The Relativistic Theory of Fluctuation Electromagnetic Interactions of Moving Neutral Particles with a Flat Surface G

The Relativistic Theory of Fluctuation Electromagnetic Interactions of Moving Neutral Particles with a Flat Surface G. V. Dedkov and A. A. Kyasov Kabardino-Balkar State University, ul. Chernyshevskogo 173, Nalchik, 360004 Russia e-mail: [email protected] Received January 28, 2003

Abstract—The problem concerning the fluctuation electromagnetic interaction of a neutral particle moving parallel to the boundary of a semi-infinite homogeneous isotropic medium characterized by a permittivity and permeability dependent on the frequency is considered in terms of the fluctuation electrodynamic theory within the relativistic formalism. It is assumed that, in the general case, the particle and the medium have different temperatures. Within the proposed approach, general expressions are derived both for conservative (normal to the boundary of the medium) and nonconservative (tangential) forces of the interaction between the particle and the medium and for the thermal heating (cooling) rate of the particle. In the nonrelativistic limit (c ∞), the derived relationships coincide with nonrelativistic analogs available in the literature. It is demonstrated that the tangential force acting on the particle can be either accelerating or decelerating. There can occur a situation when the hot moving particle will be heated and the cold medium will be cooled. The interaction of the high- conductivity medium with a high-conductivity particle is analyzed numerically. The asymptotics of the radiative contributions to the heat flux and the tangential force is investigated. It is shown that the inclusion of the relativistic effects leads to a substantial increase in the tangential force and the heat flux at distances greater than 1 µm (as compared to the nonrelativistic case); however, the corresponding dependences exhibit a monotonic decreasing behavior over the entire range of studied distances (from zero to several hundreds of microns). © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION of the fluctuation force. Dorofeyev et al. [22] investi- The fluctuation electromagnetic interaction between gated a similar problem but restricted their consider- neutral particles and a medium is governed by quantum ation to the case of a nonrelativistic relative velocity V and thermal fluctuations of the polarization and magne- and obtained (to the first order in V) the following dependence of the tangential force on the velocity V: tization inducing fluctuation electromagnetic fields 2 both inside and outside the particles and the medium. Fx ~ V/c , where c is the velocity of light in free space. The fluctuation electromagnetic interaction is associ- The related problem of calculating the tangential ated with conservative van der Waals forces [1Ð3], con- forces of the fluctuation electromagnetic interaction servative Casimir forces [3Ð6], tangential (nonconser- between two semi-infinite media separated by a plane vative) forces [7Ð14], heat exchange effects [15Ð18], vacuum space has been treated in a number of earlier and a number of other interesting phenomena [6]. works [8Ð10]. However, the relationships deduced in The purpose of this work is to develop a consistent those works are inconsistent with each other and with relativistic theory for calculating the heating rate and their nonrelativistic analogs at c ∞ [11, 12], the fluctuation force acting on a small-sized neutral because, in this limit, they lead to a zero tangential spherical particle moving parallel to a flat boundary of force within an approximation linear in velocity. It a polarization medium under vacuum. Consideration is should be noted that Pendry [11] and Volokitin and Per- given to the conservative (normal) and nonconservative sson [12] took into account the retardation effects but (tangential) components of the fluctuation electromag- restricted their consideration to the case of nonrelativ- netic interaction force. istic relative motion. After appropriate transformations The current state of the art in the study of the non- of the expression for the tangential stress between relativistic problem was analyzed in our recent review semi-infinitive media, Volokitin and Persson [12] [14]. The results of relativistic investigations were derived a relationship describing the interaction of a briefly outlined in [19, 20]. In the framework of the rig- small-sized particle and a flat surface. orous relativistic approach, the dynamic force of fluctu- Unlike the relationships derived in the aforemen- ation electromagnetic interaction between a small- tioned works, the expressions obtained in our recent sized particle and a surface was calculated for the first studies [19Ð21] completely coincide with nonrelativis- time in our previous work [21], in which we derived tic formulas at c ∞ not only within an approxima- general formulas for normal and tangential components tion linear in velocity V. As a result, considerable

1816 DEDKOV, KYASOV

z' vacuum (Fig. 1). The half-space z ≤ 0 is filled with a ε y' medium characterized by a complex permittivity and K' complex permeability µ dependent on the frequency ω. It is assumed that the particle is nonmagnetic in the V intrinsic coordinate system and possesses an electric z α ω ε ω x' dipole polarizability ( ) and (or) a permittivity 1( ). K y The real and imaginary parts of these functions (and z0 Vacuum their derivatives) are designated by one and two primes, respectively. The laboratory coordinate system K is related to the surface at rest, and the coordinate system x µ ω ε(ω) K' is related to the moving particle. In the coordinate Medium ( ), system K, the z axis is perpendicular to the interface and the x axis coincides with the direction of the particle ӷ velocity V. Under the assumption that z0 R (where Fig. 1. Coordinate systems K and K' related to the surface at R is the characteristic particle size), the particle can be rest and a moving particle. considered a point fluctuating electric dipole with the moment d(t). advances have been achieved in the description of It should be noted that, within the relativistic dynamic fluctuation electromagnetic interactions, even approach, the nonmagnetic particle having only the though the relativistic solution of the problem of inter- electric dipole moment in the coordinate system K' pos- action between a small-sized particle and a surface can- sesses the magnetic dipole moment m(t) in the coordi- not be directly extended to the case of a different geom- nate system K. Therefore, in the coordinate system K, etry and, in particular, to the geometry of two semi-infi- the vectors of the electric and magnetic polarizations nite media (and vice versa). induced by the moving particle can be written in the form Our calculation technique is based on the equations ()δ,,, ()δ()δ()() of classical and fluctuation electrodynamics (including P xyzt = xVtÐ y zzÐ 0 d t , (1) spontaneous and induced components of the fields and ()δ,,, ()δ()δ()() currents caused by fluctuations) and the formalism of M xyzt = xVtÐ y zzÐ 0 m t . (2) fluctuation dissipative relationships. It should be noted Correspondingly, in the coordinate system K' (in the that no additional simplifying assumptions were made general case), we have in the course of our calculations. ()δ,,, ()δ()δ() () This paper is organized as follows. In Section 2, we P' x' y' z' t' = x' y' z' d' t' , (3) formulate the problem and derive general expressions M'()δx',,,y' z' t' = ()δx' ()δy' ()z' m'()t '. (4) for calculating the forces and thermal effects. In Sec- tion 3, the essence of the method used is demonstrated The coordinates (x, y, z, t) and (x', y', z', t') in both sys- within the nonrelativistic approach (the results obtained tems are related by the Lorentz transformations are necessary for providing deeper insight into the rel- x' ==γ ()x Ð βct , t' γ ()t Ð βx/c , ativistic approach). The regular solution of the Maxwell (5) equations for the electromagnetic field components y' ==y, z' z, induced by a moving particle in the vacuum region is where β = V/c and γ = (1 Ð β2)Ð1/2. The formulas of the found in Section 4. The relativistic relationships for Lorentz transformations for the moments d(t) and m(t) correlators of the physical quantities and components can be derived from similar formulas for the polariza- of the retarded Green’s functions for radiation in the tion vectors P and M specified by the components par- medium are deduced in Section 5. The final formulas allel and perpendicular to the direction of the particular describing the fluctuation medium and the thermal velocity V [23]: heating rate of the particle are given in Section 6. The results of numerical calculations are presented in Sec- 1 P|| ==P||' , P⊥ γ P⊥' + ---[]VM' , (6) tion 7. The main conclusions are summarized in Sec- c tion 8. Computational details are described in the Appendix. Note that, in this work, we use the Gaussian 1 system of units. M|| ==M||' , M⊥ γ M⊥' Ð ---[]VP' . (7) c By using relationships (1)Ð(7) and the identity δ(αx) = 2. FORMULATION OF THE PROBLEM δ(x)/|α| and assuming that m'(t') = 0, we obtain AND BASIC EXPRESSIONS () γ Ð1 () () ' () () '() Let us consider a small-sized neutral spherical par- dx t = dx t' , dy t = dy t' , dz t = dz t' , ticle that moves parallel to the boundary of a semi-infi- (8) () () β ' () () β '() nite medium at a distance z0 from this boundary under mx t = 0, my t = dy t' , mz t = Ð dz t' .

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 THE RELATIVISTIC THEORY OF FLUCTUATION ELECTROMAGNETIC INTERACTIONS 1817

Note that the variables t and t' in expressions (8) are and the relationship between the volume elements in related by transformation (5). The formulas necessary these systems d3r = γÐ1d3r', we obtain for the transformations of the moments d and m for the 〈〉3 γ 2[]〈〉3 case when the particle velocity has an arbitrary direc- ∫ j'E' d r' = ∫ jE d r Ð FxV , (14) tion and m'(t') ≠ 0 are given in [24]. The fluctuation force acting on the particle can be where determined from the generalized formula for the 3 1 3 Lorentz force: F = 〈〉ρE d r + --- 〈〉[]jH d r. (15) x ∫ x c∫ x 3 1 3 F = ∫ 〈〉ρE d r + ---∫ 〈〉[]jH d r, (9) On the other hand, with allowance made for relation- c ships (3), (4), ρ = ÐdivP, j = ∂P/∂t + ccurlM, and m' = where ρ and j are the charge and current densities of the 0, we have particle, respectively; E and H are the electric and mag- ∂ netic field strengths, respectively; and the angle brack- 〈〉3 P' 3 ∫ j'E' d r' = ∫ ------∂ -E' d r' ets 〈…〉 indicate the complete quantum-statistical aver- t' (16) aging. The integrals are taken over the particle volume. 〈〉δ()δ()δ()ú 3 〈〉ú As is usually done in the fluctuation electromagnetic = ∫ x' y' z' d'E' d r' = d'E' . theory, all vector quantities are treated as Heisenberg γÐ1 operators. By using relationships (8) and dt' = dt, expression (16) can be rearranged to give By definition, we have ρ = ÐdivP and j = ∂P/∂t + ccurlM, where P and M are determined by relation- 3 2 〈〉j'E' d r' ==〈〉γdú'E' []〈〉βdúE Ð 〈〉[]dúH x . (17) ships (1) and (2), respectively. Substitution of the rele- ∫ vant expressions into relationship (9) gives Finally, from formulas (14), (16), and (17), we derive e 3 1 the following expression: F ==∫ 〈〉ρE d r 〈〉∇()dE + --- 〈〉[]d∂H/∂t , (10) c 〈〉3 〈〉βú 〈〉[]ú ∫ jE d r = FxV + dE Ð dH x (18) m 1 3 1 F ==--- 〈〉jH d r 〈〉∇()mH Ð --- 〈〉[]d∂H/∂t . (11) ≡ F VdQ+ /dt. c∫ c x By summing expressions (10) and (11), we obtain It is evident that the derivative dQ/dt can be treated as the heating (cooling) rate of the particle. It should be FF==e + Fm 〈〉∇()dE+ mH . (12) emphasized that all the quantities entering into expression (18) refer to the laboratory coordinate system K. Then, we explicitly separate the contributions of spon- By analogy with the transformation of formula (9) for taneous (sp) and induced (in) fluctuations: the fluctuation force, the derivative dQ/dt can be written sp in sp in in the more compact form F = 〈〉∇()d E + m H (13) in sp in sp V + 〈〉∇()d E + m H . Qú ==〈〉dúE Ð --- 〈〉[]dúH x 〈〉dúEm+ ú H c (19) Equation (13) is most convenient for subsequent calcu- sp in sp in in sp in sp lations of the fluctuation force. It should be noted that = 〈〉dú E + mú H + 〈〉dú E + mú H . the differentiation in this equation should be carried out prior to the substitution of the current coordinates of the In the nonrelativistic case, it is obvious that Qú = 〈〉dúE . particle (V, t, 0, z0). The normal component of the force Hence, if the particle is considered a system of coupled F describes the dynamic van der Waals interaction. charges, it is easy to show that the work done by the Now, we analyze the work done by the fluctuation fluctuation electric field on the particle in a unit time is electromagnetic field on the particle in a unit time. This equal to the sum of the power of the tangential force work is determined by the dissipation integral applied to the center of mass of the particle and the rate 3 of heating that leads to an increase in the root-mean- ∫ 〈〉jE d r . Taking into account the Lorentz transfor- square velocities of charges constituting the particle. mations of the current density, the charge density, and This is a direct consequence of the energy conservation the electric field strength in the coordinate systems K law (in the absence of radiation) and K', dW 〈〉3 Ð------==∫ jE d r FxVdQ+ /dt. (20) γ ()ρ β dt jx' ==jx Ð c , jy' jy, The left-hand side of this equation represents the ' ρ γρ() β jz ==jz, ' Ð jx , energy loss rate of the fluctuation electromagnetic field. The heating of a molecule (nanoparticle) results in an γ ()β γ ()β Ex' ==Ex, Ey' Eγ Ð Hz , Ez' =Ez + Hy , increase in its temperature, whereas the heating of a

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1818 DEDKOV, KYASOV neutral atom can be interpreted as a Lamb shift of the sional wave vector k = (kx, ky) and the frequency. The levels due to the interaction with the surface. induced electric field Ein is expressed through the scalar Relationships (18) and (20) are of fundamental potential ϕ, which is the solution of the Poisson equa- importance for understanding the interaction between tion ∇2ϕ = Ð4πρ with additional continuity conditions the particle and the surface. In particular, we ignored for the scalar potential ϕ and the normal component of the term dQ/dt in relationship (20) in our earlier work the electric induction at the boundary z = 0. The quan- [25] and, as a consequence, obtained an incorrect ρ ρ asymptotic relationship for the tangential force at tity is defined by the equation = ÐdivP and, with due T 0. On the other hand, the tangential force for regard for expression (1), can also be represented as a charged particles and multipoles with a permanent Fourier transform. The corresponding calculations are moment, is quite correctly expressed in terms of the dis- described in detail in [26, 27]. After performing the sipation integral, because dQ/dt = 0. Virtually the same statistical averaging in expressions for 〈∇dspEin〉 and error was made by Dorofeyev et al. [22], who deter- 〈∇dinEsp〉, we obtain the relationships for the normal mined the tangential force from the series expansion of and tangential components of the fluctuation force. The the dissipation integral in powers of the velocity. In this former component is conveniently related to the con- case, to the first order in the velocity (for the force F ), x servative (van der Waals) potential of attraction for the the corresponding term of the expansion is proportional surface with the use of the expression F = Ð∂U/∂z. As to V2. In contrast to identity (18), this approach makes it z impossible to separate explicitly the power of the tangen- a result, we have [14] tial force from the heating rate of the particle and to ប obtain more general results when the dependence on the (), ω Ð2kz0 Uz0 V = -----2∫d dkxdkyke velocity exhibits a nonlinear behavior. π (21) ú ≠ × { ()α, ω ()∆ω []()∆ω ()ω At Q 0, the heat flux is localized in the space, CT1 '' ' Ð Vkx + ' + Vkx because it is associated with the volume of the small ()∆, ω ()αω []}()αω ()ω particle. However, the dissipation of the electromag- + CT2 '' ' Ð Vkx + ' + Vkx , netic field energy on the surface and in the bulk of the 〈〉()sp∇ in 〈〉()in∇ sp medium at rest leads to heating of the medium even at Fx = d Ex + d Ex Qú = 0 (due to the motion of charges and multipoles ប 2 ω Ð2kz0 with a permanent moment). In this case, the thermal = Ð------d dkxdkykxke π2 ∫ effect arises in the medium at a later stage owing to the (22) × { ()α, ω ()∆ω []()ω ∆ ()ω relaxation motion of multipoles to which the transla- CT1 '' '' + Vkx Ð '' Ð Vkx ú tional energy of the particle is transferred. At Q < 0, + CT()∆, ω ''()αω []}''()ω + Vk Ð α''()ω Ð Vk , when the heat flux is directed from the particle to the 2 x x surface and the particle is retarded, the total heating of where the medium is determined by the right-hand side of Eq. (20), which accounts for both types of dissipation បω of the fluctuation field energy. Mathematically, this is (), ω  CT = coth------, equivalent to the dissipation integral taken over the vol- 2kT ume of the medium at rest. εω()Ð 1 ∆ω()== ∆' + i∆'' ------Therefore, in order to calculate the fluctuation force εω()+ 1 and the heat flux, it is necessary to perform statistical averaging in expressions (13) and (18). Unlike the com- is the dielectric-response function of the surface and T putational schemes used in [8Ð10], our technique is 1 clear physically and efficient computationally. is the particle temperature, which, in the general case, differs from the surface temperature T2 (for the neutral atom, T = 0). 3. THE NONRELATIVISTIC CASE 1 It is expedient to consider the simpler nonrelativistic Note that integration over all the variables is carried ∞ problem with the aim of demonstrating the general out in the interval (0, ) and the negative sign corre- scheme of calculations and comparing the obtained sponds to the decelerating (dissipative) tangential results with relativistic formulas. In this case, the terms force. involving the contribution of the magnetic field should be omitted from expressions (13) and (18). The dipole For subsequent comparison with relativistic formu- moments can be represented as Fourier integral trans- las, relationship (22) can be conveniently rewritten in forms with the time factor exp(Ðiωt), and the electric an equivalent form, which can be obtained by trans- field components can be written in the form of expan- forming the frequency arguments in the first term in the sions in terms of the components of the two-dimen- curly brackets with allowance made for the analytical

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 THE RELATIVISTIC THEORY OF FLUCTUATION ELECTROMAGNETIC INTERACTIONS 1819 properties of the integrands. After manipulations, we The heat ﬂux Qú can be calculated in the same man- obtain ner. As a result, we have [14]

2ប Ð2kz0 sp in in sp ω Qú ==〈〉dúE 〈〉dú E + 〈〉dú E Fx = Ð------2 ∫d dkxdkykxke π (23) ប 2 ω Ð2kz0 ×Ξω[]()Ξω, (), = Ð------∫d dkxdkyke V Ð ÐV , π2 (27) × { , ω α ω []ω∆ ω ω∆ ω where CT()1 ''() ''(+Vkx )'' + ''() Ð Vkx Ð CT()∆, ω ''()ω [()αω + Vk ''()ω + Vk Ξω(), V 2 x x ()αω (ω )]} ∆ ()αω ()ω [](), ω (), ω + Ð Vkx '' Ð Vkx . = '' '' + kxV CT2 Ð CT1 + kxV . By analogy with expression (23), relationship (27) can From relationships (21)Ð(23), it is easy to derive all be rearranged to the equivalent form standard formulas that describe both the attraction of the spherical particle to the surface and the tangential 2ប Ð2kz0 Qú = Ð------dωdk dk ke ∆''()ω force. Specifically, in the limit of low velocities, we π2 ∫ x y have ×ω{()α()ω [ (), ω + Vkx '' + Vkx CT1 + Vkx (28) ()]ω, ω ()α()ω ∞ Ð CT2 + Ð Vkx '' Ð Vkx 3បV d∆''()ω F = Ð------dω[]αωnT(), ω Ð nT(), ω ()------× []}CT(), ω Ð Vk Ð CT(), ω . x π 5 ∫ 1 2 dω 1 x 2 2 z0 0 (24) ∞ By expanding expression (27) in terms of the velocity ប and retaining the first two terms, we obtain ωα ()∆ω ()ω (), ω [](), ω Ð,------∫d '' '' nT2 1 + nT2 kT2 ∞ 0 ប Qú = Ð------dωωα''()∆ω ''()ω []nT(), ω Ð nT(), ω 1 π 3∫ 1 2 (), ω z0 nT = ------បω/kT . 0 e Ð 1 ∞ 3បV 2  d2∆'' ω ()ωα, ω Ð ------5-∫d CT1 ''------2- (29) For T = T = T, from expression (24), we deduce the 8πz  dω 1 2 0 0 formula [28] dα'' d2α''  Ð CT()∆, ω '' 2------+ ω------. 2 ω 2 ∞ d dω  3បV d F = ------α''()∆ω ''()ω ------nT()ω, ω d , (25) x π 5∫ dω 2 z0 Even a cursory examination of formulas (23), (24), 0 (28), and (29) demonstrates that, in general, the tangential force and the heat flux between the particle and the according to which the dissipative force at T 0 is surface can be different in sign. Polevoœ [9] was the first absent to the first order in the velocity. In a more gen- to call attention to this fact in the study of the tangential eral case, from relationship (23) in the limit T1 0forces between semi-infinite media. Now, we turn to an and T2 0, we obtain analysis of the relativistic problem.

∞ ∞ 4. SOLUTION OF THE MAXWELL EQUATIONS 4ប Ð2kz0 Fx = ------∫kxdkx∫dkyke FOR INDUCED COMPONENTS π2 OF THE ELECTROMAGNETIC FIELD 0 0 (26) kxV The relativistic problem can be efﬁciently solved ×ω∆()αω ()ω within the formalism of the Hertz vectors Pe(r, t) and ∫ d '' '' Ð Vkx . Pm(r, t). The Fourier transformers of these vectors obey 0 the equations [29]

2 From expression (26), at V 0, we derive the rela- ω e 4π ∆ + ------εω()µω()Pω ()z = Ð------Pω ()z , (30) 3 7 2 k εω() k tionship Fx ~ V /.z0 c

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1820 DEDKOV, KYASOV

2 Here, ω m 4π ∆ + ------εω()µω()Pω ()z = Ð------Mω ()z , 2 k µω() k 2π c Θ()z = ------exp[]Ðq ()zz+ , (31) q 0 0 ∂2 0 ∆ = Ðk 2 Ð k 2 + ------. x y ∂ 2 in () Θ() ()∆()∆ z Hx, ωk z = z Ðkx kxmx + kymy e Ð em --- The boundary conditions corresponding to the for- (37) ω2 mulated problem for the Hertz vectors and the solutions ∆ ∆ Ð ikxq0mz m + ------mx e of Eqs. (30) and (31) are given in Appendix A. The c2 amplitudes of the electric and magnetic ﬁelds can be ω expressed through the Hertz vectors with the use of the i ∆ ky ∆ ∆ Ð ------Ð em -----()kxdx + kydy ++ikydz e q0dy m , relationships [29] c q0

ω2 in () Θ() ()∆()∆ --- e εω()µω() e Hy, ωk z = z Ðky kxmx + kymy e Ð em Eωk = graddiv Pωk + ------Pωk c2 (32) ω2 ω ∆ ∆ i m Ð ikyq0mz m + ------my e (38) + ------µω()curlPω , 2 c k c iωk + ------Ð∆ -----x()k d + k d ++ik d ∆ q d ∆ , c em q x x y y x z e 0 x m ω2 0 m εω()µω() m Hωk = graddiv Pωk + -----2- Pωk c in ikxmx + ikymy 2 2 (33) H , ω ()z = Θ()z ------()k ∆ Ð q ∆ z k q em 0 e iω e 0 Ð ------εω()curlPω . (39) c k ω + k2m ∆ +,----()k d ∆ Ð k d ∆ z m c x y m y x m From expressions (32), (33), and (A3)Ð(A8), we obtain (for the vacuum region z ≥ 0) where the projections di and mi (i = x, y, z) are represented by formulas (A9), in () Θ() ()∆()∆ 2 2 2 1/2 2 2 2 1/2 Ex, ωk z = z Ðkx kxdx + kydy m Ð em --- ()ω ()εωµωω q0 = k Ð /c , q = k Ð ()() /c , 2 2 2 2 (40) ω k = kx + ky , ∆ ∆ Ð ikxq0dz e + -----2-dx m c µω()q Ð q ∆ ()ω = ------0 , (41) ω m µω()q + q i ∆ ky ()∆ ∆ 0 + ------Ð em ----- kxmx + kymy ++ikymz m q0my e , c q0 εω() ∆ ()ω q0 Ð q (34) e = ------εω() , (42) q0 + q

in 2 E , ω ()z = Θ()z Ðk ()∆k d + k d ()Ð ∆ --- ()εω()µω() y k y x x y y m em ∆ ()ω 2q0 Ð 1 em = (------εω() )µω()() -. (43) q0 + q q0 + q ω2 ∆ ∆ ∆ ∆ ∆ Ð ikyq0dz e + -----2-dy m Note also that the functions m, e, and em are related c by the expression ω i ∆ kx ()∆ ∆ 2∆ 2∆ 2∆ Ð ------Ð em ----- kxmx + kymy ++ikxmz m q0mx e , k em Ð q0 e Ð0.q0 m = (44) c q0 (35) 5. CORRELATORS OF THE PHYSICAL in () Θ()ikxdx + ikydy()2∆ 2∆ QUANTITIES Ez, ωk z = z ------k em Ð q0 m q0 When determining the contributions of spontaneous (36) ω and induced ﬂuctuations to the force and the heating 2 ∆ i ()∆ ∆ + k dz e +.------ikxmy e Ð ikymx e rate, it is necessary to express the Fourier components c of the ﬁelds and dipole moments of the particle in the

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 THE RELATIVISTIC THEORY OF FLUCTUATION ELECTROMAGNETIC INTERACTIONS 1821 coordinate system K. For the induced ﬁeld components, between din and Esp. In the rest system K' of the particle, we have this relationship can be written as

in 1 in t' E ()r, t = ------dωdk dk Eω ()z 3∫ x y k in() α()τ sp()ττ ()2π (45) d' t' = ∫ t' Ð ' E' r0' ' d '. × ()()ω Ð∞ exp ikx xk+ y y Ð t , Next, we rewrite this relationship with due regard for the in(), 1 ω in () sp H r t = ------∫d dkxdkyHωk z Fourier transform of the expansion of E' in terms of ()2π 3 (46) spatial and time variables with the use of expressions (5) × ()()ω and (8) and the Lorentz transformations for the ﬁeld exp ikx xk+ y y Ð t . amplitudes and derive the sought formulas Making allowance for expressions (5) and (8), the γ Ð1 spontaneous components of the vectors dsp(t) and in ω αγω()()sp () dx = ------∫d dkxdky Ð Vkx Exωk z0 msp(t) can be written in the form ()2π 3

+∞ × exp()Ði()ω Ð Vk t , ω x sp() γ d 'sp()γγω Ð1 Ðiωt dx t = ∫ ------dx e , γ 2π in ω αγω()() Ð∞ dy = ------3∫d dkxdky Ð Vkx ()2π ∞ + sp sp sp dω sp Ðiωt × []E ω ()βz Ð H ω ()z exp()Ði()ω Ð Vk t , d ()t = γ ------d' ()γω e , y k 0 z k 0 x y ∫ 2π y Ð∞ γ in ω αγω()() dz = ------3∫d dkxdky Ð Vkx +∞ ()2π sp dω sp Ðiωt () γ ' ()γω sp sp (49) dz t = ∫ ------π-dz e , × []() β () ()()ω 2 (47) Ezωk z0 + Hyωk z0 exp Ði Ð Vkx t , Ð∞ in sp() mx = 0, mx t = 0, βγ +∞ in ω αγω()() my = ------d dkxdky Ð Vkx sp dω sp Ðiωt 3∫ m ()t = γβ ------d' ()γω e , ()2π y ∫ 2π z sp sp Ð∞ × []() β () ()()ω Ezωk z0 + Hyωk z0 exp Ði Ð Vkx t , +∞ sp dω sp Ðiωt in βγ m ()t = Ð γβ ------d' ()γω e . m = Ð------dωdk dk αγω()()Ð Vk z ∫ π y z 3∫ x y x 2 ()2π Ð∞ × []sp ()βsp () ()()ω It should be noted that the left-hand and right-hand sides Eyωk z0 Ð Hzωk z0 exp Ði Ð Vkx t . of relationships (47) involve the “laboratory” time t. Substitution of formulas (49) and the spontaneous field Upon substituting formulas (45)–(47) into the first components of the surface into the second terms in terms of expressions (13) and (19), there appears a cor- expressions (13) and (19) leads to the appearance of relator of the dipole moment of the particle in its own correlators of the electric and magnetic field ampli- rest system. This correlator can be represented in the tudes. By virtue of the stationarity of the fluctuations, following form [21]: these correlators can be expressed through the spectral densities [30, 31]: 〈〉sp()γω sp()γω() di' ' dk' Ð Vkx 〈〉sp sp ()π 3δω()δ ω () πប γ បω (48) Ui, ωk V j, ω'k' = 2 + ' kx + kx' 2 δ δω ω α γω ' (50) = ------γ - ik ()' + Ð Vkx ''()' coth------. 2kT1 ×δ()()sp sp ky + ky' Ui V j ωk, In order to calculate the second terms in formulas (13) sp sp sp sp and (19), the induced components of the dipole where Ui = Ei and V i = Bi (i = x, y, z). moment of the particle should be expressed in the laboratory coordinate system. For this purpose, we will In turn, the spectral densities of the amplitudes can use a standard nonrelativistic integral relationship be expressed through the components of the retarded

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1822 DEDKOV, KYASOV × { ()α, γω()()γω() Green’s function for radiation in the medium with the CT1 + Vkx '' + Vkx use of the relationships [2, 27] ×χ[]()+ ()∆ω, ()ω χ()+ ()∆ω, ()ω 2 e k e' + m k m' sp sp ω ()E ()z E ()z' ω = i------CT(), ω i k k 2 ()α, γω()()γω() 2c (51) + CT1 Ð Vkx '' Ð Vkx × []()ω ()ω ()Ð ()Ð Dik k; zz' Ð Dki* k; z'z , ×χ[]()∆ω, ()ω χ ()∆ω, ()ω e k e' + m k m' sp sp 1 + CT()α, ω '()γω()+ Vk (55) ()B ()z B ()z' ω = i---CT(), ω curl curl' 2 x i k k 2 il km (52) ()+ ()+ × []()ω ()ω ×χ[]()∆ω, k ''()ω + χ ()∆ω, k '' ()ω Dlm k; zz' Ð Dml* k; z'z , e e m m ()α, ω ()γω() + CT2 ' Ð Vkx sp sp ω ()E ()z B ()z' ω = Ð------CT(), ω curl' ()Ð ()Ð i k k 2c km ×χ[]}()∆ω, k ''()ω + χ ()∆ω, k '' ()ω (53) e e m m × []D ()ωk; zz' Ð D* ()ωk; z'z . im mi បγ Ð ------dωdk dk cos()2q˜ z π2 ∫ x y 0 0 The calculation of the Green’s functions is k < ω/c described in detail in Appendix B. Substitution of the ×∆{},,∆ ∆˜ ∆˜ Green’s functions into formula (50) with allowance e m e m , made for relationships (51)Ð(53) makes it possible to determine all the required correlators. បγ () ú ω exp Ð2q0z0 Q = Ð------2 ∫ d dkxdky------π q0 6. FLUCTUATION FORCE AND THE HEATING k > ω/c × {[](), ω (), ω RATE OF A PARTICLE CT1 + Vkx Ð CT2 The results obtained in Sections 4 and 5 and Appen- ×ω()α+ Vk ''()γω()+ Vk dices A and B allow statistical averaging in relation- x x ships (13) and (19). In this case, the formulas derived in ×χ[]()+ ()∆ω, ''()ω χ()+ ()∆ω, ' ()ω [20, 21] for the fluctuation force can be simplified after e k e + m k m the frequency arguments of the integrands are trans- + []CT(), ω Ð Vk Ð CT(), ω (56) formed in the same manner as in the derivation of 1 x 2 expressions (23) and (28). As a result, the components ×ω()α Ð Vk ''()γω() Ð Vk of the fluctuation force and the heating rate of the par- x x ticle are represented by the following relationships: ×χ[]}()Ð ()∆ω, ()ω χ()Ð ()∆ω, ()ω e k e'' + m k m' បγ exp()Ð2q z ------ω ------0 0 បγ sin()2q˜ z Fx = Ð 2 ∫ d dkxdkykx ω 0 0 π q0 Ð ------2 ∫ d dkxdky Ð------k > ω/c π q˜ 0 k < ω/c × [()αCT(), ω Ð CT(), γω()+ Vk ''()γω()+ Vk 2 1 x x ×∆{},,∆ ∆˜ ∆˜ e m e m . ×χ[]()+ ()∆ω, ''()ω χ()+ ()∆ω, '' ()ω e k e + m k m Here, we introduced the designations Ð ()αCT(), ω Ð CT(), γω()Ð Vk ''()γω()Ð Vk (54) ()2 ω2 2 1/2 ()ω2 2 2 1/2 2 1 x x q0 ==k Ð /c , q˜ 0 /c Ð k , ×χ[]]()Ð ()∆ω, ''()ω χ()Ð ()∆ω, '' ()ω ()2 εµω2 2 1/2 ˜ ()εµω2 2 2 1/2 e k e + m k m qk==Ð /c , q /c Ð k , 2 () ()± 2 2 2 2 2 2 ()ω ± Vk បγ sin 2q˜ 0z0 χ ω, β ω x ω e (k ) = 2()1k Ð kx ()Ð /k c + ------, + ------2 ∫ d dkxdkykx------2 π q0 c k < ω/c 2 ×∆{},,∆ ∆˜ ∆˜ ()± ()ω ± Vk e m e m , χ ()ω, β2 2()ω2 2 2 x m k = 2 ky 1 Ð /k c + ------. c2 បγ ω () Fz = Ð------d dkxdky exp Ð2q0z0 It should be noted that the functions ∆˜ and ∆˜ are π2 ∫ e m k > ω/c defined by formulas (41) and (42) with the replacement

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 THE RELATIVISTIC THEORY OF FLUCTUATION ELECTROMAGNETIC INTERACTIONS 1823 () q(q0) q˜ q˜ 0 and integration over the frequency, as with the Fresnel reflection coefficients (RP and RS) of is the case in expressions (21) and (22), is performed in electromagnetic waves with the P and S polarizations the interval from 0 to ∞. The integrals taken over the for which the electric vector lies either in the same plane frequencies ω > ck and ω < ck correspond to the contri- with the wave vector and the vector normal to the surface butions from the radiative and nonradiative modes of (P wave) or in the perpendicular plane (S wave). Note the fluctuation electromagnetic field, respectively. that the P and S waves are predominantly electrostatic and magnetic in character, respectively. For a small-sized spherical particle of radius R, the 7. DISCUSSION AND RESULTS polarizability can be approximated in the following two OF NUMERICAL CALCULATIONS ways: First and foremost, let us consider relationships (54)Ð 3 εω()Ð 1 (56) in the nonrelativistic limit c ∞. In this case, we αω()= R ------, (61a) ()± ()± εω()+ 2 χ ω 2 χ ω have e ( , k) = 2k and m ( , k) = 0. Hence, it follows that relationships (54) and (56) transform into αω()= R3 expressions (23) and (28) and relationship (55) trans- (61b) 31()Ð κRcoth()κR forms into formula (21) with allowance made for the × 1 + ------, ∂ ∂ 2 expression Fz = Ð U/ z. 2()εω()Ð 1 ()1 Ð κRcoth()κR + εω()κ()R Note that an analysis of the relativistic motion of the κ2 π π 1/3 particle with due regard for the retardation effect is of where = (4/ )(9 /4) /rs and rs is the parameter (in practical importance. For numerical calculations, we atomic units) of the jellium model. Relationship (61b) consider contact between homogeneous nonmagnetic is derived within the ThomasÐFermi approximation for metals (µ = 1) when the bulk dielectric functions are a metallic cluster and accounts for the screening of the defined by the expression ε(ω) = 1 + 4πσi/ω ≡ 1 + ai, electron gas [33]. Setting R = 10 nm (typical radius of κ ӷ where σ is the frequency-independent conductivity. Let a tip of probe microscopes), we obtain R 1 for nor- 2 mal metals. Consequently, from expressions (61a) and us introduce new variables: t = kc/ω and u2 = |1 Ð t |, (61b) in the low-frequency limit, we have where u2 = t2 Ð 1 for nonradiative modes (ω < kc) and 3R3 3R3 2 2 ω α''()ω ≈ ------= ------, (62a) u = 1 Ð t for radiative modes ( > kc). On this basis, a 4πσ from formulas (41) and (42), we obtain the following relationships for the imaginary parts of the response 3 R3 1 3 R3 ω3 functions: α''()ω ≈ ------= ------. (62b) 2()κR 3 2()κR ()πσ 3 (a) For nonradiative modes [the upper signs in rela- a 4 tionships (57) and (58)], we have By omitting the relativistic terms of the order of β2 in relationships (54) and (56) and also in the formulas for u 2 2 ∆''()ua, = Ð------[a u()2u Ð 1 ()± ()± e 2 4 2 2 χ (ω, k) and χ (ω, k) in the case when the polariz- aa[]u + ()2u + 1 e m (57) ability is given by expression (62a), we obtain the fol- ()2 2 − ()()2 2 2 ] +2Ð a u()2u + 1 + aA1 + 2a u ++2u 1 A2 , lowing expressions: ∞ 2u2 2u 3ប ω 5 ∆'' ()ua, = Ð ------± ------A , (58) 3R V  ω ()± 2 m 2 Fx = ------∫---- d ∫ 1 u du a a 4π2σ c 0 0 ≤ u < ∞, 0 ≤≤u 1 2 4 2 1/2 × []()ω , ()ω a + u Ð u expÐ2 uz0/c sin Ð 2 uz0/c (63) A1 = ------, (59) 2 × {()θ θ []()∆± 2 , ∆ , A()1 Ð A()2 2u + 3 e''()3ua+ m''()ua 1/2 a2 + u4 + u2 θ []}()∆± 2 , ∆ , A = ------. (60) Ð B()1 2u + 1 e''()ua+ m''()ua , 2 2 ∞ 3ប ω 3 (b) For radiative modes [the lower signs in relation- ú 3R ω2 ω Q = Ð------∫---- d ∫ du ships (57) and (58)], formulas (57) and (58) are modi- 2π2σ c fied by the replacement 2u2 + 1 2u2 Ð 1. Compared 0 0 ≤ u < ∞, 0 ≤≤u 1 × []()ω , ()ω to the relevant expressions deduced in our earlier work expÐ2 uz0/c sin Ð 2 uz0/c [32], expressions (57)Ð(60) are derived without invok- × {()θ θ [()∆± 2 , ∆ , ing the additional simplifying assumption a ӷ 1. A()1 Ð A()2 2u + 1 e''()ua+ m''()ua It should be noted that, in the designations used in ∆ ∆ β2 ± 2 ()]()∆± 2 , ∆ , [11Ð13], the response functions e and m coincide + ()21 u u + 5 e''()5ua+ m''()ua (64)

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1824 DEDKOV, KYASOV

10–1 β2 θ ± 2 []}()∆± 2 , ∆ , Ð 4B()11 ()1u u e''()ua+ m''()ua , (a) 1 xex Ax()==------, Bx() ------, x x 2 e Ð 1 ()e Ð 1 –3 1 10 បω θ 12, = ------. kT12,

, arb. units ω . Here, the upper signs, the functions exp(Ð2 uz0/c) in the Q 10–5 3 square brackets of the integrands in expressions (63) and (64), and the integration interval 0 ≤ u < ∞ correspond to the contribution from the nonradiative modes of the 2 electromagnetic ﬁeld, whereas the contribution from –7 the radiative modes of the electromagnetic ﬁeld is cal- 10 ω 0 5 10 15 20 culated for the lower signs, the functions sin(Ð2 uz0/c), z0, nm and the integration interval 0 ≤ u ≤ 1. After introducing the dimensionless parameters α = πσប ប (b) 4 /kT and b = 2kT(z0/ c), we transform the double 10 integrals in relationships (63) and (64) into the univer- sal functions of α and b, which can be determined by 1 numerical integration (see below). For normal metals 2 (σ = 1017 sÐ1) at temperatures T ≤ 300 K, we have α ≥ 10–1 ≤ µ 3 30 000 and b 0.26z0 ( m). The parameter b serves as a measure of the retardation effect, which is more pro-

. , arb. units nounced at smaller distances from the surface with an Q 10–3 increase in the temperature. In the absence of retarda- 4 tion, we set c = ∞ and b = 0. In passing to the limit ∞ ∆ c , formulas (57) and (58) are reduced to e'' = 2/a and ∆'' = 0, respectively. 10–5 m 0246 It follows from expression (64) that, even under iso- z , µm 0 thermal conditions (T1 = T2 = T), the moving particle is 0.4 heated by the surface (Qú > 0). However, this effect is ú 2 ≠ (c) relativistic, because Q ~ (V/c) . At T1 T2, there is a more significant velocity-independent contribution to 0.3 the heat flux that leads to heating of the particle at T1 > ú ú 1 T2 (Q < 0) and its cooling at T1 < T2 (Q > 0). 0.2 Relationship (63) predicts a negative (decelerating) tangential force acting on the moving particle. How- . , arb. units ever, the general expression (56) admits of the occur- Q 3 rence of an accelerating force. The same inferences fol- 0.1 low from the nonrelativistic formulas (27) and (31). Physically, the fluctuation acceleration becomes possi- 2 ble owing to the internal degrees of freedom of the neutral particle. By contrast, the tangential forces can be 0 20 40 60 80 100 only decelerating when charged particles or multipoles µ z0, m with a permanent moment execute motion. More detailed discussion of this problem is beyond the scope Fig. 2. Dependences of the heat flux on the distance at α 4 α × 4 of the present work, because it would require consider- (a) (1) = 10 and T = 300 K, (2) = 2 10 and T = 300 K, ation of dielectric functions characterized by a reso- (3) α = 104 and T = 1200 K and (b) (1Ð3) α = 104 and T = 300 K. (b) (1) The sum of the radiative and nonradiative nance structure of the absorption peaks of the particle contributions, (2) the contribution of nonradiative modes, and surface materials. (3) the contribution of radiative modes, and (4) the nonrela- For a small retardation b = 2kT(z /បc) Ӷ 1, from tivistic approximation. The magnitude of the heat flux (in 0 erg/s) can be obtained by multiplying the ordinate by the relationships (63) and (64), it is easy to derive analyti- quantity κប/σ(kT/ប)3R3, where κ = (a) 2 × 1019 and cal expressions for the radiative contributions, because (b, c) 2 × 1010. the inner integrals are well approximated by functions

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 THE RELATIVISTIC THEORY OF FLUCTUATION ELECTROMAGNETIC INTERACTIONS 1825

1 104 (a) (b) 10–2 102 1 2 10–4 3 1 3 , arb. units , arb. units x –6 –2 10 x 10 5 4 F

F 2

10–8 10–4 1 10–10 10–6 0 5 10 15 20 0246 µ z0, nm z0, m 0.08 (c) 0.06 1

0.04 2 , arb. units

x 0.02 F 3 0

–0.02 0 50 100 µ z0, m

α Fig. 3. Dependences of the tangential (decelerating) force on the distance for different parameters , particle temperatures T1, and medium temperatures T . (a) (1) α = 104 and T = 300 K, (2) α = 104 and T = 1200 K, (3) α = 2 × 104 and T = 300 K at particle 2 α 4 temperatures T1 = T and medium temperature T2 = 0. (b) (1) The nonrelativistic approximation at = 10 and T1 = T2 = T = 300 K; (2) the radiative contribution, (3) the nonradiative contribution, and (4) the sum of the radiative and nonradiative contributions at α 4 α 4 = 10 , T1 = 300 K, and T2 = 0; and (5) the curve corresponding to = 10 and T1 = T2 = T = 300 K. (c) (1) The sum of the radiative and nonradiative contributions, (2) the radiative contribution, and (3) the nonradiative contribution at α = 104, T = 300 K, and 1 κ 3 σ T2 = 0. The magnitude of the tangential force (in dyn) can be obtained by multiplying the ordinate by the quantity (R VkT/ ), where κ = (a) 2.4 × 1032 and (b, c) 2.4 × 1017.

α 1/2 ωប ú of the type C(x/ ) , where x = /kT and C is a numer- at small distances. However, the quantities Fx and Q at ical constant. After integration, we obtain larger distances considerably increase compared to

3 15/2 those obtained in the nonrelativistic approximation (see ()rad បVR z kT F ≈ Ð720------0 ------, (65) Figs. 2b, 3b). The presence of the portions in which the x 3/2 6 ប () σ c ()rad ú rad contributions Fx and Q linearly increase with an

3 15/2 increase in the distance z0 according to formulas (65) ()rad បR z kT 2 Qú ≈ Ð170------0 ------()1 Ð β . (66) and (66) leads to their local predominance over the 3/2 4 ប σ c contributions of nonradiative modes in the range µ ≤ ≤ µ ()rad µ ≤ ≤ µ Figures 2 and 3 present the results of numerical calcula- 10 m z0 60 m for Fx and 10 m z0 90 m ú ()rad tions of the tangential force Fx and the heat flux Q from for Qú (Figs. 2c, 3c). At the same time, the total depen- relationships (63) and (64) for small (0 < z0 < 20 nm, dences of the tangential force and the heat flux exhibit a µ µ Figs. 2a, 3a), medium (0.1 m < z0 < 5 m, Figs. 2b, 3b), monotonic decreasing behavior over the entire range of µ µ and large (5 m < z0 < 100 m, Figs. 2c, 3c) distances distances under investigation. The calculations also dem- from the surface. The retardation effects are negligible onstrate that the fluctuation electromagnetic interaction is

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1826 DEDKOV, KYASOV enhanced with an increase in the resistivity of a mate- 8. CONCLUSIONS rial (see, for example, Figs. 2a, 3a). Thus, we developed a consistent relativistic theory The above inferences are in agreement with the of dynamic fluctuation electromagnetic interaction of a results of our preliminary calculations [32] and the data small-sized particle with a flat surface. This theory pro- obtained by Pendry [17], who noted that the contribu- vides correct passage to nonrelativistic formulas in the tion of the S mode to the heat flux is negligible (for non- limit c ∞. Within the proposed approach, general magnetic materials) at nanometer distances. In contrast, relationships were derived for the dissipative-loss inte- Volokitin and Persson [13, 18] argued that, even at dis- gral, the thermal heating (cooling) rate, and the power tances of 1Ð10 nm, the inclusion of the S mode and the of the tangential force. The derived relationships retardation effects (for normal metals) leads to a sub- account for the difference between the temperatures of stantial increase in the heat flux and the tangential force the particle and the surface and (in the general form) as compared to those calculated within the nonrelativ- their dielectric and magnetic properties. istic approximation (accounting only for the contribu- The results of the numerical calculations for high- tion of the P mode). In this respect, we note that, in their conductivity materials demonstrated that the largest calculations of the tangential force, Volokitin and Pers- values of the tangential forces and the heat fluxes son [13] disregarded the contribution of the magnetic- (larger at lower conductivities) are obtained at nanom- polarization current (jm = ccurlM). However, this cur- eter distances between the particle and the surface at rent in the laboratory coordinate system is observed which the nonradiative modes of the electromagnetic field make a dominant contribution. For distances even for a particle possessing only the fluctuation elec- µ tric dipole moment in the intrinsic coordinate system exceeding 1 m, the tangential force and the heat flux (see Section 3). Moreover, it should also be noted that, are considerably larger than those calculated in the non- in [17, 18], the heat flux was calculated only in the spe- relativistic approximation. This can be explained by the relativistic retardation effect and an increase in the con- cific case of a zero relative velocity. tribution of the radiative modes, which leads to a Let us return to the results of our calculations. The decrease (in magnitude) in the exponents of the corre- ú ∞ sponding dependences. The range of distances at which asymptotic dependence Q (z0) calculated at z0 the contribution of the radiative modes is dominant does not exhibit distance-independent contributions depends on the product z0T and reaches several tens of characteristic of black-body radiation. This is quite µ microns. For z0 > 100 m, the nonradiative modes again consistent with the fact that the heat flux is associated make a dominant contribution. The tangential force and just with the fluctuation electromagnetic interaction the heat flux, which are associated with the fluctuation between the particle and the surface and, hence, should electromagnetic interaction, monotonically decrease ∞ disappear in the limit z0 when the objects do not with an increase in the distance between the particle interact. A comparison of the heat flux Qú (z ) with the and the surface over the entire range of distances under 0 investigation (0Ð1 mm). The inclusion of the screening black-body radiation intensity Qú BR in vacuum shows results in a substantial weakening of the fluctuation ú ú ӷ electromagnetic interaction due to the decrease in the that Q (z0)/QBR = p ~ R and p 1 even at very large particle polarizability. In the case when the tempera- distances from the surface. Specifically, according to tures of the particle and the surface are equal to each σ the data presented in Fig. 2, at T1 = 300 K, T2 = 0, = other, the heat flux is directed toward the particle and is 17 Ð1 × 8 2 10 s , and R = 1 nm, we have p = 2 10 at z0 = 5 nm proportional to the dynamic factor (V/c) . × 4 µ and p = 2 10 at z0 = 1 m. Only at distances of several hundreds of microns do we obtain p ≤ 1. APPENDIX A ú The tangential force Fx and the heat flux Q numerically estimated using the polarizability determined by formula (62b) are appreciably smaller than those 1. FOURIER COMPONENTS OF THE ELECTRIC AND MAGNETIC POLARIZATION VECTORS obtained with formula (62a), because formula (62b) involves the small truncating factor (κR)Ð1(ω/4πσ)2. As a From relationships (1) and (2), with due regard for consequence, the dependences calculated with formula formulas (5) and (8), we obtain the following expres- (62b) are similar to those depicted in Figs. 2 and 3, but sions for the projections of the Fourier transforms of the the magnitudes of the quantities are less by a factor of electric and magnetic polarization vectors: ~102(κR)Ð1αÐ2 Ӷ 1 and the formulas analogous to () γδ()sp()γγω()Ð1 expressions (63)Ð(66) include an additional temperature Px, ωk z = zzÐ 0 dx Ð Vkx , factor T2. Therefore, the inclusion of the screening leads ú () γδ()sp()γω() to a drastic decrease in the calculated values of Fx and Q . Py, ωk z = zzÐ 0 dy Ð Vkx ,

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 THE RELATIVISTIC THEORY OF FLUCTUATION ELECTROMAGNETIC INTERACTIONS 1827 ε () γδ()sp()γω() m q0 Ð q P , ω z = zzÐ d Ð Vk , Π , ω ()z = Γ()z ------m ()ω , (A6) z k 0 z x (A1) x k εq + q x () 0 Mx, ωk z = 0, εq Ð q Π m () Γ()------0 - ()ω ( ) γβδ()sp()γω() y, ωk z = z ε my , (A7) My, ωk z = zzÐ 0 dz Ð Vkx , q0 + q

sp 2 , ω () γβδ()()γω() ()εµ Mz k z = Ð zzÐ 0 dy Ð Vkx , m 2q0 Ð 1 Π , ω ()z = Γ()z i------z k q ()µεq + q ()q + q where d sp (…) are the Fourier components of the spon- 0 0 0 i µ taneous dipole moment of the particle in the rest system × ()()ω ()ω q0 Ð q ()ω (A8) kxmx + kymy +,------µ -mz of coordinates K' (i = x, y, z). q0 + q 2π Γ()z = ------exp()Ðq ()zz+ , 2. BOUNDARY CONDITIONS q 0 0 FOR THE HERTZ VECTORS 0 ()ω γsp()γγω()Ð1 The necessary relationships can be obtained from dx = dx Ð Vkx , the continuity conditions of the tangential compo- ()ω γsp()γω() nents of the electric and magnetic ﬁelds at z = 0. The dy = dy Ð Vkx , relevant equations should be written separately for sp e m d ()ω = γd ()γω()Ð Vk , each of the Hertz vectors P and P . With the use of z z x (A9) expression (33), it is easy to show that the following m ()ω = 0, quantities should be continuous: x ()ω γβsp()γω() εω()µω()Π()em, (), my = dz Ð Vkx , i i = xy, m ()ω = Ð γβd sp()γω()Ð Vk . ∂Πe z y x ik Π e ++ik Π e ------z-, x x y y ∂z It should be noted that, in relationships (A9), all projections of the spontaneous dipole moment of the particle m sp 2 2 m m ∂Π ω ω ω Π Π z di ( ') [where ' = ( Ð kxV)/1 Ð V /c and i = x and ikx x ++iky ------, (A2) ∂z y] are the Fourier transforms in the rest system K'. ∂Πe εω()------i- ()i = xy, , εω()Πe, ∂z z APPENDIX B: CALCULATION OF THE RETARDED GREEN’S ∂Πm FUNCTIONS OF RADIATION IN A MEDIUM µω()------i- ()i = xy, . ∂z The retarded Green’s function of electromagnetic radiation in a homogeneous isotropic medium obeys the standard equation [2, 31] 3. SOLUTION OF EQUATIONS (30) AND (31) ω2 The solution of Eqs. (30) and (31) for the induced εω()µω()δ ()ω,, curlik curlkl Ð ------il Dlj rr' components of the Hertz vectors under additional con- c2 (B1) tinuity conditions (A2) for the vacuum region (z, z' ≥ 0) πបµω()δδ() is given by the expressions = Ð4 ij rrÐ ' . µ Π e () Γ() q0 Ð q ()ω The solution of Eq. (B1) does not differ radically x, ωk z = z µ------dx , (A3) q0 + q from the solution of Eqs. (30) and (31). Let us use the analogy with the equation for the electric ﬁeld induced µ by a point dipole d (ω) localized at the point r = r'; that Π e () Γ() q0 Ð q ()ω i y, ωk z = z µ------dy , (A4) is, q0 + q 2 2()εµ ω e 2q0 Ð 1 curlcurl Ð ------εω()µω()Eω()r Π , ω ()z = Γ()z i------2 z k q ()µεq + q ()q + q c 0 0 0 (A5) (B2) εq Ð q ω2 × ()()ω ()ω ------0 - ()ω = 4πµ() ω ------δ()rrÐ ' d()ω . kxdx + kydy +,ε dz 2 q0 + q c

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1828 DEDKOV, KYASOV

A comparison of Eqs. (B1) and (B2) shows that the 12. A. I. Volokitin and B. N. J. Persson, J. Phys.: Condens. retarded Green’s function is equivalent to the electric Matter 11, 345 (1999). ﬁeld of the point dipole with the dipole moment 13. A. I. Volokitin and B. N. J. Persson, Phys. Rev. B 65, 115419 (2002). ប 2 ()ω c δ 14. G. V. Dedkov and A. A. Kyasov, Fiz. Tverd. Tela (St. di = Ð------ij. (B3) ω2 Petersburg) 44 (10), 1729 (2002) [Phys. Solid State 44, 1809 (2002)]. On this basis, with allowance made for relation- 15. D. Polder and M. A. van Hove, Phys. Rev. B 4, 3303 ships (34)Ð(36) and (B1)Ð(B3), we directly obtain all (1971). ω components Dij( k; z, z') of the renormalized Green’s function in the vacuum region (z, z' ≥ 0): 16. M. L. Levin, V. G. Polevoœ, and S. M. Rytov, Zh. Éksp. Teor. Fiz. 79, 2087 (1980) [Sov. Phys. JETP 52, 1054 (1980)]. D = ÐΦ()zz, ' []k 2()∆1 Ð ξ2 ()ω + ()ξk 2∆ ()ω , (B4) xx x e y m 17. J. B. Pendry, J. Phys.: Condens. Matter 11, 6621 (1999). D = ÐΦ()zz, ' []k 2()∆1 Ð ξ2 ()ω + ()ξk 2∆ ()ω , (B5) 18. A. I. Volokitin and B. N. J. Persson, Phys. Rev. B 63, yy y e x m 205404 (2001). Φ , 2∆ ()ω 19. A. A. Kyasov and G. V. Dedkov, Phys. Low-Dimens. Dzz = Ð ()zz' k e , (B6) Semicond. Struct. 5/6, 58 (2002). Φ(), 20. G. V. Dedkov and A. A. Kyasov, Pis’ma Zh. Tekh. Fiz. 29 Dxy ==Dyx Ð zz' kxky (B7) (1), 36 (2003) [Tech. Phys. Lett. 29, 16 (2003)]. × []()∆ξ2 ()ω ξ2∆ ()ω 1 Ð e Ð m , 21. A. A. Kyasov and G. V. Dedkov, Nucl. Instrum. Methods Phys. Res. B 195, 247 (2002). Φ(), ∆ ()ω Dzx ==ÐDxz Ð zz' ikxq0 e , (B8) 22. I. A. Dorofeyev, H. Fuchs, B. Gotsmann, and J. Jersch, Phys. Rev. B 64, 35403 (2001). D ==ÐD ÐΦ()zz, ' ik q ∆ ()ω , zy yz y 0 e 23. W. Pauli, The Theory of Relativity (Pergamon, Oxford, πប 2 (B9) 1958; Nauka, Moscow, 1983). Φ(), 2 c ()()ξ ω zz' ==------exp Ðq0 zz+ ' , /ck. q ω2 24. V. V. Batygin and I. N. Toptygin, Problems in Electrody- 0 namics, 2nd ed. (Nauka, Moscow, 1970; Academic, Lon- don, 1978). REFERENCES 25. G. V. Dedkov and A. A. Kyasov, Phys. Lett. A 259, 38 (1999); Pis’ma Zh. Tekh. Fiz. 25 (12), 10 (1999) [Tech. 1. E. M. Lifshitz, Zh. Éksp. Teor. Fiz. 29, 94 (1955) [Sov. Phys. Lett. 25, 466 (1999)]. Phys. JETP 2, 73 (1955)]. 26. G. V. Dedkov and A. A. Kyasov, Pis’ma Zh. Tekh. Fiz. 28 2. Yu. S. Barash, Van der Waals Forces (Nauka, Moscow, (8), 79 (2002) [Tech. Phys. Lett. 28, 346 (2002)]. 1988). 27. A. A. Kyasov and G. V. Dedkov, Surf. Sci. 463, 11 (2000). 3. I. E. Dzyaloshinskiœ, E. M. Lifshitz, and L. P. Pitaevskiœ, Usp. Fiz. Nauk 73, 381 (1961) [Sov. Phys. Usp. 4, 153 28. M. S. Tomassone and A. Widom, Phys. Rev. B 56, 4938 (1961)]. (1997). 4. H. B. G. Casimir, Proc. K. Ned. Acad. Wet. 51, 793 29. M. B. Vinogradova, O. V. Rudenko, and A. P. Sukho- (1948). rukov, The Theory of Waves (Nauka, Moscow, 1982). 5. V. M. Mostepanenko and N. N. Trunov, The Casimir 30. L. D. Landau and E. M. Lifshitz, Statistical Physics, 3rd Effect and Its Applications (Clarendon, Oxford, 1997). ed. (Nauka, Moscow, 1976; Pergamon, Oxford, 1980), Part 1. 6. M. Kardar and R. Golestanian, Rev. Mod. Phys. 71, 1233 (1999). 31. E. M. Lifshitz and L. P. Pitaevskiœ, Course of Theoretical Physics, Vol. 5: Statistical Physics (Nauka, Moscow, 7. W. L. Schaich and J. Harris, J. Phys. F: Met. Phys. 11, 65 1978; Pergamon, New York, 1980), Part 2. (1981). 32. G. V. Dedkov and A. A. Kyasov, Pis’ma Zh. Tekh. Fiz. 28 8. L. S. Levitov, Europhys. Lett. 8, 488 (1989). (23), 50 (2002) [Tech. Phys. Lett. 28, 997 (2002)]. 9. V. G. Polevoœ, Zh. Éksp. Teor. Fiz. 98, 1990 (1990) [Sov. 33. M. B. Smirnov and V. P. Krainov, Laser Phys. 9, 943 Phys. JETP 71, 1119 (1990)]. (1999). 10. V. E. Mkrtchian, Phys. Lett. 207, 299 (1995). 11. J. B. Pendry, J. Phys.: Condens. Matter 9, 10301 (1997). Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

REVIEWS

Band Structure and Properties of Superconducting MgB2 and Related Compounds (A Review) A. L. Ivanovskiœ Institute of Solid-State Chemistry, Ural Division, Russian Academy of Sciences, Pervomaœskaya ul. 91, Yekaterinburg, 620219 Russia e-mail: [email protected] Received March 13, 2003

Abstract—This paper presents an overview of the current state of the art in research into the electronic structure and properties of a new superconductor, namely, MgB2, and a large number of related compounds by computational methods of the band theory. Consideration is given to the speciﬁc features of the surface states of magnesium diboride, the electron and hole doping effects in this compound, and the concentration dependences of the band structure and the properties of Mg1 − xMexB2 and MgB2 Ð yXy solid solutions and a number of superstructures. The electronic properties of AlB2-like phases, boron, higher borides, a series of ternary layered boron-containing phases, and compounds with structures of the antiperovskite type (MgCNi3 and others) are discussed in terms of their superconducting characteristics. The results obtained in modeling nanotubes and fullerene-like nanoparticles based on MgB2 and related borides are analyzed. © 2003 MAIK “Nauka/Interpe- riodica”.

≤ 1. INTRODUCTION tor (Table 1). Low-temperature superconductivity (Tc The discovery of the critical transition (T ~ 39 K) in 0.5Ð4.5 K) is observed in a number of binary d metal c borides [2, 7] and a group of ternary (LnRuB and layered magnesium diboride in 2001 [1] and subse- 2 quent systematic investigations of this diboride and a LnRh4B4) and pseudoternary [(Ln1 − x Lny' )Rh4B4] large number of related compounds have made a con- borides [8]. High critical temperatures Tc (~ 16Ð23 K) siderable contribution to the development of the con- were revealed for intermetallic borocarbides [9, 10], cept of superconductivity in anisotropic materials and namely, layered quaternary phases of the LuNi2B2C led to the revision of many previously obtained results type (see reviews [11, 12]). and models. Since 2001, the properties of the MgB2 diboride The critical temperature of MgB2 is nearly twice as have been investigated thoroughly. In particular, it has high as the maximum critical temperature Tc of classi- been established that the characteristics of this com- cal binary superconductors, such as transition-metal pound are stable in magnetic fields and under irradia- compounds and alloys with high-symmetry structures tion. The critical transport currents in MgB2 are large in of the B1 (NbN, Tc ~ 17.3 K) or A15 (Nb3Ge, Tc ~ 23 K magnitude and relatively insensitive to contact with [2]) type, in which the superconductivity is governed intercrystalline boundaries. These properties and a by the electron state of the d metals. number of other superconductivity characteristics of Among the anisotropic superconductors, magnesium MgB2 are very attractive for practical applications (see diboride MgB2 bears the closest similarity to quasi-two- reviews [13Ð16]). dimensional intercalated graphite compounds (LiC6Ð8, A large number of investigations have been per- KC8, etc.). Note that the superconductivity in the interca- formed in the field of materials science of supercon- lated graphite compounds (Tc ~ 0.55Ð5.0 K) is due to ducting MgB2. This compound has been prepared in the interaction of carbon π bands with phonon modes of form of single crystals, porous and compact ceramics, planar networks composed of carbon atoms [3Ð5]. A films, long wires, tapes, and composites (see [13–16]). similar mechanism based on the dominant role played Moreover, MgB2-based nanostructures, such as nano- by carbon π bands is also characteristic of supercon- structured thin films [17] and compact nanoceramics ducting fullerides [6]. However, the type of occupation [18], have been synthesized recently. At present, of the energy bands (partially occupied σ bands, see researchers have continued to actively search for new below) in MgB2 differs radically from that in the above superconducting phases related to MgB2. compounds. Apart from the solution of problems in materials sci- It should be noted that, among the boron-containing ence, basic concepts regarding the nature of supercon- phases, the MgB2 diboride is not the sole superconduc- ductivity in the MgB2 diboride and related compounds

1830 IVANOVSKIŒ

Table 1. Critical temperatures and structural types of superconducting binary semiborides Me2B, monoborides MeB, diborides MeB2, and higher borides of metals [14]

Boride Tc, K Structure Boride Tc, K Structure

Ta2B 3.12 MoB2.5 8.1 AlB2 θ Mo2B 4.75Ð5.97* -CuAl2 ReB1.8Ð2 4.5Ð6.3 θ W2B 3.10Ð3.22 -CuAl2 YB6 7.1 CaB6

Re2B 2.8 LaB6 5.7 CaB6

ZrB 2.8Ð3.4 ThB6 0.74 CaB6 α NbB 8.25 -TlI NbB6 3.0

MoB 0.5 ScB12 0.39 UB12

HfB 3.1 YB12 4.7 UB12 α TaB 4.0 -TlI ZrB12 5.82 UB12

MgB2 40 AlB2 LuB12 0.48 UB12

BeB2.75 0.7

ZrB2 0Ð5.5 AlB2

NbB2 0Ð0.62 AlB2

NbB2.5 6.4 AlB2 * According to the data obtained by different authors [14].

have been rapidly developed, to a large extent, owing to 2. MAGNESIUM DIBORIDE MgB2: recent progress in research into the electronic proper- ENERGY BANDS, INTERATOMIC ties of these phases. The obtained data on the electronic INTERACTIONS, AND PROPERTIES structure are widely used to interpret or to calculate different physicochemical properties of these objects. As is known, the MgB2 diboride has a hexagonal In my opinion, the amount of new data obtained 1 structure (AlB2 structural type, space group D Ð since 2001 on the electronic structure of metal borides 6h and related phases is comparable to the amount of P6/mm). The unit cell contains one formula unit (Z = 1). information amassed throughout history. Boron atoms are located at the centers of magnesium trigonal prisms, which are shared by all faces and form a In this review, an attempt is made to generalize the three-dimensional packing. The Mg and B atoms have data obtained in 2001 and 2002 on the electronic the [MgB12Mg8] and [BMg6B3] coordination polyhedra, structure of a new superconductor, namely, MgB2, with the use of theoretical quantum calculations and respectively. The coordination number is equal to 20 for spectroscopic methods. Consideration is given to the the Mg atom and 9 for the B atom. In the unit cell, the electron and hole doping effects in the MgB2 diboride, atoms occupy the following positions: 1 Mg(a) (0, 0, 0) the band structure, and the properties of Mg1 − xMxB2 and 2 B(d) (1/3, 2/3, 1/2) and (2/3, 1/3, 1/2). The unit cell and MgB2 Ð yXy solid solutions and a number of super- parameters are a = 0.30834 nm, c = 0.35213 nm, and structures. This paper also reports on the results c/a = 1.142. The interatomic distances are as follows: obtained in simulating the electronic properties of the 2 2 MgB2 surface and nanostructures, such as nanotubes B−B (a/3 ), 0.1780 nm; BÐMg (a /3 + c /4), and fullerene-like nanoclusters based on MgB2 and 0.2503 nm; MgÐMg (a in a layer), 0.3083 nm; and MgÐ related diborides. New data on the electronic states of Mg (between layers), 0.3520 nm [14]. The MgB2 struc- AlB2-like phases, boron, higher borides, a number of ternary layered boron-containing phases, and com- ture can also be represented as consisting of planar hexagonal magnesium networks and graphite-like boron pounds with an antiperovskite structure (MgCNi3 and others) are discussed in terms of their superconducting networks alternating in the order …Mg/B2/Mg/B2… characteristics. (Fig. 1).

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 BAND STRUCTURE AND PROPERTIES OF SUPERCONDUCTING MgB2 1831

III

Fig. 1. Crystal structure of hexagonal MgB2 (at the top) and structures of a nonchiral armchair (11, 11) MgB2 nanotube and an Mg30B60 cage (fullerene-like) molecule for possible atomic conﬁgurations: the metallic shell is located (I) outside and (II) inside the shell formed by boron atoms.

2.1. Band Structure of Magnesium Diboride duction band is predominantly determined by the B pπ orbitals. Prior to the discovery of superconductivity in MgB2, The ﬁrst ab initio calculations performed in the the electronic structure of this compound was investi- framework of the full-potential linearized mufﬁn-tin gated by the semiempirical method of linear combina- orbital (FLMTO) method [22] revealed that the energy tion of atomic orbitals in the basis set of sp orbitals bands of MgB2 are formed as a result of strong BÐB [19], the tight-binding band method [20], and the non- interactions in the planes of graphite-like networks. It self-consistent orthogonalized plane wave method [21]. was found that the density of states is characterized by It was noted the valence bands of MgB2 and graphite the B 2p peak at the Fermi level EF. The interatomic are similar to each other. The main contribution to the interactions in MgB2 were analyzed and compared with valence band of magnesium diboride is made by a com- those in isostructural diborides AlB2 and TiB2. Among bination of boron pσ, π states. Charge transfer occurs in the known metal diborides, titanium diboride TiB2 pos- the direction Mg B. The bottom edge of the con- sesses extreme thermal and strength characteristics [7].

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1832 IVANOVSKIŒ

4 4 The basic results obtained in these investigations can be 2 2 summarized as follows. 0 0 (1) The valence band of MgB2 (Fig. 2) is determined σ –2 –2 by the B 2p states, which form four (2px, y) and two π –4 –4 (2pz) bands. Their dispersion curves E(k) differ signif- –6 –6 icantly. (a) (d) –8 –8 (2) For the B 2px, y bands, the dispersion E(k) is max- ÉÉMK A L ÉÉMK A L Γ imum in the kx, y ( ÐK) direction. These bands reflect 4 4 the distribution of electron states of boron in graphite- 2 2 like networks, are of the quasi-two-dimensional (2D) Γ 0 0 type, and form flat regions aligned along the kz ( ÐA) –2 –2 direction. Two B 2px, y bands intersect the Fermi level, –4 –4 appreciably contribute to the total density of states at the Fermi level N(E ) (~ 30%, Fig. 3), and are respon- Energy, eV –6 –6 F (b) (e) sible for the metal-like properties of the diboride. The –8 –8 ÉÉMK A L ÉÉMK A L Fermi level EF is located in the region of bonding states. 4 4 (3) One of the most important features of the elec- 2 2 tronic spectrum of MgB2 is the energy position of the B Γ 0 0 2px, y bands. At the point of the Brillouin zone, these –2 –2 bands lie above the Fermi level and form hole-type cylindrical elements of the Fermi surface (Fig. 4). –4 –4 (4) The B 2p states (of the three-dimensional type) –6 –6 z (c) (f) are oriented perpendicularly to the boron networks and –8 –8 ÉÉMK A L ÉÉMK A L are responsible for the weak interlayer π bonds. Both π bands intersect the Fermi level and are characterized by Γ Fig. 2. Energy bands of (a) MgB2, (b) BeB2, (c) ScB2, the maximum dispersion along the kz ( ÐA) direction. (d) CaB2, (e) AlB2, and (f) YB2 according to FLMTO cal- The Mg s, Mg p, and B s states are admixed to the sys- culations [29]. tem of B 2p bands in the vicinity of the valence band edge and in the conduction band. It was shown that, for the boron networks, the overlap These results provided the basis for the development of the BÐB wave functions decreases in the following of modern models explaining the nature of superconductivity in the MgB2 diboride. order: MgB2 AlB2 TiB2. The effect of interpla- nar covalent MeÐB bonds is most pronounced in TiB2 and weakest in MgB2. The cohesion energies Ecoh 2.2. Electronic Structure and Some Properties of MgB2 (characterizing the total energy effect of interatomic There exist a number of alternative models proposed bonds in the system) and the energies of individual − for explaining the superconducting properties of MgB2. (B B, MeÐB, MeÐMe) bonds were calculated by the For example, Hirsch [49] interpreted this effect in the method proposed in [23]. It was demonstrated that the framework of the hole-type superconductivity model maximum contribution to the cohesion energy Ecoh is [50], according to which carriers are paired as a result made by the BÐB bonds (68%); the contributions of the of the Coulomb interaction. However, the methods pre- other bonds are considerably smaller (BÐMg, 23%; dicted in [49] for increasing the superconducting tran- MgÐMg, 9%). In the case of the transition MgB2 sition temperature of magnesium diboride (doping with AlB2, an increase in the cohesion energy Ecoh is associ- Al or hydrostatic compression of the lattice) turned out ated with the substantial enhancement of the MeÐMe to be in direct opposition to both experimental data and interactions and the relative constancy of the energies the theoretical conclusions based on the electronÐ of the MeÐMe and MeÐB bonds. This is the fundamen- phonon mechanism of superconductivity [13Ð16]. Zaœ- tal difference between the phases under consideration tsev [51] developed the concept of strong cationÐanion and diborides of d metals, for which the cohesion interactions. Volkova et al. [52] proposed a scheme of correlations between the critical temperature T and energy E is governed by the interlayer MeÐB interac- c coh crystal chemical parameters for the family of AlB -like tions (see below). 2 diborides. Within the proposed approach, the supercon- After the discovery of superconductivity, the spe- ductivity of MgB2 is explained by the lattice instability cific features of the band structure of MgB2 were inves- of this phase. tigated in a large number of works with the use of vir- Kortus et al. [26] were the first to attempt to estimate tually all known band calculation methods [24Ð48]. the critical temperature Tc for MgB2 semiquantitatively

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 BAND STRUCTURE AND PROPERTIES OF SUPERCONDUCTING MgB2 1833

0.3

pz (a) (b) p (p ) 0.2 x y pz – (px + py)/2

0.1

0.2

0.1

–0.1 –15 –10 –5 0 –10 –5 0 Energy, eV Energy, eV

Fig. 3. Partial densities of B pxy (dashed lines) and B pz (dotted lines) states and their differences (solid lines) for (a) MgB2, (b) BeB2, (c) TiB2, and (d) TaB2 [80]. Γ by modeling the band structure of the diboride within the electronÐphonon interaction formalism. These authors emphasized that strong interactions (λ ~ 0.9Ð1.0) of the near-Fermi B σ bands with high-frequency (~300, 700 cmÐ1) phonon modes of graphite-like boron layers play a decisive role. The effects of the electronÐ phonon interaction were discussed in more detail in Γ [25, 28, 33, 38Ð42, 45, 48]. In particular, the calculations performed by Rosner et al. [45] showed that the Γ electronÐphonon coupling constants λ for different K groups of states (σ and π) are strongly anisotropic: M λσ ~ 1.3 and λπ ~ 0.4. The ab initio calculations of the M phonon spectra of magnesium diboride [53Ð56] proved the dominant role of optical phonons corresponding to the vibrational Eg modes in the plane of boron layers Γ (Fig. 5). H The problem concerning the determination of the type and magnitude of the superconducting gap in mag- Γ L L nesium diboride was discussed in a number of works [38, 39, 42, 45Ð48]. The calculations carried out by Choi et al. [47] revealed the presence of a two-gap Γ σ π structure in the and sections of the Fermi surface of K MgB2 (Fig. 4). The gap widths (at T = 4 K) were esti- M mated at ~ 6.4Ð7.2 and 1.2Ð3.7 meV, respectively. The M total superconducting gap is of the s type. The temperature dependences of the superconducting gaps due to σ π the and states exhibit substantially different behav- Γ iors. This makes it possible to explain the observed anomalies in the low-temperature heat capacity of Fig. 4. Fermi surface of MgB2 [47].

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1834 IVANOVSKIŒ

Γ∆T KT 'MASHSΣΓ ' LRADOS films containing boron oxides are formed on the surface 120 of samples after exposure to air. MgB2 100 A number of fine features in the electronic structure B 1g of MgB2 were revealed in experimental NMR investi- 80 gations of the spinÐlattice relaxation and the Knight E2g shift [76Ð79] and in theoretical calculations of these , meV 60 parameters, hyperfine interactions, and electric-field ω A2u gradients at B and Mg nuclei [80, 81]. In particular, 40 E Medvedeva et al. [80] established that the electric-field 1u gradient at B nuclei in MgB reflects strong anisotropy 20 2 [ζζ1/2] of the spatial distribution of valence states and that the [ζζ0] [ζ00] [00ζ] [ζ01/2] 0 change in the electric-field gradient in a series of isostructural metal diborides (Table 2) depends primarily on the hybridization of the pÐd states. Fig. 5. Phonon spectrum of MgB2: dispersion curves and densities of states [55]. Considerable attention has been given to the theoretical interpretation of the effect of pressure treatment of MgB on the superconducting properties and also to MgB2 [57Ð59]. The superconducting gap widths esti- 2 mated in [47] are comparable to the experimental val- calculations of the mechanical characteristics of this ues obtained using tunneling microscopy, point-contact diboride [24, 27, 30Ð32, 37, 38, 80, 82Ð87]. It is known spectroscopy, and high-resolution photoelectron spec- (see review [14]) that the critical temperature Tc of MgB2 troscopy [60Ð66]. According to these studies, the under external compression decreases as dTc/dP ~ Ð1.6 K/GPa. The evolution of the band structure of a com- energy spectrum of MgB2 involves two superconducting gaps with widths of 5.5Ð8.0 and 1.5Ð3.5 meV. pressed magnesium diboride crystal is illustrated in Fig. 7. As can be seen from this figure, the compression In addition to theoretical investigations [24Ð48], the leads to an insignificant decrease in the concentration specific features of the electronic structure of MgB of σ holes (by ~ 0.005 at P = 10 GPa) owing to a change 2 Γ were experimentally studied in detail using photoelec- in the position of the B 2px, y bands along the ÐA direc- tron [62, 63, 67], x-ray emission, x-ray absorption [68Ð tion in the Brillouin zone and to a weak change in the 73], and electron energy loss [74, 75] spectroscopy. The density of states at the Fermi level N(EF) (a decrease by experimental data obtained are in excellent agreement ~ 0.02 states/eV). At the same time, the Hopfield parameter η increases with an increase in the pressure with the results of calculations (Fig. 6), which confirms η the general concept regarding the correlation between (d /dP ~ 0.55%/GPa [80]). The main reason for the the superconducting properties of MgB and the rela- decrease in the critical temperature under pressure is 2 the change in the phonon frequencies. tive arrangement of the σ and π bands and the occurrence of the σ hole states. The spectral lines also exhibit The results of band calculations were also used to a number of resonances, which indicates that protective interpret the de HaasÐvan Alphen effect [42, 86].

21 Table 2. Tensor components Vzz (10 V/m) of the electric-ﬁeld gradient at boron nuclei for MgB2 and a number of isostructural diborides of sÐd metals according to theoretical and experimental data [80]

el lat B B Diboride **Vzz Vzz Vzz Vzz exp

MgB2 Ð1.94 0.06 1.88 1.69 BeB2 Ð2.43 0.33 2.10 Ð AlB2 Ð1.17 0.18 0.99 1.08 ScB2 Ð0.75 0.13 0.60 Ð TiB2 Ð0.66 0.31 0.35 0.37 VB2 Ð0.76 0.38 0.38 0.43 CrB2 Ð1.01 0.42 0.59 0.63 MoB2 Ð0.55 0.32 0.23 0.23 TaB2 Ð0.21 0.25 0.04 0.02 el lat *Vzz and Vzz are the electronic and lattice contributions to the electric-ﬁeld gradient tensor, respectively. The experimental data are obtained by different authors (see [80]).

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 BAND STRUCTURE AND PROPERTIES OF SUPERCONDUCTING MgB2 1835

(a)

B 2p DOS EF Mg 3s DOS EF B Kα XES Mg L2,3 XES

–15 –10 –5 0 –15 –10 –5 0 (b)

B Kα XES Mg L2,3 XES

175 180 185 35 40 45 50 (c) B 2p DOS Mg 3s DOS B 1s XAS Mg 2p XAS Intensity/DOS, arb. units

–5 0 5 10 (d) –5 0 5

Mg 2p XAS B 1s TEY

185 190 195 40 45 50 55 Energy, eV Energy, eV

Fig. 6. (a, c) Theoretical and (b, d) experimental emission and absorption spectra of MgB2 (B K line at the left and Mg L line at the right) [68].

2.3. Electronic Properties of the MgB2 Surface to that of bulk materials. The band structure in the vicinity of the Fermi level E can be different depending on The considerable interest expressed in the specific F the type of outer monolayer. This region involves B σ features of surface electron states of superconducting and B π bands for the B-t surface and only π bands for MgB is particularly associated with the development 2 the Mg-t surface. The relaxation effects are rather weak: of methods for the reproducible synthesis of thin super- they amount to Ð2.1 and Ð3.7% between two outer conducting films for use in different electronic devices monolayers for the B-t and Mg-t surfaces, respectively, (see [13–16]). Magnesium boride films have been pre- and virtually disappear at a depth of five or six monolay- pared using two main techniques [87Ð95], which ers [110]. involve ex situ processes (synthesis of boron films followed by their saturation with magnesium) or in situ methods (direct preparation of MgB2 films). The mor- 0.02 phology of MgB2 films (including epitaxially oriented px + py films [96–100]) and a number their properties, such as pz the conductivity, the Hall effect, the surface resistivity, 0.01 and optical characteristics [87Ð108], have been studied in sufficient detail to date. Strong anisotropy of the conductivity was found and the upper critical fields were 0 determined for oriented films. The methods of the band theory were applied to –0.01 describe the electronic properties of both magnesium- terminated (Mg-t) and boron-terminated (B-t) ideal hex- PDOS change, states/eV agonal MgB (0001) surfaces [109Ð112]. The calcula- –0.02 2 –0.8 –0.4 0 0.4 0.8 tions revealed a number of surface resonances and a pro- Energy, eV nounced increase in the density of near-Fermi states for outer layers (especially for the B-t surface; Fig. 8). Fig. 7. Changes in the partial densities of B 2pxy (solid line) According to [109], this behavior is favorable for an and B 2pz (dashed line) states of MgB2 under a pressure of increase in the critical temperature of films as compared 10 GPa [80].

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1836 IVANOVSKIŒ

1.5 related superconductors. Within the former direction, 1 EF researchers have sought to extend the class of possible 1.0 superconductors through the doping or preparation of 0.5 different solid solutions and superstructures based on the MgB2 phase. The latter direction involves searching 0 for the superconductivity effect among the large variety of binary or multicomponent phases whose structural 1.0 2 or chemical units are similar to those of MgB2. The syn- 0.5 thesis procedures, crystallographic data for the synthesized materials, and a number of materials science

Density of states, arb. units 0 aspects regarding these objects are described in reviews [13Ð16]. Let us now consider the results obtained in –12 –10 –8 –6 –4 –2 0 2 4 6 Energy, eV modeling the band structure of the above systems. Before proceeding further, we recall that, in the Fig. 8. Densities of states of (1) boron and (2) magnesium early stages of investigations, the rigid-band model was outer monolayers for the MgB2(0001) surface (solid lines). Differential densities of states corresponding to outer and frequently used in the design of many experiments and inner monolayers (dashed lines) are given. LDA calcula- in interpreting the data obtained for impurity systems tions for a model of a 13-atom supercell [109]. and solid solutions. By postulating the identity of the densities of states for doped systems and the initial matrix (MgB ), investigators only took into account According to the calculations performed by Serve- 2 dio et al. [112] in the framework of the KorringaÐ that different dopants can serve as heteroelectronic or isoelectronic impurities and, hence, either lead or do KohnÐRostoker method, the MgB2 surface is characterized by a number of surface states corresponding to fine not lead to a change in the electron concentration in the features of the angle-resolved photoelectron spectra of system. Within this model, hole dopants (for example, magnesium diboride single crystals [113]. The esti- Be impurities in the boron sublattice or Li, Na, and Cu mates made by Li et al. [111] for the energies of forma- impurities in the Mg sublattice) should decrease the tion of the surfaces terminated by different atoms electron concentration in the system and lead to an showed that the system involving an outer monolayer increase in the density of hole states in the σ band and composed of more reactive Mg atoms has a higher sta- to a shift of the Fermi level EF toward the high-energy bility. The work functions of the B-t and Mg-t surfaces range with an increase in the density of states at the were calculated to be 5.95 and 4.25 eV, respectively. Fermi level N(EF). Both factors are favorable for super- It should be emphasized that, in [109Ð112], the conductivity (see above). Contrastingly, electron diboride surface was modeled by the (0001) face of a dopants (Al and d metal impurities in the Mg sublattice perfect crystal. In actual fact, the situation is more com- or C, N, and O impurities in the boron sublattice) affect plex. Vasques et al. [114] carried out special x-ray pho- the occupation of the σ band, result in a decrease in the toelectron spectroscopic investigations of the elemental density of states at the Fermi level N(EF), and should composition of the MgB2 surface and found that at least have a negative effect on the superconducting proper- three regions with different compositions are formed in ties. The role of isoelectronic substitutions (Be, Ca the vicinity of the surface. The first region directly on Mg) remains unclear in terms of the rigid-band model. the surface involves an oxide film and is appreciably enriched in magnesium (Mg/B ~ 0.80). By contrast, the It should be noted that, for the majority of systems, near-surface region is depleted in magnesium (Mg/B ~ the properties predicted on the basis of the aforemen- 0.34); in this case, the nonstoichiometry with respect to tioned rigid-band model appear to be in error. For metal favors a decrease in the electron density in layers example, the experimental data on the critical tempera- of boron atoms. Only the third region is characterized tures Tc of Mg0.97Me0.03B2 alloys [115] indicate that the by the stoichiometric ratio (Mg/B = 0.50) correspond- critical temperature of the alloy with Me = Mn ing to the bulk magnesium diboride. decreases to 33.1 K, whereas the critical temperatures of the alloys with Me = Fe, Co, Ni, and Zn are equal to 37.8, 35.7, 37.8, and 38.4 K, respectively. However, 3. DOPING EFFECTS, SOLID SOLUTIONS according to the rigid-band model, the critical temper- AND SUPERSTRUCTURES BASED ON MgB2, ature of Mg1 − xMexB2 alloys should monotonically AND RELATED TERNARY PHASES decrease with an increase in the atomic number Z of the Immediately after the discovery of superconductiv- Me impurity. A number of other examples are given in ity in MgB2, theoretical and experimental investiga- reviews [13Ð16]. Numerical calculations made it possi- tions were undertaken with the aim of modifying the ble to propose a more correct interpretation for the properties of magnesium diboride and searching for effects caused by the doping of MgB2.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 BAND STRUCTURE AND PROPERTIES OF SUPERCONDUCTING MgB2 1837

1.5 1 Sc Ti V Cr Mn 40 NaB2 20 1.0 Mg0.5Li0.5B2 Mg0.5Na0.5B2 0 90 Fe Co Ni Cu Zn Mg0.75Li0.25B2 Mg0.75Na0.25B2 Mg0.75B2 60 0.5 Mg0.75Cu0.25B2 ), eV F CaB 30 E 2 Mg0.75Be0.25B2 – Mg0.75Ca0.25B2 Γ Mg0.75Zn0.25B2 ( MgB 0 σ 2 2 E 40 0 VCrMnFeCo BeB2 MgB1.75C0.25 20 ~ YB2 ~ ScB2 –1.0 MgB1.75N0.25 Density of states, arb. units MgB1.75O0.25 0 –1.5 –2.0 AlB2 –1.0 –0.5 0 0.5 1.0 20 n, e/cell

40 –1.0 0 –0.5 –1.0 0 –0.5 –1.0 0 –0.5 –1.0 0 –0.5 –1.0 0 –0.5 σ Fig. 9. Dependence of the energy of the B (pxy) band at the E, Ry Γ point of the Brillouin zone (with respect to the energy of the corresponding band of MgB2) on the electron concen- Fig. 10. Densities of states of 3d metal impurities in MgB2. tration n (e/cell) for a number of binary and ternary AlB2- The results of the spin-polarized calculations [117] are like diborides according to the FLMTO calculations [116]. given at the bottom.

3.1. Doping of the Metal Sublattice in MgB2 The tendency toward the formation of electronic For the most part, attempts to control the electron and magnetic states of 3d metal impurities (Me = Sc, Ti, V, …, Cu, Zn) in cation positions of MgB as a func- and hole concentrations in MgB2 were reduced to the 2 introduction of different dopants into the cation sublat- tion of the atomic number Z of metals was analyzed by tice of magnesium diboride. Experimental data demon- Singh and Joseph [117], who investigated the strate (see [13Ð16]) that the replacement of Mg by other Mg0.97Me0.03B2 systems in the framework of the Kor- elements (Li, Na, Be, Al, d metals) leads to a decrease ringaÐKohnÐRostoker coherent-potential approximation in the critical temperature Tc of the system (except for (KKRÐCPA). The specific features in the localization systems with Zn and, possibly, Li), a change in the lat- of impurity states in the band spectrum of the matrix tice constants, and, in a number of cases, phase transi- are illustrated in Fig. 10. These features are character- tions attended by a change in the type of crystal struc- istic of 3d impurities in many metals, alloys, and ture. chemical compounds (see [118Ð125]). In particular, it The first systematic investigations into the regulari- can be seen from Fig. 10 that an increase in the atomic ties of chemical modification of the electronic structure number Z of the impurity leads to occupation of the of MgB2 upon doping of the cation sublattice with Be, Me 3d band, which gradually shifts toward the low- Ca, Li, Na, Zn, and Cu impurities were carried out by energy range. This band for Me = Cr and Mn is located the first-principles FLMTO method in [116]. It was in the near-Fermi region of the matrix and is responsi- found that, for Mg − Me B systems (where Me = Be, 1 x x 2 ble for the high density of states at the Fermi level Ba, Li, Na, Cu, Zn), the main features in the band spec- N(EF) and the spin splittings with the formation of tra are identical to those in the band spectrum of MgB2 and the energy positions and the occupation of the near- local magnetic moments at Me atoms, which are max- µ µ Fermi σ bands (Fig. 9) and the density of states at the imum for Cr (2.43 B) and Mn (2.87 B). For Cu and Fermi level N(EF) are determined by the type and con- Zn, the d states are completely occupied and form nar- centration of impurities. row quasi-atomic bands below the Fermi level EF.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1838 IVANOVSKIŒ

40 period), and reaches a maximum for Mg0.97Zn0.03B2. This behavior correlates with the dependence of the 35 density of states at the Fermi level N(EF) on the atomic number of the metal impurity and is in reasonable 30 agreement with the experimental data [115] (Fig. 11). The observed dependence of the superconducting prop-

, K 25 c erties on the atomic number Z of the impurity in the T Mg0.97Me0.03B2 alloys is explained in [117] by the non- 20 monotonic ﬁlling of the upper bands of the alloys and the magnetic effects (for the impurities in the middle of 15 the 3d period) preventing pairing.

10 In addition to the MgB2 : Me impurity systems, the Total electronic structure of Mg1 − xMexB2 alloys was investi- 12.0 gated over a wide range of Mg/Me ratios for d metals. 10.5 Joseph and Singh [126] studied the band structure of ≤ ≤ 9.0 Mg1 − xTaxB2 alloys (0 x 1) by the KKRÐCPA method

) B p λ F 2.7 and also estimated the values of and Tc for the compo- E ( sitions given in Table 3. As the Ta content increases to

N 2.6 40 at. %, the critical temperature Tc decreases to zero 2.5 and the Mg − Ta B alloys at 40 at. % ≤ x ≤ 80 at. % are 60 1 x x 2 Me d not superconductors. At x > 80 at. % Ta, the critical 30 temperature Tc increases and reaches ~ 1.8 K for TaB2. 0 This behavior was explained by the nonmonotonic MgB2 Sc Ti V Cr Mn Fe Co Ni Cu Zn change in the contributions of the B 2p states to the density of states at the Fermi level N(EF) as a function of Fig. 11. Dependences of the critical temperature Tc (at the the ratio Mg/Ta (Table 3), i.e., by the same factors that top) and the total and partial densities of states at the Fermi are responsible for the conduction mechanism charac- level N(E ) (at the bottom) on the Me impurity type for F teristic of MgB2. However, the substitution Mg Ta Mg0.97Me0.03B2 alloys. Circles are the results of calculations, and squares are experimental data [117]. leads to a radical change in the structure and composition of the near-Fermi bands and in the topology of the Fermi surface (see below). As a result, the d states of According to estimates, the dependence of the criti- the transition metal are dominant for the Mg1 − xTaxB2 cal temperature Tc on the atomic number Z of the metal alloys enriched in tantalum and, hence, the explanation offered in [126] for the change in the critical tempera- in Mg0.97Me0.03B2 alloys exhibits a nonmonotonic ture T of the Mg − Ta B alloys turns out to be contro- behavior. The critical temperature T decreases (with c 1 x x 2 c versial. respect to Tc for MgB2) for Me atoms in the middle of the 3d period, reaches a minimum for Mg Cr B , The concentration dependences of the energy of the 0.97 0.03 2 σ then noticeably increases (for elements at the end of the bands for Mg1 − xAlxB2 and Mg1 − xNaxB2 solid solutions (x = 0, 1/3, 2/3, 1.0) were investigated by Suzuki et al. [127] in terms of the supercell model. It was found Table 3. Lattice parameters (au), total densities of states at that, for all compositions of the Mg1 − xNaxB2 solid solu- Γ the Fermi level N(EF), and contributions of B 2p states to the tions, the top of the B pxy band at the point lies above total density of states (1/Ry spin) for Mg1 − xTaxB2 alloys the Fermi level EF; i.e., the electronic criterion for according to KKRÐCPA calculations [117] superconductivity holds. For the Mg1 − xAlxB2 solid Alloy ac/aN(E )B 2p solutions, this condition is satisﬁed at x < 0.06 and, at F larger x, the hole states at the center of the Brillouin zone disappear. The dependences of the concentration MgB2 5.834 1.141 3.331 2.834 σ nh of holes on the solid solution composition are rep- × 23 Ð3 Mg0.8Ta0.2B2 8.833 1.124 3.116 1.785 resented in the form nh = (0.8 + 0.8x) 10 cm for the × Mg Ta B 5.831 1.106 0.466 0.616 Mg1 − xAlxB2 solid solutions and nh = (0.8 Ð 1.4x) 0.6 0.4 2 23 Ð3 10 cm for the Mg1 − xNaxB2 solid solutions. The Mg0.4Ta0.6B2 5.830 1.088 1.734 0.663 concentration of holes is maximum (~1.6 × 1022 cmÐ3) for NaB . In the series NaB Mg − Na B Mg0.8Ta0.2B2 5.829 1.070 2.979 0.833 2 2 1 x x 2 MgB2 Mg1 − xAlxB2 AlB2, the cohesion energy TaB2 5.828 1.052 4.980 1.455 changes almost monotonically.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 BAND STRUCTURE AND PROPERTIES OF SUPERCONDUCTING MgB2 1839

Mehl et al. [128] performed calculations for the 1 2 Al 1.58 Å MgB2 : (Cu, C) system and made the inference that the Mg double doping (Mg Cu, B C) can lead to an B 1.79 Å increase in the critical temperature T to ~ 50 K. Al c B Mg 1.79 Å B B 1.58 Å Al 3.2. Doping of the Boron Sublattice in MgB2 3 Al 1.58 Å 4 The MgB2ÐC system has been studied most exten- B 1.83 Å sively. It was established that, upon substitution of car- 2/3Al Mg 1.72 Å 1/3Mg 1.62 Å bon for boron, Mg(B − C ) single-phase compounds 1 x x 2 B 1.78 Å are formed in a limited range of carbon concentrations B 1.72 Å Mg 1.78 Å (at x < 0.2) and an increase in the ratio C/B results in a Mg 1.83 Å rapid decrease in the critical temperature T and an B 1.62 Å c B 1.58 Å increase in the superconducting transition range [129Ð 2/3Al Al 1/3Mg 135]. The band calculations for the MgB2 diboride doped with C, N, and O atoms (in the boron sublattice) σ Fig. 12. (1Ð4) Variants of superstructures in the [29, 116] demonstrated that the band is completely Mg − Al B system [141]. occupied (Fig. 9), whereas the density of states at the 1 x x 2 Fermi level N(EF) in the series MgB1.75C0.25 MgB1.75N0.25 MgB1.75O0.25 changes nonmonotoni- each composition of the Mg1 − xAlxB2 system was based ∆ cally and increases at the end of the series. This effect on the parameter E(x) = E1(x) Ð E2(x), where E1(x) and is associated with the occupation of antibonding states. E2(x) are the energy of states of the superstructure and Note also the results obtained in band calculations for the corresponding disordered solid solution, respec- ternary phases, namely, layered borocarbide MgB2C2 tively. It was revealed that the atomic segregation into [28] and layered boronitride Mg3BN3 [136] (the struc- layers is energetically favorable not only for the stoichi- ture of the latter phase involves layers formed by dif- ometric composition x = 0.5 [138Ð140] but also for a ferent metalloids B and N). It was shown that the number of other compositions. The transformation of energy spectrum of MgB2C2 exhibits a metal-like the system into a heterophase state was explained in nature and MgB2BN3 is a semiconductor with a band [141] within a model allowing for the energy estimates gap of ~0.88 eV. and the contribution of the conﬁgurational entropy. As follows from this model, the critical points of the decomposition of the solid solutions correspond to the 3.3. Superstructures compositions with x = 0.25 and 0.75. It should be noted The effects of long-range atomic order for impurity that the inclusion of the superstructural ordering of the Mg1 − xAlxB2 solid solutions does not affect the interpre- elements in MgB2 (formation of superstructures) are of tation of the decrease in the critical temperature Tc with interest from the viewpoint of revealing factors that can σ stabilize metastable ternary alloys, which (for example, an increase in x due to the occupation of the hole bands and the decrease in the density of states at the in the MgB2ÐCaB2 system [116]), as expected, should possess improved superconducting properties. More- Fermi level N(EF). over, investigations of these effects can elucidate the mechanism of decomposition of MgxMeyB2 solid solutions at different concentrations. 3.4. Ternary AlB2-Like Phases At present, the above effects are best understood for Among ternary layered phases, lithium borocarbide the MgÐAlÐB system. It was found that the Mg1 − xAlxB2 LiBC has attracted the most attention as a possible ana- solid solutions are unstable at 0.09 < x < 0.25 and x ~ 0.7, log of the superconducting magnesium diboride. The at which the system is in a heterogeneous state [137Ð structure of the LiBC lithium borocarbide can be repre- 139]. According to [138Ð140], superstructures with sented as a result of the substitution of the carbon atom ordered motifs in the arrangement of Al atoms along the for one boron atom and Mg Li in the MgB2 unit c axis of the crystal are formed at x ~ 0.5. cell [142]. The packing of borocarbide networks is of A microscopic model of the superstructure forma- the … A/B/A/B … type. The nearest environment of tion was proposed by Barabash and Stroud [141] on the boron atoms consists of carbon atoms and vice versa. basis of energy calculations (the Vienna Ab Initio Sim- Unlike MgB2, stoichiometric LiBC is an insulator. As ulation Package with inclusion of the lattice relax- follows from the semiempirical calculation [143] for ation). These authors calculated a number of supercells hypothetical borocarbide composed of charged layers used as models of superstructures with allowance made (BC)Ð1 with empty cation positions, the band gap π for the possible segregation of Al and Mg atoms into between the bounding and antibonding (pz) bands is their “own” atomic layers (Fig. 12). The analysis of approximately equal to 4.3 eV.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1840 IVANOVSKIŒ

2.0 However, up to now, attempts to reveal the predicted 1 superconducting properties of LixBC have not been 2 successful [148, 149]. In particular, this failure was 3 explained by irregular packing of BC layers [149]. The 1.5 study of an LiBC single crystal by polarized Raman spectroscopy proved that the crystal structure has the symmetry group P3m1 and involves puckered BC 1.0 planes [150]. Mehl et al. [128] studied a hypothetical ternary crystal, copper borocarbide CuBC, which is isoelectronic and isostructural to MgB . According to the esti- 0.5 2 mates, the maximum critical temperature Tc of this Density of states, arb. units crystal can be as high as 50 K. Medvedeva et al. [151] searched for possible super- 0 –15 –10 –5 0 5 conducting analogs of the MgB2 diboride among ter- E, eV nary borides of the YCrB4 and Y2ReB6 structural types, which are characterized by two-dimensional motifs in Fig. 13. Densities of states of LixBc at x = (1) 1.0, (2) 0.75, the arrangement of boron atoms. Note that, in AlB2-like and (3) 0.5. Vertical lines indicate the Fermi level [144]. diborides, boron networks are formed by B6 regular hexagons. Similar boron networks involve B5 penta- According to the full-potential nonorthogonal- gons and B7 heptagons in the YCrB4 structure (space orbital linearized muffin-tin orbital (LMTO) calcula- group Pbam) and all three types of polygons (B5, B6, tions [144], LiBC is an indirect-band-gap (ΓÐH transi- B7) in the Y2ReB6 structure (space group Pbam). The tion) semiconductor with a band gap of ~ 1.0 eV. The calculations demonstrated that the YCrB4 boride is a energy spectrum of LiBC changes depending on the narrow-band-gap semiconductor (with a band gap of packing of BC layers: for a hypothetical crystal with the 0.05 eV) and its transformation into a metal-like state interlayer packing … A/A/A … (the total energy of this (due to hole or electron doping) should be associated crystal differs from that of the real phase by ~0.35 meV), with the change in the type of occupation of the Cr 3d Γ bands [151]. The Fermi level in the energy spectrum of the pz band (along the ÐK direction) appears to be partially occupied and the phase becomes a semimetal. the metal-like Y2ReB6 boride lies in the region of the lower edge of the pseudogap between the bonding and The energy spectrum of a virtual crystal LiB2 [where B is an intermediate atom (C + B)/2 with Z = 5.5] is sim- antibonding B 2p states. However, their density in the ilar to that of MgB . vicinity of the Fermi level EF is very low and the Re 5d 2 states make a maximum contribution to the density of The cation nonstoichiometry (hole doping; compo- states at the Fermi level N(EF). Consequently, by anal- sitions LixBC at x = 0.75, 0.50) leads to partial deple- ogy with Re3B, in which the critical transition occurs at tion of the σ bands (Fig. 13). The frequencies of the Tc = 4.7 K [152], only low-temperature superconductiv- mode E2g for LixBC and MgB2 are sufficiently close to ity can be expected for the Y2ReB6 boride. each other [144]. A very high critical temperature Tc ~ 115 K for the Li0.5BC borocarbide was determined λ 4. BAND STRUCTURE AND PROPERTIES OF s, from the electronÐphonon coupling constant ~ 1.75 p, AND d METAL DIBORIDES AND RELATED (λ ~ 0.82 for MgB ). Interpolation to the composition 2 AlB2-LIKE PHASES. NONSTOICHIOMETRY range 0 < x < 0.5 demonstrated that the LixBC borocar- EFFECTS bide can have a critical temperature T ≥ 40 K at x ≥ c 4.1. AlB -Like Metal Diborides 0.25. In order to obtain the required concentration of σ 2 holes, it was proposed to use field doping similar to that Before the discovery of superconductivity in the applied in [145] to achieve the superconducting state in MgB2 diboride, the available data on the superconduct- a hole-doped C60/Br fulleride. The calculations carried ing properties of other metal diborides were very scarce out by Dewhurst et al. [146] showed that the electronÐ [7, 14, 153Ð157]. According to the first systematic λ investigation of MeB compounds (Me = Ti, Zr, Hf, V, phonon coupling constant for LixBC monotonically 2 increases with a decrease in x. In this case, the critical Nb, Ta, Cr) [157], they had a critical temperature Tc = 0.7 K. temperature Tc ~ 65 K was predicted for Li0.5BC. An et al. [147] performed ab initio calculations of the dis- An analysis of recent results revealed a rather persion of the phonon modes and the Eliashberg func- ambiguous pattern. There are data on sufficiently high tion and noted that a small part (<2%) of the phonon critical temperatures Tc for ZrB2 (5.5 K) [158], TaB2 modes is involved in extremely strong interaction. (9.5 K) [159], and NbB2 (5.2 K) [160]. It is worth not-

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 BAND STRUCTURE AND PROPERTIES OF SUPERCONDUCTING MgB2 1841

1.5 CrB2 MoB2 WB2 1.0 0.5 0 1.5 VB2 NbB2 TaB2 1.0 0.5 0 1.5 TiB2 ZrB2 HfB2 1.0 0.5 0 –16–12 –8 –4 0 ScB2 YB2 Density of states, arb. units 1.5 E, eV 1.0 0.5 0 1.5 CaB2 SrB2 1.0 0.5 0 –16–12 –8 –4 0 –16–12 –8 –4 0 E, eV

Fig. 14. Densities of states of metal diborides with an AlB2-like structure [170]. ing that, although different research groups have inves- ever, to the best of my knowledge, there has been no tigated identical series of diborides (for example, TiB2, work done to reproduce the results obtained in [165]. ZrB2, HfB2, VB2, NbB2, and TaB2 in [158] and ZrB2, A number of theoretical and experimental system- NbB2, and TaB2 in [159], each group discovered its own atic investigations into the band structure of stable and superconductor (ZrB2 [158] or TaB2 [159]), whereas all metastable Li, Na, Be, Ca, Se, Al, Sc, Y, Ti, Zr, Hf, V, the other MeB2 phases were considered to be nonsuper- Nb, Ta, Cr, Mo, W, Tc, Ru, Rh, Pd, Ag, and Au conducting. Subsequent investigations of the TaB2 diborides have been performed in recent years [28, 29, diboride showed that the superconducting transition in 34, 36, 116, 127, 166Ð170]. The total densities of states this compound is not observed down to T ~ 1.5 K. The calculated by the self-consistent full-potential linear- estimated parameters of the electronÐphonon interac- ized augmented-plane-wave (FLAPW) method [170] tion allowed the conclusion that the ZrB2 diboride for the majority of stable MeB2 phases are presented in should be in a nonsuperconducting state [160], whereas Fig. 14. The electronic structure of a number of compounds (for the most part, diborides of Group IV and V the superconducting transitions can be observed at Tc ~ 0.1 K in the TaB diboride [161, 162] and T ~ 3 K in d metals) was repeatedly studied earlier (these results 2 c are discussed in monograph [155]). the NbB2 diboride [163]. In the majority of the above works, the most atten- The critical transitions were revealed in Re3B (4.7 K) tion was focused on the ﬁne features in the near-Fermi [152] and a new phase of beryllium boride (0.72 K, states and their inﬂuence on the superconducting prop- BeB2.75, the unit cell contains 110.5 atoms) [164]. The erties. The absence of superconductivity in the BeB2 occurrence of inclusions of a superconducting titanium boride was explained by the low-energy shift of the σ boride phase of unknown composition with a maximum bands and the absence of hole states at the Γ point critical temperature Tc ~ 110Ð115 K in TiBk diffuse lay- (Fig. 2). This leads to a change in the topology of the ers on metallic titanium was reported in [165]. How- Fermi surface: cylinders (along the ΓÐA) direction)

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1842 IVANOVSKIŒ

∆ Table 4. Elastic moduli B0 (Mbar), their ﬁrst derivatives B0' , heats of formation H (kJ/mol), cohesion energies Ecoh (Ry/atom), width ratios of the occupied (V0) and valence (Vv) bands, melting temperatures Tm (K), densities of states at the Fermi level γ 2 N(EF) (1/Ry), and low-temperature susceptibility coefﬁcients (mg/mol K ) for 3dÐ5d metal diborides according to LMTO calculations [169] and experimental data

* ∆ ∆ γ Phase B0 B0 B0' Ð H Ð H* Ecoh V0/Vv Tm N(EF)

ScB2 1.19 Ð 1.85 231.52 307 1.17 0.87 2523 11.99 2.08 TiB2 2.13 2.02 2.1 308.38 323 1.32 1.01 3253Ð3498 4.27 0.74 VB2 1.75 Ð 1.67 208.14 206 1.01 1.10 2673Ð3020 15.86 2.48 CrB2 1.56 Ð 1.68 181.93 133 0.98 1.11 2473 34.88 6.05 MnB2 2.1 Ð 1.65 140.04 120 0.88 1.09 2261 108.94 18.89 FeB2 2.3 Ð 1.69 113.71 94.14 0.79 1.10 Ð 37.28 6.46 YB2 1.41 Ð 2.05 101.68 105 0.88 0.86 2373 11.68 2.03 ZrB2 1.95 2.15 1.94 296.81 322.59 1.25 1.05 3313Ð3518 3.84 0.67 NbB2 1.01 Ð 1.67 192.73 197 0.94 1.17 3173Ð3309 13.95 2.42 MoB2 1.6 Ð 1.71 131.61 Ð 0.83 1.23 2373Ð2648 20.01 3.47 HfB2 2.16 2.1 1.35 244.20 335.98 1.21 1.05 3373Ð3653 3.688 0.64 TaB2 1.81 Ð 1.78 179.15 209.20 0.98 1.12 3310Ð3473 12.92 2.24 * Experimental data (see references in [169]). transform into cones. The occupation of the σ bands in chanical parameters (such as the elastic moduli, their the AlB2 boride (due to an increase in the concentration derivatives, etc.) were calculated for a number of of valence electrons) is responsible for its nonsupercon- diborides (Table 4). Important information on the cohe- ducting state. For the ScB2 and YB2 borides, the pxy sion characteristics and the energies of formation was bands at the A point lie above the Fermi level EF; how- obtained for MeB2 diborides. These data made it possi- ever, the concentration of hole carriers is insignificant. ble to reveal the fundamental regularities of the interre- In the case of the TiB2, ZrB2, and HfB2 diborides, the lation between the electronic properties and the conditions for the stabilization of the structural type (AlB ) Fermi level EF is located in the pseudogap (Fig. 14). An 2 increase in the electron concentration (diborides of for metal diborides. Specifically, in the framework of Group V and VI d metals) results in occupation of the d the first-principles band calculations [170], the theoret- band, and an increase in the density of states at the ical heats of formation of the MeB2 phases were deter- Fermi level N(EF) can lead to an increase in the critical mined from the formula temperature Tc. It is important that, unlike MgB2, for ∆EEMe= ()+ 2E()B Ð EMe()B , the aforementioned diborides, the d states of the metal 2 make a substantial contribution to the density of states where E(Me), E(B), and E(MeB2) are the total energies α at the Fermi level N(EF) [166Ð170]. (per unit cell) of the metal, elemental boron ( -B12), An analysis of the band structure of the metastable and diboride, respectively. The cohesion energy was Li, Na, Ca, Al, Ag, and Au diborides demonstrates that calculated from the relationship CaB corresponds to the optimum configuration and the () () () 2 Ec = Ec Me ++2Ec B Ec MeB2 , type of occupation of the σ and π bands, which are where characteristic of the MgB2 diboride. The estimated enthalpy of formation of CaB (Ð0.12 eV/formula unit) (), (), (), 2 Ec Me B = EMeB Ð Eat Me B , [116] indicates that synthesis of this compound is quite possible. The phase can be stabilized by doping (with Ec()MeB2 isoelectronic or hole dopants, such as Mg, Li, and Na). = E ()MeB Ð2E ()Me Ð E ()B Ð E ().MeB Apart from the parameters of the electronic struc- c 2 at at c 2 ture, the results of the calculations carried out in [166Ð Here, Eat(Me) and Eat(B) are the total energies of metal 170] made it possible to analyze a large number of and boron free atoms, respectively. Some of the results physicochemical properties for diborides. In particular, obtained are compared with the experimental data in consideration was given to the conditions required for Fig. 15. It can be seen from this figure that the stability the appearance of magnetism. Moreover, the low-tem- of the diboride phases directly depends on the metal perature electronic heat capacity coefficients γ, the lat- valence (occupancy of the valence bands) and is maxi- tice constants, and the thermodynamic and thermome- mum for the Ti, Zr, and Hf diborides. For these com-

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 BAND STRUCTURE AND PROPERTIES OF SUPERCONDUCTING MgB2 1843 pounds, the optimum condition of the band occupation 4 is satisﬁed: all the bonding states are occupied and all Be the antibonding states are vacant. An increase or Na–Al decrease in the electron concentration for diborides of 3 Mg(exp) the Group VÐVII or III metals leads to occupation of 2 Ca–Cr antibonding bands or depletion of the bonding bands, 2 Sc–Cr(exp) which results in a decrease in the stability of these Sr–Ag phases. Y–Nb(exp) 1 Hf–W, Au Hf–Ta(exp) 4.2. Nonstoichiometry Effects in Diborides: 0 Cation Vacancies Unlike many interstitial phases (carbides, nitrides, –1

oxides of metals [7, 19]), the majority of diborides do Heat of formation, eV/AB not exhibit nonstoichiometry effects: under equilibrium conditions, the deviation of the phase composition from –2 the “ideal” ratio B/Me = 2 does not exceed tenths of an atomic percent [153Ð155]. As a consequence, the pres- –3 ence of vacancies in the metal or boron sublattices, as a 246810 rule, is ignored in the discussion of experimental data Valence of A ion and in theoretical models. Fig. 15. Theoretical and experimental heats of formation of In this respect, of particular interest are the results of AlB2-like metal diborides as a function of the valence of recent experiments on the high-pressure synthesis of metal A [170]. Ta1 − xB2 and Nb1 − xB2 nonstoichiometric (in the metal sublattice) diborides with extended regions of homoge- stoichiometric phases). The presence of Nb vacancies neity 0 < x < 0.48 [171]. It was found that the presence in NbB gives rise to a new peak in the density of states of Me vacancies can lead to a considerable change in 2 the superconducting properties of phases. For example, and brings about a change in the contributions of the Nb a change in the atomic ratio B/Nb results in an increase 4d and B 2p states in the vicinity of the Fermi level EF. The states of vacancies are located below the Fermi in the critical temperature Tc, which reaches a maxi- level EF and do not affect the density of states at the mum (~9 K) for the composition Nb~1.76B2. Cooper et al. [172] also observed superconductivity in a number Fermi level N(EF). The nonstoichiometry effect is most pronounced for the ZrB2 diboride. In this case, lattice of metal-rich borides of nominal compositions NbB2.5 (64 K) and MoB (8.1 K). defects lead to the appearance of an intense peak in the 2.5 density of states in the vicinity of the pseudogap. As a The first investigation into the influence of Me result, the density of states at the Fermi level N(E ) for ᮀ F vacancies ( ) on the band spectrum of the Me0.75B2 Zr B increases drastically and the contribution of the diborides (Me = Nb, Zr, Y) was performed by the 0.75 2 ᮀ B 2p states increases by approximately one order of FLMTO method with 12-atom supercells (Me3 B8) magnitude (Table 5). Therefore, the presence of Zr [173]. It was established that the effect of Me vacancies vacancies can be one of the factors responsible for the on the band structure of diborides can differ fundamen- critical transition experimentally observed in ZrB2 tally depending on the type of metal sublattice (or the [157]. type of occupation of the energy bands). For Nb0.75B2 The opposite effect of Me vacancies on the band and Ta0.75B2, the density of states at the Fermi level structure is observed for the YB2 diboride. The presence N(EF) insignificantly decreases (as compared to that for

∆ Table 5. Densities of states at the Fermi level N(EF) (states/eV formula unit) and energies of formation ( H), cohesion (Ecoh), and formation of Me vacancies (Evf) for stoichiometric and nonstoichiometric diborides of Nb, Zr, and Y (in Ry/formula unit) [173] N(E ) F ∆ Diboride Ecoh HEvf total Ms Mp Md Mf Bs Bp

NbB2 1.012 0.002 0.012 0.653 0.036 0.009 0.125 1.82 0.27 Nb0.75B2 0.993 0.009 0.021 0.544 0.022 0.010 0.146 1.57 0.19 0.08 ZrB2 0.300 0.001 0.003 0.170 0.009 0.000 0.042 1.72 0.35 Zr0.75B2 1.220 0.005 0.027 0.426 0.019 0.018 0.331 1.46 0.22 0.13 YB2 0.900 0.008 0.018 0.364 0.014 0.002 0.136 1.42 0.26 Y0.75B2 0.409 0.004 0.012 0.121 0.004 0.002 0.080 1.23 0.16 0.10

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1844 IVANOVSKIŒ of vacancies results in depletion of the bonding states. Satta et al. [34] examined calcium disilicide. Under For Y0.75B2, the Fermi level is located at a local mini- equilibrium conditions, CaSi2 is a nonsuperconducting mum in the density of states and the value of N(EF) and semimetal with a rhombohedral structure. The com- the contributions of the Y 4d and B 2p states decrease pound undergoes a polymorphic transition to a phase of sharply. Consequently, no superconductivity can be the AlB2 structural type upon heat treatment under pres- expected in nonstoichiometric yttrium diborides. sure. It was found that the band structure of this phase Shein et al. [173] estimated the changes in the sta- involves the Ca d states in the vicinity of the conduction π bility of MeB2 phases due to the presence of defects. band edge. The * antibonding bands are partially For this purpose, the authors calculated the energies of occupied, which explains the structural instability of ∆ cohesion and formation ( H) for MeB2 and Me0.75B2 the layered phase under normal conditions. According phases. The energy of formation ∆H was calculated to [34], the replacement of Si atoms (rSi = 1.34 Å) by from the expressions atoms of a smaller radius (Be; rBe = 1.13 Å; CaSiBe hypothetical compound) should induce a “chemical ∆ MeB2 ()Me B MeB2 H = Etot + 2Etot Ð Etot , pressure” favorable for forming a structure of the AlB2 type in the system. A further factor stabilizing this ∆ Me0.75B2 ()Me B Me0.75B2 π H = 0.75Etot + 2Etot Ð Etot , phase is the partial depletion of the * states upon a decrease in the electron concentration, which should be Me B reduced by ~ 0.7 e in the system. where Etot and Etot are the total energies of the pure metal and α-boron crystals, respectively. As can be The band calculations for CaGa2 demonstrated that

MeB2 the σ and π bands intersect at the Γ point and lie below seen from Table 5, the energies of formation ∆H σ ∆ ∆ the Fermi level EF [174]. The hole states at a low den- decrease in the order H(ZrB2) > H(NbB2) > sity are observed in the vicinity of the A point. As a result, ∆H(YB ), which is in agreement with the results of 2 the Fermi surfaces for MgB2 and CaGa2 differ in topol- other calculations and the available experimental data ogy: hole-type cylinders (lying along the ΓÐA direction on the enthalpy of formation of stoichiometric for MgB2) degenerate into cones in CaGa2. A similar diborides [7, 153Ð155, 166Ð170]. The presence of Me band structure was obtained for isoelectronic and isos- vacancies leads to a substantial decrease in the stability tructural BeB2 [175], which is not a superconductor. In of borides. An important energy parameter that charac- CaGa , the Ca states make a dominant contribution terizes the probability of generating vacancies is the 2 energy of their formation: (~61%) to the density of states at the Fermi level. For the AlB2-like beryllides ZrBe2 and HfBe2 [174, MeB2 Me0.75B2 Me Eνf = Etot ÐEtot Ð 0.25Etot . 176], the top of the valence band is determined by the strong hybrid interactions of the Zr (Hf) d and Be sp An analysis of the results presented in Table 5 shows states. The Be σ bands have a noticeable dispersion that the energy of formation Eνf of Me vacancies in along the ΓÐA direction, and the hole states are absent. NbB2 is less than that in ZrB2 and YB2. Therefore, it is According to the type of energy bands, their composi- more difﬁcult to introduce vacancies into ZrB2 and the tion, and the occupancy, the Zr and Hf beryllides are effects of nonstoichiometry (in the Me sublattice) are similar to isostructural and isoelectronic ScB2 and YB2, more characteristic of diborides of the Group III and V in which superconductivity is not observed down to T < d metals. 1.4 K. The ﬁrst representatives of a new class of chemi- 4.3. Nonboride AlB2-Like Phases cally more complex layered (AlB2-like) phases, At present, more than one hundred binary phases namely, Sr(GaxSi1 − x)2 [177], Ca(Al0.5Si0.5) [178], and with structures of the AlB type are known. Among (Ca,Sr,Ba)(GaxSi1 − x) [179, 180], have been synthe- 2 sized recently. Their hexagonal sublattice is formed by these are sÐd metal diborides and MeX2 phases, in which graphite-like networks are formed by different alkaline-earth metal atoms, and the graphite-like net- elements X = Be, Si, Ga, Hg, Zn, Cd, Al, Cu, Ag, and works are composed of Ga and Si or Al and Si atoms. Au. As was noted above, the σ(p ) hole bands in the The superconducting transitions in the silicides occur at x, y the critical temperatures T ~ 3.3Ð7.7 K. The exception planes of boron networks play a decisive role in the c is the Ba(AlxSi1 − x)2 silicide, which is not a supercon- pairing mechanism of superconductivity in MgB2. It is quite evident that, when searching for potential super- ductor at temperatures above 2 K. Investigation of (Ca,Sr,Ba)(AlxSi1 − x)2 phases with a variable Al/Si con- conductors, the aforementioned analogies with MgB2 can be expected for nonboride AlB -like phases with tent (0.6 < x < 1.2) indicated that the critical temperatures 2 T are maximum at the stoichiometric ratio 1 : 1 : 1 [181]. networks formed by sp elements. In recent years, band c calculations have been performed for a number of these An analysis of the transport and magnetic properties systems that involve networks composed of silicon, [177Ð180] made it possible to assign these silicides to beryllium, and gallium. type-II superconductors, and the data on the Seebeck

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 BAND STRUCTURE AND PROPERTIES OF SUPERCONDUCTING MgB2 1845 coefficients suggest that electrons are the majority car- 1 riers in (Ca,Sr,Ba)(AlxSi1 − x)2 phases [181]. The first band calculations were performed by the FLMTO method for the Sr(GaxSi1 − x)2 phases at x = EF 0.375, 0.5, and 0.625 [182] and all the known 1 : 1 : 1 0 ternary phases [Me(A0.5Si0.5)2; Me = Ca, Se, Ba; A = Al, Ga] [183]. The main features of the band structure can be illustrated by using Ca(Al0.5Si0.5)2 as an example (Fig. 16). The valence band of the silicide is governed σ by the Al and Si 3p states, which form four (3px, y) and π two (3pz) bands. For the 3px, y bands, the dispersion is –5 Γ maximum along the kx, y ( ÐK) direction. These bands reflect a two-dimensional distribution of the Al and Si , eV 2 p states in graphite-like networks and form pseudoflat E Γ regions aligned along the kz ( ÐA) direction. The px, y bands contribute to the density of states over the whole width of the valence band with a maximum at approxi- EF mately 2.6 eV below the Fermi level EF. The Al and Si 0 3pz states (responsible for the interlayer bonds) have a Γ considerable dispersion along the kz ( ÐA) direction. The σ and π bands intersect at the Γ point of the Bril- louin zone. It is important that the Al and Si 3p bands lie below the Fermi level EF and do not contain hole states, as is the case in the aluminum diboride [isoelec- –5 tronic to Ca(Al0.5Si0.5)2], which is not a superconductor. The main contribution (~59%) to the density of states at the Fermi level N(EF) for the Ca(Al0.5Si0.5)2 silicide is Γ MKΓ A L H A made by the Ca 3d states, which determine the n-type conductivity of this phase. The contributions of the Al Fig. 16. Energy bands of (1) Ca(Al0.5Si0.5)2 and and Si 3p states to the density of states at the Fermi (2) Sr(Ga0.5Si0.5)2 according to FLMTO calculations. level N(EF) do not exceed ~ 9 and 10%, respectively.

In the series Ca(Al0.5Si0.5)2 Sr(Al0.5Si0.5)2 N(EF) for the Me(Ga0.5Si0.5)2 silicides. Note that the Me Ba(Al0.5Si0.5)2, the change in the type of alkaline-earth d states are dominant in the near-Fermi region. metal leads to a change in the position and dispersion of the lower cation d bands: the Sr and Ba d states form It can be seen that, compared to the band structure quasi-ﬂat bands (aligned along the LÐH direction), of the MgB2 superconductor, the band structure of the which coincide with the Fermi level EF. This results in ternary AlB2-like silicides has a number of fundamental the appearance of sharp peaks in the density of the Sr differences. These differences are as follows: (i) occu- σ and Ba d states with an admixture of Al and Si p orbit- pation of px, y bonding bands and the absence of hole als, which are separated by the pseudogap from the states, (ii) an enhancement of the covalent interaction bounding p bands. In the case of the transition from between the (Al,Si) [or (Ga,Si)] and alkaline-earth Ca(Al0.5Si0.5)2 to Ba(Al0.5Si0.5)2, the density of states at metal layers, and (iii) a change in the orbital composi- the Fermi level N(EF) increases by a factor of more than tion of the density of states N(EF), to which the d states two. It should be noted that the increase in the density of alkaline-earth metals make the dominant contribu- of states at the Fermi level N(EF) is caused by the tion. The last feature is characteristic of a large number increase in the contributions from virtually all valence of low-temperature superconducting compounds of p states of the atoms involved in the silicide. and d elements (NbN, V3Si, etc.) [2], for which the critical temperatures Tc can be adequately described by the As a whole, the band structures of Me(Al0.5Si0.5)2 〈ω〉 λ 〈ω〉 McMillan equation Tc ~ exp{ f( )}, where are and Me(Ga0.5Si0.5)2 are similar to each other. The main differences reside in the increase in the dispersion of averaged phonon frequencies (inversely proportional to λ 〈 2〉 〈 ω2〉 the σ and π bands along the AÐLÐH direction and the the atomic masses), = N(EF) I / M is the elec- decrease in the band gap (by ~ 1.0Ð0.9 eV) between the tronÐphonon coupling constant, 〈I2〉 are the electronÐ 2 s and p bands for Me(Ga0.5Si0.5)2. The change in the phonon matrix elements, and the quantities 〈Mω 〉 do type of alkaline-earth metal (Ca Sr Ba) leads not depend on the atomic masses and are determined by to an increase in the density of states at the Fermi level the force constants.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1846 IVANOVSKIŒ

E, eV increase in the electron concentration from 8.75 (at x = –49.45 0.375) to 9.25 e/formula unit (at x = 0.625) results in a drastic change in the profile of the near-Fermi density –49.50 of states (predominantly, owing to the Sr 4d states) and a nonmonotonic dependence of the density of states at –49.55 4 the Fermi level N(EF) with a minimum at x = 0.5. There- fore, the simplified rigid-band model cannot be applied –49.60 to describe these systems. 3 –49.65 2 1 5. ENERGY BANDS –49.70 AND SUPERCONDUCTIVITY OF BORON 45678 AND HIGHER BORIDES , Å3 V An analysis of different classes of binary (semiborides Me B, monoborides MeB, diborides MeB , tet- Fig. 17. Dependences of the total energy E (eV) on the unit 2 2 3 cell volume V (Å ) for (1) α-B and hypothetical structures raborides MeB4, and a number of higher borides, such 12 ᮀ of elemental boron (2) BB12, (3) B12, and (4) face-cen- as hexaborides MeB6, dodecaborides MeB12, and tered cubic B (see text). MeB66-like borides), ternary, and quaternary borides (see review [14]) has demonstrated that the supercon- According to the experimental data [177Ð181], the ducting properties, as a rule, are observed for phases with a relatively low boron content (B/Me ≤ 2.0Ð2.5, critical temperature Tc of the Me(Al0.5Si0.5)2 silicides monotonically decreases with an increase in the atomic Table 1). The structure of these phases contains boron number of cations. On the other hand, the critical tem- in the form of isolated atoms or linear and planar elements (chains or networks of boron atoms). perature Tc of the Me(Ga0.5Si0.5)2 silicides changes insignificantly (in the range 3.9–5.1 K) and is maxi- The superconducting properties are less pronounced in higher borides (B/Me ≥ 6), whose structure is formed mum (5.1 K) for the Sr(Ga0.5Si0.5)2 silicide. As follows from the calculations performed in [182, 183], the den- by stable polyhedral groupings of boron atoms, namely, octahedra B6 (MeB6), icosahedra B12 (MeB12), or their sity of states at the Fermi level N(EF) [like the contributions of the Me d and Si, Al, and Ga p states to the value combinations (MeB66). For example, among the large number of metal hexaborides involving B polyhedra of N(EF)] for the Me(Al0.5Si0.5)2 and Me(Ga0.5Si0.5)2 sili- 6 cides monotonically increases with a change in the and metal dodecaborides containing B12 polyhedra, the alkaline-earth metal from Ca to Ba (i.e., changes oppo- low-temperature superconductivity is found only for sitely to the critical temperature Tc). Furthermore, the eight phases, including four MeB6 phases (Me = Y, La, densities of states N(EF) for Me(Al0.5Si0.5)2 are higher Th, Nd) and four MeB12 phases (Me = Sc, Y, Zr, Lu) [14]. than those for Me(Ga0.5Si0.5)2, which also disagrees It should be noted that the most stable crystalline α β with the observed ratio of their critical temperatures. modifications of elemental boron ( -B12, -B105), in Consequently, the direct correlation Tc ~ N(EF) pro- which the main structural units are boron polyhedra posed in [180] for strictly stoichiometric (1 : 1 : 1) com- (icosahedra or giant icosahedra B84), are semiconduc- positions Me(Al0.5Si0.5)2 is invalid. Apparently, the tors under normal conditions [155, 184Ð186]. Recent main factor responsible for the change in the critical experiments on compression under ultrahigh pressures β temperature Tc for the Me(Al0.5Si0.5)2 compounds is the have revealed that rhombohedral -B105 transforms into variation in the phonon frequencies dependent on a superconducting state (Tc ~ 11.2 K) at pressures above atomic mass. One more reason for the change in the 250 GPa [187]. Papaconstantopoulos and Mehl [188] density of states [and, hence, in the value of N(EF)] is explained this effect as resulting from the pressure- the disordering of Al (Ga) and Si atoms in planar net- induced phase transition of β-boron to a phase with a works. As a consequence, alkaline-earth metal atoms simple face-centered cubic structure. The band struc- reside in a chemically nonequivalent trigonalÐprismatic ture of face-centered cubic boron was calculated within environment, which in part leads to splitting of the the local-electron-density (LDA) approach, and the near-Fermi bands and a decrease in the density of states estimated critical temperature Tc was in reasonable at the Fermi level N(EF). This effect should be most pro- agreement with the experimental data [187]. nounced for the Sr and Ba silicides, because the value According to another scenario of the transformation of N(EF) for these compounds in the ordered state is of β-boron under high pressures [189], the metalliza- determined by narrow intense peaks in the density of tion is caused by the distortion of the crystal structure states. The possible chemical disordering in planar net- and (or) partial destruction of its constituent B12 poly- works was also noted in [183]. hedra, during which part of the boron atoms occupy The band calculations for the Sr(GaxSi1 − x)2 phases intericosahedral sites. The calculations performed for with a variable Ga/Si content [182] showed that an hypothetical boron crystals simulated through the

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 BAND STRUCTURE AND PROPERTIES OF SUPERCONDUCTING MgB2 1847

ᮀ removal of Me atoms from the MeB12 lattice ( B12, 1 where ᮀ is a metal vacancy) or the substitution of boron atoms for Me atoms (BB12) indicated that the energy 4 spectra of these phases exhibit a metal-like nature and a considerable density of the near-Fermi states. The 2 E solution of the equation of state for the aforementioned F0 alternative structures of superconducting boron (Fig. 17) suggests that the energy stability of the poly- –2 hedral structure is higher than the stability of the simple –4 face-centered cubic boron postulated in [188]. The mechanism of structural distortion proposed in [189] –6 was experimentally confirmed by Sanz et al. [190], –8 who found that, as the pressure P increases, the initial structure of β-boron is retained to P ~ 100 GPa (in this –10 case, distortions become stronger) and then transforms into an amorphous state. , eV 2 E Among superconducting hexaborides, the band 4 spectrum was studied for YB6 in [155, 191]. This phase belongs to borides of the CaB6 structural type (space 2 1 EF group Oh ÐPm3m), whose lattice can be represented as 0 a simple cubic lattice (CsCl type) in which cations –2 occupy cesium sites and the B6 octahedra are centered at chlorine sites. The unit cell contains seven atoms –4 (Z = 1) with the following coordinates: Me (a) 0, 0, 0 –6 and 6B (f) 1/2, 1/2, z. The structure involves two types of BÐB interatomic distances that correspond to –8 interoctahedral and intraoctahedral nonequivalent BÐB –10 bonds. Γ Γ The valence band of YB6 (with a width of ~ 11.8 eV) XWX K L contains ten occupied bands involving the hybrid B 2s and B 2p states, which form the above system of BÐB Fig. 18. Energy bands of (1) ZrB12 and (2) YB12. bonds between and inside B6 clusters. A similar band structure is observed for CaB6-like alkali and alkaline- earth metal hexaborides, which are either narrow-band- the appreciable decrease in the contributions of the Y 4d gap semiconductors or semimetals [155, 192Ð195]. For states from 0.798 states/eV cell (~71%) for YB6 to example, the band calculations for metastable cubic 0.538 states/eV cell (~35%) for YB12 [189]. MgB6 (the optimized lattice parameter was determined to be a = 0.4115 nm) demonstrated that this hypotheti- The energy spectra of the superconducting YB12 and cal hexaboride is a narrow-band-gap semiconductor ZrB dodecaborides (UB type, space group O 5 Ð with an indirect transition (band gap, ~ 0.2 eV; E 12 12 h R transition) [191]. The upper occupied band (forming Fm3m, the unit cell contains 52 atoms) were studied by a flat region aligned along the XÐΓ direction) contains the FLMTO method in [189]. Their valence bands involve a complex system of energy bands responsible B 2px, y orbitals that participate in the interoctahedral interactions. The lower empty band has a considerable for the formation of different types of interatomic k dispersion and involves contributions from cation bonds in the crystal. The bands with dominant contribu- states. tions from boron states can be divided into three types. Bands of the first type are composed of the B 2s and B For superconducting YB6, the upper completely p bonding states, which participate in the formation of occupied B px, y bands are characterized by an apprecia- Γ three-center BÐB bonds in planes of the icosahedron ble k dispersion along the ÐX direction and reflect the faces. These bonds are responsible for the stabilization formation of YÐB bonds. Moreover, there is a partially occupied band with a considerable contribution from of individual B12 polyhedra and depend only slightly on delocalized Y d states. Analysis of the composition of their type of packing in the crystal (symmetry of the B12 the density of states at the Fermi level N(EF) demon- sublattice) (Fig. 18). Similar bands are revealed in poly- strates that the decrease in the critical temperatures Tc morphic modifications of elemental boron, whose lat- for the series of higher yttrium borides from 7.1 K for tices are formed by B12 icosahedra [155, 184Ð187], and YB6 to 4.7 K for YB12 can be qualitatively explained by virtually retain their form in YB12 and ZrB12.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1848 IVANOVSKIŒ

The second type of bands involves B 2s and B p gen-free superconductor. Moreover, MgCNi3 is the first bonding states responsible for the formation of interi- superconductor among ternary carbide (nitride) phases. cosahedral B12ÐB12 bonds. Bands of the third type are Ternary carbide (nitride) phases of the general formula formed by the B 2s and B 2p states hybridized with X(C,N)Me3, where Me is a metal and X is an s or d ele- outer Y (Zr) s, p, and d states. The width of the valence ment, exhibit a variety of interesting physical properties band in YB12 is approximately equal to 12.98 eV. The that differ fundamentally depending on their composi- valence band involves two groups of completely occu- tion (see review [197]). For example, ternary carbides pied B 2s and B 2p bands separated by a pseudogap. of the XC(Mn,Fe)3 type (X = Al, Ga, In, Ge, Sn, d met- The band widths are equal to 2.82 and 8.89 eV, respec- als) are magnets; possess ferromagnetic, antiferromag- tively. Near-Fermi bands of the hybrid YÐB type pos- netic, and more complex magnetic orderings; and sess a considerable energy dispersion (Fig. 18). undergo magnetic phase transitions dependent on the temperature. Perovskite-like phases based on alkaline- Upon the transition YB12 ZrB12, the band structure, as a whole, changes insignificantly. The main earth metals, such as XNCa3 (X = Ge, Sn, Pb, P, As, Sb, effect is associated with the band occupation due to an Bi), have an ionic character of chemical bonding [198Ð increase in the electron concentration. According to 200]. Among them, (Bi,Pb)NCa3 is the so-called ionic [189], an increase in the critical temperature Tc from metal [201]. The AsNMg3 and SbNMg3 intermetallic 4.7 K for YB12 to 5.8 K for ZrB12 [14] can be explained compounds [202], which are ionic semiconductors by the increase in the contributions of the Me 4d states [203], have been synthesized recently. On the other hand, phases based on Group IIIÐV metals [AlCSc , from 0.538 states/eV cell for YB12 to 0.743 states/eV cell 3 SnCSc , X(C,N)Ti , where X = Al, Ga, In, etc.] exhibit for ZrB12 to the density of states at the Fermi level 3 3 metallic properties and are high-valence compounds N(EF). Note that the Fermi level EF for YB12 and ZrB12 is located in the region of a plateau in the profile of the [204Ð207]. Therefore, the MgCNi3 compound is a density of states between the bonding and antibonding phase intermediate between the groups of perovskite- bands formed by the B 2s and B 2p states [189]. The like superconductors (oxides) and nonsuperconducting oxygen-free X(C,N)Me phases. change in the metal sublattice type (YB12 ZrB12) 3 affects both the total profile of the density of states for Among its chemical analogs, the MgCNi3 com- these phases and the value of N(EF) only slightly. The pound is most similar to the aforementioned intermetal- density of states at the Fermi level N(EF) increases by lic borocarbides, including the LuNi2B2C and YNi2B2C no more than ~ 16%, and the dominant contribution to nickel-containing phases. However, intermetallic boro- the density of states N(EF) is made by the Me 4d states. carbides are magnetic superconductors and have an In [189], the inference was drawn that doping of binary anisotropic (quasi-two-dimensional) structure. It dodecaborides (for example, when preparing should be emphasized that the nickel content in inter- YxZr1 − xB12 solid solutions) for the purpose of optimiz- metallic borocarbides is considerably lower than that in ing their superconducting properties, which is very effi- the MgCNi3 compound. cient for controlling the critical temperature Tc of other Among low-temperature superconductors, the borides (see above), will be inefficient for MeB 12 MgCNi3 compound also occupies a special place, phases. However, the specific features of the electronic because this phase is rich in nickel, which is a magnetic spectrum of the MeB12 phases indicate that their super- metal. The occurrence of superconductivity in this sys- conducting properties are stable to changes in the tem is very unusual; this makes it similar to the so- chemical composition of the system. In other words, called ferromagnetic superconductors discovered the synthesis conditions for MeB12 low-temperature recently: UGe2, URhGe [208Ð210], ZnS2 [211], and superconductors should not impose rigid requirements Sr2RuO4 oxide characterized by surface ferromagnetic on the stoichiometry of the prepared samples, unlike ordering [212, 213]. A number of theories based on the the synthesis of the MgB2 or YNi2B2C superconductors paramagnon and magnon exchange models and other (see reviews [11, 13Ð16]). approaches have been proposed for explaining the nature of ferromagnetic superconductivity [214Ð217]. Consequently, the MgCNi compound (as a phase close 6. ELECTRONIC PROPERTIES OF NEW 3 SUPERCONDUCTORS WITH STRUCTURES to the boundary of the magnetic-instability region) can OF THE ANTIPEROVSKITE TYPE be treated as an intermediate phase between the groups of classical nonmagnetic and ferromagnetic supercon- A further promising direction in the search for new ductors. superconductors is associated with the recent discovery Owing to the above specific features, particular of superconductivity (Tc ~ 8 K) in the MgCNi3 perovs- interest has been expressed by researchers in the kite-like intermetallic compound [196]. Several factors MgCNi3 compound, primarily, as a superconductor in attach special significance to this discovery. the vicinity of the critical point of a ferromagnetic tran- Among its structural analogs, namely, cubic perovs- sition. A sufficiently large number of works have been kite-like phases, the MgCNi3 compound is the first oxy- devoted to investigating the basic physical (predomi-

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 BAND STRUCTURE AND PROPERTIES OF SUPERCONDUCTING MgB2 1849

MgCNi3 B (a) MgCNi2Co 1 2 3 (100) XPS VB

Intensity, arb. units 1'

15 MgCNi (111) 3 1' (b) MgCNi Co B 2 1 Mg C Ni(Co) 2 C 10 3

Fig. 19. Structure of the MgCNi3 antiperovskite. 5 nantly, low-temperature) properties of this compound DOS, states/eV cell [196, 218Ð227]. These properties and the main proce- 0 dures for synthesizing the MgCNi3 compound and a –4 –2 0 number of MgCNi3-based solid solutions are described in review [228]. Now, we consider the results obtained Ni 10 in modeling the MgCNi3 compound and related phases in the framework of the band theory. The band calculations for the electronic spectrum of 5 the ideal MgCNi3 antiperovskite (Fig. 19) were carried out in [229Ð245]. It was found that the Mg states only slightly participate in the formation of the valence band 0 (the total width is approximately equal to 7.2 eV), 1.5 which is determined by the contributions of the C 2p B' Mg and Ni 3d orbitals. The C states lie below the Ni states DOS, states/eV cell 1.0 C and are partially mixed with them. The most important feature of the spectrum is a narrow intense peak in the 0.5 density of states with a maximum located ~ 45 meV below the Fermi level EF (Fig. 20). This peak is associ- 0 ated with the quasi-flat Ni 3d band (aligned along the –5 0 XÐM and MÐΓ directions in the Brillouin zone). The Fermi level EF is located at the high-energy slope of the Fig. 20. (a) Photoelectron spectra and (b) total densities of above peak. The density of states at the Fermi level states of MgCNi3 and MgCNi2Co. Partial densities of states of MgCNi3 are given at the bottom [242]. N(EF) is comparable to those for other BCS superconductors that have a cubic structure and are formed with the participation of d metals [2]. The maximum contri- ner et al. [234] calculated the band spectrum of a bution (~ 88%) to the density of states at the Fermi level ᮀ2+ CNi3 hypothetical crystal. It was revealed that the N(EF) is made by the Ni 3d states. The Ni 3d bands have a substantial dispersion E(k) in the vicinity of the Fermi difference between the band spectra of this crystal and MgCNi is reduced to a change in the position of the level EF and are responsible for the high velocity of car- 3 riers (2 × 105 m sÐ1 [231]) and the metallic properties of Fermi level EF. the antiperovskite. Two bands intersecting the Fermi level EF form the The nature of the near-Fermi peak that reflects the Fermi surface of the MgCNi3 compound (Fig. 21). The Van Hove singularity of the Ni 3d bands is considered Fermi surface involves electron-type spheroids in the in detail in [232]. The Ni 3d holes forming supercon- Γ ducting pairs have a very large mass. The contributions vicinity of the point and small sheets aligned along to the Van Hove singularity are made only by Ni atoms the faces and corners of the Brillouin zone. The flatter (and, partially, C atoms) located in the xy planes. This Ni 3d band exhibits lobe-type features with the center means that the wave functions can be considered “lay- at the X point on the faces of the Brillouin zone and ered.” In order to analyze this effect in more detail, Ros- cigar-shaped elements along the ΓÐR direction.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1850 IVANOVSKIŒ estimates made in [231], coincidence is achieved when approximately 0.5 holes are introduced into the system. In this case, the density of states at the Fermi level N(EF) increases by a factor of more than two and the parameter S turns out to be close to unity. The calculations for an Mg0.5Li0.5CNi3 hypothetical phase showed that this phase is in a ferromagnetic state [234]. Similar investigations were carried out for Mg1 − xNaxCNi3 solid solutions [234]. Wang et al. [240] attempted to take into account Fig. 21. Fermi surface of MgcNi3 [225]. strong electronÐelectron correlations of the Ni 3d states in the framework of one-center correction for Coulomb interactions (LDA + U method). In the LDA approxi- The interatomic interactions in the MgCNi com- mation (FLAPW method), the magnetic moments 3 µ pound were analyzed using the tight-binding method in (MM) of Ni atoms are negligible (~ 0.014 B), whereas terms of the crystal orbital overlap populations [239]. the LDA + U method leads to magnetic moments of µ The crystal orbital overlap populations were deter- ~0.661 B; in the density of states, the narrow peak due mined to be equal to 0.298, 0.027, and 0.039 e/bond for to the Ni 3d states in the vicinity of the Fermi level EF the NiÐC, NiÐNi, and NiÐMg interactions, respectively. disappears. Voelker and Sigrist [241] proposed to con- This implies that the NiÐC bonds play a dominant role. sider the MgCNi3 compound as a multiband supercon- In [231, 232, 234], considerable attention was given ductor in which, apart from the standard mechanism of to the interrelation between the superconductivity and s-wave pairing, there exists an unusual type of states the magnetic instability of MgCNi3. The Stoner param- characterized by a speciﬁc order parameter of phase eter S [231] was estimated at ~0.43, which is far from relations between particular bands. the ferromagnetic limit (~1). The calculations within An important problem associated with elucidating the rigid-ion model (at the Debye temperature for pure the microscopic nature of the superconducting and nickel Θ ~ 455 K and the constant λ ~ 0.89) indicated D magnetic properties of the MgCNi3 compound is to a strong electronÐphonon interaction. determine the effect of hole or electron doping on its In the absence of carbon (at the center of an Ni6 characteristics. Possible variants of a change in the octahedron), the structure of the MgCNi3 compound electron concentration in MgCNi3 through the intro- coincides with the face-centered cubic structure of Ni, duction of structural vacancies (nonstoichiometry in in which one out of four Ni atoms is replaced by an Mg the Mg or C sublattices) or the replacement of atoms in ᮀ atom. The band calculations for an Mg Ni3 hypotheti- intermetallic sublattices by atoms of a different sort ᮀ λ cal phase demonstrated that S(Mg Ni3) = 1.54 and = were modeled using band calculations. 0.41 [231]. Consequently, upon hole doping of In [229, 230, 234Ð236, 242, 243], the doping of the MgCNi3, the Fermi level EF can coincide with the max- Ni sublattice was analyzed with the use of 3d elements imum of the near-Fermi Ni 3d peak. According to the (Cu, Co, Fe, Mn) as impurities (Me Ni). For ter-

Table 6. Total densities of states at the Fermi level N(EF) (states/eV), Stoner parameters S, electronÐphonon coupling constants λ (at T = 300 and 400 K) [233], energies of magnetization ∆E (meV), and atomic magnetic moments MM [235] for antiperovskite phases (derivatives of superconducting MgCNi3) λ λ ∆ M µ Phase N(EF) S (300 K) (400 K) Phase EN(EF)* MM ( B)

MgCNi3 5.56 0.67 1.36 0.76 MgCNi2Co +0.2 1.2 Ð MgBNi3 5.26 0.60 0.44 0.25 MgCNiCo2 0.0 0.7 Ð MgB0.5C0.5Ni3 4.61 0.53 0.78 1.11 MgCCo3 Ð30.4 2.0 0.39 (Co) MgC0.5N0.5Ni3 2.84 0.34 0.59 0.33 MgCNi2Fe 0.0 0.5 Ð MgNNi3 4.01 0.51 1.16 0.65 MgCNiFe2 Ð271.6 2.9 0.08 (Ni) LiCNi3 20.43 2.43 2.86 1.61 1.19 (Fe) Li0.5Mg0.5CNi3 15.27 1.87 2.15 1.21 MgCFe3 Ð379.1 3.4 1.42 (Fe) Mg0.5Al0.5CNi3 1.81 0.21 0.39 0.22 MgC(FeCoNi) Ð60.6 2.6 0.97 (Fe) AlCNi3 2.61 0.29 0.52 0.29 0.24 (Co) 0.03 (Ni) Me * N (EF) are the contributions of transition-metal atoms to the total density of states at the Fermi level N(EF).

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 BAND STRUCTURE AND PROPERTIES OF SUPERCONDUCTING MgB2 1851 nary substituted intermetallic compounds MgCCo3 and pounds with an antiperovskite structure. In particular, MgCCu3, it was found that the ground state of MgCCo3 among boron-containing phases, ScBNi3 and InBNi3 is magnetic and the Fermi level EF in MgCCu3 shifts are the closest isoelectronic and isostructural analogs of toward the region of antibonding states with a drastic the MgCNi3 compound. In [231], it was shown that the decrease in the density of states at the Fermi level Fermi level EF for the ScBNi3 and InBNi3 phases is N(EF). The doping of MgCNi3 with Fe and Co atoms shifted (compared to the Fermi level EF in MgCNi3) results in suppression of the superconductivity. Note toward the high-energy range and the density of states that the suppression is more efficient upon introduction at the Fermi level N(EF) decreases sharply. It has been of iron (Table 6). In [235], the considerable decrease in assumed that the superconductivity in boron-contain- the critical temperature Tc for MgCNi3 Ð xCox (0 < x < 2) ing antiperovskites can be induced either upon intro- was explained by the decrease in the density of states at duction of hole dopants or in the presence of structural the Fermi level N(EF) with an increase in the cobalt vacancies. Kumary et al. [245] evaluated the energy content (i.e., with a decrease in the electron concentra- states for a number of hypothetical antiperovskites with tion) and the transformation of the near-Fermi Ni 3d the aim of predicting conditions for their subsequent and Co 3d bands. The doping effect was also studied synthesis. By substituting Ca, Sr, or Ba for Mg and Mn, using the x-ray photoelectron and C Kα (1s 2p), Ni Fe, or Co for Ni in the MgCNi3 compound, it was L2, 3 (3d 2p), and Co L2, 3 (4s 2p) x-ray emis- revealed that MgCMn3, MgCFe3, and MgCCo3 should sion spectra of MgC1.45Ni3, MgC1.55Ni3, and be stable phases. According to [245], the CaCNi3, MgC1.45Ni1 − xCox [242]. The magnetic moments of V, MgCFe3, and SrCCo3 phases should possess magnetic Cr, Mn, Fe, and Co impurities at the Ni site in MgCNi3 properties, whereas SrCNi3, BaCNi3, and CaCCo3 were estimated within the simplified Hartree–Fock should be nonmagnetic materials. approximation [243]. It was revealed that Co is in a nonmagnetic state and the magnetic moments for the By assessing the general role of the discovery of other metals fall in the range 0.4Ð3.0 µ . superconductivity in the MgCNi3 compound, He et al. B [196] noted that this can be an important step toward The doping and nonstoichiometry effects in the C reinitiating the class of intermetallic compounds as and Mg sublattices were considered in [230, 231, 233, promising superconductors in the context of a 234, 244]. Dugdale and Jarlborg [231] offered an expla- “renewed chemical paradigm.” nation for the decrease in the critical temperature Tc of MgC1 − xNi3 with an increase in the deviation x from carbon stoichiometry. An increase in x leads to a sub- 7. MODELING OF THE STRUCTURE stantial increase in the density of states at the Fermi AND ELECTRONIC PROPERTIES OF MgB2 level N(EF) and the Stoner parameter S and an enhance- NANOMATERIALS AND RELATED BORIDES ment of spin fluctuations, which result in the breaking of Cooper pairs and the suppression of superconductiv- In the search for new superconductors promising for ity. When modeling the doping of the C and Mg sublat- practical applications, two groups of nanomaterials tices with B, N [233], Li, and Al [244] atoms, it was have particularly attracted the attention of researchers. assumed that since the near-Fermi edge of the elec- One of these groups contains molecular crystals based tronic spectrum of MgCNi3 is predominantly formed on quasi-zero-dimensional (0D) fullerene-like nano- by the Ni 3d states, the replacement of atoms in other particles, the so-called fullerides. The second group (Mg or C) sublattices does not appreciably affect the involves quasi-one-dimensional (1D) nanotubular sys- density of states in the vicinity of the Fermi level EF and tems. The most known representatives of these systems the main effect is reduced to a shift of the Fermi level are carbon nanotubes (CNTs) [246Ð254], which, as was found in [255, 256], undergo a superconducting transi- EF depending on the electron concentration. The calculations (Table 6) demonstrated that the introduction of tion (Tc ~ 0.55 K). The theoretical interpretation of the B and N impurities into the C sublattice is accompanied observed effect was proposed by Gonzales [257]. New λ hybrid (0D + 1D) carbon nanostructures (the so-called by a decrease in the values of N(EF) and . This suggests that similar attempts to improve the supercon- peapods), which consist of fullerene molecules encap- ducting properties of the initial intermetallic compound sulated inside C60@C nanotubes, have been considered will be inefficient. The substitution of aluminum (elec- promising high-temperature superconductors [258]. tron donor) for magnesium also leads to a decrease in λ As is known, the majority of non-carbon nanotubes the values of N(EF) and , and the presence of lithium were synthesized (and predicted) for materials that, like (hole dopant) at the Mg sites results in the transforma- carbon, have layered (2D) crystalline modifications tion of the system into a magnetic state. (see review [259]). For this reason, AlB2-like phases In addition to the band spectra of the MgCNi3 com- can provide a basis for designing a new class of inor- pound and MgCNi3-based solid solutions, the band cal- ganic nanotubes that can possess superconducting culations were performed for a number of actually properties [260, 261]. In this respect, the MgB2 com- existing or hypothetical related intermetallic compound is of special interest as a phase close to the

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1852 IVANOVSKIŒ

Table 7. Total energies Etot (eV) and indices of interatomic bonds (crystal orbital overlap populations, COOPs) for (6, 6), (11, 11), and (20, 0) MeB2 single-walled nanotubes COOP, e Nanotube Me ÐEtot BÐB MeÐB (MeÐMe)

(6,6)MeB2 (I)* Mg 1263.27 0.956 0.095 Al 1425.88 0.842 0.154 Sc 1405.85 0.828 0.161 Ti 1610.41 0.606 0.236

(6,6)MeB2 (II) Mg 1274.55 0.928 0.007 Al 1427.05 0.967 0.043 Sc 1428.08 0.780 0.003 Ti 1636.04 0.752 0.072

(11,11)MeB2 (I) Mg 2320.33 0.938 0.040 (0.063) Al 2620.04 0.884 0.122 (0.040) Sc 2578.46 0.845 0.141 (Ð0.03) Ti 2945.32 0.662 0.206 (0.120)

(11,11)MeB2 (II) Mg 2340.17 0.933 0.050 (0.178) Al 2636.75 0.920 0.057 (0.274) Sc 2597.82 0.850 0.035 (0.141) Ti 2965.25 0.758 0.087 (0.390)

(20,0)MeB2 (I) Mg 4198.78 0.857 0.088 (0.127) Al 4693.19 0.844 0.190 (0.230)

(20,0)MeB2 (II) Mg 4202.57 0.852 0.054 (0.126) Al 4674.83 0.884 0.124 (0.164) * Nanotube conﬁgurations: (I) the metallic cylinder is located outside the boron tube and (II) the metallic cylinder is located inside a boron tube (Fig. 1).

boundary of lattice instability of AlB2-like structures The first calculations of the band structure for a num- [163]. ber of cylindrical structures—prototypes of AlB2 nano- Note that elemental boron has no direct analogs in tubes—were performed by Quandt et al. [260]. It was carbon nanostructures (nanotubes and fullerene-like revealed that all tubes possess metallic conductivity. molecules). Possible boron nanostructures formed from Ivanovskaya et al. [271, 272] applied the tight-bind- small-sized boron clusters, for example, pentagonal ing band method to analyze the electronic structure and and hexagonal pyramids B6 and B7, were theoretically the nature of interatomic interactions in armchair (6, 6) modeled in [262Ð270]. This made it possible to deter- and (11, 11) and zigzag (20, 0) MeB2 single-walled mine the mechanisms and conditions of formation and nanotubes (Me = Mg, Al, Sc, Ti); (3, 3)@(6, 6), (6, the relative stabilities of a series of new (non-carbon) 6)@(12, 12), and (3, 3)@(6, 6)@(12, 12) MgB2 multi- quasi-planar tubular and spherical boron nanostruc- walled nanotubes; and tubes prepared from lithium tures. borocarbide, which is assumed to be a promising superconductor (see above). 7.1. Nanotubes Based on Metal Diborides and LiBC A single-walled nanotube of the formal stoichiome- The structural and energy parameters of a number of try MeB2 is modeled by two coaxial cylinders formed MgB2 and ZrB2 hypothetical nanotubes were deter- upon the rolling-up of adjacent (in a crystal) planar net- mined by the molecular dynamics technique [261]. It works, namely, an Me hexagonal network and a graph- was found that armchair diboride nanotubes (the struc- ite-like boron network. Two types of tube configura- tural classification of nanotubes is described in [246– tions are possible: a metallic cylinder can be located 249]) with the minimum strain energy have a higher either outside (I) or inside (II) of a tube composed of stability. It is believed that diborides can form single- boron atoms (Fig. 1). The estimates of the total energy walled and multi-walled nanotubes, for example, (7, for the alternative configurations of the MeB2 nano- 7)@(11, 11)@(15, 15) MeB2 nanotubes. tubes (Table 7) suggest a higher stability of tubular

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 BAND STRUCTURE AND PROPERTIES OF SUPERCONDUCTING MgB2 1853 structures that contain the metallic cylinder inside the DOS, arb. units tube composed of boron atoms and are characterized by 1 stronger BÐB and MeÐMe bonds. 100 The MgB2 nanotubes have a metal-like electronic spectrum in which the Fermi level EF lies in the region of the B 2p states (Fig. 22). The Mg 3s and Mg 3p states insignificantly contribute to the valence band. The nan- 0 otube geometry (the nanotube diameter and the orientation of BÐB bonds with respect to the c axis of the tubu- 2 lene) has a considerable effect on the density of states. 100 In particular, for the armchair (11, 11) nanotube (a number of BÐB bonds are perpendicular to the c axis), the Fermi level EF coincides with a local minimum in the density of states. In contrast, for the zigzag (20, 0) 0 nanotube (a number of BÐB bonds are aligned along the 3 c axis), the Fermi level EF is located in the region of a maximum in the density of states with a high density of 100 B 2p states. It is obvious that these MgB2 nanotubes should be more attractive for searching for materials with improved electrical (specifically, superconduct- 0 ing) properties. –20 –10 0 E, eV According to [272], with an increase in the number of walls [(3, 3) (3, 3)@(6, 6) (6, 6)@(12, 12) Fig. 22. Total densities of states of (1) (6, 6), (2) (11, 11), and (3) (20, 0) MgB2 nanotubes with type-II configurations (3, 3)@(6, 6)@(12, 12)] in MgB2 multi-walled nano- (Fig. 1 [271]). Vertical lines indicate the Fermi level. tubes, the metallic conductivity is retained and the valence band, as a whole, changes insignificantly. The main effect is associated with the change in the density whereas the (20, 0) LiBC nanotube possesses semicon- of near-Fermi states, namely, the increase in the density ductor properties. With a decrease in the number of of B 2p states in the vicinity of the Fermi level EF. The electrons (LiBC Li0.5BC), all borocarbide nano- dominant contribution to this region is made by the tubes exhibit metallic properties. In the LiBC and states of boron atoms of an outer tube with a maximum Li0.5BC nanotubes, the CÐB interactions play a domi- diameter. As the number of walls in the MgB2 multi- nant role and the other bonds (LiÐLi, LiÐB, LiÐC) make walled nanotube increases, the valence band becomes a negligible contribution. similar to that of the crystal. The evolution of the band structure of MeB nano- At present, experimental data on the synthesis of 2 nanotubes based on MgB and other diborides are tubes as a function of the metal type (in the series 2 unavailable in the literature. Judging from the modern MgB2 AlB2 ScB2 TiB2) is governed by two factors: (i) an increase in the number of electrons in practice of synthesizing inorganic nanotubular struc- the unit cell (i.e., the occupancy of bands) and (ii) tures (see review [259]), we can assume that a possible changes in the spectrum of nanotubes due to the change way of preparing MgB2 nanotubes is the so-called tem- in the MeÐMe and MeÐB interactions. For example, plate method, which has already been applied with advantage to synthesize nanotubular composites, i.e., upon the transition MgB2 AlB2, the Al 3s and Al 3p states make a considerable contribution to the band of multi-walled nanotubes containing walls of different occupied B 2s and B 2p states. In the case of ScB and chemical compositions. According to this method, sta- 2 ble carbon tubes used as a matrix are covered with lay- TiB2 nanotubes, there arise intense peaks in the density of Sc (Ti) 3d states in the vicinity of the Fermi level E . ers of different compounds. In the given case, it is clear F that the main problem consists in designing a particular In this series of nanotubes, the BÐB bonds play a deci- technique for applying in situ magnesium diboride lay- sive role and the populations of MeÐMe and MeÐB ers to the surface of carbon nanotubes. bonds increase noticeably. The calculations performed for the (6, 6), (11, 11), A variant of this technique involves the formation of and (20, 0) borocarbide nanotubes based on LiBC and an elemental-boron nanolayer on carbon nanotubes, Li0.5BC demonstrated that a change in the composition followed by treatment of boronÐcarbon nanotubular of the nonmetallic cylinder (B2 BC) results in the composites in magnesium vapors. A similar procedure appearance of a band gap (for particular atomic config- is successfully used for preparing superconducting urations of nanotubes) between the occupied and empty MgB2 wires through annealing amorphous-boron fila- C 2p and B 2p bands [271]. Specifically, the (6, 6) and ments in Mg vapors [273]. The use of recently pro- (11, 11) LiBC nanotubes possess metal-like properties, duced crystalline-boron nanowires [274] as a matrix

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1854 IVANOVSKIŒ can prove promising for the development of techniques Chernozatonskiœ [261] proposed models of the so- for synthesizing nanotubes from metal borides. called terminal structures for closed MeB2 nanotubes. Their caps are represented as truncated bifullerenes in Another method of preparing MeB2 nanotubes is the technique of rolling-up films [275–277]). This tech- the form of double hemispheres formed by metal and nique allows one to prepare nanotubes from thin films boron atoms and involving topological defects in Me of the corresponding materials. In this case, the thick- and B shells. ness of the nanotube walls, i.e., the number of walls in multi-walled nanotubes, depends only on the method of applying the film to a substrate. The technique is free of 7.2. Fullerene-Like Nanoclusters Based restrictions on the chemical composition of individual on Metal Diborides walls in nanotubes and can be used to fabricate tubular composites from multilayer films of different composi- The structural models of quasi-zero-dimensional tions. Considering the progress in the preparation of molecular forms of diborides—fullerene-like cage particles—were discussed in [261, 278]. Molecules thin MgB2 films (see [13–16]), the use of this approach in the synthesis of nanotubes seems to be quite justified. Men ± 2B2n consist of concentric polyhedra, namely, the inner polyhedron Men ± 2 and the outer polyhedron B2n. The above synthesis procedures can be used for pre- According to the estimates of their energy states, the paring both pure MeB2 multi-walled nanotubes and structures that correspond to the stoichiometric compo- nanotubular composites containing tubular layers of sitions B/Me = 2 and contain two metal vacancies in different metal diborides (MeB2@Me'B2, etc.). In the two Me pentagons have a higher stability. latter case, it can be expected that such materials will exhibit unusual properties due to the presence of inter- Bamburov et al. [278] calculated the electronic faces between adjacent tubes of different chemical structure and the parameters of interatomic bonds for a nature. number of polyhedral molecules MenB2n (n = 10, 30, In [272], the modeling of the first representative of 90, 120, 160) based on diborides of s (MgB2), p (AlB2), composite nanotubes was carried out using a and d (ScB2, TiB2) metals (Fig. 1) and also more com- (6, 6)AlB2@(12, 12)MgB2 double-walled nanotube as plex cage nanoparticles (of the carbon onion type) com- an example. This nanotube has a metal-like spectrum. posed of two concentric MenB2n molecules of identical The B 2p states of the outer (12, 12) MgB2 tube are (Me10B20@Me90B180) and mixed compositions located in the vicinity of the Fermi level E . An increase F [Me10B20@BMe90' 180 (Me, Me' = Mg, Al)]. in the electron concentration [with respect to a pure (6, 6)MgB2@(12, 12)MgB2 nanotube] leads to an For all the particles, atomic configurations with outer increase in the band occupation. As a result, the Fermi boron polyhedra have a higher stability. The exception is level EF is located in the vicinity of a local minimum in provided by AlnB2n molecules. The optimization of their the density of states and the density of states at the structure leads to a partial destruction of the inner Aln Fermi level N(EF) is reduced substantially. polyhedron and “diffusion” of part of the aluminum Shein et al. [182] considered the electronic proper- atoms through the outer boron shell with the formation of distorted three-layer structures of the Al B Al ties of tubular structures formed by Sr(Ga Si − ) and n Ð x 2n x x 1 x 2 type. Ca(AlxSi1 − x)2 ternary silicides also belonging to AlB2- like layered superconductors. These authors modeled With an increase in the diameter, the total energy Etot nonchiral (11, 11) and (20, 0) nanotubes of the formal of cage molecules varies in a different way depending stoichiometry Ca(Al Si ) = CaAlSi and 0.5 0.5 2 on the metal type. For MgnB2n and ScnB2n, the total Sr(Ga0.5Si0.5)2 = SrGaSi. The nanotubes retain the energy increases, which suggests that their stability metal-like properties inherent in the crystalline phases. increases with a decrease in the radius of curvature (i.e., However, depending on the geometry and composition when the molecular geometry becomes similar to the of the nanotubes, their near-Fermi densities of states geometry of a planar layer). Among the TinB2n mole- can differ substantially up to the appearance of a cules under consideration, the Ti30B60 molecule appears pseudogap for the (11, 11) SrGaSi nanotube. In silicide to be most stable. nanotubes, a partial electron transfer Ca AlSi and Sr GaSi takes place, bonds in AlSi and GaSi play It seems likely that, in the immediate future, theoret- a dominant role, bonds in Ca and Sr cylinders are com- ical research on boride nanostructures will deal with the parable, and covalent intertubular bonds (of the van der development of structural models and analysis of the Waals type [246Ð249]) between dissimilar coaxial cyl- electronic properties (including the superconducting inders, as in multi-walled carbon nanotubes, are weak. properties) of new materials, such as different associ- The configurations of silicide nanotubes involving ates of quasi-zero-dimensional and quasi-one-dimen- alkaline-earth metal cylinders inside AlSi or GaSi tubes sional boride nanostructures, for example, in the form have a higher stability. In turn, the stability of SrGaSi of molecular crystals composed of fullerene-like mole- nanotubes is higher than that of CaAlSi nanotubes. cules or bundles of boride nanotubes.

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PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

METALS AND SUPERCONDUCTORS

Nonlinear Waves in Bismuth V. G. Skobov* and A. S. Chernov** * Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia ** Moscow State Institute of Engineering Physics, Kashirskoe sh. 31, Moscow, 115409 Russia Received January 21, 2003; in ﬁnal form, March 18, 2003

Abstract—The propagation of short radio waves in a bismuth crystal in a constant magnetic field H aligned parallel to the bisecting axis oriented normally to the surface of the crystal plate is investigated theoretically. In this geometry, spatial inhomogeneity of the wave field has a weak effect on electrons and a strong effect on holes. It is demonstrated that, in a certain range of magnetic field strengths H, the bismuth crystal is characterized by two modes, namely, a helicon and a doppleron, whose damping is governed by cyclotron absorption of holes. For small amplitudes of the wave field in a linear regime, the damping lengths of both modes are relatively short due to cyclotron absorption. In a nonlinear regime, the magnetic field of the radio wave captures holes responsible for cyclotron absorption. As a result, the absorption is suppressed and the damping lengths of the helicon and the doppleron increase drastically. The excitation of these modes in the bismuth plate results in the fact that the dependence of the impedance of the plate on the magnetic field strength H exhibits resonance behavior and the transmittance of the plate increases by more than two orders of magnitude. It is shown that this effect should manifest itself at frequencies of the order of a few megahertzes in relatively weak magnetic fields (of the order of a few tens of oersteds). © 2003 MAIK “Nauka/Interperiodica”.

1. The propagation of radio waves through metals in 8]). In the frequency range of interest (ω Ӷ ν), the con- a linear regime can differ significantly from that in a duction is governed by carrier scattering, the propaga- nonlinear regime. For example, Voloshin et al. [1] tion of waves becomes impossible, and the skin effect showed that the capture of electrons by a magnetic field is observed in bismuth in the linear regime. In weaker of a large-amplitude wave in cadmium leads to strong magnetic fields, the conductivity increases, the skin suppression of the cyclotron absorption and decreases depth decreases, and the nonlocal effects begin to play the damping of the hole doppleron, which results in a an increasingly important role. This implies that, in sharp increase in the amplitude of impedance oscilla- magnetic fields in which the maximum displacement of tions. More recently, it was demonstrated that the cap- carriers over the cyclotron period appears to be larger ture of carriers by a wave field suppresses collisionless than the skin depth, the collisionless cyclotron absorp- damping of helicons and dopplerons in many metals tion becomes dominant. This range of field strengths is and increases their transparency with respect to these of primary interest to us, because it is in this range that modes [2Ð4]. Moreover, suppression of the collision- the nonlinear effects most clearly manifest themselves. less absorption can lead to the propagation of specific nonlinear waves [3Ð6] that have no analogs in a linear 2. Let us consider a geometry in which a constant regime. magnetic field H and the normal to the surface of the bismuth plate are aligned parallel to the bisecting axis It should be noted that the aforementioned works of the crystal [9]. For this orientation of the magnetic have been concerned with the wave properties of typi- field H, the cross-sectional areas of all three electron cal metals with high conductivity and nonlinearity ellipsoids in the plane perpendicular to the vector H are observed at relatively low frequencies, i.e., in the long- one order of magnitude less than the cross-sectional wavelength and medium-wavelength ranges. The car- area of the hole ellipsoid. As a result, the cyclotron rier concentration in bismuth is five orders of magni- mass of electrons turns out to be one order of magnitude less than that in typical metals. Consequently, the tude less than that of holes and the maximum displace- nonlinear regime in bismuth can be achieved at higher ment of electrons over the cyclotron period is approxi- frequencies. In this work, we theoretically investigated mately 30 times shorter than the maximum displace- the propagation of short radio waves with a large ampli- ment of holes. Hence, there is a sufficiently wide range tude in bismuth. of wavelengths in which the radio wavelength in a In a strong constant magnetic field H, the Alfvén metal is considerably less than the maximum displace- velocity in the bismuth electronÐhole plasma is higher ment of holes but is substantially greater than the max- than the Fermi velocity of carriers. In this case, the non- imum displacement of electrons. In this range of wave local effects are insignificant and the Alfvén waves can vectors k, the nonlocal effects do not affect the electron propagate at frequencies ω exceeding the characteristic component of the transverse conductivity but contrib- frequency ν of carrier collisions (see, for example, [7, ute significantly to the hole component. Consequently,

NONLINEAR WAVES IN BISMUTH 1861 the contribution of electrons to the conductivity can be where adequately described in the local approximation, σ ≡σσ σ ()e σ ()h whereas the contribution of holes should be calculated Ð xx Ð i yx = Ð + Ð , (9) with allowance made for the spatial inhomogeneity of σ the wave field. αβ are the conductivity tensor components, and the superscripts e and h refer to the electron and hole con- The dependence of the hole energy ε on the hole σ tributions to the conductivity Ð, respectively. Since the momentum p can be represented by the equation vectors k and H are aligned with one of the major axes 2 2 2 of the hole ellipsoid, the expression for the nonlocal px py + pz σ ε()p = ------+ ------, conductivity Ð has the same form as in the case of the 2m1 2m2 (1) spherical Fermi surface [10]: m ==0.54m, m 0.06m. () 1 2 σ ()h (),,ω nec Ft Ð k H = i------()ω ν ω-, (10) Here, the x axis is aligned parallel to the trigonal axis of H 1 + + i / c the crystal and m is the mass of a free electron. The ε ε 2 cross-sectional area of the hole ellipsoid (p) = F in the () 3 t Ð 1t + 1 π Ft = ------2 1 +,------ln------Ð i (11) plane pz = const can be written in the form 2t 2t t Ð 1 p 2 q kcp eH ()ε , π ε z t ===------, q ------, ω ------, (12) S F pz = 2 m1m2F Ð ------, (2) ()ω ν ω c 2m2 1 + + i / c eH mcc ε where F is the Fermi energy of holes. where e is the magnitude of the elementary charge, c is the velocity of light, ω is the circular wave frequency, As follows from expression (2), the cyclotron mass ν of holes is determined by the relationship and is the frequency of hole collisions with scatterers. The only difference between our case and the case of ∂ the Fermi sphere is that the formula for q involves not ≡ 1 S Ӎ mc ------π∂ε------= m1m2 0.18m, (3) the radius p of the Fermi sphere but the parameter p 2 F F determining the displacement of holes at the reference the longitudinal component pz of the hole momentum at point of the ellipsoid over the cyclotron period. the reference point of the ellipsoid can be obtained from Note that the radio frequencies are small compared the expression to the hole collision frequency ν. For this reason, here- after, the dependence of the conductivity σ on the fre- p = 2m ε , (4) Ð z max 2 F quency ω will be ignored. Moreover, we assume that ν ω and the magnitude of the momentum p can be found the magnetic field H is strong enough for the ratio / c from the formula to be small; that is, ν ≡ 1 ∂S ε γ ≡ ------Ӷ 1. (13) p ------= 2m1 F. (5) ω 2π ∂ c pz max ε Under this condition, the expression for the hole con- The Fermi energy F and the hole concentration n are related by the expression ductivity takes the form 2/3 ()h nec q π2ប3 σ = i------F ------. (14) ε 1 3 n Ð H()1 + iγ 1 + iγ F = ---------. (6) 2m m 1 2 Our interest is in the situation where the radio wave- π For bismuth, we have n = 3 × 1017 cmÐ3. Substitution of length 2 /k in bismuth is small compared to the maxi- this concentration and the masses m and m into rela- mum displacement of holes over the cyclotron period 1 2 2πcp(eH) (q ӷ 1) but is large compared to the maxi- tionships (5) and (6) gives mum displacement of electrons over the same period. ε ==210× Ð14 erg, p 0.45× 10Ð20 g cm/s. (7) This situation is quite possible, because, in the geome- F try under consideration, the maximum displacement of In this work, we are interested in circularly polar- electrons is approximately 30 times shorter than that of () ized waves propagating in bismuth. The sense of rota- holes. In this case, the dependence of σ e on k is insig- tion of the field of these waves coincides with that of Ð σ ()e electrons in the magnetic field (i.e., we deal with the nificant and the relationship for Ð can be reduced to “minus” polarization). The dispersion relation for these the local Hall conductivity: modes has the form σ ()e nec 2 2 π ωσ (), ω Ð = Ði------. (15) k c = 4 i Ð k , (8) H

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1862 SKOBOV, CHERNOV As a result, the dispersion relation (8) can be rewritten the imaginary term. As a result, the biquadratic equa- ± ± in the form of the following equation with respect to the tion has the roots qH and qD, which can be written as dimensionless variable q: 6 ()3 1/2 Dq()= 0, (16) qH ------11+ Ð h , (24) h3/2 where 6 3 1/2 2 ------() Dq()= q Ð ξsq(), (17) qD = Ð 3/2 11Ð Ð h . (25) h 1 q () Both roots are real in the field range H < H0 (where h < sq = 1 Ð ------γ-F------γ- , (18) 1 + i 1 + i 1) and complex in the field range H > H0. Hence, it follows that the quantity H has the meaning of the upper 2 0 4πωnp cc limit of the range of magnetic field strengths at which ξ = ------. (19) eH3 the wave can propagate. For H > H0, both modes are characterized by strong damping. On this basis, hereaf- Now, we analyze the solutions to Eq. (16). For this ter, we will consider only the range H < H0. purpose, we consider the range of field strengths H cor- | | ξ ӷ Now, we replace q by qH in the imaginary term in responding to 1. In this range, the roots of Eq. (17) expression (23) and solve the appropriate equation. As are large compared to unity. Approximate solutions can a result, we obtain be obtained using an asymptotic expression for the function F at q ӷ 1 in the form 1/2 6 3π q = ------11+ Ð h 1 Ð i---q . (26) 3πi 3 1 3/2 4 H F Ӎ Ð------+ -----. (20) h 4 q 2 q With due regard for expression (12) relating q to k, the On the right-hand side of expression (20), the first term complex wave vector k1 can be represented in the form associated with the collisionless cyclotron absorption 2πωne 1/2 of the wave by holes is dominant. The collisional terms k1 = ------proportional to the small quantity γ make an insignifi- cH cant contribution. The second right-hand term in 1/2 (27) expression (20) corresponds to the nondissipative com- H 3π × 11+ Ð ------1 Ð i---q . ponent of the hole conductivity, which, at q ӷ 1, H 4 H appears to be small compared to the electron Hall con- 0 ductivity. From the physical standpoint, this result can This root of the equation is related to the helicon, whose be explained by the fact that holes move in a wave field spectrum is determined by the Hall conductivity and which changes many times along their trajectory over damping is due to cyclotron absorption of the wave by the course of one cyclotron period. Therefore, on aver- holes. As was noted above, the region of existence of age, the wave field has a relatively weak effect on holes. the helicon in bismuth is limited by the upper threshold At the same time, the wave field is nearly uniform for of the magnetic field. This distinguishes helicons in bis- electrons and they efficiently interact with this field. muth from those in alkali metals, which have no upper By substituting expression (20) into relationship (18), threshold. Another important difference is that helicon we can write the dispersion relation as follows: damping due to cyclotron absorption is enhanced with an increase in the field strength H. 2 3 iπ q = ξ 1 +.------q Ð 1 (21) Similarly, we can find the second root 24 q 1/2 6 3π q = Ð------11Ð Ð h 1 + i---q . (28) This equation has two roots q1 and q2 with positive 2 3/2 4 D imaginary parts. Let us introduce the designations h This root is associated with the Doppler-shifted cyclo- πω 2 1/3 H np c tron resonance of holes. With a decrease in q2, the hole h ==------, H0 ------, (22) H0 3e contribution to the Hall conductivity increases and becomes larger than the electron contribution as q and rewrite the dispersion relation (21) in the form approaches unity; as a result, the Hall conductivity reverses sign. Therefore, the dispersion relation pos- 2 12 3 iπ q Ð0.------1 Ð ----- 1 Ð ----- q = (23) 3 2 sesses the root q Ӎ Ð3 at h3 Ӷ 1. Since this root is h q 4 2 associated with the Doppler-shifted cyclotron reso- The solution to this equation will be sought using the nance, the corresponding mode can be referred to as the method of successive approximations. First, we omit doppleron.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 NONLINEAR WAVES IN BISMUTH 1863

It is significant that both roots lie in the range q2 > 1, 800 which corresponds to cyclotron absorption. As a consequence, the imaginary parts of these roots are relatively large in magnitude (of the order of unity). Therefore, 600 both modes are damped through a distance of the order of several displacements of holes over the cyclotron –1 400 period. Thus, it is unlikely that these modes can be ', cm k observed in bismuth in a linear regime. 1' 1 3. The situation can be different in a nonlinear 200 2' 2 regime. For large amplitudes of the exciting field, the magnetic field of the radio wave can capture holes responsible for cyclotron absorption. The captured 0 holes execute oscillations in the direction of wave prop- 0.2 0.4 0.6 0.8 1.0 agation, which prevents absorption of the wave energy. H/H0 The frequency Ω of these oscillations can be defined by the formula [1] Fig. 1. Dependences of the wave vectors on the magnetic field strength H for (1, 1') helicon and (2, 2') doppleron. 1/2 2 Ð1 Hω Sp()∂ S ∂S Ωω= ------z------. (29) c π 2∂ H ∂ pz ± pz Figure 1 shows the dependences k12' , = Rek1, 2(H/H0) pz = pc in the range of magnetic strengths H from 0.3H0 to H0. Here, Hω is the amplitude of the magnetic field of the The calculation was performed at the frequency ω/2π = wave and 10 MHz, collisional frequency ν = 108 sÐ1 (the mean ω free path of holes is approximately equal to 2 mm), and m2 c pc = ------(30) exciting-field amplitude Hω = 2 Oe. For this frequency k ω , we have the threshold field H0 = 30 Oe and the quan- γ ≈ γ Ӷ is the longitudinal momentum of the holes responsible tity 0.03 at H = H0; hence, the condition 1 is well for cyclotron absorption. By substituting the function satisfied in the above range of field strengths H. The S(p ) given by expression (2) and the longitudinal ξ ӷ ξ z inequality 1 is also satisfied at H < H0, because = momentum (30) into formula (29), we obtain 3 12(H0/H) . In Fig. 1, curves 1, 1' and 2, 2' refer to the 1/4 1/2 Hωm 2 1/2 helicon and the doppleron, respectively. Curves 1 and 2 Ωω= ------1 ()q Ð 1 . (31) c H m were obtained from the numerical solution of the exact 2 dispersion relation, and curves 1' and 2' were con- In the case when the frequency of oscillation of the cap- structed according to formulas (33)Ð(36) based on the tured holes is considerably higher than the frequency ν asymptotic expression for the hole conductivity at large of hole collisions, the cyclotron absorption decreases values of q. It can be seen that curves 1 and 1' almost by a factor of Ω/ν. This means that the function F(t) in coincide with each other, whereas curve 2' lies slightly σ ()h above curve 2. This is quite reasonable, because the dop- relationship (10) for the hole conductivity Ð can be pleron root is less than the helicon root and the use of the replaced by the function σ ()h asymptotic formula for the hole conductivity Ð at 2 3 t Ð 1t + 1 ν 2 ӷ F ()t = ------1 +.------ln------Ð iπ---- sgnt (32) q 1 leads to a larger error for the doppleron. However, n 2 Ω 2t 2t t Ð 1 even for the doppleron, the error is less than 10%. '' Therefore, in a strongly nonlinear regime, the roots q1, 2 The curves k1 = Imk1(H/H0) for the helicon damp- can be written in the following form: ing are plotted in Fig. 2. Curves 1 and 2 represent the results of numerical calculations for a strongly nonlin- 6 3 1/2 ν Ω []()Γ ear regime (at H = 0.3H0 and Hω = 2 Oe, the ratio / q1 = ------11+ Ð h 1 Ð i H , (33) h3/2 is less than 0.1 and decreases with an increase in the magnetic field strength H) and a linear regime, respec- 6 []3()Γ 1/2 tively. The dependence of k'' on H for a nonlinear q2 = Ð------11Ð Ð h 1 Ð i D , (34) 2 h3/2 regime also fits curve 1 well. It can be seen that, at the given parameters, the capture of holes by the magnetic πγ 1/2 Γ qH H field of the radio wave very strongly suppresses cyclo- H = ------------, (35) 4 3Hω tron damping at H < H0. πγ 1/2 4. Let us now consider the case of excitation of a Γ qD H D = ------Ð------. (36) helicon and a doppleron in bismuth. The spatial distri- 4 3Hω bution of the electric field E(z) in a semi-infinite metal

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1864 SKOBOV, CHERNOV where 120 2 () dD() q D' qi = ------. (40) dq qq= –1 80 i The surface impedance Z for a semi-inﬁnite metal is '', cm

k 1 proportional to the ratio of E(0) to E'(0); that is, 40 4πiω E()0 Z = ------. (41) 2 () c E'0 0 0.4 0.6 0.8 1.0 For a plate of thickness d, the impedance is determined by a similar expression whose numerator involves the H/H0 electric field strength Ed(0) at the surface of the metal plate. This quantity can be expressed through the heli- Fig. 2. Dependences of the helicon damping k1'' on the mag- con and doppleron components that have passed netic field strength H for (1) nonlinear and (2) linear through the plate. In the case when the multiple specu- regimes. lar reflections of carriers from the surfaces are ignored, the quantity Ed(0) upon antisymmetric excitation (the plate is placed in a radio-frequency coil) is equal to the upon specular reflection of carriers from the surface can difference between the strength of the field excited on be represented by the expression [11] one surface of the plate and the strength of the field that ∞ has passed through the plate on the other surface: E'0() exp()ikz dk Ez()= Ð------, (37) () () () π ∫ 2 2 Ed 0 = E 0 Ð Ed, (42) k Ð 4πiωσ()k /c Ð∞ where E(0) and E(d) are the values of function (39) at where E'(0) is the derivative dE/dz at the metal surface z = 0 and d, respectively. Consequently, the field for z = 0. strength at the plate surface can be written in the form Next, we change over to integration with respect to 2 () the dimensionless variable q and rewrite integral (37) in () ()cp 1 Ð exp ikiz Ed 0 = Ð2iE'0------∑ ------() -. (43) the form eH D' qi i = 1 ∞ E'0()cp dq eH Making allowance for the multiple reflections of carri- Ez()= Ð------exp iq------z , (38) ers from the plate surface, each term in sum (43) π eH ∫ Dq() cp Ð∞ includes an additional factor, that is, []()Ð1 where D(q) is given by formula (17). 1 + exp ikiz . The integrand in expression (38) has poles at the As a result, the expression for the function E (0) takes points q and q and the branch point q = 1 + iγ. If the d 1 2 the form contour of integration is closed in the upper half-plane q, integral (38) can be represented as the sum of two 2 () () ()cp 1 1 Ð exp ikiz pole residues and the integral over banks of the cut Ed 0 = Ð2iE'0------∑ ------()------()-. (44) eH D' qi 1 + exp ikiz passing from the branch point to infinity. The residues i = 1 at the poles q and q represent the helicon and dopple- 1 2 Consequently, the impedance Z of the plate can be ron fields, and the integral over the banks of the cut is d the GantmakherÐKaner nonexponential component, written in the following form: which rapidly decays with an increase in the distance. 2 πω 1 Ð exp()ik z We are interested in the penetration of the radio-fre- 8 p 1 i Zd = ------∑ ------()------()-. (45) quency field through the bismuth plate, i.e., the electric ceH D' qi 1 + exp ikiz field strength at considerable distances from the surface i = 1 (at z = d, where d is the plate thickness). For these dis- The results of numerical calculations of the surface resis- tances, the amplitude of the GantmakherÐKaner com- tance R = ReZd for the above parameters and thickness d ponent is negligible compared to the helicon and dop- = 0.05 cm are presented in Fig. 3. The dependence R(H) pleron amplitudes. Therefore, at distances z ӷ cp/eH, exhibits maxima for magnetic field strengths H at which the field E(z) consists of helicon and doppleron compo- the plate thickness is a multiple of an odd number of hel- nents: icon or doppleron half-waves and a standing wave is excited in the sample. Mathematically, this manifests 2 () itself in a drastic decrease in the denominator in the cor- () ()cp exp ikiz Ez = Ð2iE'0------∑ ------()-, (39) responding term in expression (45). An analysis demon- eH D' qi i = 1 strates that the first maximum is associated with the

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 NONLINEAR WAVES IN BISMUTH 1865

5 REFERENCES 1. I. F. Voloshin, G. A. Vugal’ter, V. Ya. Demikhovskiœ, et al., Zh. Éksp. Teor. Fiz. 73 (4), 1503 (1977) [Sov. Phys. JETP 46, 790 (1977)]. 3 3

10 2. V. G. Skobov and A. S. Chernov, Pis’ma Zh. Éksp. Teor. × Fiz. 61 (12), 980 (1995) [JETP Lett. 61, 1012 (1995)].

cR 3. V. G. Skobov and A. S. Chernov, Zh. Éksp. Teor. Fiz. 109 (3), 992 (1996) [JETP 82, 535 (1996)]. 1 4. V. G. Skobov and A. S. Chernov, Zh. Éksp. Teor. Fiz. 114 (2), 725 (1998) [JETP 87, 396 (1998)]. 0 0.4 0.6 0.8 1.0 5. A. S. Chernov and V. G. Skobov, Phys. Lett. A 205, 81 (1995). H/H0 6. V. G. Skobov and A. S. Chernov, Zh. Éksp. Teor. Fiz. 119 Fig. 3. Dependence of the surface resistance R(H) of a bis- (2), 388 (2001) [JETP 92, 338 (2001)]. muth plate in the strongly nonlinear regime at a frequency 7. E. A. Kaner and V. G. Skobov, Adv. Phys. 17, 605 Ӎ of 10 MHz (H0 30 Oe). (1968). 8. V. S. Édel’man, Usp. Fiz. Nauk 102 (1), 55 (1970) [Sov. Phys.ÐUsp. 13, 583 (1970)]. excitation of the doppleron, the second and third maxima are due to the excitation of the helicon, and the 9. A. P. Cracknell and K. C. Wong, The Fermi Surface: Its Concept, Determination, and Use in the Physics of Met- fourth maximum is a superposition of the helicon and als (Clarendon, Oxford, 1973; Atomizdat, Moscow, doppleron signals. 1978). The nonlinear effect described above appears to be 10. O. V. Konstantinov and V. I. Perel’, Zh. Éksp. Teor. Fiz. extremely strong; speciﬁcally, the curve R(H) in the lin- 38, 161 (1960) [Sov. Phys. JETP 11, 117 (1960)]. ear case virtually merges with the abscissa axis in 11. G. E. Reuter and E. H. Sondheimer, Proc. R. Soc. Lon- Fig. 3. Thus, the suppression of the cyclotron absorp- don, Ser. A 195, 336 (1946). tion in the nonlinear regime leads to the fact that the transmittance of the plate at the maxima increases by more than two orders of magnitude. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1866Ð1873. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1776Ð1783. Original Russian Text Copyright © 2003 by Shaœkhutdinov, Balaev, Popkov, Petrov. METALS AND SUPERCONDUCTORS

Transport and Magnetic Properties of Y3/4Lu1/4Ba2Cu3O7 + Y3Fe5O12 Composites Representing a Josephson-Type SuperconductorÐFerrimagnetÐ Superconductor Weak-Link Network K. A. Shaœkhutdinov, D. A. Balaev, S. I. Popkov, and M. I. Petrov Kirensky Institute of Physics, Siberian Division, Russian Academy of Sciences, Akademgorodok, Krasnoyarsk, 660036 Russia e-mail: [email protected] Received February 25, 2003

Abstract—Y3/4Lu1/4Ba2Cu3O7 + Y3Fe5O12 composites with different volume ratios of the starting components were synthesized. The composites model an SÐFÐS Josephson junction network, where S stands for a superconductor and F, for a ferrimagnet. A study of the transport characteristics of the composites revealed ρ that the temperature behavior of the electrical resistivity (T) below the superconducting transition point TC is different in two regions separated by a temperature T . Below T , the currentÐvoltage characteristics of m m ρ the composites are nonlinear, while in the interval from TC to Tm the values of (T) do not depend on the transport current j and magnetic ﬁeld H. This behavior of ρ(T, j ) and ρ(T, H) is assigned to speciﬁc features of the tunneling of superconducting carriers through the ferrimagnetic layers separating HTSC grains in the composite. Magnetic measurements showed the diamagnetic response of HTSC grains to be lower in composites with a ferrimagnet. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION HTSC + BaPb0.9Fe0.1O3, HTSC + BaPb0.9Ni0.1O3 [16]) Two-phase composite materials based on high-tem- and found that the superconducting properties of such perature superconductors (HTSCs) are of considerable composites are strongly suppressed by the interaction interest from both a practical [1–8] and a scientific [2, of the spins of superconducting carriers with magnetic 3, 5, 6, 9Ð14] viewpoint. The latter is due to these moments during their crossing of the nonsuperconduct- objects representing a Josephson-coupled weak-link ing layer. Investigation of the transport properties of the network. The second, nonsuperconducting, component composite HTSC + paramagnetic insulator (NiTiO3) of the composite (insulator, semiconductor, normal [17, 18] revealed not only a strong suppression of metal) acts as an artificially created weak link between superconducting properties but also an anomalous HTSC grains. By properly varying the volume concen- behavior of the temperature dependences of electrical tration ratio of the starting components in such com- resistivity. Within the temperature interval from the posites, one can vary the effective extent (or “strength”) onset of the superconducting transition in the HTSC of the weak link within a broad range. grains (at TC) to a certain temperature Tm, the electrical resistivity does not depend on the magnitude of the Obviously enough, a single Josephson junction transport current and applied magnetic field (the IÐV would be an ideal object for investigation, but the small curves are linear) and becomes a function of these coherence length and the strong chemical reactivity of quantities only below Tm, a feature characteristic of HTSCs make fabrication and study of such structures a Josephson-type weak links. This anomalous behavior difficult task. However, as shown earlier [10, 11, 14, of the temperature dependence of electrical resistivity 15], the transport characteristics (the temperature ρ of the HTSC + NiTiO3 composites was attributed to dependences of electrical resistivity (T) and of the exchange interaction of superconducting-carrier spins critical current jC(T), the currentÐvoltage curves) of with the magnetic moments of nickel in the paramag- HTSC-based two-phase composite materials reflect the netic phase. This interaction results in an anomaly in main features of superconducting carrier flow through the temperature behavior of electrical resistivity at the a single Josephson junction of a certain effective exten- melting temperature of the Abrikosov vortex lattice in sion in space, while the relative simplicity of fabrication of such materials makes them attractive objects for HTSC grains [17, 18]. studies and for possible applications [7, 8]. The next step in investigating composites with mag- We recently studied the transport properties of com- netically active nonsuperconducting components was a posites with magnetic impurities in a nonsuperconduct- study of HTSC + ferrimagnet composites. Single SÐFÐS ing composite (HTSC + Cu1 Ð xNixO (0 < x < 0.06) [15], structures (S stands for a superconductor, and F, for a

TRANSPORT AND MAGNETIC PROPERTIES 1867 ferro- or ferrimagnet) have become a subject of intense two phases only, with the 1-2-3 and garnet structures; current research, both theoretical [19Ð26] and experi- no foreign reflections were present (within the accuracy mental [27Ð29], because they exhibit interesting phe- of the x-ray analysis). The relative reflection intensities nomena, such as a nonmonotonic dependence of the due to the Y3/4Lu1/4Ba2Cu3O7 and Y3Fe5O12, Y3Al5O12 critical current on temperature [25], a manifestation of phases were in agreement with the volume content of π coupling [19, 24, 25], a reduction of superconducting the starting components in the composites. A study of properties, and a characteristic behavior of magnetore- the temperature dependences of the composite magne- sistance [27]. Of the large family of ferro- and ferri- tization M(T) showed the superconducting-transition magnetic compounds, we chose Y3Fe5O12 (classical onset temperature TC to be the same for all samples, yttriumÐiron garnet), an insulator which interacts 93.5 K, which coincides with the value of TC of the weakly enough with the 1-2-3 HTSC structure. To starting HTSC Y3/4Lu1/4Ba2Cu3O7. The diamagnetic establish the effect of ferrimagnetic ordering in the response decreases with increasing volume content of insulating spacer on the transport properties of com- the nonsuperconducting component (Y3Fe5O12, posites, we also prepared and studied HTSC + Y3Al5O12) in the composite. Thus, x-ray diffraction and Y3Al5O12 reference composites, because the Y3Al5O12 magnetic measurements permit the conclusion that compound is isostructural to Y3Fe5O12 and nonmag- HTSC-based composite materials prepared by fast sin- netic. tering are indeed two-phase materials with no foreign phases present. 2. EXPERIMENTAL 3.2. Transport Properties of the Composites The Y3/4Lu1/4Ba2Cu3O7 polycrystalline HTSC was prepared using standard ceramic technology. The non- Figure 1 shows plots of the electrical resistivity ρ(T) superconducting components of the composites, of the composite samples obtained in the 4.2- to 300-K namely, Y3Fe5O12 and Y3Al5O12, were synthesized from temperature interval. The abrupt decrease in resistivity Y2O3 and Fe3O4, Al(OH)3, accordingly, at a tempera- occurring at TC = 93.5 K signals the transition of HTSC ture of 1250°C for 48 h with three intermediate grind- grains to the superconducting state. Above the transi- ings. After that, the starting components of the compos- tion point, the ρ(T) curves are quasi-semiconducting in ites to be obtained were taken in correct proportion, character. The ratio ρ(93.5 K)/ρ(300 K) grows with ground thoroughly in an agate mortar, and pelletized. increasing volume content of the nonsuperconducting The pellets were put into preliminarily heated boats and component in the composites. This behavior of the ρ(T) placed in a furnace heated to 910°C for 2 min. They relations suggests that the transport current flows were subsequently kept in another furnace for 3 h at t = through both HTSC and nonsuperconducting grains. 350°C and cooled with the furnace. This method of ρ Below TC, the course of the (T) dependences is rapid sintering [11] was employed to prepare composite determined by the superconducting transition in samples with different volume contents of the HTSC Josephson-type weak links [10, 11, 14Ð18]. In the ref- and nonsuperconducting components Y Fe O and 3 5 12 erence composites HTSC + Y3Al5O12, this part of the Y3Al5O12. We denote our composite materials by S + ρ(T) curves exhibits a strong dependence on transport VYIG and S + VYAlG, where S stands for the current j and external magnetic field H. Figure 2 dis- Y3/4Lu1/4Ba2Cu3O7 HTSC and V is the volume concen- plays ρ(T, j) and ρ(T, H) dependences (Figs. 2a, 2b, tration of Y3Fe5O12 (YIG) or Y3Al5O12 (YAlG). accordingly) obtained on the S + 15YAlG sample. We ρ The transport characteristics [ (T) and IÐV curves] readily see that the temperature TC0 at which the elec- were measured by the standard four-probe technique. trical resistivity ρ becomes negligibly small (<10Ð6 Ω The critical current was derived from the initial part of cm) decreases as the transport current j and magnetic the IÐV curve using the standard 1 µV/cm criterion field H grow. This behavior of the ρ(T, j) and ρ(T, H) [30]. The measurement of the transport characteristics dependences is typical of the Josephson-type supercon- in magnetic fields of up to 500 Oe was performed with ductorÐinsulatorÐsuperconductor (SÐIÐS) weak-link a copper solenoid and in magnetic fields above 500 Oe, network; it was observed earlier on composite samples with a superconducting coil. The transport current j was HTSC + CuO [10] and HTSC + MgTiO3 [17] and has passed through the sample perpendicular to the mag- been accounted for in terms of the thermally activated netic field H. The magnetic measurements were con- phase slippage (TAPS) [32] and thermally activated ducted on an automated vibrating-sample magnetome- vortex flow mechanisms [33]. The pattern of the ρ(T, j) ter [31]. dependence observed in the HTSC + Y3Al5O12 insulator composites under study has also been described by the TAPS mechanism [32], but its relevant analysis is 3. RESULTS AND DISCUSSION beyond the scope of the present communication. 3.1. Characterization of the Composites According to the thermally activated vortex flow ∆ X-ray diffraction measurements of the model [33], the resistive-transition width TC0 of SÐIÐ Y Lu Ba Cu O + Y Fe O and Y Lu Ba Cu O + S-type Josephson weak links as a function of the 3/4 1/4 2 3 7 3 5 12 3/4 1/4 2 3 7 ∆ Y3Al5O12 composite samples revealed the existence of applied magnetic field H is defined as TC0 = TC0(H =

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1868 SHAŒKHUTDINOV et al.

120 )

100 H (a) ( 0 2 C

T 20 1 S + 15YIG 10 = 0) – cm

60 H 80 ( Ω 0

C 02040 cm T

, m 2/3, Oe2/3 ρ H Ω

, m 5

ρ 3 20 4 2 40 1 0 S + 7.5YIG 100 (b)

100 cm 4 60 Ω 5

, m 6

ρ 3 2 S + 15YAlG 20 1 cm 60 Ω

, m 0 20 40 60 80 100 ρ T, K

20 S + 7.5YAlG Fig. 2. Temperature dependences of electrical resistivity ρ (T) of samples of HTSC + 15 vol % Y3Al5O12 (S + 0 100 200 300 15YAlG) obtained at various values of (a) transport current , K j and (b) applied magnetic ﬁeld H. (a) (1) j = 3.4, (2) 34, T (3) 170, (4) 340, and (5) 1000 mA/cm2; (b) (1) H = 0, (2) 37, (3) 150, (4) 292, (5) 2000, and (6) 60 000 Oe. Mea- Fig. 1. Temperature dependences of electrical resistivity suring current j = 3.4 mA/cm2. Inset: ∆T = T (H = 0) Ð ρ 2/3 C0 C0 (T) of samples of HTSC + Y3Fe5O12 and HTSC + TC0(H) relations plotted vs. H for R = 0 for (1) S + Y3Al5O12 obtained in the range 4.2Ð300 K. 7.5YAlG and (2) S + 15YAlG.

2/3 0, R = 0) Ð TC0(H, R = 0) and scales as H . This depen- observed only below a certain temperature Tm. The dence was observed experimentally on polycrystalline value of Tm depends on the volume concentration of the HTSCs [33] and HTSC + CuO composite materials [10]. ferrimagnet in the composite, i.e., on the effective ∆ The inset to Fig. 2a plots TC0 = TC0(H = 0, R = 0) Ð extension of weak links in space [10, 11]. For the S + 2/3 ≈ T (H, R = 0) vs. H for samples S + 7.5YAlG and S + 15YIG sample, we have Tm 40 K; for S + 7.5YIG, C0 ≈ ≈ 15YAlG. These dependences are seen to be close to lin- Tm 55 K; and for S + 3.75YIG, Tm 65 K. The IÐV ear in fields of up to 300 Oe. Thus, the magnetoresistive curves of the samples are linear in the TCÐTm range (Fig. properties of the reference composites HTSC + Y3Al5O12 5). For the S + 15YIG sample, there is no critical cur- are similar in character to those of the HTSC + insulator rent jC at T = 4.2 K; for the S + 7.5YIG sample, 2 composites [10] and polycrystalline HTSCs [33]. jC(4.2 K) = 0.025 A/cm ; and for S + 3.75YIG, 2 jC(4.2 K) = 1.12 A/cm . The superconducting proper- HTSC + ferrimagnet composites exhibit a radically ties of the HTSC + ferrimagnet composites are strongly different pattern of transport properties (Figs. 3, 4). suppressed at relatively small values of the transport Below the temperature TC at which Y3/4Lu1/4Ba2Cu3O7 current and of the magnetic field. For instance, the tem- is transferred to the superconducting state (93.5 K), perature behavior of the electrical resistivity ρ(T) of the there is an interval in which the electrical resistivity S + 15YIG sample (Fig. 4) at a threshold transport cur- ρ 2 (T) behaves in a semiconducting manner, as is the case rent jcr = 0.6 A/cm (H = 0) or at a threshold external above TC. The strong dependence of electrical resistiv- magnetic field Hcr = 1900 Oe (measuring current den- ity ρ on transport current j and magnetic field H, which sity j = 0.015 A/cm2) continues to have the semicon- is typical of a Josephson-type weak-link network, is ducting character observed in the TmÐTC interval.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 TRANSPORT AND MAGNETIC PROPERTIES 1869

60 200 6 (a) 4 (a) 5 150 4 40

3 cm

2 Ω

Ω 100 1 , m , m ρ

ρ 2 20 50 1

0 0 (b) 5 (b) 70 6 4 200 3 5 150

50 cm

2 cm

Ω Ω 4

, m 100 , m ρ 1 ρ 30 3 2 50 10 1

0 20 40 60 80 100 0 20 40 60 80 100 T, K T, K Fig. 4. Temperature dependences of electrical resistivity ρ(T) ρ Fig. 3. Temperature dependences of electrical resistivity (T) of a sample of HTSC + 15 vol % Y3Fe5O12 (S + 15YIG) of a sample of HTSC + 7.5 vol % Y3Fe5O12 (S + 7.5YIG) obtained at various values of (a) transport current j and obtained at various values of (a) transport current j and (b) applied magnetic ﬁeld H. (a) (1) j = 1.5, (2) 7.7, (3) 15, (b) applied magnetic ﬁeld H. (a) (1) j = 10, (2) 50, (3) 130, (4) 77, (5) 150, and (6) 600 mA/cm2; (b) (1) H = 0, (2) 40, and (4) 510 mA/cm2; (b) (1) H = 0, (2) 40, (3) 80, (4) 150, (3) 80, (4) 150, and (5) 1900 Oe. Measuring current j = (5) 290, and (6) 2100 Oe. Measuring current j = 10 mA/cm2. 15 mA/cm2.

Thus, the transport properties of HTSC + ferrimag- the IÐV curves are nonlinear, while in the TCÐTm inter- net composites differ radically from those of the refer- val the values of ρ(T) do not depend on the applied ≈ ence composites HTSC + Y3Al5O12 nonmagnetic insu- transport current j. The values of Tm are 5.2 K for the lator. The data presented in Figs. 1, 3, and 4 show that Gd thickness 4 nm and ≈4 K for the thickness ≈8 nm. the IÐV curves of the HTSC + ferrimagnet Josephson junction network are nonlinear only in the temperature We propose two possible explanations to account interval below T , whereas for the reference compos- for the unusual transport properties of the HTSC + fer- m rimagnet composites. ites HTSC + Y3Al5O12 the IÐV curves remain nonlinear up to TC. The unusual pattern of the ρ(T, j) and ρ(T, H) depen- We believe that this effect cannot be due to the non- dences for the HTSC + NiTiO3 paramagnetic insulator superconducting boundaries being distributed in thick- composites was qualitatively interpreted in [17, 18] in ness or to the technology of preparation of the compos- terms of a hypothesis assuming the formation of an ites, because the magnetoresistive properties of the ref- Abrikosov vortex lattice in HTSC grains. Indeed, the erence composites Y3/4Lu1/4Ba2Cu3O7 + Y3Al5O12 are magnetically active component of the composite well described in terms of the TAPS mechanism [32] induces a certain effective field that penetrates in the and thermally activated vortex flow [33]. Moreover, our form of Abrikosov vortices into the HTSC grains to the results correlate with the data obtained in a study [27] Josephson penetration depth (~1000 Å [34]). This field of single superconductorÐferromagnetÐsuperconductor at the HTSC/Y3Fe5O12 interface is approximately equal 6 Josephson junctions, where niobium served as the in magnitude to the effective field (10 Oe for Y3Fe5O12 superconductor and gadolinium, as the ferromagnet. [35]). In flowing through the composite, the transport The ρ(T) dependences of a single Nb/Al/Gd/Al/Nb current inevitably crosses both the HTSC grains and the Josephson junction reported on in [27] also exhibit two ferrimagnet. Therefore, the current should also flow ≈ different regions below the TC of niobium ( 7.6 K) sep- through the region of the HTSC grains where the Abri- arated by a temperature Tm. At temperatures below Tm, kosov vortices formed. Assuming that the temperature

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1870 SHAŒKHUTDINOV et al.

12 mal ﬂuctuations) into the expression for the IÐV charac- (a) teristic in the TAPS model [32] results in a linear U(I) dependence. The vanishing of the critical current I1 in the SÐFÐS Josephson junction network predicted theo- 8 retically [21] for a single junction transforms the linear

cm IÐV curves into nonlinear curves at the temperature Tm. / 77 K Thus, theory [21] and the TAPS mechanism [32] can 20 K ρ , mV provide a qualitative interpretation for the (T) behav-

U 4 ior of the HTSC + ferrimagnet composites. Unfortu- nately, theory [21] yields only a very rough estimate, 4.2 K because ferrimagnetic ordering is not taken into consideration. Moreover, the results obtained for HTSCs and 0 low-temperature superconductors may differ from one 100 (b) another, a point mentioned in [25].

80 4.2 K 3.3. Magnetic Properties of HTSC + Ferrimagnet Composites cm 60 20 K Figure 6 presents ﬁeld dependences of the magnetization M(H) of samples with different volume contents , mV/

U 40 of nonsuperconducting components, namely, S + 15YIG (on the right) and S + 15YAlG (on the left) mea- 20 77 K sured in the 4.2- to 100-K temperature interval. The hysteresis loops of the sample with Y3Al5O12 are typical of HTSCs [34]. The M(H) plots of the sample with 0 200 400 600 Y3Fe5O12 actually represent a superposition of hystere- j, mA/cm2 sis loops of the superconductor and the ferrimagnet, which is clearly seen from the M(H) graphs recorded at Fig. 5. IÐV characteristics of samples (a) S + 7.5YIG and T = 61 and 100 K (Fig. 6). Thus, magnetic measure- (b) S + 15YIG obtained at various temperatures. ments provide additional evidence for the existence of only two phases in the composite, namely, Y3/4Lu1/4Ba2Cu3O7 and Y3Fe5O12. Tm observed by us is the melting point of the Abrikosov Figure 7 specifies the diamagnetic responses M(H) vortex lattice, this hypothesis can account for the obtained at T = 4.2 K from the HTSC phase in the com- unusual character of the ρ(T) dependences of the posites S + 15YIG and S + 15YAlG. To select the dia- HTSC + ferrimagnet composites. In the TCÐTm interval, magnetic response from the HTSC phase in a compos- vortices move without pinning and the resistivity does ite containing a ferrimagnet, the response due to the not depend on the transport current and magnetic field. Y3Fe5O12 ferrimagnet phase was subtracted from the Below Tm, the vortices are pinned inside the HTSC integral M(H) curve (Fig. 6, T = 4.2 K) with allowance grains, which results in the IÐV curves being nonlinear for the temperature dependence M(T) of this response. [36]. An increase in the volume concentration of a fer- As seen from Fig. 7, the diamagnetic response of the rimagnet in the composite corresponds to an increase in HTSC phase present in the composite with the ferri- the HTSC/Y3Fe5O12 interface area, which brings about magnetic insulator (S + 15YIG) is smaller than that for an increase in the number of Abrikosov vortices and a the composite with the nonmagnetic insulator (S + decrease in the Tm temperature. 15YAlG). Because the technology of preparation and Another explanation of the results obtained is con- the volume concentrations of the components of the ceivable. A theoretical study was made in [21] of the composites are the same in both cases, this decrease in stationary Josephson effect with ordered magnetic the diamagnetic response can be accounted for by sup- moments localized in the barrier. It is believed [21] that pression of the superconducting properties in the sur- the Josephson superconducting current vanishes at a face layers of HTSC grains under the influence of the certain critical barrier thickness because of the Cooper ferrimagnet. Indeed, the magnetically active compo- pairs being scattered by magnetic moments in the bar- nent Y3Fe5O12 of the composite induces a magnetic rier. On the other hand, the TAPS effect [32] accounts field, which penetrates into the HTSC grains and for the experimentally observed vanishing of the criti- destroys the superconducting state. cal current in the SÐIÐS structure at temperatures close As follows from Fig. 7, the superconductor with to the transition temperature of the superconducting suppressed superconducting properties takes up 30% of banks due to the temperature-induced voltage fluctua- the volume. Assuming the HTSC grains in the compos- tions at the junction. Substitution of I1 = 0 (I1 is the crit- ite to have spherical shape with an average diameter of ical current of the SÐIÐS junction in the absence of ther- 1.5 µm (as derived from scanning electron microscopy

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 TRANSPORT AND MAGNETIC PROPERTIES 1871

T = 4.2 K 4

0 , emu/g M

–4 S + 15 YAlG S + 15 YIG

T = 33 K

0 , emu/g M

–2

T = 61 K

0 , emu/g M

–1

1 T = 100 K

0 , emu/g M

–1

–3 –1 0 1 3 –3 –1 0 1 3 H, kOe H, kOe

Fig. 6. M(H) plots of samples S + 15YAlG and S + 15YIG obtained at various temperatures.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1872 SHAŒKHUTDINOV et al.

8 posites, which represent a superconductorÐferrimag- netÐsuperconductor Josephson junction network. T = 4.2 K Besides a strong suppression of transport properties (as 1 compared to the HTSC + Y Al O nonmagnetic-insu- 4 3 5 12 2 lator reference composites), the HTSC + ferrimagnet composites exhibit radically different temperature dependences of the electrical resistivity. Below the 0 transition temperature in the HTSC grains, the ρ(T)

, emu/g dependence can be divided into two dissimilar ranges M by a temperature Tm. At temperatures below Tm, the IÐ V curves of the HTSC + ferrimagnet composites are –4 nonlinear (a feature characteristic of Josephson-type weak links), whereas within the TCÐTm interval the behavior of ρ(T) does not depend on the transport current –8 and magnetic ﬁeld. The resistive state of the HTSC + –4 –2 0 2 4 Y3Al5O12 reference composites can be readily H, kOe explained in terms of the thermally activated phase slippage (TAPS) mechanism [32] acting in the SÐIÐS Fig. 7. Diamagnetic response M(H) measured for different Josephson junction network. HTSC phases in the composites at 4.2 K. (1) S + 15YAlG Magnetic measurements carried out on the compos- reference composites and (2) S + 15YIG composites. ites showed that the diamagnetic response due to the HTSC phase in composites with a ferrimagnetic insulator is smaller than that of the HTSC phase in composites with a nonmagnetic insulator. This decrease in the 0 diamagnetic response can be assigned to suppression of the superconducting properties of HTSC grains in a –1 3 surface layer of thickness equal to the penetration depth of the magnetic ﬁeld from the ferrimagnetic component of the composite. –2

, emu/g 2

M H = 1 kOe ACKNOWLEDGMENTS –3 The authors are indebted to A.D. Balaev for his 1 assistance in the magnetic measurements and to A.F. Bovina for x-ray diffraction analysis of the sam- –4 0 20 40 60 80 100 ples. T, K This study was supported jointly by the Russian Foundation for Basic Research and Krasnoyarsk Sci- ence Foundation (project Enisei, no. 02-02-97711) and the Lavrent’ev Competition for Young Scientists (grant, Fig. 8. Diamagnetic response M(T) measured at H = 1 kOe SD RAS, 2002). for different HTSC phases in the composites. (1) S + 15YAlG reference composites, (2) S + 15YIG composites, and (3) difference between relations 1 and 2. REFERENCES 1. G. Xiao, F. H. Stretz, M. Z. Cieplak, et al., Phys. Rev. B data), estimation of the HTSC layer “suppressed” by its 38 (1), 776 (1988). proximity to the ferrimagnet yields a thickness of 2. B. R. Weinberger, L. Lynds, D. M. Potrepka, et al., Phys- ≈800 Å, a ﬁgure that correlates with the penetration ica C (Amsterdam) 161, 91 (1989). depth of ~1000 Å for YBa2Cu3O7 [34]. A similar esti- 3. J. Koshy, K. V. Paulose, M. K. Jayaraj, and A. D. Damo- mate (≈800 Å) is obtained from the temperature depen- daran, Phys. Rev. B 47 (22), 15304 (1993). dences of magnetization M(T) (Fig. 8). Figure 8 pre- 4. E. Bruneel and S. Hoste, Int. J. Inorg. Mater., No. 1, 385 sents the diamagnetic response M(T) due to the HTSC (1999). phases in the S + 15YIG and S + 15YAlG composites, 5. D. Berling, B. Loegel, A. Mehdaouli, et al., Supercond. as well as the difference between these curves. Sci. Technol. 11 (11), 1292 (1998). 6. M. I. Petrov, D. A. Balaev, D. M. Gohfeld, et al., Physica C (Amsterdam) 314, 51 (1999). 4. CONLUSIONS 7. D. A. Balaev, D. M. Gokhfel’d, S. I. Popkov, et al., Thus, we have studied the transport and magnetic Pis’ma Zh. Tekh. Fiz. 27 (22), 45 (2001) [Tech. Phys. properties of the Y3/4Lu1/4Ba2Cu3O7 + Y3Fe5O12 com- Lett. 27, 952 (2001)].

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 TRANSPORT AND MAGNETIC PROPERTIES 1873

8. A. G. Mamalis, S. G. Ovchinnikov, M. I. Petrov, et al., 22. S. V. Kuplevakhskiœ and I. I. Fal’ko, Fiz. Met. Metall- Physica C (Amsterdam) 364Ð365, 174 (2001). oved. 62 (1), 13 (1986). 9. J. J. Calabrese, M. A. Dubson, and J. C. Garland, J. Appl. 23. A. S. Borukhovich, Usp. Fiz. Nauk 169 (7), 737 (1999) Phys. 72 (7), 2958 (1999). [Phys. Usp. 42, 653 (1999)]. 10. M. I. Petrov, D. A. Balaev, K. A. Shaœkhutdinov, and 24. M. Fogelstrom, Phys. Rev. B 62 (17), 11812 (2000). K. S. Aleksandrov, Fiz. Tverd. Tela (St. Petersburg) 41 25. Y. Tanaka and S. Kashivaya, J. Phys. Soc. Jpn. 69, 1152 (6), 969 (1999) [Phys. Solid State 41, 881 (1999)]. (2000). 11. M. I. Petrov, D. A. Balaev, K. A. Shaihutdinov, and 26. Yu. Izyumov, Yu. Proshin, and M. G. Khusainov, Usp. K. S. Aleksandrov, Supercond. Sci. Technol. 14 (9), 798 Fiz. Nauk 172 (2), 113 (2002) [Phys. Usp. 45, 109 (2001). (2002)]. 12. J. Jung, M. A.-K. Mohamed, I. Isaak, and L. Friedrich, 27. O. Bourgeois, P. Gandit, A. Sulpice, et al., Phys. Rev. B Phys. Rev. B 49 (17), 12188 (1994). 63, 064517 (2002). 13. B. I. Smirnov, T. S. Orlova, and N. Kudymov, Fiz. Tverd. 28. M. Schock, C. Surgers, and H. von Lohneysen, Eur. Tela (St. Petersburg) 36, 3542 (1994) [Phys. Solid State Phys. J. 14, 1 (2000). 36, 1883 (1994)]. 29. V. V. Ryazanov, V. A. Oboznov, A. Yu. Rusanov, et al., 14. M. I. Petrov, D. A. Balaev, S. V. Ospishchev, et al., Phys. Phys. Rev. Lett. 86 (11), 2427 (2001). Lett. A 237 (1Ð2), 85 (1997). 30. A. Barone and G. Paterno, Physics and Applications of 15. M. I. Petrov, D. A. Balaev, K. A. Shaœkhutdinov, and the Josephson Effect (Wiley, New York, 1982; Mir, Mos- S. G. Ovchinnikov, Fiz. Tverd. Tela (St. Petersburg) 40 cow, 1984). (9), 1599 (1998) [Phys. Solid State 40, 1451 (1998)]. 31. A. D. Balaev, Yu. V. Boyarshinov, M. M. Karpenko, and 16. M. I. Petrov, D. A. Balaev, S. V. Ospishchev, and K. S. B. P. Khrustalev, Prib. Tekh. Éksp., No. 3, 167 (1985). Aleksandrov, Fiz. Tverd. Tela (St. Petersburg) 42 (5), 32. V. Ambegaokar and B. Halperin, Phys. Rev. Lett. 22, 791 (2000) [Phys. Solid State 42, 810 (2000)]. 1364 (1969). 17. M. I. Petrov, D. A. Balaev, K. A. Shaœkhutdinov, and 33. M. Tinkham, Phys. Rev. Lett. 61 (14), 1658 (1988). S. I. Popkov, Pis’ma Zh. Éksp. Teor. Fiz. 75 (3), 166 (2002) [JETP Lett. 75, 138 (2002)]. 34. Physical Properties of High Temperature Superconduc- tors, Ed. by D. M. Ginzberg (World Sci., Singapore, 18. M. I. Petrov, D. A. Balaev, and K. A. Shaihutdinov, Phys- 1989; Mir, Moscow, 1990). ica C (Amsterdam) 361, 45 (2001). 35. S. Krupicka,ˇ Physik der Ferrite und der Verwandten 19. L. N. Bulaevskiœ, V. V. Kuziœ, and A. A. Sobyanin, Pis’ma Magnetischen Oxide (Vieweg, Braunschweig, 1973; Zh. Éksp. Teor. Fiz. 25 (7), 314 (1977) [JETP Lett. 25, Mir, Moscow, 1976), Vol. 1. 290 (1977)]. 36. M. Charalambous, J. Chaussy, and P. Lejay, Phys. Rev. 20. L. N. Bulaevskii, V. V. Kuzii, and S. V. Panyukov, Solid B 45 (9), 5091 (1992). State Commun. 44 (4), 539 (1982). 21. S. V. Kuplevakhskiœ and I. I. Fal’ko, Fiz. Nizk. Temp. 10, 691 (1984) [Sov. J. Low Temp. Phys. 10, 361 (1984)]. Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1874Ð1883. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1784Ð1793. Original Russian Text Copyright © 2003 by I. G. Kuleev, I. I. Kuleev, Arapova, Sabirzyanova.

SEMICONDUCTORS AND DIELECTRICS

Density of States in the Impurity d Band and Inelastic Electron Scattering by a System of Mixed-Valence Iron Ions in HgSe : Fe Crystals I. G. Kuleev, I. I. Kuleev, I. Yu. Arapova, and L. D. Sabirzyanova Institute of Metal Physics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoœ 18, Yekaterinburg, 620219 Russia e-mail: [email protected] Received February 19, 2003

Abstract—The resistivity and the Hall coefficient of HgSe : Fe crystals with various iron content are experimentally studied in the temperature range 1.3 ≤ T ≤ 300 K and in magnetic fields up to 60 kOe. The temperature dependences of the density and mobility of conduction electrons in these crystals are determined. The influence of spatial charge ordering in the system of mixed-valence iron ions on impurity states in HgSe : Fe crystals is considered. The density of states in the impurity d band is theoretically analyzed, and inelastic electron scattering in which bi- and trivalent iron ions are recharged is discussed. It is shown that the experimentally detected features in the dependence of the density and mobility of electrons on temperature and iron impurity content can be explained by the influence of Coulomb correlations in the mixed-valence iron ion system on the structure of the impurity d band. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION states. In the extreme case of strong interimpurity or sÐ Recently, much attention has been paid to the phys- d hybridization overcoming the Coulomb interaction, a ical properties of mixed-valence (MV) systems, in state with variable valence arises in the system of impu- which dopants or host ions can be in at least two differ- rities. In this case, all impurity lattice sites feature an ent charge states [1Ð5]. Coulomb repulsion between identical fractional charge and the spatial distribution ions in identical charge states causes spatial correla- of impurity charges is uniform. In HgSe : Fe crystals, tions in their arrangement. Examples of MV systems the d level width caused by sÐd hybridization is smaller are lanthanum manganites (La Sr MnO ), mercury than 0.1 meV (judging from the electron spin resonance 1 Ð x x 3 3+ selenide doped with transition 3d elements (iron, chro- (ESR) data for Fe ions [11Ð13]), while the Coulomb mium, cobalt), gallium arsenide delta-doped with sili- correlation energy of d holes is two orders of magnitude con or tin, and 2D structures made on the basis of the higher. Therefore, the local nature of d states persists at latter. Mercury selenide crystals doped with transition low temperatures, and the conditions necessary for ini- elements represent a convenient model system for tiation of spatial correlations in the MV iron ion system studying the effects associated with spatial charge in HgSe : Fe compounds are met. When analyzing the ordering in systems of mixed-valence impurities. The kinetic effects in HgSe : Fe crystals at low tempera- influence of spatial charge ordering in the MV system tures, we use the short-range correlation model [7, 8, on kinetic effects is most pronounced in HgSe : Fe crys- 14Ð16] based on the assumption that the energy of tals at low temperatures [5Ð10]. One of the most interimpurity Coulomb correlations in the MV iron ion remarkable anomalies is a significant increase in the system significantly exceeds the sÐd hybridization electron mobility observed in the range of liquid- broadening of the impurity level [5, 11Ð13]. It should helium temperatures with increasing iron content [6]. be noted that, although the sÐd hybridization interac- Iron impurities in HgSe : Fe crystals form a resonant tion is weak, it can play an important role in inelastic ε ≅ scattering of conduction electrons, resulting in iron ion d level at the energy d 210 meV reckoned from the conduction band bottom; i.e., this level is positioned in recharging [10]. the continuous energy spectrum of the perfect crystal. The Fe3+ÐFe2+ mixed-valence state arises in these When considering resonant d states, two competing ≅ × 18 Ð3 crystals at contents NFe > N* 4.5 10 cm , when processes [5] should be taken into account: (i) electron ε ≅ delocalization due to hybridization of localized impu- the Fermi level reaches the iron d level ( d 210 meV) and is fixed at it [5]. As the iron impurity content rity states with both band and other impurity states and 3+ (ii) localization of electrons (d holes, i.e., positive increases further, the Fe donor concentration remains 2+ charges on iron ions) caused by their Coulomb repul- unchanged (N+ = N*) and only the content of Fe ions sion tending to retain the local nature of impurity d (neutral in the lattice) increases (N0 = NFe Ð N+). In the

DENSITY OF STATES IN THE IMPURITY d BAND 1875 system of Fe2+ and Fe3+ ions having the same energy, 6 positive charges on iron ions (d holes) can be redistrib- uted over the lattice sites occupied by iron ions. For this reason, the Coulomb repulsion of d holes in the MV 5 system causes spatial correlations in their arrangement. –3 The higher the iron content NFe, the larger the number 4 of lattice sites for the d hole redistribution and the cm higher the degree of ordering of the correlated Fe3+ ion 18 1

, 10 3 system (CIS). Experimental studies of HgSe : Fe crys- e 2 tals (see reviews [5, 7]) have shown that anomalous run n 3 of the dependences of kinetic effects on the iron impu- 4 rity concentration and temperature T is caused by atyp- 2 ical scattering of electrons by the spatially correlated system of MV iron ions. 1 1 10 100 1000 This study is devoted to the influence of spatial , 1018 cm–3 charge ordering in the MV system on the structure of the NFe impurity d band and on electron transport in HgSe : Fe crystals. The conventional approach based on the Fig. 1. Dependence of the electron density ne on the iron impurity content NFe in HgSe : Fe crystals at T = 4.2 K: Anderson model [1, 17], taking into account electronÐ (1) experimental data from [6], (2) [11], (3) [12], and (4) electron correlations on one impurity, does not allow this study. The solid line is calculated using formulas (20) one to solve this problem, since the Coulomb interac- and (21). tion between impurities in identical charge states is disregarded in this model. In this paper, we suggest a sim- ≤ ≤ ple method which makes it possible to consider the 1.3 T 300 K in magnetic fields of up to 60 kOe. The influence of the Coulomb interimpurity correlations on temperature dependences of the density ne and mobility the structure of the impurity d band and on the kinetic µ of band electrons were also studied. effects in HgSe : Fe crystals at low temperatures. It Figure 1 shows the dependence of the electron den- should be noted that the interaction between oppositely sity ne on the iron content NFe. We can see that the elec- charged donors and acceptors in zero-gap compensated ≈ × tron density is stabilized at the level ne (4.7Ð4.8) HgCdTe-type semiconductors brings about the forma- 1018 cmÐ3 at N ≥ 5 × 1018 cmÐ3, which conforms to the tion of a Coulomb pseudogap at the Fermi level, i.e., a Fe available published data [5, 6]. This n (N ) dependence minimum density of impurity states with g (ε ) > g (ε ) e Fe i F e F indicates that the Fermi level is fixed at the iron donor [18, 19]. The analysis of the ESR data for Fe3+ ions in level. A certain spread in the values of ne is caused by HgSe : Fe crystals carried out in [10, 20] within the intrinsic defects, whose concentration is generally ~(1Ð weak Coulomb correlation model showed that there × 18 Ð3 was a minimum in the density of d states at the Fermi 2) 10 cm [5, 21]. According to the currently available data [6, 7], the energy of the unperturbed iron level; however, it was unclear whether a gap or a ≅ pseudogap was formed at the Fermi level. donor level is Ed0 210 meV, which corresponds to ne = ≅ × 18 Ð3 In Section 2, the results of measurements of the N* 4.5 10 cm . Below, we show that the fact that resistivity and Hall coefficient for HgSe : Fe crystals the electron density exceeds N* can be explained by the with various iron contents are presented. In Section 3, formation of the impurity d band and an increase in the the correlation potential is analyzed and the parameters Fermi level due to spatial charge ordering in the system defining the impurity d band structure are determined. of MV iron ions. It is shown that, due to the CIS ordering in HgSe : Fe in Figure 2 shows the temperature dependences of the the presence of strong Coulomb correlations, a correla- electron density ne in HgSe : Fe crystals of various iron tion gap appears in the density of impurity d states content. We can see that the electron density slightly between the populated (Fe2+) and empty (Fe3+) states, depends on temperature for all the samples. We note ε where gd( ) = 0, and resonance electron scattering is that ne increases for samples with NFe > 3N* and totally suppressed. In Section 4, the density of impurity decreases for samples with N* < NFe < 2N* as the tem- d states is calculated. Section 5 is devoted to a theoret- perature is increased. A similar behavior of the ne(T) ical analysis of inelastic d-hole scattering in which Fe2+ dependences for HgSe : Fe crystals was detected in [6], 3+ and Fe ions are recharged. where the experimental and calculated values of ne were compared. When calculating ne(T), the density of states in the conduction band for the Kane nonparabolic 2. EXPERIMENTAL RESULTS model was used. An analysis carried out in [6] showed ρ The resistivity and the Hall coefficient R of HgSe : that the calculated and measured values of ne(T) can be × 18 ≤ ≤ × ε Fe crystals of various iron content (2 10 NFe 2 brought into conformity by assuming that Fe varies 1021 cmÐ3) were measured in the temperature range with temperature. We show that these features can be

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1876 KULEEV et al.

6.0 0.30 1 2 5.5 3 2 0.25 4 –3 V s/cm cm 0.20 –4 18 5.0 , 10 , 10

e 1 –1 n 0.15 µ 2 4.5 3 4 0.10 5 4.0 0 25 50 75 100 0 10 20 30 40 50 T, K T, K Fig. 3. Temperature dependences of the inverse mobility µÐ1 − Fig. 2. Temperature dependences of the electron density n in for HgSe : Fe crystals with different values of N (1018 cm 3): 18 Ð3e Fe HgSe : Fe crystals of various iron content NFe (10 cm ): (1) 5, (2) 20, (3) 40, (4) 50, and (5) 100. Dashed and solid (1) 5, (2) 10, (3) 50, and (4) 100. The solid lines are calcu- lines are the µÐ1(T) dependences calculated using formulated using formulas (16)Ð(22). las (23)Ð(25) and (25)Ð(28), respectively.

3+ explained by the inﬂuence of spatial correlations in the tials of other Fe ions [16], and to the ﬁeld Ue(Ri) of system of MV iron ions on the structure of the impurity the homogeneous gas of conduction electrons: d band. () () () Figure 3 shows the temperature dependences of the URi = Us Ri + Ue Ri , inverse mobility µÐ1 of electrons in HgSe : Fe crystals ≥ × 18 Ð3 () ()()() with an iron content NFe 5 10 cm in the temper- Us Ri = ∑VRij g++ Rij Ð 1 , (1) ature range 1.3 ≤ T ≤ 50 K. For all samples at very low ji≠ ≤ temperatures (T 10 K), there are temperature ranges () πε ()2 in which the electron mobility µ(T) is independent of Ue Ri = Ð4 k rs/R+ , temperature; the width of these intervals increases with 3+ where V(Rij) is the potential produced by the jth Fe NFe. In the temperature range 10 < T < 35 K, the experimental data follow the law µÐ1(T) ~ Tν, where ν < 1. In ion at the point Ri, rs is the ThomasÐFermi screening ε 2 χ Section 5, it is shown that the exponent ν is close to 0.5. length, k = e / R+ is the Coulomb interaction energy µÐ1 Ð1/3 At T > 35 K, a more pronounced increase in with between ions spaced at an average distance R+ = N+ µÐ1 temperature is observed [ (T) ~ T], which, as follows (N is the Fe3+ ion concentration, χ is the permittivity), from [7, 15], is caused by the contribution of electronÐ + and gαβ(r) are the partial pair correlation functions phonon scattering. characterizing the spatial distribution of MV iron ions Below, we theoretically analyze the impurity d band in the Fe2+ÐFe3+ system [22, 23]. structure of the system of MV iron ions and inelastic d- hole scattering in which iron ions are recharged. In the case of the screened Coulomb interaction of impurity ions, we have

∞ 3. CORRELATION POTENTIAL 2e2 q2dq sin()qr U ()r = ------()S ()q Ð 1 ------, (2) AND THE STRUCTURE s χπ∫ 2 Ð2 ++ q + r qr OF THE IMPURITY d BAND 0 s We consider the influence of spatial charge ordering 3+ where S++(q) is the structure factor of Fe ions. Here- in the system of MV iron ions on the impurity d band after, we use the structure factors Sαβ(q) for the pene- structure in HgSe : Fe crystals. With no ordering 3+ trating solid sphere model, which are defined in [16]. effects, the random fields of Fe ions cause an energy The potential U (r) redistributes d holes in the Fe2+Ð spread of the d states and the formation of an impurity s Fe3+ system so that the potential energy is minimal at band with a width close to 10 meV. The Fermi level sep- 3+ arates empty d states (Fe3+) from populated ones (Fe2+). the sites of Fe donors. 3+ When ordering takes place, any Fe ion at a point Ri is Let us consider the change in the Coulomb interac- 3+ subjected to the correlation potential Us(Ri), which is tion energy caused by ordering of the Fe ions. The ∆ 3+ the result of the self-consistent influence of the poten- energy gain Ec(NFe) per Fe ion for the ordered distri-

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 DENSITY OF STATES IN THE IMPURITY d BAND 1877 bution (in comparison with the random distribution) Us(y) can be written in terms of Us(0) as [14, 24] 10 ∆ ∆ Ek 1 () 5 Ec ==------Ð---Us 0 , N+ 2 ∞ (3) –2.0 –1.0 –0.5 0 0.5 1.0 2.0 y 2e2 q2dq U ()0 = ------()S ()q Ð 1 . –5 E s χπ∫ 2 Ð2 ++ k q + r 0 s –10

For the Coulomb interaction of Fe3+ ions, the quantity –15 ∆ Ej Ec is immediately expressed in terms of the packing parameter η as [25] –20 I d 0.5Ð 0.1η + 0.05η2 U ()0 = Ð4πε ------------, (4) j' ijk s kR 12+ η +  II j' ijk π 3 where η = ---n d and d = r is the solid-sphere diame- 3+ 2+ 6 + c Fe Fe ter. For the screened Coulomb interaction, the quantity Fig. 4. Potential Us(y) and the energy diagram of inelastic Us(0) can be analytically expressed in terms of the transitions in a one-dimensional short-range-order cluster parameters η and λ = d/r as [14, 26] of Fe3+ and Fe2+ ions at different values of the iron content s 18 Ð3 χ NFe (10 cm ): (1) 5 and (2) 50 ( = 20). The dashed lines are the approximate Us(y) dependences calculated using () πε d {}()λ λÐ2 formulas (7) and (8) for the corresponding iron contents. Us 0 = Ð4 k------G Ð , ∆E is the energy of the inelastic transition of a d hole from R+ ji point Ri to point Rj equal to the energy difference for two λ ()λ conﬁgurations (I and II) of the short-range-order cluster. ()λ L G = ------η[]()λ ()λ () λ-, 12 L + S exp (5) L()λ = 12η[]()λ1+ 0.5η ++12η , quadratic terms in this expansion, we obtain from formula (2) S()λ = ()1 Ð η 2λ3 + 6η()λ1 Ð η 2 () () ∆ () Us r = Us 0 + Us r , +18η2λ Ð 12η()12+ η . ∆ ()≅α α()2 ()2 Us r NFe r = * NFe y , (7) In [8], by comparing the experimental and calculated 2 η α*()αN ==()N d , yr/d. values of the mobility, the empirical (NFe) dependence Fe Fe was determined to be We invoke the Laplace transformation similarly to as was done in [26]. For the screened Coulomb interac- η N α ηη Fe tion, we express the coefﬁcient (NFe) in terms of the = ∞ 1 Ð.expÐη------(6) η ∞ N+ packing parameter and the screening length rs: ∞ η 2 4 At high iron impurity contents, the function (NFe tends e q dq α()N = ------()1 Ð S ()q to a constant: η(N ∞) = η∞. However, the relation Fe πχ∫ 2 Ð2 ++ Fe 3 q + r η ≈ 0 s (NFe) 0.74N0/NFe was derived in [16] from the bal- (8) πε ance equation for d holes and neutral centers (with con- 4 k 1 () = ------+ ------Us 0 . centration N0) in a short-range cluster surrounding an 3R 2 6r 2 3+ + s Fe ion at N0 < N+. This relation is asymptotically exact in the limit N0 0. It follows from empirical equa- As can be seen from Fig. 4, Eqs. (7) and (8) yield a good tion (6) that, for N Ӷ N , we have η(N ) ≈ 2η∞N /N , approximation for calculating the correlation potential 0 + Fe 0 Fe ∆U (y) in the entire correlation sphere (y ≤ 1). which corresponds to the value η∞ ≈ 0.37 in Eq. (6). For s this reason, we use this value of η∞ in what follows. Formulas (4)Ð(6) allow one to analyze the change in ∆ the Coulomb energy Ec associated with ordering of At small values of r (r < rc), the correlation potential the Fe3+ ions as a function of the iron content and the Us(r) can be expanded in powers of r. Keeping only the screening length. Such an analysis was carried out in

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1878 KULEEV et al.

2+ V1, meV Therefore, the energy levels Edk of Fe ions that are 20 neutral in the lattice are distributed over the d0 band of impurity states. The width of this band is determined by the energy necessary to transfer a d hole from the 15 center (Ri = 0) to the boundary of the correlation sphere (Rk = rc): () () 10 WU= s rc Ð Us 0 . (10)

To determine the energy Ed0 of the unperturbed d level 1 (or the positions of the band edges with respect to the 5 3 2 unperturbed level), the potential of the central Fe3+ ion 6 5 4 should be included and the point r0 at which the derivative of the total potential with respect to the radius vec- –1.5 –1.0 –0.5 0 0.5 1.0 1.5 y tor is zero should be determined. The total potential V1(r) = Us(r) + V(r) is shown in Fig. 5 for various iron impurity contents. –5 We can see from Fig. 5 that the behavior of the Fig. 5. Potential V (y) at different values of the iron content potential V1(r) qualitatively changes as the degree of 18 Ð3 1 NFe (10 cm ): (1) 5, (2) 5.2, (3) 5.3, (4) 6, (5) 10, and spatial ordering in the system of MV iron ions (6) 50. The dashed lines are the V1(y) dependences calcu- increases: V1(r) monotonically decreases with increas- lated from the approximated formulas (7) and (8) at NFe = 50 × 1018 cmÐ3. ing distance in the region of weak correlations η ≤ ≤ × 18 Ð3 [ (NFe) 0.125 or NFe 5.2 10 cm ], remaining positive over the entire range of values of r. As the iron ∆ × [14, 15]. We note that the steep increase in Ec in the content NFe increases and becomes higher than 5.2 × 19 Ð3 18 Ð3 η range N* < NFe < 1 10 cm (see [14, Fig. 5]) is 10 cm [ (NFe) > 0.125], the V1(r) dependence exhib- caused by the fact that the distances between d holes its a minimum, whose depth increases with the iron that are the closest to each other increase initially most content. When passing to the region of strong spatial η × 18 Ð3 significantly with charge ordering in the system of MV correlations [ (NFe) > 0.185, NFe > 5.8 10 cm ], the iron ions. This yields a maximum gain in the Coulomb potential V1(r0) becomes negative. As the iron content energy and causes the formation of correlation spheres increases further, the depth of the V1(r0) minimum surrounding each Fe3+ ion, which contain no other pos- 19 Ð3 increases and, at NFe > 10 cm , becomes approxi- itive charges [14]. This circumstance makes it possible mately constant, which is characteristic of a strongly to use the model of solid spheres to describe the corre- 3+ correlated Coulomb liquid. In this case, the value of y0 lation properties of the system of Fe ions statistically. is approximately 0.65. It is clear that the force acting on In this model, Fe3+ ions cannot be arranged at distances a probing charge placed at the point r0 is zero shorter than the correlation sphere radius rc = d (d is the (dV (r)/dr = 0); therefore, the energy of the d level diameter of a solid sphere). The degree of ordering in 1 0 remains unperturbed. The upper (E1) and lower (E2) such a system is characterized by the packing parame- boundaries of the impurity d band are given by ter η equal to the ratio of the volume occupied by the 0 ∆ () solid spheres to the total volume of the system. E1 = Ed0 + Us r0 , (11) The potential Us(Rj) prevents d-hole jumps from E2 = E1 Ð W. Fe3+ onto Fe2+ ions, thereby promoting their localiza- | | tion at charged centers (Fig. 4). Due to the random dis- Figure 6 shows the dependences of Us(0) , W, and 2+ α tribution of Fe ions over lattice sites and, therefore, in * on the iron content. We can see that the impurity d0 short-range-order clusters, the probability of multipar- band width tends to zero as NFe N* (N0 0), ӷ ticle transitions of d holes is low and the contribution while at high iron contents (NFe N*) the band width from these transitions can be neglected at low tempera- tends to a constant which exceeds 14.6 meV for χ = 27 χ α tures. Here, we restrict our consideration to single-par- and 18.6 meV for = 20. The coefficient *(NFe) tends ӷ ticle transitions. Let us assume that a d-hole transition to zero as NFe N*; when NFe N*, this coefficient in a short-range-order cluster from the Fe3+ ion located becomes a constant equal to 12.2 meV for χ = 27 and 2+ χ | | at the point Ri onto an Fe ion at the point Rj proceeds 15.5 meV for = 20. The parameter Us(0) exhibits a in the static field produced by the other Fe3+ ions. The similar run: at high iron contents, it becomes a constant inelasticity energy ∆E of such a transition is equal to which is larger than 15.9 meV for χ = 27 and 20 meV jk χ the energy difference of the two configurations (Fig. 4): for = 20. We note that correlation potential Us(r) is a potential well for d holes and a potential barrier for ∆ () () Eik ==Ei Ð Ek URi Ð URk . (9) electrons. Hence, a conduction electron must overcome

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 DENSITY OF STATES IN THE IMPURITY d BAND 1879

| | 3+ the potential barrier Us(0) to be trapped by an Fe donor center. As can be seen from Figs. 4 and 6, reso- 20 1 nant trapping of a conduction electron onto the d level 2 3+ ӷ of an Fe ion at NFe N* requires a high activation 3 energy U (0) ~ 20 meV, in contrast to scattering with 15 2a

s *, meV α

recharging of d centers [10]. Thus, Coulomb interimpu- , rity correlations in the system of MV iron ions suppress W 10 , resonant electron scattering. The role of resonant scat- | (0) tering of conduction electrons involving the d states in s U HgSe : Fe compounds at low temperatures has been | 5 actively discussed in the literature [5, 13, 20]. Inclusion of this scattering yielded catastrophically low values of the electron mobility in HgSe : Fe [5]. 0 N* 10 20 30 40 50 2+ 18 –3 Since iron ions substitute for Hg ions at lattice NFe, 10 cm sites, there is a minimum length of d-hole jumps, ∆ ∆ | | Rik > Rmin (Rmin ~ a0, a0 is the lattice constant). If Fig. 6. Dependences of the parameters (1) Us(0) , (2, 2a) W, and (3) α* on the iron impurity content calculated for the Coulomb interimpurity correlations at the temperature of η χ χ sample preparation (T ~ 103 K) are taken into account, parameters (1Ð3) ∞ = 0.37 and = 20 and (2a) = 27. ∆ ∆ ∆ we obtain Rmin ~ 2a0. Therefore, Eki > Emin = . If follows from expressions (7) and (8) that order in the system of correlated Fe3+ ions becomes 3+ ∆α≈ ()∆N ()R 2. (12) “frozen,” and, in a certain sense, the system of Fe ions Fe min transfers to a metallic-glass state [28] (a strongly corre- ∆ 3+ The quantity tends to zero as NFe N* and lated Coulomb liquid of Fe ions). × 19 Ð3 becomes constant at NFe > 3 10 cm . In this concen- ∆ ∆ Ð7 χ tration range, ~ 9.3 K at Rmin ~ 10 cm and = 20 4. SPATIAL CORRELATIONS AND THE DENSITY and ∆ ~ 7.3 K for χ = 27. The estimate of ∆ obtained in OF STATES IN THE IMPURITY d BAND [27] within the model of a linear chain of Fe3+ ions is of The density of states in the impurity d band is calcu- the same order of magnitude. The existence of a mini- lated in two steps. First, we determine the density of mum energy for d-hole jumps means that the populated 2+ states in the d band in the coordinate representation band of d0 states (Fe ) is separated from the empty G (r) for a known distribution of charges in the short- 3+ ∆ 0 band of d+ states (Fe ) by the finite energy . Thus, range-order cluster of a given Fe3+ ion. Then, using for- spatial charge ordering in the system of MV iron ions mulas (7)Ð(11), we determine the distribution of d0 removes the degeneracy of the d0 and d+ states. In this states in the correlation sphere according to the above system, a new mechanism of inelastic electron scatter- analysis of the correlation potential, ing becomes possible at finite temperatures, in which bi- and trivalent iron ions are recharged [10]. Since the E Ð E = α()N r2, (14) ӷ 1 dj Fe band of d+ states at NFe N* is much narrower than the and find the density of states in the d band in the energy d0 band, we approximate the former band by a sharp d 0 level with the energy representation using the relation ∆ G ()r dr= G ()E dE . Ed+ = E1 + . (13) 0 0 dj dj Thus, ordering of the Fe3+ ions in the MV system from which we obtain results in the formation of a correlation gap between the () () G0 Edj = G0 r dr/dEdj, populated and empty bands of d states rather than of a (15) []()α()1/2 Éfros–Shklovskiœ Coulomb pseudogap [18], as was rE= 1 Ð Edj / NFe . assumed in [13, 20]. Based on the analysis performed above, we can conclude that the interimpurity Coulomb To determine G0(r), we construct a correlation 3+ correlations totally suppress resonant scattering: the sphere of radius rc around each Fe ion, average over ε density of d states at the Fermi level gd( F) is zero, and the ensemble of correlation spheres, and come to a ∆ ӷ Γ Γ d, d is the d-level width due to sÐd hybridization; description of the system of MV iron ions within the Γ d < 0.1 meV, according to [11Ð13]). The behavior of continuous-medium model. To analyze the distribution ӷ the carrier mobility in HgSe : Fe at NFe N* and T of charged and neutral particles in a short-range-order 0 also becomes clear (Fig. 3). At temperatures T < ∆, d- cluster, it is convenient to use the partial radial distribu- 2 hole jumps are frozen and the electron mobility tion functions Nαβ(r) = 4πr Nβgαβ(r) (see [16, Fig. 2]). becomes temperature-independent, as was observed The quantity Nαβ(r)dr yields the number of particles of experimentally (see Fig. 3 and [6, 27]). The short-range type β(0, +) within a spherical layer of radius r and

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1880 KULEEV et al.

5 with the density N0. It is easy to verify that, at r < R+/2, the function G0(r) is equal to the radial distribution

4 function N+0(r); at R+/2 < r < R+/2 , it decreases in

proportion to r(3R+/4 Ð r); and in the interval R+/2 < 3 1 ) r < R+ 3 /2, it decreases quadratically with distance E ( 0 and vanishes at r = R 3 /2.

G + 2 Thus, in the case of a spherically symmetric charge 2 distribution in the averaged short-range-order cluster in 1 the range 0 < y < 1/2, we have 3 () π 2 2 4 G0 r = 4 NFed y . (16) 0 202 206 210 214 216 Using Eq. (15) in the energy representation, we E, meV obtain

Fig. 7. Density of states in the d band in the energy represen- 3 E1 Ð Edj 0 18 Ð3 G ()E = 2πd N ------, tation for various values of the iron content NFe (10 cm ): 0 dj Fe()α ()3/2 (1) 50, (2) 20, (3) 10, and (4) 6; η∞ = 0.37 and χ = 27. Dashed * NFe (16a) lines are calculated from formula (16a). ≤ ()1 α Edj Ed = E1 Ð 0.25 *. thickness r + dr if a particle of type α(0, +) is placed at In the range 0.5 < y < 1, based on the results for an SC its center. In the case of a random distribution of parti- lattice, we assume that, at r < R/2 (where R is the distance to a charged center located in the spherical layer), αβ cles, we have g (r) = 1 and the number of particles in 3+ Ω all d0 states correlate in position with the central Fe a volume r is equal to the product of the concentration and the volume. Since d holes and neutral centers can ion, while at r > R/2, the d0 states correlate with the ion only exchange sites, the d hole sites in the correlation in the spherical layer d < r < 2d. Therefore, averaging sphere will be occupied by neutral centers. Therefore, over the distribution of charged centers in the spherical ≤ layer d < r < 2d can be carried out as follows: at N0 N1+, the local density of d holes in the correlation sphere is zero and the density of neutral centers is 2d defined by the total iron concentration (see [16, Fig. 2]). 〈〉, 1 , π 2 G0(r ) = G0()rR = ------〈〉- ∫ G0()rRg++()4R R dR, The reverse situation arises in the region of the peak g++ corresponding to the first coordination shell: the den- d sity of neutral centers is zero, while the density of d 2d 〈〉 π 2 () holes is determined by the total iron concentration. For g++ = ∫ 4 R g++ R dr, this reason, both the function g (r) describing correla- ++ d tions in the system of charged centers (++) and the (17) function g+0(r) describing the correlation in the system (+0) significantly deviate from unity, corresponding to  π 2 2 < (), 4 NFed y , yR/2d a random distribution of particles in the short-range- G0 rR =  (18) order cluster. Thus, the Coulomb repulsion of d holes in 0, yR> /2d. the system of MV iron impurities gives rise not only to 3+ To simplify further calculations, we restrict our analy- spatial ordering of charged centers (Fe ions) but also sis to the first-order approximation in the expansion in to correlations in the system (+0). powers of the system density and replace g++(R) by a step function: g (R) = 0 at R < d and g (R) = 1 at R > To determine the function G0(r), we construct a ++ ++ WignerÐSeitz cell in the correlation sphere; this cell is d. In this case, Eq. (17) takes the form a sphere of radius r = d/2. All neutral centers lying 2d 1 2 inside this sphere are correlated in position with the G ()r = ------G ()rR, 4πR dR, 3+ 0 ∆ ∫ 0 central Fe ion; therefore, the function G0(r) is equal to V c d (19) the radial distribution function N+0(r). At r > d/2, the 4π 3 neutral centers correlate not only with the central ion ∆V = 7------d , but also with the iron ions lying in the spherical layer c 3 * ∆ d < r < r1 . In this case, the function G0(r) decreases where Vc is the volume of the spherical layer d < r < and then vanishes at r = d. As an illustration, we con- 2d. Expressions (14)Ð(19) allow us to determine the 3+ sider the case of a simple cubic (SC) lattice of Fe ions density of states in the d0 band for various iron contents over which neutral centers are uniformly distributed in the energy representation (Fig. 7). It is noteworthy

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 DENSITY OF STATES IN THE IMPURITY d BAND 1881 that the density of states has a square-root dependence less than 0.1%. Therefore, we calculated ne from on energy [see relation (16a)] in the energy range 0 < expression (21), which is valid for a degenerate elec- ≤ E1 Ð Edj 4 meV. We can see from formulas (14) and tron gas. The results of the calculation are shown in (15) that this feature of G0(Edj) is related to the qua- Fig. 2 by solid lines and are seen to conform well to the ∆ dratic dependence of Us on the radius r within the cor- experimental data. relation sphere. The function G(Edj) peaks at the energy ≈ Thus, the above analysis of the impurity band struc- Edj Ed0. This is an indirect confirmation of the fact that ture and the approximate calculation of the density of the function G (r, R) was modeled correctly. 0 states in the d0 band have allowed us to correctly interpret The results obtained above make it possible to ana- the results of the Hall effect measurements in HgSe : Fe lyze the data from Hall effect measurements. At T = 0 K, crystals in a wide temperature range. It was shown that the Fermi level separates the populated and empty states the features in the dependence of the electron density of iron ions and is positioned in the correlation gap: on temperature and iron content can be explained in ∆ terms of the influence of Coulomb correlations in the ε ≈ ∆ () F0 = E1 Ð --- Ed0 + Us r0 . (20) system of MV iron ions on the density of states in the 2 impurity d band. Within the two-band Kane model, the electron density is ε ()ε ε 5. INELASTIC SCATTERING OF D HOLES () F0 1 + F0/ g AND THE TEMPERATURE DEPENDENCE ne NFe = N*------ε ()ε ε . (21) d0 1 + d0/ g OF THE CARRIER MOBILITY ε It is clear that the Fermi level is not fixed at the level d0 At finite temperatures, the interaction of d holes and an increase in the degree of spatial ordering with phonons or conduction electrons causes migration 2+ 3+ removes the degeneracy of the states d0 and d+ in the of d holes between Fe and Fe ions and gives rise to impurity band and causes an increase in the Fermi level. the following two effects. First, a conduction electron In this case, the electron density increases from N* to or phonon colliding with an Fe3+ ion can transfer to it ≅ × 18 Ð3 ne 4.8 10 cm . The calculated density ne(NFe) is the energy necessary for the transition of a hole to a shown in Fig. 1 by solid lines. As is evident from Fig. 1, neighboring neutral center with higher energy (Fig. 4). there is satisfactory agreement with the experimental For conduction electrons, it is this inelastic scattering data. A certain scatter of the ne values is caused by mechanism that causes recharging of d centers and intrinsic defects, whose typical density is ~1018 cmÐ3 should be taken into account. Second, as the tempera- [5, 21]. The temperature dependence of the electron ture increases, inelastic jumps of d holes cause chaoti- density at a fixed iron content is determined from the zation of the system of Fe3+ ions; the degree of spatial electrical neutrality equation (see, e.g., [29]): ordering and the solid-sphere diameter decrease. In this E case, there is a finite probability that a d hole will be at 1 the neutral centers nearest to a given donor. This effect () ()ε ε ()ε ()()ε ne T = N+F+ d+ + ∫ d dgd0 d 1 Ð F0 , can be taken into account in the approximation of so- called soft spheres [30, 31]: E2 E1 Ð1 () ()∆ ()  dT = d0 1 Ð rT/d0 . (23) ()ε ()ε  ε ()ε gd0 d = NFeG0 d ∫ d dG0 d ,  (22) Here, d is the solid-sphere diameter at T = 0 and ∆r(T) E2 0 describes the variation of the correlation sphere radius ε ε Ð1 ()ε F Ð d+ with temperature. The ∆r(T) dependence can be easily F+ d+ = 12+ exp------, kBT estimated by assuming that all the states of the d+ and d bands satisfying the condition ∆U (∆r ) ≤ (3/2)k T ε Ð ε Ð1 0 s ij B ()ε 1 d F are equiprobably populated at finite temperatures. In F0 d = 1 + --- exp------. 2 kBT this case, expression (7) immediately yields ∆r(T) ~ ()⁄ α 1/2 ∆ Here, the functions F+ and 1 Ð F0 allow for population 3kBT * . Since there is a finite energy gap of the d+ and d0 states by d holes at finite temperatures between the d+ and d0 bands, d-hole jumps are frozen at ε and gd0( d) is the density of states in the d band defined T < ∆. Therefore, the ∆r(T) dependence can be written by formulas (16a)Ð(19). The value of N+ was deter- in the form mined from the experimental data on the Hall effect at T = 1.3 K and from a given iron content. In calculating ∆ () 1/2 ()∆ rT = AT exp Ð /kBT , (24) the ne(T) dependence, we disregarded the temperature correction to the Fermi energy of electrons, since this where the coefficient A is a fitting parameter. Using ε 2 correction is of the order of (kBT/ F) and contributes Eq. (23), the temperature dependence of the electron

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1882 KULEEV et al. µ mobility can be determined from the 0(d0) dependence and the probability of the population of localized states. at T = 0 to be [32] In this case, we should take into account not only the transitions of d holes from the d level to overlying states 1 1 1 1 ∂µ + ------= ------≅ ----- + ------∆rT(), (25) of the d band [see Eq. (21)] but also the reverse pro- µ() µ()() µ 2 ∂ 0 T dT 0 µ d cesses, as well as the transitions of d holes between the where µ(d(T)) is defined by formulas (14)–(18) from d0-band states. The principle of detailed balance allows µ us to simplify the expression for the probability Γ : [32] with d(T) in place of the parameter d0 and 0 = ij µ(d ). We note that the dependence of the electron 0 Γ ()[]()()∆ mobility on the donor concentration in HgSe crystals ij = Fi 1 Ð F j Wij 1 Ð exp Ð Eij/kBT . (27) µ and the (NFe) dependence for HgSe : Fe crystals were analyzed in [32]. These dependences were described The µÐ1(T) dependence calculated using formulas (25)Ð quantitatively by using a single set of parameters with (27) is shown in Fig. 3 by solid lines. The parameter w0 inclusion of the sÐp hybridization and Bloch ampli- in Eq. (26) and the coefficient A in Eq. (24) are fitting tudes of the wave functions. Those parameters are also parameters. We can see from Fig. 3 that the calculated used here to analyze the temperature dependence of the curves conform well to the experimental data. An anal- Ð1 electron mobility. The µ (T) dependence calculated ysis showed that the transitions of d holes from the d+ using Eqs. (23)Ð(25) is shown in Fig. 3 by dashed lines. level to the d0-band states make a dominant contribu- We can see that the calculated curves fit well with the tion to ∆r(T) at low temperatures, while the transitions experimental data in the temperature range 1.3Ð35 K; of d holes between the d0-band states make this domi- within the range 10 < T < 35 K, the experimental data fol- nant contribution at T > 15 K. Thus, the inclusion of µÐ1 1/2 low the law (T) ~ T . It is obvious from formulas (7), thermally activated jumps of d holes, which decrease (14), and (15) that this dependence is associated with the degree of spatial ordering in the system of MV iron the quadratic dependence of the correlation potential ∆ ions, makes it possible to explain the temperature Us on the radius r within the correlation sphere. At dependence of the electron mobility. T > 35 K, the calculated curves lie below the experimental curves, which is due to the contribution of elec- Let us consider the contribution from inelastic scat- tronÐphonon scattering [15] being disregarded. tering of conduction electrons in which d centers are The ∆r(T) dependence can be determined more recharged to the resistivity. The inverse electron mobil- strictly by using the method suggested in [18, 33, 34]. ity, with account taken of this contribution, can be writ- In contrast to [33, 34], we consider thermally activated ten as jumps (caused by the interaction of d holes with either conduction electrons or phonons) rather than tunneling 1 1 1 ------= ------+ ------, (28) of d holes from charged to neutral centers, because the µ()T µ()dT() ∆µ ≤ ∆ ne energy of thermal chaotic motion is kBT Eij in the case under consideration. Unlike [30], we take into ∆µ τ τ Ð1 where 1/ ne = mF/e d and d is defined in [10]. As account the population of the d+ and d0 states according to [33] when considering jumps of d holes from shown in [10], the inverse relaxation time for the elec- charged to neutral centers. In this case, the quantity tron transition from state k to state k' and that for the d- ∆r(T) can be found to be hole transition from point ri to point rj are proportional to the squared overlap integral averaged over the Fermi ∆ ∆ Γ surface. Since the wave functions of d states are spread rT()= ∑ Rij ij, σ over distances < a0, this contribution for jumps with ij, ∆ ∆ rij > Rmin > 2a0 is exponentially small (see also [33]): Γ () () ∆µ |∆ | σ ij = Fi 1 Ð F j Wij Ð F j 1 Ð Fi W ji, 1/ ne ~ exp(Ð2 rij / ). For example, by estimating ∆ ∆r(T) from formula (24) at T = 20 K, we obtain that the Eij Wij = Z0w0 expÐ------exponent is smaller than minus seven. Therefore, this kBT (26) mechanism makes a very small contribution to the elec- ∆ () tron momentum relaxation in the low-temperature ∆ Us rij = Z0w0 expÐ------expÐ------, range and can be neglected. kBT kBT d Thus, the mechanism of inelastic jumps of d holes ∆ () 2+ 3+ Ð1 Us rij between Fe and Fe ions causes a decrease in the Z0 = ∫drijexpÐ------. 3+ kBT degree of spatial ordering of the system of Fe ions and 0 results in the dependence µÐ1(T) ~ T1/2 in the tempera- Formula (26) yields, in fact, the average value of ∆r ture range 10 < T < 35 K. This mechanism allows a obtained by averaging with respect to the Boltzmann dis- quantitative description of the temperature dependence tribution, which is defined by the correlation potential of the electron mobility in HgSe : Fe crystals.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 DENSITY OF STATES IN THE IMPURITY d BAND 1883

6. CONCLUSIONS 11. Z. Wilamowski, A. Mycielski, W. Janstsch, and G. Hen- dorfer, Phys. Rev. B 38 (5), 3621 (1988). The correlation potential of a system of mixed- valence iron ions has been calculated and the parame- 12. Z. Wilamowski, W. Jantsch, and G. Hendorfer, Semi- cond. Sci. Technol. 5 (3), S266 (1990). ters defining the impurity d-band structure have been determined. It was shown that spatial ordering of iron 13. Z. Wilamowski, Acta Phys. Pol. A 77, 133 (1990). ions in HgSe : Fe in the region of strong Coulomb cor- 14. I. G. Kuleev, Fiz. Tverd. Tela (St. Petersburg) 39 (2), 250 relations gives rise to a correlation gap in the density of (1997) [Phys. Solid State 39, 219 (1997)]. impurity d states, i.e., to the formation of a finite energy 15. I. G. Kuleev, Fiz. Tverd. Tela (St. Petersburg) 40 (3), 425 gap between populated (Fe2+) and empty (Fe3+) states, (1998) [Phys. Solid State 40, 389 (1998)]. as well as to total suppression of resonant electron scat- 16. I. G. Kuleev and I. Yu. Arapova, Fiz. Tverd. Tela 43 (3), tering. The density of states in the impurity d band and 403 (2001) [Phys. Solid State 43, 420 (2001)]. the inelastic transitions of d holes between Fe2+ and 17. P. W. Anderson, Phys. Rev. 124 (1), 41 (1961). Fe3+ ions due to which the bi- and trivalent iron ions are 18. B. I. Shklovskiœ and A. L. Éfros, Electronic Properties of recharged have been theoretically analyzed. It was Doped Semiconductors (Nauka, Moscow, 1979; Springer, New York, 1984). shown that the experimentally detected features in the dependences of the electron density and mobility on the 19. V. L. Nguen, M. É. Raœkh, and A. L. Éfros, Fiz. Tverd. Tela (Leningrad) 28, 1307 (1986) [Sov. Phys. Solid State temperature and iron content can be explained in terms 28, 735 (1986)]. of the influence of the Coulomb correlations in the sys- 20. Z. Wilamowski, K. Swiatek, T. Dietl, and J. Kossut, tem of mixed-valence iron ions on the impurity d-band Solid State Commun. 74 (8), 833 (1990). structure. 21. I. G. Kuleyev, N. K. Lerinman, L. D. Sabirzyanova, et al., Semicond. Sci. Technol. 12, 840 (1997). ACKNOWLEDGMENTS 22. J. M. Ziman, Models of Disorder: The Theoretical Phys- ics of Homogeneously Disordered Systems (Cambridge This study was supported by the Russian Founda- Univ. Press, Cambridge, 1979; Mir, Moscow, 1982). tion for Basic Research (projects no. 00-02-16299, 02- 23. R. Balescu, Equilibrium and Non-Equilibrium Statisti- 02-06156), the Dinastiya Foundation, and the Interna- cal Mechanics (Wiley, New York, 1975; Mir, Moscow, tional Centre for Fundamental Physics in Moscow 1978), Vol. 1. (ICFPM). 24. E. I. Khar’kov, V. I. Lysov, and V. E. Fedorov, Physics of Liquid Metals (Vishcha Shkola, Kiev, 1979). REFERENCES 25. H. D. Jones, J. Chem. Phys. 55 (6), 2640 (1971). 26. B. Firey and N. W. Ashcroft, Phys. Rev. A 15 (5), 2072 1. D. I. Khomskiœ, Usp. Fiz. Nauk 129 (3), 443 (1979) [Sov. Phys. Usp. 22, 879 (1979)]. (1977). 27. I. G. Kuleev, N. K. Lerinman, I. I. Lyapilin, et al., Fiz. 2. M. Yu. Kagan and K. I. Kugel’, Usp. Fiz. Nauk 171 (6), Tekh. Poluprovodn. (St. Petersburg) 27 (3), 519 (1993) 577 (2001) [Phys. Usp. 44, 553 (2001)]. [Semiconductors 27, 292 (1993)]. 3. J. M. D. Coey, M. Viret, and S. Molnar, Adv. Phys. 48 28. Amorphous Solids and the Liquid State, Ed. by (2), 167 (1999). N. H. March, R. A. Street, and M. Tosi (Plenum, New 4. S. K. Park, T. Ishikawa, and Y. Tokura, Phys. Rev. B 58 York, 1985). (7), 3717 (1998). 29. V. M. Askerov, Electronic Transport Phenomena in 5. I. M. Tsidil’kovskiœ, Usp. Fiz. Nauk 162 (2), 63 (1992) Semiconductors (Nauka, Moscow, 1985). [Sov. Phys. Usp. 35, 85 (1992)]. 30. I. G. Kuleev, I. I. Lyapilin, and I. M. Tsidil’kovskiœ, Fiz. 6. F. Pool, J. Kossut, U. Debska, and R. Reifenberger, Phys. Tverd. Tela (St. Petersburg) 37 (8), 2360 (1995) [Phys. Rev. B 35 (5), 3900 (1987). Solid State 37, 1291 (1995)]. 7. I. M. Tsidilkovskii and I. G. Kuleyev, Semicond. Sci. 31. P. Protapatas and N. Parlee, High Temp. Sci. 6 (1), 1 Technol. 11, 625 (1996). (1974). 8. I. G. Kuleev, I. I. Lyapilin, and I. M. Tsidil’kovskiœ, Zh. 32. I. G. Kuleev and I. I. Kuleev, Fiz. Tverd. Tela (St. Peters- Éksp. Teor. Fiz. 102 (5), 1652 (1992) [Sov. Phys. JETP burg) 45 (2), 205 (2003) [Phys. Solid State 45, 214 75, 893 (1992)]. (2003)]. 9. I. G. Kuleev, I. I. Lyapilin, A. T. Lonchakov, and 33. O. Madelung, Introduction to Solid State Theory I. M. Tsidil’kovskiœ, Fiz. Tekh. Poluprovodn. (St. Peters- (Springer, Berlin, 1978; Nauka, Moscow, 1985). burg) 28 (6), 937 (1994) [Semiconductors 28, 544 34. A. Miller and E. Abrahams, Phys. Rev. 120 (3), 745 (1994)]. (1960). 10. I. G. Kuleev, I. I. Lyapilin, A. T. Lonchakov, and I. M. Tsidil’kovskiœ, Zh. Éksp. Teor. Fiz. 106 (4), 1205 (1994) [JETP 79, 653 (1994)]. Translated by A. Kazantsev

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1884Ð1889. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1794Ð1799. Original Russian Text Copyright © 2003 by Mezhennyœ, Mil’vidskiœ, Prostomolotov. SEMICONDUCTORS AND DIELECTRICS

Simulation of the Stresses Produced in Large-Diameter Silicon Wafers during Thermal Annealing M. V. Mezhennyœ*, M. G. Mil’vidskiœ**, and A. I. Prostomolotov*** * Giredmet State Research Institute for the Rare-Metals Industry, Bol’shoœ Tolmachevskiœ per. 5, Moscow, 109017 Russia ** Institute for Chemical Problems of Microelectronics, Bol’shoœ Tolmachevskiœ per. 5, Moscow, 109017 Russia e-mail: [email protected] *** Institute for Problems of Mechanics, Russian Academy of Sciences, Moscow, 109017 Russia Received March 11, 2003

Abstract—The three-dimensional problem of stress and strain fields in dislocation-free silicon wafers 200 and 300 mm in diameter placed horizontally on three symmetrical supports and subjected to is gravitational forces and thermal stresses formulated and solved in the isotropic approximation. Under the action of gravitational forces, a wafer is shown to have the lowest total stress when the supports are positioned at a distance of 0.6Ð 0.7R from the center of the wafer. The shear-stress fields are calculated for all possible slip systems. The elastic- stress field in a 300-mm wafer is found to be induced mainly by gravitational forces even at a radial temperature gradient of 10 K. At this temperature gradient, the contribution from thermal stresses in a 200-mm wafer is comparable to the contribution from gravitational forces. At radial temperature gradients lower than 5 K, the contribution from thermal stresses can also be neglected for a 200-mm wafer. The maximum shear stresses calculated indicate that one should not neglect possible dislocation generation in the zone of contact between a wafer and the supports during high-temperature annealing of 200-mm and, especially, 300-mm wafers. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION metrically arranged supports. The wafers are subjected to the gravitational stresses caused by the wafer weight Modern commercial silicon wafers have sharply and to thermal stresses caused by a temperature gradi- increased in diameter in recent years; therefore, the ent across the wafer area. problem of elastic stresses in wafers and their possible relaxation through plastic deformation with the formation of dislocations during high-temperature thermal 2. FORMULATION OF THE PROBLEM effects has again come to the forefront. The point is that the gravitational forces induced by the weight of a The purpose of this work is to solve the three- wafer are particularly pertinent in the formation of elas- dimensional problem of finding the fields of elastic tic-stress fields as the wafer diameter increases. More- stresses and strains (in the isotropic approximation) in over, it is difficult to provide low temperature gradients wafers placed horizontally on three symmetrically across the area of a large-diameter wafer during heat arranged supports for different values of the distance of treatment. These problems became most important for the supports from the center of a wafer, namely, 0.5, the production of wafers 200 and 300 mm in diameter. 0.6, and 0.7R, where R is the wafer radius. Upon annealing, a wafer lies freely on three sym- Wafers are heat-treated in resistive furnaces [1Ð3] or metrically arranged supports (Fig. 1). The supports are with halogen incandescent lamps [4]; the latter are most taken to be perfectly elastic. The wafer can freely slide often used for rapid thermal annealing. Upon anneal- over the supports; therefore, heating of the wafer does ing, wafers are usually placed horizontally on three not induce additional stresses related to an increase in symmetrically arranged supports. Other methods of fix- the wafer dimensions with an increase in temperature ing wafers are also known: the symmetrical application as in the case of a fixed wafer. Taking into account the of four supports or the arrangement of wafers on ring symmetrical arrangement of the supports, we consid- supports. Fisher et al. [5] estimated, in a one-dimen- ered only a 120° sector of the wafer. The equations con- sional approximation, the maximum shear stresses in necting the fields of displacements u , stresses σ , and 200- and 300-mm silicon wafers caused by gravita- ε i ij strains ij were solved using the finite-element method. tional forces for all methods of fixing wafers mentioned The chosen 120° sector of the wafer was divided into a above. finite number of elements and the size of elements near In this work, we solve the three-dimensional prob- the point of support was decreased, so that the number lem of stressed dislocation-free 200- and 300-mm sili- of elements increased. The wafer was divided into five con wafers that are placed horizontally on three sym- elements according to thickness. The wafer thickness

SIMULATION OF THE STRESSES PRODUCED IN LARGE-DIAMETER SILICON WAFERS 1885

(a)

120° Support Deformed water Nondeformed water (b)

Support Si water

Fig. 2. Calculated patterns of the 300-mm wafer bending (along the z axis) under the action of gravitational forces when the supports are located at a distance of (a) 0.5 and Fig. 1. Division of a wafer into finite elements (schematic). (b) 0.7 wafer radius from the center of the wafer. was taken to be 1000 µm, and the area of the contact respect to the neutral line) in the central part of the surface of a support was ~1 mm2. It was assumed that wafer changes in magnitude and becomes negative. At the temperature field in the wafer could be determined the same time, in those peripheral regions of the wafer independently by solving the Laplace equation. where the strain is negative and minimal in magnitude, In calculations, we used the following values of the the strain also changes and becomes positive. When the elastic and thermal parameters of silicon: Poisson supports are located at a distance of 0.6 wafer radius ratio µ = 0.25, thermal expansion coefficient β = 5.2 × from the center of the wafer, the displacements are neg- 10Ð6 K−1, shear modulus G = 6.2 × 104 MPa, and elas- ative in virtually the whole wafer, excluding the regions tic modulus E = 2G(1 + µ) = 1.55 × 105 MPa (linear surrounding the contacts between the supports and the approximation of the temperature dependence of the wafer. For the 200-mm wafer, the character of changes elastic modulus) [6]. in the deformation pattern is similar. The calculated strains along the z axis in the 300-mm wafer for various arrangements of the supports 3. ANALYSIS OF STRESSES INDUCED are given in Fig. 3. The deformation pattern has a com- BY GRAVITATIONAL FORCES plex character. The maximum strains, which are The simplest estimation of the stresses can be per- observed at the farthest point from the supports at the formed by assuming that each support is acted upon by periphery of the wafer, are seen to decrease as the dis- a force equal to one-third of the wafer weight. For a tance between the support and the center of the wafer contact area of the wafer and support of 1 mm2 and a increases. A strain is considered to be positive if it is wafer thickness of 1 mm, the maximum working directed opposite to the gravitational forces (i.e., if the stresses are estimated to be 0.24 and 0.54 MPa for the wafer bends upward with respect to the neutral plane). 200- and 300-mm wafers, respectively. However, this A decrease in the wafer diameter from 300 to 200 mm estimation does not take into account the effect of the results in a decrease in the maximum strains by a factor radial arrangement of the supports. According to rough of about 4.5. estimations based on analysis of the balance of the To characterize the general stressed state of the operating forces, the location of the supports at a dis- wafers, we used the Mises stress σ [7]. Figure 4 shows tance of 0.667R from the center of the wafer minimizes σ M the M distribution in the lower plane of the wafer stresses in the bulk of the wafer. After completing such 300 mm in diameter for various radial arrangements of rough estimations, we calculated the three-dimensional σ the supports. The stress M is maximum near the sup- stress distribution in the wafers. ports. When the supports are located at 0.7R from the cen- σ Figure 2 shows the calculated patterns of a 300-mm ter of the wafer, the maximum stress is M = 1.73 MPa. wafer bending along the z axis (which characterize the When the supports are located at 0.6R from the center, σ bending flexure of the wafer) under the action of gravi- the maximum stress decreases to M = 1.37 MPa. For tational forces when the supports are located at a dis- the supports located at ~0.5R from the center of the σ tance of (a) 0.5 and (b) 0.7 wafer radius from the center wafer, the maximum stress is M = 2.76 MPa. These of the wafer. These data indicate that the character of data show that the wafer as a whole has minimum deformation of the wafer substantially changes when stresses when the supports are located at (0.6Ð0.7)R the supports are shifted radially: as the distance from the center of the wafer, which agrees well with the between the supports increases, a positive strain (with rough estimates performed earlier. Throughout the

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

1886 MEZHENNYŒ et al.

–257 (a) (a) –212 –167 –122 –77 2.8 –32 2.4 2.0 1.6 1.2 12 0.8 0.4 0

Support Support

–220 (b) (b) –183 –147 –110 –73 –37 1.38 1.15 0.93 0.70 0 0.48 0.25 0.03

Support Support –197 –163 (c) (c) –130 –63 –96 –30 1.73 4 0 1.45 1.17 0.88 0.60 0.31 0.03

Support Support

Fig. 3. Strains along the z axis caused by the action of gravitational forces in the 300-mm wafer when the supports are located at a distance of (a) 0.5, (b) 0.6, and (c) 0.7 wafer radius from the center of the wafer (numerals on the lines Fig. 4. Distribution of the Mises stresses induced by gravi- are the strains in the wafer in micrometers; positive numer- tational forces in the plane of the 300-mm wafer when the als correspond to strains directed opposite to the gravita- supports are located at a distance of (a) 0.5, (b) 0.6, and tional force, and negative values, to strains directed along (c) 0.7 wafer radius from the center of the wafer (numerical this force). values correspond to stresses in megapascals).

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

SIMULATION OF THE STRESSES PRODUCED IN LARGE-DIAMETER SILICON WAFERS 1887

– – [101] Rpin = 0.5R –– 0.25 = 0.6 [011] – Rpin R – [110] [101] –– Rpin = 0.7R [011] 0.20 – –– [111] [110] [111]–

[111]– – – 0.15 [111] [001] , MPa

τ 0.10 [010] [100] 0.05

0 Support 0123456789101112 Slip system index

Fig. 6. Maximum shear stresses induced by gravitational forces in the lower surface of the 300-mm wafer at its point Fig. 5. Arrangement of possible slip systems in a (001) sili- of contact with the support for various slip systems and var- con wafer (schematic). ious radial positions of the supports.

× σ greater part of the wafer, stresses do not exceed ~1 The shear stresses b were calculated from the for- 10Ð3 MPa. More accurate estimation of the maximum mula stresses in the contact zone requires a more correct σ σ σ model that takes into account the geometry and mate- σ σ x H ++yK z L b = M------σ 2 2 2 2 2 2 rial of the supports. The maximum value of M in the σ σ σ x ++y z H ++K L 200-mm wafer for the supports located at (0.6Ð0.7)R (1) σ σ σ from the center of the wafer is 0.89 MPa. The working × xh ++yk zl stresses in the majority of the wafer are very low and ------, σ 2 σ 2 σ 2 2 2 2 similar to those in the 300-mm wafer. Thus, the wafer x ++y z h ++k l zone near the support has maximum stress and disloca- where σ , σ , and σ are the normal stresses along the tion generation is most probable in this zone. x y z x, y, and z axes, respectively; HKL are the crystallo- To estimate the probability of dislocation generation graphic indices of the normal vector to the slip plane; in the wafer, it is necessary to take into account the geometry of possible slip systems. We consider the case of a (001) wafer, which is the most interesting from the Possible slip systems used in calculating gravitational practical viewpoint. The scheme of possible slip sys- stresses in (001) silicon wafers tems in such a wafer is given in Fig. 5. When calculating the stresses under which dislocations are generated, Slip system index Slip plane Slip direction we have to take into account the shear stress component 1 operating in the plane and direction of easy slip rather 101 1 1 1 than the total operating stresses. When calculating 2 011 shear stresses, we determined the projection of the calculated resultant vector of the three-dimensional stress 3 110 ﬁeld caused by the gravitational forces acting on a slip 4 1 1 1 1 01 plane along the easy shear direction. To analyze the propagation of dislocations, we have to answer the 5 101 question of whether the sign of stresses changes throughout the wafer thickness. In the case where the 6 011 stresses in the upper and lower surfaces of the wafer 7 101 differ in sign (the wafer contains a plane with zero 111 stress), dislocations which are nucleated in the waferÐ 8 01 1 support contact zone can propagate into the bulk of the wafer only to this zero (neutral) plane. If the stresses in 9 1 10 the upper and lower surfaces of the wafer are of the same sign, dislocations which are nucleated in the 10 1 1101 1 lower surface can propagate throughout the whole 11 wafer thickness. The depth of their penetration depends 1 1 0 on the local values of operating stresses. 12 011

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1888 MEZHENNYŒ et al.

0.20 and hkl are the crystallographic indices of a vector in the slip direction. The ﬁrst factor in Eq. (1) is the absolute value of the stress vector, the second is the cosine 0.15 of the angle between the three-dimensional stress vec- 2 tor and its projection on the slip plane (HKL), and the 0.10 third factor is the cosine of the angle between the pro- , MPa

τ jection of the three-dimensional stress vector on the slip plane and its projection on the slip direction in this 0.05 1 plane. The silicon crystal lattice contains four independent {111} easy-slip planes, each of which contains six pairwise opposite easy-slip directions 〈110〉. To esti- 0 1 2 3 4 5 6 7 8 9 10 11 12 mate the level of stresses, it is sufﬁcient to perform cal- Slip system index culations for only 12 slip systems from the 24 possible variants. The slip systems used in calculations are listed Fig. 7. Maximum shear stresses induced by gravitational in the table. forces in the lower surface of the wafers (1) 200 and (2) 300 mm in diameter at their point of contact with the Even in the isotropic approximation, in order to cal- support for various slip systems. The supports are located at culate elastic stresses and determine the shear stresses, a distance of 0.7R from the center of the wafer. we have to know the orientation of the wafer with respect to the supports. The orientation dependence of shear stresses in a slip plane is speciﬁed by the mutual crystallographic position of the plate and the three sup- 0.15 ports. When the wafer rotates about its center, the pyr- amid formed by the {111} planes (Fig. 5) also rotates around its vertex. Since the shear stresses are maximum 0.10 at the point of support, the most dangerous sections (from the viewpoint of dislocation generation) for each , MPa Top τ 0.05 Bottom slip system pass through the point of support. The line of intersection of a slip plane with the wafer surface rotates with respect to the point of support, and the cor- 0 responding maximum stress remains unchanged. Figure 6 shows the calculated maximum shear 0 0.2 0.4 0.6 0.8 1.0 stresses operating in the corresponding slip systems for t, mm the 300-mm wafer at various radial positions of the supports. The shear stresses are maximum in the waferÐ Fig. 8. Distribution of the shear stresses induced by gravita- support contact zone. Numerals on the abscissa axis tional forces over the thickness of the 300-mm wafer in the show the indices of the corresponding slip systems (see cross section that passes through the point of contact of the wafer with the center of the supporting area in the table). (111 )[011 ] slip system having the maximum Schmid fac- For comparison, Fig. 7 displays the calculated max- tor. The supports are located at a distance of 0.7R from the imum shear stresses in the contact zone for the 200- and center of the wafer. 300-mm wafers and the supports located at a distance of 0.7R from the center of the wafer. The designation of the slip systems is identical to that in Fig. 6. On average, the shear stresses in the 200-mm wafer are lower 0.035 than those in the 300-mm wafer by a factor of 1.5Ð2. It should be noted that the shear stresses near the waferÐ 0.030 support contact zone are an order of magnitude lower than the corresponding values of σ . 0.025 M Figure 8 shows the shear-stress distribution over the , MPa

M 0.020 wafer thickness in the (111 )[011 ] slip system, which σ has the maximum Schmid factor and passes through the 0.015 point of contact between the wafer and the center of the supporting area. The calculation was conducted for the 0.010 300-mm wafer and the supports located at 0.7R from 0 20 40 60 80 100 120 140 the center. This shear-stress distribution indicates that R, mm the wafer contains a plane of zero stress positioned at a σ distance of ~0.7 times the wafer thickness from the Fig. 9. Radial distribution of thermal stresses M caused by a radial temperature drop of 1 K. The wafer diameter is lower surface of the wafer. As follows from these data, 300 mm. the maximum shear stress in the lower surface of the

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 SIMULATION OF THE STRESSES PRODUCED IN LARGE-DIAMETER SILICON WAFERS 1889 wafer near the contact zone is 0.17 MPa. Such stresses formation of an elastic-stress ﬁeld even at a radial tem- can cause dislocation generation in the region adjacent perature drop of 10 K. At this temperature drop, the to the contact zone under heating of the wafers to contribution from thermal stresses in the 200-mm 1000°C and above. However, dislocations generated in wafer is comparable to the contribution from the gravi- this region can propagate into the wafer only to the neu- tational forces. If the radial temperature drop is lower tral plane and cannot reach its upper (working) surface. than 5 K, the contribution from thermal stresses can also be neglected for the 200-mm wafer. 4. ANALYSIS OF THERMOELASTIC STRESSES The distribution of elastic stresses caused by the gravitational forces in the wafers has been found to be The cause of thermoelastic stresses is a nonuniform a rather sensitive function of the radial arrangement of temperature distribution in the wafer plane, which the supports. We showed that a wafer has the lowest arises under heating. In calculations, we used the tradi- total stresses when the supports are positioned symmet- tional approach. As in the case of gravitational forces, σ rically at a distance of (0.6Ð0.7)R from the center of the we used the Mises stresses M to characterize the wafer. The calculated maximum shear stresses indicate stressed state. that one cannot neglect possible dislocation generation As an example, Fig. 9 shows the calculated radial in the zone of contact between a wafer and the supports σ distribution of thermal stresses M in the 300-mm during high-temperature annealing in 200-mm wafers wafer. The radial temperature drop was taken to be 1 K, and, especially, 300-mm wafers. and the temperature at the center of the wafer was assumed to be 1273 K. At a radial drop of 1 K, the max- σ imum thermal stress M was found to be 0.03 MPa, ACKNOWLEDGMENTS which is much less than the stresses induced by the This work was supported by the Russian Foundation gravitational forces. The thermal shear stresses are for Basic Research, project no. 01-02-16816. maximum at the center and periphery of the wafer. Unlike the gravitational stresses, these stresses are minimum near the supports. At the given temperature drop REFERENCES between the center and edge of the 300-mm wafer, the 1. H. Katsumata, H. Ito, H. Takahashi, et al., U.S. Patent maximum thermal shear stress is ~0.01 MPa. An No. 6250914 (2000). increase in the radial temperature drop causes a proportional increase in the thermoelastic stresses. The calcu- 2. M. Hatta, U.S. Patent No. 6032724 (1997). lations indicate that the maximum thermal shear 3. H.-S. Kyung, W.-S. Choi, and J.-H. Shin, U.S. Patent stresses that appear during thermal annealing are com- No. 5791895 (1996). parable with to maximum stresses caused by the gravi- 4. K. E. Johnsgard, B. S. Mattson, and J. McDiarmid, U.S. tational forces only for the 200-mm wafer and radial Patent No. 6002109 (1995). temperature drops higher than 10 K. Even at such radial 5. A. Fisher, G. Richter, W. Kurner, and P. Kucher, J. Appl. temperature drops, the stresses caused by the gravita- Phys. 87, 1543 (2000). tional forces in the 300-mm wafer are signiﬁcantly 6. O. H. Nielsen, Prop. of Silica, EMIS Datarev, Ser. 14 higher than the thermal stresses. (1987). 7. B. A. Boley and J. H. Weiner, Theory of Thermal Stresses 5. CONCLUSIONS (Wiley, New York, 1960; Mir, Moscow, 1964). Thus, we have established that the gravitational forces in the 300-mm wafer play a decisive role in the Translated by K. Shakhlevich

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1890Ð1892. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1800Ð1802. Original Russian Text Copyright © 2003 by Agekyan, Stepanov.

SEMICONDUCTORS AND DIELECTRICS

The Mechanism of Emission in the Red Photoluminescence Band of Porous Silicon V. F. Agekyan and A. Yu. Stepanov Fock Institute of Physics, St. Petersburg State University, ul. Pervogo Maya 100, Petrodvorets, St. Petersburg, 198504 Russia Received March 20, 2003

Abstract—Porous silicon (PS) exhibits several photoluminescence (PL) bands, whose spectral position and intensity depend strongly on the actual conditions of preparation of PS, its treatment, and subsequent use. The PS PL band peaking at about 1.8 eV and usually assigned to the intrinsic emission of silicon nanocrystals was studied. It was shown that the temperature-induced variation of the PL kinetics in the 80 to 300-K interval follows a complex pattern and depends noticeably on the actual point on the band proﬁle. The temperature behavior of PL decay in the 1.8-eV band is determined by the electronÐhole recombination rate within a nanocrystal and the cascade carrier transitions from small to large nanocrystals, with an attendant decrease in energy. © 2003 MAIK “Nauka/Interpe- riodica”.

As established in numerous studies, the photolumi- of specific features in the temperature dependence and nescence (PL) spectrum of porous silicon (PS) depends spectral response of the PS S-band kinetics in the range dramatically on the technique of its preparation, its sub- from 70 K to room temperature. sequent treatment, and the actual conditions and duration of its storage (see, e.g., [1Ð5]). There is, however, PS samples with a porosity of about 70% were pre- no consensus on the origin of the individual PS PL pared by anodic etching of crystalline silicon [4] and bands. Most authors assign the slowly decaying band in stored for a long period of time in standard atmospheric the visible region (the so-called S band) to the intrinsic conditions. PL was excited by a cw argon laser with a radiation of size-quantized silicon crystals [6Ð10]. The photon energy of 2.41 eV or by nitrogen laser pulses electrons and holes in semiconductor nanocrystals are with a photon energy of 3.68 eV, pulse duration of 5 ns, spaced much closer than those in bulk crystals, which and pulse repetition frequency of 100 Hz. The pulse gives rise to a very large increase in exchange coupling. excitation intensity was about 104 W cmÐ2, and the As a result, the exchange splitting of electronic levels in spectra were taken with a DFS-12 spectrometer and a silicon nanocrystals is capable of strongly affecting the Boxcar integrator. temperature dependence of the S-band kinetics [10, 11]. Calculations show that the exchange splitting in nanocrystals making up PS can be larger than 10 meV. As a result, at lower temperatures, the luminescence kinetics depends totally on the properties of the lowest 6 component in the exchange doublet, i.e., of the triplet level [11Ð14]. In this case, the PL decay time τ is long because of the triplet level being metastable. From the good agreement of the experimental τ(T) dependences 4 for the S band with theoretical calculations of the exchange splitting in silicon nanocrystals, some authors drew the conclusion that the contribution of the cascade transitions to the S-band formation at low tem- 2 peratures is insignificant. It should, however, be pointed out that agreement between experimental measurements of the PS PL kinetics with calculations has also Luminescence intensity, arb. units been reached with other models. In [15], the slow decay of the S band was described by an extended exponential 1.2 1.4 1.6 1.8 2.0 2.2 2.4 function taking into account the hopping carrier motion Photon energy, eV on the surface of a silicon nanocrystal; there are also arguments in favor of efficient tunneling migration of Fig. 1. S band in the photoluminescence of PS under pulsed electrons and holes between nanocrystals in the interval excitation measured during the first 50 µs after the pulse at 70Ð300 K [16]. This communication reports on a study room temperature (dashed line) and at 77 K (solid line).

THE MECHANISM OF EMISSION IN THE RED PHOTOLUMINESCENCE BAND 1891

300 1.0 250 200 3 s

µ 150 0.5 , τ 2 100

2 1 50 300 K 125 K 77 K

4 6 8 10 12 14 Luminescence intensity, arb. units 1 T–1 × 103, K–1 0.1 0 250 500 Fig. 2. Temperature dependence of the porous-silicon pho- τ, µs toluminescence decay time τ measured (1) on the high- energy wing, (2) at the maximum, and (3) on the low-energy Fig. 3. PS photoluminescence decay kinetics (1) on the low- wing of the S band. Dashed arrows specify the values of 1/Tc energy wing, at about 1.6 eV, and (2) at the maximum, corresponding to sharp changes in the τ(T) relations. 2.1 eV, of the S band measured at 77 K.

The S band observed in our samples at room temper- smooth decrease in τ observed to occur under sample ature under excitation with a cw argon laser peaks at heating at lower temperatures originates from the 1.8 eV and has a halfwidth of 0.35 eV; the peak position increase in the population of the upper (singlet) compo- and halfwidth vary only weakly in the temperature nent of the exchange doublet. At high temperatures, the interval studied. The spectra obtained under pulsed characteristic decay time of the electronÐhole lumines- excitation with nanosecond delay times are dominated cence decreases because of the enhanced nonradiative by a shorter wavelength radiation, which is characteris- relaxation. tic of relaxed PS samples and is related to oxides and, possibly, other silicon compounds [17Ð20], but the The kinetics curves measured at different points on spectrum gated with a microsecond scale delay con- the S-band profile support this reasoning. The decay in tains the S band only. The S band measured under the low-energy wing at T = 77 K is described by a sim- pulsed excitation within the first 50-µs gate after the ple exponential that characterizes the electronÐhole pulse is symmetric at room temperature, but at 77 K its recombination kinetics, whereas the kinetics related to high-energy wing is intensified (Fig. 1). We carried out the high-energy wing contains an additional contribu- a comprehensive study of the τ(T) relation for three tion (Fig. 3), which is apparently accounted for by the cascade migration of carriers involving a decrease in energies corresponding to the wings and the maximum τ of the S band (Fig. 2). Since the PL kinetics can be com- their energy. The (E) relation measured over the S- plex and described by more than one exponential, we band profile at room temperature turns out to be linear accepted τ as the characteristic decay time in which the PL intensity dropped e-fold. As seen from Fig. 2, each of the three graphs reveals a break with a plateau, but in 100 different temperature intervals. The temperature Tc cor- 80 1 responding to the break in the τ(T) curve is governed by the recombination dynamics in nanocrystals and by the 60 cascade carrier transitions from small to larger nanoc- s µ rystals, which involve a decrease in energy. These , 40 τ 300 mechanisms significantly depend on the nanocrystal 250 size. The break point and the plateau shift toward 2 200 higher temperatures with a decrease in the energy of the photons whose kinetics are being measured. This may 20 150 be attributed to the fact that the fast temperature- τ 100 induced decrease in for T > Tc is determined largely 1.6 1.7 1.8 1.9 2.0 by carrier migration to larger nanocrystals as a result of Photon energy, eV enhancement of the cascade transition. Such processes Fig. 4. Spectral response of the photoluminescence decay are the most efficient for carriers residing in small time τ of porous silicon measured over the S-band profile nanocrystals (the high-energy wing of the S band). The (1) at 300 and (2) 77 K.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

1892 AGEKYAN, STEPANOV

(Fig. 4). The more complex behavior of τ over the band 7. K. Ito, S. Ohyama, Y. Uehara, and S. Ushiroda, Surf. Sci. proﬁle observed at 77 K, involving a sharp change at 363, 423 (1996). about 1.9 eV, suggests that the relative contribution of 8. A. G. Gullis and L. T. Canham, Nature 333 (6342), 335 the above factors to radiative recombination at low tem- (1991). peratures differs strongly at different points on the S- 9. G. D. Sanders and Yia-Chung Chang, Phys. Rev. B 45 band proﬁle. (16), 9202 (1992). 10. A. G. Gullis, L. T. Canham, and P. D. J. Calcott, J. Appl. Phys. 82 (3), 909 (1997). ACKNOWLEDGMENTS 11. P. D. J. Calcott, K. J. Nash, L. T. Canham, et al., J. Phys.: This study was supported in part by the Ministry of Condens. Matter 5 (7), L91 (1993). Education of the Russian Federation, project no. E 02- 12. P. D. J. Calcott, K. J. Nash, L. T. Canham, et al., 3.4-426. J. Lumin. 57 (1), 257 (1993). 13. L. Pavesi and M. Ceschini, Phys. Rev. B 48 (23), 17625 (1993). REFERENCES 14. G. Mauckner, K. Tronke, T. Baier, et al., J. Appl. Phys. 1. A. Bsiesy, J. C. Vial, F. Gaspard, et al., Surf. Sci. 254 (1Ð 75 (8), 4167 (1994). 3), 195 (1991). 15. Y. Kanemitsu and T. Ogava, Surf. Rev. Lett. 3 (1), 1163 2. L. T. Canham, W. Y. Leong, T. I. Cox, et al., in Proceed- (1996). ings of the 21st International Conference on Physics of 16. T. Suemoto, K. Tanaka, and A. Nakajima, Phys. Rev. B Semiconductors, Ed. by P. Jiang and H. Z. Zhang (World 49 (16), 11005 (1994). Sci., Singapore, 1993), p. 1423. 17. L. T. Canham, A. Loni, P. D. J. Calcott, et al., Thin Solid 3. V. F. Agekyan, V. V. Emtsev, A. A. Lebedev, et al., Fiz. Films 276 (1), 112 (1996). Tekh. Poluprovodn. (St. Petersburg) 33 (12), 1462 18. V. F. Agekyan, A. A. Lebedev, Yu. V. Rud’, and (1999) [Semiconductors 33, 1315 (1999)]. Yu. A. Stepanov, Fiz. Tverd. Tela (St. Petersburg) 38 4. V. F. Agekyan, A. M. Aprelev, R. Laœkho, and (10), 2994 (1996) [Phys. Solid State 38, 1637 (1996)]. Yu. A. Stepanov, Fiz. Tverd. Tela (St. Petersburg) 42 (8), 19. L. Tsybeskov, Yu. V. Vandychev, and P. M. Fauchet, 1393 (2000) [Phys. Solid State 42, 1431 (2000)]. Phys. Rev. B 49 (11), 7821 (1994). 5. N. E. Korsunskaya, T. V. Torchinskaya, B. R. Dzhumaev, 20. N. Chiodini, F. Meinardi, F. Mozzaroni, et al., Appl. et al., Fiz. Tekh. Poluprovodn. (St. Petersburg) 31 (8), Phys. Lett. 76 (22), 3209 (2000). 908 (1997) [Semiconductors 31, 773 (1997)]. 6. S. Guha, G. Hendershot, D. Peeples, et al., Appl. Phys. Lett. 64 (5), 613 (1994). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1893Ð1897. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1803Ð1806. Original Russian Text Copyright © 2003 by Smolin, Lapshin, Pankova.

SEMICONDUCTORS AND DIELECTRICS

Crystal Structure of Di-(L-Serine) Phosphate Monohydrate [C3O3NH7]2H3PO4H2O Yu. I. Smolin*, A. E. Lapshin*, and G. A. Pankova** * Grebenshchikov Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, St. Petersburg, 199155 Russia e-mail: [email protected] ** Institute of Macromolecular Compounds, Russian Academy of Sciences, Bol’shoœ pr. 31, St. Petersburg, 119004 Russia Received February 4, 2003

Abstract—The crystal structure of di-(L-serine) phosphate monohydrate [C3O3NH7]2H3PO4H2O is determined by single-crystal x-ray diffraction. The intensities of x-ray reﬂections are measured at temperatures of 295 and 203 K. The crystal structure is reﬁned using two sets of intensities. It is established that, in the structure, symmetrically nonequivalent molecules of L-serine occur in two forms, namely, the monoprotonated positively + + Ð charged molecule CH2(OH)CH(NH3) COOH and the zwitterion CH2(OH)CH(NH3) COO , which are linked Ð with each other and with the H2PO4 ion through a hydrogen-bond system involving water molecules. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION The intensities of x-ray reflections were measured on an automated single-crystal diffractometer operat- Owing to their structural features, crystals of inor- ing in a perpendicular beam geometry with layer-by- ganic derivatives of proteinic amino acids are of great layer analysis (MoKα radiation, pyrolytic graphite interest both for searching for new pyroelectric and monochromator). The reflections were measured in the Θ ° piezoelectric materials and for understanding the role ranges h(Ð7, 7) k(0, 15) l(0, 14) at 2 max = 72 . The played by the electrical properties of proteinic amino measurements were carried out for 2037 nonzero non- acids in processes occurring in living organisms. The equivalent reflections with I > 3σ(I). The estimation of technique of growing single crystals of amino acids and the integrated intensities and correction for the back- their inorganic derivatives, tentative x-ray structure ground were performed using an algorithm of the pro- characteristics, and the temperature dependence of the file analysis [5]. piezoresponce of the crystals were described in [1]. The The stability of the crystal was checked by periodic structures of β-alanine, DL-alanine, sarcosine, and L- measurement of the intensity of the standard reflection. alanine phosphates were investigated earlier in [2, 3]. The coordinates of the non-hydrogen atoms were deter- The purpose of this study is to determine the crystal mined by direct methods with the AREN software structure of di-(L-serine) phosphate monohydrate package [6]. All the hydrogen atoms were located in a [C3O3NH7]2H3PO4H2O, which, according to the struc- series of difference syntheses alternating with the least- tural database [4], has not been characterized before. squares refinement of the positional and thermal parameters of the atoms for the F2(hkl) set with the SHELXL97 software package [7].1 The thermal param- 2. SAMPLE PREPARATION eters were refined in the anisotropic approximation for AND EXPERIMENTAL TECHNIQUE the non-hydrogen atoms and in the isotropic approximation for the hydrogen atoms. The final R1 factor cal- Crystals of [C3O3NH7]2H3PO4H2O are monoclinic, culated for all the reflections measured is equal to space group P21, a = 4.704(5) Å, b = 10.512(5) Å, c = 0.0332 and wR = 0.1047. The total number of parame- β ° 3 ρ 2 13.573(3) Å, = 98.73(5) , V = 663.4 Å , Z = 2, calc = ters refined is 257 and S = 0.97. The results of the Ð3 Ð1 1.63 g cm , λ(MoKα) = 0.71069 Å, µ = 2.68 cm , refinement, the atomic coordinates, and the equivalent F(000) = 344, and M = 326.20. Crystals of L-serine thermal parameters are given in Table 1. phosphate monohydrate were prepared by slow cooling The figure shows the projection of the crystal struc- of saturated water solutions of L-serine and phosphoric ture of L-serine phosphate monohydrate, which was acid. The temperature of the solution decreased from drawn with the ORTEP III program [8]. The selected room temperature to 8°C at a rate of 1 K/day. A color- less crystal (0.4 × 0.35 × 0.5 mm in size) suitable for x- 1 The exact determination of the parameters mentioned in this ray diffraction analysis was used in the measurements. paper is also given in [7].

1894 SMOLIN et al.

Table 1. Fractional coordinates of the atoms and their equivalent thermal parameters

2 Atom x/ay/bz/cUequiv/Uiso (Å ) P(1) 0.05339(5) 0.65189(2) 0.24138(2) 0.01682(5) O(1) Ð0.00540(19) 0.51247(9) 0.22114(8) 0.0264(2) O(2) Ð0.18764(17) 0.72715(9) 0.27598(8) 0.0247(2) O(3) 0.13069(18) 0.71983(9) 0.14711(8) 0.0244(2) O(4) 0.32675(17) 0.66183(13) 0.32538(8) 0.0306(2) O(5) 0.08244(19) 0.20332(11) 0.25228(8) 0.0280(2) O(6) 0.2747(3) 0.12542(12) 0.40126(10) 0.0395(3) C(1) 0.2688(2) 0.20107(12) 0.32377(10) 0.0226(3) C(2) 0.5315(2) 0.28586(12) 0.33227(10) 0.0215(3) C(3) 0.5730(3) 0.36569(16) 0.42652(12) 0.0306(3) O(7) 0.3217(3) 0.43459(12) 0.43828(9) 0.0338(3) N(1) 0.5079(2) 0.36553(10) 0.24097(9) 0.0215(2) O(8) 0.4607(2) 0.73740(9) Ð0.05717(8) 0.0308(2) O(9) 0.4617(2) 0.59009(10) 0.06228(8) 0.0273(2) C(4) 0.5370(2) 0.63412(10) Ð0.01567(10) 0.0185(2) C(5) 0.7309(2) 0.55024(10) Ð0.06892(9) 0.0164(2) C(6) 0.5498(2) 0.49113(11) Ð0.16007(10) 0.0202(2) O(10) 0.33541(19) 0.41453(10) Ð0.12571(8) 0.0263(2) N(2) 0.8653(2) 0.44712(9) Ð0.00126(8) 0.0190(2) O(w) 0.1285(3) 0.45925(11) 0.61932(10) 0.0386(3) H(1) 0.246(5) 0.675(3) 0.1226(19) 0.054(7) H(2) 0.487(8) 0.663(5) 0.311(3) 0.101(12) H(3) 0.127(7) 0.086(3) 0.391(2) 0.062(8) H(4) 0.693(4) 0.236(2) 0.334(2) 0.026(5) H(5) 0.628(4) 0.311(2) 0.485(2) 0.021(4) H(6) 0.755(5) 0.424(3) 0.424(2) 0.045(6) H(7) 0.315(7) 0.489(4) 0.396(2) 0.077(10) H(8) 0.511(5) 0.316(3) 0.181(2) 0.048(7) H(9) 0.345(4) 0.419(2) 0.236(2) 0.027(5) H(10) 0.650(5) 0.419(3) 0.242(2) 0.041(6) H(11) 0.897(4) 0.598(2) Ð0.089(2) 0.019(4) H(12) 0.465(3) 0.560(2) Ð0.204(2) 0.006(3) H(13) 0.666(3) 0.442(2) Ð0.197(2) 0.018(4) H(14) 0.301(5) 0.366(2) Ð0.168(2) 0.043(6) H(15) 1.014(4) 0.416(2) Ð0.021(2) 0.024(4) H(16) 0.732(5) 0.380(3) Ð0.000(2) 0.041(6) H(17) 0.920(4) 0.473(2) 0.062(2) 0.021(4) H(18) 0.213(5) 0.450(2) 0.577(2) 0.037(6) H(19) 0.134(4) 0.397(3) 0.661(2) 0.039(6) interatomic distances and bond angles are presented in ters of the atoms at this temperature differ insigniﬁ- Table 2. The structural parameters were also reﬁned cantly from those obtained for the structure measured at using a set of intensities measured at 203 K [a = room temperature, Table 2 includes only the atomic 4.688(5) Å, b = 10.482(5) Å, c = 13.524(5) Å, β = coordinates and equivalent thermal parameters for this 98.07(5)°]. Since the coordinates and thermal parame- structure.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

CRYSTAL STRUCTURE OF DI-(L-SERINE) PHOSPHATE MONOHYDRATE 1895

O(w)

Projection of the crystal structure of di-(L-serine) phosphate monohydrate onto the bc plane. The non-hydrogen atoms are shown as thermal ellipsoids. The hydrogen atoms are drawn as spheres of arbitrary radius. Dashed lines indicate hydrogen bonds.

The crystallographic data for the structure studied and bond angles in both forms of L-serine molecules have been deposited with the Cambridge Crystallo- are close to each other and fall in the predicted ranges graphic Data Center as supplementary publication no. (C–C, 1.514(2)–1.527(2) Å; C–N, 1.486(2)–1.497(2) Å) CCDC 201 068. [9, 10]. However, owing to the deprotonation of the carboxyl group in the zwitterion, the CÐO bond lengths are close to each other and equal to 1.250(2) and 1.254(2) Å, 3. RESULTS AND DISCUSSION which are the mean of the relevant bond lengths in the The structure involves L-serine molecules in the carboxyl group of the L-serine-1 molecule [1.205(2) form of a monoprotonated molecule of and 1.316(2) Å]. These two forms of L-serine mole- + cules are linked by hydrogen bonds. It follows from CH2(OH)CH(NH3) COOH (L-serine-1) and a zwitte- + Ð Ð rion of CH2(OH)CH(NH3) COO (L-serine-2). The Table 2 that the H2PO4 ion represents a distorted tetra- amine group NH2 in the L-alanine-1 molecule is addi- hedron, because the PÐOH distances [1.556 (1) and tionally protonated by a hydrogen atom of the phos- 1.587(1) Å] are considerably longer than the P–O phate ion. As a consequence, the molecule becomes bonds [1.509(1) and 1.514(1) Å]. The phosphate ions positively charged. In the zwitterion, an additional pro- are linked through a hydrogen-bond system with the tonation of the amine group occurs through the depro- surrounding L-serine molecules. The parameters of the tonation of the carboxyl group and the molecule hydrogen bonds are listed in Table 3. The water mole- remains electroneutral. The CÐC and CÐN bond lengths cule forms hydrogen bonds with the phosphate ion and

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1896 SMOLIN et al.

Table 2. Selected bond lengths (Å) and bond angles (deg)

Ð H2PO4 P(1)ÐO(1) 1.5089(10) O(1)ÐP(1)ÐO(2) 116.02(5) P(1)ÐO(2) 1.5145(9) O(1)ÐP(1)ÐO(3) 110.82(6) P(1)ÐO(3) 1.5559(10) O(2)ÐP(1)ÐO(3) 107.14(5) P(1)ÐO(4) 1.5872(9) O(1)ÐP(1)ÐO(4) 107.51(6) O(2)ÐP(1)ÐO(4) 107.81(6) O(3)ÐP(1)ÐO(4) 107.19(5) L-Serine-1 O(5)ÐC(1) 1.205(2) O(5)ÐC(1)ÐO(6) 125.5(2) O(6)ÐC(1) 1.316(2) O(5)ÐC(1)ÐC(2) 122.4(2) C(1)ÐC(2) 1.514(2) O(6)ÐC(1)ÐC(2) 112.2(2) C(2)ÐN(1) 1.486(2) N(1)ÐC(2)ÐC(1) 108.2(1) C(2)ÐC(3) 1.518(2) N(1)ÐC(2)ÐC(3) 112.0(2) C(3)ÐO(7) 1.416(2) C(1)ÐC(2)ÐC(3) 112.9(2) O(7)ÐC(3)ÐC(2) 112.0(2) L-Serine-2 O(8)ÐC(4) 1.250(2) O(8)ÐC(4)ÐO(9) 127.0(2) O(9)ÐC(4) 1.254(2) O(8)ÐC(4)ÐC(5) 116.1(1) C(4)ÐC(5) 1.527(2) O(9)ÐC(4)ÐC(5) 116.9(1) C(5)ÐN(2) 1.497(2) N(2)ÐC(5)ÐC(6) 109.6(1) C(5)ÐC(6) 1.524(2) N(2)ÐC(5)ÐC(4) 110.8(1) C(6)ÐC(10) 1.423(2) C(6)ÐC(5)ÐC(4) 108.6(1) O(10)ÐC(6)ÐC(5) 107.5(1)

Table 3. Parameters of the hydrogen bonds A–H…DAÐH(A)H…D(A) A…D(A) AHD (deg) O(3)ÐH(1)ÐO(9) 0.82(3) 1.66(3) 2.481(2) 174(3) O(7)ÐH(7)ÐO(4) 0.81(4) 2.06(4) 2.840(2) 163(3) N(1)ÐH(8)ÐO(8)iii 0.97(3) 1.90(3) 2.859(2) 172(2) N(1)ÐH(9)ÐO(1) 0.94(2) 1.90(2) 2.845(2) 174(2) N(1)ÐH(10)ÐO(1)i 0.87(3) 1.95(2) 2.808(2) 167(2) O(10)ÐH(14)ÐO(2)iv 0.76(2) 2.08(2) 2.845(2) 176(3) N(2)ÐH(15)ÐO(10)i 0.85(2) 2.23(2) 2.999(2) 151(2) N(2)ÐH(16)ÐO(8)iii 0.95(3) 1.97(3) 2.865(2) 157(2) N(2)ÐH(17)ÐO(1)i 0.90(2) 2.17(2) 3.065(2) 170(2) O(w)ÐH(18)ÐO(7) 0.75(3) 2.03(3) 2.758(2) 163(2) O(w)ÐH(19)ÐO(2)ii 0.87(3) 1.97(3) 2.816(2) 163(2) C(2)ÐH(4) 0.92(2) C(3)ÐH(5) 0.98(2) C(3)ÐH(6) 1.05(3) C(5)ÐH(11) 1.00(2) C(6)ÐH(13) 0.95(2) C(6)ÐH(12) 0.99(2) Symmetry codes: (i) 1 + x, y, z; (ii) x, y Ð 1/2, Ðz + 1; (iii) 1 Ð x, y Ð 1/2, Ðz; (iv) x, y Ð 1/2, Ðz.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 CRYSTAL STRUCTURE OF DI-(L-SERINE) PHOSPHATE MONOHYDRATE 1897 the L-serine-1 molecule and is coordinated by the 3. Yu. I. Smolin, A. E. Lapshin, and G. A. Pankova, Kristal- hydrogen atom of the carboxyl group of another mole- lografiya 48, 280 (2003) [Crystallogr. Rep. 48, 283 cule of L-serine-1. Thus, we can conclude that, in addi- (2003)]. tion to electrostatic interactions, the structure is charac- 4. F. H. Allen, Acta Crystallogr. B 58, 380 (2002). terized by an extended system of hydrogen bonds. 5. S. Oatley and S. French, Acta Crystallogr. A 38, 537 (1982). 6. V. N. Andrianov, Kristallografiya 32 (1), 228 (1987) ACKNOWLEDGMENTS [Sov. Phys. Crystallogr. 32, 130 (1987)]. 7. G. M. Sheldrick, SHELXL97: Program for the Refine- This work was supported by the Russian Foundation ment of Crystal Structures (Univ. of Göttingen, Ger- for Basic Research, project no. 01-02-17163. many, 1997). 8. M. N. Burnett and C. K. Johnson, ORTEP III: Report ORNL-6895 (Oak Ridge National Laboratory, Tennes- REFERENCES see, 1996). 9. T. J. Kistenmacher, G. A. Rand, and E. Marsh, Acta 1. V. V. Lemanov, S. N. Popov, and G. A. Pankova, Fiz. Crystallogr. B 30 (11), 2573 (1974). Tverd. Tela (St. Petersburg) 44 (10), 1840 (2002) [Phys. 10. M. N. Frey, M. S. Lenmann, T. F. Koetzle, and Solid State 44, 1929 (2002)]. W. C. Hamilton, Acta Crystallogr. B 29 (4), 877 (1973). 2. M. T. Averbuch-Pouchot, A. Durif, and J. C. Guite, Acta Crystallogr. C 44, 1968 (1988). Translated by N. Korovin

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

SEMICONDUCTORS AND DIELECTRICS

Electrical Conduction and Polarization in PbWO4 Crystals V. N. Shevchuk and I. V. Kayun Franko National University, ul. Lomonosova 8, Lviv, 79005 Ukraine e-mail: [email protected] Received March 7, 2003

Abstract—The regularities of ionÐelectron processes in an undoped PbWO4 single crystal upon transition to a quasi-equilibrium state in an external dc electric ﬁeld with a linear variation in the temperature in the range 290Ð 600 K are investigated using different methods. The total conductivity, thermally stimulated polarization current, and thermally stimulated depolarization current are measured. It is assumed that the temperature dependence of the conductivity can be described within the theory of small-radius polarons. The thermally stimulated polarization (depolarization) currents are interpreted in terms of the space-charge (peaks of the current in the range 400Ð550 K) and dipole (peaks of the current in the range 290Ð370 K) mechanisms of generation of a polarization charge in the sample. The inference is drawn that the dominant contribution to the dipole polarization is made by dipolons, namely, doubly charged (cationÐanion) vacancy pairs coupled through electrostatic interaction. The basic parameters of relaxation phenomena and charge transfer are calculated. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION range 120Ð350 K. Huang et al. [10] examined the dielectric response of PbWO4 single crystals doped The considerable scientific interest expressed in with yttrium and niobium impurities. PbWO4 crystals is associated with their practical use, in However, the nature of the electrically active defects particular, as self-activated scintillators with short that predominate in PbWO4 single crystals remains afterglow times (see, for example, [1, 2]). The opera- unclear. At present, studies of the electrical properties tional characteristics of PbWO4 crystals are deter- of these crystals and, particularly, thermal polarization mined, to a large extent, by their structural defects. effects have been reduced to accumulation of experi- However, to date, there has been no agreement among mental data. researchers regarding the nature of defects in the In this work, we elucidated the regularities of ionÐ PbWO4 structure. Although the properties of PbWO4 crystals have been investigated by different methods in electron processes in an undoped PbWO4 single crystal many works, the electrical characteristics of this com- and the behavior of electrically active defects in the pound and their changes under external actions are still PbWO4 structure upon the transition to a quasi-equilib- not clearly understood, especially in the temperature rium state with a linear variation in the temperature T or range 290Ð600 K. under isothermal conditions. For this purpose, we investigated the total conductivity σ, thermally stimu- Groenink and Binsma [3] were among the first to lated polarization (TSP) currents ITSP in an external dc investigate the electrical properties of PbWO4 crystals. electric field, and thermally stimulated depolarization They measured dc and ac electrical conductivities of (TSD) currents ITSD in the internal field of the crystal in potassium-, yttrium-, and bismuth-doped and undoped the polarized state. PbWO4 crystals grown using the Czochralski method upon heating to 1200 K in air. The electrical character- 2. EXPERIMENTAL TECHNIQUE istics of PbWO4 are also described in the monograph by Mokhosoev and Bazarova [4]. The electrical conductiv- The crystals used in the measurements were grown ity of undoped and lanthanum-, praseodymium-, by the Czochralski method along the [001] axis. Sam- × × samarium-, and terbium-doped PbWO4 samples, which ples 10 10 0.5 mm in size were cut normally to the were prepared by pressing of metal oxide powders fol- [001] axis. A dc electric field E = 102Ð104 V/cm was lowed by their sintering at temperatures of 600Ð800 K, applied to the sample placed in a capacitor measuring was measured in [5Ð7] with the aim of elucidating the cell with Aquadag electrodes and metal leads. The role played by the oxygen sublattice in charge transfer. applied field was chosen in such a way as to correspond Suszynska et al. [8, 9] analyzed the temperature depen- to a linear portion of the currentÐvoltage characteris- dence of the conductivity of undoped PbWO4 crystals tics. The experiments were performed in the tempera- immediately after the Czochralski growth and upon ture range T = 290Ð600 K in air. The conduction current annealing under vacuum or in air in the temperature I, the thermally stimulated polarization current ITSP, and

ELECTRICAL CONDUCTION AND POLARIZATION IN PbWO4 CRYSTALS 1899 the thermally stimulated depolarization current I Table 1. Activation energies ∆E for conduction in PbWO TSD σ 4 were measured under changes in the temperature at a crystals and preexponential factors 0 in different tempera- constant rate β = 0.1Ð0.2 K/s. Experimental data on the ture ranges electrical conductivity were obtained during heating ∆E, eV σ , and cooling in one heatingÐcooling cycle or several T, K 0 ΩÐ1 cmÐ1 consecutive cycles. In order to determine temperatures cooling heating at which the polarization substantially affects the dependences σ(T), we also measured the electrical con- 290Ð350 0.23 Ð Ð 1.4 × 10Ð6 ductivity of samples preliminarily cooled in an external Ð4 electric field. Weak currents were measured on a V7-30 350Ð400 0.66 0.73* Ð 3.9 × 10 voltmeter electrometer. The temperature was measured 400Ð460 0.72 0.85* Ð 3.5 × 10Ð3 using a chromelÐalumel thermocouple. The tempera- Ð2 ture in the quartz measuring cell was maintained con- 460Ð590 0.85 0.93* Ð 3.2 × 10 stant with an accuracy of ±1 K. The error in the deter- 560Ð350 Ð Ð 0.91 8.6 × 10Ð2 mination of the activation energy for conduction did not exceed ±2Ð3%. All other details of the experimental * Activation energies ∆E determined from the dependence σ procedure and the measuring setup used for studying ln( T) = f(1/T). high-resistance crystals and TSD (TSP) currents were similar to those described in [11]. σ ∆ though very close) parameters 0 and E (Table 1). In our experiments, we varied polarization condi- Table 1 also presents the data obtained upon cooling of tions, i.e., the polarization voltage Up, polarization tem- τ the sample and those determined from the curve perature Tp, and polarization time p (i.e., the time ln(σT) = f(1/T). It can be seen that the results of the required for the formation of a spatially nonuniform measurements upon cooling differ from those upon charge distribution in the sample). This made it possi- heating. The curve lnσ = f(1/T) measured during cooling ble to control the maximum intensity Im of thermally has one straight-line portion in the temperature range stimulated currents and to separate the Im peaks in the 550Ð350 K. In all cases, except for the experiment pre- TSD (TSP) spectra. The nature of polarization effects sented by curve 2 in Fig. 1b, the samples were prelimi- with due regard for the dependences of the intensity Im narily cooled in the short-circuited state in the absence on the maximum temperature Tm and the half-width of of an external electric field. the peaks under different polarization conditions in the It can be seen from Figs. 1a and 1b that, in the tem- presence of blocking electrodes was analyzed in the σ framework of the space-charge and dipole models of perature range 290Ð350 K, the curve (T) substantially polarization. depends on the relaxation effects (associated with the appearance and disappearance of polarization charges), the history of the sample prior to measurements, the 3. ELECTRICAL CONDUCTIVITY temperature range of heating in the preceding measure- Under normal conditions, the PbWO crystals are ment, and the cooling conditions. In this “sensitive” 4 range, the temperature dependence of the TSP current characterized by a band gap of 4.13 eV along the z Ð9 is characterized by a broad intense peak Im ~ 10 A at direction [12] and a permittivity of 23 [4] and, accord- ≈ ing to our measurements, exhibit a low conductivity Tm 310 K. Under identical conditions, the intensity Im (σ ≈ 10Ð11Ð10Ð14 ΩÐ1 cmÐ1). It should be noted that the depends on the number of heating cycles in the course conductivities σ obtained in different works can differ of the measurement. The maximum value of Im is observed in the first heating–cooling cycle. The behav- from each other, because these characteristics, like σ other properties, substantially depend on the method ior of the dependence (T) depicted by curve 3 in and conditions of the crystal growth, the purity of the Fig. 1a is due to partial suppression of relaxation processes through several consecutive heatingÐcooling initial materials, doping, high-temperature annealing of σ the crystal in various media, and other factors. cycles. The dependence ln = f(1/T) measured upon heating of the sample preliminarily cooled in an exter- The temperature dependence of the electrical con- nal electric field (Fig. 1b, curve 2) changes in the low- σ ductivity (T) measured during heating of the PbWO4 temperature range 290Ð350 K and does not follow the crystals in the temperature range 290Ð600 K (Fig. 1) law described by expression (1) in the range 350Ð400 can be well approximated by the formula K. This can be explained by the effect of polarization on σσ= exp()Ð∆E/kT , (1) the conduction current. It was experimentally found 0 that slow cooling of the sample in the absence of an where k is the Boltzmann constant. Detailed analysis of external field leads to less pronounced variations in the the curves in Fig. 1c shows that the dependence σ(T) initial dependence lnσ = f(1/T), even though, in this involves three portions in the temperature ranges 350Ð case, also, the relaxation leads to a decrease in the cur- 400, 400Ð520, and 520Ð600 K with different (even rent intensity after repeated heatingÐcooling cycles.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1900 SHEVCHUK, KAYUN

T, K T, K 500 400 300 500 400 300 (a) (b) 1 1 2 2 3 3 –24 –24 ) –1 cm –1 Ω , σ –28

ln ( –28

–32

–32 1.8 2.4 3.0 1.8 2.4 3.0 3 –1 3 –1 10 /T, K T, K 10 /T, K 550 500 450400 350 –18 0.93 eV (c) –16 –20 460 K –18 –22 0.85 eV

–20 ) ) 0.85 eV –1 –1 –24 400 K cm cm –22 –1 –1 Ω

Ω –26 0.73 eV , , K σ

0.72 eV T

–24 σ

ln ( 350 K

–28 ln ( –26 –30 0.66 eV –28 –32 –30 1.8 2.0 2.2 2.4 2.6 2.8 3.0 103/T, K–1

Fig. 1. Experimental temperature dependences of the total conductivity of PbWO4 crystals in the Arrhenius coordinates [the upper curve in panel (c) is obtained in the 1/TÐln(σT) coordinates] for (a) different numbers of consecutive heatingÐcooling cycles and (b) different conditions of preliminary polarization of the samples. Curves σ(T) are measured during (a) (1) the first heating, (2) heating after the first two consecutive heatingÐcooling cycles, (3) heating after several heatingÐcooling cycles and (b) (1) the first heating, (2) heating of the sample preliminarily cooled in the external electric field, and (3) cooling of the sample preliminarily cooled in the external electric field.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 ELECTRICAL CONDUCTION AND POLARIZATION IN PbWO4 CRYSTALS 1901 The crystals regain their original characteristics after holding in air at room temperature for approximately 1.8 24 h. The dependence σ(T) measured for the sample preliminarily cooled in the electric ﬁeld (Fig. 1b, curve 2) 1 2 exhibits a weak peak with a maximum at ~370 K and an 3 additional low-temperature (290Ð350 K) straight-line 1.2 portion in the Arrhenius coordinates. This indicates that, under the given experimental conditions, the relaxation processes proceed rather slowly at temperatures T < 350 K and the maximum in the polarization , A current in the range 290Ð350 K is virtually absent. 10 0.6 10

Apparently, an increase in the temperature is accompa- × nied only by an activation increase in the conductivity. TSP ∆ σ I The values of E and 0 in this temperature range differ significantly from those at higher temperatures (Table 1). This can also be judged from the inflection point of the 0 curve lnσ = f(1/T) at T ~ 350 K. The observed behavior of the dependences σ(T) can be explained by a crossover of the conduction mechanism. It seems likely that, at low temperatures, n-type carriers execute hoppings from one defect site to another. At temperatures above 0.6 350Ð400 K, the drastic increase in the intensity of the conduction current can be associated with the change not only in the mobility but also in the concentration of 320 360 400 charge carriers. Note that the inflection point in the T, K dependence lnσ = f(1/T) at approximately 450 K has also been experimentally observed for other tungstate Fig. 2. TSP spectra of PbWO4 crystals in electric fields E = crystals of similar compositions [4, 13]. (1) +1.0 × 103, (2) Ð1.0 × 103, and (3) +1.0 × 103 V/cm for Judging from the activation energies and the temper- (1) the initial sample and (2, 3) samples preliminarily ature dependence lnσ = f(1/T), the steady-state conduc- cooled from 500 to 290 K in the field E = +1.0 × 103 V/cm. tion and, in particular, its electronic component are provided by carriers that are thermally activated at local levels in the band gap. The sequential involvement of deep Maxwell relaxation time. This time can be estimated levels in charge transfer leads to a change in the slope of from the relationship the straight-line portion in the dependence lnσ = f(1/T). τ εε σ An exponential temperature dependence of the con- µ = 0/ , (2) ductivity frequently counts in favor of the formation of ε ε where is the permittivity of the crystal and 0 is the small-radius polarons. However, in our case, the forma- permittivity of free space. In our case, the Maxwell tion of polarons calls for further verification and inves- relaxation time at room temperature does not exceed tigation. It is evident that intensive polarization, espe- ~103 s. cially at 290Ð370 K, and hysteresis effects in the temperature dependence of the conduction current upon The typical TSP and TSD spectra with decomposi- heating and cooling can be explained by a considerable tion of the TSD peaks at Tm > 350 K into Gaussian com- contribution to the total conductivity of the sample ponents are shown in Figs. 2 and 3, respectively. The coming from the ionic component. relaxation processes can be separated into three main groups in the temperature ranges 290Ð350, 350Ð380, and 400Ð600 K (Fig. 3), which are associated with the 4. POLARIZATION EFFECTS corresponding peaks of the current. The resolution of In order to elucidate regularities in the relaxation the TSD method is higher than that of the TSP method. processes associated with the formation and transfor- For this reason, attention was mainly focused on the mation of a polarized (thermoelectret) state of the TSD spectra. The parameters calculated for relaxors PbWO4 crystal and their influence on the steady-state from the TSD spectra are presented in Table 2. The acti- conduction, we investigated thermally stimulated depo- vation energies Et for thermally stimulated processes larization (thermally stimulated polarization) currents. were estimated from the expression Under the assumption that deep traps are absent, the ≅ stability of the electret state can be governed by the Et 27kTm. (3)

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1902 SHEVCHUK, KAYUN

~ ~ example, the TSP spectrum (Fig. 2, curve 2) in this (a) range has at least two maxima at ~312 and ~325 K. 6 0.8 , A

9 According to the experimental data, the intensities of I' 6 10 these narrow TSP peaks in the range 290Ð350 K are ×

0.4 , A , A II' interrelated in a complex manner. In particular, the TSP 9

TSD IV 4 11 I III' 4 peak at 312 K is dominant upon initial heating. In sub-

10 0 10

× III × sequent heating cycles, the intensity of this peak 345 360 375 T, K decreases and only slightly exceeds the intensity of the TSD 1 TSD

I 2 2 2 I × 10 TSP peak at 325 K. After preliminary polarization of the sample upon cooling, the intensity of the peak at ~

~ 325 K increases more rapidly than that of the peak at

0 ~ ~ III 0 312 K. For an appropriately chosen polarization tem- 300 350 450 500 550 T, K perature Tp, the TSP peaks at 312 and 325 K can become equal to each other in intensity. 6 (b) In the range 290Ð350 K, the polarization tempera- 2.1 ture T affects the intensity I of TSD currents. As was 1 p m 2 noted above, the intensity Im of thermally stimulated 3 currents in the range 290Ð350 K depend on the prehis- tory of the sample, speciﬁcally on the number of preliminary heating cycles in the course of the measure- 4 ment, the high-temperature limit of this heat treatment, 1.4 cooling conditions, the time of relaxation in the short- , A , A 8

10 circuited state, and the time of holding the sample in air

10 prior to measurements. The dependence I (U ) for the

10 m p × ×

broad peak under consideration shows linear behavior. TSD

TSD Upon heating at a voltage +U , the height of this peak

I p I 2 for the sample preliminarily polarized at a voltage ÐUp 0.7 appears to be greater than that for the unpolarized sample by a factor of more than two. The total intensity of the peak decreases at Tp > Tm. This behavior of the TSD (TSP) currents due to variations in the experimental conditions is characteristic of relaxation processes in a system of point-defect complexes of dipole nature. It is 0 0 evident that, apart from orientation processes, the dissociation of dipole complexes and an inverse process, 300 350 namely, the association of point defects into anisotropic T, K complexes, play an important role. A comparison of the experimental data presented in Figs. 1Ð3 shows that, for Fig. 3. TSD spectra of PbWO crystals polarized under dif- 4 τ each temperature range in which the dominant depolar- ferent conditions. (a) Up = 50 V, p = 5 min, and Tp = (1) 335 and (2) 320 K. The inset shows the decomposition of the ization activation processes are associated with a par- TSD peak into individual components at T = 350Ð380 K. ticular type of electrically active defects, there is a Closed circles are experimental points. (b) (1) Tp = 290 K, straight-line portion of the dependence lnσ = f(1/T) U = 50 V, and τ = 8 min and (2, 3) T = 320 K, U = 50 V, p τ p p p with a somewhat different slope (Fig. 1c). and p = 5 min (two successive measurements under identical conditions). Most likely, the TSD peaks at T > 400 K in the range of a drastic increase in conductivity (Figs. 1a, 1b) are The frequency factors ω were evaluated from the max- attributed to space-charge polarization. This is con- ∆ imum condition ﬁrmed by the close values of energies E and Et at these temperatures. Charge carriers move toward the elec- ωβ()2 () = Et/kTm exp Et/kTm (4) trode regions, are trapped in local levels, and give rise to a nonuniform charge distribution in the sample. The for the functional dependence ITSD(T) [11] under the local levels correspond to components IÐIV of the TSD assumption that particles of one sort are involved in an peak (Fig. 3a). It can be seen that these components elementary relaxation process. comprise a complex broad high-temperature peak in the As can be seen from Figs. 2 and 3, the intensities of range 400Ð600 K. This peak is associated with the elec- the TSP and TSD currents exhibit signiﬁcant peaks tret state and shifts toward the high-temperature range with a complex structure in the range 290Ð350 K. For with an increase in the polarization temperature Tp.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 ELECTRICAL CONDUCTION AND POLARIZATION IN PbWO4 CRYSTALS 1903 It should be noted that the depolarization phenom- Table 2. Parameters calculated for the main relaxors from ena considered in this work are observed in the same the TSD spectra of PbWO4 crystals temperature ranges as the radiative processes studied ω × Ð9 Ð1 earlier by other authors (see, for example, [14Ð16]). In Peak no. Tm, K Et, eV 10 , s particular, thermoluminescence peaks were experimen- 1 305 0.71 4.7 tally revealed at temperatures of 280 and 320 K in [14]; 320, 350, 370, and 430 K in [15]; and 330 and 400 K in 2 320 0.74 3.5 [16]. In all cases, the peak observed in the temperature 3 340 0.79 3.7 range 320Ð350 K is characterized by a maximum inten- I 360 0.84 4.0 sity. II 369 0.86 3.9 Furthermore, thermally stimulated conduction cur- III 375 0.87 3.3 rents in PbWO4 crystals preliminarily exposed to high- energy radiation are also observed in the above temper- I 435 1.01 2.8 ature ranges of activation of ionÐelectron processes [17, II 458 1.07 3.2 18]. Specifically, a number of peaks of thermally stimulated currents were revealed at temperatures ~280 and III 500 1.16 2.4 ~320 K in [17] and in the range 300Ð320 K in [18]. IV 530 1.23 2.3 Moreover, in [18], the authors observed a thermoluminescence peak at ~360 K. From a comparison of the Note: TSD peaks 1Ð3 are determined from spectrum 1 in Fig. 3b. results obtained in the present work and the data available in the literature, the inference can be made that electrically active centers (at least those manifesting leads to the formation of intrinsic color centers [20]. themselves in the TSD spectra at T < 400 K) serve as However, the formation of other dipole associates trapping centers of nonequilibrium charge carriers and should not be excluded when interpreting the TSD recombination centers simultaneously. spectrum in the range 290Ð370 K, especially with due regard for the complex structure of the TSD peak and its evolution with a variation in the polarization condi- 5. DISCUSSION tions. Apart from the aforementioned vacancy pairs, The dominant relaxation process observed in the these can be triads with trapped n-type carriers for local temperature range 290Ð350 K occurs through the charge neutralization, etc. Such defect complexes of dipole mechanism. We assume that dipolons—electro- dipole nature can differ in their energy parameters and 2Ð 2+ correspond to individual TSD peaks. Single vacancies statically coupled pairs of lead (V Pb ) and oxygen (V O ) at high depolarization temperatures Td can also be doubly charged vacancies—are the most probable responsible for the TSD peaks. In particular, these com- dipole centers (responsible for the TSD or TSP peaks in plexes can be responsible for the formation of the three- the range 290Ð350 K) in PbWO4 crystals. These associ- component TSD peak shown in the inset to Fig. 3a. It is ates have no effective charge, manifest themselves only found that the intensities of components I'ÐIII' of this in the TSD (TSP) spectra, and form the dipole compo- peak can be different depending on the experimental nent of the polarization in the crystal. Similar associ- conditions. This peak is not observed in the first heat- ates in gallium garnet crystals were considered in our ingÐcooling cycle in a series of successive measure- earlier work [19]. For these crystals, the TSD peak is ments. observed at 350 K and has a complex structure. Dipole The nature of the TSP peaks can also be judged from associates of this type in PbWO4 crystals were discussed, in particular, by Lim et al. [20], who assigned their temperature positions. In the case when thermally the absorption band at 350 nm to these associates. Note stimulated polarization in a crystal can be adequately also that, according to Suszynska et al. [8, 9], the TSD described within the model of space-charge accumula- peak with a maximum at approximately 330 K can be tion due primarily to steady-state conductivity, the the- attributed to dipoles in PbWO crystals. oretical dependence of the current density at the TSP 4 peak on the applied electric field has the form [11] The above assumption regarding the formation of ()σ≅ ()∆ dipolons in PbWO4 crystals and their manifestation in JTm 0Eexp Ð1E/kTm Ð . (5) the TSD spectra in the temperature range 290Ð380 K is supported by the high intensity of peaks 1Ð3 in Fig. 3 In the framework of the above model, the tempera- (this intensity cannot be provided by impurities, ture Tm of the possible TSP peak can be easily calcu- because their concentration in the crystal is relatively lated from expression (5) for the corresponding temper- low). Moreover, this assumption is confirmed by the ature range. The calculated temperatures Tm of the fact that a portion of lead and oxygen always volatilizes peaks (at T > 350 K) differ from the experimental val- during the Czochralski growth [1]. As a result, the crys- ues by no more than 5Ð10 K. This insignificant differ- tal has a deficit of lead and oxygen ions, which also ence between the calculated and experimental values of

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1904 SHEVCHUK, KAYUN

Tm can be associated with the error in determining the temperatures. Electrons detrapped at 60 K recombine σ ∆ parameters 0 and E. In the range 290Ð350 K, the cal- with localized holes, which leads to the appearance of ≈ culated temperature Tm 235 K is approximately 80 K thermoluminescence peaks. less than the experimental value. This disagreement is It seems likely that, in PbWO4 crystals, hole con- quite reasonable because the TSP peak at ~320 K is due duction at room temperature is provided by correlated to the orientation of defect complexes of dipole nature. hoppings of charge carriers from one defect site to an At the same time, the high-temperature peaks of the adjacent defect site. The crossover of the mechanism of electric current are most likely caused by space charges charge transfer can occur at T > 350 K (Fig. 1), which arising from the drift and accumulation of equilibrium results in a drastic increase in the conduction current carriers in the electrode regions. This is confirmed by with an increase in the temperature T. Judging from the the close agreement between the calculated and exper- change in the slope of the straight-line portions of the imental temperatures Tm for the peaks in the range T > dependence lnσ = f(1/T) in the temperature range 350Ð 350 K. 600 K and from the TSD spectra (Fig. 3a), the charge The observed decrease in the intensity of the TSD transfer in the PbWO4 crystal at these temperatures is peak at ~320 K with an increase in the number of heat- controlled by several local levels in the band gap. ing cycles is consistent with the dipolon concept. Single The reorientation processes associated with the vacancies predominate at high temperatures. They have complex dipole associates proceed at temperatures no time to associate into pairs upon cooling, which below 350 K. The dissociation of these dipole com- results in a decrease in the dipolon concentration upon plexes at T > 350 K ensures appropriate conditions for repetition of heatingÐcooling cycles. the release of n-type carriers and their migration in the Let us now consider the electronic component of the external field and, consequently, leads to an activation increase in the conductivity. conductivity of PbWO4 crystals. From analyzing the experimental dependence of the electronic component By virtue of the p-type component of the conductiv- of the total conductivity of PbWO4 crystals on the oxy- ity of the PbWO4 crystal, the electroneutrality condi- gen partial pressure (the conductivity σ increases with tion can be satisfied only in the case when cation vacan- an increase in the oxygen pressure [3]) and experimen- cies or oxygen ions occupy interstices. In our opinion, tal data on the photoconductivity of these crystals in the it is more probable that the structure of the PbWO4 range of the absorption edge [21, 22], it can be assumed crystal involves not interstitial oxygen but cation that holes are majority charge carriers in the PbWO4 vacancies. These vacancies are also necessary for the compound. This inference is also confirmed by the formation of dipole centers, which were proposed in increase in the conductivity upon high-temperature this work to explain the polarization effects. annealing of the samples in air. Similar experiments with samples annealed in air were performed earlier in [8, 9]. It was found that, upon annealing, the conductiv- REFERENCES ity increases by a factor of approximately 15 [8, 9]. Our 1. L. V. Atroshchenko, S. F. Burachas, L. P. Gal’chinetskiœ, experimental results also suggest that holes whose B. V. Grinev, V. D. Ryzhikov, and N. G. Starzhinskiœ, mobility exceeds the mobility of ions are majority Stintillator Crystals and Ionizing-Radiation Detectors charge carriers in PbWO crystals. Therefore, we can on Their Basis (Naukova Dumka, Kiev, 1998). 4 2. M. Nikl, Phys. Status Solidi A 178 (2), 595 (2000). draw the conclusion that the PbWO4 crystals with mixed ionicÐelectronic conductivity exhibit p-type 3. J. A. Groenink and H. Binsma, J. Solid State Chem. 29 properties. (2), 227 (1979). 4. M. V. Mokhosoev and Zh. G. Bazarova, Complex Molyb- Laguta et al. [23] carried out an electron paramag- denum and Tungsten Oxides with Group IÐIV Elements netic resonance investigation of PbWO4 crystals and (Nauka, Moscow, 1990). 3Ð observed the formation of WO4 centers that remained 5. T. Esaka and T. Mina-ai, Solid State Ionics 57 (3Ð4), 319 stable up to 60 K. In the framework of the proposed (1992). 3Ð 6. T. Esaka, R. Tachibana, and S. Takai, Solid State Ionics model, the WO4 centers are formed as a result of 92 (1Ð2), 129 (1996). 2Ð 7. S. Takai, K. Suginura, and T. Esaka, Mater. Res. Bull. 34 autolocalization of electrons in the WO4 anionic com- (2), 193 (1999). plex. A similar autolocalization at room temperature 8. M. Suszynska, B. Macalik, and M. Nikl, Radiat. Eff. was also revealed with the use of excited-state absorp- Defects Solids 150 (1Ð4), 35 (1999). tion in the fundamental absorption range under irradia- 9. M. Suszynska, B. Macalik, and M. Nikl, J. Appl. Phys. tion with a 0.5-ps pulsed laser [24]. According to the 86 (2), 1090 (1999). 3Ð proposed model of the formation of WO4 centers, 10. H. Huang, W. Li, X. Feng, and P. Wang, Phys. Status holes are captured by deep traps (most frequently, by Solidi A 187 (2), 563 (2001). 2Ð 11. Yu. A. Gorokhovatskiœ and G. A. Bordovskiœ, Thermally V Pb centers) and can be delocalized at room and higher Stimulated Current Spectroscopy of High-Resistance

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 ELECTRICAL CONDUCTION AND POLARIZATION IN PbWO4 CRYSTALS 1905 Semiconductors and Dielectrics (Nauka, Moscow, 18. M. Martini, E. Rosetta, G. Spinolo, et al., J. Lumin. 72Ð 1991). 74, 689 (1997). 12. Y. Zhang, N. A. W. Holtzwarth, and R. T. Williams, Phys. 19. A. E. Nosenko and V. N. Shevchuk, Radiat. Eff. Defects Rev. B 57 (20), 12738 (1998). Solids 134 (1Ð4), 251 (1995). 20. O. Lim, X. Feng, Z. Man, et al., Phys. Status Solidi A 13. O. V. Ivanov, A. P. Nakhodnova, and V. N. Krivobok, Zh. 181, R1 (2000). Neorg. Khim. 27 (3), 587 (1982). 21. E. G. Reut, Fiz. Tverd. Tela (Leningrad) 23 (8), 2514 14. E. Auffrau, I. Daﬁnei, P. Lecoq, and M. Schneegans, (1981) [Sov. Phys. Solid State 23, 1476 (1981)]. Radiat. Eff. Defects Solids 135 (1Ð4), 343 (1995). 22. I. M. Sol’s’kiœ, A. S. Voloshinovs’kiœ, R. V. Gamernik, 15. M. Nikl, K. Nitsch, S. Baccaro, et al., J. Appl. Phys. 82 et al., Ukr. Fiz. Zh. 46 (8), 881 (2001). (11), 5758 (1997). 23. V. V. Laguta, J. Rosa, M. I. Zaritskii, et al., J. Phys.: Con- dens. Matter 10, 7293 (1998). 16. A. Annenkov, E. Auffrau, M. Korznik, and P. Lecoq, Phys. Status Solidi A 170 (1), 47 (1998). 24. R. T. Williams, K. B. Ucer, G. Xiong, et al., Radiat. Eff. Defects Solids 155 (1Ð4), 265 (2001). 17. L. Nagornaya, A. Apanasenko, and I. Tupitsina, in Pro- ceedings of International Conference SCINT-95, Delft, The Netherlands (Delft Univ. Press, 1996), p. 299. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1906Ð1909. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1814Ð1817. Original Russian Text Copyright © 2003 by Ulanov, Anikeenok, Zaripov, Fazlizhanov.

SEMICONDUCTORS AND DIELECTRICS

Electronic Structure of Bivalent Copper Off-Center Complexes in SrF2 Crystals from Data of EPR and ENDOR Spectroscopy V. A. Ulanov, O. A. Anikeenok, M. M. Zaripov, and I. I. Fazlizhanov Kazan Physicotechnical Institute, Russian Academy of Sciences, Sibirskiœ trakt 10/7, Kazan 29, 420029 Tatarstan, Russia e-mail: [email protected] Received March 25, 2003

Abstract—The electronic structure of bivalent copper off-center complexes in SrF2 crystals is calculated from experimental data obtained earlier by electron paramagnetic resonance (EPR) and electronÐnuclear double resonance (ENDOR) spectroscopy. The electronic structure parameters characterizing the unpaired-electron density in the vicinity of the nucleus of a copper impurity ion are determined, and the parameters of covalent bonds between an impurity copper ion and three groups of the fluorine ions nearest to this impurity are calculated. It is demonstrated that states of the ground electron configuration of the bivalent copper impurity complex involve an admixture of excited electron configurations due to electron transfer from the ligand to 4s and 4p unfilled shells of the copper ion. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION the complex under investigation. It was revealed that, at temperatures below ≈225 K, the studied complex This study is a continuation of our experimental resides in one of the wells of the adiabatic potential; works [1, 2] in which the local structure of bivalent however, at temperatures above 225 K, the frequency of copper impurity centers in SrF2 crystals was investi- transitions between wells of the adiabatic potential for gated by electron paramagnetic resonance (EPR) and this complex increases drastically. In [4], the depen- electronÐnuclear double resonance (ENDOR) spectros- dence of the dynamic properties on the hydrostatic copy. The considerable interest expressed by research- pressure was examined by stationary EPR spectros- ers in bivalent copper impurity centers of SrF2 crystals copy. It was found that an increase in the hydrostatic stems from the fact that, in these crystals, the Cu2+ cat- pressure leads to a decrease in the height of barriers ion is located at the off-center position; i.e., it is dis- between the wells of the adiabatic potential. As a result, placed by ≈1 Å from the center of the coordination cube the frequency of both tunneling and over-barrier transi- of the substituted Sr2+ cation. This leads to the formations between wells of the adiabatic potential increases 6Ð tion of a [CuF4F4] paramagnetic complex with C4v signiﬁcantly, even though the temperature remains con- symmetry and a relatively large electric dipole moment. stant. Since the crystal has cubic symmetry, there exist six equally probable directions of displacement of the Cu2+ It should be noted that complexes of this type have 2+ cation. Consequently, the crystal contains copper cen- been revealed for Ag impurity ions in SrCl2 crystals 2+ ters of six types that are similar in structure but differ in [5] and for Cu impurity ions in SrCl2 [6] and BaF2 [7] the orientation of the electric dipole moment. It is evi- crystals. All these objects are impurity complexes of dent that the dipole moment can be reoriented under JahnÐTeller origin in which the JahnÐTeller effect external action. This corresponds to a transition of the observed in the triply degenerate orbital ground state is paramagnetic center from one well of the lower sheet of accompanied by pseudo-JahnÐTeller interaction [8]. the adiabatic potential to another well that is equivalent Since these complexes are nonanuclear, they are char- to the former well. acterized by multimode vibronic interactions. There- fore, it can be assumed that the magnetic and optical The dynamics of transitions between wells of the properties of these complexes are very sensitive to var- adiabatic potential for a copper complex and the depen- ious external perturbations. Moreover, bivalent off-cen- dence of the dynamic properties on the hydrostatic ter complexes of this type exhibit both paramagnetic pressure were analyzed in [3, 4]. Hoffmann and Ulanov 6Ð and paraelastic properties due to a unique combination [3] investigated the dynamic properties of a [CuF4F4] of the JahnÐTeller effect with pseudo-JahnÐTeller inter- complex with C4v symmetry under normal pressure and actions. For the above reasons, the complexes under determined the temperature dependence of the spinÐlat- investigation are of considerable interest from theoreti- tice relaxation time T1 from the results of measure- cal and practical standpoints. ments performed by pulsed EPR spectroscopy. From analyzing this dependence, these authors estimated the The purpose of this work is to obtain more detailed 6Ð separation between the lower vibronic energy levels of information on the electronic structure of [CuF4F4]

ELECTRONIC STRUCTURE OF BIVALENT COPPER OFF-CENTER COMPLEXES 1907 impurity complexes with C4v symmetry in SrF2 crys- As a result, we found the reduction coefficients k22 = tals. This information can be derived from the parame- 0.829, k11 = 0.912, and k21 = 0.711. The parameters of ters of covalent bonds between an impurity copper ion the orbital coupling of the copper ion with any of the and the fluorine ions (ligands) surrounding this impu- λ D rity. In turn, the parameters of covalent bonds can be four nearest ligands were determined to be s = 0.092, D D obtained from the tensor components of ligand hyper- λσ = Ð0.380, and λπ = 0.227. The normalizing factors fine interactions with magnetic nuclei of the fluorine of the molecular orbitals were calculated by ignoring ligands. The tensor components of the ligand hyperfine the covalence of bonds between the Cu2+ ion and the interactions with nuclei of the fluorine ligands nearest other three ligands: N = 1.208, N = 1.097, and N = to the Cu2+ ion (according to the classification proposed 2 1 0 in [1], these ligands are included in group I) and the 1.093. The spinÐorbital coupling proved to be aniso- λ Ð1 λ Ð1 λ components of the electron Zeeman and hyperfine tropic: 1|| = Ð687 cm , 2|| = Ð758 cm , and 1⊥ = interaction tensors were determined from the EPR −721 cmÐ1. spectra in [1]. The tensor components of the ligand hyperfine interactions with nuclei of four groups of 3. ELECTRONIC STRUCTURE PARAMETERS more distant ligands (i.e., ligands of groups IIÐV [2]) CHARACTERIZING THE UNPAIRED-ELECTRON were obtained from the ENDOR spectra in [2]. In the 2+ DENSITY IN THE VICINITY OF A COPPER latter work, we also proposed a model of Cu impurity NUCLEUS centers in SrF2 crystals that allows for all the ligands involved in the interactions observed in the samples From the above results obtained earlier in [4], we studied by ENDOR spectroscopy (see [2, Fig. 4]). The derive a relationship between the parameters of the ten- parameters of the covalent bonds with the four nearest sor of the hyperfine interaction with a copper nucleus ligands were obtained in our recent work [4]. In the and the parameters of the electronic structure of the complex, which characterize the unpaired-electron present study, we determined the parameters of cova- 63 lent bonds with ligands of groups II and III. The exper- density in the vicinity of a Cu nucleus. For this pur- imentally observed interactions with nuclei of more pose, we consider the operator of the magnetic interac- distant ligands (i.e., ligands of groups IV and V) were tion of an unpaired electron with a copper nucleus [9]: adequately described by the Hamiltonians of the direct H = g β γ ប 〈〉rÐ3 magnetic dipoleÐdipole interaction, which indicates the hfi S e n 2 lack of covalent bonds of the impurity ion with these 3()Ir⋅ ()S ⋅ r Ð r ()l ⋅ I 8π (2) ligands. × ------i i i i i +,------()δS ⋅ I ()r 5 3 i i ri

2. ELECTRONIC STRUCTURE PARAMETERS where ri is the radius vector of the ith electron with CHARACTERIZING THE UNPAIRED-ELECTRON respect to the nucleus. For the term (L, S) obeying the DENSITY IN THE VICINITY OF NUCLEI Hund rule, this operator can be represented as the OF THE NEAREST LIGAND GROUP function of the total orbital and total spin angular momenta [9]: A theoretical analysis of the experimental data on  β γ ប 〈〉Ð3 ()⋅ ligand hyperfine interactions with nuclei of the nearest Hhfi = gS e n r  LI--- ligand group [4] demonstrated that the orbital function  (molecular orbital) for the ground state of the copper + []ξLL()+ 1 Ð ᏺ ()SI⋅ (3) complex is represented primarily by the d 2 2 function x Ð y of the impurity ion (the orbital functions are specified 3  Ð ---ξ[]()LS⋅ ()LI⋅ + ()LI⋅ ()LS⋅ . in the XYZ coordinate system in which the Z axis is par- 2  allel to the C4v symmetry axis of the complex and the X and Y axes are aligned with two 〈110〉 crystallographic Here, ᏺ is the dimensionless constant expressed axes perpendicular to the C4v symmetry axis). The through the unpaired-electron density at the ion energy level corresponding to the molecular orbital nucleus, |XY〉 is located 8600 cmÐ1 above the orbital ground level ξ = ()2l + 14Ð S /[]S()2l Ð 1 ()l + 3 ()2L Ð 1 , |X2 Ð Y2〉. The doubly degenerate level (|XZ〉, |YZ〉) lies Ð1 | 〉 l is the orbital angular momentum of the unpaired elec- 900 cm above the molecular orbital XY . The reduc- tron of the impurity ion (l = 2), S is the total spin angu- tion coefficients of the orbital angular momentum are lar momentum (S = 1/2), and L is the total orbital angu- determined from the following equality [4]: lar momentum of the impurity ion (L = 2). 〈|2, M L |〉2, M' Since the hyperfine interaction energy rapidly j decreases with an increase in the distance between the kMM' = ------〈||〉-. (1) dM L j dM' electron and the nucleus (due to a strong inhomogene-

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1908 ULANOV et al. ity of the nuclear magnetic field), the ligand contribu- seen from this table that the experimentally found tions to the molecular orbital can be ignored in calcula- ligand hyperfine interaction tensor components reverse tions of the parameters of the hyperfine interaction ten- sign upon changing over from the F(I) ligand to the sor. Within this approximation, we obtain the following F(II) and F(III) ligands. This implies that both mecha- 2+ equalities for a Cu impurity ion in an SrF2 crystal: nisms responsible for the nonzero spin density at nuclei of the F(II) and F(III) ligands must be included in cal- 24 2 32 2 16 2 12 a|| = P ------a1 Ð ------a2 Ð ------a3 + ------a2a3 culations of the tensor components of ligand hyperfine 7 7 7 7 interactions with nuclei of these ligands. In actual fact, (4) when the contributions from the excited electron con- ᏺ()2 2 2 ---Ð a1 + a2 Ð a3 , figurations are ignored in the calculations of the tensor components for ligands of group II, we obtain the rela- 22 4 ᏺ tionships a⊥ = P ------a1a3 +,---a1a2 Ð 2 a1a2 (5) 7 7 2 2 2 A'()II = 0.1102λ a + 0.2204λσ a Ð 0.2364λπ a , c c z s s p p where P = 2g β g β 〈rÐ3〉, a = ------1 , a = ------2 , a = S e n n 1 2 3 A' ()II ≈ A' ()II (6) N2 N2 x y c 3 λ 2 λ 2 λ 2 ------, c1 = 0.9817, c2 = 0.0745, and c3 = 0.0331 (see [4]). = 0.1102 s as Ð0.1092 σ a p + 0.1182 π a p. N1 Since the ratio of the parameters λσ and λπ is approx- For the experimental parameters a|| = 360 MHz and imately equal in magnitude to the ratio of the relevant a⊥ = 26 MHz, we obtain P = 872 MHz and N = 0.36. overlap integrals, it is evident that relationships (6) can- ≈ not provide the inequalities Az' (II) < 0 and Ax' (II) 4. ELECTRONIC STRUCTURE PARAMETERS Ay' (II) > 0. In order to achieve complete agreement CHARACTERIZING THE UNPAIRED-ELECTRON with the experimental values of the ligand hyperfine DENSITY IN THE VICINITY OF NUCLEI interaction tensor components, proper allowance must OF MORE DISTANT LIGAND GROUPS be made for additional contributions to the ligand The electronic structure parameters characterizing hyperfine interactions with ligands of this group, i.e., the unpaired-electron density in the vicinity of nuclei of for the electron transfer from the F(II) ligand to semi- more distant ligand groups can be calculated from the filled and unfilled shells of the impurity ion. The elec- relationships derived in our previous work [4] for tron transfer from a ligand of this group to the Cu2+ 3d9 ligands of group I. Note that the normalizing factors for shell corresponds to a higher order of the perturbation molecular orbitals corresponding to the ground state of theory and, hence, will not substantially affect the situ- the complex remain as before (because, in the case ation. However, as was shown earlier in [10, 11], the under consideration, the covalent contributions from electron transfer from the ligand to ns unfilled shells of ligands of groups IIÐV are disregarded). The ligand the impurity ion can make a contribution comparable to hyperfine interaction tensor components (i.e., the left- the effects of orbital overlap and the covalence of bonds hand sides of equalities (17) in [4]) can be obtained by between the impurity ion and the ligand. According to subtracting the relevant components of the tensors [10], the appropriate expressions for ligand hyperfine d A(Fi) and A (Fi) taken from the table presented in [2] interaction tensor components accounting for the elec- (for i II or III). It should be remembered that equal- tron transfer from the ligand to the Cu2+ 4s unfilled shell ities (17) in [4] are derived under the conditions where can be written in the form the nonzero spin density at the ligand nuclei is deter- (), mined by both the covalence of bonds in the complex ' 1.998G 4s 3d γ λ Az = ------∆ × 4s, 2s 2s, 4sas and the effects of overlap of 3d orbitals of the impurity 4s, 2s 5 ion with p and s orbitals of the ligands; i.e., the admix- (), 3.997G 4s 3d γ λ ture of excited electron configurations is ignored in Ð ------∆ × 4s, 2 p 2 p, 4sa p, these equalities. However, the ground electron configu- 4s, 2 p 5 ration of the impurity ion can involve such an admix- (), 1.998G 4s 3d γ λ (7) ture, in particular, due to electron transfer from the Ax' = ------4s, 2s 2s, 4sas ∆ , × 5 filled shells of the ligand to the 4s unfilled shells of the 4s 2s (), copper ion. Numerical relationships for these contribu- 1.998G 4s 3d γ λ + ------4s, 2 p 2 p, 4sa p, tions were deduced in [10]. As follows from these rela- ∆ , × 5 tionships, the relative contribution of the above mecha- 4s 2 p nism to the ligand hyperfine interaction is negligible for A' ≈ A' , a small distance between the paramagnetic ion and the y x ligand but becomes predominant in the case when this where G is the parameter related to the Coulomb inter- distance increases. The necessity of considering these action between the 3d and 4s unpaired electrons. For ∆ corrections stems from the result of analyzing the copper and fluorine ions, the energy separations 4s, 2s ∆ Ð1 parameters given in the table presented in [2]. It can be and 4s, 2p are equal to 340 000 and 190 000 cm ,

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 ELECTRONIC STRUCTURE OF BIVALENT COPPER OFF-CENTER COMPLEXES 1909 respectively, and G(4s, 3d) = 3690 cmÐ1 [10]. In the tion of the 4s electron shell is governed by the parame- case when the copper–fluorine covalent bond is rela- D D ters γ = 0.004, γσ = Ð0.122, S = 0.079213, and Sσ = tively weak, the parameters λ and λ should be s s s 2s, 4s 2p, 4s Ð0.111709. close in magnitude to the overlap integrals S2s, 4s and γ γ (2) For the F(III) ligand, the contribution of the 3d S2p0, 4s, whereas the parameters 4s, 2s and 4s, 2p can be λ ≈ electron shell to the ligand hyperfine interaction is determined from the approximate equalities 2s, 4s λ λ γ λ ≈ γ determined by the parameters s = 0.0016, σ = S2s, 4s + 4s, 2s and 2p0, 4s S2p0, 4s + 4s, 2p0 [9]. − λ D D We attempted to interpret the available experimental 0.0087, π = 0.0026, Ss = 0.000011, Sσs = 0.000461, data on ligand hyperfine interactions with nuclei of D D Sσ = Ð0.003215, and Sπ = 0.000935, whereas the the F(III) ligands under the same assumptions as for the Z FÐ(II) ions. As a result, the parallel component of the contribution of the 4s electron shell is governed by the γ γ D D parameters s = 0.02, σ = Ð0.09, Ss = 0.079, and Sσs = tensor Az' (III) for ligand hyperfine interactions with nuclei of the F(III) ligands was found to be close to the Ð0.1117. experimental value. However, the perpendicular components Ax' (III) and Ay' (III) defied description not only REFERENCES in magnitude but also in sign. In order to approach the 1. M. M. Zaripov and V. A. Ulanov, Fiz. Tverd. Tela (Len- experimental value, we had to increase the covalence ingrad) 31 (10), 251 (1989) [Sov. Phys. Solid State 31, parameter of the σ bond. In our opinion, this increase in 1796 (1989)]. the strength of the σ bond can be explained in terms of 2. I. I. Fazlizhanov, V. A. Ulanov, and M. M. Zaripov, Fiz. additional bonding with the participation of the F(I) Tverd. Tela (St. Petersburg) 43 (6), 1018 (2001) [Phys. ligand as an intermediate unit. Solid State 43, 1052 (2001)]. It should be noted that, for F(II) and F(III) ligands, 3. S. K. Hoffmann and V. A. Ulanov, J. Phys.: Condens. the calculated and experimental differences of the per- Matter 12, 1855 (2000). pendicular components (A' Ð A' ) are opposite in sign. 4. V. A. Ulanov, M. Krupski, S. K. Hoffmann, and x y M. M. Zaripov, J. Phys.: Condens. Matter 15, 1081 This disagreement indicates that the parameters of (2003). ligand hyperfine interactions with nuclei of these 5. H. Bill and R. Lacroix, in Proceedings of XVII Congress ligands are substantially affected by the electron trans- AMPERE (1973), p. 233. 2+ fer from the ligand to the Cu 4p unfilled shell. The 6. H. Bill, Phys. Lett. A 44 (2), 101 (1973). appearance of an electron in this shell is not surprising, 7. M. M. Zaripov and V. A. Ulanov, Fiz. Tverd. Tela (Len- because the potential of the crystal field acting on elec- ingrad) 31 (10), 254 (1989) [Sov. Phys. Solid State 31, trons of the impurity copper ion involves odd harmon- 1798 (1989)]. ics, which, in turn, can be responsible for an admixture 8 9 8. I. B. Bersuker, Electronic Structure and Properties of of the 3d 4p excited configuration in the 3d ground Transition Metal Compounds. Introduction to the Theory configuration. However, according to the results of our (Wiley, New York, 1996). calculations, the appearance of an electron at the 4p 9. A. Abragam and B. Bleaney, Electron Paramagnetic orbitals also becomes possible due to electron transfer Resonance of Transition Ions (Clarendon, Oxford, 1970; from the ligand. Mir, Moscow, 1973), Vol. 2. The final results obtained for ligand hyperfine inter- 10. O. A. Anikeenok and M. V. Eremin, Fiz. Tverd. Tela actions with ligands of groups II and III are as follows: (Leningrad) 23 (3), 706 (1981) [Sov. Phys. Solid State (1) For the F(II) ligand, the contribution of the 3d 23, 401 (1981)]. electron shell to the ligand hyperfine interaction is deter- 11. O. A. Anikeenok, R. M. Gumerov, and Yu. V. Yablokov, λ λ λ mined by the parameters s = 0.0061, σ = Ð0.0212, π = Fiz. Tverd. Tela (Leningrad) 26 (8), 2249 (1984) [Sov. D D D Phys. Solid State 26, 1365 (1984)]. 0.0008, Ss = 0.000094, Sσs = 0.003437, SσZ = − D 0.010874, and Sπ = 0.003942, whereas the contribu- Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1910Ð1917. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1818Ð1824. Original Russian Text Copyright © 2003 by Argunova, Sorokin, Pevzner, Balitskiœ, Gannibal, Je, Hwu, Tsai. DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

The Influence of Defects in the Crystal Structure on Helium Diffusion in Quartz T. S. Argunova1, 5, L. M. Sorokin1, B. Z. Pevzner2, V. S. Balitskiœ3, M. A. Gannibal4, J. H. Je5, Y. Hwu6, and W.-L. Tsai6 1 Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] 2 Grebenshchikov Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, St. Petersburg, 199155 Russia 3 Institute of Experimental Mineralogy, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia 4 Geological Institute, Kola Research Center, Russian Academy of Sciences, ul. Fersmana 14, Apatity, Murmanskaya oblast, 184209 Russia 5 Department of Materials Science and Engineering, Pohang University of Science and Technology, Pohang, Republic of Korea 6 Institute of Physics, Academia Sinica, Nankang, Taipei, Taiwan, Republic of China Received February 18, 2003

Abstract—The solubility and diffusion of helium in quartz crystals are investigated as functions of the distribution and density of structural defects. The types of defects in the crystals are identified and their distribution over growth sectors is determined by x-ray diffraction topography and phase radiography with a synchrotron radiation source. The effective solubility and effective diffusion coefficients for helium in quartz are estimated from the experimental data on the amount of helium extracted from samples with different contents of defects. It is revealed that the effective diffusion coefficient of helium depends on the number of dislocations.© 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION lating the linear dependence to the low-temperature 3 ≈ Investigation into the behavior of helium in the sub- range, they determined the diffusion coefficients DHe surface environment can offer a clue to the solution of 4 Ð18 2 ≈ Ð20 2 ° a number of important problems. A plausible explana- 10 cm /s and DHe 10 cm /s at 20 C. tion of the accumulation and migration of cosmogenic helium can give impetus to the development of geo- Tolstikhin et al. [4] studied crystals of natural quartz chronological methods [1]. The high helium content in aged 1.8 billion years. These crystals contained gas ground waters makes it possible to elucidate the helium microinclusions. The analysis of the isotopic helium origin and, in particular, to perform environmental composition in microinclusions demonstrated that the monitoring, because one of the sources of helium is isotopic ratios in microinclusions 1Ð2 cm apart did not grounds where radioactive waste is buried [2]. With level off throughout the above period of time. By knowledge of the migration rate of radiogenic helium assuming that equalization of the concentrations pro- in rocks, it is possible to estimate the amount of helium ceeded through the diffusion of helium atoms, these accumulated in ground waters and to determine the authors solved the diffusion equation and determined 4 Ð19 2 concentration of radioactive elements (helium sources). the diffusion coefficient (DHe = 10 cm /s) at which Since quartz is universally present in rocks, the study of the calculated and experimental ratios 3He/4He in helium diffusion in quartz is an important stage in solv- microinclusions coincide with each other. It was also ing the above problem. assumed that a number of microdefects separated by Trull et al. [3] experimentally determined the diffu- distances comparable to the unit cell size intersect each sion coefficients of 3He and 4He isotopes in quartz from microinclusion. In this case, an atom has to execute one the amount of helium extracted upon annealing of grains or several hoppings in order to find its way into a defect prepared by grinding natural quartz. From the experi- and can then almost instantaneously move along the mental data obtained in the temperature range defect. However, although the authors made the 150−600°C, these authors constructed a linear depen- assumption that the migration rates of helium atoms in dence of the logarithm of the diffusion coefficient on the a defect-free lattice and along defects differ from each reciprocal of the temperature (according to the Arrhenius other, this assumption was ignored when solving the law) and calculated the activation energies. By extrapo- diffusion equation.

THE INFLUENCE OF DEFECTS IN THE CRYSTAL STRUCTURE 1911

The structure of the outer electron shells of helium Y plate dissolved in quartz differs from that of the solvent. Hence, it is reasonable to believe that helium atoms should diffuse over interstices. The lattice of low-temperature α-quartz has a close-packed structure, which can be represented as a helical sequence of SiO4 tetra- hedra. One pitch of the helix corresponds to six tetrahe- dra. Consequently, hexagonal helical channels are X formed along the Z axis [5]. In quartz, impurity ions can move rather freely through hexagonal channels along Z +X the c axis, whereas their motion in the direction perpen- Z Z dicular to the c axis is substantially hindered. It could be expected that helium atoms will also diffuse along Y –X the channels parallel to the c axis. However, the small diffusion coefficients of helium atoms [3, 4] suggest Fig. 1. Schematic drawing of the quartz crystal grown in a that the ordered quartz lattice is virtually impenetrable helium atmosphere. to helium. This inference is confirmed by the results obtained in [6, 7]. The transverse size of a hexagonal with helium was grown in a similar solution in an auto- channel is approximately equal to 0.23 nm. According clave with helium. The sizes of the seed with ZX orien- to different estimates, the diameter of a helium atom is tation were equal to 2, 4, and 140 mm along the Z, X, equal to 0.18Ð0.26 nm [8Ð10]. Therefore, the helium and Y axes, respectively. The autoclave with a batch diffusion in an ordered quartz lattice is reduced to the was filled with a standard sodium carbonate solution of motion of atoms through channels of a comparable composition 7 wt % Na2CO3 + 0.5 wt % NaOH. For a diameter and, hence, is accompanied by the overcom- filling factor of 78%, the autoclave was covered with a ing of high energy barriers. lid and was charged with helium through a sealed-in However, geological investigations have demon- capillary connected to a gas cylinder. When the helium strated that, despite the small diffusion coefficient, pressure in the autoclave reached 7 MPa, the autoclave radiogenic helium formed in rocks finds its way into was closed hermetically and placed in a vertical furnace ground waters and is transported by them. In this with a two-section heater, which provided the required respect, the problem of helium migration in quartz calls temperature conditions. The temperatures in the lower for further investigation. and upper autoclave zones were equal to 360 and In crystals, enhanced diffusion of impurities pro- 330°C, respectively. The water pressure in the auto- ceeds over lattice dislocations. Helium migration over clave was 84 MPa. The crystal was grown for 30 days. dislocations was analyzed by Klyavin [11] in the study The growth rate of the basal face gradually increased of the new phenomenon of dynamic dislocation pipe from the lower (0.43 mm/day) to the upper diffusion. However, this phenomenon is observed only (0.53 mm/day) face of the crystal. After cooling the under plastic deformation of materials in a liquid or autoclave to room temperature, the residual helium gaseous medium whose atoms can penetrate into a pressure was of the order of 5 MPa. The quartz crystal crystal through generating and moving dislocations. grown in a helium atmosphere is drawn schematically Helium diffusion over dislocations in a stable state was in Fig. 1. theoretically investigated by Klyavin et al. [12]. It was The samples suitable for x-ray diffraction investiga- shown that helium can migrate over screw dislocations tions were prepared in the for§m of plates cut out along at T < 300 K. the (0001) and {1101 } planes. After grinding and pol- However, experimental data on enhanced migration ishing, the plate thickness was 0.35 mm. of helium over dislocations in quartz are unavailable in the literature. In order to fill this gap, we studied the solubility and the desorption kinetics of helium in syn- 2.2. Experimental Technique thetic quartz samples with different dislocation densities. Structural defects were examined using x-ray diffraction topography with a laboratory source and phase-sensitive radiography with a synchrotron radia- 2. INVESTIGATION OF DEFECTS tion source. The x-ray topographic investigation was IN THE CRYSTAL STRUCTURE performed by the Lang method in Bragg and Laue geometries with the use of CuKα and MoKα radiation. 2.1. Sample Preparation The divergence in the scattering plane was of the order Crystals of synthetic quartz were grown by the ther- of the angular separation between the spectral lines α α 1 mal gradient method [13]. The quartz crystal without and 2. The resolution in the direction normal to the helium was grown in a standard sodium carbonate solu- scattering plane was of the order of several microns in tion on a seed with XY orientation. The crystal saturated the case when the distance between the crystal and a

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

1912 ARGUNOVA et al.

(a) 2 mm – 〈 〉 g = 1210 +X

〈–X〉 〈Z〉

(b) 0.1 mm

Fig. 2. (a) Lang topogram in a Lauer geometry (MoKα radiation, 1210 reflection) for a plate cut out parallel to the (1100 ) plane from a quartz crystal saturated with helium during growth. The rectangular at the center of the plate indicates the position of the seed. (b) Phase radiograph obtained from a series of images of the crystalÐseed boundary. The plate surface is parallel to the ()1100 plane. photographic film amounted to 5–10 mm. The phase 1392 × 1040 pixels and a sensitivity of 12 bits. Optical radiographic experiments were carried out with the use lenses made it possible to change the field of vision of nonmonocrhromatic radiation from a synchrotron from 4 × 4 to 0.6 × 0.4 mm. source (Pohang, Republic of Korea). The effective µ µ source size was 60 m along the vertical and 160 m 2.3. Experimental Results along the horizontal direction. The distance from the source to the sample was 30 m. The CdWO4 crystal Figure 2a shows a typical topogram of the plate cut scintillator mounted behind the sample transformed x- out parallel to the (1100 ) plane of the quartz crystal. ray radiation into a luminous flux. The distance The radiograph of this plate is given in Fig. 2b. The between the sample and the scintillator varied from crystal was grown in a helium atmosphere. It can be several centimeters to 1 m. The images were recorded seen from the topogram that the seed is a perfect crystal on a detector with a charge-coupled device matrix. free of dislocations. However, a large number of dislo- The charge-coupled device matrix had a size of cations are observed in the crystal beginning from the

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

THE INFLUENCE OF DEFECTS IN THE CRYSTAL STRUCTURE 1913 boundary with the seed. An analysis of the topograms recorded for different reflections revealed that the dislocation density is maximum (105 cmÐ2) in the growth sector 〈Z〉 of the (0001) basal plane and in the growth sectors 〈±X〉 of the {1120 } planes of the trigonal prisms. The dislocation density between the sectors 〈Z〉 and 〈+X〉 is lower than that in the sectors by at least three orders of magnitude. The x-ray topograms of the plate cut from the crystal grown in the absence of helium is shown in Fig. 3. – – The plate surface is parallel to the (0001) plane. The g = 1101 g = 1101 dislocation density in these samples is relatively low, 1 mm and images of individual dislocations can be distin- guished ed in the topograms. This allows us to determine the Burgers vectors b of dislocations. An analysis of the topograms in Fig. 3 shows that a group of dislocations is located in the sector 〈r〉 of the small rhombo- hedron. The images were obtained for six reflections from the {1011 } planes of the rhombohedra. The dislocation lines are almost parallel to the [1011 ] direction. The images of dislocations are characterized by the following features. The dislocations are not – – observed in the image for the 1011 reflection and are g = 1011 g = 1011 clearly seen in the images for all other reflections. Since the contrast of dislocations is weak even when the condition g á b = 0 is satisfied, we can make the inference that these are edge dislocations with the Burgers vector b = a[1210 ]. An analysis of the condition for complete extinction of the image of the dislocation g á b × l = 0 (where g is the diffraction vector and l is the vector specifying the direction of the dislocation line) demonstrates that the quantity |cos∠(g, b × l)| is maximum for the 1011 reflection and minimum for the 1011 reflection. This explains why the dislocation contrast is max- – – imum for the 1011 reflection. g = 0111 g = 0111 A comparison of the topograms displayed in Figs. 2 and 3 shows that the density of dislocations in the crys- Fig. 3. Lang topograms (MoKα radiation) illustrating the tal grown in the helium atmosphere is higher than the change in the dislocation contrast with a change in the density of dislocations in the crystal grown in the direction of the diffraction vector g. The plate surface is parallel to the (0001) plane. The contrast is opposite to the orig- absence of helium. It can be seen from Fig. 2a that dis- inal. locations are generated at the boundary with the seed, but the images of dislocation sources are not distin- guishable. Figure 2b displays the higher resolution along lines of dislocations decorated by impurities [14, images obtained by phase-sensitive radiography. The 15]. Therefore, inclusions bring about the generation of image of the crystalÐseed boundary is formed from a dislocations. series of photographs recorded by stepwise moving the sample in the incident beam. The radiograph shows It can be assumed that the presence of helium in a inclusions and etch channels. The channels arose from growth chamber disturbs the physicochemical equilib- chemical etching of the plate in a 40 vol % HF solution rium of the system. The primary interaction of the solu- (at 20°C for 20 h). It is evident that the etch channels tion with the seed led to a partial dissolution of the seed are associated with the inclusions at the crystalÐseed in the undersaturated solution. The regeneration of the boundary. The direction and density of etch channels damages caused by the dissolution was accompanied (Fig. 2b) coincide with the direction and density of dis- by the formation of inclusions. locations (Fig. 2a). Earlier x-ray topographic investiga- The samples used for studying the desorption were tions revealed that etch channels in quartz are formed prepared from the crystal saturated with helium and

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1914 ARGUNOVA et al. faces of the samples ~10 × 10 × 10 and ~3 × 3 × 30 mm in size were parallel to the (0001), (1100 ), and (1120 ) 50 µm planes. After saturation, the samples were placed in (a) ampules evacuated to a pressure of ~1.333 Pa. The ampules with the samples and empty evacuated ampules were sealed. Immediately prior to analysis, impregnated samples were cut into fragments so that, during saturation, one central fragment was not in con- (b) tact with helium, whereas the other fragments had different areas in contact with helium. The crystal grown in the medium containing helium was cut perpendicularly to the Z axis into samples 3Ð4 (c) mm thick. In turn, the samples were cut into fragments corresponding to different growth sectors. A quantitative analysis of the samples for helium and an investigation into the kinetics of helium desorption were performed using mass spectrometry [16] (d) under different heating conditions (stepwise or isothermal heating and melting). For this purpose, crucibles Fig. 4. In situ phase radiographs of the sample at tempera- × Ð4 tures of (a) 20, (b) 350, (c) 420, and (d) 550°C. evacuated to a residual pressure of 1.333 10 Pa and equipped with sample holders were connected to the inlet and purification system of the mass spectrometer. The samples were dropped in turn into crucibles in could contain inclusions. In the course of desorption, which helium was accumulated at specified times and the samples were subjected to heating. Cracks should temperatures. The melting temperature of the samples appear in regions where stresses caused by the pressure was equal to 1750°C. Extracted helium was fed into of an expanding liquid contained in the formed inclu- the inlet and purification system of the mass spectro- sions exceeded the ultimate stress of the material. With meter. the aim of examining the behavior of inclusions during heating, we performed the following experiment. A The helium concentration was measured from the polished unetched quartz plate cut out parallel to the peak height on an MI-1201 mass spectrometer. Prior to analysis of each sample, the mass spectrometer was cal- (1100 ) plane was placed in a furnace and was intro- ibrated against a reference sample. The sensitivity of the duced into the incident beam of synchrotron radiation. mass spectrometer for 4He was equal to 5 × 10Ð5 A/Torr, The plate was slowly heated to 550°C in air. The and the total experimental error was 10%. The helium images of inclusions were recorded at intervals, during amount measured after each five or six experiments each of which the temperature increased by 10°C. The with the samples was approximately equal to 10Ð9 cm3. field of vision moved over the plate. In order to observe The results of the measurements were corrected to a the spatial arrangement of inclusions, the sample was temperature of 0°C and a pressure of 0.1 MPa. rotated about the axis perpendicular to the incident beam. The images were also recorded in the course of cooling of the sample and after complete cooling to 3.2. Helium Content room temperature. We studied two samples. A number The results of the investigation into the extraction of of typical images are given in Fig. 4. Neither cracks nor helium from the samples cut from the crystal grown in new inclusions that could be formed as a result of par- the absence of helium and then saturated with helium tial healing of cracks were revealed. can be summarized as follows. The helium content in Reasoning from the structural-defect distribution the initial crystal was less than 0.5 × 10Ð9 cm3/g. The determined from the topograms and radiographs, the helium content in the ampule after one- to three-month helium-containing quartz crystal was cut into frag- storage of the samples was (0.3Ð0.6) × 10Ð6 cm3/g. ments with different dislocation densities. Upon melting, the amount of released 4He was equal to (0.76Ð1.58) × 10Ð6 cm3/g. The Ostwald solubility coefficient of helium in quartz was determined to be SOs = 3. INVESTIGATION OF THE SOLUBILITY (0.6Ð1.2) × 10Ð7 cm3/g atm, which is approximately AND DIFFUSION OF HELIUM IN QUARTZ four orders of magnitude less than that in cristobalite 3.1. Sample Preparation and Experimental Technique and tridymite [17]. Since the density of dislocations in the samples was relatively low (<102 cmÐ2) and the The samples cut from the helium-free crystal were crystal did not contain inclusions, it was inferred that saturated with helium at a pressure of 3.0Ð3.25 MPa this coefficient characterizes the solubility of helium in and a temperature of 290Ð355°C for 100Ð3000 h. The the ordered quartz lattice. The amount of helium dis-

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 THE INFLUENCE OF DEFECTS IN THE CRYSTAL STRUCTURE 1915 solved in the samples did not depend on their orienta- Qt/Q∞ tion, the contact area with helium in the course of satu- 1.0 ration, or the saturation time.

3.3. Kinetics of Helium Desorption 0.6 1 The desorption kinetics was studied upon heating of 2 the samples (cut from the crystal saturated with helium 3 during growth) with different dislocation densities. The samples were rapidly heated to temperatures of 300, 0.2 ° 350, and 400 C and were isothermally held for 15Ð30 0 h. Then, the samples were melted. The time dependences of the amount of extracted helium indicate that 0246 the desorption rate initially increases: with an increase t, h1/2 in the temperature, the helium amount first increases almost linearly and then reaches saturation. For the samples with the maximum dislocation density, the Fig. 5. Dependences of Qt/Q∞ on t upon helium extrac- amount of helium desorbed upon heating is very close tion from samples with different dislocation densities ρ at to the total amount of helium extracted upon melting. temperatures of (1, 3) 350 and (2) 400°C. (1, 2) ρ ~ 105 and By contrast, the desorption from the samples cut from (3) ρ < 102 cmÐ2. the crystal regions with a low dislocation density is hindered: only an insignificant amount of dissolved helium can be extracted upon heating during the above period losses through a cracked mosaic layer on the surface. In of time. The remaining helium is released (in a consid- our experiments, all these conditions were satisfied. As erably larger amount) upon melting. follows from the results of the in situ experiment (Fig. 4), the helium desorption from inclusions can pro- Let us assume that Q∞ is the total amount of helium ceed only through the diffusion mechanism. dissolved in the sample and Qt is the amount of helium desorbed upon heating for the time t. Barker and Vaughan [17] derived the relationship describing the The experimental dependences of Qt/Q∞ on t are helium diffusion from particles with a total area S and depicted in Fig. 5. It can be seen from this figure that volume V when the diffusion coefficient D depends on the slopes of the linear portions of curves 1 and 2 are the time t; that is, appreciably larger than those of curve 3. Curves 1 and 2 correspond to the desorption of helium from the sam- 2 2 〈 〉 1 d Qt 4S ples cut from the growth sectors Z with the dislocation 5 Ð2 ------------= ------2. density ρ ~ 10 cm . Curve 3 was obtained by measur- Ddt Q∞ π V ing the extraction of helium from the crystal regions Therefore, the diffusion coefficient D can be deter- located between the growth sectors 〈Z〉 and 〈+X〉. These mined from the slope of linear portions of the depen- samples are characterized by the dislocation density ρ < 102 cmÐ2. The inference can be made that the diffu- dence of Qt/Q∞ on t . Our data demonstrate that the sion rate of helium is higher and the activation energy ratio Qt/Q∞ depends on the density of dislocations in of diffusion is lower for the samples with a high dislo- the samples. Consequently, the slope of the dependence cation density. of Qt/Q∞ on t can be treated as the effective diffusion The diffusion coefficient of helium in quartz was coefficient characterizing the dislocation pipes diffu- calculated from the data obtained for the crystal grown sion of helium. in a standard hydrothermal solution and saturated with It is expedient to compare the data obtained for sam- helium at a high temperature and a high pressure. The ples with different dislocation densities in the case samples used for investigating the extraction had the when the samples insignificantly differ in size, are free form of a cube with the characteristic sizes listed in the of internal cracks, and exhibit approximately identical table. The problem of the helium flux from a cubic sam-

Experimental conditions and typical effective diffusion coefﬁcients ° Ð6 3 ρ Ð1 Ð6 Ð1 Ð8 2 T, C Q∞, 10 cm /g d, cm , cm h, cm K, 10 s Deff, 10 cm /s 250 0.167 0.62 10 0.1 2.66 2.6 275 0.121 0.63 10 0.1 2.1 2

Note: T is the heating temperature of the sample, Q∞ is the total amount of helium extracted upon melting, d is the edge length of the cubic fragments, ρ is the dislocation density in the sample, and h = 1/ρ is the diffusion length in the defect-free lattice.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1916 ARGUNOVA et al. for increasing interstices to sizes large enough for an impurity atom to pass through. An increase in the inter- stice size due to the presence of dislocations results in a decrease in the energy of impurity migration over dislocation pipes. Since the diffusion is described by the Arrhenius law D ~ exp(ÐW/kT) (where W is the activation energy of migration, k is the Boltzmann constant, and T is the absolute temperature), a decrease in the activation energy W leads to a substantial increase in the diffusion coefﬁcient. Moreover, edge dislocations are sinks for vacancies, which, in turn, should enhance the migration of helium atoms.

4. CONCLUSIONS The main results obtained in this work can be summarized as follows. Crystals of synthetic quartz were grown in a standard hydrothermal solution in an autoclave with and without helium. The crystals contained different amounts of dislocations and inclusions, whose densities and arrangement were determined using x-ray diffraction topography and phase radiography. The desorption of helium Fig. 6. A model of the edge dislocation with l = [0001] and from different crystals was studied by mass spectrome- b = a[1210 ] in the projection onto the (0001) plane. try. It was shown that, for a crystal saturated with helium Shaded regions correspond to hexagonal channels along the during growth, the desorption rate of helium from the Z axis. The dark region at the center of the figure represents the hexagonal channel coinciding with the core of the edge samples cut from crystal regions with a high dislocation dislocation. Only the Si atoms are shown. The dashed line density is higher and the activation energy for diffusion indicates the Burgers contour. in these samples is lower than those for the samples with a considerably lower dislocation density. The helium dif- ple was solved under zero boundary conditions and the fusion coefficient calculated from the data on the helium assumption of a uniform distribution of helium over the desorption from the samples with a dislocation density of sample volume. The solution was obtained for the coef- 102 cmÐ2 was close in order of magnitude to the diffusion Deff coefficients available in the literature. A substantial ficient K = ------2 , where Deff is the effective diffusion increase in the diffusion coefficient for the samples with h 5 Ð2 coefficient and h is the diffusion length. The time depen- a dislocation density of 10 cm was explained by the dence of the flux Q(t, K) was calculated and the best increase in the size of the interstices due to the presence agreement between the calculated and experimental data of edge dislocations and by the decrease in the energy of was achieved by varying the coefficient K (see table). impurity migration over dislocation pipes. The Ostwald solubility coefficient of helium in the real quartz lattice The calculated effective diffusion coefficients Deff was determined. are close in order of magnitude to those determined by Trull et al. [3]. However, an increase in the dislocation density to 105 cmÐ2 should lead to a considerable ACKNOWLEDGMENTS decrease in the size of defect-free regions and to a We would like to thank I.N. Tolstikhin for providing change in the diffusion mechanism. In this case, the migration of helium over dislocation pipes should dom- an opportunity to participate in the studies performed inate over the interstitial diffusion. An analysis of the within the Swiss Research Program and for helpful contrast in dislocation images demonstrates that, for the consultations and S.V. Tarakanov for his assistance in most part, the studied crystals involve edge disloca- processing diffusion data. tions. An edge dislocation always creates a region of This work was supported by the Swiss Research volume expansion in the crystal lattice. In the disloca- Program “Helium Migration in Subsurface Environ- tion core, the upper horizontal rows of atoms are dis- ments” (N 7SUPJ062127) and the Russian Foundation placed upward to a greater extent than the lower hori- for Basic Research (project no. 03-02-16613). zontal rows of atoms, which results in a distortion of T.S. Argunova acknowledges the support of the Korean hexagonal helical channels whose location coincides Foundation KISTEP for the opportunity to perform with dislocation cores (Fig. 6). This leads to an increase synchrotron investigations in the framework of the in the size of the interstices. It is well known that the KoreaÐRussia Joint Research Program. J.H. Je activation energy for diffusion of helium-like rare-gas acknowledges the support of the National Research atoms in quartz involves the elastic energy necessary Laboratory Program.

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REFERENCES 10. L. Pauling, The Nature of the Chemical Bond and the Structure of Molecules and Crystals, 2nd ed. (Cornell 1. B. A. Mamyrin, G. S. Anyfriev, U. L. Kamensky, and Univ. Press, Ithaca, N.Y., 1940; Gostekhizdat, Moscow, I. N. Tolstihin, Geochem. Int. 7, 498 (1970). 1947). 2. I. N. Tolstihin, B. E. Lehmann, H. H. Loosli, and 11. O. V. Klyavin, Fiz. Tverd. Tela (St. Petersburg) 35, 513 A. Gautschi, Geochim. Cosmochim. Acta 60, 1497 (1993) [Phys. Solid State 35, 261 (1993)]. (1996). 12. O. V. Klyavin, P. P. Likhodedov, and A. N. Orlov, Fiz. 3. T. W. Trull, M. D. Kurz, and W. J. Jenkins, Earth Planet. Tverd. Tela (Leningrad) 28, 156 (1986) [Sov. Phys. Solid Sci. Lett. 103, 241 (1991). State 28, 84 (1986)]. 4. I. N. Tolstikhin, É. M. Prasolov, L. V. Khabarin, and 13. Synthesis of Minerals (VNIISIMS, Aleksandrov, 2000), I. Ya. Azbel’, in Geochemistry of Radioactive and Radio- Vol. 1. genic Isotopes (Nauka, Leningrad, 1974), pp. 79Ð90. 14. A. R. Lang and V. F. Miuscov, J. Appl. Phys. 38, 2477 5. E. Gorlich, Ceram. Int. 8, 3 (1982). (1967). 6. E. V. Kalashnikov and B. Z. Pevzner, Fiz. Tverd. Tela 15. F. Iwasaki, J. Cryst. Growth 39, 291 (1977). (St. Petersburg) 44, 283 (2002) [Phys. Solid State 44, 294 (2002)]. 16. U. L. Kamensky, I. N. Tolstikhin, and V. R. Vetrin, Geochim. Cosmochim. Acta 54, 3115 (1990). 7. G. G. Boœko and G. V. Berezhnoœ, Fiz. Khim. Stekla 29, 65 (2003). 17. R. M. Barker and D. E. W. Vaughan, Trans. Faraday Soc. 63, 2275 (1967). 8. J. F. Shackelfold, J. Non-Cryst. Solids 49, 299 (1982). 9. R. H. Doremus, Glass Science (Wiley, New York, 1973), p. 133. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1918Ð1925. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1825Ð1832. Original Russian Text Copyright © 2003 by Bublik, Matsnev, Shcherbachev, Mezhennyœ, Mil’vidskiœ, Reznik. DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Diffuse X-ray Scattering Study of the Formation of Microdefects in Heat-Treated Dislocation-Free Large-Diameter Silicon Wafers V. T. Bublik*, S. Yu. Matsnev*, K. D. Shcherbachev*, M. V. Mezhennyœ**, M. G. Mil’vidskiœ***, and V. Ya. Reznik*** * Moscow Institute of Steel and Alloys, Leninskii pr. 4, Moscow, 117936 Russia ** Giredmet State Research Institute for the Rare-Metals Industry, Bol’shoœ Tolmachevskiœ per. 5, Moscow, 109017 Russia e-mail: [email protected] *** Institute for Chemical Problems of Microelectronics, Bol’shoœ Tolmachevskiœ per. 5, Moscow, 109017 Russia Received March 11, 2003

Abstract—Diffuse x-ray scattering (DXS) is used to study the formation of microdefects (MDs) in heat-treated dislocation-free large-diameter silicon wafers with vacancies. The DXS method is shown to be efficient for investigating MDs in silicon single crystals. Specific defects, such as impurity clouds, are found to form in the silicon wafers during low-temperature annealing at 450°C. These defects are oxygen-rich regions in the solid solution with diffuse coherent interfaces. In the following stages of decomposition of the supersaturated solid solution, oxide precipitates form inside these regions and the impurity clouds disappear. As a result of the decomposition of the supersaturated solid solution of oxygen, interstitial MDs form in the silicon wafers during multistep heat treatment. These MDs lie in the {110} planes and have nonspherical displacement fields. The volume density and size of MDs forming in the silicon wafers at various stages of the decomposition are determined. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 2. EXPERIMENTAL The processes of microdefect formation in disloca- Defect formation in the early stages of decomposition-free single-crystalline Si wafers are usually stud- tion of a supersaturated solid solution of oxygen in sil- ied using optical microscopy (OM) and transmission icon during multistep heat treatment of wafers was electron microscopy (TEM). OM provides only very studied by DXS on a three-crystal x-ray diffractometer limited possibilities for investigating the nature of using the dispersion-free scheme (n; Ðn; +n) with triple microdefect (MD) formation and the transformation of reflection from a monochromator and an analyzer. A MDs upon annealing of the wafers. Our studies showed 1.5-kW x-ray tube with a copper anode was used. A slit that the application of TEM for investigating the early was placed in front of a sample to transmit the Kα1 com- stages of decomposition of a supersaturated solid solu- ponent of the radiation. The irradiated region on a sam- tion of oxygen in silicon also encounter significant dif- ple was 0.5 × 7 mm in size. The DXS intensity distribu- ficulties caused mainly by the fact that MDs forming at tion I(q), where q = G Ð H is the vector characterizing these stages induce only small distortions of the crystal the deviation of the diffraction vector G from the recip- lattice. Moreover, the use of TEM implies constraints rocal lattice vector H, was measured in the vicinity of on the minimum volume density of defects in samples. the 004 site. The vector q components in the diffraction plane are connected with the experimental geometric Investigation of the microdefect formation in the parameters by the relation early stages of decomposition of a supersaturated solid ()∆θ, ()θ ()θ∆θ ∆θ λ solution of oxygen in silicon makes it possible to reveal qz qx = 3 cos B 2 2 Ð 3 sin B / , (1) not only nucleation mechanisms but also the transfor- ∆θ ∆θ mation of the medium with defects when an internal where 2 and 3 are the angular deviations from the θ getter is created in a wafer. It is reasonable to suppose exact Bragg position B for the sample and analyzer, that valuable information can be obtained using diffuse respectively; λ is the x-ray wavelength; the Z axis is x-ray scattering (DXS). In this work, we use DXS to normal to the sample surface; and the X axis is parallel investigate the processes of microdefect formation in to the sample surface (both axes lie in the diffraction heat-treated dislocation-free silicon wafers. plane).

DIFFUSE X-RAY SCATTERING STUDY OF THE FORMATION OF MICRODEFECTS 1919

Ω 1 According to [1, 2], the DXS intensity distribution is where mn = --- (Fnbm + Fmbn), d = C11 Ð C12 Ð 2C44, with determined by the Fourier transform of the displace- 2 ment field of a defect u(q): Cij being the components of the elastic tensor of a cubic crystal in the matrix notation; b is the Burgers vector; I()q ∼ Gu() q 2. (2) and F is the vector specifying the plane of location of dislocation loops. The tensor Tij is defined as If local distortions around defects are small and () Gu(q) Ӷ 1, the differential cross section of diffuse scat- T ij = hk/V c gije j, (6) tering can be written as [1, 2] where hk and ej are the direction cosines of the vectors () Ð1 Sdif q () H and q, respectively; gij = ∑kl Cijklekel ; and Vc is 2 2 3 ()5/2 (3) the unit cell volume. 2 Ð2M H C π γ HC π γ = NF e ------∑ i i + ------1 1 , q2 V 2q The first term in parentheses in Eq. (3) corresponds i c H to the Huang scattering Sdif (q) distributed symmetri- where F is the structure amplitude, N is the number of H H scattering centers, eÐ2M is the DebyeÐWaller tempera- cally with respect to Sdif (q) = Sdif (Ðq). The second term ture factor, and C is the strength of the defect, which determines the asymmetric part of scattering, S A (q), characterizes the defect-induced change in the crystal dif ∆ and leads to a shift in the diffuse scattering distribution volume V and can be expressed in terms of the defect toward positive or negative values of q depending on parameters (shape, size, deformation sign) for a certain z the sign of ∆V. For example, for microdefects formed type of MD. For a defect producing a displacement ∆ 2 from interstitial atoms, V > 0 and the diffuse scatter- field of the Coulomb type u(r) ~ C(r0)/r , the sign of ing distribution shifts toward positive values of qz, the defect strength C(r0), which generally depends on whereas, for defects of the vacancy type (∆V < 0), it the direction of the unit vector r0 in the real space, is shifts toward negative values of q . determined by the difference ∆V between the volumes z π of the matrix and the substitutional defect: if ∆V > 0, Some of the parameters i vanish for a certain sym- then C > 0 (and vice versa). metry of defects. For example, for defects of cubic π γ π symmetry, only 1 is nonzero; in the case of tetragonal In Eq. (3), i and i are defined according to [3]: π symmetry, 3 = 0; and in the case of trigonal symmetry, π π ()2 2 = 0 [4]. Moreover, at certain indices m and n corre- 1 = Tr Pmn /3, γ sponding to high-symmetry directions, the values of i also become zero. As a result, the diffuse scattering π ()2 ADS 2 = ∑ Pnn Ð Pmm /6, intensity Sdif (q) for such directions in the reciprocal nm> space can be zero for a certain symmetry of a defect. When point defects associate with each other during π ()2 the course of structural transformations accompanying 3 = 2 ∑ Pmn /3, (4) post-solidification cooling and subsequent anneals, the nm> size of the region of strong distortions around a defect γ ()2 1 = Tr T ij /3, increases. As a result, the range of q values over which the relation I(q) ~ qÐ2 is valid becomes narrower. For strongly distorted regions (Gu(q) ӷ 1), the diffuse scat- γ ()2 2 = ∑ T ii Ð T jj /3, tering intensity can be found using the StokesÐWilson ADS > ӷ ij approximation (asymptotic scattering Sdif (q)). At q Ð1/2 q0(GC) , the cross section of asymptotic diffuse scat- γ ()2 tering is [2] 3 = ∑ T ij Ð T ji /2, > ij ADS 2 Ð2M CH S ()q = NF e ------Ψ()mn, , (7) where TrP and TrT are the traces of the dipole dif 2 4 mn ij V c q moment tensor Pmn and the tensor Tij described by Eq. (6), respectively. where m and n are the unit vectors of q and H, respectively, and Ψ is a function depending on the angle In the case of dislocation loops, the components of between q and H. Here, the function Ψ ~ 2 [2] specifies the dipole moment tensor P are expressed as follows: the anisotropy of the asymptotic scattering distribution, which depends only on the directions of the vectors H ()δΩ Ω Ω Pmn = C12 Tr mn + nnd nm + 2C44 nm, (5) and q.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1920 BUBLIK et al. The angular dependence of the diffuse scattering (3) Range of Scattering by a Defect Core intensity in the reciprocal space can be conveniently ≤ represented in the form of surfaces of equal intensity It is known that the region of r R0 (Laue scattering) (isodiffuse surfaces) or their sections made by planes is difficult to achieve experimentally, since, starting passing through reciprocal lattice sites (isodiffuse from a certain value qT, scattering by thermal lattice curves). For isodiffuse surfaces and small deviations vibrations predominates over x-ray diffuse scattering from the diffraction vector, the quantity q can be written by defects if their concentration is low [2]. The inten- as sity of the former scattering is proportional to qÐ2, which corresponds to a slope of Ð2. Thermal scattering 3 can be used as an “internal standard” to determine the 2 π γ q = const∑ i i. (8) defect concentration [6]. i One of the methods of determining the MD concen- The shape of an isodiffuse surface is substantially tration in single crystals is based on the fact that, at determined by the type of defects present, their position scattering vectors with q < q0, the Huang scattering in the lattice, and the symmetry of their displacement intensity SH(q) predominates over the thermal diffuse fields. The general shape of the isodiffuse surfaces scattering intensity ST(q) ~ qÐ2. Hence, the quantity allows us to find the symmetry of the displacement field ST(q) can be measured for a crystal under study by of a point defect and to choose one of the possible dis- using the same experimental procedure as that placement configurations [1, 2]. Sometimes, it suffices employed for measuring the intensity SH(q) at the same to consider the DXS intensity distribution along the reciprocal lattice vector H. directions parallel and normal to the diffraction vector. Using the expression for ST(q) from [7], we obtain As the magnitude of the vector q increases, the character of the decrease in the differential DXS intensity 2 T () 2 Ð2M kTH (), I(q) changes [1, 5]. In order to analyze changes in the S q = N0 F e ------2 K mn, (10) behavior of the I(q) dependence, logI = f(logq ) plots V cq are conveniently used. There are three q magnitude ranges in which the laws of the decrease in I(q) are dif- where K(m, n) is the Christoffel determinant [7], N0 is ferent. the number of unit cells in the crystal, k is the Boltz- mann constant, and T is the absolute temperature of the sample. (1) Huang Scattering Range Charniy et al. [6] derived a convenient expression for determining the MD concentration nd in a crystal X rays are scattered by weak elastic distortions of a from the intensity ratio of Huang and thermal scatter- crystal lattice caused by elastic displacement fields of ing: the defects. A defect can be considered as a point source of force. The distance r from the center of a defect is H 2 2 S ()q n ()∆V q Π()mn, much greater than the mean radius R0 of the defect. ------H = ------d ------T ------, (11) Under these conditions, I ~ qÐ2. Thus, the slope of the T () kT 2 K()mn, S qT qH log I = f(log q) curve for this scattering range is equal to Ð2. where qH and qT are the wave vectors relating to the angular ranges with predominant Huang or thermal Π 2 3 π γ (2) Asymptotic Scattering Range scattering, respectively, and (m, n) = V c ∑i i i . This expression allows us to find absolute values of the X-ray scattering is caused by rather strong distor- measured DXS intensities for the MDs under study tions of a crystal lattice and can be described in terms without regard for the thickness of the scattering layer, of the elastic-continuum theory without regard for structure amplitude, solid scattering angle, and DebyeÐ changes in the elastic moduli. A defect cannot be con- Waller factor. Hence, important parameters, such as the sidered as a point source. Here, I ~ qÐ4, which corre- symmetry, size, and concentration of MDs, can be sponds to a slope of Ð4. The value q0 at which the tran- determined directly from the experimental data on dif- sition occurs from the Huang to asymptotic scattering fuse scattering by the sample under study. (inflection point) can be used to estimate the defect strength from the relation [2] The I(q) dependences considered above for various scattering ranges relate to the differential intensity. To 1 provide an acceptable light-gathering power, we have q0 = ------. (9) to use radiation beams with a vertical divergence. In the HC presence of a vertical divergence in a three-crystal

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 DIFFUSE X-RAY SCATTERING STUDY OF THE FORMATION OF MICRODEFECTS 1921 x-ray diffractometer, the measured intensity can be Table 1. Multistep heat treatment regimes for silicon wafers written as Sample Heat treatment regime +qy max number (), () Iqx qz = ∫ I q dqy, (12) 1 As-grown state

Ðqy max 2 1000°C/15 min ϕ λ 3 1000°C/15 min + 450°C/16 h qy max = max/ , (13) 4 1000°C/15 min + 450°C/16 h + 650°C/16 h where q || (001) and q || (100) are the vector q compo- x z ° ° ° nents in the diffraction plane, the component q is nor- 5 1000 C/15 min + 450 C/16 h + 650 C/16 h y + 1000°C/4 h mal to the diffraction plane, and ϕ is the maximum max ° ° angle of deviation of the diffracted rays from the dif- 6 1000 C/15 min + 650 C/16 h fraction plane (this angle specifies the effective vertical 7 1000°C/15 min + 650°C/4 h + 1000°C/4 h ϕ divergence). The angle max is determined by the width 8 1000°C/15 min + 650°C/16 h + 1000°C/4 h of the reflection curve of an analyzer [8]. We will call 9 1000°C/15 min + 650°C/4 h + 800°C/4 h the intensity distributions I(q) along the qx and qz direc- + 1000°C/4 h tions the qx and qz sections, respectively. Thus, the func- 10 1000°C/15 min + 650°C/16 h + 800°C/4 h tion I(q) should change after integrating with respect to + 1000°C/4 h qy. For example, in the Huang scattering range, the DXS Ð1 intensity recorded experimentally decreases as q and, Table 2. Characteristics of the samples under study in the asymptotic scattering range, as qÐ3. Analysis of Changes in the the slope of the linearized logI = f(logq )curve can Microdefect density, Sample concentration give information on the nature of a defect, its mean size, cmÐ3 (optical number of optically active and the form of the defectÐmatrix interface. For quanti- microscopy data) oxygen, 10Ð17 cmÐ3 tative measurements of diffuse scattering with the aim of determining the defect concentration, the limits of 10 ~8 × 105 integration with respect to qy should be taken into 2 0.01 ~5 × 105 account. 3 0.08 ~5 × 105 We studied two series of samples, which were wafers of dislocation-free silicon single crystals (with 4 0.40 Ð vacancies) 150 mm in diameter grown by the Czochral- 5 3.62 ~3 × 109 ski technique along the 〈100〉 direction. The concentra- 6 0.03 ~5 × 105 × tion of oxygen dissolved in the samples was (7Ð8) 7 0.09 Ð 17 Ð3 Ω 10 cm , and the resistivity was 1Ð5 cm. Each series 8 0.42 Ð consisted of samples subjected to multistage heat treat- 9 ment, including low-temperature, stabilizing, and high- 9 1.60 ~3 × 10 temperature annealings. The series differed in the 10 3.42 ~2 × 1010 regime of the low-temperature annealing: for the first series, the annealing was performed at 450°C, whereas for the second, at 650°C. The heat-treatment regimes treatment according to the regimes for forming an inter- for the wafers under study are given in Table 1. As ref- nal getter (samples 5, 9, 10) caused a sharp increase (up erence samples, we used as-grown wafers. After heat to ~3 × 109Ð2 × 1010 cmÐ3) in the MD density in the treatment, the samples were chemically polished in an wafers. acid mixture HF(49 vol %) : HNO3 (70 vol %) = 1 : 6 For the as-grown sample 1, we measured the lattice for 5 min to remove a surface layer ~40 µm thick. parameter using the Bond method [9]. The measured The structure of the samples was examined by OM value (a = 0.543106 ± 0.0000008 nm) is greater than after selective chemical etching. The concentration of the lattice parameter of the standard (a = 0.543103 nm) optically active oxygen in the wafers before and after [5]. Since the crystals under study were grown in the heat treatment was estimated, using infrared spectros- vacancy mode and contained 7 × 1017 cmÐ3 oxygen copy, from the measured absorption of ~9-µm radia- atoms, this increase in the lattice parameter can be due tion. The results are listed in Table 2. to the presence of SiO2 complexes in the silicon matrix. As follows from Table 2, the density of MDs in the According to [10], the angle between the SiÐO bonds is ° as-grown samples is ~8 × 105 cmÐ3 (OM data). Heat 100 . By estimating the positive deformation caused by treatment according to the regimes 1000°C/15 min such a complex, it can be found that the observed (sample 2), 1000°C/15 min + 450°C/16 h (sample 3), increase in the lattice parameter [by ~(2Ð3) × 10Ð6 nm] and 1000°C/15 min + 650°C/16 h (sample 6) did not can be provided by ~(2Ð3) × 1017 cmÐ3 oxygen atoms in change the density of visible MDs. Multistage heat the SiO2 complexes.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1922 BUBLIK et al.

105 105

4 4 10 Sample 1 10 Sample 3 103 Sample 2 103 Sample 4 Sample 5 102 102

1 1 , count/s 10 , count/s 10 I I 100 100 10–1 10–1 10–2 10–2 10–1 100 101 10–1 100 101 µ –1 µ –1 qx, m qx, m

Fig. 1. DXS intensity distributions along the [011] direction Fig. 2. DXS intensity distributions along the [011] direction for samples 1 and 2. for samples 3, 4, and 5.

3. RESULTS AND DISCUSSION sample annealed at 450°C (Fig. 2). Analysis of isodiffuse contours (Fig. 3) in this sample indicates that it Figure 1 shows the diffuse scattering distribution contains MDs with a positive strength (the asymmetric along the qx section for samples 1 and 2 (the as-grown part of diffuse scattering, which is defined as state and the state after high-temperature annealing at []() ()⁄ 1000°C for 15 min, respectively). The curves in Fig. 1 Iq Ð IqÐ 2 and is dependent on the sign of dila- tation, is positive at q > 0; Fig. 3c). The symmetric have two pronounced segments: the segment at qx < z 0.2 µmÐ1 corresponds to the instrumental function and component of diffuse scattering []Iq()+ Iq()Ð ⁄ 2 (Fig. 3b) is specified by the symmetry of the displace- the segment of the linear logI = f(logqx ) dependence at q > 0.6 µmÐ1 with a slope close to unity. The latter ment field around MDs. The elongation of the isodif- x fuse contours in the q direction indicates that the sym- segment corresponds to the Huang scattering by defects x smaller than 0.02 µm, since the transition to asymptotic metry of the displacement field around interstitial MDs scattering (inflection point) for these defects is beyond is close to orthorhombic, which means that these MDs the measurement interval. The measurement interval in most likely have a planar shape. A simulation of the dis- our case was restricted by a very low DXS intensity, tribution of the symmetric DXS component using the and reliable measurement beyond this interval would formulas from [3, 4] shows that these defects lie along require a tenfold increase in the measurement time at the {110} planes. Using the position of the inflection ≈ µ Ð1 each point, which seems inexpedient in this case. The point q0 0.15 m in the logI = f(logqx ) curve for shift of the isodiffuse contour toward positive values of sample 3 (Fig. 2), the radius of these defects was esti- qz suggests that the defects have a positive strength. mated to be ~2.5 µm. The variation of the slope of the These MDs have a nonspherical displacement field, curve for q in the range from 0.2 to 5 µmÐ1 indicates the since they contribute to the DXS intensity along the qx presence of a size distribution of MDs generated during direction [4]. The concentration of point defects form- ° 17 Ð3 annealing at 450 C. The volume density of such MDs ing these MDs in the as-grown samples is ~10 cm . is ~3.2 × 108 cmÐ3. The concentration of optically active High-temperature annealing at 1000°C (sample 2) is oxygen in the sample remains virtually unchanged dur- usually performed to dissolve growth MDs existing in ing its annealing (Table 2). Therefore, we can assume, the crystal. Analysis of the DXS intensity distribution with a fair degree of confidence, that this annealing along the qx direction in this sample showed no appre- does not produce oxide precipitates but causes the for- ciable changes in the concentration of MDs smaller mation of extended regions of an inhomogeneous solid than 0.02 µm after the annealing (Fig. 1). The fact that solution of oxygen in silicon with a diffuse coherent the density of growth MDs changes only weakly in this interface (“impurity clouds”), which brings about case is also indicated by the OM data (Table 2). strong DXS. These regions are likely to be rich in oxy- The subsequent low-temperature annealing was car- gen. It can be assumed that the changes in the concen- ried out at 450°C for 16 h (sample 3). At this tempera- tration of point defects in these regions obey a Gaussian ture, oxygen thermodonors form intensely in the silicon distribution. Similar structural aggregates have been wafers. The OM data exhibit no significant changes in observed in germanium wafers in the early stages of the density of MDs in the samples after this annealing. decomposition of a supersaturated solid solution of However, the DXS intensity sharply increases in the lithium in germanium [11].

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 DIFFUSE X-RAY SCATTERING STUDY OF THE FORMATION OF MICRODEFECTS 1923

The subsequent annealing at 650°C for 16 h (sample 4) 3 caused signiﬁcant changes in the character of DXS (a) (Fig. 2). These changes indicate that the large cloudlike MDs disappear in the samples, to be substituted by 2 small MDs with a nonspherical displacement ﬁeld. The density of these defects exceeds the concentration of small MDs in the as-grown samples by a factor of 1.5Ð 1

2. It is difﬁcult to estimate the radius of these defects, –1 since the asymptotic scattering range for them was not m

µ 0 , reached due to a low intensity. We can only assert that z they are smaller than 0.02 µm in size. The infrared q spectroscopy data indicate that the change in the con- –1 centration of optically active oxygen is relatively significant in this sample. All these data suggest that, during annealing at 650°C, small oxygen-containing pre- –2 cipitates (nucleation centers of SiO2 particles) form in the bulk of the wafers in the regions with maximum oxygen content. –3 –2 –1 0 1 2 3 In the final high-temperature (1000°C/4h) stage of 3 the four-step heat treatment of this series of samples (sample 5), further significant structural changes occur (b) in the sample (see the corresponding DXS curve in 2 Fig. 2). The size and density of MDs abruptly increase. Analysis of the isodiffuse contours of x-ray scattering shows that the source of DXS in this sample is MDs 1 having a positive strength (obviously, SiO particles) 2 –1 and a nonspherical displacement field (Fig. 4). Unlike m

µ 0 , the impurity clouds observed in sample 3, such MDs z have a sharp incoherent interface. The shape of isodif- q fuse contours changed (Fig. 4), which may indicate that –1 the displacement vectors changed. However, the planes of defect location remain the same, namely, {110}. An estimation from the position of the inﬂection point –2 ≈ µ Ð1 (q0 0.25 m ) in the logI = f(logqx ) curve (Fig. 2) shows that the radius of these MDs is ~1.5 µm. The volume fraction of the MDs is (0.96Ð1.0) × 109 cmÐ3, –3 –2 –1 0 1 2 3 which agrees well with the OM data (Table 2). 3 The data on diffuse scattering in the samples of the second series are displayed in Figs. 5 and 6. An increase (c) in the temperature of the low-temperature step to 650°C 2 (sample 6) leads to a signiﬁcant (one order of magnitude) decrease in the concentration of the large impu- 1 rity-cloud defects characteristic of the samples annealed at 450°C (sample 3). Apart from these defects, –1 m

sample 6 also contains small MDs similar to the defects µ 0 , detected in sample 4, which is indicated by the segment z µ Ð1 q with a slope of ~2 at qx > 2 m in the curve for sample 6 in Fig. 5. The density of these MDs is identical to that –1 in sample 4, as judged from the DXS intensity. After low-temperature annealing at 650°C, samples 7 –2 and 8 were finally annealed at 1000°C for 4 h; they differed in the duration of the low-temperature step (4 and 16 h, respectively). As follows from the data shown in Fig. 5, the DXS intensity in sample 7 is significantly –3 –2 –1 0 1 2 3 lower than in sample 8. The density of MDs in sample 8 Fig. 3. DXS intensity distribution (isodiffuse contours) in is substantially higher (7.1 × 108 cmÐ3). The MDs have the vicinity of the 400 site for sample 3: (a) experimental contours, (b) symmetric component, and (c) asymmetric a positive strength and a nonspherical displacement component. The DXS intensities on isodiffuse contours field. The indistinct inflection point in the curves for vary in 1.0 count/s steps in the range 0.5Ð9.5 count/s.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1924 BUBLIK et al.

3 105 (a) Sample 3 104 Sample 4 2 Sample 6 3 10 Sample 7 1 102 Sample 8

–1 1 , count/s

I 10 m

µ 0

, 0 z 10 q –1 –1 10 10–2 10–2 10–1 100 101 –2 µ –1 qx, m

Fig. 5. DXS intensity distributions along the [011] direction –3 –2 –1 0 1 2 3 for samples 6, 7, and 8. For comparison, the DXS intensity distributions for samples 3 and 4 are also given (according 3 to Fig. 2).

(b) 2 105 Sample 9 4 1 10 Sample 10 3

–1 10 m

µ 0 2

, 10 z q 1 , count/s

I 10 –1 100 –2 10–1 10–2 10–2 10–1 100 101 µ –1 –3 –2 –1 0 1 2 3 qx, m 3 Fig. 6. DXS intensity distributions along the [011] direction for samples 9 and 10. (c) 2 µ Ð1 samples 7 and 8 in Fig. 5 at qx > 0.2 m indicates a 1 large scatter in the sizes of the MDs that form in these samples during heat treatment. The maximum MD size –1 µ

m reaches 1.2 m. Thus, an increase in the annealing time

µ 0 °

, at 650 C from 4 to 16 h results in a signiﬁcant intensi- z

q ﬁcation of the decomposition of the supersaturated –1 solid solution of oxygen to form oxide precipitates. These data correlate well with the observed decrease in the concentration of optically active oxygen in these –2 samples in the course of the three-step heat treatment (Table 2). Samples 9 and 10 were subjected to the complete –3 –2 –1 0 1 2 3 cycle of the four-step heat treatment used to form an internal getter. In contrast to the samples discussed Fig. 4. DXS intensity distribution (isodiffuse contours) in above, they were subjected to stabilizing annealing at the vicinity of the 400 site for sample 5: (a) experimental ° contours, (b) symmetric component, and (c) asymmetric 800 C for 4 h. As follows from the data shown in Fig. 6, component. The DXS intensities on isodiffuse contours the additional stabilizing annealing leads to a signiﬁ- vary in 2.0 count/s steps in the range 1Ð19 count/s. cant increase in the DXS intensity. The density of MDs

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 DIFFUSE X-RAY SCATTERING STUDY OF THE FORMATION OF MICRODEFECTS 1925 substantially increases and becomes equal to 1.2 × stages of decomposition of the supersaturated solid 109 cmÐ3 in sample 9 and 2.2 × 1010 cmÐ3 in sample 10. solution of oxygen. The scatter of the sizes of MDs sharply decreases. The MD size is ~0.4 µm in sample 10 and 0.6 µm in sample 9. Analysis of isodiffuse contours of x-ray scattering indi- ACKNOWLEDGMENTS cates MDs having a nonspherical displacement field This work was supported by the Russian Foundation with a positive strength. It should be noted that the DXS for Basic Research, project no. 02-02-16053. data on the MD density in these samples correlate well with the OM data (Table 2). REFERENCES 4. CONCLUSIONS 1. P. H. Dederichs, J. Phys. F 3, 471 (1973). 2. M. A. Krivoglaz, Theory of X-ray and Thermal Neutron Thus, we have obtained the following basic results. Scattering by Real Crystals, 2nd ed. (Plenum, New York, (1) The DXS method is efficient for studying the 1969; Naukova Dumka, Kiev, 1983). formation of MDs in silicon single crystals in the stages 3. B. C. Larson and W. Schmatz, Phys. Rev. B 10 (6), 2307 of their growth and subsequent heat treatment. This (1974). method is especially efficient for investigating the early 4. H. Trinkaus, Phys. Status Solidi B 51, 307 (1972). stages of microdefect formation, where TEM cannot 5. V. K. Ovcharov, Candidate’s Dissertation (Moscow, provide desirable information. 1998). (2) With the DXS method, we have revealed the for- 6. L. A. Charniy, K. D. Scherbachev, and V. T. Bublik, mation of specific defects of the impurity-cloud type in Phys. Status Solidi A 128 (2), 303 (1991). silicon wafers during low-temperature annealing at ° 7. W. A. Wooster, Diffuse X-ray Reflections from Crystals 450 C. These defects are regions of the solid solution (Clarendon, Oxford, 1962; Inostrannaya Literatura, enriched in oxygen (and, probably, in intrinsic point Moscow, 1963). defects) with a diffuse coherent interface. These µ 8. K. D. Shcherbachev and V. T. Bublik, Zavod. Lab. 60 (8), regions reach ~2.5 m in size and provide strong DXS. 473 (1994). In the following stages of decomposition of the supersaturated solid solution of oxygen, oxide precipitates 9. W. L. Bond, Acta Crystallogr. 13, 814 (1960). form inside these regions and these regions disappear. 10. C. Claeys and J. Vanhellemont, in Proceedings of 2nd International Autumn Meeting on Gettering and Defect (3) As a result of the decomposition of the supersat- Engineering in the Semiconductor Technology, urated solid solution of oxygen, interstitial MDs form GADEST’87, Garzau, Germany (1987), p. 3. in the silicon wafers during a multistep heat treatment. 11. I. M. Kotina, V. V. Kuryadkov, G. N. Mosina, et al., Fiz. These MDs lie in the {110} planes and have nonspher- Tverd. Tela (Leningrad) 26 (2), 436 (1984) [Sov. Phys. ical displacement fields. Precipitates have a platelike Solid State 26, 259 (1984)]. shape in the very early stages of their formation. By measuring DXS, we have determined the density and size of MDs forming in the silicon wafers at various Translated by K. Shakhlevich

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1926Ð1931. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1833Ð1838. Original Russian Text Copyright © 2003 by Bobylev, Ovid’ko.

DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Faceted Grain Boundaries in Polycrystalline Films S. V. Bobylev and I. A. Ovid’ko Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Vasil’evskiœ ostrov, Bol’shoœ pr. 61, St. Petersburg, 199178 Russia e-mail: [email protected] Received November 14, 2002; in ﬁnal form, April 3, 2003

Abstract—A theoretical model is proposed for describing a new mechanism of misfit-stress relaxation in polycrystalline films, namely, the formation of faceted grain boundaries whose facets are asymmetric tilt boundaries. The ranges of parameters (the film thickness, misfit parameter, angle between facets) in which the nucleation of faceted grain boundaries is energetically favorable are calculated. The nucleation of faceted grain boundaries is shown to be facilitated as the film thickness increases. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION strate interface is characterized by the one-dimensional misfit parameter Polycrystalline films are objects of intensive fundamental and applied research, as they are widely applied 2()a Ð a f = ------f s-, (1) in modern high-technological production processes. a + a The stability of the physical properties of such films is f s of primary importance for their technological applica- where af and as are the lattice parameters of the film and tion and depends significantly on the defects and stress substrate, respectively. fields in them (see, e.g., reviews [1Ð5] and papers [6, In this work, we analyze two physical states of the 7]). For example, a difference between the lattice film, namely, the state with a straight-line symmetric parameters in substrate and film materials causes inter- tilt boundary (Fig. 1a) and the state with a faceted nal stresses in the films. These misfit stresses signifi- boundary whose facets are asymmetric tilt boundaries cantly affect the evolution of the structure and the func- (Fig. 1b). For the sake of definiteness, we study low- tional properties of the films. When the thickness of a angle tilt boundaries simulated as ensembles of edge film reaches a certain critical value, misfit stresses, as a lattice dislocations. However, the results of our consid- rule, are partially accommodated into misfit disloca- eration can be generalized to the case of a high-angle tions at the substrate–film interface [1Ð15]. However, grain boundary simulated as a grain boundary having the presence of grain boundaries in polycrystalline and dislocations with Burgers vectors of the complete coin- nanocrystalline films causes alternative efficient mech- cidence lattice of the boundary [20]. The plane of the anisms of relaxation of misfit stresses (and in general, residual stresses of a different nature) via grain-boundary dislocations and disclinations [16Ð19]. In analyzing (a) (b) (c) these alternative mechanisms, particular attention has been given to the theoretical description of grain- y boundary defects in symmetric planar tilt boundaries. α j

In general, however, films also contain asymmetric and h faceted grain boundaries [20]. In this work, we propose a theoretical model of a new misfit-stress relaxation L mechanism which can operate in polycrystalline films H B and is related to the formation of faceted grain boundaries whose facets are asymmetric tilt boundaries. B

2. FACETED GRAIN BOUNDARIES Film IN FILMS: MODEL Substrate x Let us consider a model of the film–substrate system consisting of a bicrystal film of thickness H and a semi- Fig. 1. Dislocation structure of grain boundaries in a bicrys- infinite substrate (Fig. 1). The film and substrate are tal film: (a) symmetric plane tilt boundary, (b) faceted grain boundary, and (c) faceted grain boundary modeled as a wall assumed to be elastically isotropic solids with the same of superdislocations having Burgers vectors with alternat- shear modulus G and Poisson ratio ν. The film–sub- ing directions (schematic).

FACETED GRAIN BOUNDARIES IN POLYCRYSTALLINE FILMS 1927 symmetric tilt boundary is assumed to be normal to the (the facet elements) along its direction. Thus, a faceted free surface (Fig. 1a). Such a boundary contains M peri- grain boundary in a film subjected to a misfit-stress odically ordered edge dislocations with a Burgers vec- field is simulated as a vertical wall of N edge superdis- tor b parallel to the interface and normal to the plane of locations with alternating Burgers vectors B (differing the tilt boundary. The misorientation θ of the low-angle in direction but having the same magnitude) in the mis- symmetric tilt boundary is connected with dislocation fit-stress field (Fig. 1c). θ parameters by the Frank formula b = 2(H/M)sin( /2) Within this model, we consider the physical states [20]. (Figs. 1a, 1b) as independent states, which are actual- A faceted grain boundary consists of facets (seg- ized upon film growth. It is assumed that either a sym- ments) having the dislocation structure of an asymmet- metric or a faceted grain boundary (depending on the ric tilt boundary (Fig. 1b). For simplicity, we consider ratio of the elastic energies of the film in these physical facets of only two types, having the same structure and states) forms in the film. We do not analyze transforma- length L; the angle between adjacent facets is α, and tions between these film states or possible energy bar- their number is N. Like a symmetric boundary, a fac- riers between them. eted boundary contains M edge dislocations with a Burgers vector b that is parallel to the film–substrate A certain similarity between the formation of fac- interface. Therefore, the orientations of the crystal lat- eted grain boundaries and the faceting of the free sur- tices far from a faceted boundary are identical to those faces of crystals should be noted. However, the cause of in the case of a symmetric tilt boundary. Since the the spontaneous faceting of a crystal flat surface is an Burgers vectors b of dislocations making up a faceted orientation dependence of the surface free energy (see, grain boundary are not normal to the facet planes, each e.g., [22]), whereas the driving force of the formation facet is an asymmetric tilt boundary. of faceted grain boundaries is their involvement in the relaxation of misfit stresses. A symmetric tilt boundary and a faceted boundary in the film differ in the spatial arrangement of the dislocation ensemble. This causes the different character of 3. ENERGY CHARACTERISITCS OF GRAIN interaction between the grain boundaries under study BOUNDARIES IN THE FILM and the misfit-stress field in the film. The periodic wall of edge dislocations making up a symmetric tilt bound- The energy of the film with a faceted grain boundary ary is essentially characterized by short-range stress (Fig. 1b) is higher than the energy of the film with a fields. The stress fields of dislocations making up a symmetric boundary (Fig. 1a) by the elastic energy of periodic wall (Fig. 1a) completely compensate (shield) the superdislocations (Fig. 1c) and by the surface each other at distances exceeding the wall period H/M. energy of the boundary related to an increase in the Therefore, a symmetric tilt boundary with a periodic boundary length caused by facet formation. In the dislocation structure weakly interacts with the field of course of the facet formation, however, misfit stresses misfit stresses. relax, which should lead to a decrease in the total energy of the film. The competition of these factors Dislocations in a faceted boundary are arranged so specifies whether the formation of a faceted grain that the mutual shielding of their stress fields is sub- boundary is favorable or not as compared to the sym- stantially weakened. Consequently, a faceted boundary metric boundary. Thus, the characteristic difference ∆W is a source of long-range stress fields and strongly inter- between the energies of the faceted and symmetric acts with misfit-stress fields in the film. In particular, grain boundaries in the film consists of three parts: the dislocations in a faceted boundary can provide efficient elastic energy of superdislocations W el (which includes relaxation of misfit stresses, which decreases the total the energy of superdislocations and the energy of their elastic energy of the system as compared to the case of interactions), the surface energy of the boundary Ws, a symmetric tilt boundary. This decrease is the driving and the interaction energy of the superdislocations with force for the formation of faceted grain boundaries, misfit stresses W f: which have been detected experimentally in films (see, e.g., review [21] and references therein). In this work, ∆WW= el ++W s W f. (2) we perform a theoretical study of the conditions of formation of faceted boundaries in the context of the If ∆W < 0, faceting is energetically favorable. Note that model of the dislocation structure of such boundaries the total volume of grains in the film is the same in both shown in Fig. 1c. physical states (Figs. 1a, 1b). If facets are not long, the dislocation structure of Below, we calculate the terms in Eq. (2). The energy each of them responsible for the asymmetry can be sim- W el can be represented in the form ulated, to a first approximation, as one edge superdislocation placed at the center of a facet. The Burgers vec- N N el d 1 dÐd tor B of the superdislocation is parallel to the facet W = ∑ W + --- ∑ W . (3) i 2 ij plane, and its magnitude is equal to the sum of the pro- i = 1 ij, = 1 jections of the Burgers vectors of lattice dislocations ij≠

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

1928 BOBYLEV, OVID’KO

Here, W d is the self-energy of the ith dislocation and and (7) into Eq. (4), we obtain the pair interaction i energy W dÐd is the interaction energy between the ith and jth ij W dÐd = DB()B + B B dislocations (i, j = 1, …, N, where the ﬁrst dislocation ij ix jx iy jy is the nearest to the free surface and the dislocation h + h 2h h (8) × ln------i j Ð ------i j - , index increases as the interface is approached). Accord- 2 hi Ð h j ()h + h ing to the general procedure for calculating the interac- i j tion energy of defects [23], the energy of interaction of where D = G/2π(1 Ð ν). In our model (Fig. 1c), the com- two dislocations can be written as the work done in the ponents Biy are the same for all dislocations irrespective α nucleation of one dislocation in the stress ﬁeld of the of their index: Biy = Bcos( /2). However, the sing of the other dislocation (in the coordinate system shown in components Bix alternates with their index; i.e., Bix = Fig. 1c): (−1)iBsin(α/2). Thus, Eq. (8) can be rewritten in the form

h j () () dÐd 22 α ij+ 2 α W dÐd = Ð ()B σ i ()xy, = 0 + B σ i ()xy, = 0 dx. (4) W = DB cos --- + ()Ð1 sin --- ij ∫ jx xy jy yy ij 2 2 0 (9) h Ð h 2h h × ln------i j Ð ------i j - , Here, B and B are the components of the Burgers vec- 2 jx jy hi Ð h j () () () hi + h j tor of the jth dislocation, σ i and σ i are the compo- xy yy which is more convenient for further analysis. nents of the stress tensor of the ith dislocation, and h is j Using a calculation procedure similar to that used the distance from the free surface to the jth dislocation. dÐd d According to Fig. 1c, hj can be represented in the form above to calculate Wij , we ﬁnd the self-energy Wi of the ith dislocation at a distance hi from the free surface 2 j Ð 1 α to be h j = ------Lsin--- . (5) 2 2 hi Ð r0 d 1 ()i ()i W = Ð--- ()B σ ()xy, = 0 + B σ ()xy, = 0 dx The stress tensor components are sums of the indi- i 2 ∫ ix xy iy yy vidual contributions from each component of the Burg- 0 (10) () () () ers vector of a dislocation, i.e., σ i = σ ix + σ iy and DB22h Ð r 2h ()h Ð r xy xy xy = ------ln------i -0 Ð ------i i 0 , () () () 2 σ i σ ix σ iy 2 r0 ()2h Ð r yy = yy + yy . The components of the stress ten- i 0 sor of the ith dislocation near the free surface have the where r0 is the dislocation core radius. form [23] Substituting Eqs. (9) and (10) into Eq. (3), we obtain the elastic energy  3 3 ()ix GBix x1 x1 x2 x2 σ (),  N N xy xy = ------Ð2----2- ++4----4- 2----2- Ð 4----4- 2 4π()1 Ð ν  el DB 2 α ij+ 2 α r1 r1 r2 r2 W = ------∑ 2 ∑ cos --- + ()Ð1 sin --- (6) 2 2 2 2 4 3 i = 1 j = 1  ≠ 2 x2 x2 x2 x2 ij ------------ Ð2hi ----2 Ð16 1 6 4 ++6 2hi 6 4 Ð 8 6 , r r r r r  hi + h j 2hih j 2 2 2 2 2 × ln------Ð ------(11) h Ð h ()2 i j hi + h j GB  x x 3 x x 3 σ iy ()xy, = ------iy 6-----1 Ð 4----1- Ð 6-----2 + 4----2- () yy π()ν 2 4 2 4 2hi Ð r0 2hi hi Ð r0 4 1 Ð  r r r r ∑ + ln------Ð ------. 1 1 2 2 r ()2 (7) 0 2hi Ð r0 2 4 3 2 x x x x  +2h Ð---- + 16----2- Ð 16----2- Ð 2h 6-----2 Ð 8----2-  , i 2 4 6 i4 6  The difference between the surface energies of the r2 r2 r2 r2 r2 faceted and symmetric boundaries is s γ γ γ () 2 2 2 W ==N L Ð H NLÐ H . (12) where x1 = x Ð hi, x2 = x + hi, and rn = xn + y , with n = σ ()iy σ ()ix Here, N is the number of facets, L is the facet length, 1, 2. The components xy and yy vanish at y = 0 H is the ﬁlm thickness, and γ is the surface energy den- and, hence, are not presented here. Substituting Eqs. (6) sity of the grain boundary.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 FACETED GRAIN BOUNDARIES IN POLYCRYSTALLINE FILMS 1929

In general, the interaction energy between the ith 30 B = 2b dislocation and the misﬁt stress ﬁeld is given (by analogy with Eq. (4)) by the formula

J B = b

µ 10 h j , () () f ()σ f , σ f , W 0 Wi = Ð∫ Bix xy ()xy = 0 + Biy yy ()xy = 0 dx, (13) ∆ 0.002 0.006 0.010 f 0 –10 σ ()f σ ()f where xy and yy are the components of the misfit- stress tensor. Since the off-diagonal components of the –30 misfit-stress tensor are zero and the diagonal ones have (f) the form σ = 4πD(1 + ν)f [6], we have from Eq. (13) Fig. 2. Dependence of the difference ∆W between the energies of faceted and plane grain boundaries on the misfit h j parameter f in a film H = 700 nm thick at various values of () α W f ==Ð4B σ f dx Ð πDBcos--- ()1 + ν fh. (14) the Burgers vector of superdislocations. i iy∫ 2 i 0 Summing Eq. (14) over i and using Eq. (5), we obtain various values of the Burgers vector of a superdisloca- the energy W f: tion. The following values of the parameters are used: elastic modulus G = 100 GPa, ν = 0.3, facet length L = N α N 10 nm, angle between adjacent facets α = 90°, number f f π ()ν W ==∑ Wi Ð4 DBcos--- 1 + fh∑ i of facets N = 100 [therefore, the film thickness is H = 2 α ≈ i = 1 i = 1 NLsin( /2) 700 nm], and surface energy density γ 2 N characteristic of aluminum = 0.6 J/m [24]. The Burg- α ()2i Ð 1 α ers vector of a superdislocation is B = nb, where b is the = Ð4πDBcos--- ()1 + ν f ∑ ------Lsin--- 2 2 2 Burgers vector of a lattice dislocation. The superdislo- i = 1 (15) cation core radius is taken to be r0 = B. N The calculated ∆W( f ) dependence from Eq. (16) at π ()ν α () = Ð DB 1 + fLsin ∑ 2i Ð 1 b = 0.4 nm and H = 700 nm is given in Fig. 2. As the i = 1 misfit parameter f increases, the energy difference ∆W π ()ν 2 α is seen to decrease linearly and reach negative values, = Ð DB 1 + fLN sin . which means that the contribution of the relaxation Thus, we found all components of the difference term is predominant. Therefore, the formation of a fac- between the energies of the faceted and symmetric eted boundary becomes energetically favorable as com- boundaries, ∆W. Substituting Eqs. (11), (12), and (15) pared to a symmetric tilt boundary. The data in Fig. 2 into Eq. (2), we obtain the final expression allow the following conclusion about the effect of the Burgers vectors of superdislocations on ∆W: at small 2 N N α α values of the misfit parameter, an increase in the Burg- ∆ DB 2 ()ij+ 2 ∆ W = ------∑ 2 ∑ cos --- + Ð1 sin --- ers vector leads to an increase in W, since the elastic 2 2 2 energy of superdislocations increases in proportion to i = 1 j = 1 2 ij≠ B and the interaction energy between superdislocations and the misfit-stress field depends linearly on B. × hi + h j 2hih j However, at large values of the misfit parameter, when ln------Ð ------2 hi Ð h j ()h + h (16) misfit-stress relaxation becomes predominant, an i j increase in the Burgers vector of a superdislocation decreases ∆W. 2h Ð r 2h ()h Ð r ∑ + ln------i -0 Ð ------i i 0 The ∆W(B) dependence calculated from Eq. (16) at r ()2 0 2hi Ð r0 various values of f and the system parameters given above is shown in Fig. 3. The value of B was varied 2 from b to 5b. The plots in Fig. 3 exhibit a very strong + γ ()πNLÐ H Ð DB()1 + ν fLN sinα. dependence of ∆W on the Burgers vector of superdislocations. The maximum value of B at which the formation of a faceted boundary is still favorable as compared 4. RESULTS OF MODEL CALCULATIONS to a symmetric tilt boundary is equal to 3b at realistic Using Eq. (16), derived for the characteristic differ- values of the misfit parameter. Figure 4 shows the ence between the energies of the faceted and plane dependence of ∆W on the film thickness for various val- symmetric grain boundaries, we find the dependence of ues of the misfit parameter at B = b. These dependences ∆W on the parameters of the system. First, we deter- exhibit three possible types of behavior: (1) a faceted mine the dependence of ∆W on the misfit parameter f at boundary is energetically unfavorable over the whole

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1930 BOBYLEV, OVID’KO

f = 0.001 6 200 f = 0.001

150 f = 0.005 J 2 µ J , µ W , 100 0 ∆

W 100 300 500 700 H, nm

∆ f = 0.01 50 –2 f = 0.004 0 0.5 1.0 1.5 2.0 B, nm –6 f = 0.006 –50

Fig. 3. Dependence of the difference ∆W between the energies of faceted and plane grain boundaries on the Burgers vector of superdislocations in a film H = 700 nm thick at Fig. 4. Dependence of ∆W on the film thickness at various various values of the misfit parameter f. values of the misfit parameter.

25 0 50 100 150 H, nm J J µ µ ,

, 15 W W ∆ ∆ –0.1

0 –0.2 45 90 135 180 α, deg

Fig. 5. Dependence of ∆W on the film thickness H at f = Fig. 6. Dependence of ∆W on the angle α between facets at 0.004 and B = b. f = 0.003 and B = b. thickness range (Fig. 1b), (2) a faceted boundary is gies characterizing the states of the film with a faceted energetically favorable over the whole thickness range, boundary (Fig. 1b) and a symmetric tilt boundary and (3) a faceted boundary is energetically unfavorable (Fig. 1a). The parameters of the system that exhibit a in a thin film and becomes favorable at film thicknesses significant effect on the formation of faceted grain greater than a certain critical value. The curve at f = boundaries in films are the misfit parameter, facet 0.004 in Fig. 4 illustrates the third type of behavior. The asymmetry (which is characterized by the Burgers vec- same curve is shown in Fig. 5 on a larger scale. tor of a superdislocation), film thickness, and the angle The plot in Fig. 6 shows the dependence of ∆W of the between facets. Within the model proposed, the ranges angle α between facets (at f = 0.003, B = b, H ≅ 700 nm). of the system parameters were found in which faceted Since both the elastic and relaxation terms in ∆W grain boundaries are energetically favorable. The depend on the angle between facets (the latter term van- model agrees with the experimental data (see review ishes at α = 180°), the ∆W(α) dependence exhibits a [21] and references therein) on faceted grain bound- minimum corresponding to the most favorable angle aries observed in superconducting films. The results between facets. obtained within the model can be used to prepare polycrystalline films with a given (faceted or nonfaceted) structure of grain boundaries, which significantly 5. CONCLUSIONS affects the physical (in particular, superconducting [25, 26]) properties of films. Thus, in this work, we have theoretically studied a new mechanism of misfit-stress relaxation in polycrystalline films, namely, the formation of faceted grain ACKNOWLEDGMENTS boundaries whose facets are asymmetric tilt boundaries. We have constructed a model to describe a fac- We thank A.G. Sheœnerman for valuable discus- eted grain boundary in a film placed on a semi-infinite sions. substrate in the presence of misfit stresses. Using this This work was supported by the Russian Foundation model, we calculated the difference between the ener- for Basic Research (project no. 01-02-16853), the

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 FACETED GRAIN BOUNDARIES IN POLYCRYSTALLINE FILMS 1931

Office of US Naval Research (grant no. 00014-01-1- 13. M. Yu. Gutkin, A. E. Romanov, and E. C. Aifantis, Phys. 1020), the program “Integratsiya” (project no. B0026), Status Solidi A 151 (2), 281 (1995). the program “Structural Mechanics of Materials and 14. M. Yu. Gutkin, K. N. Mikaelyan, and I. A. Ovid’ko, Fiz. Constructions” of the Russian Academy of Sciences, Tverd. Tela (St. Petersburg) 40 (11), 2059 (1998) [Phys. and the St. Petersburg Scientific Center of the Russian Solid State 40, 1864 (1998)]; Fiz. Tverd. Tela (St. Peters- Academy of Sciences. burg) 43 (1), 42 (2001) [Phys. Solid State 43, 42 (2001)]. 15. M. Yu. Gutkin, I. A. Ovid’ko, and A. G. Sheinerman, J. REFERENCES Phys.: Condens. Matter 12 (25), 5391 (2000). 16. I. A. Ovid’ko, J. Phys.: Condens. Matter 11 (34), 6521 1. E. A. Fitzgerald, Mater. Sci. Rep. 7 (1), 87 (1991). (1999); J. Phys.: Condens. Matter 13 (4), L97 (2001). 2. J. H. van der Merve, CRC Crit. Rev. Solid State Mater. Sci. 17 (3), 187 (1991). 17. I. A. Ovid’ko, Fiz. Tverd. Tela (St. Petersburg) 41 (9), 1637 (1999) [Phys. Solid State 41, 1500 (1999)]. 3. S. C. Jain, A. H. Harker, and R. A. Cowley, Philos. Mag. A 75 (6), 1461 (1997). 18. S. V. Bobylev, I. A. Ovid’ko, and A. G. Sheinerman, Phys. Rev. B 64, 224507 (2001). 4. I. A. Ovid’ko, in Nanostructured Films and Coatings, Ed. by G. M. Chow, I. A. Ovid’ko, and T. Tsakalakos 19. A. L. Kolesnikova, I. A. Ovid’ko, and A. E. Romanov, (Kluwer, Dordrecht, 2000), p. 231. Solid State Phenom. 87, 265 (2002). 5. I. A. Ovid’ko, Rev. Adv. Mater. Sci. 1 (2), 61 (2000). 20. A. P. Sutton and R. W. Balluffi, Interfaces in Crystalline 6. Yu. A. Tkhorik and L. S. Khazan, Plastic Deformation Materials (Claredon, Oxford, 1995). and Misfit Dislocations in Heteroepitaxial Systems 21. S. E. Babcock and J. L. Vargas, Annu. Rev. Mater. Sci. (Naukova Dumka, Kiev, 1983). 25 (1), 193 (1995). 7. M. Yu. Gutkin and I. A. Ovid’ko, Defects and Mecha- 22. A. F. Andreev and Yu. A. Kosevich, Zh. Éksp. Teor. Fiz. nisms of Plasticity in Nanostructural and Noncrystalline 81, 1435 (1981) [Sov. Phys. JETP 54, 761 (1981)]. Materials (Yanus, St. Petersburg, 2001). 8. S. C. Jain, T. J. Gosling, J. R. Willis, et al., Philos. Mag. 23. T. Mura, Micromechanics of Defects in Solids (Martinus A 65 (5), 1151 (1992). Nijhoff, Dordrecht, 1987), p. 1. 9. T. J. Gosling, R. Bullough, S. C. Jain, and J. R. Willis, J. 24. G. C. Hasson and C. Goux, Scr. Metall. 5 (11), 889 Appl. Phys. 73 (12), 8267 (1993). (1971). 10. U. Jain, S. C. Jain, S. Nijs, et al., Solid State Electron. 36 25. H. Hilgenkamp, J. Mannhart, and B. Mayer, Phys. Rev. (3), 331 (1993). B 53, 14586 (1996). 11. T. J. Gosling and J. R. Willis, Philos. Mag. A 69 (1), 65 26. I. A. Ovid’ko, Mater. Sci. Eng. A 313 (1/2), 207 (2001). (1994). 12. F. Bailly, M. Barbé, and G. Cohen-Solal, J. Cryst. Growth 153, 115 (1995). Translated by K. Shakhlevich

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

DEFECTS, DISLOCATIONS, AND PHYSICS OF STRENGTH

Plasticization of NaCl Crystals under the Combined Action of Short Mechanical and Magnetic Pulses V. A. Morozov St. Petersburg State University, Universitetskiœ pr. 28, St. Petersburg, 198504 Russia e-mail: [email protected] Received January 24, 2003; in ﬁnal form, February 11, 2003

Abstract—The dynamic yield stress of NaCl crystals is found to decrease under loading with a mechanical pulse of submicrosecond duration initiated by an electron beam preceded by a pulse of a vortex electromagnetic ﬁeld (the time delay is 10Ð6 s). This effect becomes more pronounced as the intensity of loading increases. © 2003 MAIK “Nauka/Interperiodica”.

The magnetoplastic effect (MPE) was first detected mobility of individual dislocations. The samples were as the motion of individual dislocations in alkali-halide 7 × 7 × 1.5 mm in size. Pulsed loading of the samples crystals in a magnetic field in the absence of applied was performed on an electron-beam setup [8]. The load [1]. As was shown later [2Ð7], this effect can also experimental setup is shown in Fig. 1. A sample can be manifest itself on a macroscopic level as a decrease in loaded by both a single mechanical pulse and a combi- the yield stress [5], a decrease in the microhardness [2], nation of a mechanical and an electromagnetic pulse. a change in the internal friction [4], and an increase in To this end, in the former case, a sample was placed in the plastic strain rate [3, 7]. Golovin et al. [3] detected a chamber of a ferromagnetic material and, in the latter a macroscopic effect of softening of NaCl crystals case, in a chamber of a diamagnetic material (Al). A under the simultaneous action of crossed static and mechanical pulse was excited by an electron beam in microwave magnetic fields. It was found that, when either an aluminum or a steel target in the form of a turned on before reaching the yield stress, a microwave short rod 8 mm in diameter mounted in the front wall of magnetic field does not change the stress–strain curve. the chamber, which served as the anode of an electron However, after the yield stress is reached, a microwave accelerator, and the pulse was then transferred to the magnetic field changes the slope of the stress–strain sample. The sample was in acoustic contact with the curve. target through a thin layer of silicone fluid. The All the macroscopic manifestations of the MPE mechanical pulse of loading was recorded at both the indicated above were observed under either static or input and output of the sample using a piezoelectric quasistatic conditions of magnetic and mechanical transducer of lithium iodate. The pulse duration was actions on nonmagnetic crystals. It is of interest to find about 10Ð7 s. The pulse amplitude during the experi- out whether this effect manifests itself under short-term ment was varied from 15 to 50 MPa by changing the actions. This problem was studied in [8] for the motion beam current or the target material (Al, Fe). A mag- of individual dislocations in NaCl crystals under the combined action of mechanical and electromagnetic pulses created by an electron beam. (a) (b) 1 1 In dynamic experiments, as is well known, it is difficult to construct a dynamic stress–strain curve or to 7 2 7 2 choose the criterion of deviation of this curve from a 3 103 10 linear dependence. In [9], this criterion was taken to be 6 9 4 8 a change in the form of a stress wave (pulse form) trav- 5 eling through a sample. In that work, the flattening of the peak of a stress pulse was related to a nonlinear stressÐstrain curve, which, in turn, was specified by the changed internal structure of the material, including the Fig. 1. Experiments (a) with the application of mechanical dislocation structure. The experiments were carried out and magnetic pulses to the crystal and (b) for measuring the on aluminum samples in the microsecond range of magnetic field strength. (1) Vacuum chamber of an electron loading. accelerator, (2) chamber for samples, (3) electron-beam source, (4) target, (5) sample, (6) piezoelectric transducer, In this work, we examine NaCl crystals that are sim- (7) beam-current meter, (8) search coil, (9) coil load, and ilar to those used in our earlier work [8] to study the (10) oscilloscope.

PLASTICIZATION OF NaCl CRYSTALS UNDER THE COMBINED ACTION 1933

4 60 3 2 48 4' 3' 2' 36 , MPa σ 1, 1' 8 24 T µ , B 4 12

0 tl 600 1200 t, ns 1200 1260 1320 1380 1440 1500 1560 t, ns

Fig. 2. Oscillograms of pulses of (1Ð4, 1'Ð4') mechanical loading and (5) a magnetic field in the NaCl crystal. Loading (1Ð4) in the absence and (1'Ð4') in the presence of a pulsed magnetic field. tl is the time delay of the beginning of a mechanical-stress pulse in relation to the beginning of a magnetic pulse. netic-field pulse was created by an electron-beam cur- mechanical pulse in the sample is relatively far from the rent during irradiation of the target. The pulse ampli- dynamic yield stress (Fig. 2, curve 1), the magnetic tude was varied in the range (3.5Ð10) × 10Ð6 T by field does not affect the signal shape as the amplitude of changing the current, and its duration was ~3 × 10Ð7 s. the magnetic field is varied over the range (3.5Ð10) × In the case of low amplitudes of the mechanical pulse 10Ð6 T. As follows from our earlier experiments [8], (far from the yield stress of the NaCl sample) initiated individual dislocations move under these conditions. In in the aluminum target, the magnetic field was about this case, as well as in the case without applying a mag- 3.5 × 10Ð6 T. To excite mechanical pulses of similar low netic field, the amplitude of the mechanical-stress pulse amplitudes in the steel target, we had to use a larger depends linearly on the current of the electron beam (or beam current. The magnetic field was about 10Ð5 T. on the beam energy introduced into the target; see Magnetic pulses were measured using a coil placed in Fig. 3). As seen from Figs. 2 and 3, the dynamic yield the chamber where an NaCl sample was previously stress of the samples is not reached when using only located (Fig. 1). The coil was loaded to a wave imped- mechanical loading. Under the experimental conditions ance of 50 Ω. A signal from the coil was fed to a high- employed, the dynamic yield stress was at least above frequency S1-75 oscilloscope and then photographed. 70 MPa (according to our estimation, it was about To determine the magnetic field strength, the coil was 80 MPa). When the amplitude of a mechanical pulse calibrated. approaches the dynamic yield stress of a sample, the application of a vortex high-frequency magnetic field When a mechanical-stress pulse traveled along the creates conditions for decreasing the yield stress, which target, its beginning was delayed with respect to the manifests itself as a flattening of the pulse (Fig. 2, magnetic pulse. The delay time tl could be varied by curves 2'Ð4') and a deviation from the linear σ(I) depen- changing the target thickness. In our experiments, this dence (Fig. 3, curve 2). time was 1.2 × 10Ð6 s (Fig. 2). Figure 2 shows oscillograms of mechanical-stress pulses with various ampli- When the signal intensity increases, the difference tudes in the NaCl crystal with (curves 1'Ð4') and with- between the pulse amplitudes in the presence and in the out (1Ð4) a magnetic pulse. The shape of the magnetic absence of a magnetic field rises (Figs. 2, 3). The pulse is also shown (curve 5). When the amplitude of a results obtained agree with the data from [9]. However,

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

1934 MOROZOV

64 ACKNOWLEDGMENTS 1 10 2 The author is grateful to B.I. Smirnov for many use- 8 48 3 ful discussions of the results of this work. 4 T 6 µ 32 , 1 , MPa REFERENCES B σ 4 1. V. I. Al’shits, E. V. Darinskaya, T. M. Perekalina, and 16 2 A. A. Urusovskaya, Fiz. Tverd. Tela (Leningrad) 29 (2), 467 (1987) [Sov. Phys. Solid State 29, 265 (1987)]. 2. Yu. I. Golovin, R. B. Morgunov, D. V. Lopatin, and 0 400 800 1200 1600 2000 2400 A. A. Baskakov, Phys. Status Solidi A 160 (2), R3 I, A (1997). 3. Yu. I. Golovin, R. B. Morgunov, V. E. Ivanov, et al., Fig. 3. Dependences of the amplitudes of mechanical-stress Pis’ma Zh. Éksp. Teor. Fiz. 68 (5), 400 (1998) [JETP pulses σ(I) and magnetic field B(I) in the NaCl crystal on the beam current: (1) σ(I) for the case of a mechanical pulse Lett. 68, 426 (1998)]. excited in the Al target in the absence of a magnetic field, 4. N. A. Tyapunina, V. L. Krasnikov, and E. P. Belozerova, (2) the same but in the presence of a magnetic field, (3) B(I), Fiz. Tverd. Tela (St. Petersburg) 41 (6), 1035 (1999) and (4) σ(I) for the case of a mechanical pulse excited in a [Phys. Solid State 41, 942 (1999)]. steel target. 5. V. I. Al’shits, A. A. Urusovskaya, A. E. Smirnov, and N. N. Bekkauer, Fiz. Tverd. Tela (St. Petersburg) 42 (2), 270 (2000) [Phys. Solid State 42, 277 (2000)]. the change in the shape of a mechanical pulse and the 6. B. I. Smirnov, N. N. Peschanskaya, and V. I. Nikolaev, saturation of its amplitude in that work was related to Fiz. Tverd. Tela (St. Petersburg) 43 (12), 2154 (2001) changes in the internal structure of samples upon [Phys. Solid State 43, 2250 (2001)]. annealing, whereas in our work these effects were 7. Yu. A. Osip’yan, Yu. I. Golovin, R. B. Morgunov, et al., caused by a magnetic field. Fiz. Tverd. Tela (St. Petersburg) 43 (7), 1333 (2001) [Phys. Solid State 43, 1389 (2001)]. Based on the experimental data obtained, we can 8. V. I. Al’shits, E. V. Darinskaya, M. A. Legen’kov, and draw the following conclusion. Since an electromag- V. A. Morozov, Fiz. Tverd. Tela (St. Petersburg) 41 (11), netic pulse was applied to the crystal before a mechan- 2004 (1999) [Phys. Solid State 41, 1839 (1999)]. ical pulse, dislocations had time to leave impurity para- 9. Y. Yasumoto, A. Nakamura, and R. Takeuchi, Acustica magnetic centers by the instant of the mechanical pulse 30 (5), 260 (1974). was applied. This time is less than 10Ð6 s, which is two 10. V. I. Al’shits, E. V. Darinskaya, and O. L. Kazakova, Zh. orders of magnitude shorter than that obtained in [10]. Éksp. Teor. Fiz. 111 (2), 615 (1997) [JETP 84, 338 Moreover, we revealed that a weak pulsed magnetic (1997)]. field could decrease the yield stress of NaCl samples under conditions of high-rate loading. Translated by K. Shakhlevich

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

MAGNETISM AND FERROELECTRICITY

Thermodynamics of a Ladder Ferromagnet with Random Transverse Exchange P. N. Timonin Research Institute of Physics, Rostov State University, pr. Stachki 194, Rostov-on-Don, 344090 Russia e-mail: [email protected] Received November 6, 2002

Abstract—A model of a ladder ferromagnet with Ising spins that consists of two ferromagnetic chains with interaction of nearest neighbors and a random interaction between the nearest spins of different chains is considered. For properly chosen parameters of the random interaction, the description of the ground-state degeneracy and thermodynamics at T = 0 proves to be elementary in terms of the statistics of one-dimensional clusters. Analysis of the field dependences of the thermodynamic parameters has revealed a series of phase transitions associated with changes in the spin configurations of the antiferromagnetic clusters. Numerical investigation of the thermodynamics of the system under consideration at finite temperatures confirms the correctness of the analytical results obtained at T = 0. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION cidate the physical cause of the ground-state degeneracy, which is associated with the energy degeneracy of The synthesis of crystals with one-dimensional lad- spin states of some magnetic clusters with a particular der structures has stimulated investigations into their configuration of exchange interactions [3, 5, 9]. How- physical properties [1]. Quasi-one-dimensional struc- ever, the enumeration of these configurations is a non- tures with different types of disorder are of special trivial problem and further progress in the study of the interest due to both the possibility of describing real aforementioned models, for example, investigation of crystals and the large variety of physical processes ini- the nonergodic dynamics at T = 0, is lacking. tiated by disorder. The magnetic properties of quasi- one-dimensional Ising systems are best understood Therefore, it is expedient to search for and study (see, for example, the review by Liebmann [2]). other disordered quasi-one-dimensional systems with Although quasi-one-dimensional systems with short- the degenerate ground state that would be suitable for range exchange do not undergo magnetic transitions at simple description. In this regard, the application of temperatures T ≠ 0, they exhibit a number of unusual ladder models offers strong possibilities, in particular, properties at T = 0 due to disorder (such as ground-state for the experimental verification of theoretical results. degeneracy and transitions in a magnetic field) [2–5]. In In the present work, we consider a model of a ladder turn, this gives rise to singularities in the temperature ferromagnet with Ising spins that consists of two ferro- and field dependences of the physical properties at low magnetic chains with interaction of nearest neighbors temperatures. and a random interaction between the nearest spins of different chains. For properly chosen parameters of the The degeneracy of the ground state is a characteris- random interaction, the description of the ground-state tic property of spin-glass phases and mixed phases in degeneracy and thermodynamics at T = 0 proves to be which different types of magnetic order and glass dis- elementary in terms of the statistics of one-dimensional order coexist. The ground-state degeneracy is associ- clusters. Numerical investigation of the thermodynam- ated with nonergodicity of the magnetic phases in dis- ics of this system at finite temperatures confirms the ordered magnetic materials (see the review by Binder validity of the analytical results obtained at T = 0. and Young [6]). In this respect, random (quasi) one- dimensional systems with frustration and ground-state degeneracy have been attracting close research atten- 2. THERMODYNAMICS AT T = 0 tion. However, only a few of these simplest systems are The spin Hamiltonian of the model under consider- suitable for rigorous analytical thermodynamic ation has the form description. Among them are the model of a one- dimensional Ising spin glass with random binary N exchange in an external field [3–5], which is character- Ᏼ = Ð∑ [JS()S + S S ized by specific forms of the exchange distribution 1i 1i + 1 2i 2i + 1 (1) function [7], and ladder glass models with binary i = 1 ()] exchange [4, 8]. These models made it possible to elu- + JrS1iS2i + HS1i + S2i .

1936 TIMONIN

JJ The ground-state energy at H < Ja is less than the energy Ef by the value of the gain upon transition of the ≥ ferromagnetic clusters with n n0 to the antiferromag- Jf Jf –Ja netic state: ∞ δ EE= f + ∑ Nn En. (5) J J nn= 0

Fig. 1. Schematic drawing of the ladder ferromagnet with Here, Nn is the mean number of antiferromagnetic clus- random transverse exchange. ters in n cells per cell [10]: n()2 ± Nn = p 1 Ð p . (6) Here, Sα, i = 1; H is the magnetic field (H > 0); J is the constant of ferromagnetic exchange of the chains (J > Since the antiferromagnetic order of a cluster can be 0); and Jr is the constant of random exchange between achieved in two ways, the ground state is degenerate the nearest spins of different chains, which takes two with the mean entropy values, namely, J = ÐJ with the probability p and J = r a r ∞ Jf with the probability 1 Ð p. 3 SN= ∑ ln2 + δ , ()N ln---. (7) n n0 2J/ Ja Ð H n0 Therefore, for each exchange constant Jr, the system 2 nn= 0 consists of clusters with antiferromagnetic (ÐJa) and ferromagnetic (Jf) interactions between the chains. Fig- The last term in formula (7) accounts for the energy ure 1 presents a schematic drawing of the ladder ferro- degeneracy of the ferromagnetic and antiferromagnetic magnet for a particular random transverse exchange. ordering of n0 clusters in the case when 2J/(Ja Ð H) is an integral number. Let us now assume that Jf > 2J. In this case, the spins in the ground state in ferromagnetic clusters are equal The average magnetization in the ground state at to unity; therefore, it remains to determine the direc- H < Ja can be determined by subtracting the fraction of tions of the spins in the antiferromagnetic clusters. We spins in the clusters with antiferromagnetic ordering consider the change in the energy of the antiferromag- from the ferromagnetic magnetization (unity); that is, netic clusters upon the transition from a ferromagnetic ∞ n N state (S1, iS2, i = 1) to a configuration with opposite spins 0 n0 M = 1 Ð ∑ nN + δ , ()------. (8) in n consecutive cells (if the cells are not consecutive, n n0 2J/ Ja Ð H 3 the energy of the configuration increases); that is, nn= 0 δE = Ð2nJ()Ð H + 4J. (2) It should be noted that, unlike the entropy and energy n a obtained by averaging over all possible configurations δ Under the condition H > Ja, the inequality En > 0 holds of interactions, the average magnetization is deter- and the oppositely directed spins in the cells become mined as the average over the ground states with subse- energetically unfavorable. As a result, the (only) ground quent averaging over the interactions. The last term in state is a ferromagnetic state with the mean energy per relationship (8) accounts for the degeneracy of the fer- cell: romagnetic and antiferromagnetic states of n0 clusters in the case when 2J/(Ja Ð H) is an integral number. In E = Ð2JpJ+ Ð2()1 Ð p J Ð H. (3) f a f NNn actual fact, among the 3 0 states differing in the con- Under the condition H < Ja, the lowest energy is kNN observed in configurations with the largest number of figuration of n clusters, there exist 2 n0 states 0 k cells n, which coincides with the cluster size (S1, iS2, i = Ð1). This antiferromagnetic ordering is energetically with an antiferromagnetic configuration of spins in k favorable for sufficiently large-sized antiferromagnetic clusters for which the mean number of spins per site in clusters in which the number of cells is greater than or n0 clusters with antiferromagnetic ordering can be determined according to the formula equal to an integral number n0 satisfying the condition NN ()<< () n0 2J/ Ja Ð H n0 12+ J/ Ja Ð H (4) ÐNNn n0k kNN 0 2 ∑ ------2 n0 3 = ---n N , N k 3 0 n0 when 2J/(Ja Ð H) is not an integral number. If 2J/(Ja Ð k = 1 H) is an integral number, we have n0 = 2J/(Ja Ð H); in from which follows relationship (8). this case, the clusters belonging to n0 cells are characterized by equal energies of ferromagnetic and antifer- Substituting expressions (2) and (6) into formulas (5), romagnetic ordering. (7), and (8), we obtain the following explicit expres-

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 THERMODYNAMICS OF A LADDER FERROMAGNET 1937 sions for the mean energy, entropy, and magnetization at T = 0: (a) n0 0.2 E = Ð2J[]12Ð p ()1 Ð p + J []p Ð 2Ln(), p a 0 (9) [](), () Ð2H 12Ð Ln0 p Ð 1 Ð p J f , 1

n0 3 Sp= ()1 Ð p ln2 +,δ , ()()1 Ð p ln--- (10) n0 2J/ Ja Ð H 2 0.1 2

n0()2 n0P 1 Ð p M = 1 Ð Ln(), p + δ , ()------. (11) 0 n0 2J/ Ja Ð H 3 Here, 0 (b) ∞ 1.0 (), n0[]() Ln0 p ==∑ nNn p pn+ 0 1 Ð p . 0.8 nn= 0 The ﬁeld dependences of the entropy and magneti- 1 zation for the exchange constant Ja = 3J with probabil- 0.6

ities p = 0.3 and 0.8 (Fig. 2) exhibit a stepwise behavior MS with jumps at integral values of 2J/(Ja Ð H), which is 0.4 characteristic of (quasi) one-dimensional Ising magnets with a discrete disorder [2Ð4]. The physical cause of 0.2 this behavior is quite obvious: an increase in the mag- 2 netic field leads to changes in the spin configurations (from an antiferromagnetic configuration to a ferromagnetic configuration) involving progressively larger clus- 0 0.5 1.0 1.5 2.0 2.5 3.0 ters with antiferromagnetic interactions. Under the con- H/J dition H < Ja, at the points Hk = Ja Ð 2J/k (k = 1, 2, …), there occurs a series of transitions between phases in Fig. 2. Field dependences of the thermodynamic parameters at T = 0, Ja = 3J, and p = (1) 0.3 and 0.8: (a) entropy and which ferromagnetic spin ordering coexists with glass (b) magnetization. ordering. The energy described by expression (9) is continuous at the transition points. The degree of glass disorder and the ground-state degeneracy (entropy) coinciding with that defined by relationship (11) can be decrease with an increase in the magnetic field, and, obtained when the differentiation with respect to H is finally, the system becomes ferromagnetic at H > Ja. carried out before the passage to the limit N ∞: Thus, we have a typical system with disorder-induced ∂ ∂ frustration; however, unlike the models considered ear- 2M = Ð lim EN/ H. → ∞ lier [2Ð5, 7Ð9], the description of the properties of a N ladder ferromagnet with a random transverse exchange at T = 0 and Jf > 2J is quite elementary. 3. THERMODYNAMICS AT FINITE Note that the expressions obtained for the energy TEMPERATURES and magnetization satisfy the standard thermodynamic relationship An analysis of the thermodynamics in terms of the above model at finite temperatures can be performed by ∂E 2M = Ð------, the standard methods used for studying random quasi- ∂H one-dimensional systems with short-range interaction [11]. Let us introduce a partial partition function in the when 2J/(Ja Ð H) is not an integral number. In the case form when 2J/(Ja Ð H) is an integral number, this relationship (), []()βᏴ does not hold, because from expression (9) it follows Z N S1 S2 = TrN Ð 1 exp Ð . that, at the transition points, the energy has an inflection β and its derivative with respect to the magnetic field H Here, = 1/T and TrN Ð 1 indicates summation over con- does not exist. This is due to a nonuniform (with respect figurations of all spins, except for the spins of the last ∂ ∂ ∞ to H) convergence of EN/ H to its limit at N in cell (S1, S2). The quantities ZN(S1, S2) satisfy the follow- the vicinity of these values of H, which makes it impos- ing recurrence relationships: sible to differentiate the limiting value of E with respect () (), () to H. In this case the correct result for the magnetization Z N + 1 S = TrS' R SS' Z N S' , (12)

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1938 TIMONIN

1.0 (a) (a) –3.2 Ja = 0.1 J 0.8 Ja = 0.3 J Ja = 0.5 J –3.3 Ja = 0.8 J 0.6 Ja = J F W –3.4 0.4 12 3 –3.5 0.2

0 –3.6 1.0 (b) (b) Ja = 0.1 J J = 0.3 J 0.8 0.4 a Ja = 0.5 J Ja = 0.8 J 0.6 Ja = J S

W 0.4 12 3 0.2

0.2

0 –1.0 –0.5 0 0.5 1.0 σ 0 0.2 0.4 0.6 0.8 1.0 T/J

Fig. 4. Temperature dependences of the thermodynamic ω σ Fig. 3. Integrated distribution function ( ) at p = 0.5, Jf = parameters at p = 0.5, Jf = 3J and different values of Ja: 3J, Ja = (a) 0.1 and (b) 0.5J, and T = (1) 2J, (2) J, and 0.5J. (a) thermodynamic potential and (b) entropy. where R(S, S') is the transfer matrix. Below, we will ()β 2 ()σβ σ tanh Jr + tanh J N consider the case H = 0. Hence, we can write N + 1 = ------. (14) 1 + tanh()βJ tanh2 ()σ βJ (), [] β()β, r N R SS' = exp J SS' + JrS1S2 . In turn, from expression (14), it follows that, when In this case, the partial partition function can be repre- N ∞, the distribution function of the quantity σ sented in the form takes the form [11] Z () N ()σ 1 Z N S = ------1 + NS1S2 , 4 ρσ()= d σ' 〈〉δσ[]Ð f ()σ', J ρσ()' , (15) ∫ r Jr Ð1 where ZN is a standard partition function, σ Z = Tr []exp()ÐβᏴ , where f( ', Jr) is the function on the right-hand side of N N expression (14), σ and N is the mean product of the spins in a single cell, ()β 2 ()σβ tanh Jr + tanh J ÐβᏴ ()σ, ------σ ()≡ 〈〉 f Jr = 2 , (16) N = TrN S1S2e /Z N S1S2 N. ()β ()σ β 1 + tanh Jr tanh J

From relationship (12), it follows that and the angular brackets indicate averaging over the 2 ()β () β random exchange. Z N + 1 = 4cosh J cosh Jr (13) After determining the distribution function ρ(σ) × []2 ()β ()σ β 1 + tanh J tanh Jr N Z N, from relationship (15), we can calculate the mean ther-

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 THERMODYNAMICS OF A LADDER FERROMAGNET 1939 modynamic potential from expression (13); that is, suming if one wants to attain a reasonable accuracy. However, the results obtained at T > 0.2J (Fig. 4a) also ≡ T 〈〉 demonstrate that, at low temperatures, the thermody- F Ð lim ---- lnZ N N → ∞N namic potential F(T) tends to the corresponding mean []2 ()β 〈〉[]()β values E(T = 0) [expression (9)] represented in this fig- = ÐT ln 4cosh J Ð T ln cosh Jr Jr (17) ure. The behavior of the entropy also agrees with the 1 results obtained at T = 0 [expression (10)] (Fig. 4b). Ð T dσρ() σ 〈〉ln[]1 + tanh2 ()βJ tanh()σ βJ . Thus, the simple analytical relationships derived at T = ∫ r Jr 0 adequately describe the thermodynamics of the Ð1 model under consideration. Equation (15) can be solved only numerically. For this In conclusion, it should be noted that the above purpose, we transform it slightly. By continuing ρ(σ) model with such a simple but nontrivial thermodynam- through zero to the region σ2 > 1, introducing integra- ics at T = 0 that exhibits all properties of frustrated dis- tion under the averaging sign and integrating, we obtain ordered systems can serve as a test model for the verifi- the relationship cation of approximate approaches to the description of more complex low-dimensional disordered magnets. It ∂ ()σ, ρσ() g Jr ρ[]()σ, is evident that the dynamics described by this model at = ------∂σ g Jr . (18) Jr T = 0 is also nontrivial and exhibits all effects inherent σ σ in nonergodic systems. I believe that, within this model, Here, g( , Jr) is the function inverse to the function f( , the description of similar dynamic effects will also be Jr) defined by expression (16): sufficiently simple (even if it is not analytical, it will at σβÐ tanh()J least not require time-consuming numerical computa- ()σ, Ð2 () β r tions). Note also that this model can be reproduced in g Jr = tanh J ------σβ()-. 1 Ð tanh Jr practice; for example, random exchange can be created Integrating expression (18) with respect to σ, we obtain by introducing impurities between ferromagnetic chains. w()σ = 〈〉wg[]()σ, J , (19) r Jr where REFERENCES σ 1. E. Dagotto, Rep. Prog. Phys. 62 (8), 1525 (1999). ()σ σρσ() 2. R. Liebmann, Statistical Mechanics of Periodic Frus- w = ∫ d ' ' . trated Ising Systems (Springer, Berlin, 1986), Lect. Ð1 Notes Phys., Vol. 251. Unlike the distribution function ρ(σ), which can 3. A. Vilenkin, Phys. Rev. B 18 (3), 1474 (1978). involve delta functions, the integrated distribution func- 4. B. Derrida, J. Vannimenus, and Y. Pomeau, J. Phys. C 11 tion w(σ) satisfies the simpler equation (19) and has (16), 4749 (1978). weaker singularities (jumps). Consequently, the numer- 5. E. Farhi and S. Guttmann, Phys. Rev. B 48 (13), 9508 ical solution to Eq. (19) is significantly simpler and can (1993). be obtained by the iteration method. If the initial func- 6. K. Binder and A. P. Young, Rev. Mod. Phys. 58 (4), 801 σ σ (1986). tion is chosen as w0( ) = 0.5(1 + ), convergence to the solution can be reached after five to ten iterations. The 7. J. M. Luck and Th. M. Nieuwenhuizen, J. Phys. A 22 (7), σ 2151 (1989). function w( ) for p = 0.5, Jf = 3J, and several temperatures and exchange constants J is presented in Fig. 3. 8. D. C. Mattis and P. Paul, Phys. Rev. Lett. 83 (18), 3733 a (1999). The temperature dependences of the thermodynamic 9. F. Igloi, J. Phys. A 27 (10), 2995 (1994). potential F for p = 0.5, J = 3J, and different exchange f 10. D. Stauffer and A. Aharony, Introduction to Percolation constants Ja calculated according to formula (17) are Theory (Taylor and Francis, London, 1992). shown in Fig. 4a. The entropy obtained by numerical 11. D. Andelman, Phys. Rev. B 34 (9), 6214 (1986). differentiation of the thermodynamic potential F(T) is presented in Fig. 4b. Unfortunately, the computations performed for temperatures T < 0.2J are very time-con- Translated by O. Moskalev

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

MAGNETISM AND FERROELECTRICITY

Magnetic Structure of a Compensated FerromagnetÐAntiferromagnet Interface A. I. Morosov Moscow State Institute of Radio Engineering, Electronics, and Automation (Technical University), pr. Vernadskogo 78, Moscow, 119454 Russia e-mail: [email protected] Received March 25, 2003

Abstract—The magnetic structure of a compensated ferromagnet–antiferromagnet interface is considered. It is shown that the rate of decrease in the magnetic order parameters as one goes away from the interface is determined by the type of crystal structure of the layers and by the crystallographic orientation of the interface. © 2003 MAIK “Nauka/Interperiodica”.

1. A large number of studies have been devoted to causes the unidirectional anisotropy. The rotation of the the properties of ferromagnetÐantiferromagnet inter- magnetization of the ferromagnet under an external faces (see, e.g., [1]). The reason for this is that the anti- magnetic field creates a domain wall in the antiferro- ferromagnet-induced shift of ferromagnetic hysteresis magnet. The bias field of the hysteresis loop can be esti- loops from their symmetrical position (so-called unidi- mated by equating the Zeeman energy of the ferromag- rectional anisotropy, or the exchange-induced bias) has net to the energy of formation of a domain wall [3, 4] found wide use in magnetoelectronic devices. and is found to be The theory of unidirectional anisotropy for the com- waf pensated surface of an antiferromagnet was developed Bex = ------, (1) by Koon [2]. In the case of a compensated surface, an Md antiferromagnet atomic plane parallel to the interface where waf is the surface energy density of a domain wall has zero magnetic moment. If the compensated surface in the antiferromagnet and M and d are the magnetiza- is coated with a ferromagnetic layer, then, in the tion and thickness of the ferromagnetic layer, respec- absence of an external field in the exchange approxima- tively. Here, we assume that the domain-wall energy in tion, the magnetization vector of the ferromagnet is oriented perpendicular to the antiferromagnetism vector of the (two-sublattice) antiferromagnet. Exchange interaction of spins of the bottom atomic layer of the ferromagnet with the spins of the top atomic layer of the antiferromagnet results in spin canting (spin-flop), F similar to the sublattice magnetization canting induced by a magnetic field (Fig. 1) [2]. The difference is that, in the nearest neighbor interaction approximation, the exchange field of the ferromagnet acts only on the ϕ uppermost atomic spin layer of the antiferromagnet; 1 therefore, the canting angle θ decays as a function of depth in the antiferromagnet. In addition, the spins of the ferromagnet are slightly tilted in opposite directions from being parallel, which can be interpreted as the appearance of an induced antiferromagnetism vector L (Fig. 1). As in the antiferromagnet, the tilting angle ϕ decreases as a function of θ AF depth in the ferromagnet. 1 The interaction of the magnetization of the ferromagnet with the canting-induced surface magnetic moment of the antiferromagnet, as well as the interac- Fig. 1. Orientation of spins in the bottom (F) atomic layer tion of the vector L of the antiferromagnet with the of the ferromagnet and in the top (AF) atomic layer of the induced antiferromagnetism vector of the ferromagnet, antiferromagnet for the (100) cut of a simple cubic lattice.

MAGNETIC STRUCTURE 1941 the ferromagnetic layer is higher than that in the anti- The exchange interaction energy inside the ferro- ferromagnet. magnet is In this publication, we studied the law describing 2 the decay of the angles θ and ϕ as a function of depth NJfSf ()ϕ ϕ ϕ Wf = Ð------acos 1 Ð 2 + bcos2 1 --- in the respective layers. 2 

2. Since, far from the Curie and Néel temperatures, ∞ all characteristic scales of the problem at hand are of [ ()ϕ ϕ ()ϕ ϕ the atomic order of magnitude, we consider a discrete + ∑ acos j Ð j Ð 1 + acos j Ð j + 1 (3) spin lattice under the assumption that the type of crystal j ≥ 2 lattice and its parameters are the same in the ferromag-  net and antiferromagnet. ϕ ] ---+ bcos2 j , The Heisenberg exchange interaction of adjacent  spins in the ferromagnet and antiferromagnet is ϕ where j is the tilting angle of spins in the jth atomic described by the exchange integrals Jf > 0 and Jaf < 0, respectively, while the interaction between a spin in the layer of the ferromagnet (the direction of tilting is the ferromagnet and its nearest neighbor spins in the anti- same for all spins located on the same perpendicular to the interface). ferromagnet is described by the exchange integral Jf, af. The computer simulation made in [2] corresponds to The analogous energy inside the antiferromagnet is the case where all exchange energies are equal. As was 2  indicated in [2], the sublattice canting angle decays NJaf Saf ()θ θ θ Waf = Ð------acos 1 + 2 + bcos2 1 --- quickly with depth in the antiferromagnet and becomes 2  virtually equal to zero in the fifth or sixth atomic layer. ∞ The characteristic decay length of θ and ϕ depends [ ()θ θ ()θ θ on the type of crystal lattice and on the crystallographic + ∑ acos i + i Ð 1 + acos i + i + 1 (4) orientation of the interface. In particular, for a family of i ≥ 2 atomic planes parallel to the interface and for a given  spin, the ratio is of importance between the number of θ ] ---+ bcos2 i , the spin’s nearest neighbors a located in the adjacent  atomic plane and the number of its nearest neighbors b θ in the atomic plane where this spin is located (2a + b = where i is the sublattice canting angle in the ith layer z, where z is the total number of its nearest neighbors). of the antiferromagnet (the direction of canting coin- Given a and b, the Heisenberg exchange interaction cides with that in the top atomic layer; see Fig. 1). energy at the ferromagnetÐantiferromagnet interface Minimization of W with respect to the variables ϕ ≥ f j Wf, af has the form for j 2 in the context of the mean-field theory yields the following recurrence formula relating the tilting ()θ ϕ Wfaf, = ÐNa Jfaf, SfSaf sin 1 + 1 , (2) angles in adjacent layers: ()ϕ ϕ ()ϕ ϕ ϕ where Sf and Saf are the average values of the spin of an asin j Ð j Ð 1 ++0.asin j Ð j + 1 bsin2 j = (5) atom in the ferromagnet and antiferromagnet, N is the ϕ Ӷ θ For 1, we assume that number of atoms in an atomic plane, 1 is the sublattice ϕ κϕ canting angle in the top atomic plane of the antiferro- j = j Ð 1 (6) magnet, and ϕ is the tilting angle of spins in the bottom 1 and find atomic plane of the ferromagnet. We assume that the sublattice magnetizations remain unchanged in magni- b bb tude. The sign of J determines only the direction of κ = ---1+ Ð ------2+ . (7) f, af a aa the sublattice canting. At Jf, af > 0, the magnetization of the top antiferromagnetic atomic layer is parallel to the For noncubic lattices, the value of κ will also depend on magnetization of the ferromagnet, and at Jf, af < 0, it is the ratio of the exchange integrals corresponding to var- antiparallel to it [2]. ious interatomic distances. θ ϕ For small values of 1 and 1, the interaction energy A similar consideration for the antiferromagnet can be represented as the sum of two independent gives θ terms. One term, dependent on 1, is the interaction θ = Ðκθ , (8) energy of the spins of the ferromagnet with the ferro- i i Ð 1 magnetic moment induced in the antiferromagnet. The with the same value of κ as for the ferromagnet. Thus, ϕ other term, dependent on 1, is the interaction energy of in a real situation, the sublattices in adjacent atomic the spins of the antiferromagnet with the induced anti- planes of the antiferromagnet are canted in opposite ferromagnetism vector of the ferromagnet. directions.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1942 MOROSOV

Table 1. Parameters characterizing various cuts of cubic crystals Lattice type Cut ab κγ

SC (100) 1 4 5 Ð 24 = 0.101021 2 + 6 = 4.44949 SC (110) 2 2 2 Ð 3 = 0.267949 1 + 3 = 2.73205 BCC (110) 2 4 3 Ð 8 = 0.171573 2(1 + 2 ) = 4.82843

For small values of the canting and tilting angles, Minimization of the total interaction energy W of the excess energies of the ferromagnet and antiferro- the layers magnet associated with the deviation of the spin direc- ∆ ∆ tions, in view of Eqs. (6) and (8), are reduced to W = Wf ++Waf Wfaf, (12) with respect to the angles θ and ϕ gives the following ∞ ϕ 2 1 1 ∆ 2 2 []()ϕ2 ϕ ϕ a 1 expressions: Wf = NJf Sf ∑ ba+ j Ð a j j + 1 Ð ------2 j = 1 θ a Jfaf, Sf (9) 1 = -----γ------, (13) 2 Jaf Saf 2 2b a()1 Ð κ 2 2 2 = NJ S ϕ ------+ ------≡ γNJ S ϕ , f f 1 2 21()+ κ f f 1 1 Ð κ ϕ a Jfaf, Saf 1 = -----γ------. (14) 2 JfSf ∞ θ 2 ∆ 2 []()θ2 θ θ a 1 Therefore, the interaction energy of the layers W is Waf = NJaf Saf ∑ ba+ i + a i i + 1 Ð ------2 2 2 2 2 i = 1 a NJfaf, Sf Saf 1 1 (10) W = Ð------------+ ------. (15) ≡γ 2 θ 2 4γ 2 2 NJaf Saf 1 , Jaf Saf JfSf where (after certain simpliﬁcations) This energy is determined by the softest subsystem, i.e., by the layer with the lowest exchange energy of spins. 1 ϕ θ Ӷ γ = ---[]bbb+ ()+ 2a . (11) 3. The condition 1, 1 1 is equivalent to the con- | | Ӷ 2 dition of weak interlayer interaction, Jf, af Jf, Jaf. κ γ Generally (for cubic lattices, only in the case where the The values of a, b, , and for several cuts of simple exchange integral is the same for all nearest neighbors), cubic (sc) and body-centered cubic (bcc) lattices are the solution of the problem at hand is determined by listed in Table 1. The (111) cut in the sc and bcc lattices three dimensionless parameters: y = b/a, p = and the (001) cut in the bcc lattices are not compensated |J |S /|J |S , and q = |J |S /J S . surfaces. f, af f af af f, af af f f ӷ ӷ θ ϕ For the case of p 1 or q 1, we have 1 + 1 = π/2; i.e., the spins of the bottom atomic layer of the fer- θ 1 romagnet are parallel (Jf, af > 0) or antiparallel (Jf, af < 0) 1.6 to the spins in the top atomic layer of the antiferromagnet. The interaction energy in this case is W ≈ − | | a Jf, af NSfSaf. 1.2 If the inequality q/p Ӷ 1 (p/q Ӷ 1) is obeyed, one can assume that ϕ ≈ 0 (θ ≈ 0) and the number of 3 2 1 1 1 θ parameters decreases. Numerically calculated 1(p) 0.8 dependences for the case q Ӷ 1 at various values of y are shown in Fig. 2. θ π |θ | θ In the limiting case 1 = /2, the ratio 2 / 1 is less 0.4 than κ by a factor of 1.1 to 1.4 and the subsequent reduc- θ tion of i is described by the dependences obtained for small angles (with an accuracy of better than 1%). The θ 0 5 10 15 20 25 30 calculated values of i are listed in Table 2. p Formula (1) holds if the total interaction energy of the layers W per area σ of the layers is larger in magni- Fig. 2. Sublattice canting angle in the top atomic layer of the tude than w . Otherwise, a domain wall does not form. antiferromagnet as a function of the interaction energy of af the layers for (1) y = 4, (2) 2, and (3) 1. The main results of this study as follows:

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 MAGNETIC STRUCTURE 1943

Table 2. Canting angles for the ﬁrst three atomic planes of directions in the layer with a lower exchange interac- θ π the antiferromagnet at 1 = /2 tion energy. θ θ θ y 1 2 3 ACKNOWLEDGMENTS 4 1.5708 0.1125 0.0114 2 1.5708 0.2077 0.0355 The author would like to thank Yu.V. Pyl’nov for help with the calculations and École Centrale de Lille 1 1.5708 0.3661 0.0976 for providing favorable conditions for productive work. This study was supported by the CRDF and the Ministry of Education of the Russian Federation (grant (1) The sublattices of the antiferromagnet near the no. VZ-010-0) and by the PICS/RFBR (grant no. ferromagnetÐantiferromagnet interface are canted in 1573/02-02-22002). opposite directions in the adjacent atomic layers parallel to the interface. REFERENCES θ ϕ (2) The law of the decay of angles and in depth 1. J. Nogues and I. K. Schuller, J. Magn. Magn. Mater. 192 is the same for the ferromagnetic and antiferromagnetic (2), 203 (1999). layers and is determined by the crystallographic orien- 2. N. C. Koon, Phys. Rev. Lett. 78 (25), 4865 (1997). tation of the interface and by the type of crystal lattice 3. A. P. Malozemoff, Phys. Rev. B 35 (7), 3679 (1987). of the layers. The decay law is exponential in the case 4. D. Mauri, H. C. Siegmann, P. S. Bagus, and E. Kag, of small angles. J. Appl. Phys. 62 (7), 3047 (1987). (3) The main contribution to the interaction energy of the layers is connected with the deviations of the spin Translated by A. Zalesskiœ

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1944Ð1951. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1850Ð1856. Original Russian Text Copyright © 2003 by Nikitin, Tereshina.

MAGNETISM AND FERROELECTRICITY

Effect of Interstitial Atoms on the Effective Exchange Fields in Ferrimagnetic Rare-Earth and 3d Transition Metal Compounds R2Fe17 and RFe11Ti S. A. Nikitin* and I. S. Tereshina** * Moscow State University, Vorob’evy gory, Moscow, 119992 Russia e-mail: [email protected] ** Baœkov Institute of Metallurgy and Materials Sciences, Russian Academy of Sciences, Leninskiœ pr. 49, Moscow, 117911 Russia Received March 25, 2003

Abstract—The magnetic properties (magnetic ordering temperature, magnetization) of the ferrimagnetic compounds R2Fe17 and RFe11Ti, as well as of their hydrides and nitrides, were studied. The hydrogenation- and nitrogenation- induced variation of the exchange ﬁelds acting on the rare-earth (RE) ions from both the Fe sublattice and other RE ions was determined, and the dependence of the Curie temperatures of the starting compounds, their hydrides, and nitrides on the de Gennes factor was revealed. It was found that incorporation of light atoms (H, N) into the crystal lattices of RFe11Ti and R2Fe17 increases the Curie temperature TC substantially, increases the FeÐFe exchange coupling, and decreases the RÐR exchange interactions, as well as increases the RÐFe intersublattice exchange under hydrogenation and decreases it under nitrogenation, an effect that can be understood as resulting from the attendant changes in the electronic structure of these compounds and in the indirect exchange interactions. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION one of which is made up of iron atoms and the other of rare-earth atoms. Rare-earth (RE) and 3d transition metal compounds arouse considerable research interest in the physics of The magnetically active 4f subshell of a rare-earth magnetic phenomena, because these substances are ion is screened by the higher lying 5s25p6 spin-closed convenient objects for testing and refining the theoreti- electronic shells, thus precluding wave-function over- cal concepts used in describing the magnetic ordering lap of the 4f electrons with the 4f and 3d electrons of the in metallic ferro- and ferrimagnets. The electronic sys- neighboring atoms. In RE and 3d intermetallic com- tem of rare-earth compounds permits a distinct separa- pounds (including R2Fe17 and RFe11Ti), the 3d electrons tion into electrons possessing a localized magnetic of the iron atoms, which are responsible for the 3d-sub- moment and conduction electrons, and the magnetic lattice magnetism, are also localized primarily on the system can be separated into two magnetic sublattices, iron atoms; this is indicated by data on the magnetic namely, the rare-earth and 3d sublattices [1, 2]. On the properties and hyperfine fields [3, 4]. The localized other hand, many RE compounds of this type have con- model of magnetic sublattices describes the principal siderable application potential and are already employed in powerful permanent magnets, magneto- magnetic properties of these compounds fairly well by strictors, cooling agents in magnetic coolers, etc. [2]. invoking the molecular-field theory, which considers two (rare-earth, iron) magnetic sublattices coupled by The present communication considers the magnetic the 3dÐ4f exchange interaction [5Ð9]. It was shown in properties of the R2Fe17 and RFe11Ti ferrimagnetic [5Ð9] that the exchange interaction between iron atoms compounds, as well as of their hydrides and nitrides, on the iron sublattice is the strongest, whereas the which are obtained from the starting samples by insert- exchange (3dÐ4f) coupling between the iron and rare- ing interstitial atoms into their lattices. The best studied earth sublattices in these compounds is substantially to date are the R2Fe17 compounds with such interstitial weaker. In contrast to our previous studies, here we elements as nitrogen and carbon [3], whereas the attempted to take into account all three kinds of R2Fe17HX hydrides have thus far attracted less interest. exchange interactions (including those between ions on No systematic investigation has been conducted on the RE sublattice) in the hydrides and nitrides. As fol- RFe11Ti compounds with light interstitial elements. The lows from an analysis of the available publications, the interest in R2Fe17 and RFe11Ti stems from the fact that calculations for the hydrides and nitrides performed they are good model ferrimagnets (for heavy rare thus far were conducted under the assumption that the earths) with two largely localized magnetic sublattices, exchange interactions between ions on the RE sublat-

EFFECT OF INTERSTITIAL ATOMS 1945

50 0.20 LuFe11TiN 3

LuFe11TiH 2

σ 0.12 / 1

30 H LuFe11Ti , arb. units

M 0.04 10 0 2000 4000 6000 0 σ2 200 400 600 800 T, K

Fig. 1. Thermomagnetic analysis of LuFe11Ti and its Fig. 2. BelovÐArrott plots calculated for Y2Fe17 at different hydride and nitride. temperatures near the Curie point: (1) 345, (2) 340, (3) 335 K. tice may be neglected. This assumption has not been detail in [12]. The phase composition and lattice param- sufficiently substantiated. eters were studied by x-ray structural analysis (CuKα Although the exchange interaction model developed radiation) of the starting samples and their hydrides and for the R2Fe17 and RFe11Ti compounds elucidated the nitrides. effect of these interactions on the Curie temperatures The samples for the investigation of R2Fe17 (R = Y, and the temperature behavior of magnetization [7Ð9], Gd, Tb, Dy, Ho, Er, Lu) were prepared through RF the nature of the 3dÐ4f exchange coupling remains alloying of the charge of 99.95%-pure starting compo- unclear. We consider here itinerant electrons of two nents. The ingots were remelted in an electric resistance types, namely, the s electrons (appearing when the 4s furnace with a high temperature gradient and subse- and 6s electrons transfer to the conduction band) and quently cooled slowly to increase the grain size. R2Fe17 the 3d and 5d hybridized electrons, as mediators of single crystals were cut from ingots. The hydrogenation exchange interaction between the 4f and 3d electrons. and nitrogenation techniques are reported in [13]. In particular, the model proposed involves the indirect X-ray phase characterization showed all samples to be 4fÐ5dÐ3d exchange interaction [10]. The importance of practically single phase and to have the Th Ni struc- investigating this exchange interaction and of determin- 2 17 ture. The amounts of absorbed H and N in the R2Fe17HX ing the corresponding effective exchange fields Heff2 and R2Fe17NY compounds corresponded to three hydro- with which the iron sublattice acts on the RE ions gen atoms and two nitrogen atoms per formula unit, (index 1 refers subsequently to the iron sublattice, and respectively. index 2, to the RE sublattice) is accounted for by the The Curie temperatures were determined by ther- fact that Heff2 is responsible for the degree of magnetization of the rare-earth sublattice. If the energy of the momagnetic analysis. The magnetization of RFe11TiH rare-earth ion magnetic moment in the exchange field and R2Fe17H3 single crystals and magnetic field–ori- Heff2 is higher than the thermal energy, most of the rare- ented RFe11TiN and R2Fe17N2 powder samples was earth ions align with the field Heff2; this is the condition measured in the 77- to 800-K temperature region in necessary to obtain the giant magnetic anisotropy and magnetic fields of up to 13 kOe using a pendulum mag- magnetostriction observed in RE compounds [6]. The netometer. present communication reports on a study of the effects associated with the influence of interstitial hydrogen and nitrogen atoms on the effective exchange fields act- 3. EXPERIMENTAL RESULTS ing in the R2Fe17 and RFe11Ti RE compounds. We determined the Curie temperature as the point of the sharpest drop in magnetization σ experienced in the transition from the ferromagnetic to paramagnetic 2. TECHNOLOGY OF SAMPLE PREPARATION state, i.e., as the temperature at which |dσ/dT| is maxi- AND THE MEASUREMENT TECHNIQUE mum. Figure 1 illustrates temperature dependences of The starting RFe11Ti alloys (R = Y, Sm, Gd, Tb, Dy, the magnetic moment of the LuFe11Ti compound and Ho, Er, Lu) were prepared by alloying the charge of its hydride and nitride measured in a comparatively 99.95%-pure starting components in an argon atmo- weak magnetic field, H = 500 Oe. The values of TC for sphere through RF heating. The growth of single crys- these compounds were found to be 490, 546, and tals was described in [11]. The hydrogenation and 738 K, respectively. In some cases, the Curie tempera- nitrogenation techniques are reported in considerable ture was derived from BelovÐArrott plots (Fig. 2) relat-

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1946 NIKITIN, TERESHINA

Table 1. Molecular-field coefficients and effective exchange fields for the RFe Ti(H,N) compounds calculated with inclu- ≠ 11 X sion of exchange interaction between ions on the rare-earth sublattice (A2 0) ρ 3 6 6 6 Composition , g/cm TC, K Gh21, 10 Oe N21 h11, 10 Oe N11 h22, 10 Oe

YFe11Ti 7.22 540 Ð Ð Ð 6.4 6122 Ð GdFe11Ti 7.81 610 15.75 0.895 764 Ð Ð 0.382 TbFe11Ti 7.88 556 10.50 0.895 505 Ð Ð 0.382 DyFe11Ti 7.93 535 7.08 0.895 378 Ð Ð 0.382 HoFe11Ti 8.06 515 4.50 0.895 298 Ð Ð 0.382 ErFe11Ti 8.05 500 2.55 0.895 250 Ð Ð 0.382 LuFe11Ti 8.14 486 Ð Ð Ð 5.76 5446 Ð YFe11TiH 7.17 598 Ð Ð Ð 6.85 6155 Ð GdFe11TiH 7.75 658 15.75 0.962 829 Ð Ð 0.379 TbFe11TiH 7.79 602 10.50 0.962 551 Ð Ð 0.379 DyFe11TiH 7.85 577 7.08 0.962 411 Ð Ð 0.379 HoFe11TiH 7.98 553 4.50 0.962 325 Ð Ð 0.379 ErFe11TiH 7.99 537 2.55 0.962 271 Ð Ð 0.379 LuFe11TiH 8.05 520 Ð Ð Ð 5.95 5313 Ð YFe11TiN 7.10 712 Ð Ð Ð 7.76 6480 Ð GdFe11TiN 7.62 768 15.75 0.834 742 Ð Ð 0.122 TbFe11TiN 7.75 750 10.50 0.834 487 Ð Ð 0.122 DyFe11TiN 7.78 736 7.08 0.834 365 Ð Ð 0.122 HoFe11TiN 7.82 723 4.50 0.834 292 Ð Ð 0.122 ErFe11TiN 7.88 713 2.55 0.834 242 Ð Ð 0.122 LuFe11TiN 7.93 703 Ð Ð Ð 7.66 6373 Ð ing the quantity H/σ to squared specific magnetization 4. DISCUSSION OF EXPERIMENTAL DATA σ2 at temperatures close to the Curie point. Here, the ON THE DEPENDENCE OF CURIE Curie point is defined as the temperature at which the TEMPERATURES ON THE DE GENNES FACTOR H/σ vs. σ2 straight line passes through the origin. The IN THE R2Fe17 AND RFe11Ti COMPOUNDS AND THEIR HYDRIDES AND NITRIDES values of TC thus obtained practically coincide (to within 1 K) with the data derived with the use of the The effective exchange fields and magnetic ordering preceding method (from the maximum value of of RE metals originate from indirect exchange coupling |dσ/dT|). [1, 2, 14]; this accounts, in particular, for the paramagnetic Curie temperature Θ being proportional to the de Our data, which agree with the results reported ear- p lier by other authors [3, 7Ð9], show that hydrogenation Gennes factor G, i.e., the average value of the squared RE ion spin projection on the direction of the total and nitrogenation of the RFe Ti and R Fe com- 11 2 17 angular momentum, with the indirect exchange integral pounds bring about a colossal rise in the Curie temper- A serving as the coefficient of proportionality [14]: ature (Tables 1, 2) and a substantial increase in the mag- ind netic moment per Fe atom (Table 3) at T = 4.2 K. The Θ 2 Aind average increase in the Curie temperature observed to p = ---x------G. (1) 3 kB occur, for instance, in the RFe11TiH hydrides and RFe11TiN nitrides compared to the starting compounds Here, x is the RE ion concentration and kB is the Boltz- is approximately 50 and 200 K, respectively. mann constant. The de Gennes factor G can be represented as In contrast to previous studies, which made use of polycrystalline samples of the hydrides, in this study, Gg= ()Ð 1 2 JJ()+ 1 , (2) the Curie temperatures of the hydrides were derived J from measurements of the magnetic properties of sin- where gJ is the Landé factor and J is the total angular gle-crystal samples, which improved the accuracy of momentum of a rare-earth ion. Experimental investiga- Θ Curie temperature determination and permitted us to tions of the dependence of p on the atomic constants follow the dependence of TC on the RE ion spin. of RE ions and their concentrations, conducted using

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 EFFECT OF INTERSTITIAL ATOMS 1947

Table 2. Molecular-field coefficients and effective exchange fields for the R Fe (H,N,C) compounds calculated with inclu- ≠2 17 X sion of exchange interaction between ions on the rare-earth sublattice (A2 0) ρ 3 6 6 6 Composition , g/cm TC, K Gh21, 10 Oe N21 h11, 10 Oe N11 h22, 10 Oe

Y2Fe17 7.35 341 Ð Ð Ð 3.81 3076 Ð Gd2Fe17 7.91 492 15.75 0.913 768 Ð Ð 0.359 Tb2Fe17 8.07 408 10.5 0.913 503 Ð Ð 0.359 Dy2Fe17 8.15 365 7.08 0.913 376 Ð Ð 0.359 Ho2Fe17 8.25 322 4.5 0.913 298 Ð Ð 0.359 Er2Fe17 8.35 302 2.55 0.913 246 Ð Ð 0.359 Lu2Fe17 8.525 267 Ð Ð Ð 2.98 2393 Ð Y2Fe17H3 7.13 490 Ð Ð Ð 5.47 4568 Ð Gd2Fe17H3 7.88 582 15.75 1.04 881 Ð Ð 0.340 Tb2Fe17H3 7.88 495 10.5 1.04 589 Ð Ð 0.340 Dy2Fe17H3 8.06 460 7.08 1.04 434 Ð Ð 0.340 Ho2Fe17H3 8.10 426 4.5 1.04 347 Ð Ð 0.340 Er2Fe117H3 8.18 404 2.55 1.04 287 Ð Ð 0.340 Lu2Fe17H3 8.35 372 Ð Ð Ð 4.15 3412 Ð Y2Fe17N2.5 7.00 740 Ð Ð Ð 7.70 5848 Ð Gd2Fe17N2 7.61 770 15.75 0.84 751 Ð Ð 0.172 Tb2Fe17N2 7.61 722 10.5 0.84 502 Ð Ð 0.172 Dy2Fe17N2 7.77 716 7.08 0.84 371 Ð Ð 0.172 Ho2Fe17N2 7.92 705 4.5 0.84 292 Ð Ð 0.172 Er2Fe17N2 8.00 693 2.55 0.84 242 Ð Ð 0.172 Lu2Fe17N2 8.15 679 Ð Ð Ð 7.07 5296 Ð Y2Fe17C 7.05 502 Ð Ð Ð 5.48 4465 Ð Gd2Fe17C 7.74 573 15.75 0.617 536 Ð Ð 0.400 Tb2Fe17C 7.75 537 10.5 0.617 357 Ð Ð 0.400 Dy2Fe17C 7.91 515 7.08 0.617 264 Ð Ð 0.400 Ho2Fe17C 8.01 504 4.5 0.617 209 Ð Ð 0.400 Er2Fe17C 8.09 495 2.55 0.617 173 Ð Ð 0.400 Lu2Fe17C 8.25 486 Ð Ð Ð 5.31 4252 Ð

Θ high-precision measurements of p on single-crystal data, we write expressions for the temperature depen- samples of RE metals and of their alloys with one dence of the magnetization of the two oppositely ori- another and with yttrium [1, 15], showed that the ented iron (labeled 1) and rare-earth (labeled 2), sublat- Θ dependence of p on the de Gennes factors for individ- tices: ual RE ions can be expressed, in ﬁrst approximation, by the relation µ Table 3. Average values of the magnetic moments Fe at T = 4.2 K obtained for the LuFe11Ti and Lu2Fe17 com- 2 A pounds and their hydrides and nitrides Θ = ------ind-∑x G , (3) p 3 k i i µ µ B Composition Fe, B where xi is the concentration of RE ions with the de LuFe11Ti 1.77 Gennes factor G . i LuFe11TiH 1.9 The dependence of the Curie temperatures of ferri- LuFe11TiN 2.1 magnetic RE compounds on the de Gennes factor was Lu Fe 2.0 studied earlier in terms of the molecular-ﬁeld theory 2 17 developed for ferrimagnetic compounds with rare-earth Lu2Fe17H3 2.0 and iron sublattices [5Ð9]. To analyze our experimental Lu2Fe17N2 2.29

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1948 NIKITIN, TERESHINA

M M Recalling the expressions for the Curie constants C1 ------1- ==B ()y , ------2- B ()y , (4) S1 1 J2 2 and C2 M10 M20 N µ 2 g J ()J + 1 N µ 2 g 2 J ()J + 1 where M1(T) and M2(T) are the sublattice magnetic 1 B 1 1 1 2 B 2 2 2 µ C1 ==------, C2 ------, moments per mole at temperature T, M10 = N1 Bg1S1 3kB 3kB µ and M20 = N2 Bg2J2 are the sublattice magnetic one can find the contributions to the Curie temperature moments at T = 0, N1 is the number of Fe atoms per due to various exchange interactions: mole, N2 is the number of RE atoms per mole, S1 is the 2h ()g Ð 1 2 Fe atom spin, J2 is the total angular momentum of an Θ = C ------11------1 -, (10) 0 1 µ 2 RE atom, and g1 and g2 are the corresponding Landé N1 B g factors. 1 Θ 2 2 The arguments of the Brillouin functions BS1(y1) and RFe = A1h21 G, (11) B (y ) can be written as [7, 16] J2 2 Θ R = A2G, (12) µ µ 10 H 2 BS1 y1 = ------+ ------Heff1, (5) where the coefficients are kB kBT 4µ 2 µ H 2µ S A = ------B S ()S + 1 , (13) 20 ------B -2 1 2 1 1 y2 = ------+ Heff2. (6) 9kB kBT kBT µ Here, µ = µ g S ; µ = µ g J ; and H and H are 2Z22 A22 2 B 10 B 1 1 20 B 2 2 eff1 eff2 A2 ==------h22. (14) the effective exchange fields acting on the spins of the 3kB 3kB Fe and RE ions, respectively: The Curie temperature of compounds with a zero mag- Θ H = h σ + h σ , (7) netic moment of the rare-earth sublattice ( R = 0, eff1 11 1 12 2 Θ RFe = 0) is found from the condition where the exchange parameters are Θ T C0 = 0. (15) Z11 A11 Z12 A12 h11 ==------µ , h12 ------µ ; The goal of our study was to determine the exchange B B (8) fields and derive the dependence of the Curie tempera- σ σ Heff2 = h21 1 + h22 2, tures on the de Gennes factor, as well as to find the exchange fields experienced by an RE ion from both the with other RE ions and the Fe sublattice. To do this, let us Z A Z A derive the dependence of the Curie temperature of a 21 21 22 22 compound with sublattices of RE and Fe ions on the de h21 ==------µ , h22 ------µ . B B Gennes factor. Here, σ (T) and σ (T) are the average values of the iron After squaring relation (9) and making some alge- 1 2 braic transformations, we come to the following and RE ion spins, respectively; Z11 and Z22 are the numbers of nearest neighbors for RE ions on sublattices 1 expression taking into account the variation of the Curie temperature TC in an RE compound with the RE and 2; A11 and A12 are the exchange interaction integrals ≠ for Fe atoms with neighbors on sublattices 1 and 2; and sublattice magnetic moment M2 0 compared to the Curie temperature TC0 of a similar compound whose M2 A21 and A22 are those for RE atoms with neighbors on sublattices 1 and 2, respectively. is zero: ∆ Expanding the Brillouin functions in their argu- T C T C 2 ∆ ------= A1h21 + A2 T C, (16) ments y1 and y2 for a temperature slightly above the G Ӷ Ӷ magnetic ordering point TC, where y1 1 and y2 1, and solving the system of the corresponding equations where ∆ linear in M1 and M2, we obtain T C = T C Ð T C0. (17) 1 The obvious merit of Eq. (16) derived first in [5] (see T = ---()Θ Ð Θ C 2 0 R also [18]) is the explicit dependence of the Curie tem- (9) perature on the de Gennes factor and the possibility of 1 ()Θ Θ 2 ()Θ 2 Θ Θ determining the exchange fields acting on RE ions from + --- 0 + R +.4 RFe Ð 0 R 2 both the Fe sublattice (exchange parameter h21) and the This relation follows from Néel’s theory of ferrimag- surrounding RE ions (parameter A2). netism [17] and was derived and analyzed earlier in [5, Figures 3 and 4 display plots of the quantity ∆ ∆ 7Ð9]. TC TC/G as a function of TC = TC Ð TC0 derived from

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 EFFECT OF INTERSTITIAL ATOMS 1949 our data for the RFe Ti and R Fe compounds and 11 2 17 6000 2 their hydrides and nitrides, with the Curie temperatures G / 1 of lutetium compounds (LuFe11Ti, Lu2Fe17) substituted C

3+ T for TC0. It is known that the 4f electron shell of the Lu ∆ 4000 C 3 ion has no magnetic moment because of its being com- T pletely ﬁlled (4f 145s25p65d16s2). Therefore, in our case, the Lu compounds model the iron sublattice magnetiza- 2000 tion in the whole series of RE compounds of the type 0 40 80 120 160 ∆ under consideration. Figure 4 also shows the same TC, K dependences derived from the data in [3] for R Fe 2 17 ∆ ∆ Fig. 3. TC TC/G plotted vs. TC for (1) RFe11Ti, compounds in which carbon was inserted into the lat- (2) RFe TiH, and (3) RFe TiN. tice. 11 11 As is evident from Figs. 3 and 4, the experimental points derived from the Curie temperature data for the 8000 compounds under study ﬁt well to a linear relation 2 1 ∆ ∆ TC TC/G = f( TC) in full agreement with Eq. (16), which indicates the constancy of the exchange parame- 6000 ters h21 and h22 in the series of similar compounds and G the existence of noticeable exchange coupling between / 3 C rare-earth ions because A ≠ 0. The intercept of these T 4000 4

2 ∆ C

straight lines with the vertical axis and the slope of the T ∆ ∆ TC TC/G = f( TC) straight line permit one to determine the parameters h21 and A2 using Eq. (16). 2000 Equations (10)Ð(14) can now be used to calculate the effective exchange fields h21, h11, and h22 acting on the ion spins in sublattices 1 and 2; the coefficients N11, 0 50 150 250 N , and N of the molecular fields acting on the mag- ∆ 21 22 TC, K netic moments of these ions are found from the relations [7] ∆ ∆ Fig. 4. TC TC/G plotted vs. TC for (1) R2Fe17, 2 2 (2) R2Fe17H3, (3) R2Fe17N2, and (4) R2Fe17C. 2h11 g1 Ð 1 2h22 g2 Ð 1 N11 ==------µ -------, N22 ------µ -------, (18) N1 B g1 N2 B g2 (1) Introduction of hydrogen atoms into the RFe11Ti 2h ()g Ð 1 ()g Ð 1 lattices brings about an increase in the intersublattice N = ------21------1 2 -. (19) 21 µ exchange field h21 (by 7.5% per hydrogen atom). The N1 B g1g2 increase in h21 induced by hydrogen incorporation in The coefficient N22 can be derived from the parameter the R2Fe17 compounds is smaller (4.5% per hydrogen h22 using Eq. (14), and coefficient N11 can be derived atom). using Eqs. (10) and (18). Coefficient N is determined 21 (2) By contrast, insertion of nitrogen into the from parameter h21. Note that for the Fe sublattice we RFe11Ti and R2Fe17 lattices entails a decrease in the have g1 = 2; for R = Gd, whose orbital angular momen- intersublattice exchange field h21 (by 6.8 and 4% per tum L = 0, we have g2 = 2, and for RE ions with a non- ≠ nitrogen atom, respectively). zero angular momentum, we have g2 0. The coefficients N11, N21, and N22 define the molecular fields H1 (3) The exchange interactions between ions on the RE sublattice decrease upon incorporation of hydrogen and H2 acting on the magnetic moments of the iron and RE ions, respectively, and can be found from the rela- and nitrogen, the latter producing a stronger effect. tions Indeed, the RFe11Ti compounds exhibit a weak decrease in h22 (by 1%) for the RFe11TiH hydrides and H = N M + N M , (20) 1 11 1 12 2 a strong decrease (by 68%) for the RFe11TiN nitrides. Hydrogenation of the R2Fe17 compounds reduces h22 by H2 = N21M1 + N22M2, (21) 2%, whereas nitrogenation brings about a decrease by 26%. where M1 is the magnetization of the Fe sublattice and M is the magnetization of the RE ion sublattice. 2 (4) Incorporation of carbon into the R2Fe17 lattices The results of the calculations are summed up in reduces the h21 exchange field considerably (by 32% Tables 1 and 2. An analysis of the results thus obtained per carbon atom), while the exchange interaction permits the following conclusions: between ions on the RE sublattice exhibits even an

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1950 NIKITIN, TERESHINA enhancement upon carbon insertion (h22 increases by By contrast, insertion of nitrogen and carbon atoms 11%). weakens the intersublattice exchange coupling in intermetallic compounds of the 4f and 3d transition metals The results of calculations conducted for the of the type RFe11Ti and R2Fe17. The results obtained by RFe11Ti compounds under the assumption that us indicate a weakening of exchange interaction h in exchange interactions between ions on the rare-earth 21 RFe11Ti and R2Fe17 under nitrogenation and agree with sublattice can be neglected are the same in the case of the inelastic neutron scattering experiments conducted hydrogenation but are different in the case of nitrogena- on GdFe Ti and Gd Fe [24, 25], as well as with the tion (h decreases by 17%). The calculations carried 11 2 17 21 theoretical calculations carried out in [25] in the local out for R2Fe17 produced the following results: hydroge- spin density approximation within the LMTO theory, nation increases h21 by 1.5%, and nitrogenation reduces which also revealed a decrease in the GdÐFe exchange h21 by 12%, while insertion of carbon reduces this field interactions under nitrogenation. Interpretation of this by 35%. Thus, the interactions between ions on the result should take into account a number of factors, rare-earth sublattice must be taken into account both for namely, (i) the volume effect associated with the obtaining accurate numerical values of the RÐFe increase in atomic volume and in the FeÐFe and RÐFe exchange coupling and describing the variation of these interatomic distances brought about by the presence of interactions in the hydrides, nitrides, and carbides of interstitial atoms, (ii) the enhanced magnetism of the the R2Fe17 compounds. iron sublattice, (iii) the elastic strains created around an interstitial atom close in size to the octahedral voids (as Thus, we established that incorporation of light shown by neutron diffraction measurements [3], at the atoms (H, N) into the lattices of the RFe11Ti and R2Fe17 above-mentioned concentrations, only the octahedral compounds brings about a substantial increase in the voids are filled), and (iv) the change in the local elec- Curie temperature TC, an enhancement of the FeÐFe tron concentration around an interstitial atom (the exchange coupling, and a decrease in the RÐR interac- chemical effect) [25, 26]. It should also be pointed out tions in magnitude; it also results in the RÐFe intersub- that, in the R2Fe17 lattices, the octahedral positions and, lattice interaction increasing under hydrogenation and hence, the interstitial atoms lie in the basal plane and decreasing under nitrogenation, which can be well surround the RE ion, whereas in the RFe11Ti lattices accounted for by the attendant variations in the elec- they are arranged along the tetragonal axis c (which is tronic structure of these compounds and in the indirect perpendicular to the basal plane) both above and below exchange interactions. According to the theory taking the RE ion. into account spin fluctuations [19], the Curie temperature of band ferromagnets is inversely proportional to It can be maintained that the nitrogenation-induced the density of states in the subbands near the Fermi decrease in h21 is due to the chemical effect, in which level, N↑(EF) and N↓(EF). The increase in the lattice attraction between the nitrogen and RE ion valence parameters observed to occur when interstitial atoms orbitals and covalent bonding between the RE and N (hydrogen, nitrogen) are incorporated gives rise to a atoms play the main role [27, 28]. This effect should decrease in the degree of wave-function hybridization obviously entail a decrease in the concentration of 5d between the 3d electrons of Fe atoms and the 5d elec- delocalized electrons, which mediate the exchange trons of RE metal atoms and entails a decrease in both interaction between the localized RE 4f shell and Fe the densities of states N↑(EF) and N↓(EF) [20Ð23]. This atoms, and, hence, a decrease in the exchange field h21. decrease in the degree of hybridization apparently plays This mechanism is the most efficient in carbides of the an important part in suppressing the spin fluctuations R2Fe17 compounds. and the increase in the Curie temperature; the main con- Exchange interactions between RE ions (field h22) tributions to the latter are due to exchange interactions decrease under hydrogenation and nitrogenation (see between ions on the iron sublattice (the exchange field Tables 1, 2). The R Fe C carbides, however, exhibit an h ), whose magnetism is considerably enhanced by 2 17 11 increase in h , which apparently should be assigned to incorporating interstitial light atoms [20] (see Table 3). 22 Because of the localized character of the RE ion 4f sub- an additional indirect exchange between RE atoms via shell, the exchange interaction in RÐFe compounds the carbon 2p electrons. between the 4f and 3d spins is essentially indirect and Thus, the analysis of the experimental data obtained is mediated by polarized 5d electrons, which are shows that it is the specific features in the structure of hybridized with the 3d electrons of Fe [10, 22]. Hydro- the outer electronic shell of the interstitial atom that are genation fills the 3d band and magnetizes the 5d elec- responsible for the changes in the exchange interactions trons of the RE atom due to the enhanced polarization h11, h21, and h22. Incorporation of interstitial atoms pro- of the 3d electrons of the transition metal. The contri- duces additional delocalized band electrons in the 3dÐ bution of delocalized electrons to the magnetism of 5d hybridized band in the hydrides and brings about hydrogenated compounds increases, thereby increasing their decrease in number in the nitrides and carbides. the intersublattice exchange field h21, the effect This factor affects the exchange interactions because of observed by us (Tables 1, 2). changes in the concentration and density of the delocal-

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 EFFECT OF INTERSTITIAL ATOMS 1951 ized band electrons, which come primarily from the 10. M. Brooks, O. Eriksson, and B. Johansson, J. Phys.: rare-earth 5d16s2 valence electrons, 4s2 valence elec- Condens. Matter 1, 5861 (1989). trons, and iron 3d hybridized electrons. 11. I. S. Tereshina, S. A. Nikitin, T. I. Ivanova, and K. P. Skokov, J. Alloys Compd. 275Ð277, 625 (1998). 12. S. A. Nikitin, I. S. Tereshina, V. N. Verbetsky, and ACKNOWLEDGMENTS A. A. Salamova, J. Alloys Compd. 316, 46 (2001). The authors are indebted to K.P. Skokov for prepar- 13. S. A. Nikitin, E. A. Ovchenkov, I. S. Tereshina, et al., ing the starting samples and to V.N. Verbetskiœ and Metally, No. 2, 111 (1999). A.A. Salamova for preparing the hydrides and nitrides. 14. P. De Gennes, J. Phys. Radium 23 (8Ð9), 510 (1962). This study was supported by the Russian Founda- 15. S. A. Nikitin, Zh. Éksp. Teor. Fiz. 77 (1), 343 (1979) tion for Basic Research (project no. 02-02-16523) and [Sov. Phys. JETP 50, 176 (1979)]. the federal program of support for leading scientiﬁc 16. K. P. Belov and S. A. Nikitin, Phys. Status Solidi 12, 453 schools (project no. 00-15-96695). (1965). 17. L. Néel, in Antiferromagnetism (Inostrannaya Literatura, Moscow, 1956), pp. 56Ð84. REFERENCES 18. A. S. Andreenko and S. A. Nikitin, Usp. Fiz. Nauk 167 1. S. A. Nikitin, Magnetic Properties of Rare-Earth Metals (6), 605 (1997) [Phys. Usp. 40, 581 (1997)]. and Their Alloys (Mosk. Gos. Univ., Moscow, 1989). 19. P. Mohn and E. P. Wohlfarth, J. Phys. F: Met. Phys. 17 2. K. P. Belov, Rare-Earth Magnets and Their Application (12), 2421 (1987). (Nauka, Moscow, 1980). 20. O. Isnard, S. Miraglia, D. Fruchart, et al., Phys. Rev. B 3. H. Fujii and H. Sun, in Handbook of Magnetic Materi- 49 (22), 15692 (1994). als, Ed. by K. H. J. Buschow (North-Holland, Amster- dam, 1995), Vol. 9, Chap. 3, pp. 304Ð404. 21. S. S. Jaswal, Phys. Rev. B 48 (9), 6156 (1993). 4. S. A. Nikitin, V. A. Vasil’kovskiœ, N. M. Kovtun, et al., 22. G. Givord and D. Courtois, J. Magn. Magn. Mater. 196Ð Zh. Éksp. Teor. Fiz. 68, 577 (1975) [Sov. Phys. JETP 41, 197, 684 (1999). 285 (1975)]. 23. S. Jaswal, W. Yelon, G. Hadjipanayis, et al., Phys. Rev. 5. S. A. Nikitin and A. M. Bisliev, Vestn. Mosk. Gos. Univ., Lett. 67 (5), 644 (1991). No. 2, 195 (1975). 24. M. Loewenhaupt, P. Tils, D. Middleton, et al., J. Magn. 6. S. A. Nikitin and A. M. Bisliev, Fiz. Tverd. Tela (Lenin- Magn. Mater. 139, 151 (1994). grad) 15 (12), 3681 (1973) [Sov. Phys. Solid State 15, 25. T. Beuerle, M. Liebs, K. Hummler, and M. Fahnle, 2451 (1973)]. J. Magn. Magn. Mater. 132, L1 (1994). 7. K. H. J. Buschow, in Handbook of Supermagnets: Hard 26. Y. P. Li and J. M. D. Coey, Solid State Commun. 81 (6), Magnetic Materials, Ed. by G. J. Long and F. Grandjean 447 (1992). (Kluwer Academic, Dordrecht, 1991), Chap. 4, pp. 49Ð 27. A. Sakuma, J. Phys. Soc. Jpn. 61 (11), 4119 (1992). 67, NATO Adv. Study Inst. Ser., Ser. C, Vol. 331. 28. V. K. Grigorovich, Metal Bonding and Structure of Met- 8. E. Belorizky, M. A. Fremy, J. P. Gavigan, et al., J. Appl. als (Nauka, Moscow, 1988). Phys. 61 (8), 3971 (1987). 9. B.-P. Hu, H.-S. Li, J. P. Gavigan, and J. M. D. Coey, J. Phys.: Condens. Matter 1, 755 (1989). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1952Ð1956. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1857Ð1861. Original Russian Text Copyright © 2003 by Tovstolytkin, Pogorily, Lezhnenko, Matviyenko, Podyalovski, Kravchik.

MAGNETISM AND FERROELECTRICITY

Electrical and Resonance Properties of Magnetically Inhomogeneous La0.775Sr0.225MnO3 Ð d Films A. I. Tovstolytkin, A. N. Pogorily, I. V. Lezhnenko, A. I. Matviyenko, D. I. Podyalovski, and V. P. Kravchik Institute of Magnetism, National Academy of Sciences of Ukraine, Kiev, 03142 Ukraine e-mail: [email protected] Received December 17, 2002; in ﬁnal form, March 27, 2003

Abstract—Electrical properties and magnetic-resonance spectra are studied in La0.775Sr0.225MnO3 Ð δ thin films obtained on SrTiO3 single-crystal substrates through magnetron sputtering. Below 250 K, the essential influence of the intensity of an electric current flowing through a film on the electrical resistance of the sample is observed. It is experimentally shown that as the current density is increased in the low-temperature range, the semiconducting conduction transforms into metallic conduction. Arguments are advanced in favor of the hypothesis that the strong sensitivity of the film properties to external factors is caused by the coexistence of ferromagnetic metallic and charge- or orbital-ordered phases in a sample. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION bilization of the ferromagnetic metallic state [6]. How- ever, biaxial stresses, occurring, e.g., in an epitaxial Unremitting attention has been given to substituted film with a lattice parameter different from that of the manganites La1 Ð xAxMnO3 (A is an alkaline-earth ele- substrate, strengthen the tendency of electrons to local- ment) due to their unusual properties and the variety of ize. In [8], from a theoretical study on the stability of types of ordering observed with variation of the compo- ferromagnetic metallic state, it was concluded that sition, temperature, or pressure [1, 2]. Complexity in increased biaxial stresses cause an additional ordering the electronic and magnetic phase diagrams of manga- of the Mn ion orbitals, which, in turn, favors the stabi- nites is caused by strongly interacting spin, charge, and lization of an insulating antiferromagnetic state. orbital subsystems; the key role is played by manganese ions of variable valence [2, 3]. Under certain con- The results of Experimental studies performed ditions, the energies responsible for the formation of mainly on epitaxial La1 Ð xCaxMnO3 films are in qualita- different phases become comparable and an inhomoge- tive agreement with theoretical calculations [9Ð12]. In neous state characterized by the presence of various [9], it was shown that ultrathin (down to 8 nm) phases becomes more favorable in comparison to the La0.73Ca0.27MnO3 films on SrTiO3 substrates are char- homogeneous state [2Ð4]. Due to the delicacy of the acterized by a weakened magnetic moment, a lower energy balance, the behavior of the material becomes Curie temperature, and an increased electrical resis- very sensitive to external factors (magnetic and electric tance (in comparison to that of a bulk sample). Com- fields, mechanical stresses, etc.) and can be changed bined studies of the structural, electrical, magnetic, and radically under their action. resonance properties of epitaxial La2/3Ca1/3MnO3 films For a long period of time (beginning from the mid- grown on SrTiO3 substrates were performed in [10]. 1950s), the specific features of these materials were The experimental results convincingly demonstrated ≤ interpreted within the concept of double exchange that films of thickness t 50 nm have a lower TC, a between Mn3+ and Mn4+ ions, which favors the delocal- reduced magnetic moment, and higher electrical resis- ization of electrons and stabilizes the ferromagnetic tance. Studies of nuclear magnetic resonance made it ordering [5]. However, the latest studies have demon- possible to separate the phases with different magnetic strated that such an approach is oversimplified and that and electrical properties. It was shown that films simul- the electronÐlattice interaction arising from the JahnÐ taneously contain ferromagnetic metallic, ferromag- Teller instability and orbital ordering of 3d electrons of netic insulating, and nonferromagnetic insulating Mn3+ ions is of great importance [2, 3, 6]. The essential regions. Recent studies of biaxially stressed effect of the crystal lattice on the formation of the prop- La2/3Ca1/3MnO3 films showed that these films possess erties of the electron subsystem has been emphasized in anisotropic electrical properties, nonlinear currentÐ a number of theoretical and experimental studies [2, 6, voltage characteristics, and weakened ferromagnetism 7]. Theoretical calculations show that uniform com- and show evidence of antiferromagnetic ordering [11]. pression of a crystal lattice increases the probability of It was also shown in [11] that, with increasing the cur- electron transfer between adjacent sites and favors sta- rent density, the system transforms into a highly con-

ELECTRICAL AND RESONANCE PROPERTIES 1953

0.10

6 10 (002) cm

Ω 0.05 , ρ (001) Intensity, arb.units LSMO

STO 0 0 150 300 , K 20 40 60 T 2θ, deg

Fig. 1. X-ray diffraction curves of an La0.775Sr0.225MnO3 Ð δ Fig. 2. Temperature dependence of resistivity of LSMO ﬁlm (LSMO) and an SrTiO3 substrate (STO). measured at a stabilized current of 5 mA.

ducting state, which is of particular interest for applica- crystal SrTiO3 substrates oriented in the (001) plane. tions of such films in magnetoelectronic devices. The target for the preparation of thin-film samples was Much fewer studies are devoted to the effect of biax- synthesized by a standard solid-phase reaction [16, 17]. ial stresses on the properties of films of the We used films obtained at a substrate temperature of 880°C in an Ar (40%)ÐO (60%) atmosphere. The gas La1 − xSrxMnO3 system. Films of this system with x = 2 0.3 and 0.5 obtained on perovskite substrates with var- pressure in the sputtering was 10Ð2 Torr. After prepara- ious lattice parameters were studied in detail in [13]. It tion, the films were annealed for 6 h at 750°C. X-ray was shown that the stresses created by substrates could diffraction studies were performed on a DRON 3M dif- transform the ferromagnetic metallic state into an anti- fractometer (CuKα radiation). The electrical resistance ferromagnetic insulating state, which correlates with was measured by a dc four-probe method. Measure- theoretical calculations [8]. The results of studies of the ments of the resistance could be made at a fixed value magnetic, electrical, and crystallographic properties of of the current in the range from 0.01 to 10 mA. A films with 0.15 ≤ x ≤ 0.23 grown on fluorite and perovs- Radiopan spectrometer with a working frequency of kite substrates are presented in [14]. A strong influence ν ≅ 9.2 GHz was used to study magnetic-resonance of substrate structure on the properties of spectra. La1 − xSrxMnO3 films was attributed to the formation of a ferromagneticÐantiferromagnetic two-phase state. In 3. RESULTS AND DISCUSSION [15], weak stability of the ferromagnetic metallic phase in the heterostructure La0.80Sr0.20MnO3/SrTiO3 with The x-ray diffraction patterns of one of the obtained respect to external effects was reported; this manifests films and an SrTiO3 substrate shown in Fig. 1 are indic- itself, in particular, in a photoinduced metalÐsemicon- ative of a single-phase state and epitaxial growth of ductor transition. LSMO. It is seen that the film has a c-axial texture char- In this paper, we present experimental data which acterized by high-intense (00l) peaks (pseudocubic rep- demonstrate an extremely strong dependence of con- resentation). The intensity A of the (011) and (111) ductivity of La Sr MnO δ films, deposited on peaks does not exceed 0.8% of A(002). The (002) peak is 0.775 0.225 3 Ð split both for the film and for the substrate. The diffrac- SrTiO3 substrates, on the intensity of the electric current flowing through them. It will be shown that the ori- tion pattern of LSMO does not contain any reflections gin of the high sensitivity of the film properties to exter- different from those of the perovskite phase. nal factors can be a two-phase magnetic state, the exist- The temperature dependence of resistivity ρ of a ence of which is confirmed by studies of magnetic- film measured at a fixed value of the current I = 5 mA resonance spectra. is shown in Fig. 2. In the low-temperature range, the value of ρ slightly grows with increasing T, which is typical of metallic-type conductivity; then, the resistiv- 2. EXPERIMENTAL ity increases abruptly, beginning from a temperature of ≅ ρ Films La0.775Sr0.225MnO3 Ð δ (LSMO) 270 nm thick 250 K, and reaches a maximum at Tp 276 K. The (T) were obtained through magnetron sputtering on single- curve obtained at I = 5 mA agrees with similar curves

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

1954 TOVSTOLYTKIN et al.

0.4 T = 77 K T = 295 K

0.2 cm Ω , ρ

cm 0

Ω 0.1 10 , , arb. units

ρ I, mA 77 K dB 275 K / dP

250 K 200 K 100 K 0 0 6 I, mA 0 200 400 B, mT

Fig. 3. Resistivity of films as a function of current intensity Fig. 4. Magnetic resonance spectra dP/dB at T = 77 and at various temperatures. 295 K. observed for bulk samples of the same composition in by a boundary located near I = 0.5 mA. This value of which the ferromagnetic metallic phase is completely the current corresponds to the case when ρ is almost dominant at low temperatures [1, 16]. However, as we independent of T (for T ≤ 250 K), i.e., when the temper- showed previously [18], an decrease in the current ature coefficient of resistance is close to zero. For all flowing through a thin film strongly affects the resistiv- values of I larger than 0.5 mA, the resistance gradually ity, which indicates that the processes occurring in films decreases on cooling. By contrast, for I < 0.5 mA, the are more complicated in comparison to those in bulk resistance first drops on cooling from 275 K to ~250 K samples. and then begins to increase with a further decrease in T. The inset to Fig. 3 shows the ρ(I) curve obtained at Thus, in the low-temperature range, a decrease in the T = 77 K. To minimize the effect of sample heating due current results not only in a significant increase in resistance but also in a transition from the metallic to the electric current [19], the measurements were per- ρ ρ formed directly in liquid nitrogen. As seen from Fig. 3, (d /dT > 0) to the semiconducting (d /dT < 0) type of a decrease in the strength of the current results in an conductivity. increase in resistivity, which is particularly pronounced Figure 4 shows the results of our study of magnetic at small values of the current (I < 1 mA). resonance outside (at 295 K) and inside (at 77 K) the The curves in Fig. 3, showing the influence of tem- temperature range where the anomalous effect of the perature on the ρ(I) dependence, were obtained in the current on resistance takes place. At T = 295 K, the following way. At room temperature, a sample was con- absorption spectrum dP/dB obtained with the field par- nected to a stabilized current source and the current was allel to the film surface is represented by a symmetrical adjusted to a certain value. After cooling to 77 K, the line. The integrated curve is well described by a Lorent- resistance was measured as a function of temperature zian with parameters corresponding to the paramagnetic ≅ during heating. Then, the next value of the current was state of LSMO [4, 20] (resonance field Br 331 mT, line ρ ≅ set and the cycle repeated. The T = const(I) curves were width w 24 mT). The main feature of magnetic reso- plotted on the basis of the data obtained in accordance nance at T = 77 K is the occurrence of two well-resolved with the above procedure. As can be seen from Fig. 3, absorption lines. The integrated P(B) curve shown in the resistivity is virtually independent of the current at Fig. 5 is well described by a superposition of two Lorent- ≅ ≅ T = 275 K. However, at lower temperatures (T ≤ 250 K), zian lines with parameters Br1 239 mT, w1 70 mT and ≅ ≅ the ρ(I) dependences noticeably deviate from a hori- Br2 301 mT, w2 53 mT, which is indicative of the zontal line; the degree of deviation increases signifi- occurrence of two magnetic components. Since the cantly as the temperature and the current decrease. conditions of magnetic resonance in a magnetically According to the effect of the current on resistivity, one ordered phase require a smaller external field than in a can single out two ranges in Fig. 3, which are separated paramagnetic phase (for materials with magnetic order,

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 ELECTRICAL AND RESONANCE PROPERTIES 1955 the resonance field is the sum of the external field and the effective internal field), the first line obviously corresponds to the ferromagnetic phase. The position of the second line is slightly shifted relative to the resonance line of the higher temperature paramagnetic phase. However, the data obtained do not allow us to unambiguously determine whether this shift is caused by the magnetic order that might exist in the second phase or is a result of the magnetic field induced by the

first phase [20]. In any case, considered the strong cor- , arb. units relation between conduction and the degree of mag- P netic order inherent in substituted manganites, one may expect the electrical characteristics of these phases to be essentially different. By comparing the data on magnetic resonance with the results of electric measurements (Fig. 2) and with the data for bulk samples [1, 0 16], one can assert that the ferromagnetic phase has a 200 400 metallic type of conduction. The results of magnetic- B, mT resonance studies do not allow one to come to an unam- Fig. 5. Integrated magnetic resonance P(B) curve at 77 K. biguous conclusion about the properties of the second phase; however, using all of the collected experimental data, some conclusions can be made about the origin La0.775Sr0.225MnO3 Ð δ films can be easily interpreted in and features of the second phase. terms of the above hypothesis. Nonlinear currentÐvoltage characteristics in the samples of substituted manganites are usually associ- 4. CONCLUSIONS ated with the tunneling of charge carriers through insu- Thus, we have found that, at temperatures below lating or weakly conducting barriers (grain boundaries 250 K, the conduction of La0.775Sr0.225MnO3 Ð δ single- [21], amorphous regions [22], inclusions of insulating crystal films essentially depends on the magnitude of or weakly conducting phases [11, 19, 23], etc). How- the electric current; this effect becomes considerably ever, in the films studied, not only nonlinearity in the enhanced with a decrease in temperature and in the cur- U(I) curves is observed but also dependence of the type rent intensity. It has been shown that a decrease in the of conductance (metallic or semiconducting) on the current intensity in the low-temperature range results magnitude of the current flowing through a sample. In not only in a sharp increase in resistance but also in a our opinion, the whole collection of experimental data transformation from the metallic (dρ/dT > 0) to the can be explained in terms of the hypothesis that two semiconducting (dρ/dT < 0) type of conductivity. It was phases with different types of conductivity exist, the shown experimentally that the films are magnetically amount of each phase being determined by the magni- inhomogeneous in the low-temperature range. The tude of the current flowing through the sample. The lat- results obtained can be explained in terms of the est studies have shown that a similar situation takes hypothesis assuming the coexistence of a ferromag- place in manganites in the case of coexistence of metal- netic metallic and a charge- or orbital-ordered phase in lic ferromagnetism with a phase (or phases) in which the volume of a film; the volume fraction of each phase mutual ordering of the charges and/or of the orbitals of is determined by the current density. manganese ions of variable valence dominates [19, 24Ð 26]. The possibility of formation of a mixed state of such a type in films and bulk samples of La1 Ð xSrxMnO3 ACKNOWLEDGMENTS partially follows, in particular, from the results The authors would like to thank A. Belous and obtained in [27, 28]. In [28], magnetic and nuclear elas- O. V’yunov (Institute of General and Inorganic Chem- tic neutron scattering was studied in bulk single crystals istry, National Academy of Sciences, Ukraine) for man- of the La1 Ð xSrxMnO3 system, including samples with ufacturing the targets and help with the x-ray diffrac- x = 0.23, which are very close in composition to our tion studies. films. It was shown that regions with charge or orbital This study was supported by the Science and Tech- short-range order exist in a single crystal in the entire nology Center in Ukraine, project no. 1086. temperature range below TC; i.e., these regions coexist with the ferromagnetic metallic phase. Taking into consideration that increased biaxial stresses favor the stabi- REFERENCES lization of a charge- or an orbital-ordered phase [8, 13], 1. A. A. Mukhin, V. Yu. Ivanov, V. D. Travkin, et al., Pis’ma while a current flowing through the sample destroys Zh. Éksp. Teor. Fiz. 68 (4), 331 (1998) [JETP Lett. 68, this phase [2, 19, 20], the unusual properties of 356 (1998)].

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1956 TOVSTOLYTKIN et al.

2. E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344 (1Ð 16. V. G. Bar’yakhtar, A. N. Pogorily, N. A. Belous, and 3), 1 (2001). A. I. Tovstolytkin, J. Magn. Magn. Mater. 207 (1Ð3), 3. V. M. Loktev and Yu. G. Pogorelov, Fiz. Nizk. Temp. 26 118 (1999). (3), 231 (2000) [Low Temp. Phys. 26, 171 (2000)]. 17. A. I. Tovstolytkin, A. N. Pogorily, S. V. Cherepov, et al., Metalloﬁz. Noveœshie Tekhnol. 22 (11), 23 (2000). 4. N. V. Volkov, G. A. Petrakovskiœ, V. N. Vasil’ev, and K. A. Sablina, Fiz. Tverd. Tela (St. Petersburg) 44 (7), 18. A. I. Tovstolytkin, A. N. Pogorily, I. V. Lezhnenko, et al., 1290 (2002) [Phys. Solid State 44, 1350 (2002)]. in Abstracts of II International Scientiﬁc Conference on Magnetic Materials and Their Application (Belorus. 5. C. Zener, Phys. Rev. 82 (3), 403 (1951). Gos. Univ., Minsk, 2002), p. 71. 6. A. J. Millis, T. Darling, and A. Migliori, J. Appl. Phys. 19. A. Guha, N. Khare, A. K. Rayhaudhuri, and C. N. R. Rao, 83 (3), 1588 (1998). Phys. Rev. B 62 (18), R11941 (2000). 7. L. P. Gor’kov, Usp. Fiz. Nauk 168 (6), 665 (1998) 20. F. Rivadulla, M. Freita-Alvite, M. A. Lopez-Quintella, [Phys. Usp. 41, 589 (1998)]. et al., J. Appl. Phys. 91 (2), 785 (2002). 8. Z. Fang, I. V. Solovyev, and K. Terakura, Phys. Rev. Lett. 21. R. Gross, L. Alff, B. Büchner, et al., J. Magn. Magn. 84 (14), 3169 (2000). Mater. 211 (1Ð3), 150 (2000). 22. J.-M. Liu, G. L. Yuan, Q. Huang, et al., J. Phys.: Con- 9. H. W. Zandbergen, S. Freisem, T. Nojima, and J. Aarts, dens. Matter 13 (1), 11 (2001). Phys. Rev. B 60 (14), 10259 (1999). 23. A. Guha, A. K. Rayhaudhuri, A. R. Raju, and C. N. R. Rao, 10. M. Bibes, L. Balcells, S. Valencia, et al., Phys. Rev. Lett. Phys. Rev. B 62 (9), 5320 (2000). 87 (6), 067210 (2001). 24. C. W. Chang, A. K. Debnath, and J. G. Lin, J. Appl. Phys. 11. J. Klein, J. B. Philipp, G. Garbone, et al., Phys. Rev. B 91 (4), 2216 (2002). 66 (5), 052414 (2002). 25. C. N. R. Rao, A. R. Raju, V. Ponnambalam, et al., Phys. 12. A. P. Nosov and P. Strobel’, Fiz. Met. Metalloved. 93 (3), Rev. B 61 (1), 594 (2000). 50 (2002). 26. C. W. Chang and J. G. Lin, J. Appl. Phys. 90 (9), 4874 13. Y. Konishi, Z. Fang, M. Izumi, et al., J. Phys. Soc. Jpn. (2001). 68 (12), 3790 (1999). 27. J. Li, C. K. Ong, J.-M. Liu, et al., Appl. Phys. Lett. 76 (8), 1051 (2000). 14. O. Yu. Gorbenko, R. V. Demin, A. R. Kaul’, et al., Fiz. Tverd. Tela (St. Petersburg) 40 (2), 290 (1998) [Phys. 28. S. F. Dubinin, V. E. Arkhipov, Ya. M. Mukovskiœ, et al., Solid State 40, 263 (1998)]. Fiz. Met. Metalloved. 93 (3), 60 (2002). 15. H. Katsu, H. Tanaka, and T. Kawai, Appl. Phys. Lett. 76 (22), 3245 (2000). Translated by A. Zalesskiœ

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1957Ð1965. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1862Ð1869. Original Russian Text Copyright © 2003 by Belotelov, Zvezdin, Kotov, Pyatakov.

MAGNETISM AND FERROELECTRICITY

Nongyrotropic Magneto-Optical Effects in MetalÐInsulator Magnetic Multilayer Thin Films V. I. Belotelov, A. K. Zvezdin, V. A. Kotov, and A. P. Pyatakov Institute of General Physics, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia e-mail: [email protected] Received April 1, 2003

Abstract—Nongyrotropic magneto-optical effects are investigated in metalÐinsulator magnetic multilayer thin films. These effects manifest themselves in changes in the coefficients of transmission and reflection of electromagnetic radiation from the surface of a multilayer film due to the crossover of the magnetic structure from an antiferromagnetic configuration to a ferromagnetic configuration. The nongyrotropic magneto-optical effect observed in reflected light is analyzed theoretically. It is assumed that the multilayer structure is exposed to radiation of a monochromatic plane wave polarized along the direction of magnetization of the film. The magneto- optical effect is described in terms of the permittivity tensor of the multilayer medium, which depends only on the light frequency. The Boltzmann kinetic equation is treated with allowance made for spin-dependent electron scattering both inside conducting layers and at rough interfaces. Using an Fe/C multilayer as an example, it is demonstrated that the nongyrotropic magneto-optical effect is equal in order of magnitude to the equatorial Kerr effect or other strong magneto-optical effects. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION magneto-optical effects are associated with magnetic gyrotropy of the medium, whereas the effects consid- In recent years, considerable interest has been ered in the present work are not gyrotropic. This expressed by researchers in metallic magnetic multi- implies that the latter effects, like giant magnetoresis- layer films, especially in those exhibiting a giant mag- tance, are governed by the mechanism of spin-depen- netoresistance [1Ð6]. dent electron scattering; i.e., they are determined pri- As a rule, multilayer structures consist of alternating marily by the diagonal elements of the permittivity ten- layers of magnetic and nonmagnetic metals. Under sor εˆ of the multilayer medium. In other words, equilibrium conditions, adjacent magnetic layers in a conventional effects depend on the magneto-optical multilayer structure are magnetized in opposite direc- parameter Q, which is determined by the spinÐorbit tions, thus forming an antiparallel or so-called antifer- interaction of electrons, whereas the magneto-optical romagnetic configuration. In an external magnetic field, effects discussed in this work do not depend on the the magnetic state of a multilayer sample can change parameter Q (they manifest themselves to the zeroth from an antiparallel (antiferromagnetic) configuration order in Q) and arise predominantly from exchange to a parallel (ferromagnetic) configuration in which the interactions of electrons. magnetization vectors in all magnetic layers are col- linear to each other. A change in the magnetic state of Nongyrotropic magneto-optical effects (NGMOE) the multilayer film brings about variations in the condi- can be used, among other processes, in noncontact tions of spin-dependent electron scattering in the bulk probing of materials that exhibit giant magnetoresis- of conducting layers and at roughnesses of the bound- tance [10]. aries (interfaces) between the layers. In turn, this leads Jacquet and Valet [11] were the first to investigate to a change in the conductivity of the sample. The effect the nongyrotropic magneto-optical effect theoretically observed in this case can be significant. For example, and to confirm its existence in experiments for Fe/Cr the relative change in the resistivity of Co/Cu and Fe/Cr multilayers. Uran et al. [12] demonstrated that this multilayers at appropriate parameters exceeds 50% [2]. effect is clearly observed in the near-IR range and, for Apart from the giant magnetoresistance encoun- an Fe/Cr/Fe trilayer structure, amounts to approxi- tered in multilayer structures, the changes observed in mately 0.5%. their optical properties due to variations in the magnetic Theoretical treatment of the nongyrotropic mag- configuration of a multilayer, i.e., magneto-optical neto-optical effect for multilayer structures has been effects, have also attracted considerable research atten- performed in a number of works [7, 11, 13, 14]. In par- tion [7Ð9]. These effects differ from the well-known ticular, Kubrakov et al. [14] theoretically analyzed the magneto-optical Kerr and Faraday (including strong) magneto-optical effect for a monochromatic electro- effects (see, for example, [8, 9]) in that the conventional magnetic plane wave reflected from a multilayer and

1958 BELOTELOV et al. polarized along the magnetization direction. It was possible to construct an adequate and verifiable theoret- shown that the nongyrotropic magneto-optical effect ical model, are of considerable importance for under- expressed through the relative change in the intensity of standing some aspects of the optics of these nanocom- reflected radiation upon the crossover from the antifer- posites, especially those associated with spin-depen- romagnetic configuration of the multilayer to the ferro- dent transport. magnetic configuration reaches several percent. The aim of this work is to develop a theoretical It can be expected that a similar effect will be approach to the description of the optical properties of observed for metal–insulator multilayer films in which metalÐinsulator multilayer structures and to investigate nonmagnetic metal layers are replaced by insulator lay- the nongyrotropic magneto-optical effect. ers. Actually, in metal–insulator multilayer films, there also arises spin-dependent electron scattering; moreover, the spin-dependent tunneling effect manifests 2. METHOD OF DESCRIBING THE OPTICAL itself at a sufficiently small thickness of the insulator PROPERTIES OF A MULTILAYER MEDIUM layers. In the present paper, the optical properties of these multilayer structures will be described within a As was noted above, the effective-medium approach theoretical approach. cannot be applied to the multilayer films under consid- The above problem has aroused widespread interest eration. Hence, the permittivity tensor of a multilayer that goes far beyond the properties of multilayer film medium in this case should be calculated directly from structures. The optical properties of nanocomposite the kinetic equation for electrons with due regard for materials, such as nonmagnetic dielectric or metallic spin-dependent scattering and tunneling. The multi- media with embedded magnetic nanoclusters, have layer structure can be treated as a homogeneous aniso- been extensively investigated in recent years [15Ð19]. tropic medium characterized by the permittivity tensor The experimental data obtained for Cu/Al/O and εˆ [14]. Let us assume that this medium is semi-infinite CoFe/MgF granular films have demonstrated that the and occupies a half-space z > 0. Under these conditions, nongyrotropic magneto-optical effect in these materials the reflection coefficient of an s wave (the wave is reaches 0.8% and exceeds the magneto-optical effect in polarized normally to the plane of incidence) can be metal granular films by a factor of approximately two determined from the Fresnel formula [15Ð19]. Note that these magnitudes of the nongyrotropic magneto-optical effect were obtained for nanocom- 2 posites with a high metal concentration corresponding ϕ ε 2 ϕ R = cos Ð xx Ð sin , (1) to the percolation threshold. ------cosϕ + ε Ð sin2 ϕ In the majority of cases, the transport and optical xx properties of nanocomposite materials have been where ε = 1 is the permittivity above the multilayer (i.e., described in the effective-medium approximation. ϕ Within this approximation, the effective conductivity at z < 0) and is the angle of light incidence [23]. and effective permittivity of a composite are deter- From the wave equation mined as functions of the corresponding quantities for each composite component [20Ð22]. In turn, the nano- 2 2 2 Ð1 ∇ ()ε µ ω σˆ composite components are characterized by the same E +0k0 + i 0c E = , (2) tensors of conductivity and permittivity as those used for bulk media. However, in the case of granular struc- it follows that the permittivity tensor element necessary tures with giant magnetoresistance, in which the grain for determining the reflection coefficient R has the form sizes do not exceed several nanometers, the conditions ε ε µ 2ωÐ1σ of applicability of this approach are violated, because xx = + i 0c xx. (3) the mean free path of electrons in these materials is considerably greater than the mean grain size and spin- Here, ε is the part of the permittivity of the multilayer dependent scattering and tunneling of electrons become structure due to the presence of insulator layers; c is the ω µ significant. The latter circumstance is a necessary con- velocity of light; is the wave frequency; 0 is the per- σ dition for the giant magnetoresistive effect to manifest meability of free space; and xx is the conductivity (the itself in the material. diagonal element of the tensor σˆ ) relating the longitu- Thus, it is evident that the effective-medium approx- dinal current Jx, which is averaged over the period of imation, as applied to these systems, is rather contradic- the structure, to the external wave field inducing this tory and does not offer an adequate description of their current. properties. To the best of our knowledge, there has been no consistent theory proposed to date to describe the Thus, the problem is reduced to determining the σ optical properties of nanocomposite materials. In this conductivity xx [involved in formula (3)] from the respect, investigations into the optical properties of film kinetic equation with allowance made for the behavior materials with giant magnetoresistance, for which it of conduction electrons in a multilayer structure.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

NONGYROTROPIC MAGNETO-OPTICAL EFFECTS 1959

3. CALCULATION OF THE ELECTRICAL x ab (a) CONDUCTIVITY OF A MULTILAYER STRUCTURE The metal–insulator multilayer structure is an infinite periodic (along the z axis) medium consisting of magnetic conductors of thickness a separated by insu- z lator layers of thickness b. In this work, we will con- (b) sider two main configurations of this structure, namely, W Wd j WF an antiferromagnetic configuration with a period L = 2(a + b) (Fig. 1a) and a ferromagnetic configuration ↓ with a period L = (a + b) (Fig. 1d). ( ) Wa Wb Note that the current Jxj(z) in the jth conducting layer is the sum of two currents provided by electrons z with spins aligned with the x axis (↑) and in the oppo- (c) ↓ W Wd site direction ( ). At the same time, the potential energy j WF Wj of electrons in the jth layer depends on the direction of the electron spin. Therefore, electrons can move in ↑ four different potentials depending on the spin orienta- ( ) Wa tion and the magnetic configuration of the structure Wb (Figs. 1bÐ1e). The quantities characterizing the state of z an electron in the layer (potential energy W, relaxation x ab (d) τ time 0, effective mass m) will be designated by the index a when the electron spin is aligned with the layer magnetization and by the index b when the electron spin is oriented in the opposite direction. Let us assume that the plane electromagnetic wave (i) ω ϕ ϕ z E = E0exp(Ði t + ik0r) [where k0 = k0(0, sin , cos )] W W (e) and r = (0, y, z)] is incident on the multilayer structure j d W at an angle ϕ and the plane of incidence coincides with F the YZ plane. The wave is polarized along the x axis (s polarization). The choice of this direction of polariza- (↓) tion somewhat simplifies the theoretical treatment, W because, in such a geometry, there appear neither cur- b rents in the direction perpendicular to the layers nor z (f) effects associated with the spin accumulation [24]. The W Wd j WF field of this wave induces the longitudinal current Jxj(z) in the jth conducting layer of the structure. In this case, the current averaged over the period of the structure (↑) Wa

() ∑∫ J xj z dz z J = ------j (4) x L Fig. 1. Schematic diagrams of the magnetic structures of multilayer films with (a) antiferromagnetic and (d) ferro- is related to the field E0 through the Ohm law in the dif- magnetic configurations and (b, c, e, f) functions of the potential energy of electrons with “upward” and “down- ferential form ward” spins in both cases. σ J x = xxE0. (5)

σ ∂ () f Ð f Therefore, the conductivity xx can be determined as f j (), ∇ ()i , ∇ j 0 ------∂ - ++v j r f j e E p f j = Ð------τ , (6) the proportionality factor in relationship (5). t 0 j The currents Jxj(z) can be calculated from the Bolt- ν ν ν where vj = ( xj, yj, zj) is the average velocity of elec- zmann kinetic equation. Electrons are assumed to be τ × trons, 0j is the relaxation time of electrons, e = 1.6 classical particles having a coordinate r and a quasi- Ð19 momentum p. The nonequilibrium distribution func- 10 C is the elementary charge, and f0 is the FermiÐ Dirac distribution function. tion fj of electrons in the jth layer should satisfy the Boltzmann equation, which, in the relaxation time When solving the kinetic equation, we use the rep- approximation, has the form resentation fj = f0 + Yj(vj, t) under the assumption that

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1960 BELOTELOV et al.

W The changeover to the ferromagnetic case becomes 01 2 3Wd possible after the potential energies of electrons in adjacent layers are equated in the final relationships. The W + Ð F constants C j and C j [involved in formulas (8)] neces- W 2 sary for calculating the functions Ψ + (z) and Ψ Ð (z) can W1 j j be determined from the boundary conditions, which are –a – 2b –a – b – baa0 + b2a + b z written for four consecutive magnetic layers 0, 1, 2, and 3 in the bulk of the multilayer structure. For conve- Fig. 2. Function of the potential energy of an electron in the nience, we change over to the coordinate system with multilayer structure with an antiferromagnetic configura- the origin located at the interface between the first and tion. second layers (Fig. 2). As a consequence, the boundary conditions take the form Ψ ω ϕ j ~ exp(Ði t + ik0ysin ). Then, taking into account Ψ +() Ψ Ð()Ψ +() | |τ Ӷ 1 ÐabÐ = P10 1 ÐabÐ + Q01 0 Ða Ð 2b , the relationship vj 0jk0 1 (the mean free path of electrons is considerably less than the light wave- Ψ Ð() Ψ +() Ψ Ð() 1 Ðb = P12 1 Ðb + Q21 2 0 , length), Eq. (6) can be transformed into the following Ψ Ψ +() Ψ +() Ψ Ð() equation for j(vj, t): 2 0 = Q12 1 Ðb + P21 2 0 , ∂Ψ ν exp()ik z ∂ f Ψ Ð() Ψ Ð() Ψ +() j 1 Ψ xj 0 0 2 a = Q32 3 ab+ + P23 2 a , ------∂ + τ------ν j = ÐeE0------ν ∂------. (7) z j zj zj We (9) ν ϕ θ ν θ Here, xj = vjcos jsin j and zj = vjcos j are the pro- where Pjk is the coefficient of specular reflection of jections of the velocity vj onto the x and z axes, respec- electrons from the interface between the jth and kth ϕ θ tively (where j and j are the spherical coordinates of adjacent conducting layers and Qjk is the transmission τ τ ωτ Ð1 the velocity vj); j = 0j(1 Ð i 0j) ; and We is the coefficient accounting for the transmission of electrons energy of electrons. through a given interface (Fig. 2). It is assumed that the roughness of each interface is described by a random In order to simplify further calculations, we make ζ function jk(r||) such that its average over the plane is two quite obvious assumptions. First, since the condi- 〈ζ 〉 η |ν | Ӷ equal to zero: jk(r||) = 0. The parameter = tion zj < vFj c is satisfied, where 〈〉ζ 2 () () jk r|| characterizing the roughness is assumed to 2 We Ð W j v Fj = ------be identical for all interfaces. The expressions for the m j coefficients Pjk and Qjk are known as the ZimanÐSoffer is the Fermi velocity, Wj is the potential energy, and generalized formulas [25] m is the effective mass of electrons in the jth layer, the j 2η 2 Ð1 Ð1 ν θ τ ν ≈ τ P jk = R jkexp Ð,------m j j cos j following approximation holds: j + ik0 zj j . Sec- ប ∂ ∂ ≈ δ ond, we assume that f0/ We Ð (We Ð WF), where WF δ η2 is the Fermi energy and (W) is the Dirac delta func- Q = () 1 Ð R exp Ð,----- ()m ν cosθ Ð m ν cosθ 2 tion. As a result, the general solution to Eq. (7) can be jk jk ប2 j j j k k k represented in the form of two functions: (10) Ψ +() + z τ ν δ()where θ and θ are the angles of incidence and refrac- j z = eE0 C j expÐτ------ν Ð j xj We Ð WF , j k j zj tion of electrons at the interface between the jth and kth ν > layers, respectively. zj 0, The reflection coefficient Rjk can be obtained by Ψ Ð() Ð z τ ν δ()solving the quantum-mechanical problem of electron j z = eE0 C j expτ------ν Ð j xj We Ð WF , j zj tunneling through the potential barrier W(x) between ν < the jth and kth layers, which depends on the electron zj 0. spin orientation (Fig. 2). Hence, the reflection coeffi- (8) cient Rjk can be represented by the relationship Since the multilayer has a periodic structure, it is expected that the electron distribution function will also R jk be periodic. ()κκ κ κ2 () κκ()κ()κ j k + sinh a + i cosh a j Ð k (11) Without a loss in generality, we consider the poten- = ------, ()κκ κ κ2 () κκ()κ()κ tial for the antiferromagnetic configuration (Fig. 2). j k Ð sinh a + i cosh a j + k

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 NONGYROTROPIC MAGNETO-OPTICAL EFFECTS 1961 where Ψ +()ΨÐ() Ψ Ð()Ψ+() 1 ÐabÐ ==1 Ðb , 1 ÐabÐ 1 Ðb , 2m κ = ------j()θW Ð W cos , Ψ +()ΨÐa Ð 2b ==Ð()0 , Ψ Ð()0 Ψ+()a , j ប2 e j j 0 2 2 2 Ψ +() ΨÐ() (13) 2 0 = 2 a , P ===P , P P , Q Q , κ 2mk ()θ 10 12 21 23 01 21 k = ------2 We Ð Wk cos k, ប Q12 = Q32.

According to these relationships, the sought distri- 2m κ = ------j{}()θW Ð W sin2 + W Ð W bution functions can be derived using only two bound- ប2 e j j d e ary conditions:

Ð Ð Ð 2m 2 Ψ () Ψ ()Ψ () = ------k {}()θW Ð W sin + W Ð W , 1 Ðb = P12 1 ÐabÐ + Q21 2 0 , ប2 e k k d e (14) Ψ Ð() Ψ Ð()Ψ Ð() 2 a = Q12 1 ÐabÐ + P21 2 0 . Substitution of expressions (8) into formulas (14) gives and Wd is the potential barrier height for electrons in the θ θ the system of equations dielectric. Note that the angles j and k must obey the refraction law b ab+ exp Ð------Ð P exp Ð------ÐQ θ τ ν 12 τ ν 21 sin j We Ð Wk 1 z1 1 z1 ------θ- = ------. (12) sin k We Ð W j ab+ a Q12 expÐ------τ ν - P21 Ð exp------τ ν - 1 z1 2 z2 θ For Wj < Wk, there exists a critical angle j of incidence such that, if the angles of incidence are greater C Ð τ ν ()τ1 Ð P Ð ν Q than the critical angle, there occurs total external reﬂec- × 1 = 1 x1 12 2 x2 21 . tion; i.e., R = 1 and Q = 0. Ð Ðτ ν ()τ1 Ð P Ð ν Q jk jk C2 2 x2 21 1 x1 12 The periodicity of the structure and symmetry con- (15) siderations make it possible to obtain the following Ψ + Ψ Ð By solving this system of equations, we obtain the con- obvious relationships for the quantities j (z), j (z), Ð Ð Pjk, and Qjk: stants C1 and C2 in the following form:

()τ ν ()τν ()()τ ν ()τ ν ()τ ν Ð 1 x1 1 Ð P12 Ð 2 x2Q21 P21 Ð exp a/ 2 z2 + Q21 Ð 2 x2 1 Ð P21 + 1 x1Q12 C1 = ------()()τ ν []()()τ ν------()()τ ν , exp Ð ab+ / 1 z1 Q12Q21 Ð P12 Ð exp a/ 1 z1 P21 Ð exp a/ 2 z2 (16) ()τ ν ()τν ()()τ ν ()τ ν ()τ ν Ð 2 x2 1 Ð P12 Ð 1 x1Q12 P12 Ð exp a/ 1 z1 + Q12 Ð 1 x1 1 Ð P12 + 2 x2Q21 C2 = ------[]()()τ ν ()------()τ ν Q12Q21 Ð P12 Ð exp a/ 1 z1 P21 Ð exp a/ 2 z2

The constants C + and C + are related to the constants Knowing the electron distribution functions and 1 2 using the expressions taken from [26, 27], we deter- Ð Ð C1 and C2 as follows: mine the currents Jxj(z) in conducting layers: ∞ 2π + Ð a + 2b () e ρ ν ϕϕ C1 = C1 expÐ------τ ν , J xj z = ------π∫ j jdWe ∫ cos d 1 z1 4 (17) 0 0 (18) + Ð a π/2 π C = C exp ------. 2 2 τ ν ×Ψ+ 2 θ θ Ψ Ð 2 θ θ 2 z2 ∫ j sin jd j +,∫ j sin jd j π Upon substituting formulas (16) and (17) into relation- 0 /2 ρ ships (8), we obtain the sought distribution functions. where j is the density of states.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1962 BELOTELOV et al. After averaging the current in accordance with rela- medium for both configurations of the magnetization tionship (4) and taking into account expression (5), we can be written in the form obtain the conductivity 2 AF() F 2 AF() F AF() F a ω ξ ω ξ a ε = ε Ð ------p1 1 +,------p2 2 (23) σ = ------()σ ξ + σ ξ , (19) ab+ ωω()+ iγ ωω()+ iγ xx 2()ab+ 1 1 2 2 1 2 where where 3 π/2 µ ν ecmj 0 Fj 3 Ð 2 b ω = ------ξ θ θ pj π 3 1 = ------∫ C1 sin 1 cos 1 expÐ------τ ν - 6ប 2a 1 z1 0 is the plasma frequency of the jth layer and γ = τ Ð1 . × a θ j 0 j 1 Ð expÐ------τ ν - d 1 Ð 1, 1 z1 It follows from expression (23) that the frequency dependence of the effective permittivity for the multi- π (20) /2 layer medium is similar to the frequency dependence of ξ 3 Ð 2 θ θ a 2 = ------∫ C2 sin 2 cos 2 exp------τ ν - the permittivity for a nonmagnetic homogeneous metal; 2a 2 z2 0 that is, ω 2 × a θ M p 1 Ð expÐ------τ ν - d 2 Ð 1, ε = 1 Ð ------, (24) 2 z2 ωω()+ iγ and where e2m 2τ ν 3 µ ν 3 σ 1 2ρ τ ν 2 j j Fj ecm 0 F j ==---e Fj j Fj ------ω = ------. 2 3 p π 3 3 6π ប 3ប is the bulk conductivity of the jth layer [3]. By virtue Note that relationship (23) transforms into formula (24) ∂ ∂ ≈ δ of the used approximation f0/ We Ð (We Ð WF), it is at b 0, γ = γ , and W = W . ν ν a b a b necessary to set j = Fj in expressions (10), (16), (20). As can be seen from relationship (23), the difference For the antiferromagnetic configuration, it is evident in the effective permittivities for the antiferromagnetic σ ()↑ σ ()↓ and ferromagnetic multilayer structures is in the non- that xx = xx ; that is, ξ AF ≠ ξ F equality j j . According to formula (1), this leads σ σ ()↑ a ()σ AFξ AF σ AFξ AF AF ==2 xx ------1 1 + 2 2 , (21) to the magneto-optical effect, which manifests itself in ab+ the dependence of the intensity of electromagnetic radiation reflected from the multilayer structure on the con- ξ AF ξ ξ AF ξ σ AF σ σ AF σ where 1 = 1, 2 = 2, 1 = 1, and 2 = 2 figuration of its magnetization. This magneto-optical upon replacement of the indices 1 a and 2 b in effect is similar to the magneto-optical effect consid- relationships (20) (Figs. 1b, 1c). ered in our earlier work [14] for a multilayer structure In the case when the magnetization vectors are codi- consisting of magnetic and nonmagnetic metal layers. In what follows, the nongyrotropic magneto-optical σ ()↑ ≠ σ ()↓ rectional in the adjacent layers, we have xx xx and effect will be described as a relative change in the reflection coefficient; that is, σ σ ()↑ σ ()↓ a ()σ Fξ F σ Fξ F F ==xx + xx ------1 1 + 2 2 , (22) R Ð R ab+ δ = ------AF F. (25) RF where ξ F = ξ upon replacement of the indices 1 1 1 As an example, we consider a multilayer structure ξ F ξ b and 2 b, 2 = 1 upon replacement of the indices composed of Fe magnetic and C insulator layers. For σ F σ σ F σ this structure, we have WF Ð Wa = 8.2 eV, WF Ð Wb = 1 a and 2 a, and 1 = 1 and 2 = 2 upon 5.7 eV, and W Ð W = 1.0 eV [3, 28]. The effective replacement of the indices 1 b and 2 a in rela- d F masses of electrons are assumed to be identical: mj = tionships (20) (Figs. 1e, 1f). × Ð31 me, where me = 9.1 10 kg. In this case, the Fermi ν × 5 ν velocities are as follows: Fa = 8.5 10 m/s and Fb = × 5 τ × 4. NONGYROTROPIC 7.1 10 m/s. Under the assumption that 0a = 3 MAGNETO-OPTICAL EFFECT −14 τ × Ð14 10 s and 0b = 6 10 s, the mean free paths of elec- Making allowance for expressions (3) and (20)Ð trons appear to be considerably greater than the layer (22), the effective permittivity of the multilayer thicknesses.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 NONGYROTROPIC MAGNETO-OPTICAL EFFECTS 1963

Reε, Imε MR × 103 1000 1 4 2 a = 1 nm 0 3 a = 3 nm –1000 3 2

a = 5 nm –2000 1

–3000 0 4 –4000 –1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 បω, eV បω, eV Fig. 3. Dependences of the (1, 2) real and (3, 4) imaginary parts of the effective permittivity on the photon energy បω of incident light for the multilayer structure with an antiferromagnetic conﬁguration (solid lines) and the homoge- Fig. 4. Dependences of the magnetoresistive effect on the neous nonmagnetic metal (dotted lines). Structural parame- photon energy បω of incident light at different thicknesses ters: a = 1 nm, b = 0.2 nm, and η = 1 Å. a of the magnetic layer (b = 0.2 nm, η = 1 Å).

δ × 3 δ × 3 10 10 2.0 2.0

1.5 a = 1 nm 1.5 បω = 0.3 eV 1.0 1.0 a = 3 nm

បω = 0.6 eV 0.5 0.5 a = 5 nm បω = 0.8 eV 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.5 1.0 1.5 2.0 បω, eV ␩, Å

Fig. 5. Dependences of the nongyrotropic magneto-optical Fig. 6. Dependences of the nongyrotropic magneto-optical effect δ on the photon energy បω of incident light at dif- effect δ on the interface roughness of the multilayer struc- ferent thicknesses a of the magnetic layer (b = 0.2 nm, ture at different photon energies បω of incident light. Struc- η = 1 Å). tural parameters: a = 1 nm and b = 0.2 nm.

The dependences of the real and imaginary parts of and the interface roughness at different parameters are the effective permittivity on the photon energy of inci- shown in Figs. 4 and 5, respectively. The effect is absent dent light are shown in Fig. 3. in the case when the interfaces are perfect, i.e., when Figure 4 depicts the dependences of the magnetore- they are free of roughnesses (η = 0). Moreover, the non- σ σ gyrotropic magneto-optical effect virtually disappears AF Ð F sistive effect MR = ------σ - on the photon energy of when the thickness of the insulator layer increases in F order of magnitude to a nanometer. This is associated σ incident light. Here, AF(F) is the resistance of the mul- with the fact that, at b 1 nm, the coefficient of tilayer with an antiferromagnetic (ferromagnetic) con- transmission of electrons through the insulator layer is figuration. close to zero and electrons appear to be confined in The dependences of the nongyrotropic magneto- their layers. Note that, in this work, we consider only optical effect δ on the photon energy of incident light intraband electron transitions.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1964 BELOTELOV et al.

δ × 103 from an antiferromagnetic configuration to a ferromag- 3.0 netic configuration. Within the proposed approach, the permittivity tensor (dependent on the light frequency 2.5 alone) of the multilayer medium was determined by δ × 3 solving the Boltzmann kinetic equation for free elec- NGMOE 10 trons in conducting layers of the multilayer structure. 2.0 In the framework of the theoretical model used in this work, it was demonstrated that the nongyrotropic 1.5 magneto-optical effect is caused primarily by the interface roughness. The effect is absent when the interfaces 1.0 are ideally smooth and the relaxation times of electrons δ × 3 Kerr 10 in all layers are equal to each other. 0.5 The nongyrotropic magneto-optical effect under investigation can be used in noncontact probing of 0 materials exhibiting giant magnetoresistance. 0 10 20 30 40 ϕ, deg ACKNOWLEDGMENTS Fig. 7. Dependences of the nongyrotropic magneto-optical effect for the multilayer structure (a = 1 nm, b = 0.2 nm, η = This work was supported by the Russian Foundation 1.5 Å, បω = 0.3 eV) (solid line) and the equatorial magneto- for Basic Research (project nos. 01-02-16595 and 02- optical Kerr effect for the uniformly magnetized Fe sample 02-17389) and the Federal Program “Physical Proper- with refractive index n = 2.4 + i × 3.5 and magneto-optical ties of Nanostructures.” parameter Q = Ð0.034 + i × 0.003 after exposure to p-polarized light [29] (dashed line) on the angle of light incidence ϕ. REFERENCES A comparison of the dependences shown in Figs. 4 1. M. N. Baibich, J. M. Broto, A. Fert, et al., Phys. Rev. and 5 demonstrates that, although the optical properties Lett. 61, 2472 (1988). of the multilayer structure are governed by the transport 2. S. S. P. Parkin, Appl. Phys. Lett. 61, 1358 (1992). properties, there is no unique correspondence between 3. R. Q. Hood and L. M. Falikov, Phys. Rev. B 46, 8287 the magnetoresistive and nongyrotropic magneto-opti- (1992). cal effects. 4. J. Barnas, A. Fuss, R. E. Camily, et al., Phys. Rev. B 42, The calculated dependences of the nongyrotropic 8110 (1990). magneto-optical effect on the angle of light incidence 5. A. K. Zvezdin and S. N. Utochkin, Pis’ma Zh. Éksp. indicate that this effect is maximum at normal inci- Teor. Fiz. 57, 418 (1993) [JETP Lett. 57, 433 (1993)]. dence of light (Fig. 7). It should be noted that the mag- 6. X. G. Zhang and W. H. Butler, Phys. Rev. B 51, 10085 neto-optical Kerr effect [8, 9], which also manifests (1995). itself in a change in the intensity of reflected light due 7. R. Atkinson, N. F. Kubrakov, A. K. Zvezdin, and to a variation in the magnetization of the medium, is K. A. Zvezdin, J. Magn. Magn. Mater. 156, 169 (1996). characterized by a different angular dependence. As the 8. A. K. Zvezdin and V. A. Kotov, Magneto-Optics of Thin angle of incidence decreases, the magneto-optical Kerr Films (Nauka, Moscow, 1988). effect becomes weaker and disappears in the case of 9. A. Zvezdin and V. Kotov, Modern Magneto-Optics and normal incidence (Fig. 7). This can be explained by the Magneto-Optical Materials (IOP, Bristol, 1997). fact that, as was already noted, the Kerr effect is gyro- 10. V. G. Kravets, D. Bozec, J. A. D. Matthew, et al., Phys. tropic; i.e., it is determined by the off-diagonal ele- Rev. B 65, 054415 (2002). ments of the permittivity tensor, whereas the nongyro- 11. J. C. Jacquet and T. Valet, Mater. Res. Soc. Symp. Proc. tropic magneto-optical effect is related only to the diag- 384, 477 (1995). onal elements of the permittivity tensor. 12. S. Uran, M. Grimsditch, E. Fullerton, and S. D. Bader, Phys. Rev. B 57, 2705 (1998). 5. CONCLUSIONS 13. G. M. Genkin, Phys. Lett. A 241, 293 (1998). 14. N. F. Kubrakov, A. K. Zvezdin, K. A. Zvezdin, and Thus, we proposed a theoretical approach to the V. A. Kotov, Zh. Éksp. Teor. Fiz. 114 (3), 1101 (1998) description of the optical properties of metalÐinsulator [JETP 87, 600 (1998)]. magnetic multilayer films. Consideration was given to 15. I. V. Bykov, E. A. Ganshina, A. B. Granovskiœ, and the nongyrotropic magneto-optical effect, which mani- V. S. Gushchin, Fiz. Tverd. Tela (St. Petersburg) 42 (3), fests itself in a change in the coefficient of reflection of 487 (2000) [Phys. Solid State 42, 498 (2000)]. a plane electromagnetic wave (polarized along the 16. I. Bykov, E. Ganshina, V. Guschin, et al., in Moscow magnetization in layers) from the surface of a multi- Book of Abstracts: Moscow International Symposium on layer film due to the crossover of the magnetic structure Magnetism (Editorial URSS, Moscow, 2002), p. 68.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 NONGYROTROPIC MAGNETO-OPTICAL EFFECTS 1965

17. D. Bozec, V. G. Kravets, J. A. D. Matthew, and 23. M. Born and E. Wolf, Principles of Optics, 4th ed. (Per- S. M. Thompson, J. Appl. Phys. 91, 8795 (2002). gamon, Oxford, 1969; Nauka, Moscow, 1970). 18. A. Granovskiœ, V. Gushchin, I. Bykov, et al., Fiz. Tverd. 24. T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993). Tela (St. Petersburg) 45 (5), 867 (2003) [Phys. Solid 25. The Physics of Metals, Vol. 1: Electrons, Ed. by State 45, 911 (2003)]. J. M. Ziman (Cambridge Univ. Press, Cambridge, 1969; Mir, Moscow, 1972). 19. A. Granovskiœ, M. Kuzmichev, and J. P. Clerc, Zh. Éksp. 26. A. F. Abrikosov, Introduction to the Theory of Normal Teor. Fiz. 116 (5), 1762 (1999) [JETP 89, 955 (1999)]. Metals (Nauka, Moscow, 1972). 20. D. A. G. Bruggeman, Ann. Phys. (Leipzig) 24, 636 27. V. Bezak, M. Kedro, and A. Pevala, Thin Solid Films 23, (1935). 305 (1974). 21. M. Wu, H. Zhang, Xi Yao, and L. Zhang, J. Phys. D 34, 28. J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989). 889 (2001). 29. V. I. Belotelov, A. P. Pyatakov, S. A. Eremin, et al., Opt. Spektrosk. 91, 663 (2001) [Opt. Spectrosc. 91, 626 22. A. B. Granovsky, M. V. Kuzmichev, J. P. Clerc, and (2001)]. M. Inoue, in Moscow Book of Abstracts: Moscow Inter- national Symposium on Magnetism (Editorial URSS, Moscow, 2002), p. 69. Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1966Ð1971. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1870Ð1874. Original Russian Text Copyright © 2003 by Nepochatenko, Dudnik.

MAGNETISM AND FERROELECTRICITY

Thin W' and W Domain Walls in Ferroelastic Pb3(PO4)2 V. A. Nepochatenko* and E. F. Dudnik** * Belotserkovskiœ State Agricultural University, Belaya Tserkov’, 09117 Ukraine ** Dnepropetrovsk National University, Dnepropetrovsk, 49050 Ukraine e-mail: [email protected] Received February 27, 2003

Abstract—A model of ferroelastic domain walls consisting of matching interlayers of crystal lattices is proposed. The dependences of the parameters of the interlayers and of the parameters of the equations for W' and W domain walls on the crystal lattice parameters of the ferroelastic phase in Pb3(PO4)2 are determined. The problem concerning the number of possible orientational states and their interaction in a polydomain crystal is considered. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION The symmetry of interlayers cannot be lower than A model of the structure of a ferroelastic domain monoclinic [1]. Therefore, all the phases will be wall that consists of an induced phase (interlayer) described in terms of monoclinic or pseudomonoclinic symmetry. Let us introduce the following designations: matching the crystal lattices of the adjacent domains ϕ β ° β a0, b0, c0, and 0 = 0 Ð 90 are the parameters of the and a first-order phase transition between the interlayer ϕ β ° and the ferroelectric phase was proposed in [1]. This phase; a1, b1, c1, and 1 = 1 Ð 90 are the parameters of α ϕ β ° model makes it possible to determine the parameters the phase; a2, b2, c2, and 2 = 2 Ð 90 are the param- and symmetry of an interlayer for different types of eters of the interlayer; and C1, C2, and C3 are the orien- domain walls in Pb3(PO4)2, provided the orientation of tational states. a particular domain wall in the coordinate system of the α The formulated problem will be solved using an phase is known from the experiment. It was demon- orthogonal coordinate system in which the X axis is strated that the interlayer of the W' domain wall has the parallel to the c axis, the Y axis is parallel to the b axis, symmetry of the paraphase (P-type twinning), the inter- and the Z axis makes an angle ϕ with the a axis. layer of the W domain wall has the symmetry of the ferroelastic phase (F-type twinning), and the orientation of the domain walls corresponds to lost elements of the 2. STRUCTURE OF THE W' DOMAIN WALL symmetry of the β phase. It is of interest to solve a similar problem for any Let us consider a domain wall W' separating the C1 temperature of the α phase on the basis of the results and C2 orientational states in the coordinate system of obtained in [1] without invoking experimental data on the C1 orientational state. Since a first-order phase tran- the orientation of the domain walls involved and also to sition occurs in the domain wall, the interlayer matches explain the angular relationships between different the adjacent orientational states through two interphase domain walls in a polydomain crystal. boundaries whose orientation coincides with the This problem will be solved in three stages. First, we domain-wall orientation. According to the results will deduce the equations determining the orientation obtained in our earlier work [7], the equation for the and parameters of the matching interlayers for W' and interphase boundary matching two monoclinic cells ϕ β ° W domain walls at an arbitrary temperature of the α with the parameters a1, b1, c1, and 1 = 1 Ð 90 and a2, ϕ β ° phase. Second, we will perform analogous calculations b2, c2, and 2 = 2 Ð 90 in the coordinate system of the for the domain walls at 20°C in order to compare the first phase has the form obtained results with available experimental data. 2 2 2 Third, we will analyze the interaction of the orienta- X1 A11 +++Y1 A22 Z1 A33 2X1Z1 A13 = 0, (1) tional states in a polydomain crystal. where Lead orthophosphate Pb3(PO4)2 is a pure improper ferroelastic. At a temperature of 180°C, this crystal A ==1 Ð1()c /c 2, A Ð ()b /b 2, undergoes a phase transition (3mF2/m ) from the 11 2 1 22 2 1 β α rhombohedral phase to the monoclinic phase [2, 3]. c c a In this crystal, there can exist both normal W and 2 2 ϕ 2 ϕ A13 = Ð------ϕ ---- sin 1 Ð ----- sin 2 , oblique W' domain walls [4Ð6]. c1 cos 1 c1 a1

1 From relationships (7)Ð(10), we obtain the system of A33 = 1 Ð ------equations cos2 ϕ 1 2 2 Ð3,B22/B11 = × a2 2 ϕ c2 ϕ a2 ϕ ----- cos 2 + ---- sin 1 Ð ----- sin 2 . a1 c1 a1 A3 = B13/B11, S (11) In the coordinate system of the second phase, the equa- e11 = 0, tion for the interphase boundary takes the form S e13 = 0. 2 2 2 X2 B11 +++Y2 B22 Z2 B33 2X2Z2 B13 = 0, After solving this system, we obtain the following expressions: ()2 ()2 B11 ==1 Ð1c1/c2 , B22 Ð b1/b2 , 2 b2 = b1 13+/2,F (12) c c a B = Ð------1 ----1 sinϕ Ð -----1 sinϕ , (2) 13 ϕ 2 1 2 c2 cos 2 c2 a2 c2 = c1 13+/2F F, (13) 1 B = 1 Ð ------2mF 33 2 A3 = Ð------, (14) ϕ 2 2 2 cos 2 4F + m ()1 Ð F 2 2 × a1 2 ϕ c1 ϕ a1 ϕ tanϕ = ()cPtanϕ Ð A ()1 Ð c2 /c2, (15) ----- cos 1 + ---- sin 2 Ð ----- sin 1 . 2 1 3 a2 c2 a2 a = a cosϕ /()Pcosϕ , (16) Under the condition 2 1 1 2 2()2 {} c1 b0c1 A3 1 Ð c det Bij = 0 , (3) c ==---- , F ------, P =1 Ð ------2 -, c2 b1c0 c expression (2) corresponds to the equation of two intersecting planes: 2F 13+ F2 c m = ------sinϕ Ð ------1- . ϕ ()2 1 3a XBYCZ++= 0, (4) cos 1 1 Ð F 1 XBYÐ +0,CZ = (5) From expressions (12)Ð(16), we can determine the parameters of the interlayer. Then, from these data, we calculate the strain tensor P' with respect to the param- where B = ÐB22/B11 and C = B13/B11. eters of the β phase and the strain tensor D' with respect The W' domain wall is oriented along the lost two- to the parameters of the α phase. Substituting the fold axis. Therefore, the equation for this wall can be parameters of the interlayer into Eq. (1), we obtain the written in the form equation for the W' domain wall in the coordinate system of the α phase: X0 ++3Y0 A3Z0 = 0, (6) X1 ++DY1 KZ1 = 0, where A3 is the temperature-dependent parameter. 2 b0c1 13+ F (17) Since the interphase boundary has the same orienta------D ==3 , KA3 ------2 2 2 . b1c0 () tion as the domain wall, expressions (4) and (6) coin- 4F Ð A3 1 Ð F cide with each other. Consequently, we have Since equations for the W' domain wall in the two coor-

Ð3,B22/B11 = (7) dinate systems are known [expressions (6), (17)], we can determine the matrix T 1' of rotation from the coor- B13/B11 = A3. (8) dinate system of the β phase to the α phase.

The symmetry of the interlayer coincides with the The problem for the adjacent orientational state C2 symmetry of the paraphase. As a result, the spontane- can be solved in a similar way. As a result, we obtain a ous strain of the interlayer is equal to zero. Hence, we similar interlayer but rotated through an angle of 120° can write about the Z axis. This interlayer is described by the rotation matrix T ' . Taking into account the symmetry, e S = 0, (9) 2 11 these interlayers are identical to each other and a com- S mon interlayer matches the crystal lattices of the adja- e13 = 0. (10) cent domains. According to the thickness criterion of

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1968 NEPOCHATENKO, DUDNIK the ferroelastic domain wall in this model [1], we By analogy with the derivation of the equations for the obtain the domain-wall thickness d ≈ 20 Å. W' domain wall, we use expression (22) as the initial The surface density of the elastic energy of the equation and derive the system of equations domain wall can be determined from the formula Ð3/3B22/B11 = B2, d E = ---c e e , (18) B13/B11 = B3, 2 ijkl ij kl (23) S e11 = 0, where d is the domain-wall thickness, eij are the compo- e = ()c + 3a cosβ /6()a sinβ . nents of the strain tensor D', and cijkl are the elastic con- 13 2 2 2 2 2 stants [8, 9]. By solving the system of equations (23), we obtain From the matrices T ' and T ' , we calculate the 1 2 ÐTD± matrix K' specifying the mutual arrangement of the e13 = ------2 , (24) coordinate axes in the C1 and C2 orientational states 8F separated by the W' domain wall: 1 + B 2F2 T 2 () b2 = b1 ------, (25) K' = T 1' BT2' , (19) 2 1 + B2 where B is the matrix of rotation of the coordinate sys- ° 2 2 tem about the Z axis through an angle of 120 . c1 1 + B2 F c2 = ------, (26) Let us now return to Eq. (5). The interphase bound- F 1 + B 2 ary described by this equation corresponds to the W' 2 2 2 domain wall separating the C1 and C3 orientational tanϕ = ()cHtanϕ Ð B ()1 Ð c /c , (27) states. The equation for this wall 2 1 3 a cosϕ a = ------1 1 , X0 Ð 3Y0 +0A3Z0 = (20) 2 ϕ (28) H cos 2 is taken as the initial equation. As a result, according to 2()2 expressions (12)Ð(16), we obtain an interlayer similar c1 b0c1 B3 1 Ð c c ==---- , F ------, H =1 Ð ------2 -, to that used in the preceding case. c2 b1c0 c Thus, upon formation of the W' domain wall, each c 1 orientational state can take two similar orientations that 1 ϕ n = ------ϕ Ð --- tan 1, 6a1 cos 1 2 correspond to the matrices T 1' and T 2' . T ==3 ++F2 12n2 Ð ()4nF 2, D T 2 + 64F4n2. 3. STRUCTURE OF THE W DOMAIN WALL After determining the parameters of the interlayer [expressions (24)Ð(28)], we can calculate the strain ten- Let us consider a domain wall W separating the C1 sor P with respect to the parameters of the β phase and and C2 orientational states. The equation for this the strain tensor D with respect to the parameters of the domain wall in the coordinate system of the β phase α phase. Substituting the parameters of the interlayer corresponds to the lost symmetry plane: into expression (1), we obtain the equation for the W domain wall with respect to the coordinate system of X0 Ð0.3/3Y0 = (21) the ferroelastic phase:

According to [1], the matching interlayer in this X2 Ð D2Y2 +0,D3Z2 = ≠ domain wall satisﬁes the condition e13 0. Therefore, 3F 2 in order to match the interlayers corresponding to the D = ------14+,e 2 3 13 (29) C1 and C2 orientational states, it is necessary to rotate each of these interlayers together with the domain 2 2 ()2 α D3 = Ð2e13/ c Ð.4e13 1 Ð c about its own Y axis through an angle = Ð2arctane13 . By applying this operation of rotation, we deduce the From relationships (21) and (29), we determine the following equation for the W domain wall in the coor- matrix T1 of rotation from the coordinate system of the dinate system of the interlayer: α phase to the β phase. Now, we solve a similar problem for the adjacent X1 Ð 3/3B2Y1 +0,B3Z1 = orientational state C and obtain an interlayer similar to (22) 2 2 that found in the preceding case but rotated through an B2 ==14+,e13 B3 Ð2e13. angle of 120° about the Z axis of the β phase. This inter-

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 THIN W' AND W DOMAIN WALLS IN FERROELASTIC Pb3(PO4)2 1969 Table 1. Parameters of the interlayers in the domain walls W' and W at t = 20°C Parameter W' domain wall Parameter W domain wall

a2, Å 13.8938 a2, Å 13.8243 b2, Å 5.5136 b2, Å 5.6346 c2, Å 9.5491 c2, Å 9.7587 β β 2, deg 103.244 2, deg 102.903   P' × 10Ð3 Ð6.06 0 0 P × 10Ð3 15.76 0 6.16   Ð6.06 0 15.76 0 Ð1.84 Ð5.46

  D' × 10Ð3 11.29 0 Ð6.36 D × 10Ð3 33.49 0 Ð0.21   Ð31.17 0 Ð10.46 0 3.64 0.005

 T ' 0.99984 Ð0.01756 Ð0.0019 T 0.99978 0.0186 Ð0.0096 1 1  0.01753 0.99976 Ð0.0134 Ð0.01854 0.99981 0.0056 0.00213 0.01338 0.99991 0.00969 Ð0.005478 0.99994

 T ' 0.99984 0.01756 Ð0.0019 T 0.99978 Ð 0.0186 Ð0.0096 2 2  Ð 0.01753 0.99976 0.0134 0.01854 0.99981 Ð 0.0056 0.00213 Ð 0.01338 0.99991 0.00969 0.005478 0.99994

  K' Ð0.4693 Ð0.883 Ð0.0087 K Ð0.5316 Ð0.8467 0.0195   0.883 Ð0.4694 0.0052 0.8467 Ð0.5319 Ð0.0108 Ð0.0087 Ð0.0052 0.99995 0.0195 0.0108 0.99975 E', J/m2 3.7358 E, J/m2 13.338

° layer corresponds to the rotation matrix T2. Since the W analogous calculations for the domain walls at 20 C domain wall consists of two interlayers, its thickness is according to the data taken from [10]. The results two times greater than that of the W' domain wall. obtained are presented in Table 1. Therefore, the elastic energy in the W domain wall should increase signiﬁcantly. By analogy with the derivation of relationship (20), 4. INTERACTION OF THE ORIENTATIONAL STATES IN A POLYDOMAIN CRYSTAL we use the matrices T1 and T2 in order to determine the matrix K describing the mutual arrangement of the coordinate axes in the adjacent domains separated by As was shown in [11], each orientational state in a the W domain wall: single-domain sample can take two similar orientations depending on which interphase boundary forms this KT= ()T BT . (30) state. It was demonstrated that, when a single domain 1 2 wall is formed in a crystal, each orientational state can Thus, the proposed model of a domain wall makes it take four similar orientations that correspond to the possible to obtain the parameters of the interlayers; matrices T1, T2, T 1' , and T 2' . In this case, the domain equations for domain walls in different coordinate sys- walls are oriented along the symmetry elements of the tems; the matrices T 1' , T 2' , K', T1, T2, and K; the strain β phase that are lost upon the phase transition, which is tensors P', D', P, and D; and the energy of the domain consistent with the results obtained by Shuvalov [12]. walls at any temperature of the ferroelastic phase. It This inference can be extended to the case where sev- remains to verify whether this model is consistent with eral parallel single-type domain walls are observed in the experimental data. For this purpose, we carried out the crystal (stripe domain structure).

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1970 NEPOCHATENKO, DUDNIK

Table 2. Matrices of rotation of the coordinate system corresponding to different pairs of domain walls (t = 20°C) Adjacent domain walls Matrix W12' ÐW12 W12' ÐÐW13' W12' W13 W12ÐW13

 0.9993 Ð0.0361 0.0079 0.99938 Ð0.0351 0.00047 0.99997 0.0001 0.0077 0.9993 0.037 0.0002 M  0.0363 0.99917 Ð0.019 0.0351 0.99902 Ð0.0268 Ð0.00094 0.99997 Ð0.0078 Ð0.037 0.9992 0.011 Ð0.0072 0.019 0.99979 0.00047 0.0268 0.99964 Ð0.0077 0.0078 0.99994 0.0002 Ð0.011 0.99994

Let us analyze how the orientational states interact From analyzing the temperature dependence of the with each other when different domain walls are parameters of the domain walls, we obtained the fol- formed in a crystal. It is assumed that the crystal con- lowing results: (1) the equations for the W' domain wall sists of three domains: the C1 domain located at the cen- change with variations in the temperature, and the ter is separated from the C domain by the W ' domain equation for the W domain wall remains unchanged 2 12 with respect to the coordinate system of the β phase but wall on one side and from the C3 domain by the W13' changes with respect to the coordinate system of the α domain wall on the other. The parameters of the equa- phase; and (2) the energy of the domain walls is proportional to the square of the spontaneous strain. tion for the W ' domain wall with respect to the coor- 12 The difference between the structures of the domain dinate systems of the β and α phases are related by the walls leads to the difference in the orientation and expression shape of the domains. From the matrix K (Table 1), we X = T ' X . (31) found that the angle between the twofold axes of the 0i 1 1 j adjacent domains separated by the W domain wall is α ° ° ± ° The analogous expression for the W ' domain wall has = 57.86 (58 1 ) (the experimental value [5] is 13 given in parentheses), whereas the angle between the the form (100) monoclinic planes is γ = 178.7° (178.6° ± 0.1°). ' ' Similarly, from the matrix K', we determined the angles X0i = T 2 X1 j. (32) α = 117.99° (118° ± 1°) and γ = 179.42° (180°; the From relationships (31) and (32), we obtain accuracy is not speciﬁed). As can be seen, the coplanar- ity of the domains separated by the W' domain wall is not conﬁrmed. X0i = MX0' j , It follows from the analysis of the matrices M (Table 2) T MT= ' ()T ' . (33) that, upon the formation of different domain walls in 1 2 the crystal, there occurs rotation of the crystallographic The matrix M describes the rotation of one coordi- axes and domain walls. For example, the angle between nate system relative to the other coordinate system and, the traces of the domain walls W12' and W12 on the plane hence, the rotation of the equations for domain walls (100) differs from 90° by 2.2° (2° ± 0.5°) and the angle obtained above. For different variants of adjacent ' domains and domain walls, we determined the appro- between the domain walls W12 and W13 differs from priate matrices M (Table 2). 30° by 0.45° (0 ± 0.5°). This interaction is one of the factors responsible for the existence of 92° domain walls in the crystal. 5. RESULTS AND DISCUSSION The matching interlayer corresponding to the W' 6. CONCLUSIONS domain wall is a compressed paraphase (see parameters Thus, the results obtained within the model of thin ' P1 in Table 1). An almost opposite situation is domain walls allowed us to conclude that, in lead ortho- observed in the matching interlayer of the W domain phosphate, different types of domain walls differ in the wall: the paraphase is extended along the X and Y axes symmetry of the interlayer, the thickness, and the and is compressed in the direction of the Z axis; more- energy and mechanism of twinning. When one or sev- over, there arises a shear strain e13 responsible for mon- eral parallel domain walls are formed in the crystal, oclinic symmetry. The surface density of the elastic each orientational state at a given temperature can take energy E in the W domain wall is approximately four similar orientations depending on the type of 3.6 times higher than that in the W' domain wall; conse- domain wall and the orientation of the adjacent domain. quently, the W' domain wall is more energetically favor- The symmetry elements of the paraphase, which are able. lost in the course of the phase transition, become the

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 THIN W' AND W DOMAIN WALLS IN FERROELASTIC Pb3(PO4)2 1971 elements of domain twinning. If a more complex 7. E. F. Dudnik and V. A. Nepochatenko, Kristallografiya domain structure is formed in the crystal, there occur 25 (5), 984 (1980) [Sov. Phys. Crystallogr. 25, 564 additional rotations of the crystallographic axes, which (1980)]. result in an increase in the number of possible orienta- 8. Cao-Xuan An, G. Hauret, and J. P. Chapelle, C. R. Acad. tions of the domains and domain walls. Sci. Paris, Ser. B 280 (5), 543 (1975). 9. J. Torres, J. Primot, A. M. Pougnet, and J. Aubree, Fer- REFERENCES roelectrics 26 (1Ð4), 689 (1980). 1. V. A. Nepochatenko, Kristallografiya 48 (2), 324 (2003) 10. C. Joffren, J. P. Benoit, L. Deschamps, and M. Lambert, [Crystallogr. Rep. 48, 290 (2003)]. J. Phys. (Paris) 38 (2), 205 (1977). 2. K. Aizu, J. Phys. Soc. Jpn. 27, 387 (1969). 3. V. Keppler, Z. Kristallogr. 132, 228 (1970). 11. V. A. Nepochatenko, Kristallografiya 47 (3), 514 (2002) [Crystallogr. Rep. 47, 467 (2002)]. 4. L. H. Brixner, P. E. Bierstedt, W. F. Jaep, and J. R. Bark- ley, Mater. Res. Bull. 8, 497 (1973). 12. L. A. Shuvalov, Izv. Akad. Nauk SSSR, Ser. Fiz. 43 (8), 5. M. Chabin and F. Gilletta, J. Appl. Cryst. 10, 247 (1977). 1554 (1979). 6. E. F. Dudnik, E. V. Sinyakov, V. V. Gene, and S. V. Vagin, Fiz. Tverd. Tela (Leningrad) 17, 1846 (1975) [Sov. Phys. Solid State 17, 1212 (1975)]. Translated by O. Moskalev

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1972Ð1977. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1875Ð1879. Original Russian Text Copyright © 2003 by Pavlov, Raevskiœ, Sakhnenko. MAGNETISM AND FERROELECTRICITY

Specific Features in the Permittivity of Polycrystalline Ferroelectrics: The Role of Schottky Regions A. N. Pavlov, I. P. Raevskiœ, and V. P. Sakhnenko Research Institute of Physics, Rostov State University, pr. Stachki 194, Rostov-on-Don, 344104 Russia Received March 3, 2003

Abstract—This paper reports on the results of investigations into the inﬂuence of Schottky regions formed in the vicinity of crystallite boundaries in polycrystalline ferroelectrics on their permittivity. The dependence of the dielectric properties of the Schottky regions on the orientation of the spontaneous polarization in the adjacent region of the crystallite is taken into consideration. It is demonstrated that nonmonotonic dependences of the permittivity and its relative change on the crystallite size under uniaxial compression are associated with the presence of the domain structure in polycrystalline ferroelectrics. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION uration. Consequently, the permittivity of these regions is less than the permittivity of the bulk regions of the It is known that breaking of chemical bonds at crys- crystallite. However, in Schottky regions, the ferroelec- tallite boundaries of polycrystalline ferroelectrics brings about the formation of localized states of the tric material undergoes not only polarization but also acceptor type. Upon filling of these states, the crystal- polarization switching (Fig. 3a, configurations 2, 3). lite boundaries acquire a localized positive charge and These processes result in an efficient screening of the surface electric field and, correspondingly, in a high Schottky regions depleted in negative charge are ε formed in the vicinity of the charged crystallite bound- permittivity s [8]. In ferroelectric semiconductors, a aries. In these regions, the electric field is sufficiently local increase in the permittivity ε in the Schottky strong to affect the ratio of the permittivity inside the region affects the transfer of mobile charge carriers, ε Schottky region ( s) to the permittivity in the bulk which manifests itself in posistor [9], varistor, and pho- region of the crystallite (εv) [1]. In a number of works concerned with description of the dielectric properties ε of ferroelectric polycrystals and thin films, the authors examined a surface region with specific dielectric char- 6000 (a) acteristics (as compared to those of the bulk regions of 5000 the crystallite) [2, 3] but ignored the influence of the domain structure of the ferroelectrics on the properties 4000 of this region. In the present study, we consider the 3000 domain structure of polycrystalline ferroelectrics in 2000 terms of the concentration of Schottky regions with the aim of describing some anomalies in the dielectric 1000 properties of polycrystals: (i) the nonmonotonic behav- 0 ior of the permittivity ε with a decrease in the crystallite 4000 size d (Fig. 1) [4, 5] and (ii) the dependence of the sign (b) of the relative change in the permittivity ∆ε/ε on the crystallite size d under the pressure σ (Fig. 2) [6, 7]. 3000

2000 2. DESCRIPTION OF THE MODEL In [2, 3], the influence of the crystallite surface on 1000 the permittivity ε was described under the assumption that the surface layer exhibits dielectric properties with 0 ε 0.01 0.1 1 10 100 a low permittivity s. This implies that only the polariz- d, µm ing effect of the electric field of localized surface charges in a Schottky region is taken into account (see Fig. 1. Dependences of the permittivity ε on the crystallite Fig. 3a, configuration 1). In polarization regions of the size d according to the results of (a) measurements for ferroelectric material, the polarization P is close to sat- BaTiO3 [4, 5] and (b) calculations.

SPECIFIC FEATURES IN THE PERMITTIVITY OF POLYCRYSTALLINE FERROELECTRICS 1973 tovoltaic [10] effects and in asymmetry of the currentÐ ∆ε/ε voltage characteristics [11]. (a) Let us consider the permittivity ε of an unpolarized 1 crystallite in the form of a rectangular parallelepiped (with edges aligned parallel to the crystallographic axes) in a weak measuring field Em with a strength less than that of the coercive field. The main configurations 0 10 20 30 σ of the orientations of the spontaneous polarization Ps , MPa for the tetragonal phase in the crystallite bulk with respect to a particular crystallite surface in a zero mea- –1 suring field (E = 0) are depicted in Fig. 3a. Moreover, m 0.4 (b) the main configurations can be obtained from configu- 2 0.3 ration 3 (Fig. 3a) by rotating the polarization vector Ps ° ° ° through angles of 90 , 180 , and 270 , provided the 0.2 vector Ps remains parallel to the chosen surface of the crystallite. It is assumed that the measuring field is per- 0.1 pendicular to the chosen surface of the crystallite (Fig. 3b, where l is the thickness of the Schottky s 0 20 40 60 80 100 region). In this case, configurations 1 and 2 correspond –0.1 σ, MPa to c domains, whereas configuration 3 is associated 1 with a domains [2]. The dielectric properties of a poly- Fig. 2. Experimental dependences of the relative change in crystal can be described in the framework of the capac- the permittivity ∆ε/ε on the pressure σ for (a) single-crystal itor model [1–3] under the assumption that all configu- and (b) polycrystalline BaTiO3 samples [7] measured under rations shown in Fig. 3a are equally probable in the (1) uniaxial compression and (2) biaxial tension. polycrystal. Within this model, the permittivity ε of the crystallite can be represented by the expression (a) (b) 1 ε = ---()ε ++ε 4ε . (1) l 6 1 2 3 s P P P s s s Em d ε ε ε Here, the permittivities 1, 2, and 3 corresponding to 123 ls configurations 1Ð3 (Fig. 3a) are defined by the relationships Fig. 3. Schematic diagram of the orientations of (a) the spontaneous polarization Ps in the bulk of the crystallites 1 2ls d Ð 2ls and (b) the measuring field Em with respect to the crystallite ε--- = ε------+ ------ε ; surface (the designations are given in the text). i sid vid (2) < ≤ ,, 2ls d 2ls + ld, i = 123; ness ls of the Schottky region can be related by the expressions 1 2ls ld d Ð 2ls Ð ld 1 1 1 ---ε = ε------++------ε ------------ε ++ε------4 ε------, i sld vid 6d v 1 v 2 v 3 (3) qn l = ------s , (4) < ,, s 2ls +ld d, i = 123, Q

∆Pl≈ Q. (5) ε s where si (i =1Ð3) are the permittivities of the Schottky ε region for configurations 1Ð3, vi (i =1Ð3) are the per- Here, ns is the density of occupied localized states at mittivities of the bulk region of the crystallite for con- the crystallite surface, Q is the space-charge density of figurations 1Ð3, and ld is the domain size. The thickness the Schottky region, and q is the elementary electric ls of the Schottky region is determined by electroneu- charge. Under the condition 2Ps > qns > Pss, expres- trality condition (4) given below. By virtue of the local- sions (4) and (5) lead to the relationship 2P > ∆P > P . ity of the processes involved, the change observed in s s the depolarizing field due to polarization switching is Consequently, upon polarization switching Ps in the compensated for by the space-charge field of the Schot- electric field of the Schottky region, the saturation state tky region. As a result, the change in the polarization is attained for configuration 3 (Fig. 3a); however, for ∆P in the polarization switching regions and the thick- configuration 2, the saturation state is not achieved,

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1974 PAVLOV et al. ε of the dependence ε(d) in which the permittivity ε 1000 (a) remains unchanged at large values of d (Fig. 1): ε ≈ ε 800 3 v 1. (8)

600 For a thin crystallite plate (d > ld), according to the ε ε conditions ls < d < ls( v3/ s3) < ld and ls < d < ld < 400 ε ε ls( v3/ s3), we can write the expression 012345678 ε µ 1 2 d 1 v 1l D, m ε = ---ε + ---ε ------Ð ---ε ------1 ------s. (9) 2200 3 v 1 3 s32l 6 v 1 ε d (b) s s1 Expression (9) accounts for the effect of Schottky 1800 regions on the initial increase in the permittivity ε with an increase in the crystallite size d (Fig. 1). 1 ε ε 1400 Under the condition ls( v3/ s3) < d < ld, the permittiv- ε ity v3 reaches a larger value due to configuration 3 (a domains) (Fig. 3a). In this case, we obtain the relation- 1000 2 ship ε ε 2ε 2ε v 3 2ls 600 = --- v 3 Ð --- v 3------ε ------. (10) 3 3 3 s3 d ε 200 From this relationship, it follows that the permittivity 3 4 Ð6 drastically increases to ~10 Ð10 at d ~ ld ~ 10 m 0 12345678 (Fig. 1). d, µm Such a drastic increase in the permittivity ε upon changing over from coarse-grained (d ~ 104 m) to fine- Fig. 4. (a) Experimental dependence of the permittivity ε on grained (d ~ 10Ð6 m) polycrystals was explained by the thickness D of a BaTiO3 polycrystalline ferroelectric film [2] and (b) calculated dependences of the permittivity Buessem et al. [13] as resulting from the increase in the ε ε ε on the crystallite size d for s1 = s3 = (1) 100, (2) 30, and elastic stress of the polycrystalline material with a (3) 10. decrease in the crystallite size d, which, in turn, should lead to a decrease in the degree of tetragonality of the crystal lattice. However, this explanation contradicts ε 4 5 which results in the permittivity s2 ~ 10 Ð10 . The per- the experimental results obtained by Bradley [14], ε mittivity s2 was estimated from the expression according to which the degree of tetragonality of the ∆ lattice of fine-grained polycrystals is equal to that of ε ≈≈P Ps powders prepared by grinding these polycrystals. Gos- s2 ε------ε------, (6) 0Ec 0Ec wami et al. [15] explained the high permittivity ε ε 5 observed for fine-grained materials in terms of reducing where 0 is the permittivity of free space, Ec ~ 10 Ð effects of saturation of the depolarizing fields, which is 106 V/m is the strength of the coercive field in the also highly improbable due to compensating for the 2 Schottky region [8], and Ps ~ 0.1 C/m [12]. depolarizing fields by free charges. The results of our Thus, configurations 1 and 3 are characterized by a calculations of the dependence ε(d) (Fig. 1b) are in satis- saturation state. The corresponding permittivities are factory agreement with the experimental curve (Fig. 1a). ε ε estimated as s1 ~ s3 ~ 10Ð100 [1]. According to [12], These calculations were performed at the parameters Ð8 × Ð7 ε ε ε we have ε ~ ε ~ 300 for c domains and ε ~ 104 for ls = 10 m, ld = 5 10 m, s1 = 200, s2 = 9000, s3 = v1 v2 v3 ε ε ε a domains. Making allowance for the above estimates 250, v1 = v2 = 300, and v3 = 7000. The experimental ε ε ε of the permittivities si and vi, we can write the rela- dependence of the permittivity on the thickness D of tionship a BaTiO3 polycrystalline ferroelectric film (Fig. 4a) [2], unlike the curve obtained in [5] (Fig. 1a), does not ε , ε < ε ε Ӷ ε , ε s1 s3 v 1, v 2 v 3 s2. (7) exhibit nonmonotonic behavior with a sharp maximum, even though the mean crystallite size decreases with a decrease in the film thickness [2]. This can be explained 3. COMPUTATIONAL RESULTS by the fact that, in the case under consideration, the ε ε Let us now analyze the asymptotic behavior of the relationship ls( v3/ s3) < d < ld, which is a necessary permittivity ε in terms of relationships (1)Ð(3). For a condition for asymptotic equation (10) to be satisfied, thick crystallite plate (d > ld > ls), according to condi- does not hold. The theoretical dependences of the per- tion (7), we obtain the expression describing the portion mittivity ε on the crystallite size d for different values

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 SPECIFIC FEATURES IN THE PERMITTIVITY OF POLYCRYSTALLINE FERROELECTRICS 1975 ε ε of s1 and s3 (Fig. 4b, curves 1Ð3) were calculated ∆ε/ε according to relationships (1)Ð(3) at the parameters l = s (a) Ð8 × Ð7 ε ε ε 0.08 10 m, ld = 5 10 m, s2 = 9000, v1 = v2 = 300, and ε v3 = 7000. An analysis of these dependences demon- 0.06 strates that agreement with the experimental data 1 ε (Fig. 4a) can be achieved by varying the permittivity s 0.04 in the Schottky region. Now, we investigate how the pressure affects the 0.02 2 permittivity ε of ferroelectrics. Figure 5a presents the experimental results obtained for single-crystal and 0 20 40 60 80 100 120 polycrystalline BaTiO samples under hydrostatic com- 3 0.04 pression in the case when any transformation of the (b) domain structure is ruled out. In this situation, an ε ε increase in the permittivity is caused by changes in si ε and vi with a decrease in the phase transition tempera- 0.02 1 ture Tc under pressure. The calculated dependences of the relative change in the permittivity ∆ε/ε on the pres- σ Ð7 2 sure at a temperature T = Tc Ð 50 for ls = 10 m and Ð6 ld = 10 m (Fig. 5b) agree well with the experimental data (Fig. 5a). The calculations for a single crystal and 0 20406080100 a polycrystal were performed using crystallite sizes d = σ, MPa 10Ð4 and 4 × 10Ð6 m, respectively. It turned out that the relative change in the permittivity ∆ε/ε of polycrystal- Fig. 5. (a) Experimental and (b) calculated dependences of line materials is less than that of single crystals. This the relative change in the permittivity ∆ε/ε on the pressure can be explained by the fact that, for polycrystals, the σ under hydrostatic compression for (1) single-crystal and ε ε (2) polycrystalline BaTiO3 samples [7] with crystallite sizes contribution of v1 to the permittivity decreases, d = (1) 1 × 10Ð4 and (2) 4 × 10Ð6 m. ε whereas the contribution of v3 increases. It is worth ε ε noting that, unlike the quantity v1, the quantity v3 decreases with an increase in the pressure under hydro- In these calculations, the electric field strength in the ε 7 static compression (see table). The permittivities v1 Schottky region was taken to be equal to 10 V/m [2]. and ε (i =1Ð3) were calculated with the use of the ther- si Let us now elucidate how the permittivity ε is modynamic potential [16] affected by uniaxial compression and biaxial tension 1 2 2 2 1 4 4 4 (Fig. 2). Changes in the permittivity ε under these Φ = ---α()P ++P P + ---β ()P ++P P 2 1 2 3 4 1 1 2 3 actions are several times greater than those under hydrostatic pressure and, hence, cannot be explained by 2 2 2 2 2 2 1 6 6 6 the effect of pressure on the quantities ε and ε . As + β ()P P ++P P P P + ---γ ()P ++P P si vi 2 1 2 2 3 1 3 6 1 1 2 3 was shown by Dudkevich et al. [17], the domain structure of polycrystalline ferroelectrics undergoes a trans- γ ()4()2 2 4()2 2 4()2 2 + 2 P1 P2 + P3 ++P2 P1 + P3 P3 P1 + P2 formation under uniaxial or biaxial pressure. Below, we will analyze the influence of this phenomenon on the γ 2 2 2 () ε + 3P1 P2 P3 Ð E1P1 ++E2P2 E3P3 (11) permittivity . Let us consider the situation where the direction of uniaxial compression coincides with the ()σ 2 σ 2 σ 2 Ð q11 1P1 ++2P2 3P3 direction of the external measuring field Em (Fig. 3b). In ()σ ()σ2 2 ()σ2 2 ()2 2 this case, the fraction of c domains decreases (Fig. 3a, Ð q12 1 P2 + P3 ++2 P1 + P3 3 P1 + P2 configurations 1, 2) and the fraction of a domains ()σ σ σ Ð q44 4P2P3 ++5P1P3 6P1P2 . increases (Fig. 3a, configuration 3). As a consequence,

ε Calculated permittivities v for different orientations of the spontaneous polarization Ps (Fig. 3a) σ, MPa

εv 0 20 100

T = Tc Ð 50 T = Tc Ð 100 T = Tc Ð 50 T = Tc Ð 100 T = Tc Ð 50 T = Tc Ð 100 ε v1 302 162 307 164 331 170 ε v3 1424 4133 1394 3845 1290 3093

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1976 PAVLOV et al. ε ε ε ε ε ∆ε/ε s2 = 9000, s3 = 10, v1 = v2 = 300, and v3 = 7000. The calculations for a single crystal and a polycrystal were (a) performed using crystallite sizes d = 10Ð4 and 4 × 2 −6 1 10 m, respectively. As can be seen, the calculated dependences are in qualitative agreement with the experiment (Fig. 2). From relationships (12)Ð(14), we obtain the following expression for an approximate 1 description of the quantity ∆ε/ε under uniaxial compression: ∆ε 0 ------≈ +−A. (15) 0 0.2 0.4 0.6 0.8 1.0 ε A 2 Here, the signs “–” and “+” refer to thin and thick crys- –1 tallite plates, respectively. An increase in the permittivity ε for thick crystallite plates is associated with an 0.2 (b) 2 increase in the fraction of a domains under uniaxial 0.1 compression. As the crystallite size decreases, the role 0 played by the Schottky regions in the behavior of the 0 0.2 0.4 0.6 0.8 1.0 permittivity ε becomes increasingly important. There- –0.1 A fore, we can infer that the relative change ∆ε/ε in the –0.2 permittivity for thin crystallite plates is determined by 1 –0.3 the decrease in the permittivity of the Schottky regions due to a transformation of the domain structure under –0.4 uniaxial compression.

Fig. 6. Calculated dependences of the relative change in the Now, we analyze the response of the permittivity ε permittivity ∆ε/ε on the fraction of domains A under to biaxial tension in the case where the measuring ﬁeld (1) uniaxial compression and (2) biaxial tension for crystal- E (Fig. 3b) is oriented along one of the directions of lite sizes d = (1) 1 × 10Ð4 and (2) 4 × 10Ð6 m. m tension. Under these conditions, the relationships describing the effect of pressure on the permittivity ε the effect of pressure on the permittivity ε can be ade- take the form quately described by the relationships 1 A 1 A ε ---ε ---ε ε 1ε ()1ε ()2ε A = 11 + --- + 21 + --- = --- 1 1 Ð A ++--- 2 1 Ð A --- 31 + --- , (12) 6 2 6 2 6 6 3 2 (16) 1ε A 1ε () + --- 3 1 + --- + --- 3 1 Ð A , 1 2ls d Ð 2ls 3 2 3 ε--- = ε------+ ------ε , i sid vid (13) < ≤ ,, 2l d Ð 2l 2ls d 2ls + ld, i = 123, 1 s s ε--- = ε------+ ------ε , i sid vid (17) 1 2l l d Ð 2ls Ð ld --- = ------s- ++------d ------< ≤ ,, ε ε ε 2ls d 2ls + ld, i = 123, i sid vid 6d 1 Ð A 1 Ð A 1+ 0.5A (14) 2l l d Ð 2l Ð l × ------++------4 ------, 1 s d s d ε ε ε ε--- = ε------++------ε ------v 1 v 2 v 3 i sid vid 6d 2l +l < d, i = 123.,, s d × 1+ 0.5A 1+ 0.5A 1+ 0.5A 1 Ð A (18) ------ε - ++------ε -2------ε - +2------ε , Here, A is the fraction of domains in which 90° rota- v 1 v 2 v 3 v 3 ≤ tions of the polarization Ps occur under pressure (0 2l +l < d, i = 123.,, A ≤ 1). s d Figure 6 shows the theoretical dependences of the The theoretical dependences of the relative change in relative change in the permittivity ∆ε/ε on the degree of the permittivity ∆ε/ε on the degree of transformation of transformation of the domain structure. These depen- the domain structure (Fig. 6), which were calculated dences were calculated from relationships (12)Ð(14) at from relationships (16)Ð(18), agree qualitatively with × 8 × Ð7 ε the parameters ls = 7 10 m, ld = 8 10 m, s1 = 10, the experiment (Fig. 2). From relationships (16)Ð(18),

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 SPECIFIC FEATURES IN THE PERMITTIVITY OF POLYCRYSTALLINE FERROELECTRICS 1977 ε ε ∆ε/ε materials. In particular, under the condition ls( v3/ s3) < ε 0.4 d < ld, the dependence of the permittivity on the crystallite size d exhibits a nonmonotonic behavior. In this 1 case, the permittivity ε reaches a large value for a domains. As the fraction of a domains increases under 0.2 pressure, the permittivity of the bulk regions of the crystallite increases, whereas the permittivity of the Schottky regions decreases. Consequently, the relative changes in the permittivity ∆ε/ε for single-crystal and polycrystalline materials under uniaxial compression 0 02 4 6 8101214161820 are opposite in sign. d, µm 2 REFERENCES –0.2 1. F. M. Fridkin, Ferroelectric Semiconductors (Nauka, Moscow, 1976). 2. V. P. Dudkevich and E. G. Fesenko, Physics of Ferro- electric Films (Rostov. Gos. Univ., Rostov-on-Don, –0.4 1979). 3. A. K. Tagantsev, Ferroelectrics 184, 79 (1996). Fig. 7. Calculated dependences of the relative change in the 4. G. Arlt, D. Hennings, and G. de With, J. Appl. Phys. 58 permittivity ∆ε/ε on the crystallite size d under (1) uniaxial (4), 1619 (1985). compression and (2) biaxial tension for the fraction of 5. R. Waser, Integr. Ferroelectr. 15 (1), 39 (1997). domains A = 0.5. 6. I. A. Izhak, Zh. Tekh. Fiz. 27 (5), 953 (1957) [Sov. Phys. Tech. Phys. 2, 872 (1958)]. we obtain the following expression for an approximate 7. Ya. M. Ksendzov and B. A. Rotenberg, Fiz. Tverd. Tela description of the quantity ∆ε/ε under biaxial tension: (Leningrad) 1 (4), 637 (1959) [Sov. Phys. Solid State 1, 579 (1959)]. ∆ε A 8. A. N. Pavlov, I. P. Raevskiœ, and V. P. Sakhnenko, Fiz. ------≈ ±--- , (19) ε 2 Tverd. Tela (St. Petersburg) 42 (11), 2060 (2000) [Phys. Solid State 42, 2123 (2000)]. where the signs “+” and “–” refer to thin and thick crys- 9. A. N. Pavlov, Fiz. Tverd. Tela (St. Petersburg) 36 (3), tallite plates, respectively. 579 (1994) [Phys. Solid State 36, 319 (1994)]. 10. A. N. Pavlov and I. P. Raevsky, Ferroelectrics 214, 157 As follows from relationship (19), the responses of (1998). the permittivity ε to compression and tension of thick 11. A. N. Pavlov, I. P. Raevsky, M. A. Malitskaya, and and thin crystallite samples are opposite in sign. This I. A. Sizkova, Ferroelectrics 174, 35 (1995). also manifests itself in the experimental dependences of the relative change in the permittivity ∆ε/ε on the pres- 12. G. A. Smolenskiœ, V. A. Bokov, V. A. Isupov, N. N. Kraœnik, R. E. Pasynkov, and M. S. Shur, Ferro- sure (Fig. 2). Figure 7 depicts the calculated depen- ∆ε ε electrics and Antiferroelectrics (Nauka, Leningrad, dences of / on d under uniaxial compression and 1971). × Ð8 × biaxial tension for A = 0.5 (ls = 7 10 m, ld = 8 13. W. R. Buessem, L. E. Cross, and A. K. Goswami, J. Am. −7 ε ε ε ε ε 10 m, s1 = 10, s2 = 9000, s3 = 10, v1 = v2 = 300, Ceram. Soc. 49 (1), 33 (1966). ε and v3 = 7000). As the crystallite size d decreases, the 14. F. M. Bradley, J. Am. Ceram. Soc. 51 (5), 293 (1968). relative change in the permittivity ∆ε/ε reverses sign at 15. A. K. Goswami, L. E. Cross, and W. R. Buessem, J. d ~ ld under both uniaxial compression and biaxial ten- Phys. Soc. Jpn. 24 (2), 279 (1968). sion. 16. J. S. Capurso and W. A. Schulze, J. Am. Ceram. Soc. 81 (2), 347 (1998). 17. V. P. Dudkevich, V. V. Kuleshov, and A. V. Turik, Zh. 4. CONCLUSIONS Tekh. Fiz. 47 (10), 2168 (1977) [Sov. Phys. Tech. Phys. 22, 1259 (1977)]. Thus, the presence of Schottky regions and domain structure in polycrystalline ferroelectrics is responsible for a large variety of phenomena observed in these Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1978Ð1983. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1880Ð1885. Original Russian Text Copyright © 2003 by Timonin, Sakhnenko.

LATTICE DYNAMICS AND PHASE TRANSITIONS

Phase Transition in the Isotropic m-Vector Model on a Random Graph P. N. Timonin and V. P. Sakhnenko Research Institute of Physics, Rostov State University, pr. Stachki 194, Rostov-on-Don, 344090 Russia e-mail: [email protected], [email protected] Received February 19, 2003

Abstract—The thermodynamics of a phase transition is investigated within an isotropic m-vector model on a graph with sparse random connections. This model adequately describes superconductivity and superfluidity (m = 2), Heisenberg magnets (m = 3), and some structural transitions in different systems with macroscopic disorder (gels, composites, and porous media). It is demonstrated that the phase transition is characterized by classical effective-field anomalies. The thermodynamics of the phase transition is also analyzed in terms the proposed model at low temperatures. The dependences of the thermodynamic parameters on the external field, the mean coordination number, and the dimension of the order parameters are determined. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION has the meaning of the mean coordination number; at p = 1, there occurs a percolation transition [10]. The study of phase transitions in disordered structures is an important field of the physics of condensed The phase transition in the Ising model on this graph matter. This is related to the large variety of disordered was studied by Viana and Bray [11] and Kanter and structures, such as crystalline solid solutions, compos- Sompolinsky [12]. It was established that the specific ites, gels, porous media, etc. There exist many types of features in the behavior of the thermodynamic quanti- phase transitions in disordered structures owing to dif- ties in the vicinity of the temperature Tc(p) have an ferent forms of ordering elements (magnetic moments, effective-field character. Moreover, the phase diagram atomic displacements, and wave functions for a con- was examined within this model with a random densate). There also occur purely geometric percolation exchange responsible for the formation of the glass transitions due to the formation of macroscopic con- phase. The undoubted merits of these models are that nected clusters. they offer an exhaustive description of the thermody- Investigations of the phase transitions in the frame- namics in the range of applicability of the effective- work of disordered-lattice models using renormaliza- field approximation and a generalization of the meth- tion-group analysis [1] have revealed qualitative fea- ods used to other types of random graphs [3, 4]. tures in the critical behavior due to disorder, in particular, changes in the critical exponents in models with a Moreover, there are many experimental works con- second-order phase transition (see, for example, [2]). cerned with the phase transitions in disordered media Examinations of another class of models that describe that are adequately described by multicomponent order the macroscopic structural disorder in porous media, parameters (for example, superconducting, structural, gels, and composites and are, as a rule, represented by and magnetic transitions in crystalline solid solutions different types of random graphs have demonstrated and porous media [13Ð15]). The renormalization-group that, in many cases, the geometry of a random medium analysis of these transitions is limited by the disor- plays a decisive role in the thermodynamics of the dered-lattice models and the small vicinity of the tran- phase transition. In the framework of the Ising and Potts sition [1, 2]. To the best of our knowledge, description models in random graphs, it has been established that of the thermodynamics of phase transitions over wider variations in the geometric properties of the graph can ranges of variations in the temperature and disorder lead not only to changes in the critical exponents [3, 4] parameters, as well as for different types of random and the type of transition [3Ð6] but also to its splitting graphs, has been performed only in the framework of [7] or vanishing [8]. the Ising and Potts models [3Ð8]. In this respect, the Among the previously considered models of macro- purpose of the present work was to investigate the ther- scopically disordered media, the simplest model is a modynamics of a phase transition within an isotropic graph with sparse random connections [9]. Within this m-vector model in a graph with random sparse connec- model, it is assumed that any pair of N points can have tions that can describe the superconductivity and super- connections with a probability p/N. The parameter p fluidity (m = 2), Heisenberg magnets (m = 3), and some

PHASE TRANSITION IN THE ISOTROPIC m-VECTOR MODEL 1979 structural transitions in different media with macro- With this function, we can eliminate summation over scopic disorder. pairs of nodes in the replica Hamiltonian by representing it in the following form:

n 2. REPLICA FORMALISM pTN Ᏼ ≈ Ð------∏ dsαdsα' exp()Ksαsα' w(){}s , {}S The Hamiltonian of the model under consideration r 2 ∫ has the form α = 1 N N × (){}, {} pTN 1 w s' S + ------Ð HS∑ iα. Ᏼ = Ð---∑ J S S Ð HS∑ , 2 2 ij i j i i, α ij≠ i = 1 Next, we use the identity where Si are the unit m-component vectors, ∫Dρ()δρ{}s [](){}s Ð w(){}s , {}S = 1 SiSi = 1, 〈 n〉 and represent the partition function Z C in the form of H is the conjugated ﬁeld, and Jij is the constant of the random exchange interaction with the distribution a continual integral, function n 〈〉Z C () p δ()p δ() ρ(){} []βρ(){} [][]ρ(){} PJij = 1 Ð ---- Jij + ---- Jij Ð J . = ∫D s U s exp Ð NE s , N N where In order to determine the thermodynamic potential n averaged over the random exchange and other thermo- pT dynamic parameters, we will use the replica technique E[]ρ(){}s = Ð------∏ dsαdsα' exp()Ksαsα' 2 ∫ consisting in calculating the averaged nth-power parti- α = 1 tion function with subsequent passage to the limit pT n 0; that is, ×ρ()ρ{}s (){}s' + ------, 2 N n 〈〉n () β Ᏼ U[]ρ(){}s Z C = ∫∏dJijPJij ∫∏dSα, i exp Ð ∑ α. ≠ α, α jj i = 1 β ()δρHSα [](){} (){}, {} In this expression, integration with respect to the unit = ∫∏ dSαe s Ð w s S . α m-component vector Si satisﬁes the condition In the last expression, the delta function can be repre- ∫dSi = 1. sented in the form of a (continual) exponential integral. Consequently, we obtain ∞ 〈 n〉 For N , the averaged partition function Z C can be represented as U[]ρ(){}s = ∫Dρ˜ (){}s exp[]NΣρρ(), ˜ ,

βᏴ where 〈〉n Ð r Z C = ∫∏dSα, ie . α, i Σρρ(), ρ()ρ{} (){} ˜ = Ð∫∏dsα ˜ s s Here, the replica Hamiltonian has the form α N  Ᏼ ≈ pT []() r Ð ------∑ expKSiαS jα Ð 1 Ð HS∑ iα + ln∏dSα exp βHS∑ α + ρ˜ (){}S . 2N ∫ jj≠ i, α α α with summation over the replica index α ranging from 〈 n〉 β As a result, the averaged partition function Z C is rep- 1 to n and K = J. resented in the form of a continual integral of the func- According to Leone et al. [4], the calculation of the tions ρ({s}) and ρ˜ ({s}): 〈 n〉 averaged partition function Z C can be reduced to a 〈〉n ρ(){} ρ(){} []βΦ() ρ, ρ single-particle problem. For this purpose, we introduce Z C = ∫D s D ˜ s exp ÐN ˜ , the function (1) Φρρ(), ˜ = E()ρ Ð TΣρρ(), ˜ . N n (){}, {} 1 δ() ∞ w s S = ----∑ ∏ sα Ð Si, α . For N , the integral in expression (1) is deter- N Φ ρ ρ i = 1 α = 1 mined by the function ( , ˜ ) at the stationary point.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1980 TIMONIN, SAKHNENKO The functional equations for the stationary functions ρ to the limit n 0 with the assumption of the replica and ρ˜ have the form symmetry, as has been done in similar equations for the Ising model in a random graph [4, 12]. This substantially hampers the solution of Eqs. (2) and (3). How- ρ˜ (){} ' ()ρ' (){} s = p∫∏dsα exp Ksαsα s' , (2) ever, at T Tc and T 0, one can obtain explicit α analytical results in the framework of the above model.

˜ Ð1 ρ(){}s = Z exp ρ˜ ()β{}s +,H∑sα (3) 3. THERMODYNAMICS IN THE VICINITY α OF THE PHASE TRANSITION ˜ Since the hypothetical transition to the phase with where Z is the normalizing constant determined by the an order parameter M ≠ 0 is associated with the break- condition ing of symmetry, it can be assumed that this transition is a second-order transition. Therefore, in the vicinity ρ(){} of the temperature Tc and at small values of H, the solu- ∫∏dsα s = 1. α tion to Eqs. (2) and (3) differs only insigniﬁcantly from the paramagnetic solution. In this case, the functions By solving Eqs. (2) and (3), we can determine the aver- ρ({s}) and ρ˜ ({s}) should depend on a particular vec- aged thermodynamic potential tor f determining the spontaneous breaking of symme- FTN==Ð Ð1 lim nÐ1 ln 〈〉Zn lim nÐ1Φρρ(), ˜ . (4) try. By virtue of the isotropy, this dependence can be n → 0 n → 0 expressed only through scalar products of the form ρ As follows from the above consideration, ({s}) is f α = fsα. the distribution function for the single-particle parameters dependent on Si at a single node (when n 0). In Under the assumption that the vector magnitude f is particular, we obtain the order parameter small, the distribution function ρ({s}) can be represented in the form of the following expansion in terms of fα: ≡ 〈〉〈〉 ρ(){} MSi T C = lim ∫∏dsα s s1 (5) n → 0 α 2 ρ = af+ ∑ α + b ∑ f α + b ∑ f α f β and the EdwardsÐAnderson parameter 1 2 α α αβ≠

ˆ ≡ 〈〉〈〉 〈〉 ρ(){} 3 2 Q Si T : Si T C = lim ∫∏dsα s s1s2.(6) + c f α + c f α f β + c f α f β f γ (9) n → 0 1∑ 2 ∑ 3 ∑ α α αβ≠ αβγ≠≠ It is evident that Eqs. (2) and (3) at H = 0 always have a trivial “paramagnetic” solution, 4 2 2 + d1∑ f α + d2 ∑ f α f β . ρ(){}s = 1, α αβ≠ Here, we used the assumption of replica symmetry, ρ˜ (){} n () s = pRm K , according to which the distribution function ρ({s}) where (and ρ˜ ({s})) should be symmetric with respect to the m Ð 2 permutation of the replica indices. Moreover, in expan------2 sion (9), we retain only the fourth order fα terms which () ()Γm 2 () Rm K = ∫dSexp KSS' = ------Im Ð 2 K .(7) contribute to the thermodynamic potential. 2 K ------2 Substituting expansion (9) into relationships (2) and () Here, Im Ð 2 K is the Bessel function of the imaginary (3), we obtain the following equations for f: ------2 ()τ 2 β argument. The function Rm(K) satisﬁes the equation + uf f = H, (10) m Ð 1 where '' ------' Rm +0.Rm Ð Rm = (8) τ () K = 1 Ð p/ pc K , In order to describe the transition to the phase with ≠ () () () an order parameter M 0, it is necessary to ﬁnd another pc K ==Rm/Rm' Im Ð 2 K /Im K , ------solution giving a lower thermodynamic potential at 2 2 temperatures T < Tc as compared to the potential provided by the paramagnetic solution. Unfortunately, in 1 1 1 u = ------()() - ++------()()λ()------, Eqs. (2) and (3), it is impossible to change over directly mpc K Ð 1 mm+ 2 1 Ð K m + 2

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 PHASE TRANSITION IN THE ISOTROPIC m-VECTOR MODEL 1981 λ()≡ () () () K pc K Ð m/K = Im + 2 K /Im K . Differentiating expression (10) with respect to H we ------2 2 obtain the following relationship for the susceptibility: It follows from the recurrence relationships for I (K) ∂M H : H H : H m χˆ ==------χ⊥ Î Ð ------+ χ||------, ∂ 2 2 [16] that H H H ()> λ()< pc K 1, K 1. 2 Ð1 χ⊥ ==M/H, χ|| ()τ + 3uf . As a result, Eq. (10) at H = 0 has a nonzero solution for For T > T (p) and H = 0, we have τ < 0 or p > p (K): f 2 = Ðτ/u. c c ϑ() χ χ τÐ1 ≈ p The equality p = pc(K) determines the phase transi- ⊥ ==|| ------(), tion temperature as a function of the mean coordination TTÐ c p number p. The transition takes place only when p > 1; where the Curie constant is defined as at p 1, the temperature Tc(p) tends to zero, ϑ()p = () ∂τ/∂T Ð1 T ()p ≈ 2Jp()Ð 1 /()m Ð 1 . c Ð1 = pT ()p []K ()p ()p2 Ð 1 Ð ()m Ð 1 p . This stems from the fact that the graph has a macro- c c χ scopic connected cluster only when p > 1 [9]. For T < Tc(p) and H = 0, the susceptibility || also At p ∞, the phase transition temperature T (p) adheres to the CurieÐWeiss law with the Curie constant c reduced by half. increases proportionally to p; that is, In the case when p 1, the quantity ϑ(p) tends to ()≈ T c p Jp/m. a constant, ρ ϑ()p ≈ 2J/()m Ð 1 , The coefficients of ({s}) in expansion (9) at p pc and n 0 have the following form: and when p ∞, the Curie constant ϑ(p) takes the form ()λ Ð1 () 2b1 ==1 Ð , b2 pc/2 pc Ð 1 , ϑ()≈ () p T c p = pJ/m. ()λ K()12+ λ pcK 1 + pc Ð 2 From relationship (4), we obtain an equilibrium c1 ==------()λ()λ , c2 ------()()λ , 6 m + 2 1 Ð 2mpc Ð 1 1 Ð thermodynamic potential in the vicinity of the transition; that is, ()λ pc 12+ 4 c3 = ------()()λ , pT () Hf Tv f 6 pc Ð 1 1 Ð F = Ð ------lnRm K Ð ------Ð ------. 2 2m 2m2()m + 2 2 ()2 K 4c1 + 2 b1 Ð 1 Ð 1 Here, d1 = ------2 , 4 ()()[]()λ 2 K 1 Ð pc + m + 2 2KmÐ + 4 (){ []()λ v = b1 b1 Ð 1 + md1 1 Ð3pc + mm+ 2 /K 2 pc 2 2 } ()[ ()λλ d = ------[]4c ++2()b Ð 1 4()b Ð 1 Ð 3 . Ð c1 Ð 1/12 + m + 2 d2 2 Ð1/ pc Ð 2 λ2 2 1 2 4()pc Ð ()2 ] + c2 + b2 Ð 1 . It follows from relationships (5), (6), and (9) that For the entropy, we derive the following expression: Mf= /m, p pK Tv f 2 S = --- lnR ()K ++------. m () 2 ˆ 2 2 2 pc K m ()ϑm + 2 u Q = 2b2f : f/m . According to this expression, the entropy is continuous Moreover, we obtain at the transition point, whereas the heat capacity exhib- ˆ its a jump: 〈〉〈〉 I 2b1 ()2ˆ Si : Si T C = ---- + ------mf : f Ð f I , 2 m m2()m + 2 T ()p v ∆C = ------c -. 2()ϑ2 2 〈〉〈〉 〈〉 〈〉〈〉 2 ≠ m m + 2 u Si : S j = Si : Si = f : f/m , ij. T T C T C At p 1, we have Note that the equality of the two correlators in the last m2 ++4m 1 expression holds accurate only to terms of the order of u ≈ ------, NÐ1, which, in this case, are disregarded. mm()+ 1 ()m + 2 ()p Ð 1

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1982 TIMONIN, SAKHNENKO

Integrating expression (11) with respect to Sα, we ≈ 1 ()1 v ------m + 2 1 + ------obtain the normalizing constant at n 0, ()p Ð 1 2 4m p 2 2 Z˜ = e ()m Ð 1 ()5m ++10m 4 + ------4mm()+ 1 2()m + 2 and the equation for the parameter Q (also at n 0), ÐpQ 72m4 ++++188m3 245m2 212m 108 1 Ð Q = e . (12) + ------, 24()m + 1 2()m + 2 ()m + 3 ()3m + 1 It follows from expression (12) that the parameter Q has the meaning of the fraction of nodes belonging to a mac- Hence, the heat capacity satisfies the relationship ∆C ∝ 2 ∝ 2 roscopic connected cluster, because expression (12) Tc(p) (p Ð 1) . coincides with the corresponding equation of the perco- At p ∞, we obtain the following expressions: lation theory in the given graph [9]. This equation has a nontrivial positive solution when p > 1. m + 1 u ≈≈------, v mp/12, ρ mm()+ 2 The function '({S}) defined by relationship (11) has a sharp maximum at Sα + M under the condition p 1 ∆C ≈ ------1 Ð.------max()JM, H ӷ T. (13) 12 ()m + 1 2 Consequently, the range of applicability of the results Therefore, in the vicinity of the phase transition, the presented below is limited by this condition. It follows model under consideration is characterized by a classi- from expression (11) that the order parameter M is par- cal effective-field thermodynamics inherent in models allel to the field H and its magnitude can be determined with an infinite interaction range. The results obtained from the formula above are valid for f Ӷ 1. ' () ()Rm H/T MQ= + 1 Ð Q ------(). (14) 4. THERMODYNAMICS Rm H/T AT LOW TEMPERATURES The first term in formula (14) corresponds to the spon- An analytical description of the thermodynamics in taneous order parameter and differs from zero when terms of the given model can also be obtained at T p > 1. The second term describes the contribution from 0 and H ≠ 0 or M ≠ 0. In this case, the function ρ({S}), the isolated nodes. which is an approximate solution to Eqs. (2) and (3), The transverse and longitudinal susceptibilities have can be represented as the sum of the constant (1 Ð Q) the form and the function ρ'({S}) characterized by a sharp peak ≡ 〈 〉 at Sα = M Si ; that is, χ⊥ = M/H, ρ(){}S = 1 Ð Q + ρ'(){}S . 2()()() 1 Rm' H/T m Ð 1 Rm' H/T By substituting this expression into formula (2) and χ|| = ()1 Ð Q --- 1 Ð ------Ð.------T2 () HR ()H/T using the aforementioned property of the function Rm H/T m ρ '({S}), we obtain the relationship It should be noted that the longitudinal susceptibility χ|| differs from conventional paramagnetic susceptibility n only by the factor (1 Ð Q). ρ˜ (){}S ≈ p()1 Ð Q R ()K + pQexp KMS∑ α . m  α From expressions (4) and (11), we obtain the following relationship for the thermodynamic potential in In turn, upon substituting this relationship into expres- the region specified by condition (13): sion (3) and separating the constant out from the function ρ({S}), we derive the expression for the function pQ 2 2 F ≈ ÐJp()1 Ð Q + ------()1 + pQÐ p Q M Ð QH ρ'({S}), 2 Ð1 n ρ'(){}S = Z˜ exp[]p()1 Ð Q R ()K m Ð 1 p J m Ð TRln ()H/T + ------Q + ---()1 Ð Q ()2 Ð Q T ln--- . m 2 2 T  (11) × exppQexp KMS∑ α + βHS∑ α Ð 1 ,  For the entropy, we have α α and the equation for the parameter Q, m Ð 1 p J ≈ ------()()--- ˜ Ð1 []()n () S Ð Q + 1 Ð Q 2 Ð Q ln . 1 Ð Q = Z exp p 1 Ð Q Rm K . 2 2 T

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 PHASE TRANSITION IN THE ISOTROPIC m-VECTOR MODEL 1983 As in models with continuous spins, in this case, we of critical phenomena in macroscopically disordered also obtain the entropy S ∞ at T 0, because the media described by multicomponent order parameters. phase space with an ensemble of spins tends to zero. The heat capacity is positive when T 0 and ACKNOWLEDGMENTS tends to a constant, This work was supported by the International Asso- ciation of Assistance for the promotion of co-operation m Ð 1 p C ≈ ------Q +.---()1 Ð Q ()2 Ð Q with scientists from the New Independent States of the 2 2 former Soviet Union (INTAS), project no. 2001-0826. From a comparison of the analytical results with the experimental data, one can judge the degree of accu- REFERENCES racy to which the model of a graph with sparse random 1. R. B. Stinchcomb, in Phase Transitions and Critical connections describes the macroscopic disorder in a Phenomena, Ed. by C. Domb and J. L. Lebowitz (Aca- particular porous medium or composite. However, we demic, London, 1983), Vol. 7. should note that the solutions presented above are by no 2. D. E. Khmel’nitskiœ, Zh. Éksp. Teor. Fiz. 68, 1960 (1975) means the only possible solutions. The point is not only [Sov. Phys. JETP 41, 981 (1975)]. that we restricted our consideration to the case of rep- 3. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, lica-symmetric solutions. The analysis of the bifurca- cond-mat/0203227. tions of the paramagnetic solutions for the functions 4. M. Leone, A. Vazquez, A. Vespignani, and R. Zecchina, ρ({S}) and ρ˜ ({S}) demonstrated that very similar cond-mat/0203416. solutions can be obtained at T < T , specifically at tem- 5. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, c cond-mat/0204596. l peratures determined by the condition p = pc (K), (l = 6. F. Igloi and L. Turban, Phys. Rev. E 66, 036140 (2002). 2, 3, …). The same situation occurs with the Ising 7. J. Machta, Phys. Rev. Lett. 66, 169 (1991). model of the graph under study [11]. Unfortunately, we 8. H. Rieger and T. R. Kirkpatrick, Phys. Rev. B 45, 9772 failed to determine the stability region of all solutions (1992). to Eqs. (2) and (3). This requires detailed analysis of the 9. M. E. J. Newman, cond-mat/0202208. applicability of the saddle point method for continual 10. D. Stauffer and A. Aharony, Introduction to Percolation integrals; however, at present, the validity of this Theory (Taylor and Francis, London, 1992). method has not been justified. We can only assume that, 11. L. Viana and A. J. Bray, J. Phys. C 18, 3037 (1985). among the set of solutions to Eqs. (2) and (3), there 12. I. Kanter and H. Sompolinsky, Phys. Rev. Lett. 58, 164 exist solutions stable in some regions of variation in (1987). temperatures T and fields H, which describe the possi- 13. I. V. Golosovsky, I. Mirebeau, G. Andre, et al., Phys. ble metastable states in terms of this model. Elucidation Rev. Lett. 86, 5783 (2001). of the properties of these states can provide an adequate 14. E. V. Charnaya, C. Tien, K. J. Lin, et al., Phys. Rev. B 58, description of the kinetics of the given model at T < Tc 467 (1998). [17]. 15. A. V. Fokin, Yu. A. Kumzerov, N. M. Okuneva, et al., Note also that this model can be generalized by cond-mat/0205303. introducing a random exchange (or random external 16. I. S. Gradshteœn and I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Fizmatgiz, Moscow, 1963; fields) and anisotropy and by specifying the distribution Academic, New York, 1980), Chap. 8. function of the coordination numbers of the graph 17. P. N. Timonin, Zh. Éksp. Teor. Fiz. 119, 1198 (2001) nodes [3, 4]. These modifications should lead to the for- [JETP 92, 1038 (2001)]. mation of new phase states and, in a number of cases, to variations in the critical behavior of the model. This offers rich possibilities for describing the whole variety Translated by O. Moskalev

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1984Ð1989. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1886Ð1891. Original Russian Text Copyright © 2003 by Ivliev, Raevskiœ, Reznichenko, Raevskaya, Sakhnenko. LATTICE DYNAMICS AND PHASE TRANSITIONS

Phase States and Dielectric Properties of SodiumÐPotassium Niobate Solid Solutions M. P. Ivliev, I. P. Raevskiœ, L. A. Reznichenko, S. I. Raevskaya, and V. P. Sakhnenko Research Institute of Physics, Rostov State University, pr. Stachki 194, Rostov-on-Don, 344090 Russia e-mail: [email protected] Received March 11, 2003

ε Շ Շ Abstract—The temperature dependence of dielectric constant for single crystals of Na1 Ð xKxNbO3 (0.04 x 0.15) is studied for the ﬁrst time. From the shape of the ε(T) anomalies corresponding to rotational phase transitions, the type of interaction between the order parameters and the polarization is determined. A phenomenological model is developed which adequately describes the experimentally observed sequence of high-temperature (T > 300°C) phase transitions, the dielectric anomalies associated with these transitions, and the changes in the phase states under the action of external factors (pressure, electric ﬁeld). © 2003 MAIK “Nauka/Inter- periodica”.

1. INTRODUCTION The variety of lattice instabilities occurring in NN also determines the complex polymorphism of NN- Sodium niobate NaNbO (NN) belongs to the large 3 based solid solutions, among them Na K NbO family of crystals with perovskite structure and is char- 1 Ð x x 3 acterized by a rich variety of structural phase transi- (NKN), which is widely used as a piezoelectric mate- tions. At present, six disymmetric phases are known to rial. Up to the mid-1970s, quite adequate studies of the exist in NN [1]. From the crystallographic standpoint, structures of phases observed in NKN had been made the sequence of the structural transitions is caused by and the xÐT phase diagram for this system had been the instability of a cubic lattice with respect to distor- constructed [7, 8] (Fig. 1). Subsequently, the main fea- tions of two kinds: shifts of anions of the oxygen sublattice, which may be interpreted as rotation (tilting) of octahedrons, and shifts of cations from their cen- 700 trosymmetrical positions, leading to the occurrence of U ψ ferroelectric and antiferroelectric states [2Ð5]. 600 T2: 00 T2 ϕ ψ In sodium niobate, three higher temperature phase T1: 0 ϕψ ψ transitions are caused by the rotation of octahedrons T1 S: 1 2 and the next three transitions, by a combination of rota- 500 S tion and polarization of octahedrons with the formation R of two antiferroelectric complex ordered phases (P, R) 400 and of one ferroelectric (low-temperature) phase. The , °C F p00 whole set of disymmetric phases, except the P and R T P p00 ψ ψ G ϕψ ψ 0 1 2 phases, can be described by three three-component 300 1 2 order parameters, two of which, ψ and ϕ, characterize the rotation of NbO6 octahedrons corresponding to lat- 200 tice vibration modes M and R and one of which, p, 3 25 p0p K describes uniform (k = 0) polarization arising as a Q ϕ ψϕ

Q 0p p result of shifts of Nb cations from the centers of octa- 100 1 2 1 2 + ϕψ Γ ψ ϕ 0 hedrons ( 15 mode). The order parameters and P belong to the following stars of vectors k: ψ belongs to 1 0 0.05 0.10 0.15 0.20 the triple-ray star of k = --- (b + b ), and ϕ, to the single- 2 1 2 x

1 Fig. 1. Experimental xÐT phase diagram of Na K NbO ray star of k = --- (b1 + b2 + b3), where bi are vectors of 1 Ð x x 3 2 solid solutions. The notation of phases is the same as that in the reciprocal lattice; these parameters transform [3, 4, 7, 8]. Solid lines are data from [7, 8] for ceramics. τ τ Dashed lines are the phase transition lines based on the according to irreducible representations 3 and 8 (in experimental data on the dielectric properties of single crys- the notation from [6]). tals (ﬁlled circles).

PHASE STATES AND DIELECTRIC PROPERTIES OF SODIUMÐPOTASSIUM NIOBATE 1985 ε ε tures of this phase diagram were interpreted in the con- 4800 1 (a)4800 (b) text of the Landau theory [9], although some essential 3 2 details remained unexplained, which, as shown in this 3200 3200 paper, was associated with the inadequacy of the ther- 1 modynamic model suggested in [9], as well as with the inadequate (in our opinion) description of the interac- 1600 1600 tion of several critical order parameters. 3 In crystals that exhibit various types of instabilities, 0 0 the structure of the disymmetric phases is determined 350400 450 500 350 400 450 500 by the order in which the condensation of the phonon T, °C T, °C modes associated with the order parameters occurs and ε ε by the interactions between these modes. In this study, (c) the parameters characterizing the interaction of the 6000 4 1800 order parameters ϕ and ψ are obtained from the struc- 5 tural data given in [3] and information on the interac- 1600 4 tion of the rotational order parameters with polarization 4000 1400 is obtained from an analysis of the temperature depen- 470 480 490 ε T, °C dence of the dielectric constant . Studies of the dielec- 5 tric properties of compounds of this type have shown 2000 that these properties strongly depend on the method of 4 sample preparation [10]. It has been found that in single ε crystals the anomalies in (T) are pronounced much 0 stronger and the reproducibility of the results is much 350 400 450 500 better than in ceramics. In what follows, the results of T, °C ε Շ Շ measuring in Na1 Ð xKxNbO3 single crystals (0.04 x 0.15) in the temperature range 300Ð500°C with a step Fig. 2. Temperature dependence of dielectric constant ε of ° Na K NbO single crystals for different values of x: of 1Ð3 C are presented. It is precisely in thes ranges of 1≈ Ð x x ≈3 ≈ ≈ ≈ T and x that anomalies in the dielectric properties (1) 0.04, (2) 0.05, (3) 0.06, (4) 0.1, and (5) 0.15. caused by condensation of the rotational modes are ε observed. The analysis of these and other (T) anoma- mum in the ε(T) curve) is observed with increasing x. lies is made in the framework of the Landau theory. The This effect is accompanied by a noticeable decrease in data thus obtained are then used for more comprehen- ε sive (than in [9]) thermodynamic description of NKN the temperature hysteresis of (T), and, in addition, the and the construction of a reﬁned xÐT phase diagram for shape of the ε versus temperature curve is changed and these solid solutions. (particularly at x ~ 0.1) becomes similar to that typical of second-order phase transitions. 2. EXPERIMENTAL RESULTS

Crystals of Na1 Ð xKxNbO3 were grown through mass 3. PHASE DIAGRAM crystallization from the melt using NaBO2 as a solvent AND PHENOMENOLOGICAL THEORY [11]. The crystal faces were parallel to the (001) planes of the perovskite structure. The composition was deter- The phase states observed in the NKN system at x ≈ mined by a Camebax-Micro x-ray microanalyzer. Crys- 0.04Ð0.50 are described by three-component order Շ tals with x 0.15 were chosen for measurements. Opti- parameters. However, for the description of the ordered cal analysis showed that all crystals are strongly ≥ ° ≈ twinned at room temperature. The dielectric constant ε phases in the range T 300 C at x 0.04Ð0.15, we ψ ψ of platelike crystals 0.2Ð0.3 mm thick with edges 1.5Ð require two components of the order parameter ( 2, ψ ϕ 2 mm long was measured at a frequency of 100 kHz 3) in various combinations and one component for ϕ with the aid of an R5083 ac bridge in the process of and p, namely, 1 and p1 (pi denotes the polarization ψ ϕ continuous heating or cooling at a rate of 2Ð3 K/min. along the axis i; j and k characterize rotations around Aquadag electrodes were applied to the (001) natural the axes j and k). Therefore, the nonequilibrium ther- crystal faces. modynamic potential can be simpliﬁed by retaining Measurements of the ε(T) dependences for NKN ≈ only those components of the order parameters that are single crystals with x 0.04, 0.05, 0.06, 0.1, and 0.15 required for the description of the observed ordered were made for the ﬁrst time. The results are shown in Fig. 2. The following features in the behavior of ε(x, T) states. In this case, the model thermodynamic potential are worth noting. In the range of x ≈ 0.04Ð0.06, a strong has the form increase in the transition temperature to the ferroelec- ΦΦΦ Φ Φ tric state (determined from the position of the maxi- = P ++R M +int, (1)

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1986 IVLIEV et al. α ψ ϕ T 1P parameters and become nonzero decrease by 15Ð 20°C as x is increased by 1% [7, 9]). The phase diagram is obtained under the conditions α Շ α α α տ α Շ α 1M 1R, 1R = a + b 1M, a 0, b ~ 1, 2P 0, 2M > S T1 T2 U 0, α > 0, γ < 0, γ < 0, γ > 0, κ < 0, β > 0, ϕ ψ ψ 2R PM MR PR PM 2M ϕ ψ ψ 10 3 00 3 and |γ | Ӷ (α , α ). The relation between α and 1 2 3 MR 2M 2R 1R α means that first the parameter ψ and then the A C 1M B parameter ϕ become nonzero with a decrease in tem- p100 perature. The quantities a and b characterize the relative α α 1M softness of modes M3 and R25. From the analysis that α(1) α(0) 1M = 0 1M 1M follows it will be seen that the above conditions are GF quite justiﬁed and suitable for the description of the ϕ ψ ψ ψ ψ system under consideration. 1 2 3 0 2 3 α ӷ p100 p100 In the region 1P 0, the thermodynamic potential (1) describes not only the symmetric cubic phase U x ϕ ψ α ( = = p = 0, 1M > 0) but also three disymmetric x ≈ 0.06 x ≈ 0.08 x ≈ 0.13 paraelectric (p1M = 0) phases:

Fig. 3. Phase diagram described by the thermodynamic (1) phase T with ϕ = ψ = 0, ψ 2 = Ðα /2α , potential (1). Solid lines are the ﬁrst-order phase transition 2 1 2 3 1M 2M lines, dashed lines are the second-order phase transition α ()0 ≤ α ≤ lines, and dotted lines are the stability boundaries of the 1M 1M 0; phases. 2 b ()0 (2) phase T with ψ = 0, ψ ≠ 0, ϕ = ------(α Ð 1 2 3 1 2α 1M where 2R α α ()1 ≤ α ≤ α ()0 1M), 1M 1M 1M ; Φ α 2 α 4 α 6 … P = 1P p1 +++2P p1 3P p1 , () (3) phase S with ϕ ≠ 0, ψ ≠ 0, ψ ≠ 0, α ≤ α 1 , Φ α ϕ 2 α ϕ 4 … 1 2 3 1M 1M R = 1R 1 ++2R 1 , 2 ()0 aγ ()1 4α β Ð δ > 0, α ≅ Ð,--- 1 + ------MR - α = Φ α α 2 β 2M 2M M 1M b2bα 1M M = 1Mg1 ++2Mg1 1Mg2 2M ()0 2 α α β δ γ α + β g ++δ g g …, 4 2M 2R 1M Ð b M MR 1M 2M 2 M 1 2 ------α δ . 2 2R M ψ 2 ψ 2 ψ 2ψ 2 g1 ==2 + 3 , g2 2 3 , ψ ≠ ψ It should be noted that solutions of the type 2 3, 2 2 which characterize the phases S, G, and F, can be Φ = γ p g + κ p g int PM 1 1 PM 1 2 obtained only with inclusion of the eight-power invari- γ 2ϕ 2 γ ϕ 2 2 … 2 + PR p1 1 ++MR 1 g1 . ant g2 in the thermodynamic potential (1); therefore, This potential contains the terms that characterize the within the model considered in [9], it was impossible to Φ describe these phases adequately. contribution of each order parameter, L (L = P, M, R), and mixed invariants that describe the interaction The paraelectric phases are stable with respect to Φ Φ Φ between them, int. The potentials P and M are writ- polarization in the following regions: ten to within terms that allow one to describe the ﬁrst- order phase transition close to that of second order with for the cubic phase U(000), respect to the order parameter p and to analyze solu- ψ ≠ ψ χ α ≥ tions of the type 2 3, respectively. U = 1P 0; The equilibrium phase diagram for the thermody- ψ namic potential (1) is shown in Fig. 3 for various rela- for phase T2(00 3), tions between α and α . To illustrate the accordance 1M 1P α α χ α γ ψ 2 ≥ (on a qualitative level) between the 1MÐ 1P diagram T = 1P + MP 3 0; (2) and the experimental xÐT diagram (Fig. 1), we show 2 approximate values of the concentration x in Fig. 3. It for phase T (ϕ 0ψ ), α 1 1 3 was taken into account that 1P depends on x only weakly for x տ 0.06, while α and α are strongly 1M 1R χ = α ++γ ψ 2 γ ϕ 2 ≥ 0; (3) dependent on x (the temperatures at which the order T1 1P MP 3 RP 1

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 PHASE STATES AND DIELECTRIC PROPERTIES OF SODIUMÐPOTASSIUM NIOBATE 1987 ϕ ψ ψ and for phase S( 1 2 3), dynamic potential (1) suits the description of the entire range of phase transitions observed in the region stud- χ α γ ψ 2 γ ϕ 2 S = 1P ++MP 3 RP 1 ied. Therefore, we will use this thermodynamic poten- γ δ tial in our further analysis of the anomalies in dielectric κ MP M ψ 2ψ 2 ≥ (4) properties. + MR Ð ------α 2 3 0, 2 2M ψ Ӷ ψ 2 3, 4. DIELECTRIC PROPERTIES χ ε Ð1 where i = i is the reciprocal dielectric constant of As mentioned above, an increase in x in the range the ith phase. from 0.04 to 0.06 results in a strong rise in the ferroelectric phase transition temperature and a decrease in The transition from the cubic phase to the phase ε ε ϕ ψ ψ the temperature hysteresis of (T). In addition, the (T) F(p1, 1 = 0, 2, 3) is a complex first-order phase tran- dependence in the range 400Ð500°C, i.e., above the sition with the order parameter p , which destabilizes ψ ψ1 transition to a polar phase, demonstrates two well-pro- the system with respect to 2 and 3 becoming nonzero nounced kinks associated with rotational phase transi- (ψ ≠ ψ ). The softness with respect to the parameter ψ 2 3 α tions. It should be emphasized that the analogous in the region 1M ~ 0 increases the jump in the order anomaly observed in crystals with x ≈ 0.1 at T ≈ 485°C parameter at the UÐF and T ÐF transitions. However, as 2 α is much weaker (Fig. 2). one goes away from the point 1M = 0, the T2ÐF transition becomes less pronounced. First, we consider the change in the dielectric prop- The transition from the T phase to the phase G(p , erties caused by the transitions between paraelactric ϕ ψ ψ 1 1 phases. According to Eqs. (2)Ð(4), for the phase transi- 1, 2, 3) is also a complex first-order phase transition accompanied by the appearance (in addition to the tion from paraelectric phase i to the paraelectric phase ψ j characterized by an order parameter σ, the following parameter p1) of the nonzero parameter 2. Ferroelec- tric phases G and F have a common second-order tran- relation holds: sition line determined by the condition χ ==χ + dσ2, where χ Ð1 ε , (5) γ j i n n α α ()0 Ð PR 2 1M Ð 1M = ------p , from which it follows that the difference in magnitude b χ χ between j and i becomes progressively more pro- χ where nounced with decreasing i. Therefore, in the vicinity 2 of a transition to a polar phase, the dielectric anomalies 2 Ðhh+ Ð 24α α m 2 caused by a phase transition between paraelectric p = ------3P 2M , h = 4α α Ð γ , 12α α 2P 2M PM phases are much more pronounced than those observed 3P 2M far from this transition. This can be clearly seen from a α α γ α ε m = 2 2M 1P Ð PM 1M. comparison of the anomalies in (T) that accompany the T ÐT , T ÐS (x ~ 0.04Ð0.06), and UÐT (x ~ 0.1, T ≈ γ 2 1 1 2 The coefficient PR determines the relative position 485°C) phase transitions (Fig. 2). In the range 400Ð of the T2ÐT1 and FÐG interphase boundaries near the 500°C, the first kink on the ε(T) curve (Fig. 2) upon γ transition line to the ferroelectric state. Since PR > 0, cooling is caused by the T2ÐT1 phase transition and the the triple-phase point A is located to the left of the triple second kink, by the T1ÐS phase transition. The T2ÐT1 point C (Fig. 3). The same relative position of these transition is either second order or first order but very points is obtained by extrapolating the experimentally close to second order [3, 12]. The T1ÐS transition is a observed FÐG interphase boundary. It should be men- first-order phase transition at which the order parameter tioned that, due to the specific structure of the phase ψ ψ Ӷ ψ 2 arises in a jump but 2 1 [3]. The appearance of diagram in the vicinity of point A (Fig. 3), on cooling ϕ () () 1 at the T2ÐT1 phase transition results in a reduction in α α A Շ α Ӷ α C along the 1''M = const line, where 1M 1''M 1M the slope of the ε(T) curve. Consequently, according to Շ Ӷ γ ψ (i.e., x'' = const, xA x'' xC), the sequence of phases Eq. (5), PR > 0. The fact that 2 becoming nonzero in is T F, while on heating we have F G T . the T1ÐS transition is accompanied by an increase in the 1 1 ε κ The G phase arises in the region of metastability of the slope of the (T) curve suggests that PR < 0. The con- γ F phase (i.e., above the AC line) as a result of the loss dition PR > 0 means that the presence of the order ϕ ϕ of stability of the F phase with respect to 1. parameter 1 hinders the appearance of p1 and, as a con- α α sequence, the T ÐG phase transition temperature From a comparison of the 1MÐ 1P diagram (Fig. 3) 1 with the experimental xÐT diagram (Fig. 1), it is seen decreases with increasing ϕ (with a decrease in x). Sim- that they are qualitatively similar in terms of their basic ilarly, the slope of the SÐG interphase boundary with elements, such as the set of phase states and the respect to the T1ÐS boundary should become smaller κ arrangement of these states on the phase plane. This (since PR < 0), but actually it increases [7]. This means suggests that our choice of the coefficients in thermo- that the decrease in the SÐG transition temperature is

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1988 IVLIEV et al. α caused by the fact that the 1P(T, x) = 0 line shifts dra- at which the first-order-transition line transforms into a matically toward lower temperatures with decreasing x. second-order-transition line similar to that observed in PZT [4, 5, 13], one should bear in mind the following. Now, we consider variations in the dielectric prop- If we assume that such a point exists on the line of tran- erties at the transitions to a polar phase. The observed sitions to the polar phase, then it turns out that two such strong decrease in the ferroelectric phase transition points must exist because they limit the line of second- temperature with a decrease in x (x < 0.06) can be order transitions to the polar phase; however, in the caused by the following: (i) by weakening of the inter- vicinity of this line, the phase with p , ϕ = ψ = 0, and action responsible for the occurrence of polarization, ψ 1 1 ϕ2 ψ α 3 should exist rather than the phase with p1, 1 = 0, 2, which results in an increase in 1P, and (ii) by the ψ ϕ and 3 (Fig. 3). No such phase has been observed to appearance of the order parameter in a polar phase. date. The second reason becomes essential if one assumes that the line along which ϕ becomes nonzero in the For x ~ 0.15, the peak on the ε(T) curve becomes polar phase passes in such a way that the triple-phase smaller and the temperature hysteresis slightly point A lies slightly left of point B; i.e., xA ~ 0.06 increases (Fig. 2). This means that at x ~ 0.15 a first- (Fig. 3). This means that both of the above reasons order phase transition occurs which is not very close to become important simultaneously and enhance the a second-order transition (the CurieÐWeiss temperature տ ° effect. In this case, on cooling along the line x xA we is shifted from Tm by ~40 C). Thus, the experimental will have the sequence of phases S F and on heat- data obtained agree with the prediction of the theory ing, F G S, with the F G transition taking that the least pronounced transition to the polar phase () place in the region of metastability of these phases and will take place at α 0 Շ α < 0. Our estimations give being accompanied by a decrease in ε. The ε(T) curve 1M 1M α ()0 for x ~ 0.06 (Fig. 2) for heating (somewhat lower than that 1M corresponds to x ~ 0.08Ð0.09. the transition point from the polar phase) exhibits a small anomaly, which can be explained as a consequence of the phase transition from F to the G phase. 5. EFFECT OF EXTERNAL FACTORS However, extrapolation of the FÐG interphase bound- ON PHASE TRANSITIONS ary shows that the A point should be located on the boundary of the phase T near x ~ 0.07. Given the picture of phase states analyzed above, 1 one may anticipate how it will change under the action The decrease in temperature hysteresis with an of external factors. For example, a hydrostatic pressure increase in x from 0.04 to 0.06 (the maxima of ε(T) in stimulates rotational phase transitions (increasing their this range are almost of the same size) is caused by a temperature) and suppresses ferroelectric transitions decrease in the potential barrier separating the phases [14]. As a result, the UÐT2, T2ÐT1, and T1ÐS transition and by a reduction of the region of coexistence of the temperatures between the paraelectric phases increase. phases. The latter factor is caused by the slope of the SÐ In contrast, the temperatures of the transitions to a fer- G interphase boundary decreasing with increase in x roelectric state decrease and the magnitude of the jump and transforming into a plateau at x > 0.06, where the in the order parameter p also decreases; i.e., the transi- cross section of the region of coexistence of the S and tions become less pronounced. The latter circumstance G phases by the x = const path orthogonal to the phase may transform part of the line of the first-order transi- transition line becomes the smallest. tions to the ferroelectric state into a second-order-tran- In the range x ~ 0.06Ð0.15, the ferroelectric phase sition line limited by critical points and also generate an transition temperature changes insignificantly. For x ~ intermediate phase (between the T2 and F phases) with ϕ ψ ψ 0.1, the temperature hysteresis decreases markedly in the order parameters p1, 1 = 2 = 0, and 3. The trans- comparison with the case of x ~ 0.06, the maximum formations of the phase states are accompanied by value of ε increases noticeably, and the shape of the changes in the physical parameters. This is likely the ε(T) peak more closely resembles that of a second- cause of the strong dependence of the properties of fer- order phase transition (Fig. 2). This fact suggests that a roelectric niobate ceramics on the thermodynamic his- first-order, but close to a second-order, phase transition tory of the samples, since there are always residual takes place at x ~ 0.1 (the CurieÐWeiss temperature is stresses with a considerable hydrostatic component in shifted from the temperature of the ε(T) maximum by ceramics [15]. ~20°C). In addition, the ε(T) curve for x ~ 0.1 exhibits The effect of external electric fields on the phase a small, weakly pronounced discontinuity in slope at states is also a topical problem. T ~ 485°C, which, as was mentioned above, is caused by the rotational phase transition from the cubic (U) For example, a field with components (E100) transforms the U phase into the phase with parameters p phase to the T2 phase (Fig. 2). The shape and the small 1 ϕ ψ ψ magnitude of this discontinuity support the idea about and 1 = 2 = 3 = 0 (UE' phase); the T2 phase, into the the character and relative weakness of the interaction ϕ ψ ψ ' between the order parameters p and ψ. As to the possi- phase with p1, 1 = 2 = 0, and 3 (T 2E phase); the ϕ ψ ble existence of a critical point in the vicinity of x ~ 0.1 phase T1, either into the phase with p1, 1 = 2 = 0, and

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 PHASE STATES AND DIELECTRIC PROPERTIES OF SODIUMÐPOTASSIUM NIOBATE 1989

ψ (T ' phase) or the phase with p , ϕ = 0, ϕ , ψ = 0, stant in NKN single crystals made it possible to clarify 3 1E 1 1 2 2 the character of the interaction of rotational order ψ '' and 3 (T 1E phase); and the phase S, into the phase parameters with the polarization. This has allowed us to ϕ ψ ψ with p1, 1, 2, and 3 (SE' phase). In the phases F and develop a thermodynamic model which adequately G, only the value of p is changed. The phase S ' is iso- describes not only the variety of phase states, their loca- E tion on the phase plane, and structural transformations structural to the phase G. As E1 0, the phases T 1' E but also the changes in the dielectric constant caused by ϕ ψ these transformations. In addition, on the basis of this and T 1''E transform into domains of the phase T1 ( 0 3, ϕψ model, it becomes possible to make a detailed analysis 0 3). In order to decide which of these domains is more stable in the case of E ≠ 0, additional study is of the changes of the phase states in NKN under various 1 external actions, particularly, under a hydrostatic pres- required. Henceforth, T 1' E will denote the more stable sure and an electric field. phase state arising from the phase T1 in a field E1. By taking into account the character of interaction of the polarization with the rotational order parameters dis- ACKNOWLEDGMENTS cussed above, it can be shown that, with an increase in This study was supported in part by the Russian the applied field, the temperatures of the UE' ÐT 2' E , Foundation for Basic Research, project no. 02-02-17781.

T 1' E Ð,SE' SE' ÐG, T 1' E ÐF, T 1' E ÐG, T 2' E ÐF, and UE' ÐF phase transitions increase, while the temperatures of REFERENCES ' ' the T 2E ÐT 1E and FÐG transitions become lower. It is 1. H. Megaw, Ferroelectrics 7 (1Ð2), 87 (1974). worth noting that the transition temperatures to the 2. L. E. Cross and B. J. Nicholson, Philos. Mag. 46 (376), phases F and G increase to a greater extent than for the 453 (1955). remaining transitions. This indicates that the regions of 3. R. Ishida and G. Honjo, J. Phys. Jpn. 34 (11), 1279 existence of the phases UE' and T 1' E become smaller, (1973). while the regions of existence of the phases T 2' E and 4. K. S. Aleksandrov, A. T. Anistratov, B. V. Beznosikov, T ' and particularly of the phases F and G become and N. V. Fedoseeva, Phase Transitions in Crystals of 1E ABX Halogen Compounds (Nauka, Novosibirsk, 1981). larger. In sufficiently strong fields, the phase T ' dis- 3 1E 5. G. A. Smolenskiœ, V. A. Bokov, V. A. Isupov, N. N. appears due to its being “forced out” by the phases T 2' E , Kraœnik, R. E. Pasynkov, A. I. Sokolov, and N. K. Yushin, The Physics of Ferroelectric Phenomena (Nauka, Lenin- SE' , and F; the boundary between the phases SE' and G disappears, while a common boundary is formed grad, 1985). 6. O. V. Kovalev, Irreducible Representations of Space between the phases T 2' E and G. Such an approach, among other things, allows one to describe the pro- Groups (Akad. Nauk Ukr. SSR, Kiev, 1961). cesses occurring in NKN piezoelectric ceramics sub- 7. M. Ahtee and A. M. Glazer, Acta Crystallogr. A 32 (3), jected to strong electric fields in the course of polariza- 434 (1976). tion and service. 8. M. Ahtee and A. W. Hewat, Acta Crystallogr. A 34 (2), The examples considered above are part of the gen- 309 (1978). eral problem of the influence of external factors on the 9. C. N. W. Darlington, Philos. Mag. 31 (5), 1159 (1975). formation of phase states in the process of synthesis 10. I. P. Raevskiœ, M. P. Ivliev, L. A. Reznichenko, et al., Zh. and treatment of materials that exhibit several structural Tekh. Fiz. 72 (6), 120 (2002) [Tech. Phys. 47, 772 instabilities. This problem is of particular topical inter- (2002)]. est in connection with the strong dependence of the 11. I. P. Raevskiœ, L. A. Reznichenko, M. P. Ivliev, et al., Kri- physical properties of such materials on the preparation stallografiya 48 (3), 531 (2003) [Crystallogr. Rep. 48, technology [10]. This dependence is largely caused by 486 (2003)]. the fact that the instabilities that induce various order 12. C. N. W. Darlington and K. S. Knigth, Acta Crystallogr. parameters are differently affected by the external fac- B 55 (1), 24 (1999). tors. Under certain conditions, this feature can drasti- 13. V. V. Eremkin, V. G. Smotrakov, and E. G. Fesenko, Fiz. cally change the regions of existence of different Tverd. Tela (Leningrad) 31 (6), 156 (1989) [Sov. Phys. phases, the phases themselves, and, consequently, the Solid State 31, 1002 (1989)]. entire range of physical properties. The examples given 14. T. Hidaka, Phys. Rev. B 17 (11), 4363 (1978). above show that such a possibility really exists. 15. I. P. Raevskiœ, L. A. Reznichenko, and A. N. Kalitvanskiœ, Zh. Tekh. Fiz. 50 (9), 1983 (1980) [Sov. Phys. Tech. 6. CONCLUSIONS Phys. 25, 1154 (1980)]. Thus, the analysis of the obtained experimental data on the temperature dependence of the dielectric con- Translated by A. Zalesskiœ

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1990Ð1992. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1892Ð1894. Original Russian Text Copyright © 2003 by Vodop’yanov, Kozyrev, Sadof’ev.

LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Long-Wavelength Optical Phonons in the ZnTe/Zn0.8Cd0.2Te Superlattice L. K. Vodop’yanov, S. P. Kozyrev, and Yu. G. Sadof’ev Lebedev Physical Institute, Russian Academy of Sciences, Leninskiœ pr. 53, Moscow, 119991 Russia e-mail: [email protected] Received March 3, 2003

Abstract—The lattice IR reflection spectra of a ZnTe/Zn0.8Cd0.2Te superlattice measured at temperatures of 300 and 10 K are analyzed. The ZnTe/Zn0.8Cd0.2Te superlattice is grown by molecular-beam epitaxy on a GaAs substrate with a ZnTe buffer layer. It is found that the lattice IR reflection spectra of the studied structure exhibit only one reflection band. Dispersion analysis of the experimental spectrum has revealed the presence of one lattice TO mode close in frequency to the mode of pure ZnTe. This result is explained by a shift in the frequency of the lattice modes of the ZnTe and Zn0.8Cd0.2Te layers of the superlattice toward each other. In turn, this shift is caused by internal elastic stresses in the superlattice due to a mismatch between the lattice parameters of the materials of these layers. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION 2. SUPERLATTICE GROWTH AND MEASUREMENTS Although superlattices based on IIÐVI semiconductor compounds have attracted considerable research The ZnTe/CdxZn1 Ð xTe (x = 0.2) superlattices were interest, there are only a few works concerned with the grown by molecular-beam epitaxy in a Katun’ appara- study of their phonon properties. In particular, Perkow- tus. The epitaxy was performed on exactly oriented itz et al. [1] and Kozyrev [2] investigated the lattice (100) substrates of semi-insulating gallium arsenide properties of HgTe/CdTe superlattices on a through evaporation of special-purity (6N) elements for Cd0.96Zn0.04Te substrate, which are formed by compo- all the substances used. The surface of the GaAs sub- nents with nearly identical lattice parameters. The con- strate was cleaned from a layer of natural oxides by dition of matched lattice parameters provided a means ° for interpreting the lattice reflection spectra of heating to 580 C in a ultrahigh vacuum chamber during HgTe/CdTe superlattices in terms of the properties of preliminary preparation of the samples. After cooling to bulk lattice modes of the HgÐTe and CdÐTe vibrations the temperature of epitaxial growth of the ZnTe buffer layer, the substrate was placed in the growth chamber in Hg1 Ð xCdxTe alloys. The analysis of the lattice IR reflection spectra made it possible to construct a model and was then held in zinc vapors at an equivalent pres- × Ð5 µ of the real superlattice that allows for the interdiffusion sure of 3 10 Pa for about 2 min. A 1.5- to 2- m-thick of Hg and Cd atoms in the adjacent layers and the for- ZnTe buffer layer was grown at a substrate temperature mation of an HgCdTe alloy near the interface. of 350°C. The quantum-confined part of the ZnTe/CdZnTe structure was grown at a lower tempera- A more complex situation in studying the lattice ture (280°C) in order to prevent reevaporation of cad- properties occurs with superlattices based on components with mismatched lattice parameters. The lack of mium and, where possible, to provide the formation of lattice matching gives rise to elastic stresses that signifi- abrupt interfaces between the layers. The ratio of the cantly affect the physical properties of these structures. equivalent pressure of the tellurium molecular beam to In our recent work [3], we analyzed the lattice IR reflec- the total pressure of Cd and Zn molecular beams was chosen close to two in order to ensure the coexistence of tion spectra of ZnSe/Zn1 Ð xCdxSe (x = 0.2, 0.4) strained × superlattices on a GaAs substrate. Earlier [4], we demon- the superstructure consisting of a mixture of (1 2) + c(2 × 2) reconstructions and the superstructure corre- strated that solid solutions in the Zn1 Ð xCdxSe system are characterized by a single-mode transformation of the sponding to the conditions of the highest extent of sto- vibrational spectrum. Owing to the single-modality of ichiometric growth on the surface. The growth rate was Zn1 Ð xCdxSe solid solutions, the lattice IR reflection kept equal to 0.2 nm/s for the ZnTe buffer layer and spectrum of the superlattice exhibits only one lattice decreased to 0.1 nm/s before the growth of the quan- mode at an intermediate frequency rather than the two tum-confined part of the structure. The composition of expected IR-active lattice modes of the ZnSe and the layers grown was checked against the change in the Zn1 − xCdxSe layers. This result is explained by the fact oscillation period of reflections in the high-energy elec- that the lattice modes of the layers approach each other tron diffraction patterns, which correspond to the in frequency under the effect of internal elastic stresses. growth of ZnTe and CdZnTe, and also against the posi-

LONG-WAVELENGTH OPTICAL PHONONS 1991 tion of the emission lines of single quantum wells grown in a series of experiments on the calibration of 0.9 molecular beam sources. The lattice IR reﬂection spectra of the superlattice 0.8 were recorded on a laboratory grating infrared spectrometer and a Bruker IFS-66 Fourier spectrometer 0.7 with a resolution better than 1 cmÐ1.

R 0.6 3. RESULTS AND DISCUSSION 0.5 The experimental lattice IR reflection spectrum of the ZnTe/Zn0.8Cd0.2Te superlattice at 300 K is shown by 0.4 the thin line in Fig. 1. The thick line in this figure rep- Reflectivity, resents the theoretical spectrum calculated using the 0.3 dispersion analysis. It can be seen from Fig. 1 that the short-wavelength band corresponds to the lattice mode 0.2 of the GaAs substrate, whereas the spectrum of the superlattice contains only one band, which is confirmed 0.1 by the mathematical analysis. The reflection curve Ð1 exhibits frequent oscillations (with a period of 2.5 cm ) 0 associated with the interference on a 400-µm-thick 150 200 250 300 350 GaAs substrate. Wave number, cm–1 The dispersion analysis of the lattice IR reflection spectra of the superlattices was carried out using a Fig. 1. Lattice IR reflection spectra of the model structure formed by a thin film (a superlattice ZnTe/Zn0.8Cd0.2Te superlattice at a temperature of 300 K. and a buffer layer) on a semi-infinite substrate. In the The thin and thick lines represent the experimental and cal- framework of this model for a film of thickness L with culated spectra, respectively. a dielectric function ε (ω) and a substrate with a dielec- ε ω f tric function s( ) in normally incident light, the ampli- small (~6 cmÐ1), these modes can be unresolved tude reflectivity has the form [5] because of the large attenuation parameters (of approx- ()ω ()ω ()β imately 3 cmÐ1) at room temperature. In order to sepa- ()ω r1 f + r fs exp i2 r1 fs = ------()ω ()ω ()β , (1) rate the contributions from individual layers of the 1 + r1 f r fs exp i2 ZnTe/Zn0.8Cd0.2Te//GaAs superlattice, the lattice IR where reflection spectra were measured at a temperature of 10 K. The experimental lattice IR reflection spectrum 1 Ð ε ()ω ε ()ω Ð ε()ω ()ω f ()ω f s of the ZnTe/Zn0.8Cd0.2Te superlattice at 10 K is r1 f ==------, r fs ------, 1 + ε ()ω ε ()ω + ε()ω depicted by the thin line in Fig. 2. The thick line in this f f s figure represents the theoretical spectrum calculated 2πL ε ()ω using the dispersion analysis. As can be seen, the spec- and β = ------f -. trum shape does not change qualitatively. Within the λ limits of the spectral resolution of the spectrometer, we revealed only one lattice TO mode at a frequency of λ 10 000 181 cmÐ1. This value is close to the frequency of the TO Here, is the wavelength =------. The reflectivity is ω mode (182 cmÐ1) for pure ZnTe at a temperature of ω | ω |2 4.2 K. The shoulder observed in the low-frequency defined by the formula R( ) = r1fs( ) . range of the TO mode of the superlattice is not related The dielectric function ε (ω) of the film was consid- f to an IR-active mode but is caused, more likely, by the ered in the classical additive form: interference in a layer consisting of the superlattice and ω 2 buffer. This assumption is confirmed by the frequent S j tj ε ()ω = ε∞ + ∑------. (2) oscillations associated with the interference on the sub- f ω 2 ω2 ωγ j tj Ð Ð i j strate. These oscillations can be observed only in the transparency region of the substrate and multilayer From the dispersion analysis of the lattice IR reflec- structure. The mode in the vicinity of 146 cmÐ1, which tion spectrum of the superlattice at room temperature, corresponds to a vibration of the two-mode system of we obtained only one lattice mode of the superlattice at Ð1 Zn1 Ð xCdxTe solid solutions that is similar to the vibra- a frequency of 177 cm , which is close to the fre- tion in CdTe, is not observed because of its weakness. quency of the lattice TO mode (179 cmÐ1) for pure ZnTe. Since the difference between the frequencies of Let us elucidate why only one transverse optical the TO modes for the ZnTe and Zn0.8Cd0.2Te layers is mode is observed in the experiment. As was shown in

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1992 VODOP’YANOV et al.

Zn0.8Cd0.2Te layers subjected to compressive strain. Ð1 0.9 The difference in these frequencies (0.7 cm ) is considerably less than the attenuation constant (~3 cmÐ1) 0.8 and is unlikely to be found experimentally. The estimates obtained account for the manifestation of only 0.7 one TO mode in our experiments.

R 0.6 4. CONCLUSIONS

0.5 Thus, the above analysis of the lattice IR reﬂection spectra of ZnTe/Zn0.8Cd0.2Te and other superlattices based on II–VI compounds conﬁrmed our assumption 0.4

Reflectivity, that the phonon spectrum of multilayer structures depends on the nature of the chemical bonds. In partic- 0.3 ular, it was established that individual properties of each layer manifest themselves in superlattices based 0.2 on IIIÐV compounds in which short-range covalent bonds dominate. As a result, the vibrational spectra 0.1 exhibit modes typical of superlattices, such as confined, folded, and boundary phonons [9]. In multilayer struc- 0 150 200 250 300 350 tures based on IIÐVI compounds, in which the long- Wave number, cm–1 range ionic bonds dominate, the phonon frequencies of adjacent layers belong to a common mode and vibrational modes characteristic of superlattices do not man- Fig. 2. Lattice IR reflection spectra of the ifest themselves. ZnTe/Zn0.8Cd0.2Te superlattice at a temperature of 10 K. The thin and thick lines represent the experimental and calculated spectra, respectively. ACKNOWLEDGMENTS This work was supported by the Russian Foundation our earlier works [6, 7], the frequencies of the TO for Basic Research, project no. 03-02-7110. modes for bulk ZnTe and Zn0.8Cd0.2Te crystals at room temperature are equal to 179 and 173 cmÐ1, respectively. The difference in these frequencies (~6 cmÐ1), REFERENCES even if not very large, could be found experimentally. 1. S. Perkowitz, R. S. Sudharsanan, and S. Yom, J. Vac. Sci. However, it should be remembered that the frequencies Technol. A 5, 3157 (1987). of vibrational modes for the superlattice undergo shifts 2. S. P. Kozyrev, Fiz. Tverd. Tela (St. Petersburg) 36 (10), under the effect of internal elastic stresses. Now, we 3008 (1994) [Phys. Solid State 36, 1601 (1994)]. estimate these shifts within the approach used in [3, 8]. 3. V. S. Vinogradov, L. K. Vodop’yanov, S. P. Kozyrev, and The elastic strains can be calculated under the condition Yu. G. Sadof’ev, Fiz. Tverd. Tela (St. Petersburg) 43 (7), of equilibrium of two mutually strained layers forming 1310 (2001) [Phys. Solid State 43, 1365 (2001)]. the superlattice. To the best of our knowledge, no elas- 4. L. K. Vodop’yanov, S. P. Kozyrev, and Yu. G. Sadof’ev, Fiz. Tverd. Tela (St. Petersburg) 41 (6), 982 (1999) tic constants for Zn0.8Cd0.2Te are available in the literature. Therefore, they were calculated using the extrapo- [Phys. Solid State 41, 893 (1999)]. 3 5. H. W. Verleur, J. Opt. Soc. Am. 58, 1356 (1968). lation Cik(x) = Cik(0)(a/a(x)) [8], where Cik(0) and 6. L. K. Vodop’yanov, E. A. Vinogradov, and A. E. Tsurkan, Cik(x) are the elastic constants of ZnTe and the solid solution, respectively, and a and a(x) are the lattice con- Zh. Prikl. Spektrosk. 21 (2), 321 (1974). stants of ZnTe and the solid solution, respectively. For 7. E. A. Vinogradov and L. K. Vodop’yanov, Fiz. Tverd. Tela (Leningrad) 17 (11), 3161 (1975) [Sov. Phys. Solid Zn0.8Cd0.2Te, the coefficient K(x) accounting for the relative change in the frequency of TO modes polarized State 17, 2088 (1975)]. along the layers and for the strain in the same direction 8. V. S. Vinogradov, L. K. Vodop’yanov, S. P. Kozyrev, and was calculated using a linear interpolation between Yu. G. Sadof’ev, Fiz. Tverd. Tela (Leningrad) 41 (11), 1948 (1999) [Phys. Solid State 41, 1786 (1999)]. these coefficients for CdTe (x = 0) and ZnTe (x = 1). From these calculations, we obtained the following 9. H. Shin, D. Lackwood, and P. Pool, Appl. Phys. Lett. 77 ∆ω Ð1 (2), 229 (2000). shifts and final frequencies: TO = Ð2.7 cm and ω Ð1 TO = 176.3 cm for ZnTe layers subjected to tensile ∆ω Ð1 ω Ð1 strain and TO = 4 cm and TO = 177 cm for Translated by N. Korovin

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 1993Ð2000. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1895Ð1902. Original Russian Text Copyright © 2003 by Kozhevin, Yavsin, Smirnova, Kulagina, Gurevich.

LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Effect of Oxidation on the Electrical Properties of Granular Copper Nanostructures V. M. Kozhevin, D. A. Yavsin, I. P. Smirnova, M. M. Kulagina, and S. A. Gurevich Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received March 7, 2003

Abstract—The structural and electrical properties of thin granular metal ﬁlms obtained by laser electrodispersion were studied. Such structures, which consist of amorphous copper grains 5 nm in size, were established to be extremely stable against oxidation. For instance, when oxidized in air, copper grains were covered by Cu2O oxide shells about 1 nm thick after a period of a few days, after which further growth of the oxide in thickness stopped. The oxidized close-packed structures conduct current through intergrain tunneling electron transitions, whereas in partially oxidized structures the conduction involves tunneling electron hopping between conducting ensembles made up of several nanoparticles. It was shown that the size of the nanoparticles and of the conducting ensembles can be found by analyzing the temperature dependence of the conductivity and, independently, from the IÐV characteristics of the ﬁlms. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION This communication reports on a study of granular copper nanostructures prepared by laser electrodisper- Granular metals, i.e., structures made up of nano- sion [8]. This method is based essentially on laser abla- sized metal particles, can serve as a basis for develop- tion of a metal (copper) target. The conditions of abla- ing novel nanoelectronic devices [1] and new types of tion are chosen so that the molten-metal drops injected solid-state catalysts [2] and can also be employed to from the target surface into the laser torch plasma advantage in other areas of nanotechnology [3]. The become charged to the threshold of capillary instability electrical properties of such structures depend, to a con- and break up into smaller particles. It was shown in [8] siderable extent, on the size of the conducting nanopar- that under certain conditions the drops undergo cascade ticles and on their (bulk or surface) packing density. If fission to form finally copper nanoparticles. Nanoparti- the structure-averaged particle density is low, the dc cles thus formed cool rapidly; they are extracted by an conductivity of such a medium will be very low. In the electric field from the plasma zone and directed onto opposite extreme case of a very high density, where a the substrate. Depending on the deposition time, sub- sizable fraction of particles are in contact [4], fairly strate coatings with different particle densities and of extended conducting ensembles can form in the struc- different thickness can be prepared. Copper particles produced by this method typically have a small spread ture [5], with its conduction taking on a metallic pat- in size; indeed, for an average particle size of 5 nm, the tern. Of particular interest are structures with an inter- relative spread does not exceed ≈20%. After prepara- mediate density, in which the average distance between tion of the granular films in vacuum was complete, the particles is on the order of their dimensions, i.e., about structures were exposed to air and oxidized gradually, a few nanometers. In this case, the conductivity of the with the formation of oxide shells on the surface of the medium derives from electron tunneling through the copper nanoparticles. It is assumed that these shells, gaps separating the particles. The tunneling probability along with the air gaps separating the particles, act as depends on the particle size; the height, width, and barriers in electron tunneling. shape of the tunneling barriers; and the temperature of We studied the conductivity of granular copper the medium [6]. The tunneling conductivity of metallic films prepared by laser electrodispersion. The conduc- nanostructures is also governed substantially by such tivity of the films was measured in lateral geometry. factors of structural and energy disorder as the spread The conductivity of closely packed structures is shown in particle size, the pattern of the correlations in their to be extremely sensitive both to the pattern of mutual positions, and the existence of a random potential on arrangement of particles in the bulk of a film and to the the particles [7]. The above-mentioned parameters of degree of oxidation of the structure. In particular, the the structures may strongly vary depending on the tech- specific features of self-organization of particles in the nology used in their preparation, thus considerably films under study account for the very sharp transition complicating the problem of describing conduction in from tunneling to metallic conduction. The conductiv- such media. ity measurements also suggest that natural oxidation of

1994 KOZHEVIN et al. conducted earlier using atomic force microscopy [8]. The layer-by-layer formation of structures, as well as the formation of islands in the first particle layer, was suggested in [10] to be due to Coulomb interaction between charged particles impinging on the substrate and the particles deposited on its surface during the previous laser pulses. Note that, in order for TEM images to provide reliable information on the film structure, the images have to be obtained at the lowest possible electron beam current. When the current is too high, the particles are charged through electron absorption and migrated over the substrate surface, thus giving rise to a noticeable modification of the structures. – The structures prepared by laser electrodispersion in 30 nm R = 2.5 nm n = 4 × 1012 cm–2 a vacuum chamber were subsequently exposed to air, which resulted in natural oxidation. The oxygen con- Fig. 1. TEM image of a granular copper film (top view). tent in the structures was determined by XPS. The pho- Film deposition time 3 min. toelectron spectra obtained within a broad range of kinetic energies clearly revealed the oxygen line 1s1/2 in addition to the lines characteristic of adsorbed carbon copper nanoparticles in air proceeds very slowly. This and water and to those due to the presence of silicon in finding is in accord with observations made by other the substrate. In the case under study, where the surface authors [9] in structural studies. The slow oxidation of of a substrate consisting of SiO2 silicon dioxide is copper nanoparticles is apparently a consequence of coated with a fairly dense layer of copper grains, the their having amorphous structure. We also derived a presence of this line can be associated with the oxygen simple relation for the conductivity of granular films accumulated in the granular film during oxidation. The which is valid in both weak and strong fields. This rela- validity of this explanation is also supported by the tion is used to analyze the IÐV characteristics and the small electron escape depth measured by XPS (see, temperature behavior of the conductivity, which per- e.g., [11]). The oxygen concentration was derived from mits us to establish the dependence of the characteristic the intensity of this line. The measured oxygen concen- size of a conducting ensemble on the oxidation time of tration in a film deposited for 2 min and exposed subse- the structure. quently to air for 30 min was approximately 16 vol %. To exclude the contribution of the surface-adsorbed 2. STRUCTURAL PROPERTIES OF FILMS oxygen, the photoelectron spectra were also recorded after a short-time etching of the sample by Ar+ ions, a The structural properties of films were studied by procedure which removed a 0.5-nm layer from the sur- transmission electron microscopy (TEM) and x-ray face. As a result of the etching, the oxygen content photoelectron spectroscopy (XPS). The film substrates decreased to ≈10 vol %. To understand the results were silicon plates coated with a layer of thermal oxide, obtained, one has to make assumptions concerning the with the surface-oxide relief depth not exceeding 2 nm. structure of the oxide thus formed. Figure 1 presents a TEM image of a structure deposited It is known that oxidation of bulk copper samples for 3 min. As follows from an analysis of this image and develops primarily through diffusion of copper atoms the diffraction patterns of transmitted electrons, this across the growing oxide film to the surface and occurs structure consists of copper grains about 5 nm in size through the reaction Cu Cu2O CuO [12]. Oxi- and the particles are amorphous. The particle images in dation of copper particles a few nanometers to tens of the photomicrograph are seen not to overlap, which nanometers in size was shown in [9, 13] to proceed by suggests that for the given deposition time the particles the same scenario; the general conclusion obtained in are arranged in one layer on the substrate. The layer is many studies consists, however, in that the room-tem- nonuniform; indeed, it is seen to be made up of ensem- perature oxidation rate decreases with decreasing parti- bles (islands) consisting of several particles in close cle size. For copper particles a few nanometers in size, contact. The photograph also reveals a network of gaps the time needed to reach the final stage of oxidation, the separating the islands. The gaps are practically of the CuO oxide, can be very long; in fact, it can be longer same size, about 3 nm. There is also a small number of than the reasonable observation time. Thus, it may be particles (they appear as darker colored in Fig. 1) suggested that in our case, after exposure of a film to air belonging to the second layer. The particles of the sec- for a few tens of minutes to several days, the product of ond layer fall primarily into gaps between the islands. oxidation will be Cu2O, cuprous oxide. Assuming also That the second layer of particles in films deposited that nanoparticles are of spherical shape and that the by laser electrodispersion starts to fill only after the fill- thickness of the oxide shell is uniform over the struc- ing of the first layer is complete is borne out by studies ture, the above experimental data permit the conclusion

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

EFFECT OF OXIDATION ON THE ELECTRICAL PROPERTIES 1995

–2 2 (a) 10

10–4 –1 Ω 10–6 11 10–8 (b) Conductance, 10–10

10–12 Fig. 2. Formation of a continuous current channel in a struc- 2.8 3.0 3.2 3.4 3.6 3.8 4.0 ture with a partially filled second grain layer. (a) Top view t, min and (b) side view along the current path. (1) Metallic contact and (2) current path. Fig. 3. Film conductance plotted vs. deposition time t. that the thickness of the cuprous-oxide coating around varied on the same sample from 5 to 200 µm. Next, a a copper particle is, on the average, 0.8 nm. Thus, the granular copper film was deposited in the gaps between nanoparticles under study contain an amorphous-cop- the contacts. Prior to measurements, all samples were per core 4 nm in size which is coated with a cuprous- exposed to air for the same time (30 min). Measure- oxide layer 0.8 nm thick (the thickness of a cuprous- ments were conducted at room temperature. Figure 3 oxide monolayer is ~0.4 nm). displays the dependence of the film conductance on deposition time. The time threshold for the onset of conduction (determined from the sensitivity threshold 3. DEPENDENCE OF THE CONDUCTIVITY of the measuring system, ~10Ð12 ΩÐ1) is seen to be about ON FILM THICKNESS AND OXIDATION TIME 2.8 min. This time correlates well with our notions on As already mentioned, self-organization of particles the structural properties of the films and can be identi- in the course of deposition makes the structures under fied with the beginning of formation of the second par- study locally nonuniform. Monolayer films have an ticle layer. As is evident from Fig. 3, as the second layer islandlike structure. Within an island, the particles are is filled, i.e., during the time interval of deposition of in close contact with one another; as a result, the prob- three to four minutes, the conductance of the structures ability of electron transitions between particles in the rises by nine orders of magnitude. It can be readily seen island is sufficiently high. At the same time, the islands that the surface density of particles changes only by are separated by gaps about 3 nm in dimension, which ~25%. Note that, when the conductance rose to −3 ΩÐ1 practically excludes the possibility of interisland tun- ~10 , the film conduction was metallic, despite the neling. Thus, the conductivity of monolayer coatings short-time oxidation. Thus, because of the specific should be very low. Continuous current paths in the organization of particles the thin granular films under film plane can appear only when particles of the second study, the insulatorÐmetal transition occurs in them layer appear to close the gaps between the islands. The very quickly during the filling of the second layer of formation of current paths in a structure is shown sche- nanoparticles. matically in Fig. 2. The number of current paths and the Figure 4 plots the conductance of films with a par- film conductivity are determined primarily by the tially filled second layer of grains versus the time of extent to which the second layer is filled. In addition, exposure of the structure to air. We readily see that the the conductivity of a film with a partially filled second pattern of conductance variation with time depends layer of particles should obviously be very sensitive to noticeably on the degree of filling of the second layer the magnitude of the dielectric gaps between particles by grains or, what amounts to the same thing, on the ini- in islands and between particles of the first and second tial conductance of the samples. For instance, the time layer. Accordingly, the variation of the film conductiv- to reach the time-independent conductance in films ity caused by varying the magnitude of these dielectric with a high initial conductance is about 15 days, gaps can be employed to monitor the degree of oxida- whereas for films with a lower conductance (i.e., with a tion of a structure. smaller number of grains in the second layer) the stage The longitudinal conductance of the films was mea- of fast conductance variation comes to an end after two sured on specially prepared test samples. To prepare days. After the first stage of oxidation, the film conduc- these samples, a Cr layer 30 nm thick was first depos- tance stabilizes. Check measurements showed that, in ited on the oxidized silicon surface of a substrate and thin films with a relatively small number of grains in 400 × 400 µm contact pads were produced by explosive the second layer, the conductance that was reached lithography, with the distance between these pads being after the first stage of oxidation practically did not

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

1996 KOZHEVIN et al.

–2 ity of copper nanoparticles to oxidation has been 10 observed by some authors [9, 14], but in most cases the experimental conditions and the properties of the parti-

–1 –4 cles themselves were different, which does not permit

Ω 10 one to establish a one-to-one correspondence between the oxidation rate and the grain size. The extremely 10–6 high stability of copper structures to oxidation revealed in our experiments can be assigned to the fact that nanoparticles prepared by laser electrodispersion have Conductance, 10–8 amorphous structure. This conjecture is in agreement with the observation that amorphous metals are more stable to corrosion than conventional metals with poly- 10–10 0 5 10 15 20 crystalline structure [15]. τ, days

Fig. 4. Film conductance plotted vs. exposure time τ to air 4. TEMPERATURE BEHAVIOR at room temperature. OF THE CONDUCTIVITY AND CURRENTÐVOLTAGE CHARACTERISTICS OF OXIDIZED FILMS 3 Positively and negatively charged particles can appear in granular metal nanostructures both in tunneling transitions of electrons between originally neutral 1 2 adjacent grains and as a result of electron (hole) injec- tion from the electrodes. In the case of high ionization + energies or comparatively low temperatures, the system contains only singly charged particles, i.e., particles

– with one excess electron or lacking an electron. It is this case that will be considered below. If a system contains + a certain amount of charged particles, an electric ﬁeld will initiate the drift of charges resulting from tunneling

–– transitions. The current density in the ﬁlm is given by the well-known expression jen= ()ν + n ν , (1) + Ð Ð + +

where e is the electronic charge; nÐ and n+ are the den- Fig. 5. Two-stage grain ionization process: (1 2) for- sities of negatively and positively charged grains, mation of a charged-grain dipole, (2 1) recombination respectively; and ν± are the drift velocities of electrons of a charged-grain dipole, and (2 3) formation of a pair (Ð) or holes (+) in an external electric field. of independent charged grains. The density of charged grains is determined by the charge generation and recombination rates. In weak change for two to three months from the date of manu- electric fields, the thermal activation mechanism of facture of the structure. This means that the nanoparti- generation dominates; the corresponding relation for cle cores remained metallic during this long period of the density of charged particles can be derived in this time and that the thickness and composition of the case by using a statistical approach [16]. In strong oxide shells practically did not change. fields, field ionization of neutral grains can also play a Studies of the conductivity of films consisting of substantial part. To find the charged particle density two or more grain layers showed that it has a metallic within a broad range of applied fields, we consider the pattern and approaches that of a thin continuous copper generationÐrecombination balance with due account of film. The resistivity of granular films of thickness in thermal and field ionization. excess of 15 nm (consisting of three or more layers of Figure 5 is a schematic representation of the process particles) remained constant during the whole observa- of ionization of originally neutral grains. The ionization tion time period after deposition (up to three years). is seen to proceed in two stages; initially, a dipole of This effect can be attributed to the upper layer of grains two oppositely charged neighboring particles forms preventing oxidation of the lower lying ones, so that no through electron exchange between adjacent neutral oxide shells form on the grains making up the lower grains (transition 1 2), after which the dipole layers and they are in direct contact, providing metallic breaks up to form two independent charged grains conduction of the film as a whole. Note that a low activ- (transitions 2 3). If, conversely, charge migration

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

EFFECT OF OXIDATION ON THE ELECTRICAL PROPERTIES 1997 Γ brings positively and negatively charged grains to charged particles of both signs, i = 2n0ri, where n0 is neighboring positions, the charges can recombine. the density of neutral grains. In general, the rate of tunneling transitions of electrons between neighboring metal grains separated by a Electron tunneling from the newly formed nega- dielectric gap s wide can be written as [6] tively charged particle to a neighboring neutral particle or from a neutral to a neighboring positively charged γ ∆ ∆ grain produces charge migration in the bulk of the film. r ==--- exp()2χs ------A------, (2) Ð∆/kT Ð∆/kT If this migration brings charges of opposite sign to h 1 Ð e 1 Ð e neighboring grains, these charges recombine in a very where γ is a geometry factor depending on particle short time 1/r21. The recombination rate of charged par- 2 1/2 Γ ν σ shape, χ = [2mφ/ប ] is a tunneling constant, φ is the ticles is given by the relation r = 2n T , where n is the effective dielectric-barrier height, ∆ is the change in the density of positively or negatively charged particles, ν | electrostatic energy of the grain system caused by the T = dr ∆ → 0 = AdkT is the thermal velocity of charge γ motion due to thermally activated electron tunneling, given transition, and A ≡ --- exp(Ð2χs). In the 1 2 σ ≈ h and d is the recombination cross section (equal to the distance between the centers of neighboring con- transition, the electrostatic energy of the grain system ν changes by ∆ = eU Ð E , where U is the potential ducting particles). The relation for T thus obtained is 12 d 12 d valid if the particle size is fixed and the tunneling-gap difference of the neighboring grains between which the dimensions along the current paths are constant. In this transition takes place and E is the energy of the dipole 12 case, one can assume that, in the absence of an external thus formed. For the reverse transition 2 1, i.e., for field, the electron tunnels from a charged to a neighbor- the dipole recombination, we have ∆ = E Ð eU . 21 12 d ing neutral particle without any change in energy. It is Similarly, the change in energy occurring in the 2 ∆ these properties that characterize the monodisperse 3 transition is 23 = eUd Ð 2E + E12, where E is the work structures under study here, in which the tunneling gaps that has to be done for an electron to be transported are determined by the thickness of the grain oxide shell. from infinity to a neutral grain. It was shown in [6] that ≈ ∆ ∆ ≈ ∆ For only partially oxidized structures, this approach E E12; hence, 12 = Ð 21 23 = eUd Ð E. The corre- also turns out to be applicable, because the tunneling sponding transition rates calculated from Eq. (2) are transition probabilities cannot vary strongly along a chosen current path [5]. eUd Ð E r12 ==r23 A------, EeUÐ d In a steady state, the generation rate is equal to the 1 Ð exp ------Γ Γ kT charge recombination rate, i.e.; i = r. If the density of Ӷ (3) charged particles is low (n n0), this relation readily EeUÐ d transforms to r21 = A------. eU Ð E 1 Ð exp ------d -  E eUd Ð E kT nn≈ ------exp ------. (5) 0 kT + 2eU kT These relations for the transition rates can be used to d find the rate of the two-stage ionization process ri = ≈ ≈ Substituting the experimental values E 200, Ud 20, r12r23/(r21 + r23). Because the electrostatic energy of a ≈ ≈ ≈ and kT 25 meV (T 300 K) into Eq. (5) yields an esti- singly charged nanoparticle 5 nm in size is E 200 meV mate of the density of charged particles n ≈ 10Ð3n , and eU in our experiments does not exceed 20 meV, 0 d which bears out the validity of the above assumption. the ionization rate at close to room temperature (kT ≤ 25 meV) can be described by a simple relation, Note that, because the ionization process under study produces negatively and positively charged particles in () equal numbers and their recombination rates are equal, ≈ 2 eUd Ð E ri AEexp ------. (4) a stationary homogeneous system is always in the state kT of charge neutrality, nÐ = n+ = n. In general, an electron can make a tunneling transition ν at an angle ϕ to the electric field vector. To include this To find the drift velocities ± in Eq. (1), one has to ϕ consider the electron (hole) tunneling in the presence of possibility, one has to replace Ud in Eq. (4) by Udcos ϕ an external field in more detail. We assume, as before, and average the expression for ri over the angle . The averaging over angles complicates Eq. (4), but it can be that all particles are of the same size and form a close- shown that correcting the quantity r reduces in this case packed structure with tunneling gaps of a fixed width. i If the electron migrates in an external field by tunneling kT to introducing a factor on the order of ------that from a negatively charged to a neutral grain, the change kT + 2eUd in energy of the system due to the transition is ∆ = ϕ ϕ does not greatly differ from unity. Finally, knowing the eUdcos , where is the angle between the direction of ionization rate, one can calculate the generation rate of transition and the electric field vector. (A hole transition

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 1998 KOZHEVIN et al.

T, K conductance of the same film measured immediately 300 200 140 100 80 after film preparation and following exposure of the sample to air for one week, which was sufficient for the 10–9 film conductance to stabilize (Fig. 4). The measurements were conducted at a fixed voltage U = 1 V across –1 the interelectrode gap. As follows from Eq. (10), the Ω –10 10 conductance measured at a fixed voltage and at a constant energy E over the structure should exhibit a close 10–11 to activated temperature behavior, which yields a linear dependence when expressed in the lnσ vs. 1/T coordi- 2 1

Conductance, nates. In the experiment, this relation was observed to 10–12 hold for an oxidized film. The value of E derived from 1 the slope of curve 1 in Fig. 6 is 200 meV. Using an ≈ 2 πεε approximate expression for this energy, E e /3 0D 4 6 8 10 12 (ε is the effective permittivity of the medium and D is 103/ , K–1 T the particle diameter), obtained in the dipole approxi- Fig. 6. Temperature dependences of the film conductance mation [18], one comes to the characteristic size of con- measured at different instants of time: (1) after exposure to ducting particles D ≈ 5 nm, which agrees with the main air for a week and (2) immediately after film deposition. dimension of copper nanoparticles in such structures (see Section 2). Therefore, the dielectric gaps between all grains (including the grains in the first-layer islands) involves the same change in energy.) Accordingly, the in the films whose conductance stabilized in time rate of charge displacements rdr in various directions is formed uniformly over the structure. eU cosϕ r = A------d -. (6) The nonlinear temperature dependence of conduc- dr ϕ eUd cos tance shown by curve 2 in Fig. 6 obviously corresponds 1 Ð exp Ð------kT to the case of temperature-dependent E. As seen from the shape of this curve, at low temperatures, the current By averaging rdr over all directions, one can calculate paths in as-prepared films pass over ensembles of par- the directed velocity of charge motion: ticles or conducting islands of a comparatively large 2π size, for which the effective value of E is small. At high d temperatures, the current paths pass over smaller grains ν± = ------r cosϕϕd . (7) 2π ∫ dr as well. It thus follows that as-prepared films have only 0 fragments of the network of dielectric gaps that forms Integration yields after the structure has completely oxidized. 1 Equation (10) also shows that the slope of the tem- ν± = ---AeU d. (8) 4 d perature dependence of conductance plotted in the lnσÐ 1/T coordinates is determined by the difference E Ð Substituting Eqs. (5) and (8) into Eq. (1) yields the cur- eUd/L, i.e.; hence, at low voltages, the slope of the rent density: curve is dominated by the magnitude of E and starts to

1 2 E eUd Ð E decrease with increasing voltage. Indeed, as seen from j = ---Ae dn U ------exp ------. (9) 2 0 d kT + 2eU kT Fig. 7, the slope of the temperature dependence of con- d ductance of an oxidized film decreases as the voltage at The average voltage drop between neighboring grains which it is measured increases. in the direction of the field is Ud = Ud/L, where U is the voltage applied to the contacts and L is the interelec- Consider the currentÐvoltage characteristics of the trode gap. Therefore, from Eq. (9), we obtain films. As follows from Eq. (10), the dependence of the current on voltage is dominated by the factor ∝ E eUd/LEÐ eUd jU------exp------. (10) Uexp------. This relation can be conveniently ana- kT + 2eUd/L kT kTL Relation (10) defines the current–voltage characteris- lyzed by using a normalized characteristic. Introducing tics and the temperature dependence of conductivity of the parameters i = I/Imax and u = U/Umax, where Imax is granular films. 1 Another important condition for the temperature dependence of Consider first the temperature behavior of the con- the conductance to be linear consists in the smallness of the ran- ductivity and the effect of film oxidation on this rela- dom-potential amplitude on the grains [17]. This condition is also tion. Figure 6 presents temperature dependences of the met in the structures under study.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 EFFECT OF OXIDATION ON THE ELECTRICAL PROPERTIES 1999

10–7 1.0 1 0.8 10–8 2 3 0.6 4

10–9 I 5 6

, A 0.4 I 10–10 0.2 3 10–11 2 1 0 0.2 0.4 0.6 0.8 1.0 U 10–12 3 4 5 6 7 8 9 Fig. 8. Normalized currentÐvoltage characteristics mea- 103/T, K–1 sured after different film exposures to air. Experiment: × Ð7 (1) Imax = 1.6 10 A for an as-prepared film, (2) Imax = × Ð9 × Ð9 Fig. 7. Temperature dependences of the film conductance 6.8 10 A three days later, and (3) Imax = 3.1 10 A measured at three different voltages: (1) 1, (2) 10, and seven days later. Calculated curves correspond to β equal to (3) 40 V. Gap width 5 µm. (4) 0.6, (5) 0.9, and (6) 1.5.

the maximum current at the maximum voltage Umax, we 5. CONCLUSIONS obtain Thus, it has been shown that the conductivity of () granular metallic structures depends substantially on I U ed UÐ Umax i ==------exp ------the packing density and the pattern of mutual arrange- I U kTL (11) max max ment of conducting particles, as well as on the degree = uexp[]β()u Ð 1 , of their oxidation, i.e., on the presence, dimensions, and properties of dielectric barriers between individual β where = edUmax/kTL. As is evident from Eq. (11), the particles or ensembles of particles. One of the most curvature of the currentÐvoltage characteristic plotted essential conclusions obtained in this study is that gran- in the uÐi coordinates depends on the only parameter, β. ular structures prepared by laser electrodispersion and β Note that the magnitude of parameter for fixed values made up of close-packed amorphous-copper particles of Umax, L, and T is determined by the value of d only. 5 nm in size oxidize extremely slowly in air. A fairly Experimental studies of the currentÐvoltage charac- long oxidation (a few to ten days) coats all particles by teristics revealed that their curvature decreases with approximately 1-nm thick Cu2O oxide shells, after increasing exposure time of as-prepared films to air. which further growth of the oxide layer in thickness This effect is well illustrated in Fig. 8, which displays stops. The conduction in such completely oxidized several normalized characteristics of the same film structures occurs, provided the particles are closely measured at different instants of time. The best fit packed, through tunneling transitions of electrons between the theoretical and the experimental character- between individual grains. Current flow in partially istic obtained immediately after the film deposition is oxidized structures involves tunneling electron hopping reached for β = 1.5, which corresponds to a film with an between conducting ensembles, which may consist of a average distance between the centers of conducting ≈ few grains. It has been shown that the parameters of a ensembles d 18 nm. Such ensembles may consist of granular film that are responsible for conduction can be three to four contacting grains 5 nm in size. After the determined by analyzing the temperature behavior of exposure of a film to air for three days, parameter ν decreases to 0.9. In this case, d ≈ 10.8 nm, which means the conductance and, independently, from the shape of that the conducting ensembles are made up primarily of the currentÐvoltage characteristics. two grains. Oxidation of the film in air for seven days brings about a further decrease in the nonlinearity of the experimental characteristic. Its comparison with the ACKNOWLEDGMENTS theoretical curve yields β = 0.6 and d ≈ 7.2 nm, which corresponds practically to close packing of 5-nm cop- This study was supported by the Russian Founda- per nanoparticles. Thus, the shape of the currentÐvolt- tion for Basic Research (project nos. 01-02-17827, age characteristics also leads one to conclude that pro- 02-03-32609), ISTC (project no. B 678), and the Min- longed oxidation of a structure in air results in the for- istry of Science, Industry, and Technologies of the Rus- mation of uniform dielectric gaps around all grains in sian Federation (“Technology of Low-Dimensional the structure. Objects and Systems”, project no. 40.072.1.1.1178).

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 2000 KOZHEVIN et al.

REFERENCES 10. V. M. Kozhevin, D. A. Yavsin, S. A. Gurevich, et al., in Proceedings of 7th International Symposium on Nano- 1. R. H. Chen and K. K. Likharev, Appl. Phys. Lett. 72, 61 structures: Physics and Technology, St. Petersburg, Rus- (1998). sia (1999), p. 205. 2. V. M. Kozhevin, D. A. Yavsin, M. A. Zabelin, et al., in 11. M. P. Seah and W. A. Dench, Surf. Interface Anal. 1, 2 Proceedings of 10th International Symposium on Nano- (1979). structures: Physics and Technology, St. Petersburg, Rus- 12. B. V. Nekrasov, Fundamentals of General Chemistry sia (2002), p. 41. (Khimiya, Moscow, 1970), Vol. 3. 3. A. L. Buchachenko, Problems and Advances in the 13. A. Yanase, H. Matsue, K. Tanaka, and H. Komiyama, Physics, Chemistry, and Engineering Science of Nano- Surf. Sci. 219, L601 (1989). materials (Moscow, 2002), Vol. 1, p. 15. 14. M. Diociaiuti, A. Bascelli, and L. Paoletti, Vacuum 43, 4. K. Deppert and L. Samuelson, Appl. Phys. Lett. 68 (10), 575 (1992). 1409 (1996). 15. Glassy Metals, Ed. by H. S. Güntherodt and H. Beck 5. B. I. Shklovskiœ and A. L. Éfros, Electronic Properties of (Springer, Berlin, 1981; Mir, Moscow, 1983). Doped Semiconductors (Nauka, Moscow, 1979; Springer, 16. E. Z. Mekhlikhov, Zh. Éksp. Teor. Fiz. 120, 712 (2001) New York, 1984). [JETP 93, 625 (2001)]. 6. B. Abeles, P. Sheng, M. D. Coutts, and Y. Arie, Adv. 17. D. A. Zakgeœm, I. V. Rozhanskiœ, I. P. Smirnova, and Phys. 24, 407 (1975). S. A. Gurevich, Zh. Éksp. Teor. Fiz. 118, 637 (2000) 7. E. Guevas, M. Ortuno, and J. Ruiz, Phys. Rev. Lett. 71, [JETP 91, 553 (2000)]. 1871 (1993). 18. D. A. Zakgeœm, I. V. Rozhanskiœ, and S. A. Gurevich, Pis’ma Zh. Éksp. Teor. Fiz. 70, 100 (1999) [JETP Lett. 8. V. M. Kozhevin, D. A. Yavsin, V. M. Kouznetsov, et al., 70, 105 (1999)]. J. Vac. Sci. Technol. B 18, 1402 (2000). 9. R. van Wijk, P. C. Gorts, A. J. M. Mens, et al., Appl. Surf. Sci. 90, 261 (1995). Translated by G. Skrebtsov

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

Physics of the Solid State, Vol. 45, No. 10, 2003, pp. 2001Ð2011. Translated from Fizika Tverdogo Tela, Vol. 45, No. 10, 2003, pp. 1903Ð1912. Original Russian Text Copyright © 2003 by Pavlov, Lang, Korovin.

LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

An Analog of the Kubo Formula for Conductivity of Spatially Inhomogeneous Media in Spatially Inhomogeneous Electric Fields S. T. Pavlov1, 2, I. G. Lang3, and L. I. Korovin3 1Escuela de Fisica de la UAZ, Apartado Postal c-580, 98060 Zacatecas, Mexico e-mail: [email protected] 2Lebedev Institute of Physics, Russian Academy of Sciences, Leninskiœ pr. 53, Moscow, 117924 Russia 3Ioffe Physicotechnical Institute, Russian Academy of Sciences, Politekhnicheskaya ul. 26, St. Petersburg, 194021 Russia e-mail: [email protected] Received March 27, 2003

Abstract—The average densities of currents and charges induced by a weak electromagnetic field in spatially inhomogeneous systems at a finite temperature are calculated. The Kubo formula for the electrical conductivity tensor is generalized to spatially inhomogeneous systems and spatially inhomogeneous fields. The contributions containing electric fields and derivatives of the fields with respect to coordinates are separated. It is shown that semiconductor quantum wells, wires, and dots can be treated as spatially inhomogeneous systems. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION dependent electric field necessarily depends on the coordinates. Therefore, the use of relationship (1) is In recent years, great interest has been expressed by always an approximation as applied to the field E researchers in experimental and theoretical investiga- dependent on t. In the case of spatially inhomogeneous tions of low-dimensional semiconductor objects, such systems, the field can strongly depend on the coordi- as quantum wells, wires, and dots. In this respect, the nates. development of a basic theory of interaction between electromagnetic fields and spatially inhomogeneous The problem is to take into account the homogeneity systems is an important problem. of a medium and to obtain additional terms involving The Kubo formula for the electrical conductivity derivatives of the electric field with respect to coordi- tensor [1] is applicable for spatially homogeneous sys- nates in the Kubo formula. tems and electric fields independent of spatial coordi- The operator for the interaction of an electromag- nates. This formula correctly accounts for the interac- netic field with a system of charged particles is tion of charge carriers with a medium and, hence, can expressed through the vector A(r, t) and scalar ϕ(r, t) be used in solving particular problems of electrical con- potentials (see, for example, [2, p. 68]) but is not ductivity in the solid-state theory. In the present work, expressed in terms of the electric E(r, t) and magnetic we generalize the Kubo formula to the case of spatially H(r, t) fields (except for specific cases, for example, a inhomogeneous systems and spatially inhomogeneous constant electric field). Correspondingly, the current- fields. For this purpose, we preliminarily calculate the density operator j1(r, t) and the charge-density operator ρ average densities of currents and charges induced by an 1(r, t) within an approximation linear in the vector and electromagnetic field in a spatially inhomogeneous scalar potentials are not expressed through the fields 〈 〉 medium. E(r, t) and H(r, t). However, the averages j1(r, t) and 〈ρ 〉 The Kubo formula [1] was derived using the opera- 1(r, t) should be expressed in terms of the fields, tor for the interaction of charge carriers with the electric because these quantities are observables. At a finite field in the following form: temperature T, the average can be defined by the expression [1, 3, 4] U = Ð∑e r E()t , (1) K i i Sp{}exp()…ÐβᏴ 1 i 〈〉… ==------, β ------, (2) Sp{}exp()ÐβᏴ kT where ei is the charge of the ith particle, ri is the radius vector of the ith particle, and E(t) is a time-dependent where Ᏼ is the Hamiltonian without regard for the but spatially homogeneous electric field. However, it interaction of particles with a weak electromagnetic follows from the Maxwell equations that the time- field.

2002 PAVLOV et al. In our recent study [5], we deduced relationships for The fields are assumed to be classical, and the gauge of 〈 | | 〉 〈 |ρ | 〉 ϕ the averages 0 j1(r, t) 0 and 0 1(r, t) 0 at T = 0, the potentials A and is arbitrary. To generalize the when the average 〈…〉 transforms into the average problem, we assume that the system of particles resides 〈0|…|0〉 over the ground state |0〉. In the present work, in a constant magnetic field Hc that can be relatively we continue our investigation and, hence, use the des- strong. This field is described by the vector potential ignations and many of the results obtained in [5]. Ac(r), so that Hc = curlAc(r). The total Hamiltonian 〈 〉 Ᏼ The problem of expressing the averages j1(r, t) and tot can be written in the form 〈ρ (r, t)〉 in terms of electric and magnetic fields is very 1 1 e e 2 important, because the average 〈j (r, t)〉 expressed Ᏼ = ------∑ P Ð -- A ()r Ð -- Ar(), t 1 tot 2m i c c i c i through the vector and scalar potentials includes the i 〈ρ 〉 (4) term Ð(e/mc) (r) A(r, t), where e = ei is the charge of particles, m = mi s the mass of particles, c is the velocity + V()r ,,… r + e∑ϕ()r , t , of light, and ρ(r) is the charge-density operator within 1 N i the zeroth approximation in the fields [see formula (9)]. i ប ∂ ∂ This term leads to difficulties in solving particular where Pi = ( /i)( / ri ) is the generalized momentum problems, such as the reflection and absorption of light operator and V(r1, …, rN) is the potential energy, in semiconductor quantum wells. These difficulties can including the interaction between particles and the 〈 〉 〈ρ 〉 be avoided if the averages j1(r, t) and 1(r, t) are external potential. The noncommutativity of Pi with expressed in terms of the electric fields and their deriv- Ac(ri ) and A(ri , t) should be taken into account in rela- atives with respect to coordinates. tionship (4). Next, the interaction energy U of particles The question as to what kind of interaction (involv- with the electromagnetic field is separated in the total ing the vector potential or the electric field) is more Hamiltonian (4) and the interaction with the strong convenient to use was previously discussed in [6] as magnetic field is included in the main Hamiltonian; applied to the problem of light scattering in bulk crys- that is, tals. Zeyher et al. [6] were likely the first to change over Ᏼ Ᏼ tot = + U, (5) from the velocity operators vi to the coordinate operators ri of particles according to the quantum relation- where ship v = (i/ប)[Ᏼ, r ]. As a result, these authors suc- i i 1 2 ceeded in changing over from expressions containing Ᏼ = ------∑p + V()r ,,… r , 2m i 1 N the vector potential to expressions including the electric i (6) field. However, since we need to solve other problems e () of spatially inhomogeneous systems, it is necessary to pi = Pi Ð -- Ac ri , return to this question. c

We will consider the case of a finite temperature and UU= 1 + U2, use the mathematical approaches proposed in [1, 3]. Our results will be compared with the inferences made 1 3 () (), 3 ρ()ϕ(), U1 = Ð ---∫d rjrArt + ∫d r r r t , by Konstantinov and Perel’ [4], who developed the c (7) quantum theory of spatial dispersion of the electric and e 3 ρ() 2(), magnetic susceptibilities. It is assumed that charges and U2 = ------d r r A r t . 2mc∫ currents are absent at infinity and the fields E(r, t) and H(r, t) are equal to zero at t Ð∞, which corresponds Here, we also introduce the current-density operator to their adiabatic switching-on. () () jr = ∑ ji r , 2. HAMILTONIAN OF THE SYSTEM i () ()δ[]() δ() AND THE CURRENT-DENSITY ji r = e/2 rrÐ i vi + vi rrÐ i , AND CHARGE-DENSITY OPERATORS vi = pi/m, Let us consider a system composed of N particles and the charge-density operator with a charge e and a mass m in an arbitrary weak electromagnetic field characterized by the intensities E(r, t) ρ()r ==∑ ρ()r , ρ ()r eδ()rrÐ . and H(r, t). We introduce the vector A(r, t) and scalar i i i ϕ(r, t) potentials through which the fields can be i ρ expressed in the form The operators ji(r) and i(r) are related by the continuity equation 1∂Ar(), t Er(), t = Ð------Ð grad ϕ()r, t , () ρ() c ∂t div ji r +0,ú i r = (3) (8) ρ () ()ប []Ᏼ, ρ() Hr(), t = curl Ar(), t . ú i r = i/ r ,

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 AN ANALOG OF THE KUBO FORMULA 2003 〈 | | 〉 〈 |ρ | 〉 which will be used in further calculations. Note that the the averages 0 j1α(r, t) 0 and 0 1(r, t) 0 and operator U2 will not appear below, because it is qua- obtained the relationships dratic in the vector potential and its inclusion is beyond 〈| (), |〉 the scope of an approximation linear in the fields. 0 j1α r t 0 〈| (), |〉 〈| (), |〉 Additions linear in the potentials A(r, t) and ϕ(r, t) = 0 j1α r t 0 E + 0 j1α r t 0 ∂E/∂r, (12) to the current-density and charge-density operators in 〈|ρ0 ()r, t |〉0 = 〈|ρ0 ()r, t |〉0 + 〈|ρ0 ()r, t |〉0 ∂ ∂ , the Heisenberg representation can be written as fol- 1 1 E 1 E/ r lows: where the subscripts E and ∂E/∂r designate the terms containing the electric field and the derivatives of the e field with respect to the coordinates, respectively. When j α()r, t = Ð------ρ()r, t Aα()r, t 1 mc turning to consideration of the system at a finite temper- t ature, we replace the averaging 〈0|…|0〉 over the ground i state in formulas (12) by the averaging 〈…〉 defined by + --- dt'[]U ()t' , jα()r, t , ប ∫ 1 (9) expression (2). The correctness of this replacement will Ð∞ be proved by comparing our data with the results t obtained in [1, 3, 4]. ρ (), i []()ρ, (), 1 r t = ប--- ∫ dt' U1 t' r t , According to [5], we have Ð∞ i 3 〈〉j α()r, t = --- d r' where subscript 1 indicates the approximation linear in 1 E ប∫ ρ the fields; (r, t), j(r, t), and U1(t) are the operators in t (13) the interaction representation, for example, × 〈〉[](), , (), (), ∫ dt' jα r t dβ r' t' Eβ r' t' , ρ()r, t = exp()ρiᏴt/ប ()r exp()ÐiᏴt/ប ; Ð∞

i 3 and [F, Q] = FQ Ð QF. Substitution of expression (7) for 〈〉ρ (), --- 1 r t E = ប∫d r' U1 into formulas (9) gives t (14) e × 〈〉[]ρ (), , (), (), j α()r, t = Ð------ρ()r, t Aα()r, t ∫ dt' α r t dβ r' t' Eβ r' t' , 1 mc Ð∞ t i 3 [](), , (), (), e ∂aβ()r, t i 3 + ------∫d r' ∫ dt' jα r t jβ r' t' Aβ r' t' 〈〉(), ------〈〉()------បc (10) j1α r t ∂E/∂r = dβ r ∂ Ð ប ∫d r' Ð∞ mc rα c t t (15) ∂aβ()r', t' i 3 []ϕ()ρ, , (), (), × 〈〉[](), , (), Ð ---∫d r' ∫ dt' jα r t r' t' r' t' , ∫ dt' jα r t Yβγ r' t' ------, ប ∂rγ' Ð∞ Ð∞

t 〈〉ρ (), i 3 i 3 1 r t ∂E/∂r = Ð------∫d r' ρ ()r, t = ------d r' dt'[]ρ()r, t , jβ()r', t' Aβ()r', t' បc 1 បc∫ ∫ Ð∞ t (16) (11) ∂aβ()r', t' t × 〈〉[]ρ(), , (), ∫ dt' r t Yβγ r' t' ------. i 3 ∂ ' Ð ------d r' dt'[]ϕρ()ρr, t , ()r', t' ()r', t' . rγ បc∫ ∫ Ð∞ Ð∞ Here, we introduced the designations () ρ() () () 3. AVERAGES OF THE INDUCED DENSITIES dr ==r r , Yβγ r rβ jγ r , (17) OF CURRENTS AND CHARGES t In our previous work [5], we considered the problem ar(), t = Ðc ∫ Er(), t' dt'. (18) at T = 0 and operators (10) and (11) averaged over the Ð∞ ground state of the system. In [7, p. 84], it was shown that the averaging should be performed using the wave Now, we rearrange the derived relationships in such functions |0〉 of the ground state without regard for the a way as to obtain the Kubo formula for spatially homo- interaction U. In [5], we rearranged the expressions for geneous systems and an electric ﬁeld independent of

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 2004 PAVLOV et al. coordinates. For this purpose, we invoke the expression As a result, the first term contains the electric field and [1, 3] the second term involves the derivatives of the field with respect to the coordinates; that is, β i dQt()' --- 〈〉λ[]Ft(), Qt()' = ∫d ------Ft()+ iបλ , (19) t ប dt' 〈〉(), ()1 3 0 j1α r t = ∫d r' ∫ dt' Ð∞ which is valid for any pair of operators F and Q. With β (25) the use of expression (19), from formula (13), we find ×λ〈〉(), (), បλ (), ∫d jβ r' t' jα r ti+ Eβ r' t' , t 0 〈〉(), 3 j1α r t E = ∫d r' ∫ dt' () ∂aβ()r, t Ð∞ 〈〉(), 2 e 〈〉() (20) j1α r t = ------dβ r ------β mc ∂rα ∂dβ()r', t' ×λ (), បλ (), β (26) ∫d ------∂ jα r ti+ Eβ r' t' . ∂ (), t' 1 3 λ 〈〉() (), បλ aβ r' t 0 Ð ---∫d r'∫d Yβγ r' jα r i ------. c ∂rγ' It can be shown that the following equality holds: 0 It is evident that relationship (24) does not coincide ∂ jγ ()r, t ∂sβ()r, t /∂tr= Ð β------. (21) with expression (12), which is convenient only at T = 0. ∂rγ In a similar way, from expressions (14) and (16), we Substitution of formula (21) into relationship (20) and find integration over the variable r' by parts give () () 〈〉ρρ ()r, t = 〈〉()r, t 1 + 〈〉ρ ()r, t 2 , (27) 〈〉(), 1 1 1 j1α r t E t β t 3 λ 〈〉(), (), បλ (), 〈〉ρ (), ()1 3 = ∫d r' ∫ dt'∫d jβ r' t' jα r ti+ Eβ r' t' 1 r t = ∫d r' ∫ dt' Ð∞ 0 (22) Ð∞ β β (28) t ∂ (), 3 λ 〈〉(), (), បλ Eβ r' t' ×λ〈〉()ρ, (), បλ (), + ∫d r' ∫ dt'∫d Yβγ r' t' jα r ti+ ------. ∫d jβ r' t' r ti+ Eβ r' t' , ∂rγ' Ð∞ 0 0

〈 〉 ()2 Expression (15) for j1α(r, t) ∂E/∂r involves two terms. 〈〉ρ (), 1 r t The first term remains unchanged. In the second term, β we perform integration over the variable t' by parts and ∂ (), (29) 1 3 λ 〈〉()ρ(), បλ aβ r' t use expression (19). As a result, we obtain = Ð---∫d r'∫d Yβγ r' r i ------. c ∂rγ' ∂ (), 0 〈〉(), e 〈〉() aβ r t j1α r t ∂E/∂r = ------dβ r ------mc ∂rα After the derivation of relationships (24)Ð(29), we β achieved our objective. To be more specific, we sepa- ∂ (), 1 3 λ 〈〉() (), បλ aβ r' t rated the basic contributions with the superscript (1) to Ð ---∫d r'∫d Yβγ r' jα r i ------c ∂rγ' (23) the average induced densities of currents and charges 0 and showed that the additional contributions with the t β superscript (2) involve derivatives of the electric field ∂ (), 3 λ 〈〉(), (), បλ Eβ r' t' with respect to the coordinates. The reason for separat- Ð ∫d r' ∫ dt'∫d Yβγ r' t' jα r ti+ ------. ∂rγ' ing the contributions into basic and additional ones is Ð∞ 0 that, when solving problems with a spatially inhomogeneous field E(r, t), it is possible to estimate the addi- After combining expressions (22) and (23), the last tional contributions and to determine whether they terms on the right-hand side of both relationships can- should be allowed for or can be rejected. As in the case cel each other out. The resultant expression can be sep- of T = 0, the obtained relationships contain the coordi- arated into two terms, nate operators ri of particles, in contrast to the initial expressions (10) and (11), which do not involve these 〈〉(), 〈〉(), ()1 〈〉(), ()2 j1α r t = j1α r t + j1α r t . (24) operators.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 AN ANALOG OF THE KUBO FORMULA 2005

4. COMPARISON WITH THE RESULTS and, similarly, OBTAINED IN [4] 〈〉ρ (), ()2 1 r t Nakajima [3] and Konstantinov and Perel’ [4] β 〈 〉 (33) obtained expressions for j1(r, t) that differ from our 1 3 = --- d r' dλ 〈〉jβ()ρr' ()r, iបλ aβ()r', t . expressions derived in the present work. Moreover, we c∫ ∫ 〈ρ carried out similar calculations for the quantity 1(r, 0 t)〉, which was not considered in [1, 3, 4], even though Thus, there exist two types of formulas for the addi- this quantity and the average current density enter into tional contributions: formulas (26) and (29), which the Maxwell equations on equal terms. Note that the contain the derivatives of the electric field with respect authors considered a homogeneous medium in [1, 3] to the coordinates, and formulas (32) and (33), which and an inhomogeneous medium in [4]. By solving the involve the electric field. equation for the density matrix, the authors of [3, 4] Let us demonstrate how expression (26) can be derived the formula that can be obtained from expression 〈 〉 transformed into relationship (32) and expression (29) (10) if the averaging … defined by relationship (2) is can be rearranged to give relationship (33). In the last performed on the right- and left-hand sides. It is evident 〈ρ 〉 term in expression (26), we perform integration over that a similar expression for 1(r, t) can be deduced through the averaging on both sides of formula (11). the variable rγ' by parts. Then, we use the equality () ∂ () Then, the authors of [3, 4] rearranged the expression dYβγ r jγ r ------==jβ()r + rβ------jβ()r Ð rβρú ()r . (34) for the average induced current density in such a way drγ ∂rγ that the electric field E(r, t) appears but the vector potential A(r, t) is also retained in this expression. An As a result, from expression (26), we have analogous procedure performed with the initial rela- ∂ (), 〈〉(), ()2 e 〈〉() aβ r t tionship for the average induced charge density results j1α r t = ------dβ r ------∂ - in an expression similar to formula (27) in which the mc rα 〈ρ 〉(1) β contribution 1(r, t) is defined by formula (28) and (2) 1 3 the contribution 〈ρ (r, t)〉 is written in the form + --- d r' dλ 〈〉jβ()r' jα()r, iបλ aβ()r', t 1 c∫ ∫ (35) 0 ()2 〈〉ρ (), β 1 r t 1 3 β ' λρ〈〉() (), បλ (), (30) Ð ---∫d r'rβ ∫d ú r' jα r i aβ r' t . 1 3 c = --- d r' dλ 〈〉jβ()ρr' ()r, iបλ Aβ()r', t . 0 c∫ ∫ 0 In the last term, we use relationship (19) and integrate over the variable r', which leads to the formula In [3, 4], expression (25) was obtained for the contribu- 〈 〉(1) tion j1α(r, t) on the right-hand side of relationship (24) ie Ð------jα()r , ∑r βaβ()r , t and the formula បc i i i (36) 〈〉(), ()2 e 〈〉ρ() (), ∂ (), j1α r t = Ð------r Aα r t e aβ r t mc = Ð------〈〉ρ()r aα()r, t +.〈〉dβ()r ------mc ∂rα β (31) 1 3 Substituting this formula into relationship (35) gives + --- d r' dλ 〈〉jβ()r' jα()r, iបλ Aβ()r', t c∫ ∫ expression (32), which is the required result. Relation- 〈ρ 〉(2) 0 ship (29) for 1(r, t) is rearranged in a similar manner. The only difference is that, instead of formula (36), was derived for the additional contribution. we have Furthermore, in [4], the time derivative of the con- (2) tribution 〈 j α(r, t)〉 was expressed through the electric ie ρ(), (), 1 Ð0------r ∑riβaβ ri t = field with the use of expression (31). By integrating this បc derivative over the time, we obtain i and transform expression (29) into relationship (33). ()2 e Thus, a comparison of our data with the results 〈〉j α()r, t = Ð------〈〉ρ()r aα()r, t 1 mc obtained in [3, 4] shows the applicability of expres- β (32) sions (26) and (29) for the additional contributions to 1 3 the average induced current and charge densities, which + --- d r' dλ 〈〉jβ()r' jα()r, iបλ aβ()r', t c∫ ∫ contain only the derivatives of the electric field with 0 respect to the coordinates.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 2006 PAVLOV et al. 5. ANALYSIS OF THE FORMULAS field. We separate relationship (38) into two parts: FOR AN ADDITIONAL CONTRIBUTION 〈〉(), ()2 〈〉(), ()Ð 〈〉(), ()+ TO THE AVERAGE CURRENT DENSITY j1α r t = j1α r t + j1α r t , (41) The three expressions (26), (31), and (32) were ()± e ∂Aβ()r, t ∂Aα()r, t 〈 〉(2) 〈〉(), ------〈〉() ± ------derived for the quantity j1α(r, t) . Some properties of j1α r t = dβ r ------2mc ∂rα ∂rβ this quantity are given in [4]. Now, we deduce the (2) β fourth expression for 〈 j α(r, t)〉 through the deriva- 1 1 3 tives of the vector potential A(r, t) with respect to the Ð ------d r' dλ 〈〉Yβγ()r' jα()r, iបλ (42) 2c∫ ∫ coordinates. In the second term in formula (31), rela- 0 tionship (34) is used for jβ(r'). As a result, this second ∂ (), ∂ (), × Aβ r' t ± Aγ r' t term is separated into two terms. In the first term, we ------. ∂rγ' ∂rβ' carry out integration over the variable rγ' by parts. In 〈 〉(Ð) the second term, we use the relationship Since H = curlA, the quantity j1α(r, t) can be expressed through the magnetic field; that is, β i --- 〈〉λ[]FQ, = d 〈〉Qú Fi()បλ , (37) 〈〉(), ()Ð e ()(), × 〈〉ρ() ប ∫ j1α r t = Ð------Hrt r α r 0 2mc β (43) which follows from expression (19) at t' = t. By calcu- 1 3 + ------d r' dλ()Hr()', t × r' β 〈〉jβ()r' jα()r, iបλ . lating the commutator, we obtain 2c∫ ∫ 0 ∂ (), ()2 e Aβ r t 〈 〉(+) 〈〉(), 〈〉() The quantity j1α(r, t) can be expressed in terms of j1α r t = ------dβ r ------mc ∂rα the second derivatives of the vector potential with β (38) respect to the coordinates in the following way: 1 3 ∂Aβ()r', t' Ð --- d r' dλ 〈〉Yβγ()r' jα()r, iបλ ------. 2 ∫ ∫ ∂ () ∂ Aβ c rγ' 〈〉(), + e 〈〉ρ() 0 j1α r t = Ð------r rβrγ ------2mc ∂rα∂rγ β (44) In order to derive the fifth expression for the addi- 2 ∂ Aβ tional contribution to the current density, we use rela- 1 3 λ 〈〉() (), បλ + ------∫d r'rβ' rδ' ∫d jγ r' jα r i ------∂ ∂ . tionship (34) for jα(r, iបλ) in formula (31) and the fol- 2c rγ' rδ' 0 lowing modification of formula (37): Relationship (44) was obtained from expression (42) as β follows. In formula (42), we used the relationship i 〈〉λ[], 〈〉ú()បλ Yβγ(r') = rβ' jγ (r') and expression (34) for jγ(r'). Then, in ប--- FQ = ∫d FQ i . (39) 0 the term containing ∂Yγδ(r')/∂rδ' , we performed integra- ' Then, we obtain tion over the variable rδ by parts. In the term involving dúγ (r'), we used expression (39), integrated over the ∂ 〈〉(), ()2 e {}〈〉() (), variable r', and calculated the commutator. j1α rt = Ð------∂ dα r Aβ r t mc rβ 〈 It can be shown that the individual quantities j1α(r, β (40) 〉(+) 〈 〉(Ð) t) and j1α(r, t) possess the properties 1 ∂ 3 + ------d r' dλ 〈〉jβ()r' Y γ ()r, iបλ Aβ()r', t . ∂ ∫ ∫ a ()± ()± c rγ 〈〉(), 3 〈〉(), 0 div j1 r t ==0, ∫d rj1α r t 0. (45)

〈 〉(+) From expression (40), it follows that the integral of Now, we demonstrate that the contribution j1α(r, t) 〈 〉(2) j1α(r, t) over the entire space is equal to zero [4]. can be expressed through the second derivatives of the electric ﬁeld with respect to the coordinates. For this purpose, the relationship for the vector potential [fol- 6. DERIVATION OF EXPRESSIONS INVOLVING lowing from formula (3)]

A MAGNETIC FIELD t Let us derive the sixth expression for the additional Ar(), t = ar(), t Ð ct∫ d '∂ϕ()r, t' /∂r (46) 〈 〉(2) contribution j1α(r, t) and introduce the magnetic Ð∞

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 AN ANALOG OF THE KUBO FORMULA 2007 is substituted into expression (42). Then, in the quanti- 7. ELIMINATION OF THE DIAGONAL 〈 〉(±) ties j1α(r, t) , it is possible to separate the contribu- ELEMENTS OF THE ri OPERATORS tions of the scalar potential Unlike expressions (32) and (33), relationships (26) ()± ()± ()± and (29) for the additional contributions to the average 〈〉j α()r, t = 〈〉j α()r, t + 〈〉j α()r, t , (47) 1 1 E 1 ϕ induced current and charge densities involve the coordi- which are designated by the subscript ϕ. nate operators ri of particles. Actually, deﬁnitions (17) From these relationships, it immediately follows can be rewritten in the form that () δ() dr = e∑ri rrÐ i , (53) 〈〉(), ()Ð j1α r t ϕ = 0, (48) i

t 2 ()+ e ∂ ϕ()r, t' 〈〉(), 〈〉() Yβγ = ()e/2 ∑()r β j γ + j γ r β . (54) j1α r t ϕ = ∫ dt' Ð---- dβ r ------∫ i i i i m ∂rα∂β ∞ i Ð (49) β However, the averages 〈 j (r, t)〉 and 〈ρ (r, t)〉 should ∂2ϕ 1 1 3 λ 〈〉() (), បλ not depend on the position of the reference point of the + ∫d r'∫d Yβγ r' jα r i ------. ∂rβ' ∂rγ' coordinates ri. This means that relationships (26) and 0 (29) contain only off-diagonal elements of the opera- In the second term in formula (49), we doubly integrate tors ri and the diagonal elements can be eliminated. by parts: initially, over the variable rγ' and then over the Now, we demonstrate that this is the case. variable rβ' . Finally, with the use of the continuity equa- For this purpose, we transform expression (54) so tion (8) and relationship (37), we find that the second that the operator riβ appears only on the left-hand side; term in expression (49) is equal to the first term with the that is, opposite sign and () ()δប ρ() () Yβγ r = Ð i /2m βγ r + ∑riβ jiγ r . (55) 〈〉(), ()+ j1α r t ϕ = 0. (50) i With due regard for expressions (48) and (50), from Substitution of expressions (53) and (55) into rela- relationships (47) and (42), we derive tionship (26) gives ()± ()± ()2 〈〉(), 〈〉(), 〈〉α(), j1α r t = j1α r t E j1 r t ∂ (), ∂ (), β e aβ r t aα r t iប 3 ------〈〉β() ± ------λρ〈〉() (), បλ (), = d r ------∂ = ------d r' d r' jα r i div ar' t 2mc ∂rα rβ 2mc∫ ∫ β 0 (51) 1 3 2 λ 〈〉() (), បλ ∂aβ()r, t (56) Ð ------∫d r'∫d Yβγ r' j1α r i e 〈〉δ() 2c + ------∑ riβ rrÐ i ------∂ - 0 mc rα i ∂aβ()r', t ∂aγ ()r', t β × ------± ------. ∂ (), ∂ ∂ 1 3 λ 〈〉() (), បλ aβ r' t rγ' rβ' Ð ---∫d r'∫d ∑ riβ jiγ r' j1α r i ------. c ∂rγ' By using the Maxwell equation curlE = Ð(1/c)(∂H/∂t) 0 i and definition (18) of the vector a(r, t), we are again led In formula (56), the first two terms remain unchanged 〈 〉(Ð) to expression (43) for j1α(r, t) and also obtain and the operator ri in the last term is separated into two 2 parts: ()+ e ∂ aβ()r, t 〈〉(), 〈〉ρ() d nd j1α r t = Ð------r rβrγ ------2mc ∂rα∂rγ ri = ri + ri , (57) β (52) ∂2 (), where superscripts “d” and “nd” indicate the diagonal 1 3 λ 〈〉() (), បλ aβ r' t and off-diagonal contributions, respectively. The oper- + ------∫d r'∫d rβ' rγ' jγ r' jα r i ------. 2c ∂rγ' ∂rδ' d 0 ator ri is determined through the matrix elements: According to definition (18), we find that the quantity 〈| d|〉 〈| |〉〈| 〉 〈 〉(+) n ri m = n ri m nm , (58) j1α(r, t) is expressed through the second derivatives of the electric field with respect to the coordinates. As a where |n〉 are the eigenfunctions of the Hamiltonian Ᏼ. result, the sixth (and last) formula for the additional The commutativity property (2) contribution 〈 j α(r, t)〉 is determined by the combina- 1 []Ᏼ, d tion of expressions (43) and (52). ri = 0.

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 2008 PAVLOV et al. is obvious. Let us consider the contribution of the oper- Integration over the variable r' and calculation of the d commutator give the relationship ator ri to the last term in expression (56). This contri- 2 ∂aβ()r, t bution will be denoted as Iα(r, t). We perform integra- (), e 〈〉δ() Iα' r t = Ð------∑ riβ rrÐ i ------, (65) ' mc ∂rα tion over the variable rγ by parts, use the continuity i equation (8), and obtain which is canceled by the second term in expression

β (56). Then, we rearrange the remaining quantity Iα'' (r, (), ()3 λ Iα r t = Ð 1/c ∫d r'∫d t) with the use of the continuity equation (8), integrate over the variable rγ' by parts, and obtain 0 (59) β d 3 × ∑ 〈〉r β ρú ()r' jα()r, iបλ aβ()r', t . (), 1 λ i i Iα'' r t = ---∫d r'∫d i c 0 (66) d ∂aβ()r', t Since the operator ri commutes with the Hamiltonian × 〈〉() (), បλ nd()បλ ∑ jiγ r' jα r i riβ i ------. Ᏼ, relationship (59) can be rewritten in the following ∂rγ' form: i By using relationships (61), (65), and (66), we derive β the ﬁnal expression for the additional contribution to (), ()3 λ Iα r t = Ð 1/c ∫d r'∫d the average induced current density without diagonal matrix elements of the operator ri; that is, 0 (60) ()2 iប 3 〈〉j α()r, t = ------d r' ×ρ〈〉() (), បλ d ()បλ (), 1 ∫ ∑ ú i r' jα r i riβ i aβ r' t . 2mc β i ×λρ〈〉() (), បλ (), d nd ∫d r' jα r i div ar' t Substitution of the expression r β = r β Ð r β into for- i i i 0 mula (60) leads to separation of the contribution Iα(r, t) β (67) into two parts: 1 3 nd + --- d r' dλ〈∑ j γ ()r' jα()r, iបλ r β ()iបλ c∫ ∫ i i n i Iα()r, t = Iα' ()r, t + Iα ()r, t , (61) 0 ∂ (), nd () ()〉, បλ aβ r' t where Ð riβ jiγ r' jα r i ------. ∂rγ' β 3 Similarly, we obtain Iα' ()r, t = Ð()1/c d r' dλ ∫ ∫ ()2 iប 3 〈〉ρ()r, t = ------d r' 0 (62) 2mc∫ β ×ρ∑ 〈〉ú ()r' jα()r, iបλ r β()iបλ aβ()r', t , i i ×λρd 〈〉()ρr' ()r, iបλ div ar()', t i ∫ 0 β β (68) 3 3 nd Iα''()r, t = ()1/c d r' dλ 1 ∫ ∫ + --- d r' dλ〈∑ j γ ()ρr' ()r, iបλ r β ()iបλ c∫ ∫ i i 0 (63) 0 i ∂aβ()r', t ×ρ〈〉() (), បλ nd()បλ (), nd ()ρ()〉, បλ ∑ ú i r' jα r i riβ i aβ r' t . Ð riβ jiγ r' r i ------. ∂rγ' i With due regard for relationship (37), expression (62) transforms into the formula 8. THE ELECTRICAL CONDUCTIVITY TENSOR The Fourier transform of the electric ﬁeld can be

Iα' ()r, t represented as ∞ (64) ()ω ()ប 3 〈〉[]() , ρ() (), (), ω 3 (), Ði kr Ð t = Ð i/ c ∫d r'∑ jα r riβ r' aβ r' t . Eα k = ∫d rtE∫ d α r t e , (69) i Ð∞

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 AN ANALOG OF THE KUBO FORMULA 2009

Eα()r, t 2 II e kα nd ∞ σαβ()k, ω r = ------∑〈|0 r β δ()rrÐ |〉0 mω i i Ð4 3 i()kr Ð ωt (70) i = ()2π d k dωEα()k, ω e + c.c. ∫ ∫ ∞ (77) 0 ikγ 3 Ði()kr' Ð ωt 〈| (), ()nd Ð ------∫d r'∫dte ∑ 0 jα r t jiγ rrÐ ' riβ The average induced current density can be written in បω the following form: 0 i nd ()(), |〉 ∞ Ð riβ jiγ rrÐ ' jα r t 0 , 〈〉(), ()π Ð4 3 ωσ (), ω j1α r t = 2 ∫d k∫d αβ k r ∞ (71) III kβ 3 Ði()kr' Ð ωt 0 σ (), ω αβ k r = ------ω-∫d r'∫dte i()kr Ð ωt 2m (78) × Eβ()k, ω e + c.c., 0

× 〈|0 []jα()ρr, t , ()rrÐ ' |〉0 , where σαβ(k, ω|r) is the electrical conductivity tensor + dependent on the spatial coordinates (the designation following from [8]). and [F, Q]+ = FQ + QF. Expression (75) transforms into relationship (72) when the averaging 〈0|…|0〉 in formu- From expressions (25) and (67) for the basic and las (76)Ð(78) is replaced by the averaging 〈…〉 and for- additional contributions to the average induced current mula (37) is used.1 density, we obtain σ ()σ, ω ()1 ()σ, ω ()2 (), ω αβ k r = αβ k r + αβ k r , (72) 9. APPROXIMATION OF A SPATIALLY where HOMOGENOUS ELECTRIC FIELD σ ()1 (), ω In some cases, we can disregard the contributions αβ k r containing the derivatives of the electric field with ∞ β respect to the coordinates to the average induced cur- 3 λ 〈〉(), បλ (), rent and charge densities; i.e., we can assume that = ∫d r'∫dt∫d jβ rrÐ ' Ði jα r t (73) 0 0 Er(), t Ӎ E()t . (79) × exp[]Ði()kr' Ð ωt , For example, this was done in [1], even though, in the ()2 σαβ ()k, ω r strict sense, only a time-independent field E can be homogeneous in space. Under assumption (79), we β take the Fourier transform iបkβ 3 Ðikr' = ------d r'e dλρ〈〉()rrÐ ', Ðiបλ jα()r 2mω∫ ∫ ∞ 0 ()ω iωt () β (74) Eα = ∫ dte Eα t . (80) kγ 3 Ðikr' λ〈(), បλ ()nd Ð∞ + ----ω ∫d r'e ∫d ∑ jiγ rrÐ ' Ði jα r riβ 0 i Hence, we can write nd Ð r β ()Ðiបλ j γ ()rrÐ ', Ðiបλ jα()〉r . 〈〉(), i i j1α r t ប At T = 0, instead of formula (72), we have ∞ Ðiωt (81) σ (), ω = ()ωσ1/2π d αβ()ω r Eβ()ω e + c.c., αβ, 0 k r ∫ I II III (75) 0 = σαβ()σk, ω r ++αβ()σk, ω r αβ()k, ω r , where the subscript h indicates the spatially homoge- where neous field. It is evident that the following equality ∞ holds: I i 3 Ði()kr' Ð ωt σ (), ω σ ()σω (), ω αβ k r = ប---∫d r'∫dte αβ r = αβ k = 0 r . (82) 0 1 (76) In our earlier work [5], the formula for the electrical conductivity × 〈| ()ρ, ()nd tensor at T = 0 does not coincide with formula (75). This is ∑ 0 jα r t i rrÐ ' riβ 〈 | | 〉 explained by the fact that only the diagonal elements 0 ri 0 are 〈 | | 〉 i eliminated and the operator ri = ri Ð 0 ri 0 differing from the ndρ ()(), |〉 nd Ð riβ i rrÐ ' jα r t 0 , operator ri is introduced in [5].

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 2010 PAVLOV et al. Next, from expressions (72)Ð(74), we obtain sion (26) can be replaced by relationship (38). Substi- ∞ β tution of formula (86) into expression (67) gives ω σ ()ω λ 〈〉()បλ (), i t 2 αβ r = ∫dt∫d Jβ Ði jα r t e , (83) e nd 〈〉j α()r, t = Ð------∑ 〈〉()Hr× αδ()rrÐ 0 0 1 2mc i i i where we introduced the current operator ie 〈〉() ()nd × nd Ð ------∑ jα r Hβ ri ri β (87) 2បc Jα = er∑ úiα. (84) i i β e nd Expression (83) is a generalization of the Kubo formula + ------dλ∑ 〈〉()Hr× βv β jα()r, iបλ . 2c∫ i i to the case of a spatially inhomogeneous medium when 0 i the electrical conductivity tensor depends on r. nd Let us consider a spatially homogeneous medium in Note that, since the projections riα with different sub- which averages cannot depend on the coordinates r. In scripts α do not commute with each other, the condition this case, the tensor σαβ(ω|r) does not depend on r and, nd nd for the vector r × r ≠ 0 is satisfied and we have, for from relationship (83), we have i i example, ∞ β σ ()ω Ð1 λ 〈〉()បλ () iωt ()r nd × r nd = []r nd, r nd . (88) αβ = V 0 ∫dt∫d Jβ Ði Jα t e , (85) i i z ix iy 0 0 where V0 is the normalized volume. The derived for- 11. CONCLUSIONS mula coincides with the Kubo formula [1] if ω on the ω The main results obtained in this work can be sum- right-hand side is replaced by Ð and it is remembered marized as follows. It was demonstrated that the aver- that V0 = 1 in [1]. age densities of the current and charge induced by a weak electromagnetic field in spatially inhomogeneous systems at finite temperatures can be expressed in terms 10. THE CASE OF A CONSTANT MAGNETIC of electric fields and their derivatives with respect to FIELD coordinates. The contributions expressed through the We now analyze the case when the external weak electric field were referred to as the basic contributions, electromagnetic field is reduced to a magnetic field and the contributions expressed through the derivatives H(r, t) = H that is constant in space and time and the of the electric field were termed the additional contribu- electric field E(r, t) = 0. Recall that the vector potential tions. Ac(r) corresponding to the constant magnetic field Hc Six pairs of different expressions were derived for was included in Hamiltonian (6). In this section, we the additional contributions to the average induced cur- assume that Hc = 0, Ac(r) = 0, and the field H = const is rent and charge densities. Two of these expressions for so weak that only the contributions linear in field to the the current density coincide with those obtained in [4]. induced current and charge densities can be taken into However, the expressions determined in [4] contain the account. Then, according to formula (25), we have electric fields or the vector potentials rather than their 〈 〉(1) j1(r, t) = 0 for the main contribution to the induced derivatives with respect to the coordinates, which com- current density, because E(r, t) = 0. The vector poten- plicates the estimation of the additional contributions. tial can be chosen in the form In general, integration over the variable r' by parts makes it possible to eliminate the derivatives Ar()= ()1/2 ()Hr× . (86) ∂Eβ(r', t)/∂rγ' and to obtain formulas that involve only 〈 〉(+) It follows from expression (44) that j1(r, t) = 0, the fields rather than their derivatives. However, the because this expression contains the second derivatives reverse procedure, namely, the changeover from the of A(r) with respect to the coordinates. As a result, only field to the derivatives, is not necessarily possible and 〈 〉(Ð) the contribution j1(r, t) given by relationship (43) the fields are always retained in the basic contributions. turns out to be nonzero. Thus, we succeeded in express- The sixth expression (deduced in Section 6) for the ing the induced current density through the magnetic additional contribution to the average induced current field intensity at H = const in the linear-field approxi- density includes two terms. The first term with the mation. Now, the diagonal matrix elements of the oper- index (Ð) is expressed only through the magnetic field ators ri can be eliminated from the expression for H(r, t), and the second term with the index (+) is 〈 〉 j1(r, t) at H = const. For this purpose, we use relation- expressed through the second derivatives of the electric ship (67), in which the vector a(r, t) can be replaced by field with respect to the coordinates. A similar result the vector potential A(r, t), because the initial expres- was obtained for the average induced charge density. In

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 AN ANALOG OF THE KUBO FORMULA 2011 the case when the additional contributions are well width) and, instead of the small factor v/c, there expressed in terms of the derivatives of the electric appears the factor fields, the corresponding formulas necessarily involve v vλ the coordinate operators r of particles. This result can M Ӎ ------= ------. i dω 2πcd seem absurd, because the coordinate ri depends on the reference point. However, as was shown in Section 7, At wavelengths λ ӷ d, the factor M can be considerably 〈 | | 〉 the diagonal matrix elements n ri n do not enter into larger than v/c. In specific cases, it is necessary to esti- the expressions for the average induced current and mate the magnitude of this factor. In [11, 12], it was charge densities and the off-diagonal elements do not assumed that M Ӷ 1 and the additional contributions to depend on the reference point. the average induced current and charge densities were disregarded (at T = 0). The basic and additional contributions to the electri- In principle, substitution of the obtained expressions cal conductivity tensor for spatially inhomogeneous for the average current and charge densities into the systems and spatially inhomogeneous fields were cal- Maxwell equations makes it possible to determine the culated in Section 8. It was shown that only the basic true fields inside and outside semiconducting low- contributions are retained within the approximation of dimensional objects. This provides a way of calculating spatially homogeneous electric fields dependent on the coefficients of reflection and absorption of light by time (Section 9). In this approximation, the modified these objects (see, for example, [11, 12]). Kubo formula was derived for the electrical conductivity tensor dependent on the frequency ω and the coordinates r. This formula transforms into the Kubo for- ACKNOWLEDGMENTS mula [1] for spatially homogeneous systems. We are grateful to M. Harkins for careful reading of Finally, the expression for the average induced cur- the manuscript and critical remarks. rent density in the case when the weak electromagnetic field is reduced to the constant magnetic field was REFERENCES obtained in Section 10. 1. R. J. Kubo, Phys. Soc. Jpn. 12 (6), 570 (1957); R. Kubo, From relationship (74), it follows that the additional in Problems of Quantum Theory of Irreversible Pro- contributions to the electrical conductivity contain the cesses, Ed. by V. L. Bonch-Bruevich (Inostrannaya Lit- eratura, Moscow, 1961). factor kγ/ω. When the field E(r, t) is considered a plane 2. L. D. Landau and E. M. Lifshitz, The Classical Theory wave propagating at the velocity of light (under mono- of Fields, 6th ed. (Nauka, Moscow, 1973; Pergamon, chromatic irradiation) or a wave packet (under pulsed Oxford, 1975). irradiation), k Ӎ ω/c and the additional contributions 3. S. Nakajima, Proc. Phys. Soc. 69, 441 (1965). contain a small factor v/c (where v is the particle veloc- 4. O. V. Konstantinov and V. I. Perel’, Zh. Éksp. Teor. Fiz. ity in the system) as compared to the basic contribu- 37 (3), 786 (1959) [Sov. Phys. JETP 10, 560 (1959)]. tions. However, this estimate is not necessarily correct 5. I. G. Lang, L. I. Korovin, J. A. de la Cruz-Alcaz, and for spatially inhomogeneous systems, such as semicon- S. T. Pavlov, Zh. Éksp. Teor. Fiz. 123 (2), 305 (2003) ductor quantum wells, wires, or dots. The field E(r, t) [JETP 96, 268 (2003)]; cond-mat/0212549. can be treated as an external or exciting field only in the 6. R. Zeyher, H. Bilz, and M. Cardona, Solid State Com- case when the induced current and charge densities are mun. 19 (1), 57 (1976). calculated to the lowest order in the interaction of the 7. A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshinskiœ, field with the system of charged particles. Such an Methods of Quantum Field Theory in Statistical Physics (Prentice Hall, Englewood Cliffs, N.J., 1963; Dobrosvet, approximation is applicable for quantum wells if the γ Ӷ γ γ γ Moscow, 1998). condition r is satisfied, where r( ) is the radiative 8. R. Enderlien, K. Peuker, and F. Bechstedt, Phys. Status (nonradiative) reciprocal lifetime of the electronic exci- Solidi B 92 (1), 149 (1979). tation [9, 10]. 9. L. C. Andreani, F. Tassone, and F. Bassani, Solid State γ ӷ γ Commun. 77 (11), 641 (1991). Otherwise, at r , the interaction of the field with 10. L. C. Andreani, G. Panzarini, A. V. Kavokin, and particles to all orders should be taken into account. In M. R. Vladimirova, Phys. Rev. B 57 (8), 4670 (1998). this case, the field E(r, t) is the true field inside a low- 11. L. I. Korovin, I. G. Lang, D. A. Contreras-Solorio, and dimensional object. This field cannot be represented as S. T. Pavlov, Fiz. Tverd. Tela (St. Petersburg) 43 (11), a superposition of plane waves for which k = ω/c. For 2091 (2001) [Phys. Solid State 43, 2182 (2001)]; cond- example, the true field considerably changes in magni- mat/0104262. tude inside a quantum well along the z axis perpendic- 12. L. I. Korovin, I. G. Lang, D. A. Contreras-Solorio, and ular to the quantum well plane if the incident light prop- S. T. Pavlov, Fiz. Tverd. Tela (St. Petersburg) 44 (11), ω 2084 (2002) [Phys. Solid State 44, 2181 (2002)]; cond- agates along the z axis and the frequency is at reso- mat/0203390. nance with one of the discrete excitation levels of the electronic system in the well [11, 12]. In this case, the quantities kd Ӎ 1 become significant (where d is the Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003

LOW-DIMENSIONAL SYSTEMS AND SURFACE PHYSICS

Electrostatic Response and Surface Electron States of a Ca(001) Face-Centered Cubic Film G. V. Wolf and D. V. Fedorov Physicotechnical Institute, Ural Division, Russian Academy of Sciences, ul. Kirova 132, Izhevsk, 426001 Russia e-mail: [email protected] Received January 22, 2003

Abstract—The electron response of a Ca(001) face-centered cubic film to an external electrostatic field is calculated. The results of calculations are compared with the previously obtained data on the electron response of a Cu(001) film. The energy location of occupied and unoccupied excited surface states of the Ca(001) film is determined. © 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION in that the 3d electron states are completely occupied in Investigation of the electron response of an anisotro- copper atoms and completely unoccupied in calcium pic metal to an external electrostatic field in the frame- atoms. It is known that, in a copper bulk crystal, the work of the electron-density-functional theory is of Fermi level EF lies above the sÐd hybridized states and considerable scientific interest. This is associated not the dominant contribution to the density of states at the only with the necessity of interpreting the results of Fermi level EF is made by the 4s states [4]. experiments that are sensitive to changes in the surface distribution of the electron density and the potential [1] In crystalline calcium, the Fermi level EF lies below the main group of 3d electron states. However, accord- but also with the need to solve a number of general 1 problems in the inhomogeneous electron gas theory. It ing to the results of our calculations (Fig. 1), the partial seems likely that, at present, only within this approach densities of states np(EF) and nd(EF) are comparable in can the exchangeÐcorrelation effects be properly magnitude. This makes it possible to elucidate the role accounted for in the electron response of inhomoge- played by 3d electrons in the electrostatic response of neous systems with a complex band structure [2]. In the metal. principle, the electron-density-functional theory leads In this work, the electronic structures of electroneu- to exact results in the long-wavelength limit [3] and tral and charged Cu(001) and Ca(001) films were cal- offers an independent verification and evaluation of the approximations used in diagram techniques for deter- culated using a new method in the framework of the mining the low-frequency electron response. electron-density-functional theory. Within this approach, the electron screening of the metal surface is The screening of an electrostatic field by the surface efficiently included in the calculation (the method was of a particular metal is substantially affected by the described in detail in our previous work [6]). The sys- metal electronic structure, which, in turn, depends on a tem of self-consistent equations (in atomic units with large number of different factors in a complex manner. Among these are the composition and the structure of the energy expressed in rydbergs) to be solved has the the crystal, the orientation of its surface, relaxation pro- following form [7]: cesses occurring in the surface layer, etc. In order to {}Ψ∆ []ρ , (), ()Ψ, (), elucidate the influence of all these factors on the phe- Ð + V ζ; r q nk r q = En k q nk r q , nomenon under investigation, it is expedient to perform Ψ ()r, q = exp()ΨÐikR ()rR+ , q , a comparative analysis of the systems differing from nk n nk n each other by a minimum number of factors that have a Ψ (), 2 significant effect on the electron response. For this pur- ∫ nk r q dr = 1, (1) pose, in the present work, we analyzed the evolution of Ω electron states of Cu(001) and Ca(001) face-centered cubic films in an external electrostatic field. ρ(), Ψ(), 2 ζ r q = ∑ nk r q , n, k 2. COMPUTATIONAL TECHNIQUE, RESULTS, AND DISCUSSION 1 The calculations of the densities of states for a calcium bulk crystal were carried out by the self-consistent q-spin-polarized rela- The occupancies of electron states of copper and tivistic KorringaÐKohnÐRostoker (SPRÐKKR) code for calculat- calcium atoms of these isostructural metals differ only ing solid-state properties [5].

ELECTROSTATIC RESPONSE AND SURFACE ELECTRON STATES 2013

1.4 50 III II n III tot II ns 40 EF 1.0 np n d 30 ) ) E E ( (

n I n 0.6 20 I

10 0.2 0 0 –6 –4 –2 0 2 4 –7 –6 –5 –4 –3 –2 –1 0 E, eV E, eV

Fig. 1. Total (ntot) and partial (ns, np, nd) densities of states Fig. 2. Density of states of the Ca(001) five-layer film (the of the copper bulk crystal (the energy expressed in electron- energy expressed in electronvolts is reckoned from the vac- volts is reckoned from the Fermi level). uum zero). where k is the two-dimensional reduced quasi-momen- surface are more scarce. The work function calculated tum, Rn is the translational vector of the Bravais lattice for the Ca(001) film in our case differs by 0.5 eV from of the film under consideration, and q is the number of the result of the self-consistent calculation performed uncompensated electrons in the unit cell of the film Ω. within the pseudopotential approach for a Ca(001) When calculating the electron density ρζ (r, q), summa- seven-layer film by Ley et al. [14]. It should be noted that the experimental value of 2.87 eV (Table 1) was tion was performed over all states with energies obtained by Gaudart and Rivoira [15] for a polycrystal- En(k, q) lying below the Fermi level EF(q). line sample. The effective potential V[;ρ r, q] represents the sum ζ Figure 2 presents the results of our calculations of of the Coulomb and exchangeÐcorrelation contribu- the density of states for the Ca(001) five-layer film. A tions. The Coulomb contribution was determined by the comparison of these data with the calculated density of method used in our earlier work [8]. The exchangeÐcor- states for a bulk calcium crystal (Fig. 1) shows that, relation contribution was obtained within the local-den- upon changing over from the film to the bulk crystal, sity approximation in the form of a HedinÐLundqvist the energies corresponding to maxima I and II vary potential with the parameters taken from [9]. only slightly. The shift in these maxima does not exceed The lattice constants of copper and calcium films 0.2 eV. The energy position of maximum III changes were assumed to be equal to those of copper and calcium more significantly and lies 0.98 eV closer to the Fermi bulk crystals: ACu = 6.8309 au and ACa = 10.5296 au. level in the film as compared to that in the bulk crystal. It is known that calculated values of the work func- The calculation of the electrostatic response of tion W are very sensitive to changes in the electron-den- Cu(001) films [6] and examination of the electron sity distribution in the vicinity of the metal surface. The screening of the (001) surfaces of Al, Ni, and Ag semi- difference between theoretical and experimental values infinite crystals [20, 21] demonstrated that the induced of the work function serves as a severe criterion of charge is localized in a very thin surface region contain- applicability of both the model and the method used in solving the system of equations (1). The work functions calculated in this study for Cu(001) and Ca(001) five- Table 1. Work functions of Cu(001) and Ca(001) films (in eV) layer films were compared (see Table 1) with the avail- Our Other able experimental data and the results of other self-con- Film Experiment sistent calculations [10Ð19]. It turned out that our data calculation calculations on the work function of the Cu(001) film coincide very ± closely with the experimental values obtained by Del- Cu(001) 5.11 4.50 [10] 5.16 0.054 [16] char [16] and Strayer et al. [18]. As regards the theoret- 4.94 [11] 4.59 ± 0.054 [17] ical data, our value of the work function of the copper film is in best agreement with the result (5.18 eV) of the 4.91 [12] 5.10 ± 0.030 [18] precision calculation carried out by Bross and Kauz- 5.18 [13] 4.77 ± 0.050 [19] mann [13]. Unfortunately, experimental data available in the literature on the work function of the Ca(001) Ca(001) 3.38 2.90 [14] 2.87 ± 0.06 [15]

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 2014 WOLF, FEDOROV

1.0 30 q = 0.01 q = 0.01 q = 0.03 q = 0.03 20 0.5 ) ) E E ( ( n E2 E3 E4 n 10 ) –

0 ) –

E b E 4 , , q E1 q 0 ( ( a5 a2 a n 1 n a a 4 3 b –0.5 1 b b –10 2 3

–1.0 –20 –12 –10 –8 –6 –4 –2 0 –8 –6 –4 –2 0 E, eV E, eV

Fig. 3. Difference in the densities of states of the charged Fig. 4. Difference in the densities of states of the charged and electroneutral Cu(001) films. The dot-dashed line cor- and electroneutral Ca(001) films. The dot-dashed line corresponds to the energy at the Fermi level EF of the electro- responds to the energy at the Fermi level EF of the electroneutral film (the energy expressed in electronvolts is reck- neutral film (the energy expressed in electronvolts is reckoned from the vacuum zero). oned from the vacuum zero). ing only one or two atomic layers. The Stark shift C(q), ∆n(q, E) for several values of the uncompensated which is determined by the dipole moment of the charge. In these calculations, proper allowance was charge of the surface region, is nearly identical for all made only for the specific features in the curve ∆n(s)(q, bulk states [8]. Consequently, after the energy shift, the E) that are not very sensitive to changes in the number difference between the densities of states of the charged q of uncompensated electrons per unit cell of the film, and electroneutral films, which suggests a weak energy dependence n(s)(E + C(q), q) in the vicinity of the corresponding maxima in ∆nqE(), = nE()+ Cq(), q Ð nE(), 0 , (2) the curve n(s)(E, 0). The differences between the surface-state densities becomes equal to the difference between the surface- of Cu(001) and Ca(001) five-layer films are presented state densities of these films, in Figs. 3 and 4, respectively. For the Cu(001) film, the ∆ () () () positions of all minima in the curve n(q, E) at q = 0.01 ∆n s ()qE, = n s ()ECq+ (), q Ð n s ()E, 0 . (3) and 0.03 almost coincide with each other. From the above it follows that they correspond in energy to the The contribution n(s)(E + C(q), q) of the surface- maxima in the surface-state density of the electroneu- state density of the charged film leads to an insignifi- tral film. Our results are in good agreement with avail- cant change in the energy location of the specific fea- able theoretical [10] and experimental [22] data on the tures (which are associated with the maxima in the sur- energy location of Cu(001) occupied surface states. In face-state density of the electroneutral film) in the particular, the positions of minima at the energies E = ∆ 1 curve n(q, E) as compared to the energy location of Ð3.55 eV and E2 = Ð1.45 eV (the energies are reckoned the corresponding features in the curve n(s)(E, 0). In from the Fermi level of the electroneutral film) coincide order to decrease this error, we calculated the difference to within several hundredths of an electronvolt with the

Table 2. Energy location of occupied surface states of the Ca(001) ﬁlm (the energies expressed in electronvolts are reckoned from the Fermi level of the electroneutral ﬁlm)

Speciﬁc features of ∆n(q, E)

a1 a2 a3 a4 a5 Our calculation q = 0.01 0.00 Ð0.48 Ð0.89 Ð1.29 Ð2.46 q = 0.03 0.00 Ð0.46 Ð0.81 Ð1.31 Ð2.44 Calculation [14] Ð0.1 Ð Ð Ð1.4 Ð2.5 Experiment [14] Ð Ð0.6 Ð Ð1.6 Ð2.4

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 ELECTROSTATIC RESPONSE AND SURFACE ELECTRON STATES 2015 positions of the peaks in the difference between the Table 3. Energy location of unoccupied surface states of the densities of states of the surface and central layers of Ca(001) film (the energies expressed in electronvolts are the Cu(001) nine-layer film [10]. The surface-state den- reckoned from the Fermi level of the electroneutral film) sity in the vicinity of the Fermi level for the copper film is negligible. qb1 b2 b3 b4 For the Ca(001) film, the surface electron states con- 0.01 0.43 0.92 1.88 2.16 tribute significantly to the density of states at the Fermi level. In this case, the position of the local maximum in 0.03 0.51 0.91 1.87 2.14 the surface-state density n(s)(E, 0) virtually coincides with the energy at the Fermi level EF of the electroneutral film (Fig. 4, minimum a1). A comparison of our tion to a semiconductor state with zero energy gap [24, results with data available in the literature on the energy 25]. The observed changes in the bulk contribution to location of occupied surface states of the Ca(001) film the density of states in the near-Fermi region of the is given in Table 2. Ca(001) film (Figs. 1, 2) suggests that calcium films can also exhibit a similar effect. The surface contribu- It can be seen from Table 2 that, as in the case of the tion to the density of states at the Fermi level in the Cu(001) surface, the results of our calculations for the Ca(001) film, unlike the Cu(001) film, is significant and Ca(001) film are in satisfactory agreement with theoret- depends on the strength of the external electrostatic ical and experimental data obtained by other authors. field. Therefore, it is quite possible that the charge state Moreover, from these results, we can predict an of the calcium film can affect the above transition. increased density of surface states lying 0.8Ð0.9 eV below the Fermi level. An analysis of the specific features revealed in the REFERENCES ∆ curve n(q, E) makes it possible to determine the 1. P. J. Feibelman, Prog. Sufr. Sci. 12, 287 (1982). energy location of unoccupied surface states. As can be seen from Fig. 3, the curve ∆n(q, E) for the Cu(001) 2. A. Williams and U. Barth, in Theory of the Inhomoge- film is characterized by two minima at the energies E = neous Electron Gas, Ed. by S. Lundqvist and N. H. March 3 (Plenum, New York, 1983; Mir, Moscow, 1987). 0.29 eV and E4 = 1.26 eV (the energies are reckoned from the Fermi level of the electroneutral film). The 3. D. J. W. Gerdart, M. Rasolt, and R. 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PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003 2016 WOLF, FEDOROV

16. T. A. Delchar, Surf. Sci. 27, 11 (1971). 22. S. D. Kevan, N. G. Stoffel, and N. Y. Smith, Phys. Rev. 17. P. O. Gartland, S. Berge, and B. J. Slagsvold, Phys. Rev. B 31, 3348 (1985). Lett. 28, 738 (1972). 23. S. L. Hulbert, P. D. Johnson, M. Weinert, and R. F. Gar- 18. R. W. Strayer, W. Macki, and L. W. Swanson, Surf. Sci. rett, Phys. Rev. B 33, 760 (1986). 34, 225 (1973). 24. R. A. Stager and H. G. Drickamer, Phys. Rev. 131, 2524 19. G. G. Tibbets, J. M. Burkstand, and J. C. Tracy, Phys. (1963). Rev. B 15, 3652 (1977). 25. J. W. McCaffrey, J. R. Anderson, and D. A. Papaconstan- 20. J. E. Inglesﬁeld and G. A. Benesh, Phys. Rev. B 37, 6682 topoulos, Phys. Rev. B 7, 674 (1973). (1988). 21. G. C. Aers and J. E. Inglesﬁelds, Surf. Sci. 217, 367 (1989). Translated by O. Borovik-Romanova

PHYSICS OF THE SOLID STATE Vol. 45 No. 10 2003