for Ce1 Ð xLaxNi compounds in which the valence the rare-earth ion sublattice, most likely associated with + magnetic interaction, and its importance in the formation of a cerium ion approaches 3 as Ce is substituted by of low-temperature properties were observed during La. In studying the spectra of excitations of 4f electrons investigations of Ce1 Ð x(Y,La)xNi compounds [12, 16]. We will now turn to the anomalous contribution to Table 2. The Grueneisen coefficients of partial components ∆α of the anomalous contribution to the coefficient of thermal the coefficient of thermal expansion, (T), for CeNi expansion for Ce La Ni compounds (Fig. 5a). The anomalous contribution to the coefficient 1 Ð x x of thermal expansion of CeNi, ∆α(T), is adequately Sample Γ Γ* described in a wide temperature range in the case of the f lat values of the partial Grueneisen coefficient of Γ ~ 2 CeNi 2.0 2.0 (Table 2). Unlike SmB6-based compounds, the Grue- Γ Γ* Ce0.5La0.5Ni 1.5 1.0 neisen coefficients f and lat of CeNi have positive

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∆α, 10–5 K–1 contribution to the coefficient of thermal expansion, 2 which was discussed when analyzing CeNi, disappears

–1 in dilute compounds. Indeed, the description of the (a) ) 10 temperature dependence of the coefficient of thermal expansion is significantly improved in the case of a CeNi sr meV 1 dilute compound. Therefore, the main reason for the , ∆α strong variation in (T) in Ce1 Ð xLaxNi is due to the mag S modification of the spectrum of excitations of 4f elec- mbarn ( trons as a result of transition of a Ce ion from a state 5 0204060 with intermediate valence to a state with an almost E, meV localized magnetic moment. It will be recalled that the substitution with lanthanum in SmB6 did not cause the α f disappearance of either the calculated or the experi- mentally obtained extremum of the coefficient of ther- ∆α lat mal expansion. The unstable valence state of a Sm ion 0 is characteristic of all of the samarium compounds

–1 4 treated by us [14]. 4 (b) In view of the foregoing, the temperature depen- dence of the coefficient of thermal expansion of Ce0.5La0.5Ni Ce − La Ni is largely defined by the spectrum of exci- 2 1 x x tations of 4f electrons. At low temperatures, however, one cannot but take into account the component associ- , mbarn (sr meV) ated with the special features of the spectrum of lattice mag 2 S excitations. 0204060 E, meV α 4. CONCLUSIONS f The anomalous contribution to the temperature ∆α lat dependence of the thermodynamic quantities of Ce- 0 100 200 300 and Sm-based unstable valence systems in a wide tem- perature range may be described in view of the actual T, K special features of the excitation spectra of the electron Fig. 5. The temperature dependence of the experimentally and phonon subsystems. It has been found that the rea- obtained anomalous contribution to the coefficient of ther- son for the emergence of a low-temperature (T < 80 K) mal expansion for Ce1 Ð xLaxNi [16] and the calculated par- anomaly of the heat capacity and of the coefficient of tial contributions: (a) x = 0, (b) x = 0.5. The dotted line indi- thermal expansion of SmB is due to the specific spec- ∆α 6 cates the additional lattice contribution ( lat). The dashed trum of 4f-electron states and to the presence of a gap line indicates the contribution associated with the 4f-elec- α in the electron density of states in the vicinity of the tron states ( f). The solid line indicates the total calculated contribution to the coefficient of thermal expansion. The Fermi energy. At T > 100 K, a significant negative insets show the magnetic response for CeNi and anomaly of the coefficient of thermal expansion arises Ce0.5La0.5Ni. The lines in the insets indicate the result of fit- because of the strong electronÐphonon interaction and ting [25]. associated variation of the phonon frequencies. The renormalization of the phonon frequencies in CeNi rel- ative to LaNi leads to the emergence of a marked com- using the method of inelastic neutron scattering [25], it ponent at a low temperature. The anomalous contribu- was observed that a qualitative transformation of the 4f- tion to the heat capacity and the great positive anoma- electron spectrum occurs when CeNi is doped with lan- lous contribution to the coefficient of thermal thanum (insets in Figs. 5a and 5b). The reason for this expansion of CeNi are largely defined by the unusual transformation is associated with the transition of Ce spectrum of 4f electrons. The general and main reason ions from a state with intermediate valence to a state for the strong modification of the temperature depen- with an almost localized magnetic moment (Kondo dence of the coefficient of thermal expansion for Sm- state). One can see that the calculated contribution and Ce-based compounds consists in the transforma- ∆α (T) for the Ce0.5La0.5Ni compound agrees well with tion of the spectrum of 4f-electron states. the experimentally observed anomalous contribution; this calculated contribution both allows for the shift of the maximum to the region of lower temperatures and ACKNOWLEDGMENTS reflects a gradual decrease in this maximum (Fig. 5b). We are grateful to A.S. Mishchenko, K.A. Kikoin, The dilution results in the violation of coherence in the V.G. Vaks, A.A. Chernyshov, E.S. Klement’ev, and rare-earth ion sublattice; i.e., the possible additional J.-M. Mignot for valuable discussions.

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This study was supported by the Russian Founda- 16. V. N. Lazukov, P. A. Alekseev, E. S. Klement’ev, et al., tion for Basic Research (project nos. 02-02-16521 and Zh. Éksp. Teor. Fiz. 113, 1731 (1998) [JETP 86, 943 03-02-17467), the Leading Scientific Schools Program, (1998)]. and the NIKS Russian State Program. 17. D. Mandrus, J. L. Sarrao, A. Lacerda, et al., Phys. Rev. B 49, 16809 (1994). 18. P. A. Alekseev, A. S. Ivanov, B. Dorner, et al., Europhys. REFERENCES Lett. 10, 457 (1989). 1. J. M. Lawrence, P. S. Riseborough, and R. D. Parks, Rep. 19. E. S. Clementyev, M. Braden, V. N. Lazukov, et al., Progr. Phys. 44, 24 (1981). Physica B (Amsterdam) 259Ð261, 42 (1999). 2. P. Wachter, in Handbook on the Physics and Chemistry 20. B. Gorshunov, N. Sluchanko, A. Volkov, et al., Phys. of Rare Earth, Ed. by K. A. Gschneidner, Jr., L. Eyring, Rev. B 59, 1808 (1999). G. H. Lander, and G. R. Choppin (North-Holland, Amsterdam, 1994), Vol. 19, p. 177. 21. N. E. Sluchanko, V. V. Glushkov, B. P. Gorshunov, et al., 3. H.-G. Purwins, Ann. Phys. 7, 329 (1972). Phys. Rev. B 61, 9906 (2000). 4. E. Müller-Hartmann, Solid State Commun. 31, 113 22. S. Gabáni, K. Flachbart, E. Konovalova, et al., Solid (1979). State Commun. 117, 641 (2001). 5. N. E. Bickers, D. L. Cox, and J. W. Wilkins, Phys. Rev. 23. S. Curnoe and K. A. Kikoin, Phys. Rev. B 61, 15714 B 36, 2036 (1987). (2000). 6. A. S. Edelstein and N. C. Koon, Solid State Commun. 24. P. A. Alekseev, V. N. Lazukov, J.-M. Mignot, and 48, 269 (1983). I. P. Sadikov, Physica B (Amsterdam) 281Ð282, 34 7. R. Pott, R. Schefzyk, D. Wohlleben, et al., Z. Phys. B: (2000). Condens. Matter 44, 17 (1981). 25. V. N. Lazukov, P. A. Alekseev, E. S. Clementyev, et al., 8. G. Creuzet and D. Gignoux, Phys. Rev. B 33, 515 Europhys. Lett. 33 (2), 141 (1996). (1986). 26. S. Nakamura, T. Goto, M. Kasaya, and S. Kunii, J. Phys. 9. A. Lacerda, A. de Visser, P. Haen, et al., Phys. Rev. B 40, Jpn. 60, 4311 (1991). 8759 (1989). 27. D. Gignoux, C. Vettier, and J. Voiron, J. Magn. Magn. 10. R. Takke, M. Niksch, W. Assmus, et al., Z. Phys. B: Con- Mat. 70, 388 (1987). dens. Matter 44, 33 (1981). 28. I. V. Berman, N. B. Brant, V. V. Moshchalkov, et al., 11. J. M. Tarascon, Y. Isikawa, B. Chevalier, et al., J. Phys. Pis’ma Zh. Éksp. Teor. Fiz. 38, 393 (1983) [JETP Lett. (Paris) 41, 1135 (1980). 38, 477 (1983)]. 12. V. N. Lazukov, E. V. Nefedova, V. V. Sikolenko, et al., 29. P. A. Alekseev, J.-M. Mignot, V. N. Lazukov, et al., Fiz. Met. Metalloved. 93, 61 (2002). J. Solid State Chem. 133, 230 (1997). 13. J. M. Tarascon, Y. Isikawa, B. Chevalier, et al., J. Phys. (Paris) 41, 1141 (1980). 30. J. Röhler, in Handbook on the Physics and Chemistry of Rare Earth, Ed. by K. A. Gschneidner, Jr., L. Eyring, and 14. E. V. Nefedova, P. A. Alekseev, E. S. Klement’ev, et al., S. Hüfner (North-Holland, Amsterdam, 1987), Vol. 10, Zh. Éksp. Teor. Fiz. 115, 1024 (1999) [JETP 88, 565 p. 453. (1999)]. 15. D. Gignoux, F. Givord, R. Lemaire, et al., J. Less-Com- mon Met. 94, 165 (1983). Translated by H. Bronstein

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Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1122Ð1123. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1276Ð1277. Original Russian Text Copyright © 2003 by Strakhov. SOLIDS Electronic Properties

Peculiarities of Emission of Subminiature Injection Lasers V. P. Strakhov Lebedev Physics Institute, Russian Academy of Sciences, Leninskiœ pr. 53, Moscow, 119991 Russia Received December 10, 2002

Abstract—The emission spectrum of an injection GaAs laser with a four-sided resonator with a square cross section of size 13 × 13 µm2 is presented. This laser is the world’s smallest laser, having the threshold current Ith = 0.7 mA and a photon flight time in the resonator that is shorter than the thermal relaxation time T2. It is shown that the emission spectrum of the laser drastically differs from the spectrum emitted by lasers of usual size. © 2003 MAIK “Nauka/Interperiodica”.

It has been shown in paper [1] that the spontaneous produced in the active region, resulting in the unusual emission of injection lasers with a four-sided resonator development of lasing. of small size (30 × 30 µm2) saturates above the lasing threshold, and single-frequency lasing lasts up to the The possibility of the appearance of two modes in tenfold excess over the lasing threshold. the spectra of semiconductor lasers was discussed in [2]; however, this cannot explain the result pre- The study was continued with lasers of even smaller sented above. I would be grateful to V.F. Elesin and size. Lasers of a square shape with a four-sided resona- Yu.V. Kopaev if they would explain this result theoreti- tor of size 13 × 13 µm2 were fabricated. It is important cally. that the length of such a resonator is L < 37 µm, and the photon flight time in the resonator is t < 5 × 10Ð13 s, which is shorter than the thermal relaxation time T2 in semiconductors at 77 K. The lasing spectrum shown in the figure drastically differs from typical single-frequency emission spectra obtained in [1]. Here, a is the spectrum observed when the current is slightly below the threshold; b and c are the lasing spectra at the threshold Ith = 0.7 mA recorded with wide and narrow slits of a spectrometer, respectively. Lasing appears at the threshold simultaneously on all equidistant axial modes in the long-wavelength part of the gain band and c b is observed up to two- to threefold excess over Ith; then, as a rule, the laser is destroyed. The width of the emis- a sion spectrum exceeds 100 Å, and the mode interval is ∆λ ≈ 21 Å, corresponding to the resonator length L ≈ 37 µm. Note that an amplification deficit exists for the longest wavelength modes, and no lasing should be 8620 observed at these wavelengths under usual conditions. Å The width of the emission spectrum of usual lasers is 8750 8500 8500 20–30 Å, while lasers with a four-sided resonator of a 8800 larger size (30 × 30 µm2) exhibit single-mode lasing. 8800 Only in the presence of a dominating short-wavelength mode developing for the time t < 10Ð12 s is an anomaly Emission spectrum of a subminiature laser.

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PECULIARITIES OF EMISSION OF SUBMINIATURE INJECTION LASERS 1123

Unfortunately, lasers of such a small size are of 10 µm2, which can only be achieved using advanced one-time use. Stresses produced upon mounting such nanotechnology. a miniature laser with a cross section as small as − 10 6 cm2 give rise to pressures up to thousands of REFERENCES atmospheres. For this reason, we failed to study the 1. P. G. Eliseev and V. P. Strakhov, Pis’ma Zh. Éksp. Teor. temporal parameters of the lasers, although the gener- Fiz. 16, 606 (1972) [JETP Lett. 16, 428 (1972)]. ation of controllable ultrashort light pulses is 2. V. F. Elesin and Yu. V. Kopaev, Zh. Éksp. Teor. Fiz. 72, expected. It is necessary to perform experiments at 334 (1977) [Sov. Phys. JETP 45, 177 (1977)]. room temperature, but because T2 at 300 K is even shorter, the resonator size should be smaller than 10 × Translated by M. Sapozhnikov

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Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1124Ð1130. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1278Ð1285. Original Russian Text Copyright © 2003 by Kuz’min, Ovchinnikov, Singh. SOLIDS Electronic Properties

Frustrated Antiferromagnetism in the Sr2YRuO6 Double Perovskite E. V. Kuz’mina, S. G. Ovchinnikova,*, and D. J. Singhb aKirenskiœ Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia bNaval Research Laboratory, Washington, DC, USA *e-mail: [email protected] Received December 9, 2002

5+ Abstract—The spins of Ru ions in Sr2YRuO6 form a face-centered cubic lattice with antiferromagnetic near- est neighbor interaction J ≈ 25 meV. The antiferromagnetic structure of the first type experimentally observed below the Néel temperature TN = 26 K corresponds to four frustrated spins of 12 nearest neighbors. In the Heisenberg model in the spin-wave approximation, the frustrations already cause instability of the antiferro- magnetic state at T = 0 K. This state is stabilized by weak anisotropy D or exchange interaction I with the next- Ð3 nearest neighbors. Low D/J ~ I/J ~ 10 values correspond to the experimental TN and sublattice magnetic moment values. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION Another reason for our interest in the double perovskite ≈ is the appearance of superconductivity with Tc 50 K Like cuprates and manganates, perovskite-like ruth- after doping it with copper [9, 10]. A study of a possible enates have been attracting much interest of researchers magnetic mechanism of superconductivity in this sys- in recent years. Initially, this interest was caused by the tem requires understanding the magnetic properties of discovery of exotic superconductivity in Sr RuO [1]. 2 4 undoped Sr2YRuO6. This is the only oxide superconductor isostructural to cuprates that does not contain copper. Later, it was Earlier, an attempt was made to explain the small- found that other ruthenates have very interesting mag- ness of TN by frustration effects in the Ising model, but netic and electric properties. Increasing x in the the suppression of TN in the Ising model proved to be Sr2 − xCaxRuO4 system results in a complex sequence of too weak [8]. In this paper, we show that the major con- structural phase transitions, competition between ferro- tribution is made by fluctuations of transverse spin magnetic and antiferromagnetic exchange interactions, components in the Heisenberg model. If only the near- and the MottÐHubbard metalÐdielectric transition in est neighbors are taken into account, the antiferromag- netic state is unstable in the spin-wave approximation. Ca2RuO4 [2, 3]. Another ruthenate, SrRuO3, is the only ≈ Its stabilization requires including exchange with the metallic ferromagnet with TC 165 K and magnetiza- ≈ µ next-nearest neighbors I or anisotropy D. Our calcula- tion m 1.6 B per Ru ion among 4d metal oxides [4, 5]. Ð3 The Sr YRuO double perovskite has an elpasolite tions show that very small I/J ~ D/J ~ 10 values are 2 6 sufficient for obtaining the observed T and magnetic structure, which can be obtained from SrRuO by N 3 moment values. replacing each second Ru ion with nonmagnetic Y; below TN = 26 K, the face-centered cubic (FCC) lattice of Ru5+ spins experiences ordering to produce an anti- 2. THE SPECIAL FEATURES ferromagnetic structure of the first type [6, 7]. In this OF THE STRUCTURE structure, (001) ferromagnetic planes exhibit antiparal- AND EXCHANGE INTERACTION IN Sr2YRuO6 lel ordering along the c axis. As distinct from other ruthenates and cuprates, One of the reasons for our interest in the magnetic neighboring RuO6 octahedra in Sr2YRuO6 do not share properties of Sr2YRuO6 is its low TN temperature and anions (Fig. 1). This justifies applying the cluster small value of the sublattice magnetic moment per approach to the description of its magnetic structure. µ ruthenium ion, m = 1.85 B, compared with the Similarly, the electronic structure of Sr2YRuO6 is well exchange integral J ≈ 25 meV and the nominal m(S = modeled in first-principles band calculations by a sys- µ 3 5+ 3/2) = 3 B per Ru ion for the d configuration of Ru . tem of RuO6 clusters, which form an FCC lattice [8]. The m value was measured by neutron diffraction [6, 7], From the magnetic point of view, the replacement of and the J value was calculated theoretically [8]. Ru5+ magnetic by Y3+ nonmagnetic ions is diamagnetic

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FRUSTRATED ANTIFERROMAGNETISM IN THE Sr2YRuO6 DOUBLE PEROVSKITE 1125 substitution. The FCC lattice of spins in Sr2YRuO6 can therefore be treated as produced by diamagnetic dilu- tion of spins in the SrRuO3 perovskite to a 0.5 concen- tration of nonmagnetic vacancies, which are spatially ordered (Fig. 1b). The presence of vacancies consider- ably changes the exchange interaction between neigh- boring Ru spins. Whereas ferromagnetic exchange interaction is characteristic of SrRuO3, strong competi- tion between ferromagnetic and antiferromagnetic inter- (a) (b) actions is observed in Sr2RuO4 [11], Sr2 Ð xCaxRuO4 exhibits a trend toward antiferromagnetism as x Fig. 1. Ordered diamagnetic replacement of every second increases (see discussion in review [12]), and Ru ion by Y ion in (a) SrRuO3 leads to (b) the Sr2YRuO6 × ᭺ ᮀ Sr2YRuO6 is characterized by strong antiferromagnetic lattice: ( ) Ru; ( ) O; ( ) Y. interaction. It follows that exchange interactions in var- ious ruthenates vary to a greater extent than in cuprates, The order of the levels is determined by the conditions where these interactions are always antiferromagnetic. E ()T ≈ E ()E <<

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1126 KUZ’MIN et al. The FCC lattice contains z = 12 nearest neighbors 3. THE SPIN-WAVE THEORY with the exchange J(R1) = ÐJ. We also take into account OF A FRUSTRATED ANTIFERROMAGNET exchange with the next-nearest spins J(R2) = I on the ON AN FCC LATTICE assumption of ferromagnetic exchange. Exchange for ប + the next-nearest spins arises as the RuÐOÐOÐRuÐOÐOÐ The exact equation of motion ( = 1) for S f is lin- Ru coupling; it can be estimated as earized in the Tyablikov approximation: ∼ τ4 ∆3 ≤ Ð2 I σ/ 10 J0. ú+ ≈ ()〈〉z + 〈〉z + iSf ∑ J()R SfR+ Sf Ð Sf SfR+ . (4) In order to describe the antiferromagnetic state, we R introduce sublattices A (sites a) with spins upward The h = H/zJ dimensionless Hamiltonian can conve- 〈〉z 〈〉z ≡ and B (sites b) with spins downward, SA = ÐSB niently be used. For the antiferromagnetic state of the S, where the magnetization of the sublattices depends first type, taking into account (3) then allows (4) to be on temperature. For the antiferromagnetic state of the written as (λ = I/J) first type, we have ferromagnetically ordered xy planes with an antiferromagnetic alternation of the planes. Set ú+ S ()+ + ()+ + ≡ iSa = --- ∑ Sa + d Ð Sa + ∑ Sad+ + Sa lattice parameter a = 1; the length of the R1 D vectors z d d connecting the nearest neighbors is then ∆ = 1/2 , and ≡ λ that of the R2 a vectors, a = 1. Let us divide the D vec- S ()+ + + ------∑ Sa Ð Sa + a , tors into two groups, those lying in the xy planes d and 2z2 interplanar vectors d, a (5) 1 1 + S + + + + d = ±---,,±--- 0 ()xy , iSúb = Ð--- ∑()S + S + ∑()S + S 2 2 z b + d b bd+ b d d 1 1 1 1 ,,± ± () ± ,,± () λS + + d1 ==0 ------yz , d2 --- 0 --- xz . () 2 2 2 2 Ð ------∑ Sb Ð Sb + a . 2z2 a 〈〉z The distribution of the S f means in this model is as Performing the Fourier transform over the sublattices follows: + + z z z ()⋅ 〈〉 〈〉 〈〉 SA()q = 2/NS∑ a exp iq a , Sa ===S, Sa + d S, Sad+ ÐS, a 〈〉z 〈〉z 〈〉z Sa + a ==S, Sβ ÐS, Sb + d =ÐS, (3) + + ()⋅ 〈〉z 〈〉z SB()q = 2/NS∑ b exp iq b , Sbd+ ==S, Sb + a ÐS. b Because of the ferromagnetic order in the xy plane, all we obtain four antiferromagnetic bonds in this plane are frus- trated (energetically unfavorable). Eight interplanar ú+ α + β + iSA()q = S(),qSA()q + qSB()q antiferromagnetic bonds, however, give energy gain for (6) + + + the antiferromagnetic state. For this reason, frustrations iSúB()q = ÐS(),α S ()q + β S ()q decrease the mean field acting on a spin even in the q B q A molecular field approximation. Without frustrations, where α ()λ()γ the mean field is h = 2JS = 12JS ; taking frustrations q = 0.33 1 + cxcy + 0.5 1 Ð q , into account makes it h = 4JS . Without frustrations in β () q = 0.33 cx + cy cz, MF the mean-field approximation, T N = zJS(S + 1)/3, () ,, ci ==cos qi/2 , i xyz, which is much higher than the experimental TN value. A γ () decrease in TN by a factor of 3 caused by frustrations in q = 0.33 cosqx ++cosqy cosqz . the mean-field approximation does not solve the prob- lem. A similar result is obtained for the Ising model, The thermodynamic properties will be calculated using where frustrations decrease T [13]. The T value (700Ð two-time retarded commutator Green’s functions at N N finite temperatures 900 K [8]) is, however, as previously, high compared with the experimental one. In the next section, we con- 〈〉〈〉S+()q SÐ ()Ðq = G ().q, ω sider the spin-wave theory of a frustrated antiferromag- F G ω FG net to take into account transverse spin component fluc- Here, sublattice indices F and G take on two values, A tuations. and B. For simplicity, we will only consider the spin

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 FRUSTRATED ANTIFERROMAGNETISM IN THE Sr2YRuO6 DOUBLE PEROVSKITE 1127 S = 1/2. Of course, spin-wave theory also can be con- x 0), Eq. (11) yields the Néel temperature structed for an arbitrary spin S, including S = 3/2 for α λ Sr YRuO . Such a theory will, however, be fairly cum- 1 Ð1 () 2 6 τ ==---I (),λ I ()λ 2N ∑------q -. (13) bersome, whereas the main results for S and T will N 4 2 2 ε2 λ N q q() differ by unimportant multipliers of the S(S + 1) type. Equations of motion (6) allow us to easily obtain the Consider the integrands in the expressions for I1 and I2 corresponding Green’s functions in the neighborhood of the Brillouin zone points Γ = (0, 0, 0) and X = (0, 0, 2π). In the neighborhood of Γ, ()ω α ()ω α 2S + S q 2S Ð S q we have GAA ==------, GBB ------, D()q, ω D()q, ω 2 2 2 1 q + q λq 2 2 2 2 (7) α ≈ --- x y 2β q 2 Ð,------+ ------q = qx ++qy qz , 2S q 3 8 4 GAB ==GBA Ð------, D()q, ω 2 2 2 2 2 1 q + q q 2 q + λq β ≈ --- 2 Ð,------x y Ð -----z ε = ------z - , ω ω2 Ω2 Ω ε q 3 8 4 q 9 Dq()==Ð q, q S q, (8) ε ()α2 λ β2 1/2 and the integrand in I1 takes the form q = q()Ð q . α ()λ 2 Ð ()q2 + q2 /8 + λq2/4 Applying the standard procedure yields the spectral ------q - = ------x y -. (14) ε λ 2 2 1/2 density q() ()λ qz + q 1 n ()q, ω = Ð---ImG ()q, ω If only exchange J between the nearest neighbors is AA π AA taken into account, the λ = 0 spectrum in the vicinity of Γ α becomes one-dimensional with the special direction q δω() Ω z, along which A and B layer spins alternate. At λ = 0, = S 1 + -----ε Ð q q the integral I1 logarithmically diverges, which means and the transverse spin correlator that S (0) 0; that is, the antiferromagnetic state is already unstable at T = 0. The integral I2 in the vicinity 〈〉+ Ð Γ 2 ∝ CAA()q = SA()q SA()Ðq of behaves as ∫dqz/qz 1/q, that is, diverges by a ∞ power law. As a result, T 0. In the vicinity of X, exp()ω/τ N = ------n ()q, ω dω the I and I integrals exhibit similar behaviors. It fol- ∫ exp()ω/τ Ð 1 AA (9) 1 2 Ð∞ lows that, if only nearest neighbor exchange J is taken into account, the effect of frustrations is strong to the α Ω q q extent that the antiferromagnetic state is completely = S1 + -----ε coth------τ . q 2 suppressed. Precisely this is, in our view, the main rea- τ son why TN and S are small in Sr2YRuO6. The antifer- Here, = T/zJ is the dimensionless temperature. For romagnetic state can be stabilized both by exchange S = 1/2, with the next-nearest spins I and by anisotropy.

2 2 + Ð 1 ----∑C ()q ==----∑ 〈〉SαSα --- + S, (10) N AA N 2 4. THE STABILIZATION q α OF ANTIFERROMAGNETIC STATES BY NEXT-NEAREST-NEIGHBOR EXCHANGE and the equation for the order parameter S therefore reads The instability of the antiferromagnetic state in FCC lattices has long been known and treated within the α τ ε frameworks of both the spin-wave approach and the τ 1/2 τ 2 q S() q BeteÐPeierlsÐWeiss cluster approximation [14Ð16]. S()==------τ , I() ----∑-----ε coth------τ . (11) I() N q 2 The stabilization of the antiferromagnetic state by next- q nearest-neighbor exchange has been considered in At τ = 0, the hyperbolic cotangent equals one, and detail in [17, 18]. Ferromagnetic exchange I stabilizes the antiferromagnetic phase of the first type, which is α λ observed in Sr2YRuO6, and antiferromagnetic 0.5 λ Ð1 q() S()0 ==------λ , I1() 2N ∑------ε λ-. (12) exchange I stabilizes the phase of the third type. The I1() q() q Néel temperature as a function of the λ = I/J ratio was λ calculated in [17, 18] only numerically, and the TN( ) τ τ In the other limit N, S 0 (cothx 1/x as plots with a characteristic nonanalytic dependence for

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kz ferromagnetic state of the first type. This conclusion is at variance with our results. Indeed, anisotropy creates a gap in the spectrum of magnons that cuts off the diver- gences as λ tends to 0. This problem is considered in more detail in the next section.

5. THE STABILIZATION OF THE ANTIFERROMAGNETIC STATE BY ANISOTROPY The turns of the octahedra and the monoclinic dis- tortion of the Sr2YRuO6 lattice can cause anisotropy of 2 two types, namely, single-ion anisotropy of the DSz kx ky type or exchange coupling anisotropy. In our simplified model with S = 1/2, the single-ion anisotropy is absent; therefore, consider the exchange anisotropy. The Fig. 2. Brillouin zone of a face-centered cubic lattice. Hamiltonian of the system can then be written as Squares indicate dangerous directions leading to magnetic moment and Néel temperature divergences. 1 ()+ Ð ξ z z ξ ≠ H = Ð---∑ JR()S f S fR+ + RS f S fR+ , R 1. – 2 S f , R 0.5 Equation (4) now transforms into

+ z z z + iSúf ≈ ∑ J()ξR ()〈〉S S Ð 〈〉S S . 0.4 R fR+ f f fR+ R In the simplest situation, it suffices to take into account 0.3 exchange anisotropy for the nearest neighbors ignoring 0 0.02 0.04 0.06 0.08 0.10 exchange anisotropy for the next-nearest spins. This D implies that Fig. 3. Dependence of sublattice magnetic moment S on ξ ξ ∆ ==1 +1,D, a exchange anisotropy D. where D is the dimensionless anisotropy parameter. As T /J the lattice distortions are small, we can assume that N D Ӷ 1. The spin-wave theory described in Section 3 can 0.6 easily be generalized to systems with anisotropy. After α α the q q(D) renormalization, α λ, ()λ()γ 0.5 q(D )= 0.33 1 ++Dcxcy + 0.5 1 Ð d , (15) ε ()α2()βλ, 2 1/2 0.4 q()D = q D Ð q , all the other equations obtained in Section 3 remain 0.3 valid. The order parameter at T = 0 is

λ, 0.5 0.2 S()D = ------λ, -, I1()D (16) 0.1 2 α ()λ, D 0 0.02 0.04 0.06 0.08 0.10 λ, q I1()D = ----∑------ε λ, . N q()D D q Fig. 4. Dependence of the Néel temperature on exchange For the Néel temperature, we obtain anisotropy D. 0.25 τ ()λ, D = ------, N I ()λ, D λ 0 were similar to the T (D) plot (see Fig. 4 2 N (17) below). At the same time, it was claimed in [17, 18] that α λ, λ, 2 q()D anisotropy of exchange interactions per se, without tak- I2()D = ----∑------. N ε2 λ, ing exchange I into account, could not stabilize the anti- q q()D

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 FRUSTRATED ANTIFERROMAGNETISM IN THE Sr2YRuO6 DOUBLE PEROVSKITE 1129 λ If = 0 and D 0, both integrals I1 and I2 diverge, approximation variant. However, as distinct from the τ 〈〉z z which yields S = 0 and N = 0. It follows that anisotropy trivial Weiss mean field with JijSiSj Jij Si S j , per se, in the absence of next-nearest-neighbor spin which does not depend on the space dimension, the exchange, stabilizes the antiferromagnetic state in an Tyablikov approximation takes into account transverse FCC lattice. spin density fluctuations in the form of collective exci- To single out the diverging asymptotic functions, we tations, that is, spin waves. As a result, the Tyablikov analytically calculated the contributions to the integrals approximation reveals the absence of long-range order in the neighborhood of the dangerous Brillouin zone at finite temperatures in agreement with the exact Mer- points Γ and X (see Fig. 2). The high-symmetry points minÐWagner theorem [17, 18]. As long as there is long- will be denoted as follows: range magnetic order in the system and spin fluctua- tions at low temperatures can be described in terms of Γ(),000,, L ==()πππ,, , K ()3π/2,, 3π/2 0 spin waves, we can hope that the results obtained using the Tyablikov approximation will be in at least qualita- ,,π π,,π (),,π π Z(),002 W1(),02 W2 = 0 2 , tive agreement with experiment. Note that the stabilization of the antiferromagnetic ˜ ˜ X = () 2π,,00, W1 = () 2ππ,,0 , W2 = () 2π,,0 π , state of the first type takes place not at arbitrary signs of exchange and anisotropy, but only at ferromagnetic next-nearest-neighbor exchange I and anisotropy D > 0. (), π, ()π, π, (), ππ, Y = 0 2 0 , W1* = 2 0 , W2* = 0 2 . Indeed, ferromagnetic exchange I prevents frustrations and is intrasublattice. Conversely, antiferromagnetic Along several Brillouin zone directions shown by next-nearest-neighbor exchange would only strengthen ε λ squares in Fig. 2, q = 0 at D = = 0. Some points of the effect of frustrations. As far as anisotropy is con- this set are dangerous in the sense that the I1 and I2 inte- cerned, D > 0 is evidence of Ising-type anisotropy, J|| > grals diverge at D = λ = 0. Further, we will study the J⊥. In the limit D ∞, we can ignore transverse spin role played by anisotropy D on the assumption λ = 0. components and obtain the Ising model, for which frus- Consider the small volume v = (π/4)3 in the neigh- trations partially suppress the antiferromagnetic phase, borhood of Γ (recall that the total Brillouin zone vol- but TN and S remain finite [13]. At all D > 0 values, a ume is 32π3). All integrals normalized with respect to v gap appears in the spectrum of magnons, which is the will be denoted by ˜I . Expanding all cosines into series factor that stabilizes the antiferromagnetic phase. At ε and performing fairly simple calculations, we can ana- D < 0, the q(D) spectrum of magnons becomes imagi- lytically find the contributions that diverge as D 0. nary at certain wave vectors, which is evidence of anti- For instance, for I1, we obtain ferromagnetic phase instability. At D = 0, the antiferro- magnetic state with a long-range order is unstable and ˜ I1(D )≈ 0.5lnD Ð 1.9. is replaced by a state with a spin-liquid-type short- range order [19, 20]. ˜ For the integral I2 , analytic calculations give A comparison of our results with the experimental data on Sr YRuO should be performed bearing in ˜I (D )≈ 12/ D. 2 6 2 mind that the monoclinic distortion of the lattice and

Similar asymptotic behaviors (lnD for I1 and D for I2) the spin S = 3/2 can lead not only to exchange but also can also be obtained for the other dangerous Brillouin to single-ion anisotropy. The DzyaloshinskiÐMoriya zone points. As a result, we find SD() and T (D)/J. anisotropic exchange is also possible. Clearly, all aniso- N tropic interactions are weak compared with J, which λ The SD() and TN(D)/J dependences at = 0 are allows us to qualitatively compare our results with shown in Figs. 3 and 4. The curves labeled by squares experiment taking into account exchange anisotropy are described by the approximations with D Ӷ J only. It follows from (18) and (19) that, to × obtain TN = 30 K and J = 300 K, we must set D = 8 1/2 Ð4 SD()≈ ------, (18) 10 . This means that the exchange anisotropy J|| Ð J⊥ = 0.043lnD + 1.256 DJ = 0.24 K is exceedingly small. At such an anisotropy value, 4 D/1()+ 4 D ,0<

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1130 KUZ’MIN et al. non modes along several Brillouin zone directions. In 4. J. J. Randall and Ward, J. Am. Chem. Soc. 81, 2629 particular, in the vicinity of the Γ point, the spectrum (1959). becomes one-dimensional. For this reason, divergences 5. J. M. Longo, P. M. Raccah, and J. B. Goodenough, in spin-wave theory similar to divergences in low- J. Appl. Phys. 39, 1327 (1968). dimensional systems are not surprising. Very weak per- 6. P. D. Battle and W. J. Macklin, J. Solid State Chem. 54, turbations in the form of ferromagnetic next-nearest- 245 (1984). neighbor exchange or an Ising-type exchange anisot- 7. G. Cao, Y. Xin, C. S. Alexander, and J. E. Crow, Phys. ropy are sufficient for the antiferromagnetic state to be Rev. B 63, 184432 (2001). stabilized. 8. I. I. Mazin and D. J. Singh, Phys. Rev. B 56, 2556 (1997). 9. M. K. Wu, D. Y. Chen, F. Z. Chein, et al., Z. Phys. B 102, ACKNOWLEDGMENTS 37 (1997). One of the authors (S.G.O.) thanks the Naval 10. H. A. Blackstead, Abstract of the International Confer- Research Laboratory for financial support of his visit to ence on Modern Problems of Superconductivity, Yalta, the Department of the Theory of Complex Systems in Ukraine (2002). August 2001, where this work was initiated, and 11. I. I. Mazin and D. J. Singh, Phys. Rev. Lett. 82, 4324 I.I. Mazin and the other members of the department for (1999). hospitality. 12. S. G. Ovchinnikov, Usp. Fiz. Nauk 173, 27 (2003). This work was financially supported by the Russian 13. R. Liebmann, Statistical Mechanics of Periodic Frus- trated Ising Systems (Springer, Berlin, 1986). Foundation for Basic Research (project no. 00-02-16110), RFFI-KKFN “Enisei” (project no. 02-02-97705), 14. Y. Y. Li, Phys. Rev. 84, 721 (1951). INTAS (grant 01-0654), and the Quantum Macrophys- 15. J. M. Ziman, Proc. R. Soc. London, Ser. A 66, 89 (1953). ics Program of the Russian Academy of Sciences. 16. D. ter Haar and M. E. Lines, Philos. Trans. R. Soc. Lon- don, Ser. A 255, 1 (1962). 17. M. E. Lines, Proc. R. Soc. London 271, 105 (1963). REFERENCES 18. M. E. Lines, Phys. Rev. A 135, 1336 (1964). 1. Y. Maeno, H. Hasimoto, K. Yoshida, et al., Nature 372, 19. H. Shimara and S. Takida, J. Phys. Soc. Jpn. 60, 2394 532 (1994). (1991). 2. S. Nakatsnuji and Y. Maeno, Phys. Rev. Lett. 84, 2666 20. A. Barabanov and O. Starykh, J. Phys. Soc. Jpn. 61, 704 (2000). (1992). 3. S. Nakatsnuji and Y. Maeno, Phys. Rev. B 62, 6458 (2000). Translated by V. Sipachev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1131Ð1139. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1286Ð1296. Original Russian Text Copyright © 2003 by Ovchinnikova. SOLIDS Electronic Properties

Coexistence of Superconducting and Spiral Spin Orders: Models of Ruthenate M. Ya. Ovchinnikova Joint Institute of Chemical Physics, Russian Academy of Sciences, ul. Kosygina 4, Moscow, 117334 Russia e-mail: [email protected] Received November 19, 2002

Abstract—The effect of a spiral spin structure on superconducting (SC) pairing in a three-band Hubbard model related to Sr2RuO4 is analyzed in the mean-field approximation. Such a structure with incommensurate vector Q = 2π(1/3, 1/3) is the simplest one that removes the nesting instability of α and β bands. It is assumed that there is an intralayer pairing interaction between two types of neighbor sites, those with attraction in a singlet channel and with attraction in both two-singlet and triplet channels. In both cases, a mixed singletÐtriplet SC order is observed in the γ band: a d-wave singlet order is accompanied by the formation of p-wave triplet pairs − (k, −k − Q)↑↑ and (k, Ðk + Q)↓↓ with large total momenta +Q and the spin projections ±1 onto an axis perpen- dicular to the spin rotation plane of the spiral spin structure. Both the SC and normal states are states with bro- γ ken time-reversal symmetry. In contradiction to the experiment, the models give different scales of Tc for the band and for α and β bands. This fact shows that the models with intralayer interactions or with the spin struc- ture assumed are insufficient. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION symmetry of anisotropic thermal conductivity in Sr RuO in a magnetic field with the field vector lying The problem of interplay between superconducting 2 4 (SC) and spin orders still remains topical for systems in the RuO4 plane [9, 10]. However, the observed with strong electron correlations. Among such systems, anisotropy also agrees with the conventional d-wave a single-layer quasi-two-dimensional ruthenate attracts SC order. However, this assumption requires a new interpretation for the behavior of the Knight shift. considerable attention as a superconductor (Tc ~ 1.5 K) with a possible triplet type of pairing [1, 2]. One of Recently, models with pairing interaction of elec- arguments in favor of this type of pairing is the behavior trons in adjacent sites (including the interlayer interac- of the Knight shift [3]. It was also assumed that the tion of orbitals with xz and yz symmetries) have been pairing is determined by ferromagnetic (FM) fluctua- considered; the constants of these models were chosen tions that clearly manifest themselves in the parent FM to describe a simultaneous transition to the p-wave SC compound SrRuO3. The spin-triplet SC order parame- state in all three bands [18, 19]. In all SC state tests, it ∆ σ σ ∝ ter (OP) ss'(k) = (i 2 i)ss'di(k) with dz(k) kx + iky, sug- is usually assumed that the bands have already been gested in [4, 5], agrees with the fact that the Knight shift renormalized by correlations due to a strong on-site is invariant under the SC transition [3] and with an repulsion. Such a renormalization is necessary for increase in the rate of the muon spin relaxation matching the band widths obtained from LDA calcula- tions to those obtained from photoemission data. The observed for T < Tc [6]. Such an order parameter corre- sponds to a nodeless gap function on the quasi-two- renormalization mechanism is certainly determined by dimensional Fermi surface. These nodeless gaps are spin correlations or, in the static limit, by local spin naturally derived from weak-coupling theory [5]. How- structures in a system that should substantially affect ever, the experimentally observed power-law behavior, the SC order. Therefore, the study of the effect of the as T 0, of the specific heat, C(T) ∝ T2 [7]; the NMR spin order on SC pairing in the ruthenate models remains a topical problem. relaxation rate, T Ð1 ∝ T3 [8]; the thermal conductivity, 1 The situation with the SC order and magnetic prop- κ ∝ 2 (T) T [9, 10]; the penetration depth [11]; and the erties may be complicated if the normal state of the ultrasonic attenuation [12] points to the existence of RuO4 plane possesses a certain static or dynamic spin node lines in the SC gap. structure. In particular, spiral spin structures have In view of these results, other possible types of sym- recently been studied [20] as the simplest structures metry of the SC gap were discussed in [13Ð17]. In par- that describe an incommensurate peak in the spin sus- ticular, the f-wave symmetry of the gap with a horizon- ceptibility χ''(q, ω) for q ~ Q = 2π(1/3, 1/3) that was tal node plane was assumed in [13]. This type of sym- observed in inelastic neutron scattering (INS) [21, 22] metry is likely to support the observed fourfold and some features of the ARPES spectra [23] for

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1132 OVCHINNIKOVA

Sr2RuO4. The quasi-one-dimensional sheets of the done in [4]. Interaction (1) corresponds to the SC pair- Fermi surface for α and β valence bands with a full ing constants κs = 2V + J/2 and κt = 2V Ð 3J/2 in the sin- 〈 〉 population of four electrons per site in the RuO4 plane glet and triplet channels for every bond nm . The natu- are characterized by the nesting with q = Q [24, 25]. ral signs V > 0 and J > 0 are expected from the theory The spiral structure with q = Q removes the instability of strongly correlated systems. In single-band models, due to nesting simultaneously in two quasi-one-dimen- these signs correspond to κt > 0 and κs < 0 in the triplet sional bands and reduces the on-site interaction energy. and singlet channels. They correspond to attraction in The spiral structure is not the only structure that the singlet rather than in the triplet channel. In the removes this type of instability of the system. Periodic present paper, however, we consider models with either × t spin structures with a 3 3 unit cell in the RuO4 plane sign of the triplet constant κ . may also be responsible for the incommensurate peak 2. The electron structure of Sr2RuO4 is determined at q = Q observed in INS. However, the spiral state is by three almost independent bands α, β, and γ con- the simplest state that allows one to study the effect of structed from the d orbitals of Ru+2 of xz, yz, and xy the spin structure and umklapp processes on SC pair- symmetries and the corresponding combinations of the ing. In the mean-field approximation, the energy of the π orbitals of oxygen [24, 25]. A small hybridization of spiral state is indeed lower than the energies of para-, bands of xz and yz nature occurs only in their intersec- ferro-, and antiferromagnetic states [20]. The coexist- tion domain for k = ±k . According to [4, 5], the orbital ence of SC pairing and the spiral spin order, as well as x y the coexistence of antiferromagnetic (AF) and SC symmetry also significantly suppresses the interband orders in cuprates, remains an intriguing problem. scattering of Cooper pairs. Therefore, we study the SC These questions are of interest due to the facts that even order arising in each separate band and choose the most a normal state with a spiral spin structure is a state with active band from the viewpoint of SC instability. We do broken time-reversal symmetry and that a whole series not touch upon the interband scattering and the proxim- of new mixed SC states with a simultaneous formation ity phenomena discussed in [4]. of singlet and triplet pairs arises in the system. 3. In contrast to [4], we begin with a normal mean- The aim of the present paper is to study the possibil- field state with broken time-reversal symmetry, namely, ity of coexistence of a spiral spin order and supercon- the state with a local spiral spin structure characterized by a diagonal vector Q = 2π(1/3, 1/3). This is a normal ductivity in models related to Sr2RuO4. We consider models with pairing interactions of neighbor sites state with nonzero spin currents j↑ = Ðj↓ of opposite within the RuO plane. We will study the symmetry and directions for two spin polarizations that are perpendic- 4 ular to the spin rotation plane of the spiral structure. the interplay of triplet and singlet SC OPs. We will ↑ ↓ demonstrate that both types of pairs survive simulta- This means that electrons with polarizations or pre- neously in the SC state in the presence of a spiral spin dominantly occupy k states with k á Q < 0 or k á Q > 0, structure. In the models under test, one γ band is distin- respectively. According to [20], this leads to a polariza- guished as an active band with respect to the SC transi- tion asymmetry of the Fermi surface and may lead to tion. In this band, singlet d-wave pairs coexist with trip- the formation of a mixed tripletÐsinglet SC order in the let bands. Earlier [16], the possibility of a mixed SC system. order in Sr2RuO4 was presumed because the energies of states with different SC-order symmetries are close to 2. MEAN-FIELD APPROXIMATION each other. The SC order was described by the spin sus- IN A SPIRAL SPIN CONFIGURATION ceptibility with a peak at the incommensurate momen- tum. In contrast to [16], a spiral spin structure gives rise A three-band model of the RuO4 plane is described to microscopic mixing of d-wave singlet and p-wave by the Hamiltonian [25] triplet orders. First, we analyze models with large con- stants κ of pairing interactions. Then, we calculate the ⑀ † κ κ HTH==++U V, T ∑∑ νkcνkσcνkσ, phase curves Tc( ) for more realistic values of . νσ, k First, we have to stress several points. (2) 1. Supposing that the attraction between electrons 1 HU = ∑Unνn↑nνn↓ + ∑ U2---nνnnν'n Ð JSνnSν'n . has an electronic (correlation) nature, we simulate it by 4 the interaction n, ν ν' ≠ ν Here, ν = 1, 2, 3 (or α, β, γ) correspond to bands of xz, yz, and xy natures; ⑀ν and H are zero band energies VV= ∑ νnνnnνm + JνSνnSνm (1) , k U 〈〉ν, and on-cite interactions with the parameters defined nm in [25]. The interband interaction between two adjacent sites for each of the three bands ν = α, β, γ. This situation corresponds to taking into ()† consideration the lowest k harmonics in the momentum T αβ = ∑4tαβsinkx sinky c1kσc2kσ + H.c. representation of the pairing interaction Vkk', as was k, σ

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 COEXISTENCE OF SUPERCONDUCTING AND SPIRAL SPIN ORDERS 1133 is small. Thus, in the normal state with an arbitrary spin level. Therefore, one can expect that (any) spin struc- structure, there are three almost independent bands ture increases Tc as compared with the Tc in paramag- with weak mixing of α and β bands at the crossing point netic solutions. of their Fermi surfaces. We will neglect this mixing. For Since the Fermi surfaces are different for all three α β simplicity, we retain the notation and for unmixed bands, it suffices to take into account only the intraband bands of xz and yz natures. The interaction between anomalous means. Only these means are determined by bands occurs through the common value of the chemi- the contributions of a large phase volume along the cal potential and the mean fields produced by the elec- entire Fermi surface. The formation of electron pairs trons of all three bands. These fields depend on spins ()† † ν ≠ ν due to the on-site exchange interaction. The interaction ckν↑cÐkν'↓ that belong to different bands ' can be V〈nm〉 of type (1) of nearest neighbors is included in the effective only in a small domain of k near the intersec- simulation of the possible SC pairing in the system. tion of their Fermi surfaces. Therefore, we retain only In the mean-field approximation, the energy aver- anomalous means within each subband and neglect the aged over an arbitrary BCS-type state is calculated as interband scattering of pairs. Then, the expression for an explicit function the contribution HSC to the mean energy (3) has the form N() SC , θ HH= yi + H ()w j j (3) 1 SC 2 θ ----H = Uwν()0 that depends on normal (yi) and anomalous (wi, j) one- N (6) electron means. The normal OPs {yi} include on-site (l = 0) and bond (l = ex, ey) densities s 2 t 1 2 + ∑ κν()l wν()l + κν()l ∑ ------θµν, ()l . 1 + µ ν, , µ , ± ν 1 † le= x ey = 01 r ()l = ---∑c ,,νσc ,,νσ , 2 n nl+ θ σ Here, w(l) and (l) are singlet and triplet SC OPs either at the same site (l = 0) or at neighboring sites (l = ex or the mean kinetic energies l = ey). These quantities are given by the expressions ν ⑀ † T = 1/N∑ νkcνkσcνkσ , kσ 1 σ † † wν()l = ------∑------〈〉cν,,σcν,,σ (7) and local (l = 0) or nonlocal ((l = e , e ) spin densities 2N σ n nl+Ð x y n, σ dν(l) in each band ν. The local spin densities and ν []ν ∗ 〈〉iQn † d ()0 ==d ()0 e cnν↑cnν↓ (4) θνµ()l determine the spiral spin structure with the spirality (8) π 1 iµQn()+ l/2 † † vector Q = 2 (1/3, 1/3): ------∑e ()σµσ 〈〉cν,,cν,,. 2N y ss' ns nl+ s' 〈〉 [] n, σ Snσ = dν ex cosQn + ey sinQn . (5) σ σ +−()σ ± σ Previous calculations [20] have shown that the energy Here, the matrices µ are equal to z or x i y for µ ± σ of the normal state with a spin structure with such Q is = 0, 1, respectively, and x, y, z are Pauli matrices. lower than the energies of similar para-, ferro-, and The phases φ(n, l) = µQ(n + l/2) for µ = 0, ±1 in the def- antiferromagnetic mean-field solutions. Such a spin inition of the triplet OPs (8) guarantee that each term of structure removes the instability due to the nesting of α the sum in (8) for each bond 〈n, n + l〉 is independent of and β bands. Simultaneously, exchange fields also the bond number n. This is analogous to the situation 〈 ÐiµQn〉 µ ± induce a similar spin structure in the γ band. The solu- when cyclic spin components Snµe for = 1 are tions give collinear contributions to the total local spin independent of n for a state with a spiral spin structure. of the site from each band. These phases are associated with the existence of spin Three-band models that describe Sr RuO were currents in the spiral state. Below, we will show that the 2 4 ↑↑ ↓↓ µ µ derived in [24, 25]. In the case of the spiral spin struc- triplet Cooper pairs ( ) or ( ) with spin = 1 or = − ± ture, the normal state Fermi surfaces were studied 1 are moving pairs that carry large total momenta Q. in [20]. It was shown that, for a certain spin polariza- The constants κs, t of SC pairing in (6) are related to the tion, a spiral SDW order opens up a gap along half of constants V(l) and J(l) of interaction between neighbor- the unperturbed Fermi boundary of a paramagnetic ing sites in Eq. (1). solution. Other regions of the Fermi surface remain Let us take into account that the α and β bands are metallic (gapless). In addition, a number of new independent of the γ band. This is associated with the shadow Fermi boundaries arise that are attributed to fact that the symmetry of the α and β bands with respect umklapp processes. As a result, the bands are split into to the reflection about the ab plane of ruthenate bands the lower and upper Hubbard subbands; this increases is different from that of the γ band. For instance, a direct the densities of states in the subbands and on the Fermi interband mixing of these bands due to hopping within

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1134 OVCHINNIKOVA the RuO4 plane is forbidden. This property of bands channel. The second case corresponds to the attraction was discussed in [4, 17], where the authors also evalu- in both channels. ated a small interband interaction due to interlayer hop- To solve the problem, we apply the standard proce- ping. We will neglect interlayer interactions and will dure of mean-field approximation. From the explicit analyze a possible SC order in each band separately, as well as its symmetry and compatibility with the spiral dependence of the mean energy Hz()i on the one-parti- θ spin order. We will also neglect the interband scattering cle OPs zj = {yi, wi, i}, we obtain the linearized Hamil- of pairs and take into account only the intraband con- tonian stants κs and κt in (7). To reduce the number of SC OPs, ∂()µ we apply arguments typical for all strongly correlated µ H Ð N () µ HLin Ð N = ------∂ zˆi Ð zi + Hz()i Ð N, (11) systems. For any interaction V〈nm〉 in (1), a large on-site zi repulsion U > 0 suppresses the singlet s-wave OP w(0) γ in the band according to (6), so that, of the singlet where zˆi are operators corresponding the appropriate OPs, we retain only the d 2 2 wave OP; i.e., we set x Ð y means zi. The BCS-type state is an eigenstate of HLin; it allows one to calculate, in turn, the values of z . In this wγ(0) = 0 and wγ(ex) = Ðwγ(ey). This guarantees that the i pair function is orthogonal to the forbidden s-wave pair way, one obtains a self-consistent solution. function. For quasi-one-dimensional bands α and β For the state with a spiral spin structure, the most (here, α and β refer to the bands of xz and yx natures convenient basis set of the Nambu representation is a rather than to their combinations), the same on-site basis of the following Fermi operators for each band ν: interaction suppresses all singlet OPs, i.e., those of both s and d symmetries. As a result, we set wα β (0) = † † † ( ) b ν = {}cν ↑,,cν,,↓ cν,,()↑ ,cν,,↓ . (12) ± i k k kQ+ Ð kQ+ Ðk i wα(ex) = wβ(ey) = 0 because both combinations wα(ex) wα(ey) are nonorthogonal to the on-site pair function Here, i = 1, …, 4, and the quasimomentum k runs over w(0) for quasi-one-dimensional bands with a broken the domain F that constitutes half of the total momen- tetragonal symmetry. tum space and is bounded by the conditions In the BCS approximation, an interaction of type (1) ∈ ()< may induce an SC order only under the condition that kF: kQ+ /2 Q 0. (13) s t π some of the constants κν or κν in (6) are negative. One For a vector Q with Qx = Qy = 2 /3, Eq. (13) implies may assume that such an attraction is of correlational or that the components kx and ky range within the limits kinematic origin or is attributed to the hybrid nature of the site orbitals composed of the d and pπ orbitals of Qx Qx Ðπ Ð ------<

st() st() st() st() st() ⑀ µ ⑀ µ κ κ κ κ κ h11 ==ν()k1 Ð , h22 ν()k2 Ð , α ()ex ====β ()ey γ ()ex γ ()ey . (9) 1∂ ∂ By the same analogy with the tÐJ model, one could h12 = Ð--- H/ dν, expect that these constants have the following signs: 2 κs κt () (15) = Ð < 0. However, in view of the expected triplet h14 = Ak()1 + B0 k1 , type of the SC order in Sr2RuO4 [1], we extend the cal- culations to the two limiting cases of triplet constants of h23 ==ÐAk()2 + B0(),k2 h13() 24 B±1(),k different signs h33 ===Ðh22, h44 Ðh11, h34 Ðh12, κs κt < κs κt < I: ==Ð 0, II: 0. (10) where The first case corresponds to the attraction of particles () only in the singlet channel and repulsion in the triplet k1 ==k, k2 kQ+ , k =k1 + k2 /2

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 COEXISTENCE OF SUPERCONDUCTING AND SPIRAL SPIN ORDERS 1135 and the functions A and Bµ are given by The required BCS-type spiral state is determined by χ† the occupation of one-particle eigenstates λνk corre- s Ak()= ∑ κν()l wν()l coskl, sponding to the energy levels Eλ(k), le= , e x y (16) χ† † λνk = bikSiλ(),k t (19) µ κν θµν B ()k = ∑ ()l ()l sinkl. hij()k S jλ()k = Siλ()k Eλ().k , le= x ey Here, the omitted index ν is implied. The matrices For the γ band, the d symmetry of the singlet OP γ γ Siλ(k) of the eigenvectors and the Fermi populations requires the antisymmetry A (kx, ky) = ÐA (ky, kx) of f(Eλν ) of levels determine the normal and anomalous functions with respect to the replacement k k . k x y OPs (7) and (8). This completes the self-consistency The solution also yields identical values for the triplet procedure. OPs θµ for µ = ±1. Thus, we actually have only three real OPs for the γ band: (),,θθ ,, 3. RESULTS zi ==wd 0 i, i 123, 1 Since a complete solution with SC order is easily w = ---[]we()Ð we(), obtained for large pairing constants, we first consider d 2 x y models with large ks(t) and then calculate the phase 1 s s θ []θ θ curves Tc(k ) for more realistic models with small k 0 = --- 0()ex Ð 0()ey , 2 (17) and kt. The results are obtained for two types of models in (10), with attraction in the singlet channel alone θ 1 []θ θ (case I) and in both singlet and triplet channels (case II). = --- ∑ µ()eµ Ð µ()ey . 4 In the first case, the α and β bands do not display any µ = ±1 SC order. The reason is that the singlet d-wave order, We apply the same symmetry with respect to the just as the s-wave one, is suppressed by an on-site replacement x y to the solutions in the α and β repulsion in bands with inequivalent hopping integrals bands subject to the simultaneous replacement α α α β β in the x and y directions: t ӷ t or t ӷ t (see the β (xz yz). The solution yields identical values θµ x y y x for two projections µ = ±1 in the α and β bands as well. parameters of the three-band model [25]). In contrast to As a result, we retain the following triplet OPs for the the α and β bands, a mixed-type SC order arises in the α and β bands that correspond to a nonzero triplet con- γ band of a system with a spiral spin configuration. The stant in (9): d-wave singlet order is accompanied by the formation t of triplet pairs even for κγ > 0. Figure 1 shows the tem- θµα, ()e ==Ð01.θµβ, (),e µ , (18) x y perature dependence of the singlet and triplet OPs (17) If the initial values of the OP satisfy Eqs. (17) and (18), in the γ band for κs = Ðκt = Ð0.6 eV. The values of the θ θ ӷ subsequent iterations of the self-consistency procedure triplet OPs obtained satisfy the relation +1, γ = Ð1, γ preserve the same symmetry of the solution. θ 0, γ. Taking into account the definition of triplet OPs There was one more simplification. In fact, an inter- in (17) and (18) and their momentum representation, action of type (1) yields contributions to both parts one can conclude that coupled triplet pairs of particles γ ↑↑ ↓↓ H ()y and H of the mean energy (3). We may in the band arise mainly in the form ( ) or ( ) with N i SC the total quasimomenta ÐQ or Q, respectively. This fact assume that the first contribution, which depends on the distinguishes between triplet SC orders in the spiral normal means—the charge and spin densities—has state and in an isotropic Fermi liquid, where only Coo- already been taken into account in the renormalized per-type pairs (k ↑, Ðk ↑) or (k ↓, Ðk ↓) with zero total band energies ⑀ν(k) whose parameters were chosen ear- momentum are possible. Moreover, unlike the isotropic lier [25] for the correct description of the observed model, triplet pairs can arise in the state with the spin magnetic quantum oscillations. Thus, we retain only structure even for a positive triplet constant κt > 0, 〈 〉 the part of V in H that depends on anomalous means. which corresponds to repulsion in the triplet channel, The definitions of triplet OPs (8) allow one to deal with due to the coupling between singlet and triplet OPs. real solutions. They have a certain symmetry with respect to the reflection in the diagonal plane (z, x = y) The Cooper pairs with nonzero total momenta Q that contains the spirality vector Q under a simulta- were first predicted in the Fulde–Ferrel–Larkin– neous permutation of bands xz yz , as well as a Ovchinnikov (FFLO) states [26]. The latter have spatial symmetry with respect to the reflection in the plane (z, variations of the OPs and can actually exist only if the x = Ðy) (Q ÐQ) with the change σ Ðσ. scale of 2π/Q is much smaller than the coherence length

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1136 OVCHINNIKOVA acter and manifests itself within finite domains or finite 0.03 time intervals, the asymmetry phenomena are sup- pressed. 1 0.02 Thus, in states with the spiral structure due to the

SC OP nesting of α and β bands, the attraction solely in the sin- 0.01 2 glet channel gives rise to both singlet and triplet pairs. Figure 1 also represents the specific heat of the system. 3 The finite limit of C(T)/T as T 0 is attributed to the 0 contribution of the α and β bands that remain in the nor- mal state. This fact does not agree with the observed behavior of C(T)/T ∝ T as T 0. 10 T

/ Now, let us consider a model of the second type with C attraction in both channels: κs = κt < 0 in (10). For large 5 |κs| and |κt| and small T, an SC order arises in all three bands. For equal values of constants (9) on each bond with a large hopping integral, the SC order in the γ band α β 0 0.005 0.010 0.015 is more clearly pronounced than that in the and bands. For κs ≤ 0.65 eV and T ~ 10Ð3 eV, the SC order , eV T in the α and β bands vanishes. The ensemble of coupled Fig. 1. Total specific heat divided by T, C(T)/T, and super- pairs in the γ band mainly consists of d-wave singlet γ conducting order parameters in the band as functions of ↑ ↓ s temperature for a model with large interaction constants Cooper pairs {k , Ðk } and moving triplet pairs κs κt θ θ {k ↑, Ð(k + Q) ↑} and {k ↓, Ð(k Ð Q) ↓} with the total = Ð = Ð0.6 eV. Curves 1Ð3 refer to wd, Ð , and Ð 0, respectively, that are defined by Eqs. (17). momenta ÐQ or Q, respectively. In the α and β bands, only a triplet order is possible ξ. In our case of coexisting spiral spin and mixed SC for kt < 0. The corresponding transition temperatures αβ orders, only the anomalous triplet components γ t are significantly lower than Tc in the band: T c (k = γ  ks) Ӷ T . The triplet p-wave SC order in the α and β 〈〉† † ∝ i µ ()θ c cνnσcνmσ exp--- Qn+ m ν, 2σ, 2 bands is mainly attributed to the formation of triplet pairs {k ↑, Ðk ↓}t with zero total momenta. They corre- ± µ σ θα θβ ≠ mne==xy(), 2 , spond to the OP 0 ()ex = Ð0()ey 0. Figure 2 shows are characterized by the spatial phase modulation the temperature dependence of the SC OP in the α, β, according to Eq. (8). At the same time, the leading sin- and γ bands for large constants κs = κt = Ð0.8 eV. Note θ θ glet component wν in (7) is constant and is independent that the relation 0()l > ±1()l for the triplet compo- of the number of bond 〈nm〉. Therefore, there is no con- nents for l = e in the α and β bands differs from the ξ x(y) straint on the relation between and 1/Q in these solu- θ Ӷ θ γ tions. relation 0()l ±1()l for the OP in the band. The The emergence of coupled pairs with large total differences in Tc and the symmetry of the SC order in the α, β, and γ bands are associated with the quasi-one- momenta of 2kF, equal to the nesting vector, is also sub- stantiated in the new theory of high-temperature super- dimensional or quasi-two-dimensional character of the conductivity (HTSC) [27]. In [27], such pairs are bands and with different densities of states on the Fermi assumed to be singlet and are associated with the stripe level. structure. In our model, such moving pairs are triplet Thus, for models with intralayer interaction of and are synchronized by the static spin structure. Note neighboring sites, only the γ band is characterized by that a photoemission technique capable of distinguish- γ large Tc. Models with different scales of Tc in the band ing between the spin polarizations of photoelectrons α β could also distinguish between the polarization asym- and in the and bands are characterized by a two-step metries of photospectra. The latter are associated with behavior of the specific heat (Fig. 2). This fact contra- the spiral spin structure that breaks the time-reversal dicts the observed dependence C(T)/T which is indica- symmetry in both the normal and the SC state. In [20], it tive of the simultaneous SC transition in all three bands. is demonstrated that electrons from different segments of Nevertheless, the solutions obtained are instructive the Fermi boundary—with kQ < 0 (or kQ > 0)—are because they show the possibility of new mixed types characterized by different spin polarizations k ↑ (or of SC order that are compatible with the spiral spin k ↓). When the spin order has a local or dynamic char- order in a correlated system.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 COEXISTENCE OF SUPERCONDUCTING AND SPIRAL SPIN ORDERS 1137

0.06 1

0.04 1 SC OP 0.02 2 4 5 3 π

0 / y 0 k

10 T / C 5

0 0.02 0.04 –1 0 1 T, eV π kx/ Fig. 2. Same as in Fig. 1, but for a model with κs = Ðκt = Fig. 3. Contour map of the gap function (23) for the same −0.8 eV. Curves 1Ð3 refer to the same SC OPs for the γ band model as in Fig. 1 in the entire domain of the momentum α π θ space kxy() < . Solid (dashed) lines correspond to posi- as those in Fig. 1; curves 4 and 5 correspond to 0 ()ex = tive (negative) values of the gap function G(kx, ky) defined −θβ θα θβ 0()ey and ±1()ex = б1()ey . by Eq. (22).

γ θ Pairing potentials in the active band with the SC B2u) + 0(Eux Ð Euy)}. In contrast to the paramagnetic OPs (17) can be represented as state, in the spiral state, the coupled pairs (↑↑), (↓↓) are moving pairs with the total momenta +−Q . The spin cur-

SC † † rents j↑↑ = Ðj↓↓ associated with the motion of pairs have H = {[]Ak()+ B ()k c ↑c ↓ Lin ∑ 0 k Ðk the same sign as the spin currents j↑ = Ðj↓ in the normal (20) kG∈ state with the spiral spin structure. Recall that the spins † † † † are projected here onto an axis perpendicular to the spin + B ()k []c , ↑c , ↑ + c , ↓c , ↓ + H.c.}. 1 kQÐ /2 ÐkQÐ /2 kQ+ /2 ÐkQ+ /2 rotation plane in the spiral structure. Here, k runs over the entire region G of the phase space The SC band, as a function of k, which corresponds γ (in contrast to representation (14), where k runs over to the pairing potential (21) for the band, is deter- half of the entire region G), and the omitted band index mined by the real matrix element ν = γ is implied. The functions A(k) and Bµ(k) are η SC η defined by Eqs. (16) and are given by Gk()= Ðk H k (22)

κs () between electron and hole quasiparticles η†, η of the Ak()= wd coskx Ð cosky , (21) upper Hubbard band of the normal spiral state. κtθ () Bµ()k = µ sinkx Ð sinky . Figure 3 represents a level map of the gap function G(k) for the γ band for κs = Ðkt < 0. This function is anti- These functions have the following symmetry: symmetric with respect to the change kx k y but A(kx, ky) = ÐA(ky, kx) and B(kx, ky) = ÐB(ky, kx); hence, the does not possess the inversion symmetry. However, a diagonal kx = ky along the vector Q is a node line for the hypothetical photoemission experiment for the γ band SC gap in the solution with a mixed d-wave singlet and would give two different gaps |G(k)| and |G(Ðk)| for p-wave triplet SC order. In the case of an isotropic nor- every k for different polarizations (↑ and ↓) of photo- mal state without a spiral spin structure (dν = 0, Q = 0), electrons. Note that, for a quasimomentum k that varies the pairing potential (21) would correspond to the along the known Fermi boundary of the γ band, the superposition of contributions corresponding to differ- function G(k) is close to the d-wave function propor- ent representations of a tetragonal point group classi- tional to coskx Ð cosky. This fact corresponds to the four- fied in [5]. In the notation of [5], this superposition can fold anisotropy of thermal conductivity in a longitudi- θ schematically be represented as { 1(A1u Ð B1u Ð A2u + nal (in the ab plane) magnetic field [9, 10].

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1138 OVCHINNIKOVA of the upper and lower Hubbard subbands of the γ band. 1.0 At the SC-transition point T = Tc , where Eq. (23) is sat- 1 isfied, the solution of the corresponding homogeneous 0.5 equations

SC OP 2 ()δ κ γ 2 2 2 0 ij Ð Rij j z˜ j ==0, z˜ j z j / z1 ++z2 z3 (25) 3 –0.5 gives relative normalized values z˜ j of the SC OPs. Fig- ure 4 shows the phase curves T (κs) and the relative val- 0.06 κ = κ c t s ues of the OPs z˜ for T T as functions of κs for two κ = –κ j c t s t c signs of the triplet constant κ in (10). The symbols refer T 0.04 to the values of z˜ j for T ~ 0.6Tc obtained from full mean-field calculations for the models with large con- 0.02 stants |κs| discussed above. These values agree with those obtained from Eqs. (25). The models with realis- Ð4 tic Tc ~ 10 eV display the same symmetry properties 0 0.2 0.4 0.6 0.8 1.0 as the models with large κs and T . The transition tem- κ c s peratures Tc = 1.5 K observed in Sr2RuO4 correspond to κs κs κs the constants = Ð0.145 eV or = Ð0.12 eV, respec- Fig. 4. Phase curves Tc( ) and relative normalized values of the superconducting OPs (25) and (17) in the γ band as tively, for two types (10) of models. In the first case, functions of the interaction constants in the singlet channel. when kt = Ðks > 0, the SC transition occurs only in the γ Solid (dashed) curves refer to models with κt = +κs and κt = band. In the second case, when kt = ks < 0, transition −κs α β , respectively. Curves 1, 2, and 3 correspond to z˜i , i = 1, temperatures in the and bands are estimated to be −3 2, 3, obtained from the solution of homogeneous equa- Tc ~ 10 K. Actually, this means a normal metallic state tions (25) (T Tc). Symbols (circles and squares) cor- of the α and β bands for models of both types (10). respond to the same quantities at T = 0.6Tc obtained from the full mean-field solution for models represented in Figs. 1 and 2, respectively. 4. CONCLUSIONS Thus, models with intralayer pairing interactions of Up to now, the results referred to models with unre- neighboring sites admit the SC order only in the γ band. alistically large interaction constants and temperatures This fact does not agree with the situation in supercon- Tc. For more realistic models that correspond to small ducting Sr2RuO4. Nevertheless, the models considered Tc and constants (10) of both types, we calculated the are instructive in that they demonstrate the possibility κs κs phase curves Tc( ). The function Tc( ) for the of new mixed types of SC order when a correlated sys- SC-transition temperature in the γ band is determined tem possesses a spin structure. We have shown that the by the equation obtained by a linear expansion of SC order in the γ band may coexist with a spiral spin α β Eqs. (7) and (8) in terms of the arising SC OPs wγ and order due to the nesting of the and bands in s Sr RuO . The mixed d-wave singlet and p-wave triplet θλµ. This equation for T (κ ) has the form 2 4 c SC orders emerge from the pairing interactions of adja- det δ Ð0.R κ = (23) cent sites on the basis of the normal state with a spiral ij ij j structure with the nesting vector Q = 2π(1/3, 1/3). For two types of constants of pairing interactions—with Here, i and j number the SC OPs {zi} defined by attraction in both (singlet and triplet) channels or only Eq. (17), and the matrix Ri is given by in the singlet channel—the main coupled pairs in the s system are the singlet d-wave pairs ()kk, Ð ↑↓ and the fE()Ð λ Ð fE()λ 1 i j ' t Rij = ----∑∑Mλλ'Mλλ'------, (24) moving triplet pairs ()kQÐÐ/2, kQÐ /2 ↑↑ and N Eλ + Eλ' k λλ, ' t ()kQ+Ð/2, kQ+ /2 ↓↓ with large total quasimomenta where Eλ = Eλ(k) and f(Eλ) are the normal-state energies +−Q and the spin projections µ = ±1 onto an axis per- and the Fermi populations of one-electron levels, pendicular to the spin rotation plane of the spiral struc- respectively. The matrices Mi(k), i = 1, 2, 3, correspond- ture. The predominant d-wave SC order in the γ band is ing to the SC OPs (17) are given in the Appendix, while consistent with the observed fourfold anisotropy of κ κ κs κt κt the constants j in (23) are given by j = { , , }j. thermal conductivity in Sr2RuO4 in a longitudinal (in λ λ The indices , ' = 1, 2 number the normal-state levels the plane of RuO4) magnetic field [9, 10]. The problems

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 COEXISTENCE OF SUPERCONDUCTING AND SPIRAL SPIN ORDERS 1139 of SC order in the α and β bands and of the Knight shift 3. K. Ishida, H. Mukuda, Y. Kitaoka, et al., Nature 396, 658 behavior remain unsolved within the models consid- (1998). ered. The extension of the calculations to other periodic 4. M. E. Zhitomirsky and T. M. Rice, Phys. Rev. Lett. 87, spin structures requires the consideration of other, in 057001 (2001). particular, interlayer interactions. The calculation of the 5. M. Sigrist, D. Agterberg, A. Furusaki, et al., cond- behavior of the spin susceptibility under the SC transi- mat/9902214. tion should clear up the following question: Can the 6. K. M. Luke, Y. Fudamoto, K. M. Kojima, et al., Nature triplet pairs accompanying the d-wave SC order guar- 394, 558 (1998). antee the invariance of the Knight shift under the SC 7. S. Nishizaki, Y. Maeno, and Z. Mao, J. Low Temp. Phys. transition? 117, 1581 (1999); J. Phys. Soc. Jpn. 69, 572 (2000). 8. K. Ishida, H. Makuda, Y. Kitaoka, et al., Phys. Rev. Lett. 84, 5387 (2000). ACKNOWLEDGMENTS 9. M. A. Tanatar, M. Suzuki, S. Nagai, et al., Phys. Rev. I am grateful to A.A. Ovchinnikov† and V.Ya. Kriv- Lett. 86, 2649 (2001); M. A. Tanatar, S. Nagai, Z. Q. Mao, nov for useful remarks and help, as well as to R. Werner et al., Phys. Rev. B 63, 064505 (2001). for kindly submitting the preprints of [15] before their 10. K. Izawa, H. Takahashi, M. Yamaguchi, et al., Phys. Rev. publication. Lett. 86, 2653 (2001). 11. I. Bonalde, B. D. Yanoff, M. B. Salamon, et al., Phys. This work was supported by the Russian Foundation Rev. Lett. 85, 4775 (2000). for Basic Research, project nos. 00-03-32981 and 12. C. Lupien, W. A. MacFarlane, C. Proust, et al., Phys. 00-15-97334. Rev. Lett. 86, 5986 (2001); cond-mat/0101319. 13. H. Won and K. Maki, Europhys. Lett. 52, 427 (2000). APPENDIX 14. I. Eremin, D. Manske, C. Joas, and K. M. Bennemann, cond-mat/0102074. The matrices Mi in (24) are given by 15. R. Werner, Phys. Rev. B 67, 014505 (2003); Phys. Rev. B 67, 014506 (2003); R. Werner and V. J. Emery,  Phys. Rev. B 67, 014504 (2003). i cs i sc Mλλ'()k = M , (26) 16. M. Eschrig, J. Ferrer, and M. Fogelström, cond- scÐ csÐ λλ' mat/0101208; Phys. Rev. B 63, 220509 (2001). 17. D. F. Agterberg, T. M. Rice, and M. Sigrist, Phys. Rev.  Lett. 78, 3374 (1997). 1 0 c ()k M = d 1 , 18. J. F. Annett, G. Litak, B. L. Györffy, and K. I. Wysokin-  Ð0cd()k2 ski, Phys. Rev. B 66, 134514 (2002). 19. G. Litak, J. F. Annett, B. L. Györffy, and K. I. Wysokin-  ski, cond-mat/0203601. 2 0 s ()k M = p 1 , 20. A. A. Ovchinnikov and M. Ya. Ovchinnikova, cond-  Ð0sd()k2 mat/0201536; A. A. Ovchinnikov and M. Ya. Ovchinni- kova, Zh. Éksp. Teor. Fiz. 122, 101 (2002) [JETP 95, 87  (2002)]. 3 () 10 21. Y. Sidis, M. Braden, P. Bourges, et al., Phys. Rev. Lett. M = s p k . 01 83, 3320 (1999). 22. M. Braden, O. Friedt, Y. Sidis, et al., cond-mat/0107579. Here, s = sinϕ, c = cosϕ, and ϕ = ϕ(k) for band ν are deter- 23. A. Damascelli, D. H. Lu, K. M. Shen, et al., Phys. Rev. Lett. 85, 5194 (2000). mined by the equation tan()2ϕ = Ð[∂H /∂dν][⑀(k ) Ð 1 24. I. I. Mazin and D. J. Singh, Phys. Rev. Lett. 79, 733 ⑀ Ð1 (k2)] . Other functions are given by cd(k) = (coskx Ð (1997); Phys. Rev. Lett. 82, 4324 (1999). cosky)/2, sp(k) = (sinkx Ð sinky)/2; k1 = k, k2 = k + Q, and 25. A. Liebsch and A. Lichtenstein, Phys. Rev. Lett. 84, k = k + Q/2. 1591 (2000). 26. P. Fulde and R. A. Ferrell, Phys. Rev. A 135, 550 (1964); A. I. Larkin and Yu. N. Ovchinnikov, Zh. Éksp. Teor. Fiz. REFERENCES 47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1964)]. 27. V. I. Belyavsky and Yu. V. Kopaev, cond-mat/0203138; 1. Y. Maeno, T. M. Rice, and M. Sigrist, Phys. Today 54 V. I. Belyavskiœ and Yu. V. Kopaev, Zh. Éksp. Teor. Fiz. (1), 42 (2001). 121, 175 (2002) [JETP 94, 149 (2002)]; V. I. Belyavskiœ, 2. T. M. Rice and M. Sigrist, J. Phys.: Condens. Matter 7, V. V. Kopaev, and Yu. V. Kopaev, Zh. Éksp. Teor. Fiz. 118, L643 (1995). 941 (2000) [JETP 91, 817 (2000)].

† Deceased. Translated by I. Nikitin

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1140Ð1148. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1297Ð1307. Original Russian Text Copyright © 2003 by Isaev, Protogenov. SOLIDS Electronic Properties

Structures of Order Parameters in Inhomogeneous Phase States of Strongly Correlated Systems L. S. Isaev and A. P. Protogenov* Nizhni Novgorod State University, Nizhni Novgorod, 603950 Russia Institute of Applied Physics, Russian Academy of Sciences, Nizhni Novgorod, 603950 Russia *e-mail: [email protected]

Abstract—The structures of order parameters which determine the bounds of the phase states within the frame- work of the CP1 GinzburgÐLandau model are considered. Using the formulation of this model [1] in terms of the gauged order parameters (the unit vector field n, density ρ2 and momentum c of particles), we found that some universal properties of phases and field configurations are determined by the Hopf invariant Q and its gen- eralizations. At a sufficiently high level of doping, it is found that, outside the superconducting phase, the charge distributions in the form of loops may be more preferable than those in the form of stripes. It is shown that, in phase with its mutual linking number L < Q, the transition to an inhomogeneous superconducting state with nonzero total momentum of pairs takes place. A universal mechanism of breaking of the topological coherence of the superconducting state due to a decrease of the charge density is discussed. © 2003 MAIK “Nauka/Inter- periodica”.

1. INTRODUCTION order parameter is a spinor realizing a two-dimensional Among the challenging problems of cooperative representation of the braid group which arises due to phenomena in planar systems near Mott transition, classification of the quantum states upon permutations there are such that, at first glance, may not be associated of particles in (2 + 1)-dimensional systems. Consider- ing factorization with respect to the center of this non- with the appearance of high-temperature superconduct- 1 ing states in doped antiferromagnetic insulators. For Abelian group, we obtain the gauged CP GinzburgÐ example, we are interested in the origins of qualita- Landau model. Note that the order parameter of this tively similar cooperative behavior in various com- model is two-dimensional [1, 11]. Only in this case can pounds and very rich content of their phase diagram, as we introduce the unit vector field which describes the well as in origin of the emergence of inhomogeneous distribution of one-half spin degrees of freedom in the states typical of such systems [2Ð6]. Low-dimensional long-wavelength limit as well as use the Hopf invariant structures in the distribution of spin [2, 3] and charge for classifying n-field configurations and consider cor- [4Ð6] degrees of freedom exist in the state which pre- rectly the phases with different distribution of charge cedes a high-temperature superconducting phase. degrees of freedom. For the above reasons, we use the Keeping this property in mind [7], we should choose a generalized n-field model which, after the exact map- model for describing the aforementioned phases that ping of the CP1 GinzburgÐLandau model [1], includes contains them as limiting cases. The mean-field Gin- the Faddeev term [12]. Because of the non-Abelian zburgÐLandau theory with the appropriate choice of the gauge theory origin of this significant part of the model, order parameters may be used for understanding gen- we hope that the obtained answers are universal and eral problems of such kind. A key question in this uni- will give a deeper insight into the problems under con- versal approach is the method with the aid of which the sideration. order parameters encode simultaneously the content The Hopf invariant describes the degree of linking and the distribution of charge and spin degrees of free- or knotting of the filament manifolds where the field of dom of excitations in various phase states. the unit vector n is defined. The study of the behavior The recent progress in solving analogous problems of the vortex filament tangle is a separate problem and in the non-Abelian field theory [8] and its development attracts attention for several reasons. First, at small dis- in the physics of condensed matter [1] showed that the tances, the topological order associated with the linking CP1 GinzburgÐLandau model is preferable. The two- exists against the background of the disorder caused by component order parameter of this model is used for arbitrary motion of separate parts of the system of solving the problems of two-gap superconductivity [9]. entangled vortex filaments. Thus, unlike point particles, In the theory of electroweak interaction [10], this the properties of the tangle are determined by the parameter has the sense of the Higgs doublet of the behavior of its fragments in the ultraviolet and infrared standard model. In this paper, we will assume that the limits. Since the coordinates of the vortex core are

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STRUCTURES OF ORDER PARAMETERS IN INHOMOGENEOUS PHASE STATES 1141 canonically conjugated, the indicated circumstances used before [1, 6, 9] in the context of two-gap supercon- are manifested by noncommutativity of these variables ductivity, as well as in the standard model of the non- depending on the degree of linking. Second, a soft Abelian field theory [8, 10]. In the present paper, we medium such as the tangle of linked filaments is a hot consider the states in planar systems and suppose that problem in the physics condensed matter and beyond, Ψ has the sense of the order parameter realizing two- in particular, in connection with the DNA problem. dimensional non-Abelian representations of the braid When the coupling constants are transformed into the group used for classifying quantum states upon permu- sought functions, the appearance of a double helix as tations of particles in systems with two spatial dimen- the solution of the equations of motion in the soft vari- sions. Realizing Abelian projection [20], the vector Ak ant [13] of the n-field model is its general property. compensates the local choice of the phase of the func- Ψ In the present paper, we consider some properties of tion . The last terms in (1) describe the GinzburgÐ Landau potential V(Ψ , Ψ ) and the self-energy of the the field configuration in the CP1 GinzburgÐLandau 1 2 model defined in the following section. The main goal gauge field. of the paper is to find the bounds of free energy in the It has been shown recently [1] that there is an exact superconducting state and in the inhomogeneous phase mapping of the model (1), (2) into the following version with broken antiferromagnetic order, as well as to of the n-field model: describe the properties of the charge density distribu- tions corresponding to this state. Considering the 3 1 2 2 2 Fd= x ---ρ ()∂ n + ()∂ ρ nonsuperconducting phase in the soft version of the ∫ 4 k k model [13], we analyze the contribution to the free (3) energy from the charge density distributions in the form 1 2 2 2 + ------ρ c ++.()F Ð H V()ρ, n of loops and stripes. In the third and fourth sections, 16 ik ik 3 we discuss, along with the results from brief publica- tions [14Ð16], the properties of the inhomogeneous superconducting state with nonzero total momentum of The free energy in Eq. (3) is defined by a scalar—the ρ2 a pairs and compare this with the LOFF states [17, 18] density of particles , the field of the unit vector n = χ σaχ χ χ χ σa and with the results from the recently proposed [19] (where = (1* , 2* ), and is the Pauli BCS-like model with two types of particles. We also matrix), and the field of the momentum c = J/ρ2 = 2(j Ð pay attention to the dependence of the bounds of phase 4A). The total current J contains the paramagnetic part states on the generalized (2 + 1)D Hopf invariant in the j = i[(χ ∇χ* Ð c.c.) + (χ ∇χ* Ð c.c.)] and the diamag- case of S2 × S1 S2 and S1 × S1 × S1 S2 mapping 1 1 2 2 netic term Ð4A. Equation (3) is written with the use of classes and on the external magnetic field. In the Con- ∂ ∂ ∂ × clusions, we discuss some open problems. The Appen- the following notation: Fik = ick Ð kci, Hik = n á [ in ∂ ξ dix gives the proof of the inequality which determines kn], and dimensionless units of the length L = ( 1 + the relation between the contributions to the free energy ξ ξ ប 2)/2 (with the coherence length a = /2mbα ), the of n- and c-field configurations. momentum ប/L (as the unit of momentum c), the parti- cle density c2/(512πe2L2) (per unit mass for the param- 2. CP1 GINZBURGÐLANDAU MODEL etrization of Ψα in the form (2)), and the energy γ/L with γ = (cប/e)2/512π. We will use the GinzburgÐLandau model In formulation (3), the GinzburgÐLandau functional 2 2 depends on gauged order parameters ρ , c, and n. They 3 1 2e Fd= x ∑------ប∂ + i------A Ψα characterize spatial distributions of the charge and spin ∫ 2m k c k α degrees of freedom with or without current. The func- (1) tions χα determine the orientation of the unit vector n 2 which describes (in the long-wavelength limit) the 2 cα 4 B + ∑ Ð bα Ψα + ----- Ψα + ------properties of the magnetic order. In addition, functions 2 8π α χα define the value of the paramagnetic part of the cur- rent. Comparing different forms of representation of with a two-component order parameter, the CP1 GinzburgÐLandau model, we note that vortex field configurations Ψα in the model (1), (2) are equiv- ϕ i α alent to textures of the field n in terms of the model (3). Ψα ==2mρχα, χα χα e , (2) We must also note that the ansatz (2) has the sense of factorization of the longitudinal ρ and transversal χα |χ |2 |χ |2 which satisfies the condition 1 + 2 = 1. This cou- degrees of freedom. In the superconducting state, the χ pling of two components α takes place in the complex composition of spin j and charge degrees of freedom is projective space CP1, for which the given model is important, since the current contains diamagnetic U(1) defined. The model (1), (2) with different masses was gauge component Ð4A.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1142 ISAEV, PROTOGENOV In the soft variant of the extended model of n field (3), linearity of the orientation of neighboring spins in the the multipliers of the first term describe the distribu- following way: tions of spin stiffness and the square of the inverse ⋅∂[]∂ × ∂ ≡ ∂ length of the density screening. It is seen from this Hik = n in kn iak Ð kai. example that the competition of the order parameters ρ, ∝ | |3/4 The dependence Fmin Q (5) with the boundary n, and c may account for the coexistence of the phase conditions n (0, 0, 1) in the space infinity was ver- states with different ordering of charge and spin ified in [22–24] in simulation of the n-field configura- degrees of freedom. We enumerate the limiting cases of tions. Such a boundary condition means that the space the model (3) in inhomogeneous (n ≠ const) situations: ޒ3 of the n-field definition effectively compacts into a (1) a state with broken antiferromagnetic order: c = three-dimensional sphere S3. Thus, the unit vector n 0, ρ = const; accomplishes the mapping of the S3 sphere into the (2) a state with quasi-one-dimensional density dis- 2 ρ ≠ space of the two-dimensional sphere S . Let the vector tributions: c = 0, const; n be directed to some general point of a two-dimen- (3) an inhomogeneous superconducting state: c ≠ 0, sional sphere. We are interested in answering the fol- ρ = const; lowing question: what is the pullback of this point in the (4) c ≠ 0, ρ ≠ const. space S3 or, in other words, what set of points from the In the case of n = const and c ≠ 0, ρ ≠ const, func- domain of definition of the vector n(x, y, z) contributes tional (1) is equivalent to the one-component Ginz- to a given point of the target two-dimensional sphere? burgÐLandau model. Since the space S3 is compact and its dimensionality is greater by unity than that of the sphere S2, the pullbacks 2 3. THE BOUNDS OF THE FREE ENERGY of points on the sphere S are closed and, in the general case, linked lines on the S3 sphere. 3.1. A Phase State with Broken Antiferromagnetic Order The Hopf invariant Q (6) describes the degree of linking or knotting of these lines. It belongs to a set of Let us consider the first case in the above list. In this integers ޚ to which the considered homotopy group limit, the free energy is π 2 ޚ 3(S ) = is equal. In particular, for two once-linked Fd= 3 xg[]()∂ n 2 + g ()n ⋅ []∂ n × ∂ n 2 . (4) circles, Q = 1; for one of the simplest knots (a trefoil), ∫ 1 k 2 i k Q = 6; etc. As a result, the n-field configurations are We supposed that, in the phase under consideration, divided into classes corresponding to the values of the ρ ρ Hopf invariant. We should emphasize once again that the constant value = 0 can be found from the mini- mum of the potential V, and we introduced the notation linked or knotted configurations may be numbered by the Hopf index only in the case of the CP1 GinzburgÐ gi for the coupling constants. The properties of the model (4) were studied in detail in [21Ð27]. The analy- Landau model with its two-component order parame- π M sis of the dimensionality shows that the first term in ter, because at M > 1 the homotopy group 3(CP ) = 0 Eq. (4) is proportional to the characteristic size RQ of is trivial [11]. the n-field configurations, and the second term is inversely proportional to this scale. Therefore, the 3.2. Quasi-One-Dimensional Density Distributions energy (4) has a minimum which is achieved at RQ = Let us consider the states outside the superconduct- g2/g1 . This explains why the second term in Fad- ing phase from the second line of the list of limiting deevÐNiemi model (4) allows us to avoid Derrik’s cases, to which the CP1 GinzburgÐLandau model leads. restriction of the existence of three-dimensional static In this soft version of the model, the functional (3) has configurations with finite size. In the infrared limit, this the form term characterizes the mean degree of noncollinearity 〈 | × | 〉 3 0 S1 á [S2 S3] 0 in the orientation of three spins Fd= ∫ x located at the sites of the quadratic plaquette. (7) 1 2 2 2 2 2 d 4 It was shown [25Ð27] that the lower energy bound in × ---ρ ()∂ n ++()∂ ρ H Ð bρ +.---ρ the model (4) 4 k k ik 2 F ≥ 32π2 Q 3/4 (5) In Eq. (7), the positive constant b corresponds to the phase with broken antiferromagnetic order. is determined by the Hopf invariant The state with the broken antiferromagnetic order 1 3 ε ∂ considered above has a lower energy than the “soft” Q = ------∫d x iklai kal. (6) 16π2 state we are interested in now. The latter may be meta- stable [28]. In this section, we will consider only such In this equation, the vector ai denotes the gauge poten- states and compare their contribution to the GinzburgÐ tial which parametrizes the mean degree of the noncol- Landau energy without studying the problems of their

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 STRUCTURES OF ORDER PARAMETERS IN INHOMOGENEOUS PHASE STATES 1143 ρ relaxation, the critical sizes of nuclei of different Here 0 = b/d , r0 is the ring radius, R is its width, π phases, etc., which are of separate interest. 2Ly = 2 r0 is the stripe length, and Lx = R is its width. Under the condition that the electron spin and Since configurations (8) and (9) do not depend on the charge are transferred from one of four sites of some third coordinate, we will assume that the size in this plaquettes to the dopant reservoir, the terms with Hik in direction is limited by the length Lz and also that R < r0. Eq. (1) characterize (in the infrared limit) the mean The calculation of energy (7) with the aid of (8) and 〈 | × | 〉 degree of noncollinearity 0 S1 á [S2 S3] 0 in the ori- (9) yields the following results for the contribution to Ӷ entation of three spins, which remain in the sites of a the free energy from the ring Fr (at R r0) and the quadratic lattice plaquettes. Therewith, the deficit of the stripe Fxy: ρ2 ρ2 charge density h relates to the density , describing 2 2 r0R the distribution of the exchange integral in (1), by the F = πρ L ---- 1 + ----- , (10) r 0 z Rξ2 ρ2 ρ2 relation + h = const. From the long-wavelength 2 2 3 point of view, the distribution of the spin density ρ in 2 r0 R R 3 R F = πρ L ---- 1 +++------n Ð ---b ----- .(11) a limited region with an exponential law of decrease at xy 0 z R ξ2 r 0 4 r the boundary (for example, for a distribution in a circle 0 0 with radius r0 with an exponential decrease over a Here, length R Ӷ r ) will be accompanied by a quasi-one- 0 r ==πr ,1/ξ2 2[]n Ð ()11/8Ð b , ρ2 0 0 0 dimensional distribution of the charge density h along the boundary of this region, i.e., along a ring with thick- n0 is a certain characteristic value of the “multiplier” ∂ 2 Ð2 ness R and radius r0. From this, it is seen that studying ( kn) in (7), which is of the order c1R , whereas b = 2 Ð2δ δ spatial configurations of the density field ρ in planar c2R T, where ci ~ 1 and T = (Tc Ð T)/Tc. In these equa- systems makes it possible to find the form of one- tions, we omitted the term H2 from Eq. (7) since we dimensional distributions of the density ik with the aid of the above-mentioned holographic pro- consider that it is approximately the same for both types jection. of distributions. πρ2 It has been known for a long time that, in such a The exact equation for Fr (in units 0 Lz) contains phase state, the distributions of the charge density the term ∂ ρ 2 1 ( k ) have the form of stripes. Due to the gradient δ []() () ∂ ρ 2 ρ Fr = 2 I3 x0 Ð x0 I2 x0 term ( k ) in (7), quasi-one-dimensional field con- figurations are really preferable. It seems almost obvi- 2 bR []() () ous that a density distribution in the form of rings give + ------I1 x0 2 Ð x0 2I0 x0 2 the smallest contribution to the energy. Let us find the 2 contribution to the free energy (7) from quasi-one- 2 ρ2 R []() () dimensional density distributions in the form of + ----- I1 x0 Ð x0 I0 x0 , rings and stripes and compare the computation results λ2 with the experimental data. We will choose the follow- where ing trial functions for the field ρ configurations in the form of a ring and a stripe: ∞ R 1 m Ðx2 x ==----, ----- 2()n Ð b , I ()z =x e dx. ρρ []()2 2 0 2 0 m ∫ = 0 exp Ð rrÐ 0 /2R (8) r0 λ z and However, this value is already exponentially small λ λ2 δ Ð7 at R/r0 ~ 1/4 and R ~ with 1/ = 2(n0 Ð b): Fr ~ 10 . x2 ρρ= expÐ------For the optimum width R = ξ (at R Ӷ r ), the differ- 0 2 0 2Lx ∆ πρ2 ence in free energies F = Fxy Ð Fr in units 0 Lz has ≤ the form 1, yLy, (9)  3 ×  ()yLÐ 2 ∆F = 1 + c Ð ---c δT. y > 1 4 2 exp Ð,------2 - y Ly.  2Lx From this equation, one can see that, at 4(1 + c1)/3c2 < 1 in the temperature range 1 We suppose that the characteristic size of the stripe is substan- []() << tially greater than the lattice scale. In this case, the use of a phe- 141Ð + c1 /3c2 T c TTc, nomenological approach of the mean-field GinzburgÐLandau the- ory is justified. bordering on the critical temperature Tc of the transition

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1144 ISAEV, PROTOGENOV One can see that configurations in the form of stripes are more preferable in the range 0.65 < n0/b < ρ2 0.68, where Fxy < Fr. Normalizing the density 0 to the 2 π ξ ρ2 1 particle number N by the condition N = 2 r0 Lzm 0 , we obtain a relation between the parameters n and b, 3 0 which we write in the form

xb= n 0 Ð,0.65b π where x = Nd/( 2 m Lzr0). Thus, for the bounds of the δ range considered above, where n0 ~ b ~ T, we have 2 T(x) = Tc(1 Ð Ax ) with a certain constant A. Therefore, Schematic representation of closed (1) and open (2) quasi- inside the region belonging to the phase state with bro- one-dimensional structures of the charge density (see [29]) ken antiferromagnetic order, there is a narrower region, around antiferromagnetic dielectric nanoclusters (3). located between the parabolas T(x), where the charge structures have the form of stripes. to the state with the spin pseudogap, rings are prefera- ble (see figure). In the temperature range 3.3. The Inhomogeneous Superconducting State TT< []141Ð ()+ c /3c , Let us consider a superconducting state with finite c 1 2 value of the total current J which exists against the stripes are the main configurations. As is known, one background of a certain n-field distribution, assuming ρ ρ may approach Tc keeping the temperature constant by that = 0 = const. In this case, the free energy is increasing the level of doping. The recent paper [29] gave evidence of the existence of ring-shaped charge FF= n + Fc Ð Fint structures obtained just under such experimental condi- (13) 3 ()()∂ 2 2 1 2 2 tions. In a certain sense, the tunneling microscope in = d x kn + Hik +.---c + Fik Ð 2FikHik ∫ 4 this experiment [29] collects data from a two-dimen- sional slice of knots [21]. The negative sign of the interaction energy Fint of c Let us make several remarks concerning the spin and n fields appears because of diamagnetism of the con- 2 density ρ distribution in the disk surrounded by a ring sidered state. As a result, the coupling constant g2 = 1 of charge distribution. The above considerations referred 2 the term Hik decreases due to renormalization in such to the case when the spin disorder arose only in the ≠ region directly adjacent to the disk edge, and, as a a way that energy of the superconducting state c 0 is result, we had an antiferromagnetic phase inside. If the smaller than the minimum value in inequality (5). To find the exact lower bound of the free energy in the super- antiferromagnetic order is broken everywhere in the ≠ disk, it is necessary to consider the corresponding dis- conducting state c 0, we will use the auxiliary ine- tribution of the density ρ2 in the form of a disk in order quality 4/3 to compare its contribution with Fr. When we consider F5/6F1/2 ≥ ()32π2 L , (14) the contribution of the density distributions ρ2 to the n c free energy in the form of a disk, we get a double gain where the invariant (in comparison with rings) due to the existence of one edge and have a loss due to the area. The calculation 1 3 ε ∂ L = ------2∫d x iklci kal (15) shows that at small R/r0 the distributions in the form of 16π rings appear to be more preferable. determines the degree of the mutual linking [30, 31] of Let us consider the dependence of the critical tem- the current lines and the lines of the magnetic field H = perature Tc on the level of doping. To do this, we present [∇ × a]. Like Q, it is the integral of motion [31, 32] in the the relation of Fr to Fxy in the following form: considered barotropic state. The proof of inequality (14) [14] is given in the Appendix. F 1 ------r = ------, (12) Linking indices, which characterize the correlations Fxy 1 + BR/r0 of spin and charge degrees of freedom, form the follow- where ing matrix:  1 1 n0 Ð 3b/4 3n0 Ð 0.68b 1 3 ε αα∂ β QL' B ==--- 1 + ------. Kαβ ==------∫d x ikl i kal . (16) 2 2 ()4n Ð 0.65b π2  n0 Ð 11/8Ð b 0 16 LQ'

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 STRUCTURES OF ORDER PARAMETERS IN INHOMOGENEOUS PHASE STATES 1145 1 ≡ 2 ≡ In this symmetric matrix (L = L') with ai ai and ai One can see from Eq. (19) that, for all numbers L < ci, the integral might also be determined by the asymp- Q, the energy of the ground state is smaller than that in totic linking number [30]. Let us pay attention to a cir- model (4), for which inequality (5) is valid. The origin cumstance which will be important below. Being nor- of the energy decrease may be understood by compar- malized to the charge density, unlike the unit vector n, ing the values of different terms in Eq. (13). Even under the vector of momentum c = J/ρ2 belongs to the non- the conditions of a considerable paramagnetic contribu- compact manifold. Because of this, the Hopf numbers tion j to the current, the diamagnetic interaction in the defined with the aid of this vector in (16) are not inte- superconducting state for all classes of states with L < gers in general: (L, Q') ∉ ޚ. In the superconducting Q reduces in (13) its own energy Fc of the current and state, where Abelian U(1) gauge symmetry is broken part of the energy Fn associated with dynamics of the and the charge is not conserved, the numbers L and Q' n field. In the state under consideration, the total play the role of continuous interpolation parameters momentum of superconducting pairs c is not equal which unite the compressed and uncompressed (Kαβ ∈ zero. In this respect, the inhomogeneous state with cur- ޚ) phases under consideration. From this point of view, rent is analogous [34] to the state proposed in [17, 18]. the superconducting states with Kαβ ∈ ޚ and Kαβ ∉ ޚ belong to one and the same class of universality [33]. 4. THE PROPERTIES OF PHASE STATES To find the lower bound of the functional (13), along The phase state with a broken antiferromagnetic with Eq. (14), we will use the SchwarzÐCauchyÐBuny- ∂ ρ 2 ≠ order at ( k ) 0 is a background on which the transi- akovski inequality tion to the inhomogeneous superconducting phase with F ≠ 0 occurs. It is convenient to discuss the character- 1/2 1/2 ik ≤ ⋅ ≤ 2 Fint 2 Fik 2 Hik 2 2Fc Fn , (17) istics of this transition upon a change in the density ρ , beginning with the superconducting state. In this phase, the constant value of the charge density, related to the || || ≡ []3 2 1/2 where Fik 2 ∫d xFik . Note that the equality on breakage of the gauge invariance U(1), plays the role of the right-hand side of Eq. (17) is achieved in the ultra- the tuning parameter of the system. ρ violet limit, when the size of linked vortex configura- Let the parameter 0 change in some range. Since all tions is small enough. Substituting the boundary value terms in Eq. (3) are of the same order, the momentum c Fint into (13), we obtain and, consequently, the index of the mutual linking L ρ decrease when 0 increases. In this case, for sufficiently 2 FF≥ = ()F1/2 Ð F1/2 . (18) small L, the smallest superconducting gap decreases min n c with an increase in Q against the background of a large value of 32π2|Q|3/4 of the spin pseudogap. The Hopf configuration with Q = 1, for which the ρ lower limit occurs in Eq. (5), represents two linked When 0 decreases, the following effect takes place. rings with radius R and Ð1/2 ∝ ρÐ1 ᏾ Being proportional to g1 0 , the radius of com- ޒ3 3 2 38 8 2 pactification S grows until it exceeds some ()F ==2π R ----- + ----- 32π . n min 2 4 critical value ᏾ . At ᏾ > ᏾ , the Hopf mapping is not R R cr cr R = 1 stable [27] relative to small perturbations of linked vor- We will assume that, in our case of c ≠ 0, there are tex field configurations. As a result, the U(2) symmetry configurations for which the equality in Eq. (14) is which is associated with identical Hopf mapping valid. Let us emphasize an important circumstance appears to be spontaneously broken. This means that which will be discussed more thoroughly in the next the topological configurations of field n and c, instead section. For small values of ρ and, therefore, for large of being spread out over the whole space S3, localize values of the field c (since all terms in (13) are of the around a particular point (the base point of the stereo- same order), we encounter the instability of linked con- graphic projection ޒ3 S3) and collapse to localized figurations with respect to small perturbations. This structures [27]. We can see that there is an optimal ρ leads to the restriction of values F from above. Keep- value of 0 and, consequently, values of the character- c | | | | ing in mind this remark and using in Eq. (18) for Fc the istic momentum c and the relation L / Q for which lower bound there arises the greatest gain upon the transition to the superconducting state. 1/2 ()π2 4/3 Ð5/6 Fc = 32 Fn L , Until now, the vector A has characterized the degrees of freedom, associated with the internal charge π2| |3/4 we obtain from Eq. (14) and relation Fn = 32 Q , for gauge symmetry U(1). If the external electromagnetic the states with Q ≠ 0, that field is applied, the vector A equals the sum of internal and external gauge potentials. In the external magnetic F ≥ 32π2 Q 3/4()1 Ð L/ Q 2. (19) field, due to diamagnetism of the superconducting

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1146 ISAEV, PROTOGENOV state, the momentum c decreases. Similar to the case of (0.1Ð1) eV, this gives several tens or hundreds of ρ increasing 0, this leads to suppression of the supercon- degrees. Taking into account the relation between the ducting gap. Playing the role of a smooth tuning param- dimensionality of the systems at their dynamic and sta- eter, the external magnetic field determines the bound- tistical descriptions, we note that the (2 + 1)-dimen- ary conditions of the problem. As a result, the answer to sional case k = 2 with the open ends of excitation world the question of completeness of the Meissner screening lines is equivalent (after identification of the ends) to depends on the results of the competition of contribu- the compact statistical (3 + 0)-dimensional example of tions from a paramagnetic (spin) j and diamagnetic the Hopf linking with Q = 1. (charge) Ð4A parts of the total current J. The (3 + 0)-dimensional and (2 + 1)-dimensional Similar to energy distribution in the fractional quan- situations differ by the topology of the regions of the tum Hall effect with the filling factor ν = p/q and p, q ∈ field definition n [38Ð40]. When the system is periodic ޚ, the energy gain in Eq. (19) depends on the relation in one of the space variables and also when calculating |L|/|Q|. The Hopf invariant Q ∈ ޚ numbers vacuums the partition function in planar systems, one of three [35] and is equivalent to the degree of degeneracy q of coordinates—Matsubara variable—is a periodic vari- able. This means that instead of the sphere S3, we deal the ground state. The index L plays the role of the filling × degree p of incompressible charged fluid state in the with the topology of a three-dimensional torus T2 1 = fractional quantum Hall effect. From this point of view, S2 × S1 or T3 = S1 × S1 × S1 and with the corresponding the multiplier (1 Ð |L|/|Q|) in Eq. (19) is equivalent to mapping classes. The content of Hopf invariant in this the filling factor 1 Ð ν for holes. The distinction of our case appears to be richer [38Ð40]. For a three-dimen- system from the states in the fractional quantum Hall sional torus T3, the Hopf invariant is defined modulo 2q, effect is that the superconducting state is compressible where q is the greatest common divisor of the numbers ∈ ޚ and here (as was mentioned above) the effective num- {q1, q2, q3} . Here, qi is the degree of mapping ber L of the charge degrees of freedom is not an integer T 2 S2, where T2 is the section of T3 with the fixed in the general case. The configurations of fields n and ith coordinate. Four integral numbers {qi, Q}, where Q c = a with the integers L = Q, satisfying the relation of is defined by modulo value 2q, give us the complete self-duality Fn = Fc, correspond to the minimum value homotopic classification of mappings T3 S2 with of free energy (13). In this limit, Kαβ is proportional to π 2 2 ޚ 1[Mapq(S S )] = 2q and a fixed degree q [38Ð  11 40]. The geometrical meaning of this modified Hopf the matrix  , which was used in [36] to describe invariant (an integer from the range {0, 2q Ð 1}) is the 11 same. It is a linking index of the preimages of two × the topological order in the theory of fractional quan- generic points in T3 S2. The cases T2 1 and T3 are tum Hall effect with the filling factor 1 Ð ν at ν = 1/2. characterized physically by different boundary condi- tions. The boundary conditions change if an angular The boundary conditions which determine the velocity of the rotation of the neutral superfluid phase momentum c and the topological invariants L, Q 3 depend not only on the values of the tuning parameter in He increases [38, 39] or an external magnetic field ρ in our charged system grows. Restricting the Hopf 0 of the model and the external . Their 2 × 1 3 physical sense and value also depend on the dimension- invariant change, the transition T T promotes ality of the manifold for which the model is defined. In the appearance of the incompressible phase. the (3 + 0)-dimensional case of the free energy (3), the Hopf invariant (6) is analogous to the ChernÐSimons 5. CONCLUSION action One can see from the above analysis that the gain of the free energy upon the transition to the superconduct- k 2 ε α ∂ α ≠ ------π∫dtd x µνλ µ ν λ. ing state with c 0 arises when there is a coherent phase 4 associated with the spin degrees of freedom. This phase is characterized by a pseudogap (5) and a topological This term in the action of (2 + 1)-dimensional systems ρ2 describes the dependence of the contribution of non- order associated with linking. If the density 0 is rather linear modes to the free energy on the statistical param- large, the momentum c is small and the transition to the eter k. The coefficient k in the ChernÐSimons action has superconducting state is not preferable. According to the geometrical sense of the braiding number of the our classification, this second state is characterized by excitation world lines. In particular, when semifermion changing values of the order parameters ρ and n. The ∂ ρ 2 excitations (semions) permutate and return to the initial energy loss due to the term ( k ) may be reduced positions on the plane, the world lines braid twice and because of the development of one-dimensional struc- k = 2. Therewith, the statistical correlations of nonlinear tures. Whether these one-dimensional charge structures modes have the character of attraction, and for the val- will be open, forming stripes, or closed almost one- ues k ~ 2 their greatest contribution to the energy is of dimensional structures in the form of rings depends on ρ the order of several percent [37]. For the energy scale the parameters of the potential V( , n3). In the phase

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 STRUCTURES OF ORDER PARAMETERS IN INHOMOGENEOUS PHASE STATES 1147 n3 = const with neutral spin currents, the answer will The superconductivity in cluster systems may evi- depend on the value and the sign of the multiplier b(n3) dently be studied on the basis of approaches beyond the in the potential mean field theory. For example, we may use the exact Richardson solution and Bethe ansatz equation [41], as 2 d 4 V()ρ, n = Ð bρ + ---ρ . well as the methods of the conformal field theory [42]. 3 2 The exact solution of the ground state problem under the condition of a finite value of the total momentum of If b > 0, then far from Tc charge structures with open pairs in such an approach is one of the important prob- ends are preferable [13], and in the case T Tc we lems. Since the conformal nature of the dimensionality should prefer rings. The first experiments that verified 3/4 [30] in (5) and (14) influences the character of the the existence of charge structures in the form of rings in scale which enters into energy dependent response the underdoped phase of planar systems were described functions (which is proportional to T3/4 [43]), it should in [29]. also be studied carefully. On the “temperature–charge density” phase dia- gram, the superconducting phase occupies only part of ACKNOWLEDGMENTS the region belonging to the phase with the broken anti- ferromagnetic order. The bounds of its existence on the The authors are grateful to A.G. Abanov, S.A. Bra- phase diagram, associated with the characteristic values zovsky, S. Davis, L.D. Faddeev, S.M. Girvin, V.E. Krav- ρ2 ρ tsov, E.A. Kuznetsov, B. Lake, A.I. Larkin, A.G. Litvak, of the density 0 , are determined for great 0 by the C. Renner, H. Takagi, V.A. Verbus, G.E. Volovik, ρ inequality (14) [14], while for small 0 these bounds Yu. Lu, and Y.-S. Wu for advice and useful discussions. depend on the critical size of the knot [27], beginning This paper was supported in part by the Russian from which instability of the Hopf mapping arises. Foundation for Basic Research, project no. 01-02-17225. Comparing the results of this paper based on consid- eration of the local fields with the conclusions follow- ing from the BCS-like model [19] with two species of APPENDIX fermions, we pay attention to the following qualitative The proof of Eq. (14) uses the following chain of coincidence. One can conclude from Eq. (3) that the inequalities: parameter ρ determines the value of the coupling con- 0 < ⋅ ≤ 1/6 []∇ × ⋅ stant. Therefore, the appearance of the solutions (differ- L c 6 H 6/5 6 c 2 H 6/5 ent from the standard BCS model) for the supercon- ≤ 1/6 []∇ × 2/3 1/3 ducting gap in the paper [19] with a finite value of the 6 c 2 H 1 H 2 (20) coupling constant is analogous to the existence of the ρ ≤ ()π2 Ð4/3 1/2 2/3 1/6 ()π2 Ð4/3 1/2 5/6 threshold for small values of 0 in this paper. 32 Fc Fn Fn = 32 Fc Fn . In contrast to the model [19], the states considered || || ≡ ()3 p 1/p in the present paper are significantly inhomogeneous. Here, H p ∫d x H . At the first and third steps, ≠ ρ ≠ The analysis of the state Fik 0, const is still an we used the Hölder inequality open problem. Here, we only mention that the super- ⋅ ≤ ⋅ fg f p g q conducting current with the amplitude c0, flowing 2 2 ∇ around the rings (8), gives the additional term c R to with 1/p + 1/q = 1. Under the condition á c = 0, we 0 employ at the second step the Ladyzhenskaya inequal- the multiplier in Eq. (10). This explains, in particular, ity [40, 44]: why the superconducting region on the “temperature– doping level” phase diagram is shifted to the line c ≤ 61/6 []∇ × c . δT(x) = 0 of the transition to the state with the spin 6 2 pseudogap. Indeed, in this case, The fourth step in the set of inequalities arises after the comparison of the terms ||[∇ × c]|| and ||H|| with the 2 d 4 V ()ρ, n = Ð b ρ + ---ρ , terms Fn and Fc in Eq. (13). The last line also shows eff 3 eff 2 separate contributions from n and c parts of the free energy (13) to the finite result (14). Using a chain of with Hölder and Ladyzhenskaya inequalities, analogously one may find that ()2 constδ beff ==bnÐ 0 + c0 ------T. 2 1/2 5/6 ≥ ()π2 4/3 R Fn Fc 16 L'. Therefore, the finite value of the momentum c of super- The coefficient in this inequality differs from (14) conducting pairs decreases δT. In addition, the contri- due to the coefficient 1/4 (because of the charge 2e of bution to the free energy in the inhomogeneous state superconducting pairs) of the first term of the free ∂ ρ 2 ≠ due to ( k ) 0 decreases the gain in Eq. (19). energy Fc in (13).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1148 ISAEV, PROTOGENOV

REFERENCES 24. J. Hietarinta and P. Salo, Phys. Lett. B 451, 60 (1999). 1. E. Babaev, L. D. Faddeev, and A. J. Niemi, Phys. Rev. B 25. A. F. Vakulenko and L. V. Kapitanskiœ, Dokl. Akad. Nauk 65, 100512 (2002). SSSR 246, 840 (1979) [Sov. Phys. Dokl. 24, 433 (1979)]. 2. B. Lake, G. Aeppli, K. N. Clausen, et al., Science 291, 1759 (2001). 26. A. Kundu and Yu. P. Rubakov, J. Phys. A 15, 269 (1982). 3. B. Lake, H. M. Rönnow, N. B. Christensen, et al., Nature 27. R. S. Ward, Nonlinearity 12, 1 (1999); hep-th/9811176. 415, 299 (2002). 28. H. Takagi, http://agenda.ictp.trieste.it/agenda/current/ 4. J. E. Hoffman, E. W. Hudson, K. M. Lang, et al., Science fullAgenda.php?email=0&ida=a01152. 295, 466 (2002). 29. V. I. Arnold and B. A. Khesin, Topological Methods in 5. S. H. Pan, J. P. O’Neal, R. L. Badzey, et al., Nature 413, Hydrodynamics (Springer, New York, 1998), Appl. 282 (2001). Math. Sci., Vol. 125, Chap. 3. 6. S. J. L. Billinge, E. S. Bozin, M. Gutmann, and H. Tak- 30. H. K. Moffatt, J. Fluid Mech. 106, 117 (1969). agi, cond-mat/0005032. 31. V. E. Zakharov and E. A. Kuznetsov, Usp. Fiz. Nauk 167, 1137 (1997) [Phys. Usp. 40, 1087 (1997)]. 7. S. Caprara, C. Castellani, C. Di Castro, et al., cond- mat/9907265. 32. N. Read and D. Green, Phys. Rev. B 61, 10267 (2000). 8. L. D. Faddeev and A. J. Niemi, Phys. Rev. Lett. 82, 1624 33. A. I. Larkin, private communication. (1999); Phys. Lett. B 525, 195 (2002). 34. P. van Baal and A. Wipf, hep-th/0105141. 9. E. Babaev, Phys. Rev. Lett. 88, 177002 (2002). 35. X.-G. Wen, cond-mat/9506066. 10. Y. M. Cho, Phys. Rev. Lett. 87, 252001 (2001); hep- 36. A. P. Protogenov, Usp. Fiz. Nauk 162, 1 (1992). th/0110076. 37. L. A. Abramyan, V. A. Verbus, and A. P. Protogenov, 11. A. G. Abanov and P. W. Wiegmann, hep-th/0105213. Pis’ma Zh. Éksp. Teor. Fiz. 69, 839 (1999) [JETP Lett. 12. L. D. Faddeev, Preprint No. IAS-75-QS70 (Princeton, 69, 887 (1999)]; A. P. Protogenov, Pis’ma Zh. Éksp. 1975). Teor. Fiz. 73, 292 (2001) [JETP Lett. 73, 255 (2001)]. 13. M. Lübke, S. M. Nasir, A. Niemi, and K. Torokoff, Phys. 38. V. M. H. Ruutu, U. Parts, J. H. Koivuniemi, et al., Pis’ma Lett. B 534, 195 (2002). Zh. Éksp. Teor. Fiz. 60, 659 (1994) [JETP Lett. 60, 671 (1994)]. 14. A. P. Protogenov and V. A. Verbus, Pis’ma Zh. Éksp. 39. Yu. G. Makhlin and T. Sh. Misirpashaev, Pis’ma Zh. Teor. Fiz. 76, 60 (2002) [JETP Lett. 76, 53 (2002)]. Éksp. Teor. Fiz. 61, 48 (1995) [JETP Lett. 61, 49 15. A. P. Protogenov, cond-mat/0205133. (1995)]. 16. L. S. Isaev and A. P. Protogenov, cond-mat/0210295. 40. A. G. Abanov, private communication. 17. A. I. Larkin and Yu. N. Ovchinnikov, Zh. Éksp. Teor. Fiz. 41. J. von Delft and R. Poghossian, cond-mat/0106405. 47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1964)]. 42. G. Sierra, Nucl. Phys. B 572, 517 (2000); hep- 18. P. Fulde and R. A. Ferrell, Phys. Rev. A 135, 550 (1964). th/0111114. 19. W. V. Liu and F. Wilczek, cond-mat/0208052. 43. P. Coleman, C. Pepin, Q. Si, and R. Ramazashvili, 20. P. van Baal, hep-th/0109148. J. Phys.: Condens. Matter 13, 723 (2001). 21. L. D. Faddeev and A. J. Niemi, Nature 387, 58 (1997). 44. O. A. Ladyzhenskaya, The Mathematical Theory of Vis- 22. J. Gladikowski and M. Hellmund, Phys. Rev. D 56, 5194 cous Incompressible Flow, 2nd ed. (Gordon and Breach, (1997). New York, 1969; Nauka, Moscow, 1970). 23. R. A. Battye and P. M. Sutcliffe, Phys. Rev. Lett. 81, 4798 (1998). Translated by L. Isaev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 1149Ð1162. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1308Ð1322. Original Russian Text Copyright © 2003 by Elesin, Kopaev. SOLIDS Electronic Properties

Stark Ladder Laser with a Coherent Electron Subsystem V. F. Elesina,* and Yu.V. Kopaevb aMoscow State Institute of Engineering Physics, Kashirskoe sh. 31, Moscow, 115409 Russia *e-mail: [email protected] bLebedev Institute of Physics, Russian Academy of Sciences, Leninskiœ pr. 53, Moscow, 119991 Russia Received February 6, 2003

Abstract—A theory of generation in a two-subband “Stark ladder” with a coherent electron subsystem is devel- oped. In the proposed model, electrons reach the upper level of a due to resonant tunneling and pass to the lower level of the well (vertical transitions), emitting a photon បω, then tunnel resonantly to the upper level of a neighboring well, performing a radiative transition, and so on until electrons leave the lower level of the last well. A static electric field applied to the superlattice shifts the levels so that the lower level of the nth well coincides with the upper level of the (n + 1)th well. Analytic expressions are derived for the wave functions and polarization currents of an N-well structure. The possibility of bulk oscillation of the N-well structure in the optimal mode with an efficiency close to unity, weak reflection, and a linear dependence of the power on the pumping current is demonstrated. The total generation power is proportional to the number of wells. For struc- tures with an even number of wells, the energy of electrons from the emitter must simply coincide with the res- onance energy for any laser fields; i.e., the energy tuning which is necessary in a single-well structure is not required. Universal relations are derived for parameters of the N-well structure, which ensure the simultaneous fulfillment of resonance conditions in all the wells. The possibility of coherent lasing in a one-subband Stark ladder with a lower gain is also indicated. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION the reflection from the structure is equal to zero, and In 1972, Kazarinov and Suris [1] proposed a new amplification line broadening due to the field is absent. type of a semiconductor laser in which radiative transi- A natural question arises: Is effective lasing possible tions occur between size-quantization levels (sub- for a structure consisting of N tunnel-coupled QWs? bands). This idea was implemented in 1994 in nano- This problem is closely related to the problem of Stark structures (so-called cascade lasers) [2, 3], in which the ladder lasing, for which a certain difficulty exists. main elements are two quantum wells (QWs) with According to Bastard et al. [8], Stark ladder lasing is a working levels in each well [2] (“diagonal transitions”) surface effect. The reason is that radiative transitions in or a single QW with two working levels [3, 4] (“vertical the sample are compensated, and the gain is determined transitions”). by the population inversion between the levels at the left and right boundaries. The main advantage of cascade lasers is the possi- bility of wavelength tuning from the infrared to the sub- However, this conclusion is valid if the electron sub- millimeter range. Cascade lasers are characterized by system is incoherent and the concept of radiative tran- unipolarity, identical masses of subbands, etc. Another sition probability per unit time is applicable. The situa- fundamental feature is the coherent nature of resonant tion in a coherent system is fundamentally different. tunneling ensuring pumping. This leads to the possibil- Under certain conditions, an electron supplied to a QW ity of lasing without participation of dissipative pro- performs coherently radiative transitions irrespective of cesses. Indeed, an electron reaches the upper level as a the difference in the occupancies of the levels [6]. Con- result of coherent resonant tunneling; passes to the sequently, we can expect that lasing takes place in the lower level, emitting a photon; and leaves the well, thus entire volume of the N-well structure. interrupting the process of interaction with the electro- It should be noted that radiative transitions in the magnetic field (in conventional lasers, lasing is inter- one-subband model occur between the Stark levels in rupted due to dissipative processes such as the emission neighboring QWs (diagonal transitions). It was shown of a phonon [5]). Such a single-quantum-well laser in [4, 9] that diagonal transitions lead to a much lower (referred to as a coherent laser) was proposed and ana- gain as compared to transitions between two energy lyzed theoretically in [6]. It was shown that a strong levels in the same QW (vertical transitions). For this field can be generated in the absence of population reason, we consider here a superlattice with vertical inversion. The optimal operating conditions were deter- radiative transitions (see figure). After the transition of mined, for which the efficiency is close to unity [6, 7], an electron from the upper to the lower level of the nth

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1150 ELESIN, KOPAEV α 2 ture; the results obtained in this section are used in the subsequent analysis. The wave functions of an N-well structure, which are valid both for weak and strong 1 α fields, are determined in Section 4. A detailed analysis ε ε(1) R 2R of a two-well structure is carried out in Sections 5 and 6, while a three-well structure is treated in Section 7. 2 2 α q ω Structures with an even number of wells N ≥ 4 are ana- ε(1) ε(2) 1R 2R lyzed in Section 8. An analysis of resonance conditions ω 3 α is carried out in the Appendix. ε(2) ε(3) –a 0 1R 2R x ω 4 α 2. DESCRIPTION OF THE MODEL: ε(3) ε(4) 1 1R 2R BASIC EQUATIONS ω Let us consider the following model of a one-dimen- ε(4) ε ω 1R – n sional N-well structure (see figure), which is a general- ization of the models described in [7, 10]. The figure shows schematically the structure in the form of a set of δ-functional barriers positioned at points x = an, n = −1, 0, 1, …, N Ð 1. The parameters of the QWs are chosen ε ε Schematic diagram of a four-well structure (N = 4). so that the energies 2R and 1R of the two lower levels of an isolated well differ by a value approximately equal to the electromagnetic field frequency ω (ប = 1): ε ε ω QW, accompanied by the emission of a photon, the 2R Ð 1R = . The application of a constant external electron performs resonant tunneling to the upper level electric field shifts the energy of the nth well relative to ω of the neighboring QW, and so on. This is the model of the (n Ð 1)th well by quantity E0 equal to . The energy () a coherent laser in which an electron passes through N ε n Ð 1 of the lower level of the (n Ð 1)th well coincides QWs, emitting photons in each QW. It is well known 1R ε()n that an electron in a cascade laser [2, 3] after each radi- with the energy 2R of the upper level of the nth well. ative transition gets into the relaxation region, where its The steady-state electron flux with a density propor- state is prepared for the next radiative transition. tional to q2 and with energy ε approximately equal to () This study is aimed at the development of the theory ε 1 is supplied to the system from the left (x = Ð∞). of a coherent laser for N QWs with vertical transitions. 2R It should be noted that some preliminary results were The electromagnetic field, which is regarded as clas- obtained in [10]. However, the formalism used in [10] sical, for solving the system of 4N × 4N algebraic equations is quite cumbersome, which did not permit one to carry E ()zt, = Ekzsin()cos() ωt + ϕ , out a comprehensive analysis and to clarify the optimal x lasing conditions. is emitted during electron transitions from levels “2” to We succeeded in deriving exact analytic solutions to levels “1” of the QWs. We assume that the field is polar- this system of equations for 2, 3, 4, and N wells. It is ized at right angles to the plane of a well (i.e., along the shown that optimal bulk lasing of a structure with x axis), while the wave vector lies in the plane (along N wells can be obtained with an efficiency close to the z axis). The optical resonator of length L separates unity, a small reflectance, and a linear dependence of the modes. We confine our analysis to single-mode las- the power on the pumping current. For structures with ing. The equations for the stationary amplitude E and an even number of wells, the energy of supplied elec- phase ϕ (frequency ω) can be written in the form [5] trons must simply coincide with the resonant energy for any fields. In other words, the energy tuning, which is π inevitable in a structure with a single well [6, 7], is not E 2 ()N ------τ- = Ð------κ Jc , (1) required. Universal relations are derived for the param- 2 0 eters of an N-well structure, ensuring the fulfillment of the resonance conditions in all the wells simulta- π ()ωΩ 2 () neously. Ð E = Ð------κ Js N , (2) The article has the following structure. In Section 2, the model is described and the steady-state solution to ()N Ð 1 a the time-dependent Schrödinger equation is given; the ()N 1 J , = ------dxJ , (),x (3) boundary conditions and expressions for currents are cs Na ∫ cs also derived. Section 3 is devoted to a single-well struc- Ða

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 STARK LADDER LASER WITH A COHERENT ELECTRON SUBSYSTEM 1151 where Jc and Js are the reduced polarization currents of the boundary conditions will be specified below after π Ψ coinciding in phase with the field (Jc) and shifted by /2 obtaining a solution for (x, t). relative to the field (Js). Currents Jc and Js describe tran- Following [6], we seek the steady-state solution to τ sition between energy levels. Here, 0 is the photon life- Eq. (5) in the form time in the resonator, κ is the dielectric constant, Ω is the natural frequency of the resonator, and c is the Ψ()xt, = exp[]Ðit()ε + mω Ð E ()n + 1 velocity of light (we will henceforth assume that c = 1). ∑∑ 0 Currents J (x) and J (x) can be determined using the n m (9) c s ×ψ , well-known expression nm(),Nx ∂Ψ m = 012, ± , ± , …,1≤≤nN. , Ψ∗ Jxt()= Ðie------Ð c.c. , (4) ∂x ψ , Functions nm()xN describe the states with quasi- where the wave function Ψ(x, t) of the system obeys the energies ε + mω in the nth well and satisfy the follow- Schrödinger equation ing system of equations: ∂Ψ ∂2Ψ 2ψ i------= Ð------++Ux()Ψ Vˆ ()xt, Ψ. (5) d m 2 []ψε ω () ------∂t ∂ + m Ð E0 n + 1 nm + 2 x dx (10) Here, dψ , dψ , = V ------nmÐ -1 Ð.------nm+ 1 N dx dx Ux()= α δ()xa+ + ∑ αδ()xanÐ 2 (6) n = 0 It is well known that the main contribution to lasing + α δ()xaNÐ ()Ð 1 Ð E ϕ(),x comes from two resonance levels with the energy dif- 1 0 ference equal to frequency ω. In the present case, for the nth well, these levels are the upper level with energy ϕ Θ () ()x = ∑ (),xnÐ Ð 1 a ε()n ε()n 2R and the lower level with energy 1R . The corre- n ψ ψ (7) sponding wave functions are n2(x) and n1(x), so that 1, x > 0, wave function (9) can be reduced to two terms in each Θ()x =  0, x < 0; well: Ψ , ψ () []()ε α α α α ()xt = n2 x exp Ðit Ð E0()n Ð 1 j are the barrier intensities, 2, 1, and being the ψ () []()εω barrier heights of the left, extreme right, and internal + n1 x exp Ðit Ð Ð E0()n Ð 1 , (11) ប wells, respectively; and 2m = = 1. The last term in ()≤≤() ≤≤ Eq. (5), i.e., n Ð 2 ax nÐ 1 a,1nN. ψ ψ ∂Ψ Functions n2 and n1 satisfy the system of equa- Vˆ ()xt, = 2eiA ------, tions x ∂x d2ψ dψ describes the interaction between electrons and the []ψε Ð E ()n Ð 1 + ------n-2 = V------n-1, (12) 0 n2 ∂ 2 dx electromagnetic field and Ax is the vector potential in x the Coulomb gauge, which differs from zero in the nanostructure region. Expressing A in terms of the field 2ψ ψ x []ψεω d n1 d n2 amplitude, we can write the last term in the form Ð Ð E0()n Ð 1 n1 + ------= ÐV------(13) dx2 dx iωt Ðiωt ∂Ψ eE Vˆ Ψ ==Ve()Ð e ------, V Ð------. (8) ∂x ω with the following boundary conditions (see [6]): It should be noted that we have omitted in this α 1 dψ ()Ða ψ ()1Ða Ð ------2 + ------12 = q, expression the term quadratic in Ax. This approxima- 12 ip ip dx tion, which is usually employed in the theory of lasers, 2 2 is also valid here if parameter V/p is small; i.e., ψ ψ ψ 12()0 === 0, 11()Ða 0, 22()a 0, V eE --- = ------Ӷ 1, dψ ()0 dψ ()0 p pω ------22 Ð ------11 = αψ (),0 dx dx 11 where p is the electron momentum. Equation (5) should be supplemented with boundary conditions. The form …………………………………

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1152 ELESIN, KOPAEV ψ ()() ψ ()() N1 N Ð 2 a ==0, N2 N Ð 1 a 0, (14) field. The strong-field parameter will be determined below. α We seek the solution to system (16) in the form ψ ()()1 N1 N Ð 1 a 1 Ð ------ψ γx ψ γx ip1 1 ==A1e , 2 A2e . (18) 1 dψ ()()N Ð 1 a The eigenvalues γ satisfy the equation Ð ------N1 - = 0, ip dx 1 2 ω γ 4 γ 2ε V Ð ε2 ωε +02 + ------+ Ð = ψ ()() ψ ()() 2 d N2 N Ð 2 a dNN Ð11, Ð 2 a ------Ð ------dx dx and are given by αψ ()() = N Ð11, N Ð 2 a . 2 2 2 V Ð ω V Ð ω 2 γ , = ±i ε +,------Ð ------+ εV The boundary conditions describe the electron flux 12 2 2 from x = Ð∞, their reflection and departure to x = ∞, and (19) the continuity conditions for the wave functions as well 2 ω 2 ω 2 γ ± ε V Ð V Ð ε 2 as the jump in their derivatives at the boundaries of 34, = i ++------+ V . 2 2 the QWs. The general solution to Eqs. (16) has the form For N = 1, Eqs. (12) and (13) and boundary condi- tions (14) are transformed into their counterparts for a 4 ψ j ()γ , single well [7]. It should be noted that the boundary l()x ==∑ Al exp j x , l 12. (20) conditions prohibit the departure of electrons from the j = 1 energy levels of internal wells as well as from the lower j j level of the first well and the upper level of the last well. Coefficients A1 and A2 are connected through the rela- This situation optimizes lasing and is realized in practice. tions following from Eqs. (16), Using the form of wave function (11), we can γ V express current (4) in terms of functions ψ (x): A j ==ε A j , ε Ð------j -. (21) nm 1 j 2 j εωγ 2 Ð + j N , ψ j Jc()Nx = ∑ Jnc(),x Substituting l from Eq. (20) and A1 from Eq. (21) n = 1 (15) into boundary conditions (17), we arrive at the system of algebraic equations for the coefficients, dψ dψ J ()x = Ðie ψ* ------n1 + ψ* ------n2 Ð.c.c. nc n2 dx n1 dx N 4 j ()γ ()β j ∑ A2 exp Ð ja 1 Ð j ==q, ∑ A2 0, j = 1 j = 1 3. SINGLE-WELL STRUCTURE N ε A j exp()Ðγ a = 0, (22) For N = 1, the general system of equations and ∑ j 2 j boundary conditions described in Section 2 is trans- j = 1 formed into the corresponding system for a single well: 4 ε j ()β˜ ∑ j A2 1 Ð j = 0, εψ ψ ψ ()ψεω ψ ψ j = 1 2 + 2'' ==V 1' , Ð 1 + 1'' ÐV 2' , (16) where dψ ψ ≡ ψ , ψ ≡ ψ , ψ' ≡ ------, 2 12 1 11 dx α γ α γ β 2 Ð j β˜ 1 + j j ==------, j ------, α ψ' ip2 ip1 ψ ()2 2()Ða ψ 2 Ða 1 Ð ------+ ------==q, 2()0 0, ip2 ip2 2 ε 2 εω (17) p2 ==, p1 Ð . α ψ' ψ ψ 1 1()0 1()Ða == 0, 1()10 Ð ------Ð0.------The solution to system (22) can be written in the form ip1 ip1 ()j 2q ()kl++1ε ε ∆˜ Following [6], we will find the exact solution to sys- A2 = ∆------()∑ Ð1 k l kl. (23) 1 1 tem (16), (17) without using perturbation theory in the klj≠≠

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 STARK LADDER LASER WITH A COHERENT ELECTRON SUBSYSTEM 1153 ∆ Here, 1(1) is the determinant of system (22), It can easily be seen from these expressions that the electromagnetic field starts influencing the resonant ∆ () ()ij+ ∆˜ ∆ tunneling significantly if 1 1 = ∑ Ð1 ij kl, (24) 2 2 ijkl≠≠≠ 2 p p λ2 > Γ Γ or V˜ > ------1 2 . (30) where 1 2 α2α 2 2 1 2 ˜ ˜ ˜ ∆ij = exp()γ a ()1 Ð β j Ð exp()γ a ()1 Ð βi , Since α/p ӷ 1, condition (30) holds simultaneously with j i (25) ∆ ()γ ()β ()γ ()β ˜ Ӷ ij = expÐ ja 1 Ð j Ð exp Ð ia 1 Ð i . the inequality V 1. A field satisfying condition (30) will be referred to as a strong field. The expressions for γ and ε can be simplified by j j 2 taking into account the smallness of ratio V/p: Assuming that ratio V˜ α/p is small, we derive coef- ()j 2 2 V V ficients A2 from relations (23): γ , ≈ ±ip 1 + ------, γ , ≈ ±ip 1 Ð ------, (26) 12 22ω 34 12ω ()3 ()4 2q ε ε ∆˜ A2 ==ÐA2 ∆------()1 2 12, iω iVp 2 1 ε ≈ +−------ε ≈ ±------2 ε ε ------12, , 34, , 1 2 = 2, α (31) Vp ω ˜ ()1 ()2 8qp2i 1 1 V A ==A ------. (27) 2 2 ∆ 2 2 2 ()1 p1 p 2 2V p ε ε ≈ -----2, V˜ = ------1 Ӷ 1. 1 3 p ω2 Using relations (31), (20), and (15), we obtain current (3) 1 in the form These results are also valid for an N-well structure and ε will be used in the subsequent analysis. We assume that Jc()1 = Ð 4ie 1M12K(),1 one more inequality, which is universal for any N, is satisfied: ()1 ()3 K()1 = []A ()1 A (1 )+ c.c. α 2 2 -----j ӷ 1. (28) 2 ˜ p j ()4q p α ε ε 2Im∆12 = ------2 1 1 2 -, (32) This inequality corresponds to a small resonance level p2 ∆()1 2 Γ ε 1 width j as compared to the resonance energy jR. It is well known that resonant tunneling exhibits most a clearly its remarkable properties precisely when ine- 1 M = --- dxp[ sin()p x cos()p x qualities (28) are satisfied. 12 a∫ 1 2 1 It was shown in [6] that determinant (24) and coef- 0 j ()()] 8 ficients Am (23) can be represented in the form of Ð p2 sin p1 x cos p2 x = ------. 2 3a expansions in V˜ . Omitting terms on the order of V˜ Ӷ 2 Substituting relations (25) and (29), carrying out some 1 and V˜ α/p Ӷ 1, we arrive at the following expression ∆ transformations, and assuming that α = 2 2 α and for 1(1): 2 1 Γ = Γ = Γ for the optimal mode [7], we obtain α α 1 2 ∆ () εε ∆˜ ∆ ˜ 28 1 2 1 1 = Ð 1 212 34 Ð V ------2 2 Γ2 η 64e p p p1 Q ()1 η 1 2 Jc()1 ==E------, ()1 ------, α α ∆˜ 2 3ω2a2 (33) ε ε 1 2 ∆˜ () ()1 = 1 2------1 1 , ()2 2 2 2 2 2 2 2 p1 p2 ∆˜ ()1 ≡λf ()1, ξ = ()+ Γ Ð ξ + 4Γ ξ . ∆˜ () λ2 ()ξξ Γ ()Γ 1 1 = Ð + i 1 + i 2 , The equation for the laser field can be derived from 2 Eq. (1): 16 p p V˜ 2 p3 (29) λ2 ==------1 2 -, Γ ------j , 2 j α2 Q˜ Γ2 4πτ Qη()1 a a j 1 ==------, Q˜ ------0 -. (34) ∆˜ 2 κ ε 2 ξεε ()1 jR ==p jR, Ð 2R, πα πα The basic quantity determining the dependence of 1 2 2 current J (1) and of the laser field corresponding to p1R ==------α , p2R ------α . c 1 + 1a 1 + 2a wavelength λ on ξ is the square of resonance determi-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1154 ELESIN, KOPAEV nant f(1, ξ). The minimum of f(1, ξ) corresponds to the arrive at the following system of inhomogeneous alge- maximum of the current and of the laser field. The braic equations for A j : equation for the extreme values of ξ, i.e., n2 4 4 df()1, ξ 2 2 2 ------== 0 ξξ()+ Γ Ð λ , j ()γ ()β j dξ ∑ A12 exp Ð ja 1 Ð j ==q, ∑ A12 0, j = 1 j = 1 ξ has two solutions. The first solution, 1 = 0, corresponds to the minimum of f for λ < Γ and to its maximum for 4 ε j ()γ λ > Γ. The second solution, ∑ j A12 exp Ð ja = 0, ξ2 λ2 Γ2 λΓ> j = 1 2 = Ð , , (35) 4 gives a minimum of f(1, ξ) for a strong field for λ > Γ. ()ε j j ∑ j A12 Ð A22 = 0, Using relations (34), we obtain the laser power P(1) j = 1 ˜ which is a linear function of pumping current Q for 4 ξ ξ = 2, j ()γ ∑ A22 exp ja = 0, (40) 2 j = 1 2 Q˜ P()1 ==λ ------, (36) 4 4 ()j γ j γ ε α j ε ∑ A22 j Ð A12 j j Ð A12 j = 0, and a root function of Q˜ for ξ = ξ = 0, 1 j = 1 ……………………………………… P()1 = Γ().Q˜ Ð Γ 4 ξ j ε []γ ()()β˜ The physical meaning of solution 2 becomes clear ∑ AN2 j exp j N Ð 1 a 1 Ð j = 0. after calculating the reflectance j = 1 2 ()ξ2 + Γ2 Ð λ2 The solution to this system of equations for an R()ξ ==------, R()ξ 0. (37) ˜ 2 2 N-well structure can be written in compact form. In par- ∆()1 ticular, for the first well, we have ε ε ξ It follows hence that energy 2 = 2R + 2 coincides with j 1 + j 2q the resonant energy of the structure in an ac field. A ()N = ()Ð1 ------[]d DN()Ð P Π()N , (41) 12 ∆()N j j

4. GENERAL SOLUTION d = ∑ ()Ð1 klj++ε ε Γ , (42) FOR AN N-WELL STRUCTURE j k l kl klj≠≠ Let us consider a structure with N QWs. The wave function satisfies the system of equations (12), (13) with ()klj++ε ε ∆α P j = ∑ Ð1 k l kl, (43) boundary conditions (14). In analogy with the case of a ≠≠ single well, we seek the solution in the form (see Sec- klj tion 3) ∆()N = Ð ∆()1 DN()+ ∆()1 Π().N (44) ˜ 4 Determinants D(N) and Π(N) satisfy the recurrence ψ j ()γ n1 = ∑ An1 exp j x , relations j = 1 (38) Dn()= Ð ∆Dn()Ð 1 + ∆Π(),n Ð 1 (45) 4 ˜ ψ j ()γ n2 = ∑ An2 exp j x , Π()n = Ð∆˜ ()D ()∆n Ð 1 + ˜ Π()n Ð 1 , n ≥ 3. (46) j = 1 ˜ where γ is defined by expression (19) or (26) (for The remaining “elementary” determinants have the j form ˜ Ӷ j V 1), and coefficients Anm are connected through γ γ γ γ the relation Ð 1a Ð 2a Ð 3a Ð 4a m1e m2e m3e m4e j ε j An1 = j An2. (39) 1111 ∆()1 = , (47) γ γ γ γ ψ j ε Ð 1a ε Ð 2a ε Ð 3a ε Ð 4a Substituting nm from Eqs. (38) into boundary condi- 1e 2e 3e 4e j j ε ε ε ε tions (14) and expressing An1 in terms of An2 , we 1 2 3 4

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γ γ γ γ nants with other lower, upper, and upper and lower Ð 1a Ð 2a Ð 3a Ð 4a Π m1e m2e m3e m4e lines, respectively. Determinants D(2) and (2) corre- spond to the last well of the structure. In order to sim- 1111 ∆()1 = , (48) j γ γ γ γ plify the notation, we wrote the expression for A12()N ˜ ε Ð 1a ε Ð 2a ε Ð 3a ε Ð 4a 1e 2e 3e 4e for the first well only, assuming that the internal wells ε n ε n ε n ε n are identical. These coefficients are sufficient for calcu- 1 1 2 2 3 3 4 4 lating the reduced current in the first well. There is no need to find the currents in the remaining wells since γ γ γ γ 1 2 3 4 these currents are identical in the model considered γ a γ a γ a γ a here and for steady-state solution (9). Nevertheless, the e 1 e 2 e 3 e 4 ∆()2 = , (49) proof for a two-well structure will be carried out ε ε ε ε 1 2 3 4 through direct calculations. γ γ γ γ The expressions for the determinants and coeffi- ε 1a ε 2a ε 3a ε 4a 1e 2e 3e 4e cients can be simplified if we take into account the 2 smallness of parameters V˜ , p/α, and V˜ α/p: γ γ γ γ 1 2 3 4 ≡εε ∆˜ δ γ a γ a γ a γ a D()2 D = 1 2 12 34, e 1 e 2 e 3 e 4 ∆()2 = , (50) α ˜ ε ε ε ε Π ≡ Πε≈ ε ∆˜ Γ˜ ˜ 28 2 1 2 3 4 ()2 1 234 12 Ð iV ------, γ γ γ γ p1 ε 1a ε 2a ε 3a ε 4a 1n1e 2n2e 3n3e 4n4e (55) 28α ∆()1 ≈ εε ∆ Γ Ð iV˜ ------2 , 1 234 12 p 1111 1 γ γ γ γ αα 1a 2a 3a 4a α 28 e e e e ∆ ≈ ε ε ∆ ∆ ˜ ------2 ∆˜ ()2 = , (51) ()1 1 234 12 Ð iV , ε ε ε ε ˜ p1 ˜ 1 2 3 4 γ a γ a γ a γ a ∆ε≈ ε Γ˜ δ ε 1 ε 2 ε 3 ε 4 1 2 12 34, 1n1e 2n2e 3n3e 4n4e ∆˜ ≈ ε ε Γ˜ Γ˜ ˜ 28 p2 1 212 34 Ð V ------, 1111 p1 γ a γ a γ a γ a e 1 e 2 e 3 e 4 ˜ α ∆˜ ()2 = , (52) ∆ε≈ ε ∆12δ , ε ε ε ε 1 2 34 1 2 3 4 ˜ γ γ γ γ α 28α p 1a 2a 3a 4a ˜ ˜ ˜ ˜ 2 ε ε ε ε ∆ ≈ ε ε ∆12Γ34 Ð V ------, (56) 1e 2e 3e 4e 1 2 ˜ p1 α γ γ γ γ −4 p2 −4 p2 1 2 3 4 d12, ==+------, P12, +------, γ γ γ γ p1 p1 1a 2a 3a 4a D()2 ≡ D = e e e e , (53) ε ε ε ε ε ε Γ ± ˜ 22 p2 1 2 3 4 d34, = 1 212 V ------, ε ε ε ε p1 ˜ 1 ˜ 2 ˜ 3 ˜ 4 α ε ε ∆˜ α ± ˜ 22 p2 P34, = 1 212 V ------. 1111 p1 γ γ γ γ 1a 2a 3a 4a Here, the following definitions have been introduced: Π()2 ≡ Π = e e e e . (54) ε ε ε ε ∆α ()γ ()γ 1 2 3 4 ij = ni exp Ð ja Ð n j exp Ð ia , ε˜ ε˜ ε˜ ε˜ 1 2 3 4 ∆˜ α ()γ ()γ ij = n j exp ja Ð ni exp ia , Here, we have introduced the following notation for the δ γ ()γ γ ()γ determinants: ∆(1) is the determinant for the first well, ij = i exp ja Ð j exp ia , ∆()1 is that with a different lower line, and ∆(2) = Γ ==exp()Ðγ a Ð exp()Ðγ a Γ˜ *, (57) ∆˜(3) = … ∆(n) = ∆ are determinants for the internal ij j i ij ˜ ˜ ε ε ()γ wells (2 < n ≤ N), where ∆ , ∆ , and ∆ are the determi- ˜ j = jm˜ j exp ja , ˜ ˜ JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1156 ELESIN, KOPAEV

m ==1 Ð1β , m˜ Ð β˜ , Taking into account relations (61) and (62), we can j j j j write the wave functions for the first well in the form αγ n j = + j. ψ ()3 () () 12()2 = 2iA12 2 sin p2 x , Using relations (55) and (56), we can derive simpli- (66) fied expressions for the remaining determinants and ψ ε ()1 () () 11()2 = 2i 1 A12 2 sin p1 x . j coefficients Anm for an N-well structure. Substituting these expressions into Eq. (15), we obtain the reduced current in the first well: 5. TWO-WELL STRUCTURE Many basic features of a multiwell structure are 0 1 manifested even for N = 2. In view of the relative sim- J ()2 = --- J ()2, x dx 1c a ∫ 1c plicity of this structure, we will analyze it in detail. (67) Wave functions (38) for N = 2 assume the form Ða ()1 () 4 ε []* 3 = Ð4ie 1M12 A12 ()2 A12 (2 )+ c.c. . ψ j ()γ k2()2 = ∑ Ak2()2 exp j x , j = 1 (58) Using relations (61) and (62), we arrive at the fol- 4 lowing expression: ψ j ε ()γ , k1()2 ==∑ Ak2()2 j exp j x , k 12. j = 1 8QΦ J ()2 = 4ieε M ()ε ε 3------2 ψ ψ 1c 1 12 1 2 ∆ 2 Here, k2(2) ( k1(2)) is the wave function of the kth ()2 p1 well (k = 1, 2) of the upper (lower) level of the two-well (68) structure. In accordance with relations (41)Ð(44), corre- α Φ ∆˜ * 216 1 p2 1i 12 j × ∆˜ Φ ˜ sponding coefficients A (2) and determinant ∆(2) are i 12 0 + V ------+.c.c. 12 p1 given by

j 1 + j 2q It can easily be seen that the first term in this expression A ()2 = ()Ð 1 ------()d DPÐ Π , (59) 12 ∆()2 j j vanishes, and a contribution comes only from the sec- 2 ond term proportional to the square of the field V˜ and ∆()2 = Ð ∆()1 D + ∆()1 Π. (60) ˜ to the imaginary part of ∆˜ , Using approximate expressions for D, Π, ∆(1), and 12 ∆()1 (see relations (55) and (56)), we obtain 2 ˜ ˜ ˜ ˜ eε M ()ε ε 16α QΦ()∆12* Ð ∆12 J ()2 = Ð------1 12 1 2 1 -, ()1 ()2 ε ε 2q 8ip2∆˜ Φ 1c ∆ 2 A12 ()2 ==A12 ()2 Ð 1 2------∆ ------12 2, (61) p1 ()2 ()2 p1 (69) 64 p2 () () Φ˜ Φ Φ 3 4 = ------1 2. A12 ()2 = ÐA12 ()2 p1 α 2 2q 216 p (62) = Ð()ε ε ------Φ ∆˜ + V˜ ------2 -1Φ , 1 2 ∆()2 0 12 2 1 This fundamental result indicates that the attenua- p1 tion responsible for radiative transitions between the energy levels in the first well is induced by the current 2216 p ∆()2 = ()ε ε ∆˜ ∆ Φ Ð V˜ ------2Φ . (63) ˜ 2 1 2 12 34 0 p 4 of the second (right) well and is proportional to V . The 1 result is also valid for any number of wells N ≥ 2, but 2()N Ð 1 Here, we have introduced the notation the attenuation will be proportional to V˜ . Since α ˜ there are no dissipative processes in a coherent system, Φ ==∆ Γ34 Ð4δ Γ [αsin()p a cos()p a 0 12 34 12 1 2 (64) the attenuation in the extreme right well, which is ()α () ()()] + p1 cos p1 sin p2a + p2 cos p2a sin p1a , “transported” by the current, is responsible for lasing. For this reason, a description of coherent systems Φ α () () 1 = sin p1a + p1 cos p1a , requires that the boundary conditions be taken into Φ = αsin()p a + p cos()p a , account correctly. 2 2 2 2 (65) α α Let us analyze current Jc1(2) in the first well of a Φ 2∆˜ Φ 1∆ Φ 4 = ----- 12 2 Ð ----- 34 1. two-well structure. The key quantity is the resonance p2 p1 determinant ∆(2) defined by expression (63). The basic

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 STARK LADDER LASER WITH A COHERENT ELECTRON SUBSYSTEM 1157 difference between ∆(2) and ∆(1) (see relations (29)) relations (74)Ð(76) are valid for values of ξ from the lies in the emergence of a new resonant determinant interval 0 < ξ < δε. Φ 0 (64). In this approximation, current (69) in the first well The equality assumes the form Φ Γ Γ η() 0 = 0 (70) ˜ 3 1 2Q 2 J1c()2 = V ------2 , ∆˜ ()2, ξ leads to the equation for the spectrum of two tunnel- (77) coupled QWs. Equation (70) has two solutions: an anti- 3 64 p symmetric solution with an unshifted energy and a η()2 = 4e 2M ------2, symmetric solution. The energy of the latter solution is 12 25 shifted downwards and the levels split (see the Appen- 2 dix). The upper level in the first well and the lower level 5()ε ε α α ∆()2, ξ = ------1 2 1 2∆˜ (),2, ξ in the second well are also displaced downwards on the 2 2 α α p1 p2 energy scale due to finite heights 2 and 1 of extreme (78) barriers. Their shift is determined by the equations Γ2 Γ ∆˜ , ε ξξ2 λ2 1 3 1ξ2 2λ2 ()2 = Ð Ð ----- +.i------Ð --- ∆˜ ∆ 2 2 5 Re 12 ==0, Re 34 0. (71) Obviously, the resonance conditions It follows from relations (77) that the dependence of the current on the electron energy ξ (and, hence, on the ε()1 ε()1 ω ε()2 ε()2 ω laser field) is determined only by the square of the mod- 2R Ð 1R ==, 2R Ð 1R (72) ∆˜ , ξ 2 are satisfied only if the levels are shifted identically. In ulus of the reduced determinant ()2 : particular, this means that a sharp resonance is possible , ξ ∆˜ , ξ 2 only for a symmetric solution to Eq. (70). It will be f ()2 = ()2 proved below (see the Appendix) that resonance condi- Γ2 2 Γ2 2 (79) ξ2ξ2 λ2 1 9 1ξ2 2λ2 tions (72) and Eqs. (70) and (71) can be satisfied simul- = Ð Ð ----- + ------Ð --- . taneously if the following relations hold for any α: 2 4 5 4α α The optimal values of the energy ξ of supplied elec- α ==------, α --- , α + α =α . (73) 2 5 1 5 1 2 trons can be determined from the condition for the min- imum of f(2, ξ) (or for the maximum of λ2(ξ)): These relations are preserved for multiwell structures 2 2 also; in this case, α and α are the depths of the first df ()2, ξ 2 2 Γ 2 1 ------= 2 ξξÐ λ Ð -----1 and last wells, respectively. dξ 2 (80) Assuming that relations (73) and (72) are satisfied 2 22 2 Γ 9 22 2 2 and that the energy ε of supplied electrons is close to +2ξ ξ Ð λ Ð -----1 +0.---Γ ξ Ð ---λ = ε 1 the resonance energy 2R, we can write the resonance 2 2 5 determinant and Φ in the form k This equation has three solutions: aα aα ∆˜ 1()ξ Γ ∆ 2()ξ Γ Γ2 12 ==Ð------2- + i 1 , 34 Ð------2- + i 2 , (74) 2 1 2 5 ξ ==0, ξ , --- 2λ Ð ------p1 p2 1 23 3 4 Φ ≈εε5ξ, ξ = Ð , (75) 0 2R 13λ2Γ2 13Γ4 −+ λ4 Ð ------1 +,------1 (81) 5 16 Φ ≈ Φ p1 1 2 p2, 2 = Ð----- , 2 λ2 ξ2 ≈ ξ2 ≈ λ2 λ ӷ Γ 2 -----, 3 , 1. 2 3 5α 3 Φ = ------1 ξ + ---Γ , Solutions ξ and ξ correspond to a minimum of 4 2 5 1 1 3 p1 f(2, ξ), while ξ corresponds to a maximum of this (76) 2 3 Γ function, so that Γ 2 p j Γ ≈ 1 j = ------, 2 ----- . α2 2 9 2 4 2 81 2 4 a j f ()0 ==------Γ λ , f ()λ ------Γ λ , 25 1 100 1 In relations (75) and (76), the terms on the order of 2 (82) ξ δε δε ≈ 2 α λ 4 6 / have been omitted and p / a is the energy f ----- ≈ ------λ . shift exceeding considerably the level width Γ. Thus, 3 27

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1158 ELESIN, KOPAEV In contrast to a single-well structure, the minimum in the second well: ξ λ of f(2) for 1 = 0 exists for any . It should be noted that ξ ≈ λ this minimum is lower than that for 3 . In addition, a 1 ()2q 2 it is more convenient since it does not require electron J ()2 ==--- dxJ ()2, x Ðie------ξ 2c ∫ 2c 2 energy tuning. It is solution 1 that is of greatest impor- a ∆()2 tance. 0 (88) 2 α The corresponding values of current J (2) for ξ = 42D2 8 1 p2 1c 1 × ------M ()∆˜ * Ð ∆˜ . ξ ≈ λ 3 12 12 12 0 and 3 are ˜ 2 V p1

, ξ Qe22M12 9 , λ J1c()2 1 = 0 ==------J1c().2 (83) Substituting the approximate value 9V˜ 4

A comparison of current J1c(2, 0) (83) in the first 82p λ D ≈ Ð------2 (89) well with current J1c(1, ) (33) in a single-well structure 2 ˜ shows that these currents are virtually identical: V 8 , , λ into expression (88), we find that J2c(2) coincides with J1c()20 = ---J1c().1 (84) 9 current J1c(2) (83) in the first well. Thus, reduced cur- ξ ξ rent (3) of a two-well structure is equal to the reduced It should be noted that solutions 1 and 3 lead to a current in the first well: Jc(2) = J1c(2). Since currents linear dependence of the lasing power on the pumping J (2, 0) and J (1, λ) coincide in accordance with rela- current, while ξ leads to a root dependence. 1c c 2 tion (84) (to within a factor of 8/9), the field gener- Let us now determine the wave functions and the ated by the two-well structure for ξ = 0 is given by current for the second well, which will enable us to formula (36) (with Q˜ (8/9)Q˜ ). The total power prove the equality of currents in the first and second P(2) is naturally doubled: wells and to calculate the populations of the levels. ()j ()j ψ Coefficients A21 and A22 of wave functions 22(2) 2Q˜ ψ P()2 = 2------. (90) and 21(2) can be determined from Eqs. (40) for N = 2. 9 We can write approximate expressions with the above accuracy: 6. LEVEL POPULATIONS AND REFLECTANCE () () i()2q D 4α p 1 2 ≈ 2 1 2 FOR A TWO-WELL STRUCTURE A22 = A22 ------∆ ------, ()2 p1 (85) It would be interesting to compare the level popula- (), 2qD 22α p 34 2 ˜ 1 2 tions nij, A = Ð------ε ε ∆ exp()γ , a ± iV˜ ------, 22 ∆()2 1 2 12 43 2 p1 a 1 2 where the matrix is given by n = --- dx ψ . (91) ij a∫ ij 0 1111 γ a γ a γ a γ a ξ ε 1 ε 2 ε 3 ε 4 Calculations made for 1 = 0 lead to the following 1e 2e 3e 4e D2 = . (86) results: ε ε ε ε 1 2 3 4 ε n ε n ε n ε n 2 Γ 2 1 1 2 2 3 3 4 4 4λ 1 n12 ==------Γ n11, n22 -----λ n21, Accordingly, the wave functions assume the form 1 (92) n ==4n , n 4n . ∆˜ 12 21 22 11 ψ , 8qD2 12 []() 22()2 x = Ði------2 -sin p2 xaÐ , V˜ ∆()2 It can be seen that the mutual population of the levels in (87) α same well changes upon an increase in the field of ψ , 8 1 p22qD2 2 () wavelength λ. At the same time, the ratio of populations 21()2 x = Ði------sin p1 x . V˜ p2∆()2 of the upper level in the first well and the lower level in 1 λ ӷ Γ the second well is preserved. For 1, we have ӷ Ӷ Using these functions, we can find the reduced current n12 n11, while n22 n21; i.e., the first well exhibits

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 STARK LADDER LASER WITH A COHERENT ELECTRON SUBSYSTEM 1159 population inversion, while the second well does not. It with the reflected fraction (1/9). A similar conservation follows hence that the inverse population condition is law holds for ξ = λ also. not required for coherent lasing. This conclusion is con- ξ λ ξ firmed for 3 = also. Indeed, in the case of tuning = λ ӷ Γ, the level populations in each well are identical: 7. THREE-WELL STRUCTURE It will be shown below that the properties of struc- n12 ==n11, n22 n21,4n11 =n22. (93) tures with even and odd numbers of wells differ consid- erably. By way of the simplest example of an odd N, we It should be noted that a two-well structure for ξ = 0 and consider a three-well structure. The expressions for ξ = λ behaves as a single-well structure for ξ = λ in the coefficients of the first well in the three-well structure sense that the ratio of populations of “extreme levels” can be obtained from the general formulas (41)Ð(44): (the upper level of the first well and the lower level of the second well) is independent of the field. j 1 + j 2q A ()3 = ()Ð 1 ------[]d D()3 Ð P Π()3 , (97) 12 ∆()3 j j It was proved earlier for N = 1 [7] (see Section 3) that the optimal values of energy ξ correspond to a res- onance of the nanostructure in the field in which the ∆()3 = Ð∆()1 D()3 + ∆()1 Π(),3 reflection from the structure becomes minimal. Let us ˜ (98) D()3 ==Ð ∆D + ∆Π, Π()3 Ð ∆˜ D + ∆˜ Π. find the reflectance for a two-well structure. The reflec- ˜ ˜ tance is defined as Using the smallness of parameters V˜ , p/α, and ψ , 2 2 12()2 Ða ˜ R = ------1Ð . (94) V α/ p and approximate expressions for ∆, D, and Π, q we obtain

ψ Using formula (66) for 12(2, x), we obtain the reflec- 2 α 8iα p tance ≈ ()ε ε 2δ ∆˜ Φ ˜ ∆˜ 1 2 D()3 Ð 1 2 34 12 0 + V 12------2 - , p1 iΓ Φ 2 R = ------1 3 Ð 1 ,  ˜ Π ≈ ()ε ε 2 ∆˜ Γ˜ Φ ˜ 28ip2 5∆()2 ()3 Ð 1 2  12 34 0 + V ------(99)  p1 2 (95) Φ p1 Φ ∆˜ ˜ 264 p2Φ ξξ()Γ λ2 3 = Ð----- 0 12 + V ------2 1 = 5 + i 1 Ð 4 . 2 α α α 464iαα p  1 p1 × Ð2∆˜ Φ + -----1∆˜ Γ˜ Ð V˜ ------1 -2 . 12 2 p 12 34 3 1 p1  The reflectance in the ξ = 0 and ξ2 = λ2 modes is given, respectively, by Using these expressions, we derive from Eq. (97)

, 1 , λ 49 R()20 ==---, R()2 ------. (96) ()3 ()4 2q 3 2 28ip 9 81 () ()ε ε ∆˜ Φ ˜ 2 A12 ()3 ==ÐA12 3 ------∆ 1 2 12 0 + V ------()3  p1 ξ (100) It follows hence that the mode with = 0 is much more 2 ξ λ α α Φ α 464iαα p  effective (90%) than that with = (40%). ×∆˜ 1 0 ∆ ∆˜ Φ ˜ 1 2  12------Ð 2 12 12 2 Ð V ------3 - , p1  Let us prove that the law of conservation of the num- p1 ber of transmitted and reflected electrons holds. For this purpose, we find the number of electrons performing a () () 4 p transition between the energy levels in the first well: 1 2 2q ()ε ε 2 2 A12 ()3 ==A12 ()3 ------∆ 1 2 ------()3 p1 a 2π/ω  α Φ 1 ˜ ˜ 216ip2 2 1 1 ˜ (),, × 2i∆12Φ Φ Ð V ------Φ ------Ð α∆12 ------∫dx ∫ dtJ1c 2 xtEt(). 0 2 2 (101) 2π  p1 p1 0 0 2 2 464iα α p  ξ ˜ 1 2  It is found that, for = 0, it is equal to 8/9 of the flux + V ------3 - , supplied to the structure, which gives unity together p1 

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1160 ELESIN, KOPAEV ξ Φ  formula (68)). Indeed, for = 0 and 0 = 0, the second ∆ ()ε ε 3 ∆˜ ∆ Φ2 ˜ 216 p2 ()3 = 1 2 Ð 12 34 0 + V ------ξ λ ˜ 4  p1 term is zero, while for ~ it gives V . ξ Solution 1 = 0 corresponding to a maximum of f(3) ˜ 44α α Φ  is not optimal for N = 3. Indeed, substituting ξ = 0 into ×Φ[]Φ Ð 2∆˜ ∆ Φ Φ + V˜ ------1 2 -  1 0 4 12 34 1 2 2 formula (104) for current, we obtain p1  1 25α α J ()3 ∝ ------, (105) ()ε ε 3 1 2∆˜ 1c ˜ 3 = 1 2 ------2 2 (),3 (102) V p1 p2 2 2 ˜ ∝ 1/2 ∆˜ ()3 = Ðξ ()ξξ + iΓ ()+ iΓ i.e., the root dependence V Q (see Section 3), as 1 2 in the case of a single well. λ229ξ2 ξΓ˜ 4 λ4 The optimal solution is ξ = λ, for which J (3) ∝ + ------+ i Ð ------, 4 1c 25 25 2 1/V˜ and V˜ ∝ Q; the situation is similar to that in a sin- 3 4 ξ λ Γ˜ = ---Γ + ------()Γ + Γ . gle-well structure. Since the solution with = was 5 1 25 1 2 analyzed in detail for N = 1 [6, 7] (see Section 3), we will consider below only even structures. This is justi- It should be noted that, in the case of a three-well fied the more so that the optimal situation in this case is 4 ξ structure, we must retain the fourth-order term V˜ in that with = 0, which can be realized in experiments more easily. the expression for ∆˜ ()3 . Let us begin our analysis with the resonance deter- 8. EVEN STRUCTURES WITH N ≥ 4 ∆˜ 2 minant f(3) = ()3 . It can easily be demonstrated that We will start with a four-well structure. In order to ξ ξ f(3) has four extrema i (we give i for large values of find determinant ∆(4) as well as D(4) and Π(4), i.e., ξ, λ ӷ Γ): ∆()4 = Ð ∆()1 D()4 + ∆()1 Π(),4 2 2 4 2 2 29 2 2 2 ˜ ξ ====0, ξ ------λ , ξ ------λ , ξ λ , (103) D()4 = Ð ∆D()3 + ∆Π(),3 (106) 1 2 25 3 50 4 ˜ Π()4 = Ð ∆˜ D()3 + ∆˜ Π(),3 ξ ξ which correspond to maxima for 1 and 3 and to min- ˜ ξ ξ ima for 2 and 4. we will use formula (99). In the approximation adopted here, we have ξ ξ λ Thus, solutions 1 = 0 and 4 = in the three-well structure correspond to a maximum and a minimum of  f(3), respectively. Consequently, the three-well struc- ∆ ()ε ε 2 ∆˜ ∆ Φ3 ()4 = Ð 1 2  12 34 0 ture is a combination of one- and two-well structures. It  ξ behaves analogously to a structure with N = 1 for 1 = 2 0 and ξ = λ and to a structure with N = 2 for ξ = λ. 216 p2 2 464 p2 + V˜ ------[] Φ Φ Ð 4 Φ ∆ ˜ Φ Φ + V˜ ------(107) p 0 4 0 12 1 2 2 Let us determine the current in the first well of a 1 p1 three-well structure: α α  × Φ αΦ 1 2Φ Φ Φ Φ Φ 2 5 2 0 4 + 2------1 2 + 4 1 2 4 . ()2q ()ε ε 2  1 2 ()∆˜ Φ3Φ p1 p2  J1c()3 = ------2i 12 0 2 + c.c. ∆()3 2 p 1 The structure of the determinant has a clear physical α Φ Φ ˜ 2Φ232 p2 1 1 2 ()∆˜ ∆˜ * (104) meaning. The first term is the product of the determi- + V 0------i 12 Ð 12 p2 nants for the outer wells and three determinants of col- 1 lectivized energy levels of the inner wells. It is impor- ˜ 4Φ2Φ2α ()∆˜ ∆˜ * 2 + V 2 1 1i 12 Ð 12 . tant that the second term with V˜ also contains deter- Φ ξ minant 0. In addition, we must retain the terms with The last term in this expression corresponds to = 0 4 6 V˜ as in ∆(3). since for ξ ~ λ the omitted terms are proportional to V˜ . It can be proved that solution ξ = 0 corresponds to It can be seen from Eq. (104) that the first term is 1 a minimum of |∆(4)|2 and is optimal. This property is equal to zero, as in the case of a structure with N = 2. universal for structures with an even N (in particular, ˜ 4 ˜ The last two terms are proportional to V Im∆12 (see for N = 2).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 STARK LADDER LASER WITH A COHERENT ELECTRON SUBSYSTEM 1161 ξ Φ For the optimal solution 1 = 0, 0 = 0, determinant tons in each well. The electrons supplied due to reso- ∆ j nant tunneling perform N transitions from the upper to (4) and coefficients A12()4 assume the form the lower levels, preserving their phase, this process being independent of the level occupancy. Conse- ∆ ()ε ε 4 ˜ 48 p2Φ Φ˜ ()4 = Ð 1 2 V ------4 , (108) quently, the gain and lasing in such a system are volume p1 effects in contrast to the incoherent case [8]. The total 2 lasing power is proportional to the number N of the () 2q()ε ε 8 p A 3 ()4 = Ð------1 2 ------24α Φ Φ˜ , wells. 12 ∆()4 2 1 1 p1 Thus, we can assume that lasing at vertical transi- (109) tions (see the Introduction) in a superlattice with a ()ε ε 2 ()1 2q 1 2 8ip2∆˜ Φ Φ˜ strong constant field (Stark ladder) is possible. Pro- A12 ()4 = Ð------12 2 . ∆()4 p2 ceeding from the results obtained in [9], we can expect 1 Stark ladder lasing with diagonal transitions also. It should be noted that, in the expression for A3 ()4 , we It has been found that effective lasing requires the 12 fulfillment of the following conditions: observance of ˜ have omitted the term proportional to i∆12 , which is resonance in each well, the choice of optimal energy of compensated in the current in analogy with structures supplied electrons, the sharpness of the electron energy with N = 2 or 3. Comparing relations (108) and (109) distribution ∆ε, and, finally, the fulfillment of the coher- Φ with the corresponding formulas (for 0 = 0) (61)Ð(63) ence conditions for the electron subsystem. for a two-well structure, we can derive the following We have proved that the resonance conditions can relations: be satisfied by choosing the barrier parameters (see the ∆ ∆ ()Φε ε ˜ Appendix). For structures with an even N, the problem ()4 = ()2 1 2 , (110) of choosing energy can be solved quite easily: the ε ε ()1 ()1 ()3 ()3 energy must be equal to the resonance energy ( = 2R) A12 ()4 ==A12 (),2 A12 ()4 A12 ().2 (111) for any fields. Thus, the coefficients for the first well of a four-well As regards the width ∆ε, the optimal situation is structure coincide with the coefficients for a two-well attained for ∆ε ≤ Γ. If ∆ε > Γ, the lasing parameters structure. Consequently, the reduced current of a struc- (efficiency, reflectance, and current) naturally decrease. ture with N = 4 is given by However, the equality ∆ε ≈ Γ can be attained in differ- ent ways, e.g., by using energy filters based on QWs. J ()40, = J ().20, (112) c 1c Naturally, the requirement of coherence is the most Property (111) is preserved for any even N, so that the stringent: the time τϕ of coherence breakdown must be reduced current for such a structure has the form longer than the time TN of passage through the struc- ΓÐ1 λ Γ J ()N, 0 = J ().20, (113) ture. This time can be estimated as TN = N for < c 1c λÐ1 λ Γ or TN = N for > . In all cases, the length of the Generalizing formula (90), we find that the total power structure is finite: of an N-well structure is N < τ Γτ, λ. (115) ˜ max ph ph 2Q τ Γ PN()= N------, (114) Parameter ph for high-quality QWs can in principle 9 be much greater than unity. For example, in [11], a i.e., is proportional to the number N of wells. coherent effect of superfluorescence on ten wells (i.e., Γτ Obviously, the reflectance remains unchanged (R = ϕ > 10) was observed. 1/9) and independent of N since the reflection is deter- Condition (115) becomes less stringent if we take mined by the wave function for the first well. into account the fact that the fields being generated can be strong; i.e., λ ӷ Γ. The use of quantum wires and dots weakens condition (115) still further. The techno- 9. CONCLUSIONS logical breakthrough in producing ensembles of We have proved the possibility of coherent lasing of extremely high quality quantum dots [12] renders them a structure consisting of any number of wells in the quite promising for designing lasers with a coherent absence of dissipative processes. It is well known (see, subsystem on their basis. for example, [5]) that the emission of a photon requires attenuation determined by the interaction with phonons in bulk systems. In the structure studied by us here, ACKNOWLEDGMENTS attenuation is associated with the departure of electrons This study was carried out in the framework of the from the lower level of the extreme right well. The Federal Special Program (project no. A0133) and sup- steady-state intersubband current “transports” this ported by the Ministry of Industry and Science of the attenuation to all wells and causes the emission of pho- Russian Federation (project no. 99-1140) and the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1162 ELESIN, KOPAEV project “Construction of the Theory of Interaction of together with formulas (A.2)Ð(A.4) leads to the rela- Strong Electromagnetic Fields with the Electron Sys- tions tem of Resonance-Tunnel Diodes and Lasers.” 4α α α ==------, α --- . (A.6) 2 5 1 5 APPENDIX In order to satisfy the resonance condition in a mul- It can easily be proved that the resonance condition (72) tiwell structure, it is necessary that the energy differ- for the second well is also observed if relation (A.6) ence between the resonance levels of the wells be the holds; α can assume any value. It should be noted that, same and equal to the frequency ω of the electromag- if we choose netic field. If the wells were isolated, these conditions 4α 9 α 9 could be satisfied by choosing identical wells. How- α ==------+ ------, α --- Ð ------, (A.7) ever, in order to ensure the passage of current, the 2 5 10 1 5 10 α α α heights 2, 1, and of the barriers must be finite. This leads to splitting of degenerate levels and to a level shift the resonance conditions are observed to within the in the outer wells, which is described by Eqs. (70) terms quadratic in 1/α. and (71). We will first prove that these equations are satisfied simultaneously when the last equality in (73) REFERENCES holds. It follows from Eqs. (71) that 1. A. F. Kazarinov and R. A. Suris, Fiz. Tekh. Poluprovodn. () α ()α (Leningrad) 6, 135 (1972) [Sov. Phys. Semicond. 6, 109 p1 cos p1a = Ð 1 sin p1 , (A.1) (1972)]. () α ()α p2 cos p2a = Ð 2 sin p2 . 2. J. Fainst, F. Capasso, D. L. Sivco, et al., Science 264, 553 (1994). Substituting these relations into Eq. (70), we obtain 3. J. Fainst, F. Capasso, C. Sirtori, et al., Appl. Phys. Lett. 66, 538 (1995). Φ ()()αα()α 4. V. F. Elesin and Yu. V. Kopaev, Solid State Commun. 96, 0 ==4sin p1a sinp2a Ð 1 Ð 2 0. (A.2) 987 (1995); V. F. Elesin and Yu. V. Kopaev, Zh. Éksp. Teor. Fiz. 108, 2186 (1995) [JETP 81, 1192 (1995)]. This leads to the last equality in (73). 5. V. M. Galitskiœ and V. F. Elesin, Resonance Interaction We can now find the resonance values of energy for of Electromagnetic Fields with Semiconductors (Éner- Eq. (70): goatomizdat, Moscow, 1986). 6. V. F. Elesin, Zh. Éksp. Teor. Fiz. 112, 483 (1997) [JETP 85, 264 (1997)]. π2 εAS π2 2 εS 10 7. V. F. Elesin, Zh. Éksp. Teor. Fiz. 122, 131 (2002) [JETP 1R ==/a , 1R -----1 Ð ------. (A.3) a2 αa 95, 114 (2002)]. 8. G. Bastard, Y. A. Brum, and R. Rerrera, Solid State Phys. 44, 229 (1991). εAS εS Energies 1R and 1R correspond to the antisymmetric 9. V. F. Elesin, V. V. Kapaev, Yu. V. Kopaev, and A. V. Tsu- and symmetric solutions, respectively. The energy of the kanov, Pis’ma Zh. Éksp. Teor. Fiz. 66, 709 (1997) [JETP upper level in the first well is given by (see Section 3) Lett. 66, 742 (1997)]. 10. V. F. Elesin and A. V. Tsukanov, Fiz. Tekh. Poluprovodn. 2 (St. Petersburg) 34, 1404 (2000) [Semiconductors 34, ()1 4π 2 ε = ------1 Ð ------. (A.4) 1351 (2000)]; V. F. Elesin and A. V. Tsukanov, Phys. 2R 2 α a 2a Low-Dimens. Semicond. Struct. 11/12, 125 (2000). 11. S. Haas, T. Stroucken, M. Hübner, et al., Phys. Rev. B The resonance condition (72) 57, 14860 (1998). 12. D. Bimberg, M. Grundmann, F. Heinrichsdorff, et al., Thin Solid Films 367, 235 (2000). π2 ε ε ω 3 2R Ð 1R ==------2 (A.5) a Translated by N. Wadhwa

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 985Ð992. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1123Ð1130. Original Russian Text Copyright © 2003 by Bliokh, Kontorovich. GRAVITATION, ASTROPHYSICS

On the Evolution and Gravitational Collapse of a Toroidal Vortex K. Yu. Bliokh and V. M. Kontorovich* Institute of Radio Astronomy, National Academy of Sciences of Ukraine, Krasnoznamennaya ul. 4, Kharkov, 61002 Ukraine *e-mail: [email protected] Received December 23, 2002

Abstract—The evolution and collapse of a gaseous toroidal vortex under the action of self-gravitation are con- sidered using the Hamiltonian mechanics approach. It is shown that evolution occurs in three main stages sep- arated by characteristic time scales. First, a compression along the small radius to a quasi-equilibrium state takes place, followed by a slower compression along the large radius to a more stable compact vortex object. In the latter stage, the possibility of effective scattering and ejection of particles along the vortex axis (jet formation) is detected. As a result, mass, energy, and momentum losses take place, and the vortex collapses. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION A possible relation between this problem and the existence of occluding toroids1 in the vicinity of central Vortices, which are traditionally described in the compact objects in the active galactic nuclei is men- framework of incompressible fluid dynamics [1, 2], are tioned [14]. special objects of application of Hamiltonian methods. In the epoch of solitons, the interest in vortices as allied 2. FIRST STAGE: localized formations has grown considerably. The cur- EVOLUTION OF A THIN VORTEX rent state of this problem is described in reviews and articles [3Ð7] and in the references cited therein. We assume that the shape of a vortex at the initial stage is a thin toroid (Fig. 1) whose radii satisfy the ine- Compressibility and self-gravitation of such objects quality may play a significant role in astrophysical applica- r Ӷ R. (1) tions [8Ð10]. In some cases (e.g., in the vicinity of com- pact objects at the centers of galaxies), the effect of the (An analog of such a vortex in fluid dynamics is a Max- external gravitational field on these objects may be sig- wellian vortex [1, 2].) We will henceforth assume that nificant. Both of these possibilities will be taken into the vortex evolution at the initial stage occurs without a change in the toroidal shape. Thus, in addition to a rota- account in this study. tional degree of freedom, the system possesses two Another extremely important problem in contem- translational degrees of freedom corresponding to porary astrophysics is associated with the origin of changes in r and R (we disregard the translational cosmic jets arising, according to prevailing concepts, motion of the vortex as a whole). It should be noted that, in view of condition (1), the motion in r and rota- in accretion disks of various origins and scales (from tion constitute a local compression of a rotating cylin- galactic [11] to stellar [12]). The interpretation of such der, while the motion in R indicates the collapse of a jets is a nontrivial problem. In spite of considerable thin ring. advances made in this direction in the framework of We write the system Hamiltonian in the form magnetohydrodynamics (MHD) [13], serious difficul- ties still remain since solutions were obtained in a spe- 2 1 2 2 pϕ cial geometry and strong magnetic fields are required. H = ------p ++p ----- + Ur()., R (2) 2M r R 2 It will be shown below that the formation of (unidirec- r tional) jets is a natural consequence of the evolution of self-gravitating vortices and is an indispensable con- 1 We do not touch upon the classical problem of stability of toroids rotating as a single entity, which dates back to Poincaré and dition of their collapse. Jets are also generated in zero Dyson. The modern state of affairs in the framework of the gen- magnetic fields. eral theory of relativity and references can be found in [15].

1063-7761/03/9606-0985 $24.00 © 2003 MAIK “Nauka/Interperiodica”

986 BLIOKH, KONTOROVICH In the present case of a thin toroid, we have χ = M/2πR, ϕ r whence

M2 r R Ur(), R = G------ln--- + c ()R . (8) πR R 1 In order to determine the dependence of potential energy U on large radius R, we consider the second equation of motion, Eq. (4b). It must describe the grav- itational contraction of a thin ring of radius R. The force Fig. 1. Thin toroidal vortex of the Maxwellian type. acting on a test particle located on an infinitely thin ring is given by

π Here, M is the total mass of the vortex, p are the ()ϑ s Mm d /2 ∞ momenta corresponding to coordinates s, ϕ is the cyclic FR ==ÐG------∫------. (9) 2πR2 sin()ϑ/2 coordinate of rotation, and U is the gravitational poten- 0 tial energy of the system. Hamilton equations corre- sponding to Hamiltonian (2) have the form This formula can be derived by direct integration of the contributions from the interaction of the particle with p p pϕ all elements of the ring. In order to avoid divergence for r ú R ϕú ϑ rú ===-----, R ------, ------2, 0, we must take into account the finite thickness M M Mr of the ring. For this purpose, we truncate the diverging (3) 2 part of expression (9), replacing the integration domain pϕ ∂U ∂U π ϑ π ϑ α α pú ===------Ð ------, pú ------, púϕ 0. (0, ) by ( c, ), where c = r/R ( ~ 1 is a numerical r 3 ∂ R ∂ Mr r R factor). This gives

This leads to the equations of motion for the transla- M2 α˜ r Ur(), R = G------ln------+ c (),r (10) tional degrees of freedom, 2πR R 2 2 pϕ 1 ∂U where α˜ = α/e. Setting rúú = ------Ð ------, (4a) M2r3 M ∂r c2 = 0 1 ∂U Rúú = Ð------, (4b) in Eq. (10) and M ∂R 2 GM α˜ and to the integral of motion for the rotational degree of c1()R = ------π - ln freedom: R in Eq. (8), we note that formulas (8) and (10) can be 2ϕ pϕ ==Mr const. (5) reduced to the same form (the difference will be only in the coefficient 1/2). Such a difference is insignificant This integral of motion expresses the angular momen- for our analysis, and we assume the true numerical π tum conservation law (the quantity 2 pϕ/M in fluid coefficient in formula (8). dynamics corresponds to vorticity). We can now write the Hamiltonian (2) of a thin tor- Let us now define function U(r, R). We consider first oidal vortex: Eq. (4a) describing the evolution of a rotating cylinder of radius r. The gravitational force acting on a test par- 2 2 1 2 2 pϕ M α˜ r ticle on the surface of the cylinder is given by H = ------pr ++pR ----- + G------ln------. (11) 2M r2 πR R 2χm F = ÐG------, (6) r r In this case, the equations of motion (4) assume the form where G is the gravitational constant, χ is the mass of 2 the cylinder per unit length, and m is the mass of the test pϕ M rúú = ------Ð G------, (12a) particle. Thus, the gravitational force appearing on the 2 3 πRr right-hand side of Eq. (4a) has the form M r 1 ∂U 2χ M αr Ð------= ÐG------. (7) Rúú = ÐG------ln------. (12b) M ∂r r πR2 R

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In view of condition (1), we have Let us determine some of the most important parameters of the vortex, characterizing it at the sec- r r --- ln--- Ӷ 1. ond stage (15). Using the angular momentum conser- R R vation law (5) and (13), we obtain the following expres- sion for the velocity of particles on the vortex surface: Then, the force of gravitational attraction along r (12a) 2 is much stronger than the force of gravitational contrac- pϕ GM GM r tion along R (12b). This allows us to divide the evolu- v = ------∼∼------. (16) Mr πR π pϕ R tion of the system over different scales: fast (in r) and slow (in R). It is natural to assume that fast evolution It can be seen that the rotational velocity of a vortex, (for practically constant R) leads to the establishment of which is in equilibrium in r, is determined only by its equilibrium in Eq. (12a). In this case, the force of grav- mass and the large radius and increases upon contrac- itational attraction is compensated by the centrifugal tion along R.2 rotational force: When the vortex reaches the end of the first stage, 2 the velocity attains the value pϕ M ------Ð0G------= , (13) 2 2 3 πRr ∼ CM M r v fin ------(17) π pϕ whence and becomes much larger than the initial velocity v0: 2 π pϕ R r = ------. v fin R0 r0 R0 eq 3 ------∼∼∼------ӷ 1. (18) GM v 0 Rc Rc r0 This expression determines the equilibrium small In analogy with relation (14), we can estimate the radius as a function of the large radius, req = r(R), while time of the vortex contraction along the large radius as the inequality r > req corresponds to the criterion of R Ð R gravitational instability with the Jeans scale req. The ∼ 0 c t2 ------[], latter becomes obvious if we assume that the toroid FR req()R0 R0 mass is M ≈ ρπr2R, where ρ is its density [8Ð10]. (19) π 2()π 2 3 As a rough estimate of the time of vortex contraction R0 R0 Ð pϕ/GM ӷ = ------t1, along small radius r to the quasi-equilibrium state, we 2 3 GMln()απpϕ/GM R can use the expression 0 where FR is the force appearing on the right-hand side t ∼ r0 Ð req()R0 of Eq. (12b). 1 ------, Fr()r0 R0 Let us also consider the distribution of the kinetic (14) energy acquired by the vortex over the degrees of free- 2 4 3 M R Ð pϕr πMR /G dom. We assume that the substance in the vortex is = ------0 0 0 , 2 3 2 π initially almost free and its potential energy and pϕ Ð GM r0/ R0 2 π kinetic energy (11) are small as compared to GM / Rc. where r0, R0, and Fr are the initial values of the small Then, at the critical stage (15) of the collapse, the and large radii of the vortex and the force appearing on potential and kinetic energies of the substance can be the right-hand side of Eq. (12a), respectively. estimated as If inequality (1) holds at the initial instant, the toroi- GM2 GM2 G2M3 dal vortex will experience, in accordance with the equa- U ∼ Ð------lnα˜ ∼ Ð------= Ð------, fin π π 2 2 tion of motion (12b) and relation (13), a slow contrac- Rc Rc π pϕ tion along both radii until they become on the same (20) 2 2 5 order of magnitude: ()rot pϕ G M T ∼ ------= ------. fin 2 π2 2 π 2 2 2MRc 2 pϕ ∼ pϕ req Ӷ req R ==Rc ------=------r0. (15) GM3 R (It should be noted that this result is in accordance with the virial theorem for U ~ RÐ1; i.e., E = ÐT < 0.) Since At this stage, the initial assumptions (1) are violated, the total energy is an integral of motion and its initial and the description used above becomes inapplicable. It is impossible in this case to divide the vortex evolution 2 It should be noted that this is due to the fact that momentum pϕ is over two translational degrees of freedom r and R, and canceled out in relation (16). This is a consequence of the equilib- the vortex should be described as a single compact rium condition (13), whose form is determined in turn by the object with a complex internal structure. form of potential (11).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 988 BLIOKH, KONTOROVICH energy of the vortex. Hamiltonian (21) corresponds to the equation of motion

2 4 pϕ GM Rúú = ------Ð ------. (22) M2 R3 R2

This equation has an equilibrium position, when

2 2 4 pϕ GM 4 pϕ ------Ð0or------==Rc1 ------, (23) M2 R3 R2 GM3

where Rc1 plays the role of the Jeans scale as before. Thus, this object is in equilibrium for a radius on the same order of magnitude as that at which the first stage of evolution terminates (cf. relation (15)). This means that, in the problem on the collapse of a thin toroidal vortex, there is no need to consider the evolution of a compact vortex separately. We can assume that equilib- Fig. 2. Cross section of a compact spheroidal vortex of the rium sets in immediately after the ring acquires param- Hill vortex type. eters (15)Ð(18). On the other hand, we can consider the problem of energy is close to zero, relations (11) and (20) imply contraction of a vortex, which resembles a Hill vortex that the kinetic energy of translational motion is of the from the very outset (Fig. 2), but is initially far from same order of magnitude. Thus, we can conclude that equilibrium. Then, we ultimately arrive at an equilib- the kinetic energy released during the contraction is rium compact vortex with a radius on the same order of distributed uniformly (in order of magnitude) between magnitude as the radius in (23). The rotational velocity the rotational and translational degrees of freedom. of particles in this case is of the order of velocity (16): This fact will be important for the subsequent analysis; 2 in particular, this means that, if the rotational velocity ∼ GM v fin ------, (24) of a particle of the substance is less than doubled at ran- 2 pϕ dom, it is sufficient for the particle detachment and escape from the system. where

v R 3. SECOND STAGE: ------fin ∼ ------o ӷ 1. EVOLUTION OF A COMPACT VORTEX v o Rc1

Let us try to imagine the scenario of contraction of Similarly, it can be easily proved that, upon the a gravitating vortex, when it is a compact object topo- establishment of equilibrium (23), at least half the logically equivalent to a toroid. We can expect that, released potential energy is transformed into the kinetic under the action of gravitational forces, it will approach energy of rotation (the remaining part being trans- a certain spheroidal configuration resembling a Hill formed into heat). vortex [1, 2] (Fig. 2). If we consider such an object as an estimate, we can assume that it possesses a rotational and a translational degree of freedom. The latter is 4. SCATTERING AND DETACHMENT determined by a change in its radius R. The rotational OF PARTICLES radius of particles (which was the independent quantity Thus, after various possible stages of evolution, a r in the previous section) is now approximately equal to toroidal vortex is transformed into a compact object R/2. The Hamiltonian of such a vortex can be written in with characteristic parameters (15), (17), and (18) the form (or (23) and (24)) (see Fig. 2). The rotation velocity of 2 2 2 the substance in it is much higher than in the initial vor- p 2 pϕ GM H = ------R + ------Ð ------. (21) tex. It is worth noting that flows of matter passing 2M MR2 R through the vortex in the vicinity of its axis are closely spaced. This means that effective scattering of particles Here, we have assumed that the form of the second term may take place in this region. Such a scattering will is the same as in relation (11) with r = R/2 and have obviously increase the velocity of a certain fraction of taken the potential energy of a sphere for the potential particles. In accordance with the arguments given at the

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ξ ξ (a) (b)

0 ρ 0 ρ

ξ ξ (c) (d)

0 ρ 0 ρ

ξ ξ (e) (f)

0 ρ 0 ρ

ρ ξ Fig. 3. Typical finite trajectories of a particle in the gravitational field of the ring. In the plane ( = r/r0, = z/r0), the particle moves ρ ρ ξ ρ around two attracting centers formed as a result of the dissection of the ring. The initial conditions are = 1 + 0, = 0, ú = 0, and ξú = 2 (the value of ξú is chosen to coincide with the “orbital velocity” in Eq. (12a), which is valid in the vicinity of attracting ρ ρ ρ centers). (a) 0 = 0.08, rotation around a single center; (b) 0 = 0.17, “dovetail”-type motion around a single center; (c) 0 = 0.22, ρ motion of the double “figure-of-eight” type around two centers; (d) 0 = 0.42, motion of the “figure-of-eight” type around two cen- ρ ρ ters; (e) 0 = 0.81, motion of the “dovetail” type around two centers; (f) 0 = 2, rotation around two centers. end of the previous two sections, a less than double of a compact vortex acquire a sufficient energy as a increase in the rotational velocity of particles is enough result of scattering and are ejected from the vortex. for gathering a kinetic energy sufficient for the detach- Thus, a directional jet carrying away the matter from ment. Consequently, we can expect that a certain frac- the center of the vortex can be formed. (Here, we dis- tion of particles from the flows passing along the axis regard the change in the vortex configuration that

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 990 BLIOKH, KONTOROVICH may take place as a result of the mass loss due to such Differentiating this relation with respect to time, we ejection.) obtain Let us consider one more argument illustrating the 1 pϕ above scenario. In the Appendix, we will consider the ú ∼ ()ú ú R ------púϕ Ð ------3/2 ME + EM . (27) motion of a test particle in the gravitational field of a ÐME 2()ÐME ring (thin toroid) with a fixed radius (Fig. 3). For low We assume that the particle flux is comparatively energies, the particle rotates in a small-radius orbit small and this process occurs at a much lower rate than wound around the ring (Fig. 3a). This motion corre- the rate of establishment of equilibrium of the compact sponds to a thin vortex (the possible first stage of the vortex. In this case, we can estimate the change in the evolution). As the particle energy increases, various characteristics of the vortex carried away by the particle complex trajectories appear; however, the orientations flow as of these trajectories do not correspond to vortex motion and we will not consider such trajectories here. Finally, v 2 J starting from a certain energy value, the particle passes Mú = ÐJ, Eú ∼ J------∼ Ð-----E, 2 M to almost closed trajectories of a figure-of-eight shape (28) (Fig. 3d). The rotational radius of particles becomes on Rv J púϕ ∼ ÐJ------∼ Ð----- pϕ, the order of the ring radius (15), which corresponds 2 M precisely to the final sage of vortex contraction. The kinetic energy of a particle in such trajectories is close where J is the mass flux in the ejected jet of matter. Sub- to the energy required for the detachment of particles. stituting relations (28) into (27) and taking into account The motion of particles in “figures-of-eight” will lead relation (26), we obtain to their effective collisions and scattering in the vicinity 3 J Mú of the vortex axis. Rú ∼ ÐR----- = R-----. (29) We can state that the toroid contraction has qualita- M M tively the same consequences for moving particles as an The solution to this equation has the form increase in their energy for a fixed size of the toroid. Obviously, a tendency ultimately leading to the detach- Mt() β Rt() = R()0 ------. (30) ment of a fraction of particles exists, the most favorable M()0 conditions for this effect being created in the vicinity of the vortex axis. In the long run, this leads to the emer- Here, β ~ 1 is a certain positive constant (emerging due gence of an axial (unilateral) jet carrying away the to the fact that we obtained above only order-of-magni- energy, mass, and angular momentum of the vortex. As tude estimates for the vortex parameters), and the initial a result, the vortex contraction will continue (resulting instant of time corresponds to the arrival of the vortex in collapse), the contraction rate dR/dt being deter- at the compact equilibrium state and to the beginning of mined by the vortex mass loss rate (particle flux in the the effective scattering and detachment of particles. jet; see below). Thus, the vortex collapse and the emer- The time dependence of the vortex mass M(t) is deter- gence of a jet are correlated unambiguously. mined for the specific mechanism of particle scattering. In the general case, the mass flux of matter, J = ÐMú , is 5. VORTEX COLLAPSE a function of the main vortex parameters: mass, energy, and angular momentum. If we assume in the simplest Let us consider the consequences of the ejection of case that the flux of matter is proportional to the vortex particles from a vortex according to the scenario pro- mass and weakly depends on other parameters posed in the previous section. The particle flow carries (J = kM), Eqs. (28)Ð(30) will lead to the exponential away the mass, energy, and angular momentum of the laws vortex. The latter quantities can be estimated as β Mt() ==M()0 eÐkt, Rt() R()0 eÐ kt. (31) Mv 2 MRv E ∼ Ð------, pϕ ∼ ------, (25) 2 2 Thus, Eqs. (29)Ð(31) show that the scattering of par- ticles and ejection of matter indeed lead to the collapse where all the quantities correspond to an equilibrium of a compact vortex. compact vortex (see Section 3) and E = ÐT (see above). The characteristic time scale of the collapse is Relations (25) lead to defined as pϕ ∼ ú R ∼ ------. (26) tcol M/M . (32) ÐME In accordance with the above assumptions, the collapse must be slow as compared to the characteristic time of 3 The existence of flows of matter of the figure-of-eight type also the vortex contraction to the equilibrium state, which follows from the hydrodynamic model of a Maxwell vortex (see, ӷ for example, [2]). corresponds to tcol t2.

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6. GENERALIZATIONS ACKNOWLEDGMENTS We can easily generalize the above analysis to the This study was partly supported by INTAS (grant case when a system contains a massive body at its cen- no. 00-00292). ter and when the vortex rotates about its axis. These fac- tors lead to the emergence of additional terms in Eq. (12): APPENDIX

2 M αr M˜ pϑ Rúú = ÐG------ln------Ð G----- + ------. (33) Motion of Particles in the Gravitational Field πR2 R R2 M2 R3 of the Ring ˜ We assume that the gravitational field of a thin tor- Here, M is the mass of the central object and pϑ = oid is close to the field of an infinitely thin ring of the 2 ϕ MR ϑú = const is the angular momentum associated same mass. Let the ring radius be r0 and r, , and z be with the rotation of the toroid about its axis. The sup- the cylindrical coordinates, z = 0 corresponding to the plementary terms do not affect in any way the evolution plane of the ring. The gravitational potential of the ring of the system along the small radius. Their effect on the is given by vortex evolution along R can be divided into the follow- 2π ing limiting cases. ϕ , GM d Ur() z = Ð------π ∫ ------. ˜ 2 2 2 3 Ӷ π 3 2 2 2 2 ϕ 1. If max(GM /,Rc pϑ /M Rc ) GM/ Rc , the 0 z ++r r0 Ð 2rr0 cos effect of these terms can be disregarded, and the entire ξ dynamic analysis carried out in Sections 2 and 3, as Introducing the dimensionless variables = z/r0 and well as the corresponding conclusions, remains in ρ = r/r and time τ = t GM/2πr3 , we obtain the equa- force. However, the presence of the central mass in the 0 0 region of the most probable intersection of particle tions of motion of a test particle in this potential: flows may affect their scattering and detachment. 2π ()ρϕÐ cos 2 3 ρ'' = Ð ------dϕ, 2. If πM˜ /M Ӷ 1 and π pϑ /GM R ӷ 1, the rotation ∫ 3/2 c ()ξ2 ++ρ2 12Ð ρϕcos of the toroid about its axis arrests contraction before it 0 reaches its critical stage r ~ R ~ Rc. The equilibrium 2π state corresponds to the large radius defined by the rela- ξ ζ'' = Ð ∫ ------dϕ, tion ()ξ2 ρ2 ρϕ3/2 0 ++12Ð cos 3 2 GM R π pϑ where the primes indicate differentiation with respect ------lnα ------= 1 2 3 to τ, and we assume that ϕú = 0 for a particle. π pϑ GM R Figure 3 shows typical results of numerical calcula- and to the small radius defined by substituting the large tions based on these equations for a finite motion. The radius into relation (18). In this case, the probability of trajectories lie in the (ρ, ξ) plane and are given in effective scattering and detachment of particles at the increasing order of the particle energy. It can easily be middle of the vortex virtually vanishes and, hence, col- seen that, in trajectories of the “dovetail” type (see lapse does not take place. Figs. 3b and 3e), a particle moves practically along the same curve in opposite directions; consequently, such π ˜ ӷ π 2 3 Ӷ 3. If M /M 1 and pϑ /GM Rc 1, the revolu- motion cannot be maintained in the framework of col- tion of a vortex around its axis is insignificant, and the lective motion of particles since the latter motion would central mass enhances the contraction. The vortex con- inevitably lead to collisions and strong scattering. In tracts to a compact object and its subsequent behavior addition, the trajectories in Fig. 3c and 3f cannot exist depends on the scenario of direct interaction of the ver- for a collective vortex motion of particles since differ- tex with the central mass. Naturally, the scattering of ent segments of a trajectory correspond to opposite particles and the possible vortex collapse in this case directions of vorticity. Thus, only the trajectories in also depend to a considerable extent on the interaction Figs. 3a and 3d can exist in the framework of collective of the matter with the central mass. vortex motion of particles. The trajectory in Fig. 3a is the cross section of a thin toroidal vortex of the Max- π ˜ ӷ π 2 3 ӷ 4. If M /M 1 and pϑ /GM Rc 1, the last two wellian vortex type considered in Section 2, while the terms on the right-hand side of Eq. (33) compete. If the “figure-of-eight” in Fig. 3d can appear during motion first term is greater than the second (the attraction of the of particles in a compact vortex of the type of a Hill vor- central object prevails), the situation corresponds to tex emerging at the late stage of contraction (see Sec- case 3; for the opposite relation, we have case 2. tion 3).

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REFERENCES 8. D. Linden-Bell, NATO ASI Ser., Ser. B 156, 155 (1986). 1. H. Lamb, Hydrodynamics, 6th ed. (Cambridge Univ. 9. B. F. Schutz, NATO ASI Ser., Ser. B 156, 123 (1986). Press, Cambridge, 1932; Gostekhizdat, Moscow, 1947). 10. J. L. Tassoul, Theory of Rotating Stars (Princeton Univ. 2. M. A. Lavrent’ev and B. V. Shabat, Problems of Hydro- Press, Princeton, N.J., 1978; Mir, Moscow, 1982). dynamics and Their Mathematical Models (Nauka, Mos- 11. Physics of Extragalactic Radio-Sources, Ed. by R. D. Dag- cow, 1973). kesamanskiœ (Mir, Moscow, 1987). 3. P. G. Saffman, Vortex Dynamics (Cambridge Univ. Press, Cambridge, 1995; Nauchnyœ Mir, Moscow, 2000). 12. C. J. Lada, Annu. Rev. Astron. Astrophys. 23, 267 (1985). 4. V. I. Petviashvili and O. A. Pokhotelov, Single Waves in the Plasma and Atmosphere (Énergoatomizdat, Moscow, 13. G. V. Ustyugova, R. V. E. Lavelace, M. M. Romanova, 1989). et al., Astrophys. J. 541, L21 (2000). 5. Yu. A. Stepanyants and A. P. Fabrikant, Wave Propaga- 14. R. Antonucci, Annu. Rev. Astron. Astrophys. 31, 473 tion in Shear Flows (Nauka, Moscow, 1996). (1993). 6. E. A. Kuznetsov and V. P. Ruban, Zh. Éksp. Teor. Fiz. 15. M. Ansorg, A. Kleinwachter, and R. Meinel, gr-qc/0211040. 118, 893 (2000) [JETP 91, 775 (2000)]. 7. V. F. Kop’ev and S. A. Chernyshev, Usp. Fiz. Nauk 170, 713 (2000) [Phys.ÐUsp. 43, 663 (2000)]. Translated by N. Wadhwa

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003

Journal of Experimental and Theoretical Physics, Vol. 96, No. 6, 2003, pp. 993Ð1005. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 123, No. 6, 2003, pp. 1131Ð1144. Original Russian Text Copyright © 2003 by Kozlovskii. ATOMS, SPECTRA, RADIATION

Quantum Dynamics and Statistics of a Bose Condensate Generated by an Atomic Laser A. V. Kozlovskii Lebedev Institute of Physics, Russian Academy of Sciences, Moscow, 119991 Russia e-mail: [email protected] Received March 20, 2002

Abstract—A self-consistent quantum theory is developed for an atomic laser utilizing cooling of atoms in a trap by the method of stimulated evaporation. The model describes the pumping and extraction of the atomic field from a trap upon its interaction with independent atomic reservoirs. The stimulated collisions between atoms in the trap, which produce a Bose condensate in the lower state of the trap, are considered. The interaction of atoms with a phonon field causes spontaneous transitions between the discrete states of the trap. Calculations performed for the three- and four-level models of the trap showed the possibility of generation of a strongly squeezed sub-Poisson Bose condensate. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION laser based on the mean field approximation [16–18] does not allow one to study the quantum statistics of the The recent successful experiments on the produc- field because in this case the preliminarily specified tion of a Bose condensate in traps showed the possibil- phenomenological statistics of the atomic gas are used. ity of creation of coherent sources of material waves Only a completely quantum-mechanical theory permits (atomic lasers) [1, 2]. As the resonator of an atomic one to investigate quantum-statistical effects and to laser, a parabolic magnetic trap is used, the discrete determine the coherent properties of the atomic-laser energy levels of the trap pumped by an external source field. The quantum-mechanical theories of the atomic of cooled atoms representing the resonator modes. The laser being currently developed use a model with a conditions for producing a Bose condensate are real- finite number of atomic-field modes in the trap. Such an ized at least for the lower state of the trap (the conden- approach allows one to study the effect of collisions sate mode). The excitation of atoms from their inner between atoms inside the trap on the dynamics and sta- states to the electronic states, in which atoms can be tistics of a Bose condensate generated by the atomic captured by a trap or extracted from it, is performed by laser. This model of the laser corresponds to the exper- a radio- or microwave electromagnetic field. The con- imental conditions under which processes of stimulated tinuous and pulsed lasing regimes are achieved in cooling (evaporation) rapidly deplete the upper energy atomic lasers by the same means, but at different values levels of the trap. In this paper, the self-consistent the- of the parameters of the electromagnetic field used for ory of Bose condensation is developed for an atomic pumping and extraction of the atomic field from the gas in a trap, taking into account collisions between trap. The lasing dynamics of the atomic laser is deter- atoms under thermally nonequilibrium conditions. The mined by the balance between the introduction of theory is based on the solution of the control equation atoms to the trap and the extraction of the Bose conden- for the density matrix simultaneously with the system sate from the trap taking into account the cooling rate of generalized HartreeÐFock equations for the wave (population of the lower states of the trap), which is functions of the atomic field in the states of the trap. also stimulated by the external electromagnetic field. A The numerical calculations of the dynamics and sta- fundamental property of the atomic laser is the coher- tistics of an atomic field are performed in the approxi- ence of the field produced. The coherence and quan- mation of the eigenfunctions of the trap for the three- tum-statistical properties of the field are studied in the and four-level models of a laser. The model describes quantum optics of atomic fields [3–11]. The modern pumping processes, the extraction of the field from the phenomenological semiclassical theories [12Ð18] and ground state of the trap, and spontaneous transitions, as quantum-mechanical theories [3Ð11, 19Ð29], which well as the collision redistribution of atoms among the consider models of the atomic laser with different states of the trap. The lasing dynamics of an atomic schemes of pumping, cooling, and extraction of the laser is analyzed for different pump rates and different field from the trap, predict the presence of the lasing frequencies of collisions between atoms, and the rates threshold, saturation, and a high degree of coherence of of spontaneous transitions of atoms between the trap the Bose condensate generated by the atomic laser [3, 9, levels are calculated. It is shown that the Bose conden- 10, 23]. Note that the semiclassical theory of the atomic sate generated by an atomic laser exists in a squeezed

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994 KOZLOVSKII with sub-Poisson fluctuations of the The above operators satisfy the boson permutation number of atoms. For large Bose condensates with the relations average number of atoms of the order of 106, the ratio []δΨ , Ψ† of the dispersion of the number of atoms to their aver- α()r α'()r' = αα, '(),rrÐ ' age number (the Fano factor for the atomic field) can αα, ,, achieve 0.5. ' = spout, (3) []ΨΨ , Ψ []† , Ψ† The model of an atomic laser is presented in Sec- α()r α'()r' ==α()r α'()r' 0. tion 2. The effective many-particle Hamiltonian is con- sidered for the open system of colliding Bose atoms in The quantum-mechanical average number of atoms a trap, which describes their interaction with the atomic captured by the trap is pump and loss reservoirs, as well as the phonon field 3 〈〉Ψ† Ψ 〈〉† inducing spontaneous transitions between the discrete ∫d r s()r s()r ==∑ a j a j N. (4) energy levels of the trap. Section 3 is devoted to the j analysis of the self-consistent dynamics of atomic fields. Approximations are considered which are used Along with the atomic reservoirs described above, we for calculations of the wave functions of the atomic will consider the reservoir of a phonon field, which is a field in the trap simultaneously with the solution to the source of spontaneous transitions between the states of equation for the reduced density matrix. In Section 4, the trap (see also [27, 28]). In addition, we will consider the control equation is obtained, within the framework an electromagnetic field involved in the processes of of the BornÐMarkoff approximation, for the density extraction of the field from the trap and trap pumping, operator in the model of an atomic laser with a finite as well as in the process of stimulated cooling of atoms number of levels in the approximation of the eigen- in the trap [1, 2]. states of the trap. The approximation is considered at We consider the effective many-particle Hamilto- which collisions of atoms in the trap do not affect the nian of the problem, which contains the free-energy spatial distribution of the atomic field. The results of terms and the operators of interaction of the system numerical calculations of the dynamics and statistics of with reservoirs, as well as operators describing the the Bose gas are presented in Section 5. Conclusions interaction between atoms in the trap, in the form are formulated in Section 6.

HH= s ++∑HRβ ∑V sRβ +V coll. (5) 2. MODEL β β OF AN EVAPORATION-BASED ATOMIC LASER The Hamiltonian (4) describing the atomic Bose con- We will describe the open system of atoms in a trap, densate in the Hartree approximation can be written, which interact with reservoirs, by the method of sec- using the field operators (1) and (2), as a sum of the fol- ondary quantization of atomic fields using the creation lowing terms: and annihilation operators, which are defined as the energy of atoms in the trap,

3 Ψ φ 3 φ Ψ Hs = ∫d r s()r ==∑a j j(),r a j ∫d r *j ()r s(),r (1) j ប2 (6) × Ð------∇Ψ†()r ∇Ψ ()r +;Ψ†()r V ()r Ψ ()r ∞ 2m s s s tr s Ψα()r = ∑ bαλψαλ(),r the energy operators of the atomic pump reservoir and λ = 0 (2) the reservoir of extraction of atoms from the trap (β = 3 ψ Ψ α , p, out), bαλ ==∫d r αλ* ()r α(),r p out, 3 HRβ = ∫d r Ψ Ψ† where s(r) and s()r are the annihilation and cre- 2 (7) ation operators for an atom at the point r in the trap, ប † † × Ð------∇Ψβ()r ∇Ψβ()r +;Ψβ()r V β()r Ψβ()r † 2m respectively, and Ψα(r) and Ψα()r are the annihilation and creation operators, respectively, for particles in α the energy operator of the electromagnetic field applied pump reservoirs ( = p) and in reservoirs performing to the trap (β = EM), the extraction of the atomic field from the trap (α = φ ψ out). The c-numerical functions j()r and αλ(r) enter- បω † ing the right-hand sides of Eqs. (1) and (2) determine HR, EM = ∑ jbEM, jbEM, j; (8) the spatial distributions of the fields. j

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 QUANTUM DYNAMICS AND STATISTICS OF A BOSE CONDENSATE 995 and the free energy of the phonon field (β = phon), herent pump reservoir, which is, in this particular case, ∞ in thermodynamic equilibrium. We will assume below បω † that the interaction of atoms with the fields performing HR, phon = ∑ phon, λbphon, λbphon, λ. (9) pumping, the extraction of the condensate, and stimu- λ = 0 lated evaporation are weak (low Rabi frequencies); i.e., The terms describing the interaction of the system of we will consider the case of a cw atomic laser [1]. At atoms in the trap with the reservoirs have the form the same time, we will take into account the time dependence of the coupling constants. ប 3 Ψ† Λ , Ψ V sRout = ∫d r out()r out()r t s()r + h.c., (10) Using the field operators (1) and (2), we can write the Hamiltonian (5) in the form ប 3 Ψ† Λ , Ψ V sRp = ∫d r s()r p()r t p()r + h.c., (11) ∞ បω † បω † ∞ H = ∑ ja j a j + ∑ ∑ αλbαλbαλ ប 3 Ψ† j α = p,,out phon λ = 0 V sRphon = ∑ ∫d r s()r (12) ∞ λ = 0  ប()Γ † † × Λ Ψ + ∑ ∑ λ,,ijbphon, λaia j + h.c. bphon, λ sp, λ()r s(r )+ h.c.,  λ = 0 ij> 1 3 3 Ψ† Ψ† V coll = ---∫d r∫d r' s()r s()r' 2 (13) † + ∑ប[]κλ, ()t b , λa + h.c. (14) × Ψ Ψ i P i U()rrÐ ' s()r s().r' i Λ The functions entering (10)Ð(12) are the coupling  constants of the atomic field of the trap with the fields ប()µ † + ∑ λ, j()t bout, λa j + h.c.  of reservoirs.  j We will consider the electromagnetic field applied to the trap classically. In this case, the operator (8) rep- 1 † † resents a c-numerical constant. + ---∑បg ,,,a a a a , 2 ijkl i j k l Neutral atoms forming the Bose condensate are ijkl located in a parabolic trap with the potential where the energy of the jth state of the trap is m 2 2 ω 3 V tr()r = ---- ∑ αrα, បω φ φ 2 j = ∫d r *j ()r K()r j(),r α = xyz,, 2 (15) where m is the atom mass. The effective interaction ប 2 K()r = Ð ------∇ + V (),r between atoms in the trap is described by the pseudo- 2m tr potential and បωαλ is the energy of the harmonic oscillator λ in δ() π ប2 α V()rrÐ ' ==u rrÐ ' , u 4 a0 /m, the reservoir . (where a is the collision length for the scattering of The coupling constants characterizing the interac- 0 tion with reservoirs are defined as S waves) and by the dipoleÐdipole two-particle interac- 3 tion potential V dd()rrÐ ' , i.e., µ Ψ Λ , φ λ, i()t = ∫d r out* , λ()r out()r t i(),r (16) U()rrÐ ' = V()rrÐ ' + V dd().rrÐ ' κ 3 Ψ Λ , φ The terms in (11) and (10) containing the operators λ, j()t = ∫d r p, λ()r p()r t *j (),r (17) Ψ Ψ p(r) and out(r) are related to the reservoirs from which the trap levels are pumped by external sources Γ 3 φ Λ φ λ,,ij = ∫d r *j ()r sp, λ()r i().r (18) and the atoms are extracted from the trap. In particular, the atomic field can be extracted during the interaction The coupling constants for elastic and inelastic colli- of atoms with a resonance radio-frequency field, which sions between atoms in the trap are induces electronic transitions between the Zeeman sub- levels of the atoms [1, 2]. In this case, Λ (r, t) = u 3 φ φ φ φ out gijkl,,, = ---∫d r i*()r *j ()r k()r l()r Ω(r, t), where Ω(r, t) = m á B(r, t)/ប is the Rabi fre- 2 quency of an atom in the magnetic field B(r, t), m is the (19) 1 3 3 transition magnetic moment, and Ψ (r) is the creation + --- d r' d rφ*()r φ*()r' V ()rrÐ ' φ ()r φ ().r' out 2∫ ∫ i j dd k l operator for a free atom extracted from the trap. The pump operator (11) is defined similarly with the help of The first two terms in (14) are the energy of atoms Ψ the annihilation operator p(r) for an atom in the inco- in the trap and the intrinsic energy of oscillators in res-

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 996 KOZLOVSKII ervoirs, respectively. The next three terms in (14), cor- 3. SPATIAL DISTRIBUTION responding to VsRβ in (4), are the sum of potentials of OF ATOMS IN TRAP MODES interaction of atoms in the trap with reservoirs, which We will use the variational principle to derive equa- cause the spontaneous decay from the discrete energy φ tions determining the spatial dependences { j(r)} of levels of the trap, the extraction of atoms from the trap, field operators (1). We will require the stationarity of and pumping. The last term is the interaction potential the functional Vcoll for elastic and inelastic binary collisions between atoms captured by the trap. []φ , φ 3 〈〉 E j()r *j ()r = ∫d r H()r In this paper, we consider the model of an atomic φ laser in which atoms are cooled during evaporation at each instant upon variation of { j(r)}. By imposing [1, 2]. Preliminarily cooled atoms enter a trap of reser- the orthonormality condition voirs, which are in thermodynamic equilibrium. It is assumed that atoms in the trap can be in four energy 3 φ φ δ ∫d r i*()r j()r = ij, states and are characterized by a set of creation (annihi- † for a discrete spectrum on the basis of functions {φ (r)}, lation) operators ai (ai), where i = 0, 1, 2, 3. It is also j assumed that the high-lying energy states of the trap are we will use the Lagrange method of multipliers. In this weakly populated due to stimulated evaporation per- case, the variational equation will take the form formed with the help of a radio-frequency electromag- netic field applied to the trap [1, 2]. A Bose condensate δE Ð0,∑ε δ d3rφ*()r φ ()r = (22) is obtained in the lower energy state |0〉, from which the j ∫ j j κ j accumulated condensate enters, at the rate out, a reser- voir of the continuous spectrum of vacuum states (out- which gives the equations for the level energy eigenval- φ put laser radiation). Different methods for the extrac- ues {ej} and functions { j(r)}, whose form can be tion of the Bose condensate from a trap were consid- found from the relation ered in papers [17, 25]. Each of the states of the trap is δ 3 pumped independently at the rate pi from the corre------d r 〈〉H()r = 0, (23) δϕ ∫ sponding incoherent reservoirs with the average occu- *j pump pation numbers Ni . Dissipation processes related to following from (22), where the exchange of atoms in the trap with the correspond- 〈〉〈〉[]φ φ ing reservoirs with the average occupation numbers H()r = H j()r *j ()r decay γ Ni occur at the rates i. The pumping of an atomic (24) ε φ φ 〈〉 laser from thermal reservoirs was considered in papers Ð ∑ j j()r *j ()r n j . [10, 19Ð21]. j For binary collisions between atoms in the trap in a Let us define now the mean value of the total Hamil- particular case of four levels considered below, the last tonian (4) as term in (14) represents the interaction energy of collid- ing atoms in the dipole approximation and consists of 〈〉H[]φ ()r, t , φ*()r, t = Sp()σ ()t H[]φ ()r , φ*()r , the elastic and inelastic contributions j j total j j σ where total is the total density operator of the system V coll = V elast + V inelast, (20) and reservoirs interacting with the system. Upon aver- aging with a time-dependent density operator, one-par- where φ ticle eigenfunctions { j(r, t)} and eigenvalues {ej(t)} acquire a parametric dependence on time. By using the V elast = V jj + V ij assumption of a weak interaction between the system 3 and reservoirs (the Born approximation), we consider ប †2 2 ប † † = ∑ g jjjja j a j + ∑ gijijai a j aia j, the total density operator in the form of a sum j = 0 ij, = 0, ij< σ ρ ∆ρ()1 total()t = ()t f 0 + (),t (25) ប † † 2 ប †2 ρ σ V inelast = g0211a0a2a1 + g1102a1 a0a2 where (t) = SpR( total(t)) is the reduced density opera- tor of the system representing the trace of the total den- ប † † ប † † (21) + g0312a1a2a3a0 + g1203a0a3a1a2 sity operator over the variables of reservoirs. The quan- † † 2 †2 tity f in (25) is the product of the density operators of + បg a a a + បg a a a , 0 1322 1 3 2 2213 2 1 3 independent reservoirs in thermodynamic equilibrium. In the interaction representation, the last term in (25) in gijkl = gijkl* . the first approximation over the potential of interaction

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 QUANTUM DYNAMICS AND STATISTICS OF A BOSE CONDENSATE 997 〈 〉 VsR = ∑β V sRβ of the system with reservoirs has the changes in the time of average values nj(t) and 〈〉†() †() () () form a j t ai t ak t al t entering into these equations. t The distribution of the Bose condensate density can ∆ρ()1 1 [], ρ be calculated using different approximate equations, ()t = -----∫ V sR()t' ()t' f 0 dt', (26) iប which follow directly from the general equation (28). 0 By considering a system of atoms in a trap as a canon- where […, …] is a commutator. By performing func- ical ensemble in thermodynamic equilibrium and tional differentiation in the variational equation (23) assuming that the number of atoms in the trap is fixed and using (14)Ð(19), we obtain the system of coupled [27Ð35], we obtain for the density operator, which is eigenvalue differential equations of the type (for any j) stationary in this case, L []φ ()r, t +0.L []φ ()r, t = (27) exp[]Ðβ()H Ð µN coh j irr j ρ = ------0 , (31) can Sp[]exp()ÐβH The coherent part of Eq. (27) appears after the calcula- 0 tion of the quantum-mechanical mean for the terms where containing the system operators with the help of the ε † ≡ ε β ≡ 1 first term in (25). The terms determining the change of H0 = ∑ ja j a j ∑ j N j, ------, the wave functions of atoms in the trap caused by the kBT j j interaction with reservoirs (irreversible processes) are contained in the second term in Eq. (27). For coherent and µ is the chemical potential. terms, we obtain The quantum-mechanical mean for the operators O of the system is defined in this case as L []φ ()r, t = K ()r, t φ ()r, t 〈〉n ()t + φ()r, t coh j j j j ∑ k 〈〉 ()ρ ()ρ O T = Sp canO /Sp can . (32) ikl,, (28) 3 By using (31) and (32), we obtain from (28) the system × d r'φ*()r', t U()rrÐ ' φ ()r', t 〈〉a†()t a†()t a ()t a ()t , ε φ ∫ i l j i k l of equations for { j, j} φ 〈〉 where K j()r j()r n j T

2 3 ប 2 φ φ* φ 〈〉 K ()r, t = Ð ------∇ + V ()r Ð ε ().t + j()r ∫d r' j ()r' U()rrÐ ' j()r' n j()n j Ð 1 T j 2m tr j (33) 3 The irreversible terms in (27), which appear after the φ φ φ 〈〉〈〉 + ∑ j()r ∫d r' i*()r' U()rrÐ ' i()r' ni T n j T = 0. calculation of the mean using the total density opera- ij≠ tor (25), prove to be, within the framework of our model, proportional to the mean values of the reservoir The value of the chemical potential µ can be deter- operators. Since we assume in our calculations that the mined for any fixed number of atoms in the trap reservoirs are incoherent or are in thermodynamic equi- because the condition librium, it is easy to see that 〈〉{}[] βε()µ Ð1 NN==∑ expj Ð Ð 1 (34) † 〈〉bβλ, ==〈〉bβλ, 0, j 〈〉〈〉† † is fulfilled in the case of thermal equilibrium. bβλ, bβ', λ' ==bβλ, bβ', λ' 0, (29) The calculation of the spatial distribution of the 〈〉δ† δ ()()α Bose condensate is greatly simplified if we assume that bβλ, bβ', λ' = λλ, ' ββ, ' Nλ + 1 , the temperature of the canonical ensemble T = 0 and † ()α 〈〉δbβλ, bβ , λ = λλ, δββ, Nλ . only the lower state of the trap with the wave function ' ' ' ' φ ε µ 0 is populated, and 0 = . Because the number of par- By virtue of these relations, the irreversible terms ticles in the canonical ensemble is fixed, the creation † † in (27) proportional to bβ λ, 〈〉bβλ, , 〈〉bβλ, bβ , λ , , ' ' (annihilation) operators a0 (a0 ) should be replaced by † † 〈〉bβλ, bβ , λ are zero. Therefore, in the BornÐMarkoff ' ' c numbers equal to N0 . Then, we obtain from (28) approximation for the interaction of the system with 3 reservoirs, we have φ φ φ K0()r + N0∫d r' 0*()r' U()rrÐ ' 0()r' 0()r = 0,(35) L []φ ()r, t = 0. (30) irr j where This means that the irreversible processes under con- ប2 sideration lead to the appearance of additional terms in K ()r ==Ð ------∇2 + V ()r Ð µ, ε µ. equations for the wave functions and introduce 0 2m tr 0

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 998 KOZLOVSKII Equation (35) is reduced to the usual GrossÐPitaevskii persion nature and affect the time dependence of non- equation if we assume that the effective interaction diagonal matrix elements and, hence, determine the between atoms of the type degree of coherence of the first-order field. U()rrÐ ' = uδ()rrÐ ' + V ()rrÐ ' The equation of motion for the reduced density dd ρ σ operator of the system (t) = SpR( total) is a sum of contains only the first term (pseudopotential). terms responsible for the irreversible processes of As the crudest approximation for the density distri- pumping and extraction of atoms from the trap and of bution, we can consider the equation losses and spontaneous transitions in atoms. In the φ ε φ interaction representation, we have T()r j()r = j (),r i 2 ρ []ρ, ប ú()t = --- V coll ∇2 ប T()r = Ð ------+ V tr()r 2m t (36) 1 for the eigenfunctions and energy eigenvalues of the Ð -----∑ Sp {}[]V β()t , []V β()t' , ρ()t' f dt'. ប2 ∫ R SR SR 0 trap. It is assumed that collisions of atoms in the trap β have no effect on the spatial distribution of the atomic 0 field. In this representation, the time dependences of the cre- The approximation of the trap eigenstates was used ation (annihilation) operators related to the system and in the quantum-mechanical models of an atomic laser reservoirs are determined by the equations [3Ð11], the coupling parameters (16), (17), and (19) ()ω † † ()ω being constant in time due to the assumption that the O j()t ==O j exp Ði Ojt , O j()t O j exp i Ojt , atomic-field distributions for all modes of the trap were ω where Oj is the eigenfrequency of the jth oscillator of independent of time. the field. By substituting expression (25) for the total density operator into (36), we will keep in the obtained 4. CONTROL EQUATION equation the terms up to second order inclusive in the FOR THE REDUCED DENSITY OPERATOR interaction potential. Then, we will use the expressions OF THE SYSTEM for the interaction potentials of the type (14) in the equation obtained. Assuming that the coupling con- By excluding variable reservoirs in a standard way stants in (14) are independent of coordinates and using using the BornÐMarkoff approximation, we obtained the Born and Markoff approximations [36], we obtain the “control” equation for the reduced density operator the control equation of the system. The presence of the reservoirs leads to the appearance of the irreversible processes of dissipa- i ρú [], ρ tion and extraction of the field from the trap, pumping, = Ðប--- V coll and spontaneous decay in the equation of motion. In the self-consistent model of the nonequilibrium Bose gas µ 1 decay † decay considered here, the coupling parameters λ, i(t) and Ð ---∑{γ []N j Da[]+ ()N j + 1 Da[] κ 2 j j j λ, i(t) entering the Hamiltonian (14), which are deter- j mined by integrating the wave functions of the field []}ρpump []† ()pump [] over spatial variables [expressions (16)Ð(19)], depend + p j N j Daj + N j + 1 Daj (37) on time in the general case. Because the self-consistent scheme assumes the calculation of the wave function φ 1 phon † j(r, t) by solving Eqs. (27) and (28) at each instant of + ---∑{γ , [N j Ð 1 j Da[]a 2 sp j j Ð 1 j time, the coupling parameters also depend on time. The j > 0 simplest approximation of the trap eigenstates used at ()phon [† ]]}ρ present in the models of an atomic laser [3Ð11, 19Ð29] + N jj, Ð 1 + 1 Daj Ð 1a j leads in general to the violation of the self-consistency ρ ≡ in the calculation of the dynamics of the nonequilib- for the reduced density operator, where D[O] ρ † † ρ ρ † rium Bose gas. In this paper, we consider the time 2O O Ð O O Ð O O for the corresponding opera- dependence of the coupling parameters describing tor O. The terms in the control equation that are propor- pumping processes and losses in the trap caused by the tional to the frequency shift caused by the interaction of time dependence of the wave functions of the trap the system with reservoirs are omitted in (37), and they modes. are assumed to be small below. Equation (37) was derived using the stationarity of random processes Because all the operators (20), (21) entering the 〈〉τ 〈〉τ Hamiltonian are bilinear in the creation and annihila- O1()O2()0 = O1()0 O2()Ð , tion operators, the contribution to the coherent (unitary) component of the evolution of diagonal matrix ele- where O1(t) and O2(t) are the operators of random pro- ments is zero. All elastic collision processes have a dis- cesses related to reservoirs.

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The loss and pump rates entering (37) have the form 1 {γ [ phon []† + --- ∑ sp, j N jj, Ð 1 Daj Ð 1a j ∞ ∞ 2 γ  j > 0 i ----Re=  dτexp()µiω τ * ,,()0 µ ,,()τ ∑ ∫ i out ji out ji ()phon [† ]]}ρ 2  + N jj, Ð 1 + 1 Daj Ð 1a j , j = 0 0  (38) where × ()〈〉τ † 〈〉† τ ∫ bout, j()bout, j()0 Ð bout, j()0 bout, j() , eff eff γ decay pump  p1 N1 = 1 N1 + p1 N1 , > i 0, γ eff()γeff ()decay ()pump 1 N1 + 1 = 1 N1 + 1 + p1 N1 + 1 ∞ ∞  | 〉 pi τ ()κω τ κ τ are the effective pump and loss rates for the 1 state. ----= ∑ R e ∫d exp i i *pji,,()0 pji,,() 2  The superscript “eff” will be omitted below. j = 0 0 (39) In the model under study, the stimulated cooling of  atoms in the trap is performed by the evaporation × []〈〉τ † 〈〉† τ ∫ b pj, ()b pj, ()0 Ð b pj, ()0 b pj, () , method, where the atoms are removed from the states  |2〉 and |3〉 by a radio-frequency field that changes their electronic states. Assuming that the rates of extraction and the average occupation numbers of the pump and of atoms in the upper states from the trap are high, i.e., loss reservoirs are γ , γ ӷ p ,,,κ Ω γ , , ∞ ∞ 2 3 1 out j sp i (43) decay 2   τ ()µω τ * i ==123,,, j 12,, Ni = ---γ- ∑ Re ∫d exp i i pji,,()0 i  j = 0 0 for the density operator, we perform the adiabatic (40) | 〉 | 〉  exclusion of modes 2 and 3 . ×µ τ 〈〉† τ ∫ pji,,() bout, j()0 bout, j() , Using the Hamiltonian (14), we write the stochastic  HeisenbergÐLangevin equations for the operators a2

∞ and a3 in the interaction representation (see, for exam- ∞  ple, [35] in the form pump 2 τ ()κω τ Ni = ---- ∑ Re∫d exp i i *pji,,()0 pi  i γ j = 0 0 aú = ---[]a , V Ð -----2a + γ B (),t (44) (41) 2 ប 2 coll 2 2 2 2 †  ×κ ,,()τ 〈〉b , ()0 b , ()τ . ∫ pji pj pj i γ  aú = ---[]a , V Ð -----3a + γ B (),t (45) 3 ប 3 coll 2 3 3 3 pump γ decay Note that the quantities pi, Ni , and i, Ni can be where the operators B2 and B3 of random sources satisfy both positive and negative, depending on the reservoir the conditions properties and the interaction dynamics. 〈〉 〈〉† As mentioned above, we will consider a simplified Bi()t R ==0, Bi ()t Bi()t' R 0, model of an atomic laser that takes into account the four 〈〉† δ() , lower states of a trap. We assume that pumping is per- Bi()t Bi ()t' R ==ttÐ ' , i 23. formed only into the first |1〉 excited state of the trap, ≠ i.e., pj = 0 if j 1. The Bose condensate is extracted By using in (44) and (45) the terms Vcoll from (20) from the lower |0〉 state to vacuum. In this case, Eq. (37) and (21) and neglecting the dispersion terms of the type ∆ ∆ can be written in the form i 2a2, i 3a3, which are caused by inelastic collisions, i.e., assuming that ∆ , ∆3 Ӷ γ , γ , we can easily obtain i 2 2 3 ρú ---[] ρ = Ðប V coll, γ aú = Ð -----2a Ð ig a†a2 + γ B (),t (46) 2 2 2 0211 0 1 2 2 1 {}ργ []decay []† ()decay [] + ---∑ j Ni Daj + Ni + 1 Daj γ 2 aú = Ð -----3a Ð ig a a†a + γ B ().t (47) j 3 2 3 0312 2 0 1 3 3 (42) 1κ []ρ1[]ρeff eff []γ† eff eff [] According to the assumption of adiabaticity, we + --- out Da0 + --- p1 N1 Daj + 1 N1 Da1 2 2 have da2/dt = 0 and da3/dt = 0. By solving algebraic

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1000 KOZLOVSKII equations, we find Note also that condition (43), which means that the loss rates for nonlasing modes of the field are much ad 2g0211 † 2 2 higher than the pump rate, the rate of the coherent out- a2 = Ð i------γ -a0a1 + ------B2(),t (48) 2 γ put of the lasing mode, and the effective rates of inelas- 2 tic collisions, provides the low population of modes |2〉 | 〉 ad 4g0312g0211 † 3 and 3 . a3 = Ði------γ γ -a0a1 3 2 The diagonal matrix elements of the reduced density (49) matrix ρ ()t ≡ 〈n0n |ρ(t)|n n 〉 have the form n0n1 1 0 1 2 2g0211 + ------B3()t Ð i------B2().t γ γ γ ρú = []κ + γ ()N + 1 3 3 2 n0n1 out 0 0 By substituting adiabatic values (48) and (49) into the × []()ρn + 1 , Ð n ρ γ ρ γ ρ 0 n0 + 1 n1 0 n0n1 terms 2D[a2] /2 and 3D[a3] /2 in Eq. (42) and aver- aging over the reservoirs, we obtain the terms + p N []n ρ , Ð ()ρn + 1 0 0 0 n0 Ð 1 n1 0 n0n1 Ω Ω 1 † 2 2 †2 3 ------Da[]ρa , ------Da[]ρa + γ ()N1 + 1 []()ρn + 1 , Ð n ρ 2 0 1 2 0 1 1 1 n0 n1 + 1 1 n0n1

+ p N []n ρ , Ð ()ρn + 1 (52) of the control equation, where 1 1 1 n0 n1 Ð 1 1 n0n1 2 2 2 4 g0211 16 g0211 g0312 + γ {()N01 + 1 []n ()n + 1 ρ , Ð n ()n + 1 ρ Ω ==------, Ω ------. (50) sp 0 1 n0 Ð 1 n1 + 1 1 0 n0n1 1 γ 2 γ 2γ 2 2 3 []}()ρ ()ρ + N01 n1 n0 Ð 1 n + 1, n Ð 1 Ð n0 n1 + 1 n n As a result, the “control” equation for irreversible pro- 0 1 0 1 cesses can be written in the form + Ω [n ()n + 1 ()ρn + 2 , 1 0 1 1 n0 Ð 1 n1 + 2 ()()ρ] κ + γ ()N + 1 p Ð n0 + 1 n1 n1 Ð 1 n n ρ out 0 0 [] 0 []† 0 1 ú = ------Da0 + ----- N0 Da0 irr  2 2 + Ω [()n Ð 1 n ()n + 1 ()n + 2 ()ρn + 3 , 2 0 0 1 1 1 n0 Ð 2 n1 + 3 γ 1 p1 † Ð ()n + 2 ()n + 1 n ()n Ð 1 ()ρn Ð 2 ]. + -----()N + 1 Da[]+ ----- N Da[] 0 0 1 1 1 n0n1 2 1 1 2 1 1 (51) γ The quantum-mechanical means of the operators of the sp † † number of atoms in the lower energy state of the trap + ------{}()N01 + 1 Da[]a + N01 Da[]a 2 0 1 1 0 (lasing mode) and of dispersion (fluctuation) of the number of atoms are calculated using the diagonal ele- Ω Ω  ments of the density matrix: 1 []† 2 2 []†2 3 ρ + ------Da0a1 + ------Da0 a1  . 2 2  〈〉n ()t ==Sp()a†a ρ()t ∑ n ρ (),t (53) γ BC 0 0 0 n0n1 Here, sp is the rate of spontaneous transitions between , | 〉 | 〉 ω n0 n1 the modes 1 and 0 of the trap and N01 = N( phon, λ) = 2 † † 2 〈〉† បω ប ω ω បω 〈〉()∆n ()t ≡ Varn ==Sp()a a a a ρ()t 〈〉n bphon, λbphon, λ , phon, λ = ( 1 Ð 0) = 01 is the i i i i i i i average number of phonons in the thermal reservoir of (54) spontaneous decay at the frequency of the |1〉 |0〉 = ∑ ()n Ð 〈〉n 2ρ (),t i = 01., i i n0n1 transition in the trap. The spontaneous decay is ana- n , n lyzed under the assumption that the density of states of 0 1 oscillators in the phonon reservoir weakly depends on The Fano factors for the Bose condensate in the ground the frequency [27, 28]. The parameters N and N state and in the first excited state of the trap are defined 0 1 as in (51) are the average numbers of atoms in reservoirs | 〉 | 〉 2 related to the 0 and 1 states, respectively. They are F ==〈〉()∆n / 〈〉n , i 01., (55) calculated in the general case from expressions (40) i i i and (41). We will neglect below the pump rates for all modes The rates Ω and Ω defined in (50) represent the except the |1〉 mode. The transitions of atoms from the 1 2 | 〉 effective rates of inelastic collisions involved in the cre- lasing 0 mode to the thermostat will also be neglected. ation of the Bose condensate in the ground state of the We also assume that the Bose condensate is extracted trap and corresponding, therefore, to stimulated transi- from the lower state of the trap to vacuum at the rate κ ӷ γ tion in a usual laser. out 0(N0 + 1).

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5. QUANTUM DYNAMICS AND STATISTICS 60 OF AN ATOMIC LASER (a) The self-consistent theory of an atomic laser devel- 〈 ∆ 2〉 oped above assumes that Eqs. (27) and (28) for the ( n0) φ wave functions { j(r, t)} of the field representing a sys- 40 tem of coupled eigenvalue equations are solved by the iteration method simultaneously with the solution of the control equation (51) for the reduced density oper- ator for an open system. Such a coupled system of 20 〈 〉 n0 equations can be solved only numerically. Its solution seems impossible at present because it is too cumber- some. In this paper, we calculated numerically the den- F0 sity matrix of the system in the approximation of the 0 eigenstates of the trap, when the dissipation and pump 0 0.10.2 0.3 rates are independent of time, i.e., in the approximation used in [3Ð9, 11, 19Ð24, 27, 28]. 40 We solved numerically the system of coupled differ- (b) ential equations (52) for matrix elements ρ ()t using n0n1 30 ρ δ δ the initial conditions n n ()0 = n , 0 n , 0 , i.e., when 〈 ∆ 2〉 0 1 0 1 ( n1) atoms were initially absent in the trap. 20 The generation dynamics of the atomic field for the three-level model of an atomic laser considered here depends qualitatively on the relations between the rate 〈n 〉 | 〉 10 1 p1 of pumping of the 1 state of the atomic trap, the κ extraction rate out for the Bose condensate in the | 〉 Ω ground 0 state, the rate 1 of stimulated transitions F | 〉 | 〉 | 〉 γ 0 1 from the 1 state to the 0 and 2 states, and the rate sp of spontaneous transitions from the |1〉 state to the |0〉 0 0.10.2 0.3 state due to the interaction with the reservoir. Under the γ assumption that the rate sp is much lower than all other 7 rates mentioned above, the lasing regimes of an atomic (c) laser can be divided into two characteristic types. If 6 Ω Ω κ p1 > 1 and p1, 1 ~ out, then two stages of lasing dynamics are typical (see Figs. 1a and 1b). At first, the 5 |1〉 state is populated, and the number of particles of a F0 slowly accumulated Bose condensate in the |0〉 state is 4 small, the fluctuations in the number of particles drasti- F cally increasing up to values that are typical for a ran- 3 1 〈〉()∆ 2 ≈ 〈 〉 〈 〉 dom thermal field: nBC ( n0 + 1) n0 . At the 2 〈 〉 next stage, the number n1 of atoms decreases, whereas 〈 〉 〈 ∆ 2〉 1 n0 increases and fluctuations ( n0) decrease, 〈 〉 approaching the Poisson value equal to n0 ss in the sta- 0 0.05 0.10 0.15 0.20 tionary state. For p < Ω and p , Ω ӷ κ , the station- 1 1 1 1 out κ t ary Bose condensate in the |0〉 state can be in a weakly out squeezed sub-Poisson state F0 < 1 (Fig. 1c). Fig. 1. (a) Dependences of the average number n0 of atoms Ӷ Ω Ω ӷ κ of the Bose condensate, the dispersion (fluctuations) When p1 1 and p1, p1, 1 out, another regime 〈 ∆ 2〉 of generation of the Bose condensate is realized ( n0) of the number of atoms, and the Fano factor F0 = 〈 ∆ 2〉 〈 〉 κ (Fig. 2). There is no stage of population of the |1〉 state; ( n0) / n0 on the reduced time outt for the pump rate κ Ω κ a drastic increase in the fluctuations of the Bose con- p1 = 100 out, the rate of collision transitions 1 = out, the densate in the |0〉 state is also absent. The fluctuations average number of particles in the reservoir N1 = 1, and of the Bose condensate can achieve the sub-Poisson γ Ӷ Ω 〈 〉 〈 ∆ 2〉 sp 1. (b) Dynamics of n0 , ( n1) , and F1 = values of the squeezed state (Fig. 2a). The population n1 〈 ∆ 2〉 〈 〉 | 〉 ( n1) / n1 for the first excited state of the trap for the of the 1 state proves to be low at any time until the same parameters as in Fig. 1a. (c) Comparison of the establishment of the , and the fluctua- dynamics of the Fano factors F0 and F1 for the same param- 〈 ∆ 2〉 tions ( n1) of the number of particles become essen- eters as in Fig. 1a.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1002 KOZLOVSKII Ω ӷ κ tially sub-Poisson: F1 < 1. In this regime, p1, 1 out, 1.0 and irrespective of the values of transition rates, the 〈 〉 (a) population n1 ss is equal to 0.333 for the dispersion of the number of particles 〈(∆n )2〉 = 0.667〈n 〉 (Fig. 2b). 0.9 1 ss 1 ss Systematic calculations showed that the condition for the creation of a Bose condensate in a squeezed sub- 0.8 Ω ӷ ӷ κ ӷ γ Poisson state is the relation 1 p1 out 1, irre-

spective of the value of N1 . The degree of squeezing 0.7 F increases with increasing number of bosons in the con- 1 〈 〉 ӷ Ω ӷ κ densate, n0 ss 1, when 1, p2 out. For the values of the laser parameters satisfying the inequalities Ω ӷ 0.6 1 p ӷ κ ӷ γ , at which 〈n 〉 ~ 106, the maximum sup- F 1 out 1 0 ss 0 pression of fluctuations achieves almost half the level 0.5 of the shot noise. In this case, the Fano factor can 0 0.51.0 1.5 ≈ achieve the value F0, ss 0.54 (Fig. 2a). ӷ Ω The calculations showed that, for p1 > N1 1, 1.0 (b) κ ӷ γ p1 N1 > out 1, the stationary average number of atoms in the Bose condensate can be estimated from the 0.8 expression F1 κ p1 1 1 0.6 〈〉≈ ------out n0 ss κ N1 Ð --- + --- + ------Ω - . (56) 2 out 2 4 1 0.4 〈 〉 n0 In [23], the control equation for the density operator in the model of an atomic laser, similar to that considered here, was transformed to the FokkerÐPlanck equation 0.2 〈 ∆ 2〉 ( n0) for the quasi-probability P function in the phase space of the amplitude and phase of the atomic field. The sto- 0 chastic differential equations for the number of parti- cles and phases of the fields in the trap modes, which 0 0.51.0 1.5 were obtained from the FokkerÐPlanck equation, were solved in [23] under stationary conditions for average 14 〈 〉 ӷ values. In the limit n0 1, the semiclassical average 12 number of atoms in the lower state of the trap was (c) found in [23, 26] in a form similar to (56). However, the 10 sign of the square root, which was arbitrary within the framework of calculations performed in these papers, 8 〈 〉 was chosen to be negative. Our exact quantum-mechan- n0 ical calculations confirm the validity of expression (56), 6 where the sign of the root is positive. Ω 4 At the same time, for p1 ~ N1 , 1, the average num- 〈(∆n )2〉 ber of atoms in the Bose condensate is described by the 2 0 〈 〉 η κ η ≈ expression n0 ss = p1 N1 / out, where 2. This 0 expression agrees qualitatively with calculations per- formed in [3] for a similar laser scheme, according to 〈 〉 κ Ω ӷ –2 which n0 ss = 2p1 N1 /3 out for the case 1 > p1, N1 0 0.5 1.0 1.5 2.5 κ out considered in [3]. The lasing threshold for the laser κ × 5 outt 10 κ we considered is determined by the relation p1 N1 > out. It was assumed in calculations discussed above that Fig. 2. (a) Dynamics of the Fano factors F0 and F1 for p1 = 6κ Ω 9κ γ Ӷ κ the rate of spontaneous transitions between trap modes 10 out, 1 = 10 out, N1 = 1, and sp out. (b) Dynam- is much lower than the rates of other processes. The cal- 〈 〉 〈 ∆ 2〉 ics of n1 , ( n0) , and F1 for the same parameters as in culations performed under conditions when the rate of Ω ӷ ӷ κ ӷ γ Fig. 2a: 1 p1 out sp. (c) Dynamics of n0 and spontaneous transitions is comparable with the rate of 〈 ∆ 2〉 ( n0) for the same parameters as in Fig. 2a. extraction of the coherent condensate from the trap

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γ κ 7 ( sp = 0.5 out for N01 = 1) are presented in Fig. 3. A comparison of the parameters characterizing the con- densate under such conditions with the case of the 6 〈 ∆ 2〉 absence of spontaneous decay (γ Ӷ κ ) shows that ( n0) sp out 5 spontaneous transitions have no effect on the stationary 〈 〉 〈 〉 mean value n0 ss but substantially enhance fluctuations 4 n0 〈 ∆ 2〉 ( n0) ss of the Bose condensate and change the lasing dynamics of an atomic laser. 3 F0 The calculations of the lasing dynamics presented 2 above were performed within the framework of the Ω Ӷ κ three-level model assuming that 2 out. To estimate 1 the effect of high-lying energy levels of the trap on the dynamics and statistics of the generated condensate, we 0 Ω Ω performed calculations for the case 1 ~ 2, i.e., for the four-level model of an atomic laser. Figure 4 shows the 012345 κ time dependences of the Fano factor for the four-level outt Ω Ω scheme at 1 = 2 and for the three-level scheme at Ω Fig. 3. Effect of spontaneous transitions on the dynamics of 2 = 0. The comparison shows that, even when the an atomic laser. The solid curves show the dynamics of the upper levels are substantially populated, the fluctua- characteristics of the Bose condensate for the parameters of γ κ tions of the Bose condensate in the ground state of the spontaneous transitions sp = 0.001 out and N01 = 0.001. γ trap increase only slightly (by several percent) at high The dashed curves show the same quantities for sp = rates of collision transitions. κ κ 0.5 out and N01 = 1. The other parameters are p1 = 2 out, Ω κ Our calculations showed that a Bose condensate 1 = 0.1 out, and N1 = 1. with minimal sub-Poisson fluctuations of the number of particles can be produced by the stimulated cooling of atoms, when the populations of the upper levels of the F0 trap are always much lower than the population of the 1.1 |0〉 ground state of the trap and of the |1〉 state through which pumping is performed. 1.0 The fluctuations of the number of particles in a Bose condensate were studied in [27Ð35] in thermal equilib- 0.9 rium for a fixed number of particles in a trap within the framework of a standard description of an ideal gas 0.8 with the help of a canonical or a microcanonical ensem- ble. The numerical and analytic calculations of fluctua- tions in a canonical ensemble of noninteracting parti- 0.7 cles were performed in [31]. It was shown that the fluctuations of the Bose condensate consisting of 0.6 102−106 atoms in the trap approached almost linearly to zero at T/T 0, where T is the critical temperature 0.5 c c 0 5 10 15 20 of the Bose condensation. The results obtained in [30, κ × 3 32, 34, 35] for a microcanonical ensemble (the isolated outt 10 state of atoms in the trap) agree qualitatively with data [31], fluctuations in a microcanonical ensemble Fig. 4. Comparison of the dynamics of the Fano factor for always being greater than those in a canonical ensem- the three- and four-level schemes of an atomic laser for p1 = 800κ , Ω = Ω = 1.6 × 104κ , N = 1, γ /κ , ble. It was found in [35] that, for small values of T/Tc, out 1 2 out 1 sp out Ӷ Ω the magnitude of fluctuations was independent of the N01 1 (four-level scheme: the solid curve) and 2 = 0 total number N of particles. Similar results were (three-level scheme: the dashed curve); other parameters obtained in paper [32] for both canonical and microca- are the same. nonical ensembles. A canonical ensemble of particles with spontaneous The authors of [35] have drawn other conclusions. transitions between the trap states was studied in [27Ð29]. The HartreeÐFockÐBogolubovÐPopov theory used in Quantum-mechanical calculations performed in these this paper, which takes into account collisions between papers also showed the achievement of the Fock state of atoms in a trap at a fixed number of atoms and constant an ideal Bose condensate at T/Tc 0. temperature, shows that the quantum statistics of the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 96 No. 6 2003 1004 KOZLOVSKII atomic field tends to the Poisson statistics, which is typ- results obtained in the paper suggest the existence of ical for the of a Bose condensate when atomic lasers capable of generating relatively small T/Tc 0. The calculations [33] were performed Bose condensates in the states that are close to the Fock under conditions of thermodynamic equilibrium (T ≠ 0) state. Such lasers can be considered as sources of indi- for a large canonical ensemble of atoms taking into vidual groups of ultracold atoms with a prescribed account collisions between atoms in a trap. A compari- exact number of atoms, which are required for a num- son of the results obtained in this paper with the data for ber of experiments of current interest in the optics of the canonical and microcanonical ensembles of an ideal atoms and photons. gas suggests that the Fock state with the sub-Poisson fluctuations of the number of particles can be obtained in thermal equilibrium only for an ideal gas when the ACKNOWLEDGMENTS number of particles is fixed. If the number of particles I thank A.N. Oraevskiœ for useful discussions. depends on other parameters of the system (a large canonical ensemble), an ideal Bose gas exhibits ther- mal (random) fluctuations at any temperature. How- REFERENCES ever, collisions between atoms at low temperatures 1. M.-O. Mewes, M. R. Andrews, D. M. Kurn, et al., Phys. (T < Tc) reduce fluctuations to the Poisson level, which Rev. Lett. 78, 582 (1997). is typical for the coherent state of the field. 2. I. Bloch, T. W. Hansch, and T. Eslinger, Phys. Rev. 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