Electron density of states and spectrum of disordered s-d model

Structurally disordered s-d model of magnetism in metals is investigated. The simplified model of binary alloy structure is offered, the structural correlation functions are calculated. The self-consistent equations for calculation of a spectrum, magnetization, electronic spin polarization and te...

Повний опис

Збережено в:

Бібліографічні деталі
Дата:	2001
Автори:	Rudavskii, Yu.K., Ponedilok, G.V., Dorosh, L.A.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2001
Назва видання:	Condensed Matter Physics
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/119772
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Electron density of states and spectrum of disordered s-d model / Yu.K. Rudavskii, G.V. Ponedilok, L.A. Dorosh // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 141-148. — Бібліогр.: 6 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	nasplib_isofts_kiev_ua-123456789-119772
record_format	dspace
spelling	nasplib_isofts_kiev_ua-123456789-1197722025-06-03T16:30:15Z Electron density of states and spectrum of disordered s-d model Спектр та густина електронних станів невпорядкованої s - d моделі Rudavskii, Yu.K. Ponedilok, G.V. Dorosh, L.A. Structurally disordered s-d model of magnetism in metals is investigated. The simplified model of binary alloy structure is offered, the structural correlation functions are calculated. The self-consistent equations for calculation of a spectrum, magnetization, electronic spin polarization and temperature of phase transition are obtained. Досліджується структурно невпорядкована s-d модель магнетизму металiв. Запропонована спрощена модель структури бiнарної сумiшi, порахованi структурні кореляційні функції. Отриманi рівняння для самоузгодженого розрахунку спектра, густини одноелектронних станiв, намагнiченостi, електронної спінової поляризацiї i температури фазового переходу 2001 Article Electron density of states and spectrum of disordered s-d model / Yu.K. Rudavskii, G.V. Ponedilok, L.A. Dorosh // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 141-148. — Бібліогр.: 6 назв. — англ. 1607-324X PACS: 75.30.Ds, 75.50.K DOI:10.5488/CMP.4.1.141 https://nasplib.isofts.kiev.ua/handle/123456789/119772 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Structurally disordered s-d model of magnetism in metals is investigated. The simplified model of binary alloy structure is offered, the structural correlation functions are calculated. The self-consistent equations for calculation of a spectrum, magnetization, electronic spin polarization and temperature of phase 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. Electron density of states and spectrum of disordered s-d model Condensed Matter Physics
author_facet	Rudavskii, Yu.K. Ponedilok, G.V. Dorosh, L.A.
author_sort	Rudavskii, Yu.K.
title	Electron density of states and spectrum of disordered s-d model
title_short	Electron density of states and spectrum of disordered s-d model
title_full	Electron density of states and spectrum of disordered s-d model
title_fullStr	Electron density of states and spectrum of disordered s-d model
title_full_unstemmed	Electron density of states and spectrum of disordered s-d model
title_sort	electron density of states and spectrum of disordered s-d model
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2001
url	https://nasplib.isofts.kiev.ua/handle/123456789/119772
citation_txt	Electron density of states and spectrum of disordered s-d model / Yu.K. Rudavskii, G.V. Ponedilok, L.A. Dorosh // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 141-148. — Бібліогр.: 6 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT rudavskiiyuk electrondensityofstatesandspectrumofdisorderedsdmodel AT ponedilokgv electrondensityofstatesandspectrumofdisorderedsdmodel AT doroshla electrondensityofstatesandspectrumofdisorderedsdmodel AT rudavskiiyuk spektrtagustinaelektronnihstanívnevporâdkovanoísdmodelí AT ponedilokgv spektrtagustinaelektronnihstanívnevporâdkovanoísdmodelí AT doroshla spektrtagustinaelektronnihstanívnevporâdkovanoísdmodelí
first_indexed	2025-12-01T23:02:39Z
last_indexed	2025-12-01T23:02:39Z
_version_	1850348834806300672
fulltext	Condensed Matter Physics, 2001, Vol. 4, No. 1(25), pp. 141–148 Electron density of states and spectrum of disordered s -d model Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh Lviv National Polytechnic University, 12 S.Bandera Str., 79013 Lviv, Ukraine Received August 23, 2000, in final form November 1, 2000 Structurally disordered s-d model of magnetism in metals is investigated. The simplified model of binary alloy structure is offered, the structural corre- lation functions are calculated. The self-consistent equations for calculation of a spectrum, magnetization, electronic spin polarization and temperature of phase transition are obtained. Key words: amorphous systems, magnetization, polarization PACS: 75.30.Ds, 75.50.K 1. Introduction The exchange s-d model is used to describe the alloy properties, that contains transitional and rare-earth elements. The exchange s-d model offered by Shubin and Vonsovskii was further developed by Ziner, Turov, Kassuj, Iosid and others, and it found a further application in different fields of solid state physics. First the electric and magnetic properties description of transitional d-metals was suggested and at present it is a basic magnetism theory of the rare-earth metals, magnetic quasi- conductors [1] and dissolved alloys of transitional metals [3]. From the theoretical point of view, the importance of the s-d model is determined by a lot of wonderful and non-trivial effects that it contains: formation of spin polarons, ferrons, fluctuons in magnetic quasi-conductors [1,4], occurrence of the spin glass state in metallic alloys [2], Kondo effect etc. 2. Spin-electron model of amorphous magnet An amorphous system of N atoms in volume V ⊂ R 3 is considered, part of which has got localized magnetic moments (further – magnetic alloy subsystem), and other atoms do not have localized magnetic moments (non-magnetic subsystem of amor- phous alloy). Atomic coordinates (R1, ...,RN) = R N ∈ V accept random values. Quantitative relation between magnetic and non-magnetic atoms is described by c© Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh 141 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh concentration c (0 6 c 6 1). The microscopic model of amorphous magnet consists of two quantum subsystems i.e., a subsystem of the localized atomic spins and a subsystem of conductivity electrons. Hamiltonian of model has the form Ĥ = Ĥs + Ĥel + Ĥel−s. (1) The first term describes the energy of the subsystem of localized spins, which are in the external magnetic field h, and the pair couple by the Heisenberg exchange interaction Ĥs = 1 2 N cS(S + 1)J(\|R\| = 0)− µh ∑ 16j6N ĉjS z j − 1 2 ∑ q∈Λ JqSqS−q . (2) Here µ is the moment of magnetic atoms, and Sj is the spin operator of the magnetic atom placed in point Rj ∈ V . The subsystem of conductivity electrons is described in the pseudopotential ap- proach Ĥel = ∑ k∈Λ ∑ σ=±1 Ekσa+k,σak,σ + ∑ k,q∈Λ ∑ σ±1 Wqa + k,σak−q,σ . (3) Here a+kσ(akσ) are Fermi operators of creation (annihilation) of electrons in states {k, σ}. The index σ takes values respective to two possible projections of the electron spin on the axis of quantization OZ. In the paper we also employed notation σ = (↑ , ↓). The wave vector k ∈ Λ, where Λ = {k : k = ∑ 16α63 2π V −1/3nαeα, nα ∈ Z, (eα, eβ) = δαβ} is momentum quasi-continuous space. The spectrum of free electronic gas in presence of external magnetic field h is Ekσ = ~ 2k2 2m − κσµBh. (4) Scattering potential of the conductivity electrons on the ions is given by Wq = 1 N N∑ j=1 e−iqRjwq , wq = N V ∫ V drw(\|r\|) e−iqr , (5) where w(\|r\|) is an electron-ions pseudopotential. For local pseudopotentials, the matrix element of scattering potential of electrons on ions wq depends only on the module of impulse transfer. The Hamiltonian of spin-electron interaction takes the form Ĥel−s = − 1√ N ∑ q∈Λ Iq { Sz qσ̂ z −q + 1 2 [ S+ q σ̂ − −q + S− q σ̂ + −q ]} . (6) In expression (6), which is similar to (2), the operator Sα q = 1√ N ∑ 16j6N ĉjS α j e−iqRj , α = z,+,− (7) 142 Electron density of states and spectrum of disordered s-d model where S± j = Sx j ± iSy j . In the case of lattice systems (Rj ∈ Z 3) the operator (7) is the exact Fourier-transform of operators Sα j . In formula (6) the bilinear combinations of electrons creation and annihilation operators are defined σ̂z q = ∑ σ=↑,↓ κσ 2 n̂q,σ, n̂qσ = ∑ k∈Λ a+k,σ ak+q,σ, σ̂+ q = ∑ k∈Λ a+k,↑ ak+q,↓, σ̂− q = ∑ k∈Λ a+k,↓ ak+q,↑. (8) The Fourier-components n̂q of the density operator are n̂q = ∫ dr n̂(r) e−iqr; n̂(r) = 1 V ∑ q∈Λ n̂q e iqr. (9) Hermitian operator σ̂z q is the Fourier-component of the operator of the electronic spin polarization density. It is convenient to describe the structure of amorphous system by correlation functions Qn(k1, . . . ,kn; c) = 〈ĉk1 . . . ĉkn 〉irr conf . (10) Here the Fourier-coefficient of the atomic density fluctuations is ĉk = 1√ N N∑ j=1 ĉj e −ikRj , k 6= 0. The structure correlation functions are reduced to such a form Q1(k; c) = 0, Q2(k1,k2; c) = c (1− c+ c S2(k1,−k1)) δk1+k2,0 , . . . . (11) The correlation functions of the atomic density fluctuations Sm(k1, . . . ,km) are impossible to calculate from the first principles, and were accepted as phenomeno- logical values. Let us suppose that fluctuations of atomic density ̺k are described by the Gaus- sian law. Probability distribution function of fluctuations of the atomic density is P (. . . , ̺k, . . .) = ∏ k 6=0 1√ 2πS2(k) exp [ − ∑ k 6=0 ̺k̺−k S2(k) ] . (12) Dispersion of Gaussian distribution S2(k) = ∫ (d̺k) ̺k ̺−k P (. . . , ̺k, . . .) (13) is the pair structural system factor. Integration in formula (13) over ̺k with distri- bution function (12) is equivalent to the operation of configurational averaging. The structural functions of higher order as follows: Sm(k1, . . . ,km) def = 〈̺k1 , . . . ̺km 〉R = ∫ (d̺k) ̺k1 · · · ̺km P (. . . , ̺k, . . .). (14) 143 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh When m is even (m = 2n), then S2n(k1, . . . ,k2n) = ∑ {p} S2(ki1)δki1 +kj1 ,0 S2(ki2)δki2 +kj2 ,0 · · ·S2(kin)δkin+kjn ,0. (15) In this formula the sum is taken over for all possible decomposition wave numbers k1, . . . ,k2n on pairs. 3. Dispersion law and density of electron states The Green function Gσ(k,q\|ω) = 〈〈ak,σ\|a+q,σ′〉〉 ω is needed to evaluate the distri- bution of electrons in momentum space nk,σ = i lim ε→0 +∞∫ −∞ Gσ(k,k\|ω+iε)−Gσ(k,k\|ω−iε) eβω + 1 dω. (16) Where β = (kBT ) −1 is the inverse temperature in energy units. For Hamilton operator (1) the equation of motion for one-electron Green function has the form (~ω − Ek,σ)Gσ(k,q\|ω) = 1 2π δk,q + ∑ p∈Λ WpG σ(k− p,q\|ω) − 1 2 √ N ∑ p∈Λ Ip {κσL σ(p,k− p;q\|ω) +Mσ(p,k− p;q\|ω)} , σ = ±1 (↑, ↓). (17) The equation (17) contains the Green functions of higher order Lσ(p, l;q\|ω) = 〈〈Sz pal,σ\|a+q,σ〉〉ω, Mσ(p, l;q\|ω) = 〈〈S−σ p al,−σ\|a+q,σ〉〉ω. (18) We introduce here the decoupling procedure Lσ(p,k− p;q\|ω) → 〈Sz p〉〈〈ak−p,σ\|a+q,σ〉〉ω, Mσ(p,k− p;q\|ω) ≡ 0. (19) The thermodynamic average value is 〈S z p〉 = yN−1/2 N∑ j=1 e−ipRj , where y ≡ 〈Sz j 〉 is the magnetization of atom. Then from equation (17) we obtain (~ω − Ek,σ)Gσ(k,q\|ω) = 1 2π δk,q + ∑ p∈Λ,p6=0 W̃p,σG σ(k− p,q\|ω). (20) Here the effective scattering pseudopotential W̃p,σ ≡ 1 N n∑ j=1 ( wp − 1 2 Ip y κσ ) e−ipRj = 1 N n∑ j=1 w̃p,σ e −ipRj . (21) 144 Electron density of states and spectrum of disordered s-d model We write the following equation to calculate the configurational averaging of one-electron Green function (~ω − Ek−p,σ)〈W̃p,σG σ(k− p,q\|ω)〉 conf = = 1 2π δk−p,q〈W̃p,σ〉conf + ∑ p′∈Λ 〈W̃p,σW̃p′,σG σ(k− p− p′,q\|ω)〉 conf . (22) In the latter formula we made approximation 〈W̃ W̃G〉 → 〈W̃ W̃ 〉〈G〉. (23) The value 〈W̃p,σ〉conf = 0 because in initial Hamiltonian in the sum over p the term with p = 0 is absent. The configurational averaged value is equal to 〈W̃p,σW̃p′,σ〉conf = 1 N S2(p)w̃p,σw̃p′,σδp,−p′, where S2(p) = 〈̺p̺−p〉 is pair structural factor. From the equation for the averaged Green function in such an approach, the final expression is obtained G(k,q\|ω) ≡ 〈Gσ(k,q\|ω)〉 conf = δk,q 2π 1 (~ω − Ek,σ − Σσ(k\|ω)) . (24) The self-energy part of the one-electron Green function in this approximation is Σσ(k\|ω) = 1 N ∑ p∈Λ S2(p)\|w̃p,σ\|2 ~ω − Ek−p,σ . Equation for electron spectrum in Stonner subband with spin orientation σ takes the form ~ω = Ek,σ + 1 N ∑ p∈Λ S2(p)\|w̃p,σ\|2 ~ω − Ek−p,σ . (25) The electron energy in the long-wavelength region (small k) E(k) = ∆(h) + ~ 2k2 2m∗ + . . . , (26) where m∗ is the effective electron mass in the external magnetic field m∗ = m 1 + 16Ω0m 2 3π2~2 ∞∫ 0 dp1 p21 S2(p1)\|w̃p1,σ\|2 (27) and ∆(h) = 2Ω0m π2~2 ∞∫ 0 dp1S2(p1)\|w̃p1,σ\|2 − κσµBh (28) 145 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh is Stonner gap in the electron spectrum. For non-perturbation case when Σσ(k\|ω) ≡ 0 the well known formula is obtained ̺σ(ω) = m3/2 √ 2π2 ~3 √ ~ω + κσµBh, (29) which describes the density of states of free electrons in the external magnetic field. In the Σσ(k\|ω) 6= 0 case the self-energy part contains the real and imaginary parts Σσ(k\|ω) = Σ′(k\|ω) + i Σ′′(k\|ω), where Σ′(k\|ω) = 1 N P ∑ p∈Λ S2(p)\|w̃p,σ\|2 ~ω − Ek−p,σ , Σ′′(k\|ω) = − π N ∑ p∈Λ S2(p)\|w̃p,σ\|2 δ (~ω − Ek−p,σ) . Equation for density of states may be written in such a form ̺σ(E) = m 2π2~ ∑ i p̃i ∣∣∣∣∣∣ −1−Ω0m 2 4π2~4 1 p̃3i ∞∫ 0 dp p S2(p)\|w̃p,σ\|2 ln ∣∣∣∣∣ Eσ − ~2 2m (p̃i − p)2 Eσ − ~2 2m (p̃i + p)2 ∣∣∣∣∣ − Ω0m 2 4π2~4 1 p̃i ∞∫ 0 dp p S2(p)\|w̃p,σ\|2 ( p/p̃i − 1 Eσ − ~2 2m (p̃i − p)2 + p/p̃i + 1 Eσ − ~2 2m (p̃i + p)2 )∣∣∣∣∣∣ −1 .(30) In this equation Eσ = E + κσµBh, and p̃i denotes simple roots of equation Eσ − ~ 2 2m p̃2 − Ω0m 4π2~2 1 p̃ ∞∫ 0 dp p S2(p)\|w̃p,σ\|2 ln ∣∣∣∣∣ Eσ − ~ 2 2m (p̃− p)2 Eσ − ~2 2m (p̃+ p)2 ∣∣∣∣∣ = 0. If we don’t employ approach (23), and start with the iteration procedure, then the expression for the averaged function of configurations will be written in series form (~ω − Ek,σ)G(k\|ω) = 1 2π { 1 + 1 N ∑ p1∈Λ \|w̃p1,σ\|2S2(p1) (~ω − Ek−p1,σ)(~ω − Ek,σ) + 1 N2 ∑ p1,p2∈Λ \|w̃p1,σ\|2\|w̃p2,σ\|2S2(p1)S2(p2) (~ω − Ek−p1,σ)(~ω − Ek−p1−p2,σ)(~ω − Ek−p2,σ)(~ω − Ek,σ) + 1 N2 ∑ p1,p3∈Λ \|w̃p1,σ\|2\|w̃p3,σ\|2S2(p1)S2(p3) (~ω − Ek−p1,σ)(~ω − Ek,σ)(~ω − Ek−p3,σ)(~ω − Ek,σ) + 1 N2 ∑ p1,p2∈Λ \|w̃p1,σ\|2\|w̃p2,σ\|2S2(p1)S2(p2) (~ω − Ek−p1,σ)(~ω − Ek−p1−p2,σ)(~ω − Ek−p1,σ)(~ω − Ek,σ) + . . . } . (31) 146 Electron density of states and spectrum of disordered s-d model We can see, that Green function is represented in the form of infinite series, each following term of which contains an additional factor \|w̃\|2 S2/N(~ω−ε)(~ω−ε) with suitable combination indexes. The issue of convergence of such series remains open, because in our problem small parameters are absent. Such series in field theory are very well known. Especially it is frequently met in the problems concerning the interaction of electrons with bosonic field (electron- phonon interaction). But in the field models, the S2(q\|ω) and w̃I,σ values have got another sense. Therefore we can use the results known from the field theory for presenting the self-energy part of one-electron Green function Σ(k\|ω) = 1 N ∑ p∈Λ S2(p)\|w̃p,σ\|2 ~ω − Ek−p,σ − Σ(k− p\|ω) . (32) The absence of a small parameter in our problem does not permit to estimate a correct approximation for self-energy part. The detailed numerical computations and studies are necessary. For self-consistent computation of the dispersion law it is necessary to find equa- tions for calculation magnetization M of system localized spins and electron spin polarizationm. Such equations were obtained in the approach of self-consistent field and their form is written in [6] M = Bs ( S T [ µh+ c S M J0 +m V N I0 ]) , 8 3 ( EF T )3/2 m = F1/2 ( µ+ ξ T ) − F1/2 ( µ− ξ T ) , 4 3 ( EF T )3/2 = F1/2 ( µ+ ξ T ) + F1/2 ( µ− ξ T ) . (33) Here F1/2(α) is the Fermi-Dirak integral. Values I0 = lim k→0 Ik, J0 = lim k→0 Jk and µ is the chemical potential, and variable ξ = µh + 1 2 c I0 SM . In the first equation of the system we employed the notation Bs(x) for Brillouin function. A solution of the system (33) makes it possible to build the dependence M and m on temperature at different parameters values J0, I0, n, S, EF. From equations (33) at T = 0, the value of electron spin polarizationm is smaller than maximal possible value which is equal to 1. It is valid for S\|I 0\|/2EF parameter considerably smaller for unit. Condition S\|I0\|/2EF ≪ 1 is also valid for all ferro- magnetic metals and semiconductors. It is so that EF due to real electrons densities is a big value: EF ∼ 104K. In this case, using the asymptotic for Fermi-Dirak func- tion Fn(α) = αn+1/(n+ 1), the analytic expression for electron magnetization at temperature T = 0 was obtained m = 1 2 [( 1 + cSI0M 2EF )3/2 − ( 1− cSI0M 2EF )3/2 ] . (34) 147 Yu.K.Rudavskii, G.V.Ponedilok, L.A.Dorosh 4. Conclusions In this paper firstly the equations for spectrum and electron states density of structurally disordered s-d model are obtained. The dispersion law of both Stonner subbands is investigated. The electron effective mass in the first Born approximation is found. The integral equation for self-energy part is obtained for special structure type. It is valid in long-wavelength region. The system of self-consistent equations for the model of magnet states calcula- tion is obtained. If I0 > 0 (ferromagnetic spin-electron interaction) then m in (34) will be positive and if I0 < 0 (non-ferromagnetic spin-electron interaction) then m is negative. In the first case, spin and electron magnetization will be co-directed and in the second case controversially directed and partly compensate one another. Particularly, when parameter SI0/2EF is close to a unit, m also approaches a unit. In case SI0/2EF > 1 value m is identically equal to unit. The last two cases may be realized in magnetic semiconductors with small density of conductivity electrons. References 1. Nagajev E.L. Physics of magnetic semiconductors. – Moscow, Nauka, 1979. 2. Fisher K.H. Spin glasses (I). // Phys. Stat. Sol. (b), 1983, vol. 116, No. 2, p. 357–414. 3. Tselvick A.M., Wiegmann P.B. Exact results in the theory of magnetic alloys. // Ad- vances in Phys., 1983, vol. 32, No. 4, p. 453–713. 4. Andrej N., Zuruja K., Löwenstein J.H. Solution of the Kondo problem. // Rew. Mod. Phys., 1983, vol. 32, No. 2, p. 331–402. 5. Yonezawa I.A., Morigaki K. Coherent potential approximation. // Prog. Theor. Phys. (Supplement), 1973, vol. 58, p. 1–76. 6. Rudavsky Yu.K., Ponedilok G.V., Dorosh L.A. Long-wave spin excitation in crystalline s-d models. // Cond. Matter Phys., 1998, vol. 1(13), p. 145–160. Спектр та густина електронних станів невпорядкованої s -d моделі Ю.Рудавський, Г.Понеділок, Л.Дорош Національний університет “Львівська політехніка” 79013 Львів, вул. С.Бандери, 12 Отримано 23 серпня 2000 р., в остаточному вигляді – 1 листопада 2000 р. Досліджується структурно невпорядкована s-d модель магнетизму металiв. Запропонована спрощена модель структури бiнарної су- мiшi, порахованi структурні кореляційні функції. Отриманi рівняння для самоузгодженого розрахунку спектра, густини одноелектронних станiв, намагнiченостi, електронної спінової поляризацiї i температу- ри фазового переходу Ключові слова: аморфнi системи, намагніченість, поляризація PACS: 75.30.Ds, 75.50.K 148

Electron density of states and spectrum of disordered s-d model

Репозитарії

Схожі ресурси