Thermodynamics of structurally disordered s–d model

Spin-electron exchange model is generalized and used for description of magnetic states of amorphous substitutional alloys with the structural disorder of the liquid type. A scheme of consistently accounting for the contributions of structural fluctuations to the thermodynamic functions and obser...

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Datum:	2005
Hauptverfasser:	Rudavskii, Yu.K., Ponedilok, G.V., Dorosh, L.A.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2005
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spelling	irk-123456789-1197422017-06-09T03:03:59Z Thermodynamics of structurally disordered s–d model Rudavskii, Yu.K. Ponedilok, G.V. Dorosh, L.A. Spin-electron exchange model is generalized and used for description of magnetic states of amorphous substitutional alloys with the structural disorder of the liquid type. A scheme of consistently accounting for the contributions of structural fluctuations to the thermodynamic functions and observable quantities is considered. Using the perturbation theory, the functional of thermodynamic potential is constructed as a functional power series. In the random phase approximation (RPA), the grand thermodynamic potential of the model is calculated. Self-consistency conditions are given, from which equations for magnetizations and critical temperature of the paramagnetic-ferromagnetic transition are obtained. Спін-електронна обмінна модель узагальнюється і застосовується для опису магнітних станів аморфних сплавів з рідиноподібним типом структурної невпорядкованості. Розглянута схема послідовного врахування вкладу структурних флуктуацій у термодинамічні функції та спостережувані величини. За теорією збурень побудовано функціонал термодинамічного потенціалу у формі функціонального степеневого ряду. У наближенні хаотичних фаз (RPA) розраховано великий термодинамічний потенціал моделі. Записані співвідношення самоузгодження, з яких знаходяться рівняння для намагніченостей та критичної температури переходу “парамагнетик-феромагнетик”. 2005 Article Thermodynamics of structurally disordered s–d model / Yu.K. Rudavskii, G.V. Ponedilok, L.A. Dorosh // Condensed Matter Physics. — 2005. — Т. 8, № 3(43). — С. 579–602. — Бібліогр.: 18 назв. — англ. 1607-324X PACS: 72.15.C, 73.20.H, 75.10, 75.30.E, 82.65.Y DOI:10.5488/CMP.8.3.579 http://dspace.nbuv.gov.ua/handle/123456789/119742 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	Spin-electron exchange model is generalized and used for description of magnetic states of amorphous substitutional alloys with the structural disorder of the liquid type. A scheme of consistently accounting for the contributions of structural fluctuations to the thermodynamic functions and observable quantities is considered. Using the perturbation theory, the functional of thermodynamic potential is constructed as a functional power series. In the random phase approximation (RPA), the grand thermodynamic potential of the model is calculated. Self-consistency conditions are given, from which equations for magnetizations and critical temperature of the paramagnetic-ferromagnetic transition are obtained.
format	Article
author	Rudavskii, Yu.K. Ponedilok, G.V. Dorosh, L.A.
spellingShingle	Rudavskii, Yu.K. Ponedilok, G.V. Dorosh, L.A. Thermodynamics of structurally disordered s–d model Condensed Matter Physics
author_facet	Rudavskii, Yu.K. Ponedilok, G.V. Dorosh, L.A.
author_sort	Rudavskii, Yu.K.
title	Thermodynamics of structurally disordered s–d model
title_short	Thermodynamics of structurally disordered s–d model
title_full	Thermodynamics of structurally disordered s–d model
title_fullStr	Thermodynamics of structurally disordered s–d model
title_full_unstemmed	Thermodynamics of structurally disordered s–d model
title_sort	thermodynamics of structurally disordered s–d model
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2005
url	http://dspace.nbuv.gov.ua/handle/123456789/119742
citation_txt	Thermodynamics of structurally disordered s–d model / Yu.K. Rudavskii, G.V. Ponedilok, L.A. Dorosh // Condensed Matter Physics. — 2005. — Т. 8, № 3(43). — С. 579–602. — Бібліогр.: 18 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT rudavskiiyuk thermodynamicsofstructurallydisorderedsdmodel AT ponedilokgv thermodynamicsofstructurallydisorderedsdmodel AT doroshla thermodynamicsofstructurallydisorderedsdmodel
first_indexed	2025-07-08T16:31:08Z
last_indexed	2025-07-08T16:31:08Z
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fulltext	Condensed Matter Physics, 2005, Vol. 8, No. 3(43), pp. 579–602 Thermodynamics of structurally disordered s–d model Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh National University “Lvivska Politechnika”, 12 S.Bandera Str., 79013 Lviv, Ukraine Received May 5, 2004, in final form October 22, 2004 Spin-electron exchange model is generalized and used for description of magnetic states of amorphous substitutional alloys with the structural dis- order of the liquid type. A scheme of consistently accounting for the con- tributions of structural fluctuations to the thermodynamic functions and ob- servable quantities is considered. Using the perturbation theory, the func- tional of thermodynamic potential is constructed as a functional power se- ries. In the random phase approximation (RPA), the grand thermodynamic potential of the model is calculated. Self-consistency conditions are giv- en, from which equations for magnetizations and critical temperature of the paramagnetic-ferromagnetic transition are obtained. Key words: s–d-model, ferromagnetic alloys, free energy, magnetization, Curie temperature, functional integral PACS: 72.15.C, 73.20.H, 75.10, 75.30.E, 82.65.Y 1. Introduction The exchange s-d model is used for description of electric and magnetic pro- perties of compounds containing transition and rare-earth elements. Introduced by Shubin and Vonsovsky, the exchange s-d model was further developed by Zener, Turov, Kasuya, Yosida and others and found an increasing number of applications in various fields of solid state physics. Initially it was proposed for description of electric and magnetic properties of transition d-metals [1]; today it forms a basis for a theory of magnetism in rare-earth metals and magnetic semiconductors [2,3] and Kondo systems [4]. From the theoretical point of view, the importance of s-d model is confirmed by the fact that it predicts many beautiful and non-trivial effects: for- mation of spin polarons, ferrons, fluctuons in magnetic semiconductors, occurrence of spin-glass state in the dissolved metal alloys, Kondo-effect, etc. Applications of the s-d model are not restricted only to the listed objects. The pseudo-spin-electron model in which the role of spin operators is played by formal pseudo-spin operators, describing the generalized degrees of freedom of a many-body system, is used for c© Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh 579 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh description of a strongly correlated electron system and compounds in which the phenomenon of high-temperature superconductivity is observed [5,6]. Most of papers (see [1–5,7] and references therein) concern the systems with one magnetic impurity in a metal or semiconductor, or the so-called periodic s-d model. These theoretical investigations of s-d model are mainly aimed at calculating the static and dynamic correlation functions. However, only in a few theoretical works the thermodynamics of s-d model is studied, and phase transitions for this model are investigated and classified. In particular, in [8] the high-temperature expansions are constructed for free energy of the simplified model, where a transeverse part of the spin-electron interaction operator was neglected. Authors of [9] have presented a critical survey of different versions of perturbation theory which were used to calcu- late the free energy. In [10], by using a method developed in [11,12], the functional of thermodynamic potential for the generalized periodical s-d model was constructed. The present work continues the investigation [13] of the structurally disordered s-d model. In contrast to the previous work, within the perturbation theory the contribution to the thermodynamic potential from correlational spin-electron in- teractions as well as the effects caused by scattering of conductivity electrons on structure fluctuations are taken into account. 2. The spin-electron model for the amorphous binary magnet In the traditional classical sense, the s-d model of ferromagnetic crystal metals describes the interaction of conductivity electrons with localized magnetic moments of atoms by an exchange mechanism [1,2]. Thus states of conductivity electrons are described by plane waves or Bloch wave packages. To apply this model to structurally disordered ferromagnetic alloys, it is necessary to introduce certain specifications and generalizations which are essential for a correct description of this class of substances. 2.1. The Hamiltonian Amorphous binary alloy of N atoms in the V ⊂ R 3 volume is considered. Part of these atoms have magnetic moments (hereafter – a magnetic subsystem of the alloy), but the other ones do not have any (nonmagnetic subsystem of the alloy). The atom coordinates (R1, . . . ,RN) ≡ RN ∈ V can have random values. Ratio of the numbers of magnetic and nonmagnetic atoms is given by concentration c (0 6 c 6 1). The microscopic model of an amorphous magnet takes into account the pres- ence of two quantum interacting subsystems – i.e., of localized atom spins and of conduction electrons. The model Hamiltonian is given by Ĥ = Ĥs + Ĥel + Ĥel−s . (1) The first term describes the energy of the localized spin subsystem, being in the external magnetic field h and in pairs interact between themselves via the Heisenberg 580 Thermodynamics of structurally disordered s – d model exchange Ĥs = −gµBh N∑ j=1 ĉjS z j − 1 2 ∑ 16i6=j6N J(\|Ri − Rj\|)ĉi Si ĉj Sj . (2) Here ĉj is a random variable equal to 1 or 0 if the site j is occupied by a magnetic or a nonmagnetic atom, respectively; gµB is the magnetic moment of atoms, and Sj is the operator of magnetic atom spin located at Rj ∈ V . The spin operator satisfies the condition \|Sj\|2 = S(S + 1), where 1/2 6 S < ∞ is the magnitude of the atom spin. External magnetic field h is directed along the OZ axis. In terms of operators Sα q = 1√ N N∑ j=1 Sα j ĉj e−iqRj , α = x, y, z, (3) the expression (2) takes the form Ĥs = 1 2 N c S(S + 1)J(\|R\| = 0) − gµBh N∑ j=1 ĉjS z j − 1 2 ∑ q∈Λ JqSqS−q . (4) Fourier coefficients of the exchange spin-spin interaction of localized spins Jq = N V ∫ V dr J(\|r\|) e−iqr (5) obey the following relations: Jq = J−q. Electron subsystem of the alloy is described within the framework of the pseudo- potential approach. The Hamiltonian of conduction electrons in our model has the form Ĥel = ∑ k∈Λ ∑ σ=±1 Ek,σa + k,σak,σ + ∑ k,q∈Λ ∑ σ±1 Wqa + k,σak−q,σ . (6) Here a+ k,σ, (ak,σ) are the creation (annihilation) Fermi operators of electrons in quan- tum states {k, σ}. The wave vector k takes the value in quasi-continuous k-space Λ = {k : k = ∑ 16α63 2π V −1/3nαeα, nα ∈ Z, (eα, eβ) = δαβ}. The quantum numbers σ are equal to ±1, corresponding to the two possible projec- tions of electron spin on the axis of quantization OZ. The notation σ = (↑, ↓) will also be used. A spectrum of free electron gas in presence of external magnetic field h is Ek,σ = ~ 2k2 2m − κσµBh, κσ = { 1, if σ =↑ −1, if σ =↓ . (7) The matrix elements Wq = 1 N N∑ j=1 e−iqRjwq , wq = N V ∫ V drw(\|r\|) e−iqr, q ∈ Λ, (8) 581 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh characterize the processes of elastic scattering of electrons on the ions of the alloy; w(\|r\|) is a pseudo-potential of electron-ion interaction. For local pseudo-potentials the Fourier coefficients wq depend only on the absolute value of the wave vector wq = w−q. Energy of the spin-electron interaction in the coordinate representation is given by: Ĥel−s = − Ne∑ i=1 N∑ j=1 I(\|ri − Rj\|) si ĉjSj. (9) Here si is the operator of electron spin, localized at ri ∈ V ; Ne is the number of electrons in the system. Cartesian components of the electron spin operator are sα j = σ̂α/2, α = x, y, z, where σ̂α are the Pauli matrices. An integral of spin-electron exchange coupling in the case of contact interaction is described by expression I(\|Ri − rj\|) = I δ(\|Ri − rj\|). (10) The interaction parameter I can be positive or negative. If I > 0, then the coupling is called ferromagnetic; if I < 0 is antiferromagnetic. For non-contact interactions, the alternating dependence of the interaction I(\|Ri − rj\|) on the distance between electron and atom nucleus, basically, is not excluded. The Hamiltonian of the spin-electron interaction in the representation of the second quantization by plane waves reads Ĥel−s = − 1√ N ∑ q∈Λ Iq [ Sz qσ̂ z −q + 1 2 ( S+ q σ̂− −q + S− q σ̂+ −q )] . (11) Fourier coefficients of the spin-electron exchange interaction are Iq = N V ∫ V I(\|r\|) e−iqr dr, q ∈ Λ. (12) In expression (11) the operator is Sα q = 1√ N N∑ j=1 ĉjS α j e−iqRj , α = z, +,− . (13) Here S± j = Sx j ± Sy j are the components of spin flip operators. In case of lattice systems (Rj ∈ Z 3) the operator (13) is the exact Fourier transform of operators ĉjS α j . Then the wave vector q changes within first Brillouin zone. In equation (11) the bilinear combinations of electron creation and annihilation Fermi operators are defined σ̂z q = ∑ σ=↑,↓ κσ 2 n̂q,σ , n̂q,σ = ∑ k∈Λ a+ k,σ ak+q,σ , σ̂+ q = ∑ k∈Λ a+ k,↑ ak+q,↓ , σ̂− q = ∑ k∈Λ a+ k,↓ ak+q,↑ . (14) 582 Thermodynamics of structurally disordered s – d model Hermitian operator σ̂z q means a Fourier component of the operator of electron spin polarization density. Fourier coefficients of the total electron density operator n̂(r) are n̂q = n̂q,↑ + n̂q,↓ = ∑ k∈Λ ∑ σ=↑,↓ a+ k,σ ak+q,σ, (15) and N̂ = lim \|q\|→0 n̂q is the operator of the total number of electrons. 2.2. The structural correlation functions For the sake of simplicity, we shall accept that all N atoms of the binary sys- tem MxN1−x (both magnetoactive and those without localized magnetic moments) are of the same size, have identical potentials of ion-ion interaction and the same pseudo-potentials of electron-ionic interaction. We assume that magnetic exchange interactions do not effect a spatial configuration of atoms, or, at least, effect insignifi- cantly. With such assumptions it is possible to avoid identification of the distribution of atoms of different sorts by an additional sort index. For this purpose it is suf- ficient to accept that the probability of the spatial allocation of atoms at points {R1 . . .RN} ∈ V and the probability of random occupation of these points of space by atoms of different sorts are statistically independent. Such a model of binary system will be called an amorphous substitutional alloy. The probability density of the distribution of atoms of the binary system in the volume V will be described by the set of structural correlation functions Kn(ĉ1R1; . . .; ĉnRn) = Pn(R1, . . . ,Rn) Cn ( ĉ1; . . .; ĉn R1; . . . ;Rn ) , n = 1, N . (16) Here Pn is the probability density of distribution of system of atoms in space, and Cn is the conditional probability density of distribution of different sorts of atoms over randomly chosen fixed points {R1, . . . ,RN} of space. The magnitude of a random physical quantity averaged over all possible configurations F (ĉ1R1; . . . ; ĉNRN) (for instance, the free energy) is calculated over the formula F (ĉ1R1; . . . ; ĉnRn) = ∫ V n ∑ [ĉn] F (ĉ1R1; . . . ; ĉNRN) Kn(ĉ1R1; . . . ; ĉNRN) dRn. (17) The symbol (· · ·) will mean hereinafter the full configurational averaging over the normalized distribution (16). Such a configurational averaging in our model can be treated as composed of two ones (· · ·) = 〈〈(· · ·)〉c〉R , (18) where 〈(· · ·)〉 R means the averaging over the distribution of the atoms in space, and 〈(· · ·)〉c means the averaging over the occupation of the fixed points of space by atoms of different sorts. 583 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh It is convenient to describe the structure of amorphous system by the correlation functions Kn(k1; . . . ;kn) = [ 1√ N ]2−n ĉk1 . . . ĉkn , (19) where the notation (· · ·) stands for the irreducible average over the random config- urations. The values ĉk = 1√ N N∑ j=1 ĉj e−ikRj , k 6= 0. with accuracy to the factor √ N are the Fourier components of fluctuations of mag- netic atoms concentrations. In the case of lattice structure which corresponds to a crystal binary substitutional alloy 〈ĉk1 · · · ĉkn 〉c = 〈ĉj1 · · · ĉjn 〉c δk1+···+kn,0. (20) Configurational average 〈ĉj1 . . . ĉjn 〉c over the distribution of atoms of different sorts can be expressed through the irreducible averages: Pm(c) = 〈ĉj1 . . . ĉjm 〉irrc , the gen- erating functional of which is g(t; c) = ∑ m>1 Pm(c) tm m! = ln(1 − c + c et). (21) Some first cumulants are P1(c) = c, P2(c) = c(1 − c), P3(c) = c(1 − c)(1 − 2c), P4(c) = c(1 − c)(1 − 6c + 6c2), etc. The structural correlation functions are reduced to the form: K2(k1;k2) = c [1 − c + c S2(k1)] δk1+k2,0 ; (22) K3(k1;k2;k3) = [ c(1−c)(1−2c) − c2(1−c) { S2(k1) + S2(k2) + S2(k3) } + c3 S3(k1;k2;k3) ] δk1+k2+k3,0, etc. (23) Correlation functions of the atomic density fluctuations defined as Sm(k1, . . . ,km)δk1+···+km,0 def = [ 1√ N ]2−m 〈ρk1 . . .ρkm 〉irrR (24) cannot be calculated ab initio. Therefore we will suppose that they are phenomeno- logical quantities. 3. The order parameters. The effective Hamiltonian Thermodynamic and dynamic properties of the described spin-electron model shall be considered within the framework of the concept of order parameters. In this 584 Thermodynamics of structurally disordered s – d model case it is necessary to make some generalizations. It is caused by the fact that the order parameters can and in some cases must be defined as functionals of random structural parameters. Let us introduce the operators of magnetization fluctuations of a subsystem of localized atomic spins from the thermodynamic average value as Qk = 1√ N N∑ j=1 e−ikRj (ĉjSj − ĉj y ez) . (25) The thermodynamic average value of the spin of jth atom depends on a spatial configuration of the system, i.e. it is the functional of coordinates of all atoms 〈Sz j 〉 T = yj(ĉ1R1, . . . , ĉNRN). Hereinafter the symbol 〈(· · ·)〉 T = Sps Spe [ (· · ·)e−β(Ĥ−µN̂e−Ω) ] will mean a thermodynamic average over the grand canonical Gibbs distribution with full Hamiltonian of model, Ω is the thermodynamic potential, β = (kBT )−1 the inversion temperature in energy units. We shall use here the approximation, when in the equation (25) the thermodynamic average yj(ĉ1R1, . . . , ĉNRN) is replaced by its configurationally averaged value y = yj(ĉ1R1, . . . , ĉNRN) ≡ 〈Sz j 〉 T . Using the same scheme we shall define the operators π̂σ q =    σ̂z q − 〈σ̂z q〉 T , σ = z, σ̂± q , σ = ±, (26) the component π̂z q of which describes fluctuations of electron spin polarization near the thermodynamic and configurational average value. Then 〈σ̂z q〉 = ∑ σ=↑,↓ κσ 2 δq,0 ∑ k 〈a+ kσakσ〉 T = 1 2 (N↑ − N↓) δq,0, (27) where Nσ = ∑ k 〈a+ kσakσ〉 T is thermodynamic and configurational average value of the total number of conductivity electrons with a spin projection σ on the axis of quantization. We shall define the value of electron spin polarization of the unit volume (in Bohr units) as p = 1 2V (N↑ − N↓). (28) Then 〈σ̂z q〉 T = p V δq,0. (29) The equation (29) is correct under the assumption that there is only a spatially homogeneous magnetic structure of electron gas in the system, i.e. 〈σ̂z q〉 T = 0 at all values q 6= 0. Thus, the second order parameter is the quantity m, which describes 585 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh the electron spin polarization per unit volume and is determined self-consistently from the condition of a minimum of the thermodynamic potential. In terms of operators (25) and (26), the total Hamiltonian of the s-d model can be rewritten in the form Ĥ = C + Ĥ0 + Ĥint. (30) Here the non-operator part of the Hamiltonian is C = 1 2 cNS(S+1)J(0) + y p V I0 + y2 2 ∑ k Jk ĉk ĉ−k. (31) An effective one-particle Hamiltonian which describes the energy of a subsystem of free conductivity electrons and energy of a subsystem of localized spins in self- consistent fields reads Ĥ0 = Ĥ0E + Ĥ0S = ∑ k ∑ σ Ẽk,σa + k,σak,σ − N∑ j=1 h̃j ĉjS z j . (32) Effective field which acts on the spin of the atom localized at the point Rj is h̃j = gµBh + V N p I0 + c y J0 + y 2 √ N ∑ k 6=0 Jk ( cke ikRj + c−k e−ikRj ) . (33) The parameter J0 ≡ lim \|k\|→0 Jk characterizes the intensity of exchange spin-spin inter- action. For amorphous and crystal structure it is determined, respectively, by the relations J0 = N V ∫ V J(\|R\|)dR, J0 = N∑ j=1 J(\|Rj\|). For interaction only between the nearest neighbours, we have J0 = zJ , where z is the number of nearest neighbours, and J is the exchange integral of interactions between the nearest atoms. In the case of an ideal one-sort magnetic crystal (ĉj ≡ 1, Rj ∈ Z 3 for all j = 1, N) under the condition of contact spin-electron interaction the relation (33) reduces to the expression h̃j → h̃ = gµBh + pI + yJ0 for the usual homogeneous self-consistent field of magnetic crystals. Renormalized due to the effects of magnetization by a subsystem of localized atom spins, the spectrum of electron gas is as follows: Ẽk,σ = ~ 2k2 2m − κσ (µBh + cyI0) + w0. (34) The constant w0 = N V ∫ V w(\|r\|) dr 586 Thermodynamics of structurally disordered s – d model is the homogeneous effective potential caused by non-coloumbic character of the pseudo-potential and does not disappear due to neutrality condition as it takes place in the model of point ions. From the condition of thermodynamic balance, the pseudo-potential should satisfy an additional condition \|w0\| < ∞. The operator of correlation interaction in (30) is Ĥint = ∑ k ∑ q 6=0 ∑ σ=↑,↓ W̃q,σ a+ k,σak−q,σ − 1 2 ∑ k∈Λ Jk Qk Q−k − 1√ N ∑ q Iq [ Qz qπ̂ z −q+ 1 2 ( Q+ q π̂− −q+Q− q π̂+ −q )] . (35) The operator Ĥint includes the term which characterizes the energy of electron scat- tering by ions. However, now the renormalized pseudo-potential of spin-electron scattering reads W̃q,σ = 1 N N∑ j=1 e−iqRj (wq − κσ ĉj y Iq) , (36) i.e. includes the term which characterizes the exchange spin-electron interaction. It is natural that such a pseudo-potential already depends on the orientation of an electron spin with respect to the axis of quantization. 4. The functional representation for the thermodynamic potential In order to go beyond the limits of the self-consistent field approximation, we use the method of functional integration developed in [11,12]. Here, in contrast to [13], the fluctuation of spin-electron interaction and the effects related to scattering of conductivity electrons by fluctuations of structure will be taken into account. For a certain given configuration of atoms {ĉ1R1; . . . ; ĉNRN} ≡ [ĉNRN ] the grand partition function of the model is Ξ[ĉNRN ] = e−βCSp S Sp E  e−β(Ĥ0S+Ĥ0E−µN̂e) T̂τexp  − β∫ 0 dτ Ĥint(τ)     , (37) where N̂e is the operator of the total number of conductivity electrons, µ is the chemical potential of the electron subsystem. The operator Ĥint(τ) is written in the interaction representation, when the dependence of operators on parameter τ is given by Â(τ) = e(Ĥ0−µN̂e)τ Â e−(Ĥ0−µN̂e)τ . The symbol T̂τ stands for the operator of chronological ordering of imaginary “times” τ ∈ [0; β]. It is convenient to use the frequency representation in the partition function (37), having exploited the expansion into Fourier series ak,σ(τ) = 1√ β ∑ ν ak,ν,σ e−iτν , a+ k,σ(τ) = 1√ β ∑ ν a+ k,ν,σ eiτν (38) 587 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh for Fermi operators, and Qα k(τ) = 1√ β ∑ ω Qα k,ω e−iτω, σ̂α k(τ)= 1√ β ∑ ω σ̂α k,ω e−iτω, Qα k,ω = 1√ β β∫ 0 Qα k(τ) eiτω, σ̂α k,ω = 1√ β β∫ 0 σ̂α k(τ) eiτω, α = x, y, z (+,−, z). (39) for direct and inverse expansions into Fourier series of spin operators. In expansions (38)–(39) the Fermi frequencies have discrete values ν = (1 + 2n)π/β, n ∈ Z, and Bose frequencies, accordingly, have the values ω = 2πn/β, n ∈ Z. The Fourier components of the operators, being bilinear combinations of Fermi operators, read n̂k,ω,σ = 1√ β ∑ q∈Λ ∑ ν a+ q,ν,σ aq+k,ν+ω,σ, σ̂z k,ω= 1√ 2 [n̂k,ω,↑ − n̂k,ω,↓], σ̂+ k,ω = 1√ βN ∑ q∈Λ ∑ ν a+ q,ν,↑ aq+k,ν+ω,↓, σ̂− k,ω= 1√ βN ∑ q∈Λ ∑ ν a+ q,ν,↓ aq+k,ν+ω,↑. (40) We shall use the reduced notation x = (k, ω) for vector four-dimensional quasi- continuous momentum-frequency space. The contribution of an electron subsystem to the grand partition function will be calculated within the framework of the conventional thermodynamic perturbation theory [15]. Integration over the states of electron subsystem in (37) results in the following form of the grand partition function Ξ[ĉNRN ] = e−β(C+Ω 0E +∆Ω1) Sp S  e−βĤ0S T̂τexp  − β∫ 0 dτ Ĥs−s(τ)     . (41) Here Ω 0E = − 1 β ∑ k ∑ σ=↑,↓ ln [ 1 + e−β(Ẽk,σ−µ) ] is the thermodynamic potential of a subsystem of non-interacting electrons in the external self-consistent magnetic field. The value ∆Ω1 = ∑ m>2 (−1)mβ m 2 −1 m! ∑ x1,σ1 · · · ∑ x1,σm m σ1...σm m (x1; . . . ; xm) m∏ i=1 δωi,0 W̃qi,σi (42) means the effective many-particle energy of interionic interaction for the given ran- dom configuration of atoms of the model. Coefficient functions m σ1...σm m (x1; . . . ; xm) = 〈T̂τ n̂−x1,σ1 · · · n̂−xm,σm 〉irr 0E (43) are the irreducible averages over the Gibbs’ distribution with the operator Ĥ 0E , constructed on operators (40). In the absence of magnetic field, in the theory of 588 Thermodynamics of structurally disordered s – d model metals there arise the irreducible averages µm(k1; . . . ;km) = 〈T̂τ n̂−k1 · · · n̂−km 〉irr 0E , constructed on operators n̂k = ∑ σ n̂k,σ, the so-called many-tail averages [15] and dynamic irreducible averages µ̃m(x1; . . . ; xm) = 〈T̂τ n̂−x1 · · · n̂−xm 〉irr 0E , constructed on operators n̂x = ∑ σ n̂kω,σ [16]. In the partition function (41) the operator Ĥs−s(τ) describes the energy of spin- spin interaction of a subsystem of the localized magnetic moments which includes the pair direct and effective many-particle interaction. The latter arises due to integrati- on over the degrees of freedom of an electron subsystem. In the momentum-frequency representation this operator reads Ĥs−s(τ) = ∑ m>2 Ĥ(m) s−s (τ) = −β 2 ∑ x Jk Qx Q−x + ∑ m>2 βm m!N m 2 ∑ x1,α1 · · · ∑ xm,αm S α1...αm m (x1; . . . ; xm) m∏ i=1 Iki Qαi xi + ∑ m>2 m−1∑ l=1 (−1)lβm−l/2 m!N (m−l)/2 R n1...nlαl+1...αm m (x1, . . . .xm) l∏ i=1 δωi,0 × ∑ kiσi W̃kiσ m∏ j=l+1 ∑ kj ,αj Ikj Q αj kjωj . (44) The irreducible averages over the Gibbs’ distribution of a subsystem of non-interacting electrons in the external self-consistent magnetic field S α1...αm m (x1; . . . ; xm) = 〈T̂τ π̂ α1 −x1 · · · π̂αm −xm 〉irr 0E (45) and R n1...nlαl+1...αm m (x1, . . . , xm) = 〈T̂τ n̂−x1,σ1 · · · n̂−xl,σl π̂ αl+1 −xl+1 · · · π̂αm −xm 〉irr 0E (46) are easily calculated with the help of the Wick theorem. Using expression (44) one can write down the operator of the energy of pair spin-spin interaction in the momentum-frequency representation Ĥ(2) s−s(τ)=−β 2 ∑ x [{ Jk+ β2 2 I2 k χzz(k, ω) } Qz x Qz −x+β2 I2 k χ+−(k, ω)Q+ x Q− −x ] . (47) In this formula χσ1σ2(q, ω) = 1 βN ∑ k,ν Gσ1 0 (k, ν) Gσ2 0 (k − q, ν + ω), σi =↑, ↓ , χzz(q, ω) = χ↑↑(q, ω) + χ↓↓(q, ω) (48) is a dynamic spin susceptibility of an electron subsystem. Here Gσ 0 (k, ω) = (iω − Ẽk,σ)−1 is a Fourier component of the zeroth one-electron Green function Gσ 0 (k, τ) = −〈T̂τak,σ(τ)a+ k,σ(τ)〉0E = −e−τ Ẽk,σ { 1 − nk,σ, τ > 0 −nk,σ, τ < 0 , (49) 589 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh and nq,σ = [1 + e−β(Ẽq,σ−µ)]−1 is the function of Fermi-Dirac distribution introduced above. After summation over frequency, the susceptibility becomes χσ1σ2(q, ω) = 1 N ∑ k nk,σ1 − nk+q,σ2 ω + Ẽk+q,σ1 − Ẽk,σ2 . (50) Hereafter we shall take into account only the pair effective interaction and neglect the multispin interactions. Using a method developed in [11,12], the grand partition function of the model will be written as a functional integral Ξ[ĉNRN ] = e−β(Ω0+∆Ω1) ∫ (Dϕ) exp ( −1 2 ∑ x \|~ϕx\|2 −βΨ[ϕ] ) , (51) here Ω0 ≡ Ω0[ĉ NRN ] = C− 1 β ∑ k ∑ σ=↑,↓ ln [ 1+e−β(Ẽk,σ−µ) ] − 1 β N∑ j=1 ĉj ln   sinh ( βh̃j 2S+1 2 ) sinh ( βh̃j 1 2 )  . (52) A symbol of functional integration stands for ∫ (Dϕ) ≡ ∏ α=x,y,z ∏ x>0 ∞∫ −∞ dϕα 0√ 2π ∫ ∞∫ −∞ dRe ϕα x√ π dIm ϕα x√ π (53) and components of field functions are ϕα q,ω = 1√ β β∫ 0 dτ ϕα q(τ) e−iτω, α = x, y, z. (54) The functional Ψ[ϕ] = − ∑ m>1 βm−1 m! ∑ x1,α1 · · · ∑ xm,αm Mα1,...,αm (x1; . . . ; xm) m∏ i=1 √ Jαi xi ϕαi xi . (55) Here the parameters of interaction are Jα x = Jk + β2 2 I2 k χα,−α(k, ω) (56) and the kernels of the functional Mα1;...;αm (x1; . . . ; xm) = 〈T̂τ Q̂ α1 x1 · · · Q̂αm xm 〉irr0s (57) 590 Thermodynamics of structurally disordered s – d model are the irreducible averages over the Gibbs’ distribution with Hamiltonian Ĥ0s, constructed on spin operators. A few first coefficient functions read [12]: Mz(x\|ĉNRN) = Mk = δω,0 1√ N N∑ j=1 ĉje −ikRj (M1(yj) − y) , Mzz(x1, x2\|ĉNRN) = δω1,0δω2,0 1 N { N∑ j=1 ĉje −i(k1−k2)RjM2(yj) − N∑ j1=1 N∑ j2=1 ĉj1 ĉj2e −ik1Rj1 +ik2Rj2 [ M1(yj1)y + M1(yj2)y + y2 ]} , M−+(x1, x2\|ĉNRN) = δω1+ω2,0 2 N N∑ j=1 ĉje −i(k1+k2)RjM1(yj)K −+ 0 (ω2\|yj). Here the ideal spin temperature Green function is K−+ 0 (τ1 − τ2\|yj) def = 〈T̂τS − j (τ1)S + j (τ2)〉0 2〈Sz j 〉0 = ∑ ω K−+ 0 (ω\|yj)e iω(τ1−τ2) , (58) and its Fourier components are K−+ 0 (ω\|yj) = (yj − iβω)−1. We consider here only the Gaussian approximation when we take into account only F1[ϕ] = ∑ x √ Jz xMz(x\|ĈNRN) ϕz x; (59) F2[ϕ] = ∑ x1 ∑ x2 √ Jz x1 Jz x2 Mzz(x1, x2\|ĉNRN)ϕz x1 ϕz x2 + 1 2 ∑ x1 ∑ x2 √ J− x1 J+ x2 M−+(x1, x2\|ĉNRN)ϕ− x1 ϕ+ x2 (60) in the functional. Correction to the thermodynamic potential (composed with the term ∆Ω1) in Gaussian approximation reads ∆ΩG[ĉNRN ] = β ∑ k∈Λ { w2 q %k%−k + y2I2 q ckc−k } χzz(q, 0) + 1 2β ∑ x { ln ( 1+2Jz xM 0 zz(x,−x\|ĉNRN) ) +ln ( 1+2J− x M−+(x,−x\|ĉNRN) )} + ∑ x Jz x M 0 z(x\|ĉNRN)M0 z(−x\|ĉNRN) [1 + 2Jz x M0 zz(x,−x\|ĉNRN)]2 . (61) 591 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh After the configurational averaging we obtain an expression for the thermody- namic potential Ω[ĉNRN ] = Ω0 + β ∑ q∈Λ [ w2 qS2(q) + I2 qy 2 c (c − 1 + cS2(q)) ] χzz(q, 0) + N β ∑ q∈Λ Jz q,0c(c − 1 + cS2(q)) [M1(βh0) − y]2 [ 1+2cJz q,0 {√ NM2(βh0)−y(c−1+cS2(q))(2M1(βh0)+y) }]2 + 1 2β ∑ q∈Λ ln ( 1+2cJz q,0 [√ NM2(βh0)− ( c−1+cS2(q) ) y ( 2M1(βh0)+y )]) + 1 2β ∑ q,ω ln ( 1 + 4c √ NJ− qωM1(βh0)K0(ω\|ĉN) ) . Here Ω0 is the thermodynamic potential of the self-consistent field approximation, set by expression (46) without the last term. We can neglect the third term in this expression, as [M1(βh0) − y] ∼ ∑ k ; ac- cordingly, in the Gaussian approximation the term with ∑ k ∑ p drops out. To sum over frequency in the last term it is necessary to use an approximation of homo- geneous media, i.e. to accept that the susceptibility does not depend on frequency: J− q,ω ' J− q,0 = Jq + (β2/2)I2 q χ↑,↓(q, 0). Then, using the integral representation of the natural logarithm, we get ∑ ω ln ( 1 + 4c √ NM1(βh0)J − q,0 β(h0 − iω) ) = 1 2πi ∮ Γ dξ ln ∣∣∣∣∣ βh0 + 4c √ NM1(βh0)J − q,0 − βξ βh0 − βξ ∣∣∣∣∣ 1 1 − e−βξ = 1 2 [ ln ( 1 − eβh0+4c √ NM1(βh0)J− q,0 ) − ln ( 1 − eβh0 )] . Here the contour of integration Γ is chosen such that it encloses poles ξ = h0 and ξ = h0 + 4c √ NM1(βh0)J − q,0/β of analytically prolonged logarithmic function. Finally, the averaged thermodynamic potential is Ω[ĉNRN ] = Ω0 + β ∑ q∈Λ [ w2 qS2(q) + I2 qy 2 c (c − 1 + cS2(q)) ] χzz(q, 0) + 1 2β ∑ q∈Λ ln ( 1+2cJz q,0 [√ NM2(βh0)− ( c−1+cS2(q) ) y ( 2M1(βh0)+y )]) + 1 4β ∑ q ( ln { 1 − eβh0+4c √ NM1(βh0)J− q,0 } /{1 − eβh0} ) . (62) The obtained equations for magnetization of a localized spin subsystem are y [ c2NJ0+ ∑ q 6=0 JqK2(q; c)+2β ∑ q∈Λ I2 q K2(q; c)χ zz(q, 0) ] −c2NJ0M1(βh0) 592 Thermodynamics of structurally disordered s – d model + cJ0 4 ( 1 + 4c √ NJ− q,0M2(βh0) 1 − e−βh0−4c √ NJ− q,0M1(βh0) − 1 1 − e−βh0 ) + c β ∑ q∈Λ Jz q,0 ( βc √ NJ0M3(βh0)−2K2(q; c)[βcJ0M2(βh0)y+M1(βh0)+y] ) 1+2cJz q,0 [√ NM2(βh0)−K2(q; c)y(2M1(βh0)+y ] =0, and the equation for electron spin polarization is c N ∑ q∈Λ Jz q,0 (√ NM3(βh0)−2K2(q; c)M2(βh0)y ) 1+2cJz q,0 [√ NM2(βh0)−K2(q; c)y(2M1(βh0)+y) ] +c [y − M1(βh0)] + 1 4% ∑ q∈Λ ( 1 + 4c √ NJ− q,0M2(βh0) 1 − e−βh0−4c √ NJ− q,0M1(βh0) − 1 1 − e−βh0 ) =0. The equation for the chemical potential in the given approximation remains the same as in the self-consistent field approach and is given by the last equation of system (72). 5. The self-consistent field approximation We shall neglect in the Hamiltonian (30) the term Ĥint which describes the correlation spin-spin and spin-electron interactions. Then the effective Hamiltonian reads Ĥeff = C + Ĥ0 = C − N∑ j=1 ĉj h̃j Sz j + ∑ k ∑ σ=↑,↓ Ẽk,σa + k,σak,σ. (63) A random external magnetic field h̃j, renormalized spectrum of free electrons Ẽk,σ, and non-operator part of the Hamiltonian are given by the relations (33), (34) and (33). Such an approximation is equivalent to the approximation of the self-consistent mean field. Peculiar to this problem in our case, in contrast to the ideal crystal magnetics, is a structural disorder of the model which requires additional analysis and calculations. Thermodynamic potential in the approximation of the mean self-consistent field for a fixed configuration of atoms of the system Ω0[ĉ NRN ] = − 1 β ln Sps Spe e−β(C+Ĥ0−µN̂e) (64) is easy to calculate and reduces to the analytical expression (52). The last term in this formula is the thermodynamic potential of a subsystem of noninteracting magnetic atoms in a random magnetic field. Now it is necessary to average the thermodynamical potential over all possible configurations. For this purpose we shall separate a structurally dependent compo- nent of the field h̃j h̃j = h0 + ∆j, (65) 593 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh where the spatially homogeneous component of the self-consistent magnetic field is h0 = gµBh + 1 % p I0 + c y J0, (66) and the quantity ∆j = y 2 √ N ∑ k 6=0 Jk [ ĉke ikRj + ĉ−ke −ikRj ] (67) characterizes a deviation of a field at the point Rj from the average value. It is ob- vious that the configurational average ∆j = 0. We will suppose that the fluctuation ∆j is small. As a criterion of smallness of fluctuations of the magnetic field acting on the spin of the atom localized at the point Rj it is convenient to use the parameter γ = √ 1 N ∑ 16j6N ĉj∆2 j ycJ0 = vM √ 1 V 2 ∑ k ∑ q J̃k J̃q K3(k;q;−k − q) (68) which is the ratio of the mean-arithmetic value of a linear dispersion of the distribu- tion to the value of the average field. Here J̃k = Jk/J0 is the normalized coefficient of interaction, and vM = V/NM = 1/cρ is the volume per one magnetic atom. The parameter γ will be small under condition of large concentrations of magnetic atoms and long-range spin-spin interactions. The concrete numerical estimation of param- eter can be performed, having a model interaction law and an expression for the correlation function K3. Let us expand the expression (52) into a power series over the fluctuations ∆j. After cumbersome transformations, we obtain the following final expression for the configurationally averaged value of thermodynamic potential Ω0(β; c) = Ω0[ĉnRn] in the approximation of self-consistent field Ω0(β; c) = 1 2 cNS(S+1)J(0) + ypV I0 + 1 2 c2y2NJ0 + y2 2 ∑ k 6=0 Jk K2(k) − 1 β ∑ k ∑ σ=↑,↓ ln [ 1 + e−β(Ẽk,σ−µ) ] − cN β ln [ sinh ( βh0 2S+1 2 ) sinh ( 1 2 βh0 ) ] + ∑ n>1 ynβn−1Mn(βh0) Nn−1n! ∑ k1 6=0 · · · ∑ kn 6=0 Jk1 · · ·Jkn Kn+1(k1; . . .;kn;k1 − . . . − kn). (69) Functions Mn(x) satisfy the recurrent relations Mn(x) = d/dxMn−1(x), n > 1. Here M0(x) = ln [sinh (0.5x[2S+1])/ sinh (0.5x)], the first derivative is M1(x) = S BS(xS), where BS(x) is the Brillouin function. The obtained expression for thermodynamic potential of the spin-electron mod- el in mean field approximation is a generalization of results known in the theory of magnetism to the case of an amorphous binary substitutional alloy, with making use of a special, more accurate for structurally disordered systems, method of allocation 594 Thermodynamics of structurally disordered s – d model of self-consistent fields. Expression (69) contains contributions to the thermodynam- ic potential due to structural fluctuations being taken into account. We shall make the following remark about this fact. In the thermodynamic potential Ω0(β; c) does not contain contributions from spin fluctuations, since the mean field approximation is used. The terms which contain the sum over wave vectors are due to structural fluctuations being taken into account. Using the thermodynamic potential in the form (69) to explore the thermodynamic properties of spin-electron model will not be correct, and the drawn physical conclusions will be invalid. The fact is that the contributions of the same order to the thermodynamic potential will be given by the correlation energy Ĥint being taken into account, which is neglected at the present stage. 6. Magnetizations and critical temperature of spin-electron model. The self-consistent field approximation Let us find the equation for magnetization in the simplest approximation, when the contribution of structural fluctuations in thermodynamic potential (69) is ne- glected, i.e. we shall take those correlation functions Kn(k1; . . . ;kn) ≡ 0, n > 1. Actually, as noted above, such terms being taken into account in the approximation of the self-consistent field cannot be validated. From the condition of a minimum of thermodynamic potential with respect to the order parameters y and p dΩ0(β; c) dy = 0, dΩ0(β; c) dp = 0 (70) and the equation for the chemical potential of the electron subsystem of the alloy ρe = 1 V ∑ k [ 1 1 + eβ(Ẽk,↑−µ) + 1 1 + eβ(Ẽk,↓−µ) ] , (71) where ρe = Ne/V is the density of conductivity electrons, we obtain a system of equations for y, p and µ: M = Bs ( S T [ gµBh + c S M J0 + 1 2 P ne I0 ]) , 4 3 ( EF T )3/2 P = F 1 2 ( µ+ξ T ) −F 1 2 ( µ−ξ T ) , 4 3 ( EF T )3/2 = F 1 2 ( µ + ξ T ) +F 1 2 ( µ−ξ T ) . (72) The system (72) was obtained using the expression for the density of states of free electrons ρ0(E) = 3/4 · Ne √ E · E −3/2 F , where Fermi energy of electron gas is EF = (3π2)2/3 ~ 2ρ 2/3 e /2m. The function F0.5(α) is Fermi-Dirac integral. Here 595 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh ξ = µBh + c I0 S M/2, and the reduced (normalized per the value of a localized spin) magnetization of an atom is M = y/S. In the equations (72) the transition to the values convenient in calculations is made: P = 2pV/Ne is the electron spin polarization per one electron, which can take values m̃ ∈ [−1; 1] (in Bohr magne- tons). The quantity ne = Ne/N is the number of electrons per one atom. Numerical solution of system (72) enables one to investigate the dependences of magnetizations M and P on temperature at different values of parameters J0, I0, ne, S, EF. Let us examine the value of electron spin polarization at zero temperature. In this case we shall not consider the first equation of the system, replacing every- where M = 1, i.e. by the maximal value of the relative magnetization of atoms. We shall suppose also that the condition cS\|I0\|/2EF < 1 is satisfied. In general, practically for all ferromagnetic metals and wide-zone semiconductors a stronger condition cS\|I0\|/2EF ¿ 1 is obeyed. That is so due to the fact that the energy EF at actual densities of electrons is large, order of EF ∼ 104 K, and the parameter of spin-electron interaction is I0 ∼ 102 K. Under such conditions it is possible to use the asymptotics of the Fermi-Dirac function Fn(α) = αn+1/(n + 1) for large values of its argument. From the equation for chemical potential we obtain an approximate formula, which describes a shift of chemical potential at zero temperature µ0 = µ(T = 0) due to spin-electron interaction µ0 = EF  1 + 2 3 2 − (1 + cI0S 2EF ) 3 2 − (1 − cI0S 2EF ) 3 2 √ 1 + cI0S 2EF + √ 1 − cI0S 2EF   . (73) When cS\|I0\|/2EF ¿ 1 it simplifies to µ0 = EF [ 1 − c2I2 0S 2 8E2 F + · · · ] . (74) Hence, a shift of Fermi energy under the effect of spin-electron interaction in ferro- magnetic metals is small, and this effect can be neglected. From the second equation of the system (72) we obtain a simple expression for electron magnetization at zero temperature P0 ≡ P (T = 0) P0 = 1 2 [( 1+ cSI0 2EF )3/2 − ( 1−cSI0 2EF )3/2 ] . (75) Formally this expression makes sense at 0 6 cSI0 2EF 6 1. However at the upper limit of this interval the electron magnetization gets nonphysical values P0 = √ 2. It indicates that the equation (75) is applicable only at cSI0 2EF ¿ 1. If one takes into account the renormalization of the Fermi energy due to spin- electron interaction then P0 = 1 2 [( 1+ cSI0 2EF − c2S2I2 0 8E2 F )3/2 − ( 1−cSI0 2EF − c2S2I2 0 8E2 F )3/2 ] . (76) 596 Thermodynamics of structurally disordered s – d model This formula is applicable for 0 6 cSI0/2EF 6 √ 2 − 1, when at the upper limit the physically allowable value of magnetization P0 ' 4/5 is obtained. However at values cSI0/2EF close to unity, results of calculations can be doubtful due to the approximations made in obtaining the equation (76). We can see from the equation (76) that electron magnetization m̃0 is always smaller than its maximal possible value: \|P0\| = 1. This effect is of a quantum origin and could be explained by the Pauli blocking rule. For the first time such an effect has been found in the Stoner model in which, using the concept of a molecular field and Fermi statistics, the problems of occurrence of ferromagnetic ordering of conductivity electrons in ferromagnetic metals were studied. If I0 > 0 (ferromagnetic spin-electron interaction), then P in (76) will be posi- tive, otherwise if I0 < 0 (antiferromagnetic spin-electron interaction) the values of electron spin polarization P are negative. In the first case the spin and electron magnetizations will be codirected and in the second – opposite directed, i.e. they will partially compensate each other. Figure 1. Chemical potential and electron magnetization at T = 0: a) – equati- on (74), b) – equation (75), c) – equation (76). In figure 1 the dependence of chemical potential µ0 at T = 0 calculated over the equation (74) on the parameter cSI0/2EF is shown, as well as the results of the exact numerical calculations for the system of the equations (72). The right part of figure 1 contains the dependence of the electron spin polarization on the same parameter calculated over the approximate formulas (75), (76) as well as the results of exact numerical calculations. At small values of cSI0/2EF the linear behaviour of electron spin polarization is observed. It is seen that the approximate extrapolation formulas give quantitatively good values in the region 0 6 cSI0/2EF < 0.5. In figure 2 the dependence of electron spin polarization and magnetization of a subsystem of localized spins from temperature is shown at different values of the model parameters. Results are obtained by numerical solving of the system of the equations (72). The temperature is given in units of T 0 c = c S(S +1) J0/3 – of Curie temperature of Heisenberg magnet in the molecular field approximation. 597 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh Figure 2. Electron spin polarization and magnetization of a subsystem localized spins at T = 0. An important characteristic of magnetic systems is the temperatures of phase transitions. We shall find the temperature of transition from the disordered para- magnetic to the ordered ferromagnetic phase (the so-called Curie temperature Tc). For this purpose we shall put the external magnetic field equal to zero. At T → Tc we have ξ = cI0SM/2 → 0. Chemical potential of the electron subsystem is an analytical function of temperature. Let us introduce the notation µc = µ(T = Tc). We shall expand the right sides of (72) in the vicinity of temperature Tc into the ascending power series up to the linear in M and m terms: M = S(S+1) 3 Tc ( c J0 M + 1 2 P neI0 ) , 4 3 ( EF Tc )3/2 P = 2F ′ 1 2 ( µc Tc ) c S I0M 2 Tc , 4 3 ( EF Tc )3/2 = 2F 1 2 ( µc Tc ) . (77) Here we denoted F ′ n(x) = dFn(x)/dx. Two first equations form a system of equa- tions, uniform with respect to M and P . It has non-trivial solutions if the main determinant of the system is equal to zero. From this condition we obtain an equa- tion for the Curie temperature: 1 − c S(S+1) J0 3 Tc = c S(S+1) 8 T 2 c neI 2 0 ( Tc EF )3/2 F ′ 1 2 ( µc Tc ) . (78) The equation (78) is transcendent. We can make estimations for some limiting cases. So if concentration of electrons ne or parameter I0 are equal to zero, the solu- tion of equation (78) will be the critical temperature for the ideal Heisenberg model. If J0 is zero (classical s-d model of Shubin-Vonsovsky), we obtain the following 598 Thermodynamics of structurally disordered s – d model equation 1 = c S(S+1) 8 T 2 c I2 0 1 % ( Tc EF )3/2 F ′ 1 2 ( µ Tc ) . (79) Using the asymptotical expression for the Fermi-Dirac function we find Tc = c S(S+1) 8 I2 0 EF 1 % . (80) Similarly, we shall obtain an estimation for the full equation (78) Tc T o c = 1 + 9 8 1 c S(S+1) ( I0 J0 )2 1 % T o c EF . (81) Figure 3. Dependence of Curie temperature on concentration of magnetic impu- rities and ratio of Fermi energy to the spin-electron interaction integral. In figure 3 the results of numerical calculations of critical temperature are pre- sented at some values of the model parameters. Calculations are performed for the case of metal alloys, when parameter cS\|I0\|/2EF ¿ 1. 7. Nondirect spin-spin interaction in ferromagnetic phase From the equation (71) one can draw an important conclusion: in the presence of magnetic ordering in the system the indirect pair spin-spin interaction is anisotropic, with the preferential direction along the axis of quantization. In the paramagnetic phase such an interaction becomes isotropic, which is typical of the RKKY interac- tion. In general, to obtain an interaction of the RKKY type from expression (71), we put the self-consistent field equal to zero (the absence of external magnetic field and a paramagnetic phase), replace the Fourier components Ik with the constant I for all values of wave vectors (contact spin-electron interaction), put the frequency ω 599 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh equal to zero in a susceptibility χα1,α2(k; ω), and pass to the limit of absolute zero temperature. Then the Fourier components of the effective spin-spin interaction are Jeff(k) = lim T→0 βI2 2 χ(k, 0) = I2m 16%~2π2 ∑ σ=↑,↓ qvσ [ 4q2 vσ − k2 4kqvσ ln ∣∣∣∣ k + 2qvσ k − 2qvσ ∣∣∣∣ + 1 ] (82) here qvσ = √ 2m {µ − κσ(µBh + ycI0) − w0}/~. Figure 4. The typical behaviour of RKKY type interaction (figure at left side) and normalized by interaction without spin-electron effect (figure at right side) In the coordinate representation Jeff(r) = I2k2 F 16π%EF ∑ σ=↑,↓ q4 vσ sin(2qvσr) − 2qvσr cos(2qvσr) (qvσr)4 (83) we receive some complication of the known form of nondirect spin-spin interaction. Here z = c y I0/2EF, Jeff(r) means the energy in units normalized by magnitude I2k2 F 16π%EF q4 vσ and the ∆Jeff(r) = J0 eff(r) − (J↑ eff(r) + J↓ eff(r)) 2 normalized of the same magnitude too. The parameter z = 0 in case of the absence of an exterior magnetic field. In figure 4 one can see that with injection in metal of some concentrations of localized atom moments the spins of conduction electrons create oscillating polari- zation near it. The oscillations of spin density are similar to the Friedel oscillation of charge density. 600 Thermodynamics of structurally disordered s – d model 8. Conclusions From expression (81) one can see that the presence of exchange spin-electron interaction results in the increase of critical temperature. It could be explained by increasing the effective spin-spin interaction due to spin-electron interaction. The contribution of the electron subsystem to magnetization and the value of critical temperature enhance with the increase of electron concentration ne and the param- eter of exchange spin-electron interaction I0. The obtained above results for magnetizations of the subsystems of localized spins and of conductivity electrons, as well as the critical temperature of phase transition in the self-consistent field approximation qualitatively correctly display the known experimental data of a certain class of magnetic alloys. However, for a quantitative description of properties of magnetic alloys, in par- ticular, in the absence of small parameters and at small concentrations (densities) of magnetoactive atoms, it is necessary to take into account the structural and thermodynamical spin fluctuations, as done in section 4. Without the high-order fluctuations being consistently taken into account it is impossible to explain some observable properties even qualitatively. It concerns, for example, the features of a spectrum of elementary excitations in the short-wave regions (ka ∼ 1), the per- colation limit of the critical temperature, features of the system behaviour in the vicinity of a critical point and so forth. References 1. Vonsovsky S.V. Magnetizm. Moskva, Nauka, 1971 (in Russian). 2. Nagajev E.L. Fizika magnitnyh poluprovodnikov. Nauka, Moskva, 1979 (in Russian). 3. Metfessel Z., Mattis D. Magnitnyje poluprovodniki. Mir, Moskva, 1972 (in Russian). 4. Izumov Yu.A., Katsnelson M.I., Skriabin Yu.N. Magnetizm kollektivizirovannyh elek- tronov. Fizmatlit, Moskva, 1994 (in Russian). 5. Stasyuk I., Shvajka A., Tabunshchyk K., Condens. Matter Phys., 1999, 2, 109–132. 6. Stasyuk I.V. Pseudospin-electron model for strongly correlated electron system (Ther- modynamics and dynamamics). – In: AIP Conference Proceedings. Highlights in con- densed matter physics, 2003, 695, 281–290. 7. Izumov Yu.A., Kassan-Ogly F.A., Skriabin Yu.N. Polevyje metody v teorii ferromag- netizma. Nauka, Moskva, 1974 (in Russian). 8. Letfulov B.M., FTT, 1976, 18, 645. 9. Maćkowiak J., Wísniewski M., Physica A, 1997, 242, 482. 10. Công B.T., Phys. Stat. Sol. (b), 1986, 134, 569. 11. Vakarchuk I.A., Rudavskii Yu.K., TMF, 1984, 58, 445. 12. Vakarchuk I.A., Rudavskii Yu.K., Ponedilok G.V, Phys. Stat. Sol. (b), 1985, 128, 231. 13. Ponedilok G.V., Savenko V.P., Phys. Stat. Sol. (b), 1994, 184, 433. 14. Dovgopol S.P., Kroxin A.L., Mirzojev A.A., TMF, 1979, 41, 378. 15. Brovman E.G., Kagan Yu.M., UFN, 1974, 122, 3, 369. 16. Vavruch M.V., Krochmalskii T.Ye., Ukrainian Journal of Physics, 1987, 32, 648. 17. Ianarella L., Guimaraes A.P., Silva X.A., Phys. Stat. Sol. (b), 1982, 114, 255. 18. Vakarchuk I.O., Tkachuk V.M., Kuliy T.V., Phys. Stat. Sol. (b), 1998, 208, 167. 601 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh Термодинаміка структурно невпорядкованої s–d моделі Ю.Рудавський, Г.Понеділок, Л.Дорош Національний університет “Львівська політехніка”, вул. С.Бандери 12, Львів 79013, Україна Отримано 5 травня 2004 р., в остаточному вигляді – 22 жовтня 2004 р. Спін-електронна обмінна модель узагальнюється і застосовуєть- ся для опису магнітних станів аморфних сплавів з рідиноподібним типом структурної невпорядкованості. Розглянута схема послідов- ного врахування вкладу структурних флуктуацій у термодинамічні функції та спостережувані величини. За теорією збурень побу- довано функціонал термодинамічного потенціалу у формі функц- іонального степеневого ряду. У наближенні хаотичних фаз (RPA) розраховано великий термодинамічний потенціал моделі. Записані співвідношення самоузгодження, з яких знаходяться рівняння для намагніченостей та критичної температури переходу “парамагнетик- феромагнетик”. Ключові слова: s–d-модель, феромагнітні сплави, вільна енергія, намагніченість, температура Кюрі, функціональні інтеграли PACS: 72.15.C, 73.20.H, 75.10, 75.30.E, 82.65.Y 602

Thermodynamics of structurally disordered s–d model

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