Self-consistent model of strong coupling theory of electron correlations in disordered crystals
The method of the description of electron correlations in the disordered crys-tals based on a Hamiltonian of many-particle system as well as the diagram technique for calculation of Green’s functions are developed. Розвинено методу опису електронних кореляцій у невпорядкованих кри-сталах, що базуєть...
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| Опубліковано в: : | Успехи физики металлов |
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| Дата: | 2012 |
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Інститут металофізики ім. Г.В. Курдюмова НАН України
2012
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| Цитувати: | Self-consistent model of strong coupling theory of electron correlations in disordered crystals / S.P. Repetsky, O.V. Tretyak, I.G. Vyshivanaya, V.V. Shastun // Успехи физики металлов. — 2012. — Т. 13, № 2. — С. 189-223. — Бібліогр.: 38 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860068794752303104 |
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| author | Repetsky, S.P. Tretyak, O.V. Vyshivanaya, I.G. Shastun, V.V. |
| author_facet | Repetsky, S.P. Tretyak, O.V. Vyshivanaya, I.G. Shastun, V.V. |
| citation_txt | Self-consistent model of strong coupling theory of electron correlations in disordered crystals / S.P. Repetsky, O.V. Tretyak, I.G. Vyshivanaya, V.V. Shastun // Успехи физики металлов. — 2012. — Т. 13, № 2. — С. 189-223. — Бібліогр.: 38 назв. — англ. |
| collection | DSpace DC |
| container_title | Успехи физики металлов |
| description | The method of the description of electron correlations in the disordered crys-tals based on a Hamiltonian of many-particle system as well as the diagram technique for calculation of Green’s functions are developed.
Розвинено методу опису електронних кореляцій у невпорядкованих кри-сталах, що базується на Гамільтоніяні багаточастинкової системи та діяг-рамній техніці для розрахунку Ґрінових функцій.
Развит метод описания электронных корреляций в неупорядоченных кристаллах, основанный на гамильтониане многочастичной системы и диаграммной технике для расчёта функций Грина.
|
| first_indexed | 2025-12-07T17:09:32Z |
| format | Article |
| fulltext |
189
PACS numbers: 63.20.K-, 71.15.Ap, 71.20.Tx, 71.27.+a, 72.10.-d, 72.25.-b, 73.63.Fg
Self-Consistent Model of Strong Coupling Theory
of Electron Correlations in Disordered Crystals
S. P. Repetsky1, O. V. Tretyak2, I. G. Vyshivanaya2, and V. V. Shastun1
1Department of Physics, Taras Shevchenko National University of Kyiv,
Volodymyrs’ka Str., 64,
01033 Kyyiv, Ukraine
2Institute of High Technologies,
Taras Shevchenko National University of Kyiv,
Academician Glushkov Ave., 4
g,
03022 Kyyiv, Ukraine
The method of the description of electron correlations in the disordered crys-
tals based on a Hamiltonian of many-particle system as well as the diagram
technique for calculation of Green’s functions are developed. Electron states
of system are featured within the scope of the self-consistent tight-binding
multiband model. Processes of scattering of electrons on potentials of the ion
cores of different kinds and oscillations of a crystal lattice are considered.
The consecutive method for the account of the long-range Coulomb interac-
tion of electrons at different sites of a lattice is offered. The cluster expan-
sion for a density of states, a free energy and an electrical conductivity of the
disordered system is obtained. As shown, the contributions of processes of
scattering of elementary excitations on clusters decrease with increasing
number of sites in a cluster according to some small parameter. Precision of
calculation is defined by accuracy of a renormalization of vertex parts of
mass operators of electron–electron and electron–phonon interactions as well
as by small parameter of the cluster expansion. The nature of spin-dependent
electron transport in carbon nanotubes with chromium atoms adsorbed on a
surface is found out. As shown, the quantity of spin-dependent transport is
related to the relative offset of energy levels of electrons (i.e. to the Coulomb
gaps arising in a vicinity of the Fermi level) for different projections of a spin
in an external magnetic field.
Розвинено методу опису електронних кореляцій у невпорядкованих кри-
сталах, що базується на Гамільтоніяні багаточастинкової системи та діяг-
рамній техніці для розрахунку Ґрінових функцій. Електронні стани сис-
теми описано в рамках самоузгодженого багатозонного моделю сильного
зв’язку. Враховано процеси розсіяння електронів на потенціялах йонних
кістяків різного сорту та коливаннях кристалічної ґратниці. Запропоно-
Успехи физ. мет. / Usp. Fiz. Met. 2012, т. 13, сс. 189–223
Îòòèñêè äîñòóïíû íåïîñðåäñòâåííî îò èçäàòåëÿ
Ôîòîêîïèðîâàíèå ðàçðåøåíî òîëüêî
â ñîîòâåòñòâèè ñ ëèöåíçèåé
2012 ÈÌÔ (Èíñòèòóò ìåòàëëîôèçèêè
èì. Ã. Â. Êóðäþìîâà ÍÀÍ Óêðàèíû)
Íàïå÷àòàíî â Óêðàèíå.
190 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
вано послідовну методу врахування далекосяжної Кульонової взаємодії
електронів на різних вузлах ґратниці. Одержано кластерне розвинення
для густини станів, вільної енергії та електропровідности невпорядкова-
ної системи. Показано, що внески процесів розсіяння елементарних збу-
джень на кластерах зменшуються зі збільшенням числа вузлів у кластері
за деяким малим параметром. Точність розрахунку визначається точніс-
тю перенормування вершинних частин масових операторів електрон-
електронної і електрон-фононної взаємодій та малим параметром класте-
рного розвинення. З’ясовано природу спін-залежного електронного тран-
спорту вуглецевих нанорурок з атомами хрому, адсорбованими на повер-
хні. Показано, що величина спін-залежного транспорту пов’язана з відно-
сним зсувом у зовнішньому магнетному полі енергетичних рівнів елект-
ронів (Кульоновими щілинами, що виникають в області рівня Фермі) для
різних проєкцій спіну.
Развит метод описания электронных корреляций в неупорядоченных
кристаллах, основанный на гамильтониане многочастичной системы и
диаграммной технике для расчёта функций Грина. Электронные состоя-
ния системы описаны в рамках самосогласованной многозонной модели
сильной связи. Учтены процессы рассеяния электронов на потенциалах
ионных остовов разного сорта и колебаниях кристаллической решётки.
Предложен последовательный метод учёта дальнодействующего кулонов-
ского взаимодействия электронов на разных узлах решётки. Получено
кластерное разложение для плотности состояний, свободной энергии и
электропроводности неупорядоченной системы. Показано, что вклады
процессов рассеяния элементарных возбуждений на кластерах уменьша-
ются с увеличением числа узлов в кластере в соответствии с некоторым
малым параметром. Точность расчёта определяется точностью перенор-
мировки вершинных частей массовых операторов электрон-электронного
и электрон-фононного взаимодействий и малым параметром кластерного
разложения. Выяснена природа спин-зависимого электронного транспор-
та углеродных нанотрубок с атомами хрома, адсорбированными на по-
верхности. Показано, что величина спин-зависимого транспорта связана с
относительным сдвигом во внешнем магнитном поле энергетических
уровней электронов (кулоновскими щелями, возникающими в области
уровня Ферми) для разных проекций спина.
Key words: self-consistent tight-binding model, electron correlations, carbon
nanotubes doped with Cr atoms, energy spectrum of electrons and phonons,
electroconductivity, spin polarization of electric current.
(Received March 15, 2012)
1. INTRODUCTION
Progress in describing of disordered systems is strongly concerned
with development of electron theory. Substitutional alloys are best de-
scribed among disordered systems. Traditional knowledge about
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 191
physical properties of alloys is based on the Born approximation of the
scattering theory. However, this approach obviously cannot be applied
in case of a large scattering potential difference of components that
holds for the description of alloys with simple, transition and rare-
earth elements. The same difficulty relates the pseudopotential
method [1]. Because of non-local nature of pseudopotential, the prob-
lem of pseudopotential transferability exists. It is impossible to use
nuclear potentials determined by the properties of some systems to de-
scribe other systems. Due to using the theory of Vanderbilt ultra-soft
potentials [2, 3] and method of projector-augmented waves proposed
by Blochl [4, 5], fundamental progress in investigations of electronic
structure and properties of the system has been achieved. Significant
success in the study of electronic structure and properties of the sys-
tems was achieved recently because of the use of ultra-soft pseudopo-
tential Vanderbilt [2, 3] and the method of projector-augmented waves
within the density functional theory proposed by Blochl [4, 5]. This
approach was developed further because of use of the generalized gra-
dient approximation in density functional theory of multi-electron
systems developed in Perdew’s works [6–10]. In projector-augmented
waves approach, the wave function of valence states of electron (all-
electron orbital) is expressed by using the conversion through the
pseudoorbital. Pseudoorbital expands to pseudo partial waves in the
augment area. Even so, all-electron orbital in the same area is ex-
panded with the same coefficients via partial waves described by
Kohn–Sham equation. Expression for pseudo-Hamiltonian, which we
have in equation for pseudo-wave function, is derived by minimizing
the total energy functional. Using this equation and expanding pseu-
doorbital into plane waves, we can derive set of equations for expan-
sion coefficients. With this set, it is possible to get electron energy
spectrum, wave functions, and value of the total energy functional. As
shown in [10], there is a way to use this method for describing the elec-
tron structure of crystals, using VASP program package. Using clus-
ter methods of calculation and GAUSSIAN program package, this ap-
proach could be used for description molecule electronic structure.
It should be noted that simple effective calculation method of elec-
tronic structure and properties for big molecules had been proposed
recently in [10–17]. This method is based on tight-binding model and
density functional theory, which includes long-range Coulomb interac-
tion of electrons at different sites of crystal lattice. Long-range Cou-
lomb interaction of electrons at different sites is described in the local
density approximation.
However, mentioned methods [6–17] are used only for description of
ideal ordered crystals and molecules.
In disordered crystals, effects associated with localized electron
states and lattice vibrations occur. They cannot be described with a
192 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
model of an ideal crystal. In this regard, other approaches are devel-
oped too.
Essential achievement in description of properties of disordered sys-
tems is connected with application of tight-binding model in the multi-
electron scattering, including approximation of coherent potential.
Starting from Slater–Koster works [18], there was wide use of the
tight-binding model in electronic-structure calculations and in de-
scription of ideal crystals properties [19]. Later, it was generalized for
the case of disordered systems.
In Refs [20, 21], method of describing magnetic alloys electronic
structure based on functional density theory is proposed. The effective
potential in Kohn–Sham equation [22, 23] consists of atomic potential
and Pauli addition, which is expressed through magnetic field induc-
tion. Atomic potential and induction of magnetic field are expressed
through variational derivative of exchange–correlation energy by
electron density and magnetization, respectively. Calculations of elec-
tronic structure of magnetic alloy are based on already mentioned ef-
fective mass potentials using self-consistent Korringa–Kohn–Rostoker
approximation—the coherent potential, but more developed in [24–
26]. In Ref. [20], a method is proposed for calculating the parameters
of interatomic pair correlations due to the pair mixing potential, which
is expressed through the second derivative of the thermodynamic po-
tential of the alloy concentration [27]. This thermodynamic potential
is calculated within the one-site coherent potential approximation. It
should be noted that the methods developed in [18–20, 24–26] do not
include long-range Coulomb interaction of electrons at different lat-
tice sites.
For calculations of energy spectrum, free energy and electroconduc-
tivity of disordered crystals in our work, multi-scattering theory based
on Green’s functions is developed. Electron correlations in crystal are
described in multiband tight-binding model. It includes recalculation
of wave functions and atomic potentials with taking into account the
electron density redistribution because of atomic interaction. The
model includes long-range Coulomb interaction of electrons on differ-
ent sites of crystal lattice. The wave functions of noninteracting atoms
are calculated with Kohn–Sham equation using the Perdew potentials
[6–10]. Electron scattering processes on the ion core potentials of dif-
ferent kinds and on vibrations of crystal lattice are taken into account.
Calculations of two-time Green’s functions are based on temperature
Green’s functions [28]. This uses a known relation between spectral
representation for two-time and temperature Green’s function [29].
Calculation of two-time Green’s function of disordered crystal is
based on diagram techniques, which are analogous to diagram tech-
nique for homogeneous system [29]. Set of equations for temperature
Green’s function, expressions for both free energy and electroconduc-
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 193
tivity of solids are derived. Accuracy of the energy spectrum, free en-
ergy and crystal conductivity calculations is based on renormalization
of vertex parts of the electron–electron and electron–phonon mass op-
erators. Calculations based on this method for energy spectrum, free
energy, conductivity and spin-dependent transport of nanotubes doped
with Cr atoms were performed.
2. HAMILTONIAN FOR SYSTEM OF ELECTRONS AND PHONONS
IN DISORDERED CRYSTALS
Hamiltonian of disordered system (alloy, disordered semiconductor)
consists of Hamiltonian of electrons in the external ion field, the Ham-
iltonian of electron–electron interaction, the Hamiltonian of ions, and
the Hamiltonian of electron–ion interaction. Motion of ion subsystem
is reduced to the ions’ oscillations near equilibrium positions under the
influence of ions’ interaction forces and their indirect interaction
through electrons. Within the Wannier representation, the system
Hamiltonian is as follows [28]:
0 int
H H H , (1)
where the zero-order Hamiltonian,
0 0 0 0f e
H H H , (2)
consists of noninteracting electrons’ subsystem Hamiltonian in an ex-
ternal field of the A kind ions,
1 1 1 2 2 2 1 1 1 2 2 2
1 1 1
2 2 2
(0)
0 ,e n i n i n i n i
n i
n i
H h a a
, (3)
Hamiltonian of subsystem of the A kind ions,
1 2 1 2
1 1 1
2 2 2
2
1 2 1 2(0)
0
1 2 1 2
1
2 2
f
n ini A
n i
n
P
n n n ni
H u u
i i i iM
, (4)
and the electrostatic-interaction energy of ions at equilibrium posi-
tions, 0.
Perturbation Hamiltonian in (1),
int ei ef ee fi ff
H H H H H H , (5)
consists of electron–ion interaction Hamiltonian,
194 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
1 1 1 2 2 2 1 1 1 2 2 2
1 1 1
2 2 2
,ei n i n i n i n i
n i
n i
H w a a
, (6)
Hamiltonian of electrons’ interactions with oscillations of crystal lat-
tice (electron–phonon interaction),
1 1 1 2 2 2 1 1 1 2 2 2
1 1 1
2 2 2
,ef n i n i n i n i
n i
n i
H v a a
, (7)
electron–electron pair interaction Hamiltonian,
1 1 1 2 2 2
3 3 3 4 4 4 1 1 1 2 2 2 3 3 3 4 4 4
1 1 1
2 2 2
3 3 3
4 4 4
(2) ,
,
1
2
n i n i
ee n i n i n i n i n i n i
n i
n i
n i
n i
H v a a a a
, (8)
component of ion–ion interaction Hamiltonian caused by the presence
of different kinds of atoms (phonon–impurity interaction),
1 2 1 2
1 1 1
2 2 2
1 2 1 21
1 2 1 2
1
2
fi
n i
n i
n n n n
H M P P
i i i i
1 2 1 2
1 1 1
2 2 2
1 2 1 2
1 2 1 2
1
2 n i
n i
n n n n
u u
i i i i
, (9)
where
1 2 1 2 1 2 1 2
1 1
1 21
1 2
1 1
n n i i
n i A
n n
M
i i M M
,
1 2 1 2 1 2
1 2 1 2 1 2(0)
1 2 1 2 1 2
n n n n n n
i i i i i i
,
and anharmonic part of ion–ion interaction Hamiltonian (phonon–
phonon interaction),
1 2 3 1 2 3
1 1 1
2 2 2
3 3 3
1 2 3 31 2(0)
1 2 3 31 2
1
3!
ff
n i
n i
n i
n n n nn n
H u u u
i i i ii i
. (10)
In expression of ion–ion interaction Hamiltonian (10), only anhar-
monic terms of third order were taken into account.
In expressions written above, ni
a
, ni
a
are the creation and destruc-
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 195
tion operators in the state described by the Wannier function,
ni() ni, (r, ). State index includes a quantum number,
1 2, 1 2 , which defines value of spin projection on Z axis, and a set
of other quantum numbers describing spatial movements of electrons.
Here, n is the number of primitive cell, i—the sublattice-site number in
primitive cell, r—the radius vector of electron,
1 1 1 2 2 2
(0)
,n i n i
h
—matrix ele-
ments of Hamiltonian of electrons in an external field of the A kind ion
cores,
n
i
u —the atom displacement operator at the site (ni);
n
P
i
—
the operator of -projection of atom momentum onto orthogonal axes,
1 2
1 2(0)
1 2
n n
i i
,
1 2 3
1 2 3(0)
1 2 3
n n n
i i i
—force constants related to poten-
tial energy of the A kind ion interactions.
Potential energy operator of electron in a field of different-kind ion
cores can be expressed as
( ) ( )
ni
ni
ni
V v r r r ,
s
ni ni
n n
r r u u
i i
,
where r—electron radius vector, rni rn i—radius vector of atom
equilibrium position at the site (ni) in a crystal lattice,
s
n
i
u —vector
of atom static displacement from equilibrium position at site (ni).
Random addition to matrix element of one-electron Hamiltonian of a
pure crystal caused by impurity presence is as follows:
1 1 1 2 2 2 1 1 1 2 2 2, ,
ni
n i n i n i n i
ni
w w
, (11)
where
1 1 1 2 2 2 1 1 1 2 2 2, ,
ni ni
n i n i ni n i n i
w c w
,
1 1 1 2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 1 1 2 2 2, , , ,
ni ni ni Ani
n i n i n i n i n i n i n i n i
w v v v
.
Here, nic
are random numbers taking values of 1 or 0, depending on
whether the kind atom is at the site (ni) or not.
Hamiltonian of electron–phonon interaction (7) is expressed
through directional derivative of potential energy in ion-core field
along displacement-vector projections of atom,
n
i
u . In (7), the value
of
1 1 1 2 2 2,n i n i
v
is given by
196 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
1 1 1 2 2 2 1 1 1 2 2 2, ,
ni
n i n i n i n i
ni
n
v v u
i
,
where
1 1 1 2 2 2 1 1 1 2 2 2, ,
ni ni
n i n i ni n i n i
v c v
,
and
1 1 1 2 2 2,
ni
n i n i
v
is defined by (11), where v
(r rni) is replaced with
ni ni
ni
d
e v
d
r r
r r
,
where
ni
ni
ni
r r
e
r r
.
Values of
1 1 1 2 2 2,
ni
n i n i
v
in (11) describe electron scattering on atom
static displacements and are defined by
1 1 1 2 2 2 1 1 1 2 2 2, ,
ni ni s
n i n i n i n i
n
v v u
i
.
3. GREEN’S FUNCTIONS OF ELECTRONS AND PHONONS
To calculate the energy spectrum of electrons and phonons, free energy
and electrical conductivity of disordered crystal, we introduce two-
time Green’s function. We define two-time retarded ( ( , )
AB
rG t t ) and
advanced ( ( , )
AB
aG t t ) Green’s functions as follow [30]:
( , ) ( ) [ ( ), ( )]
AB
r
i
G t t t t A t B t , ( , ) ( ) [ ( ), ( )]
AB
a
i
G t t t t A t B t .
(12)
Within the Heisenberg representation, operator
/ /
( )
i t i tA t e A e H H ,
where ħ is the Planck’s constant, H H e, e—the chemical potential
of electron subsystem, Ne—the operator number of electrons:
e ni ni
ni
N a a
.
In expression (12),
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 197
[ , ]A B AB BA , (13)
where 1 for Bose operators A, B, and 1 for Fermi operators A,
B; (t)—the Heaviside step function. Brackets ... in (12) denote av-
eraging such as:
Sp( )A A , ( )e H , (14)
where is thermodynamic potential of a system, kBT, T—
temperature.
Calculation of two-time retarded and advanced Green’s functions
(12) is based on the calculation of temperature Green functions.
Known relation between the spectral representations for retarded, ad-
vanced and temperature Green’s functions is used.
Let us define the temperature Green’s function as
( , ) ( ) ( )
ABG T A B
(15)
where operator ( )A is derived from ( )A t in (12) by replacing t iħ,
( )A e Ae H H ;
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )T A B A B B A
,
1 for Bose operators A, B, and 1 for Fermi operators A, B.
We introduce the operator
( ) e e
0H H , (16)
where H H0 Hint, H0 H0 eNe. Operator () satisfies the equation
int
( )
( ) ( )H
(17)
where int int
( )H e H e
0 0H H
.
The solution of Eq. (17) provided that (0) 1 follows from the defi-
nition of (16) and has the form
int
0
( ) exp ( )T H d
. (18)
Taking into account expression (16) for the operator within the
Heisenberg representation, one can write as follows:
1
( ) ( ) ( ) ( )A A . (19)
198 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
Expression (13) for temperature Green’s function, in view of (17),
leads to equation
0 0
( , ) ( ) ( ) (1 ) (1 )
ABG T A B
, (20)
where
00
Sp( )A A , 0 0( )
0
e
H
.
Expanding the exponent () in expression (18) in a series of powers
Hint(), substituting the result into (20), and using Wick’s theorem for
calculating the temperature Green’s [28] functions of disordered crys-
tals, it is possible to construct a diagram technique similar to a homo-
geneous system [29]. The denominator in formulas (20) is derived from
the same factor in the numerator. So, Green’s functions can be ex-
pressed in a series only connected diagrams. Using the relation be-
tween the spectral representations of temperature and time Green’s
functions [29], by analytic continuation to real axis, we obtain the fol-
lowing set of equations for retarded Green functions (hereinafter, in-
dex r will be omitted) [28]:
0 0
( ) ( ) ( ) ( ) ( ) ( )
aa aa aa aa
ef ee
G G G w G
,
1
0 0 0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
uu uu uu uu uP Pu
fe ff
G G G G G M G ,
1
0 0 0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
PP PP PP PP Pu uP
fe ff
G G G M G G G , (21)
1
0 0 0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
uP uP uP PP uu uP
fe ff
G G G M G G G ,
1
0 0 0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
Pu Pu Pu uu PP Pu
fe ff
G G G G G M G ,
where ħ.
Here, ( )
aaG
, ( )
uuG , ( )
PPG , ( )
uPG , ( )
PuG are spectral represen-
tation of one-particle Green’s function of the electrons’ subsystem as
well as ‘displacement–displacement’, ‘impulse–impulse’, ‘displace-
ment–impulse’, ‘impulse–displacement’ Green’s functions of pho-
nons’ subsystem, respectively; ( )
ef
, ( )
fe
, ( )
ee
, ( )
ff
are actual
energy parts (mass operators), which describe the electron–phonon,
phonon–electron, electron–electron, and phonon–phonon interac-
tions, respectively.
Upon receipt of the expression (21), the spectral decompositions for
the two-time (12) and temperature (15) Green’s functions are used:
, ,
1
( ) ( )
2
AB AB i t
r a r a
G t G e d
,
, ,
( ) ( )
AB AB i t
r a r a
G G t e dt
,
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 199
( ) ( ) n
n
iAB AB
n
G G e
,
1/
1/
1
( ) ( )
2
niAB AB
n
G G e d
,
2 for Bose particles,
0, 1, 2,....
(2 1) for Fermi particles,
n
n
n
n
There are well-known relations between spectral representations of
Green’s functions, which follow from Green’s functions definitions:
( ), 0,
( )
( ), 0.
AB
AB r n n
n AB
a n n
G i
G
G i
Green’s functions are matrices with respect to the indices (ni) and
(ni), respectively, for the subsystem of electrons and phonons.
From the equations of motion for Green’s functions of zero ap-
proximation [30], one can be obtained:
(1) 1
0 0
( ) [ ]
aaG H
, (1) (0)
0 ,ni n i
H h
,
2 (0) 1
0
( ) [ ]
uu
A
G M ,
(0) (0)
,ni n i
,
2 1 (0) 1 (0)
0
( ) [ ]
PP
A
G M ,
(0)
0 0
( ) ( )
uP PPG i G
, 1
0 0
( ) ( )
Pu uu
AM G i G
. (22)
Provided
2
(0)
2
,
( ) ( ) 1
fe ff
ni n i
nn
M
i i
, (23)
solution of the set (21) has the form:
1
1
0
( ) ( ) ( ) ( )
aa aa
ef ee
G G w
,
1
2
1
0 2
( ) ( ) ( ) ( )
uu uu
fe ff
G G M
, (24)
2
1
2
( ) ( )
PP uuM G M G
,
where ( )
A ni nn ii
M M M
.
Upon receipt of Eqs (24), the members, which are proportional to the
second and higher powers of small parameter (23), are neglected.
Using the mentioned above diagram technique, in work [28], explicit
200 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
expression has been found for the mass operator of Green’s functions
that describe the many-particle interactions in the system.
The mass operator of Green’s function of electrons for the electron–
phonon interaction, ef(, ), is described by the diagram in Fig. 1.
Solid lines in the Fig. 1 correspond to the Green’s function of elec-
trons, ,
( , )
aa
ni n i
G
; dashed lines correspond to the Green’s function of
phonons, ,
( , )
uu
ni n i
G
. Vertex part
2 2 2
1 1 1, 2 1
( , , )
n i
ni n i
is described by
diagrams in Fig. 2, where unshaded triangle corresponds to the ex-
pression
2 2 2 2 2 2
1 1 1 1 1 10 , 2 1 , 2 1
( , , ) ( ) ( )
n i n i
ni n i ni n i
v
.
For internal points n in Figs. 1 and 2, summation is carried out.
Summation by n provides summation by ni and integration over .
Expressions corresponding each diagram attribute multiplier (1)n F,
where n is order of diagram (number of vertices 0 in the diagram), and
F is the number of lines for the Green’s function of electrons,
aaG
,
which goes out and goes into in the same vertices.
For the mass operator that describes electron–phonon interaction,
we have:
1 2 1 1 1 2 2 2
1 1 2 2
1 1 1
2 2 2
,
, ,
( ) ( )
n i n i
ef ni n i n i n i ef ni n i
n i
n i
c c
,
1 1 1 2 2 2 1 1 1 1
3 3 3
,
, ,
1
( ) coth
4 2
n i n i n i
ef ni n i ni n i
d v
i
Fig. 1. Diagram for , ,
( , )
ef ni n i ef n n
; here, ( )n ni .
Fig. 2. Diagrams for the vertex part
2 2 2 2 2
1 1 1 1 1, 2 1 ,
( , , )
n i n
ni n i n n
; here, ( )n ni .
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 201
2 2 2 2
1 1 1 2 2 2 2 2 2 1 1 1 3 3 3 4 4 4 4 4 4
*
, , , ,
( ) ( ) ( ) ( , ; )
n iuu uu aa
n i n i n i n i n i n i n i n iG G G
. (25)
For repeated indices, summation is performed.
The diagram in Fig. 3 describes the phonon–electron interaction.
Designations in Fig. 3 correspond to designations in Figs. 1 and 2.
The mass operator describing phonon–electron interaction has the
form:
, ,
( ) ( )
fe ni n i ni n i fe ni n i
c c
,
2 2 2 1 1 1 , ,
1
( ) ( )
2
ni
fe ni n i n i n i
d f v
i
1 1 1 3 3 3 3 3 3 1 1 1 2 2 2 4 4 4
* *
, , ,
( ) ( ) ( )
aa aa aa
n i n i n i n i n i n i
G G G
1 1 1 3 3 3 4 4 4 2 2 2 2 2 2 4 4 4 3 3 3 4 4 4
*
, , , ,
( ) ( ) ( ) ( , ; )
aa aa aa n i
n i n i n i n i n i n i n i n i
G G G
.(26)
Diagrams for the mass operator ee(, ) that describes electron–
electron interaction are shown in Fig. 4.
Vertex part
2 2 2 1 1 1,
, 2 1
( , , , )
n i n i
ni n i
is shown by diagrams in Fig. 5. Un-
shaded triangle in Fig. 5 corresponds to the expression
2 2 2 1 1 1 2 2 2
1 1 1
, (2) ,
0 , 2 1 , 2 1
( , , , ) ( ) ( ) ( )
n i n i ni n i
ni n i n i n i
v
,
2 2 2 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1
(2) , (2) , (2) ,
, , ,
ni n i ni n i ni n i
n i n i n i n i n i n i
v v v
.
Fig. 3. Diagram for , ,
( , )
fe ni n i fe n n
; here, ( )n ni .
Fig. 4. Diagrams for , ,
( , )
ee ni n i ee n n
; here, ( )n ni .
202 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
The mass operator that describes electron–electron interaction is as
follows:
(1) (2)
, , ,
( ) ( )
ee ni n i ee ni n i ee ni n i
,
2
1 1 2 2 1
(2) ,(1) *
, , , ,
1
( ) ( ) ( )
4
n n aa aa
ee n n n n n n n n
d f v G G
i
,
3
2 1
2
(2) ,(2)
, 1 2 ,
1
( )
2
n n
ee n n n nd d v
i
5 2 1 4 2 5 4 1
* *
1 2 , 1 2 , 1 , 1 2 , 1
( ) ( ) ( ) ( ) ( ) ( )
aa aa aa aa
n n n n n n n n
f f G G G G
5 6
6 3 3 6 4
, *
, 2 , 2 , 1 1 2 2
( ) ( ) ( , ; , )
n naa aa
n n n n n n
G G
1 4 6 3 4 1 3 6
* *
1 1 2 , 1 , 1 2 , 1 , 1 2
( ) ( ) ( ) ( ) ( ) ( )
aa aa aa aa
n n n n n n n n
f f G G G G
5 6
2 5 5 2 4
, *
, 2 , 2 , 1 2 1 2
( ) ( ) ( , ; , )
n naa aa
n n n n n n
G G
, (27)
2 2 2
1 1 1
(2) , (2) , (2) ,
, , ,
n n n n n n
n n n n n n
v v v (n ni).
Similarly, in Ref. [28], expressions are obtained for the mass opera-
tor ff() that describes phonon–phonon interactions.
Upon receipt of expressions (25)–(27), the relations arising from the
theory of functions of complex variables are used:
1
( ) coth ( ) ( 2 )
4 2
n
n n
C
z
i dz z n
i
,
1
( ) ( ) ( (2 1) )
2
n
n n
C
z
i dzf z n
i
,
1
exp 1
z z
f
,
Fig. 5. Diagrams for vertex part
2 2 2 1 1 1 2 2 1 1, ,
, 2 1 ,
( , , , )
n i n i n n
ni n i n n
; here, ( )n ni .
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 203
where (z) is the analytical function of complex z in the region covered
by a contour C.
Between the Green’s functions (12) G() defined at a constant
chemical potential e and the Green’s function GN() defined at a con-
stant number of electrons Ne, there exists a relation G() GN( ).
When renormalization of vertex parts is neglected, diagrams for the
mass operators in expressions (25)–(27) should be put as follow:
2 2 2 2 2 2 2 2
4 4 4 4 4 4 , ,
( , ; )
n i n i
n i n i n i n i
v
, 5 6 4 5
4 6
, (2) ,
, 1 1 2 2 ,
( , ; , )
n n n n
n n n n
v .
In expressions (26), (27), f() is the Fermi function; Fermi level F e
of a system is determined by the equation:
( , ) ( )
F e
Z f g d
, (28)
where Z—the average number of electrons per atom. Electron density
of states, ge(), in expression (28) is given by the formula:
, , ,
1
( ) ( )
e ni ni
i
g P g
, (29)
where ni ni
P c —the probability of filling the site (ni) with the kind
atom, —the number of atoms in a primitive cell, ( )
ni
g
—the condi-
tional partial electron density of states per one atom for state (ni)
provided the kind atom be placed at the site (nі):
,
( )
1
( ) Im ( )
aa
ni ni ni
ni
g G
.
In the last formula, configuration averaging is carried out provided
that, in site (ni), the kind atom is placed.
Equation (28) follows from the definitions of electrons’ number op-
erator (12) and Green’s function G
AB(, ) (15); ni
A a
, ni
B a
[27].
It should be noted that the first term
(1)
,ee ni n i
in the expression for
the mass operator ,
( )
ee ni n i
of electron–electron interactions is de-
scribed the Coulomb and exchange electron–electron interactions
within the Hartree–Fock approximation. The second term
(2)
,
( )
ee ni n i
,
which is caused by output beyond the Hartree–Fock approximation,
describes the electron correlations.
Expressions (24) differ from the corresponding expressions for the
Green’s function of the single-particle Hamiltonians of disordered sys-
tem only by the view of mass operators. Therefore, to calculate the
204 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
Green’s function (24), the well-known methods of the theory of disor-
dered systems [28] will be used.
Let us perform cluster expansion for Green’s functions ( )
aaG
,
( )
uuG in expressions (24) by introducing the mass operator as the sum
of one-site operators and selecting a zero one-site approximation of the
Green’s function of an effective environment. Specified expansion is a
generalization of the cluster expansion for the Green’s function ( )
aaG
of single-particle Hamiltonian. Green’s function of an effective envi-
ronment for the subsystem of electrons determine the expression [28]
1
1
0
( ) ( ) ( ) ( ) ( )
aa aa A
ef ee e
G G
, (30)
where the mass operator of the electron–phonon interaction for the A
kind atoms in the effective environment is equal to
1 1
1 1
( ) ( )
An iA
ef ef
n i
, 1 1 1 1 2 2
2 2
,
( ) ( )
An i An i An i
ef ef
n i
.
The values
1 1 2 2,
( )
An i An i
ef are determined by expression (25), in which the
Green’s functions of disordered crystal are replaced by Green’s func-
tions of the effective environment.
Green’s functions of an effective environment for phonon subsys-
tem are defined by expression:
1
1
0
( ) ( ) ( ) ( ) ( )
uu uu A
fe ff fG G
. (31)
The values of ( )ee , ( )
A
fe
, ( )
ff
in expressions (30) and (31) are de-
fined similarly to ( )
A
ef
.
In expressions (30) and (31), e(), f() are potentials of an effective
environment (coherent potentials); their values will be determined.
Green’s functions (24) satisfy the equation:
( ) ( ) ( ) ( ) ( )G G G T G , (32)
where T is a matrix of scattering, which can be represented as a series
with terms describing the scattering by clusters with different num-
bers of sites:
1 1 1 1 2 2
1 1 1 1 2 2
(2) ,
( ) ( ) ( )
...
n i n i n i
n i n i n i
T t T
. (33)
Here,
1 1 2 2 1 1 2 2 1 1 2 2 1 1(2) , 1
[ ] [ ]
n i n i n i n i n i n i n i
T I t Gt G t Gt I Gt , and
1 1n i
t is opera-
tor of scattering on the same site that is determined by the expression:
1 1 1 1 1 1 1 1 1 1
1
( ) ( )
n i n i n i n i n i
t I G
.
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 205
Value
1 1 ( )
n i
e in the formula (39) for the subsystem of electrons is given
by the expression
1 1
1 1( )
( ) ( ) ( ) ( )
n iA
ef ee ef ee e
n i
w , (34)
and, for the subsystem of phonons, by the expression
1 1
1 1
2
2
( )
( ) ( ) ( ) ( )
n iA
fe ff fe ff f
n i
M
. (35)
Coherent potentials determined from the condition
10
0
i
t satisfy
the set of coupled equations [28]:
1 1 1
1
1
0 0 0
( ) 1 ( ) ( ) ( )
i i i aa
e e e
G
1 1 1
1
0 0 0
1 ( ) ( ) ( ) ( )
i i iaa
e e e
G
, (36)
and, for the subsystem of phonons,
1 1 1
1
1
0 0 0
( ) 1 ( ) ( ) ( )
i i i uu
f f f
G
1 1 1
1
0 0 0
1 ( ) ( ) ( ) ( )
i i iuu
f f f
G
. (37)
Using (33), cluster decomposition for the Green’s function of elec-
trons and phonons of disordered crystal can be obtained. As a zero one-
site approximation in this method of cluster expansion, the approxi-
mation of coherent potential is chosen. The densities of electrons’ and
phonons’ states, free energy and electrical conductivity are presented
as infinite series with terms describing the processes of scattering on
clusters with different numbers of atoms. As shown, the contributions
of scattering processes with electrons and phonons in clusters decrease
with increasing number of atoms in the cluster by a small parameter:
1 1 2 2
2 2 1 1
0 ,0
( ) ( )
( ) ( ) ( ) ( ) ( )
n i n i
i i
n i n i
p t G t G
. (38)
Investigation of this parameter shows that it is small in a wide range of
changes of characteristics of the system (including the concentration
of components), except for narrow intervals of energy values at the
edges of the spectrum. For single-electron Hamiltonian in single-band
model, this parameter was introduced for the first time and investi-
206 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
gated in Ref. [31]. Subsequently, this parameter was investigated in
Ref. [32]. For many-particle Hamiltonian, which includes electron–
electron and electron–phonon interactions, this parameter was ex-
plored in Ref. [28].
Using (29), (32), (33), spending averaging over the distribution of
atoms of different kinds at the sites of a crystal lattice, and neglecting
the contribution of processes of electron scattering in clusters consist-
ing of three or more atoms, which are small by the above parameter,
for the density of electron states, we obtain
0 0
, , ,
1
( ) ( )
e i i
i
g P g
,
0
0 ,0
(2) 0 , (2) , 0
0 0
( ) (0 )
( )
1
Im [ ] ,
i
i i
i lj lj i
i lj i lj
lj i
g
G Gt G P G t T T G
(2) 0 , 0 1 0 0
[ ] [ ]
i lj i lj i lj iT I t Gt G t Gt I Gt
. (39)
Doing the above-configuration averaging the Green’s function
( )
uuG for the phonon density of states similarly, we obtain
0 0
, ,
1
( ) ( )
f i i
i
g P g
,
(40)
0
0 ,0
/ (2) 0 , (2) , 0
0 02
( ) (0 )
( )
2
Im [ ] ,
i
i i
i lj lj iA
i lj i lj
lj i
g
M
G Gt G P G t T T G
where ( )
uuG G .
Matrix elements of the Green’s function of the electron subsystem
of the effective environment can be calculated, using the Fourier
transformation:
1
( )(1)
,
,
1
( ) ( ) n i n ii raa
ni n i
i i
G H e
N
k r
k
k
, (41)
where
(1) (1)
0
( ) ( ) ( , ) ( , ) ( , )
A
ef ee e
H H k k k k k , (42)
N—the total number of primitive cells.
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 207
Calculation of matrix elements of Green’s functions of phonons’
subsystem of an effective environment is performed by the formula
1
( )2
,
,
1
( ) ( ) n i n iiuu
ni n i A
i i
G M e
N
k r r
k
k
, (43)
where
(0)
( ) ( ) ( , ) ( , ) ( , )
A
fe ff f
k k k k k . (44)
In formulas (41)–(44), vector k is varying within the first Brillouin
zone.
Components of matrix of force constants, which are caused by the
direct Coulomb interaction of ions, have the form:
2
5
0
4
ni n i
n i n i
n n Z Z e
i i
r r
2
3( )( )
n i n i n i n i n i n i
r r r r
r r ,
(45)
where Zni—the valence of ion, which is located at the site ( ) ( )ni n i .
Diagonal (by number (ni) of a site) elements of the matrix of force
constant are determined from the condition
0
n i
n n
i i
.
Value of
(0)(k) in expression (44) is the Fourier image of matrix,
(0)
n n
i i
, given by expression (45); here, Zni Zni Z
A, where Z
A
is
the А kind ion valence.
The mass operator ( , )
A
fe
k in expression (44) describes the interac-
tions of ions by means of electrons.
Single-centre scattering operator
1 1n i
t
in the formula (39) is given
by expression (33), in which the quantity
1 1 ( )
n i
e
describes the scatter-
ing of electrons, according to (11), (25), (27), (34), and has the form:
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1
(2) ,
, , ,
1
( ) ( )
2
imn i n i ni n i
eni n i ni n i n i n i n i i
w v Z Z
. (46)
In expression (46), ni
Z
—the number of filled electron state (ni)
provided that, at the site (ni), the kind atom is located. The value
ni
Z
is determined by the expression (28), in which the electron den-
208 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
sity of states, ge(), is replaced by conventional partial density of
states, ( )
ni
g
. The value ni
Z
is equal to
( , ) ( )
ni i F ni
Z Z f g d
. (47)
The total number of electrons in the valent states of the kind atom
is equal to
i i
Z Z
. (48)
The effective charge of the kind atom is determined by the expres-
sion
0
i i
Z Z Z
. (49)
Here,
0Z
is the number of electrons in the valent states of noninteract-
ing kind atoms.
Localized magnetic moment of the kind atom is equal to
( )
i i i
m Z Z
. (50)
The value i
Z
in the formula (47) is derived from expressions (39)
and (47) by replacing of the Green’s function to the function of effec-
tive environment.
Single-centre operator of scattering
1 1n i
t
in the formula (40) is given
by expression (33), in which the quantity
1 1 ( )
n i
f
that describes, ac-
cording to (24), (26), (35), the scattering of phonons, has the form
1 1
1 1
2
(0)
, 2
( ) ( )
n i
fni n i A nn ii nn ii
n n n n
M M
i i i i
,
(51)
where
n n
i i
is determined by expressions (45) and (44), in which
Zni Z
,
A
n iZ Z .
The value
(1) (2)
( , ) ( , ) ( , )ee ee ee k k k in expression (42) and sec-
ond component in right part of expression (46) describes the Coulomb
interaction of electrons, including long-range Coulomb interaction at
different sites in crystal lattice. The value
(1)
( , )ee k in expression (42)
and second component in right part of expression (46) describes the ex-
change interactions of electrons. The value
(2)
( , )ee k in expression (42)
describes the electron correlations. Unlike the Refs [11–17], in our
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 209
work, long-range Coulomb interaction of electrons at different sites in
crystal lattice is described with considering of arbitrary numbers of
energy bands.
The value ( , )
A
ef
k in expression (42) describes the scattering of
electrons by crystal-lattice vibrations.
In expressions (39), (40), 0lj i
P
is conditional probability to find at
the site (lj) the kind atom provided that, at the site (0i), the kind
atom. nit is the value of matrix elements of single-centre operators of
scattering for the case when, at the site (ni), the kind atom is located.
Probability of filling of site (0i) by the kind atom in expression (39)
is equal:
0 0i i
P c , (52)
where nic
is the random numbers taking values of 1 or 0 depending on
the kind atom located at the site (ni) or not.
For crystals with two kinds of atoms and two sublattice types,
2
0
A
i A aP x
for 1 sublattices of first type and
1
0
A
i A aP x
for
2 sublattices of second type; 0 0
1
B A
i i
P P ; 1 2; хA, хB 1 хA—
concentrations of components А, В in alloy, respectively; а—the pa-
rameter of long-range atomic order.
Conditional probability 0lj i
P
in (39) is defined by the expression
0 0 0 0lj i i lj i lj i
P P P c c
(53)
and is related to parameters of interatomic pair correlation 0
BB
lj i
by
next equation [25, 29]:
0
0
0
( )( )
BB
lj i
lj i lj B A B A
i
P P
P
; (54)
here, —the Kronecker symbols; 0 0
( )( )
BB B B B B
lj i lj j i i
c c c c .
4. FREE ENERGY
Thermodynamic potential of the system is defined by the expression
[29]
lnSp( )
He . (55)
Hamiltonian H is defined in expression (12). Using the formula (14),
210 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
from (55), we obtain
0
,
0
ln (1 ) , (56)
where 0—the thermodynamic potential of noninteracting electrons
and phonons.
For the thermodynamic potential (56), diagram technique can be
developed, which is similar to the diagram technique for the tempera-
ture Green’s function.
As a result, we obtain
1
0
1
Im ( ) ( ) ( , ) ( , ) ( , )
aa
ef ee
d
d f Sp w G
N
11
coth Sp ( ) ( , ) ( ) ( , ) ( , )
2 2
PP uu
ff
M G G
. (57)
The thermodynamic potential 0 in the absence of interaction in the
formula (56) is equal to
0 0 0c e f
. (58)
Thermodynamic potential of the electron subsystem is as follows:
( )
0 0
ln 1 ( )e
e e
e g d
. (59)
Thermodynamic potential of the phonon subsystem is
0 0
ln(1 ) ( )
f f
e g d
. (60)
In expressions (59), (60), g0e(), g0f() are given by formulas (39),
(41), in which ( )
aaG
, ( )
uuG are replaced by zero-order approxima-
tion Green’s functions, 0
( )
aaG
, 0
( )
uuG .
Configuration component of the thermodynamic potential in the
formula (58) depends on the distribution of atoms of different kinds
and is defined as
0c c
S , (61)
where ln
c c
S P —the configuration entropy, Pc—the distribution
function of atoms over crystal lattice sites.
Configuration entropy can be represented as in Ref. [26]:
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 211
0
0 0 0
,, 0
,
(0 ) ( )
1 1
ln ln ...
2
i lj
c i i ilj
ii i lj
lj
i lj
P
S P P P
P P
. (62)
Free energy F as a function of volume V, temperature T, number of
electrons Ne, and parameters of interatomic correlations
1 1 2 2,
( , )n i n i is
determined by the thermodynamic potential : e e
F N . In a
weak dependence approximation of the mass operators on energy of
electrons and phonons, free energy F can be represented in the form
[28]
c e f e
F Z , (63)
where e, f are given by expressions (59), (60), in which g0e(), g0f()
are replaced by ge(), gf() (see (39), (41)).
The values F, c, Sc, e, f in formulas (59)–(63) are calculated per
one atom.
5. ELECTRICAL CONDUCTIVITY
To calculate the electrical conductivity tensor, we use the Kubo for-
mula [33]:
1
0 0
( ) (0) ( )
i t te J J t i d dt
, (64)
where J is the operator of the -projection of the current density.
It follows from Eq. (64) that
Re ( ) ( ) ( )
2
J J J J
r a
i
G G
. (65)
To calculate the spectral representations, ( )
J J
rG and ( )
J J
aG , of the
retarded and the advanced Green’s functions, we use the expression
for current-density operator,
( ) ( , ) ( , )J t e t v t d
, (66)
where ( , )t and ( , )t are the field operators of creation and anni-
hilation of electron, respectively; is the operator of the -projection
of the velocity; e is the electron charge. By integrating over , we mean
integrating over the crystal volume and summing over the projections
of spin onto the z-axis; the volume of the crystal is assumed equal to
212 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
unity.
The temperature Green’s function in this case is written as
4 2 3 1
1 2 3 4
2
1 2 3 4
1
( , ) ( , , , )
J J
n n n n
n n n n
e
G v v G n n n n
NV
, (67)
where V1—volume of primitive cell; the two-particle Green’s function
is written as
1 2 3 4
1
1 2 3 4 00
( , , , ) ( ) ( ) ( ) ( ) (1 ) (1 )n n n nG n n n n Ta a a a
, (68)
( )n ni . The two-particle Green’s function (68) is described by the
diagram in Fig. 6. Numbers in Fig. 6 correspond to point numbers; e.g.,
1 corresponds to (n1i111).
Using the diagram technique for two-time temperature Green’s
function and neglecting the contributions of scattering processes on
clusters of three or more sites, for the static conductivity tensor, we
can get the expression:
2
1 1 1
, , ,1 1
(2 1) ( , , )
4
s s
ss
s s i
e f
d v K v
V
0 1 1 0 1 1 1 0 1
( , , ) ( ) ( , , ) ( )
s s s s s s
i i i
P K v t K v t
(2) 0 ,
0 0 1 1 1 1
0 ,
( , , ) ( ) ( )
s s s i lj s
i lj i
lj i
P P K v v G T
(2) , 0 (2) 0 ,
1 1 1 1 1 1 1 1
( , , ) ( ) ( ) ( , , ) ( ) ( )
s s s lj i s s s s i lj sK v v G T K v v G T
(2) , 0
1 1 1 1 1 1 1 1 1 0 1
( , , ) ( ) ( ) ( , , ) ( ) ( , , ) ( )
s s s lj i s s s s s s s
lj iK v v G T K v t K v t
(2) 0 , (2) , 0
1 1 1 1 1 1 1 0 1
( ) ( , , ) ( ) ( ) ( , , ) ( )
s s s i lj s lj i s s s s
lj i
t K v T T K v t
(2) , 0 (2) 0 ,
1 1 1 1
( ) ( , , ) ( )
lj i s s s i lj sT K v T
Fig. 6. Diagrams for the two-particle Green’s function.
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 213
0 ,0
(2) , 0 (2) , 0
1 1 1 1
( ) ( , , ) ( )
i i
lj i s s s lj i sT K v T
II
1 2 1 2 1 2
( ) ( ) ( , )d d f f G
, (69)
where 1 1 1 1
( , , ) ( ) ( )
s s aa s aa sK v G v G
, and
1 1
( ) ( )
aa aa
r
G G
, *
1 1 1
( ) ( ) ( ) ( )
aa aa aa
a rG G G
.
In formula (69), component
II
1 2
( , )G
of two-particle Green’s
function is caused by the interaction and has the form:
4 2 3 1 1 6 1 6
II
1 2 1 1
( , ) ( ) ( )
2
aa aa
n n n n rn n an n
i
G v v G G
2 5 2 5 7 4 8 32 2 2 1
( ) ( ) ( ) ( )
aa aa aa aa
rn n an n an n rn n
G G G G
7 4 8 3 1 6 2 5 2 52 1 1 2 2
( ) ( ) ( ) ( ) ( )
aa aa aa aa aa
rn n an n an n rn n an n
G G G G G
7 4 8 3 8 3 1 62 1 1 1
( ) ( ) ( ) ( )
aa aa aa aa
an n rn n an n rn n
G G G G
2 5 2 5 7 4 8 3 8 32 2 2 1 1
( ) ( ) ( ) ( ) ( )
aa aa aa aa aa
rn n an n rn n rn n an n
G G G G G
1 6 2 5 1 6 2 51 1 1 1
( ) ( ) ( ) ( )
aa aa aa aa
an n rn n rn n an n
G G G G
7 4 7 4 8 3 8 32 2 1 1
( ) ( ) ( ) ( )
aa aa aa aa
rn n an n rn n an n
G G G G
1 6 1 6 2 5 7 4 7 41 1 2 2 2
( ) ( ) ( ) ( ) ( )
aa aa aa aa aa
rn n an n rn n rn n an n
G G G G G
8 3 1 6 1 6 2 51 1 1 2
( ) ( ) ( ) ( )
aa aa aa aa
rn n rn n an n an n
G G G G
6 7
7 4 7 4 8 3 5 82 2 1 1 2 2 1
( ) ( ) ( ) ( , ; , )
n naa aa aa
rn n an n an n n n
G G G
(n ni ). (70)
Operator of -projection of the electron velocity in (69) is as follows:
,
,
( )1
( )
i i
i i
h
v
k
k
k .
To simplify the formula (69), we use approximate expression
II II
1 2 1 2
( ; ) ( ; )G G
, where
II
1 2
( ; )G
is derived from the ex-
pression (70) by replacing ( )
aaG
with ( )
aaG
.
214 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
It should be noted that the description of electron correlations in
disordered crystals would have to come from Hamiltonian system that
consists of interacting subsystems of nuclei and electrons. However, it
leads to excessively large rank of the Hamiltonian matrix (1) and major
complications in numerical calculations of electronic structure and
properties of the system. To reduce the rank of the Hamiltonian matrix
(1), the real wave functions of neutral free the A kind atom are chosen
as a basis.
Hamiltonian of the system consists of a single-electron Hamiltonian
in the ion core field, the potential energy of the pair electron–electron
interactions, Hamiltonian describing the vibrations of the lattice, and
Hamiltonian of electron–phonon interaction.
The wave functions and potential energy of an electron in the kind
atom, which is located at the site (0i1), are determined by the method
of electron density functional from Kohn–Sham equation [4, 5, 6, 7].
Potential energy of electrons in the ion core and the Hamiltonian
wave-function basis, we find from the Kohn–Sham equation for the
noninteracting atoms [22, 23]:
2
2
ext XC,
( ) ( ) ( ) ( ) ( )
2
i i
H i i i
V V V
m
r r r r r , (71)
where —the quantum number of spin projection on the z-axis;
( )lm ; l, m—quantum numbers of angular momentum, and —the
quantum number describing the value of electron energy. To reduce
the record in the expression (71), (r rni) is denoted by (r).
In expression (71), the value ext
( )V r is potential energy of an elec-
tron in the kind atom core at the site (ni);
2
( ) ( )
i
H i
e
V dv n
r r
r r
(72)
is the Coulomb potential of the electron charge.
In formula (72), electron density is as follows:
( ) ( ) ( )
i i i
n n n
r r r . (73)
Electron density with projection of spin is given by expression
*
( ) ( ) ( )
i i i i
n Z
r r r , (74)
where i
Z
—the occupation number of electron state (i), provided
that the site (ni) be occupied by the kind atom, which is calculated by
the formula (45). The expression (74) follows from the definition of the
density operator of electrons [31] and the definition of the Green’s
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 215
function (15). In meta-generalized gradient approximation obtained by
Perdew [6–10], based on the density functional theory of many-
electron systems, taking into account the spin polarization and inho-
mogeneous distribution of electron density, the result of the action of
exchange–correlation potential
MGGA
XC, XC,
( ) ( )V V
r r on the wave func-
tion can be represented as follows:
MGGA GGA
XC, XC, XC,
1
( ) ( ) ( ) ( ) ( ) ( )
2
V V
r r r r r r , (75)
MGGA MGGA
GGA XC XC
XC,
( )
e e
V
n n
r ,
MGGA
XC
XC,
( )
e
r ,
where
MGGA
XC
(2 ) 2e n
is the exchange–correlation energy density,
2
2
—kinetic energy density.
Wave functions of basis set, ( )
ni
r , on which Hamiltonian of the
system is given by the set (1), are defined from Kohn–Sham equation
(71) for the A kind atom and equal to ( ) ( , )
A v
ni i l ni lm
R Y
r r r ,
where A
i l ni
R
r r —the radial part of wave function in Eq. (71),
(). Real spherical functions, ( , )
v
lmY , are related to complex
spherical functions, ( , )
lm
Y , by relations
*
( , ) ( , )
( , )
2
c lm lm
lm
Y Y
Y
,
*
( , ) ( , )
( , )
2
s lm lm
lm
Y Y
Y
i
(m 0). (76)
Electron density with spin projection (74) can be represented as
( ) ( ) ( )
i i i
n n n
r r r , (77)
where
*
( ) ( ) ( )
i i i i
n Z
r r r (78)
is the density of electron states in the ion cores; summation is carried
out over the filled states 1
i
Z
of ion core, and
*
( ) ( ) ( )
i i i i
n Z
r r r (79)
is the density of electrons in valent states of the atom. We assume that
the potential energy of electrons in the ion core and the Hamiltonian
wave-function basis have the same form as for noninteracting atoms.
216 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
This assumption puts a requirement that the value of occupation num-
ber i
Z
in expression (79) has the same value calculated by the formula
(45).
Using formula (76), it can be shown that i
Z
in expression (79) is
equal to
1
( )
2
i i c i sZ Z Z
. (80)
Here, i
Z
is calculated by expression (45), in which should be put
(), where c, s. Above-mentioned consistency can be achieved by
using the iterative procedure, in which the value of the right side of
expression (80), which is calculated by the formula (45), at a certain
step of iteration with a given accuracy coincides with the left side of
expression (80) in the previous step.
Matrix elements
1 1 1 2 2 2
(0)
,n i n i
h
of Hamiltonian (1) in multi-band s–p–d-
tight-binding model are the matrix elements of the operator of kinetic
energy and potential energy of an electron in a potential of the ion
core, ( )
Ani
ni
ni
T v r r , on the real wave functions of the neutral A
kind atoms. Potential energy of an electron in the field of ion core,
( )
Aniv r , is determined by formulas (76)–(79), in which, instead of the
total electron density, one should put the density of electrons, ( )
Ai
n
r
(79), in states of the A kind ion core.
Diagonal (by site number (n1i1)) matrix elements are equal to
1 1
1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 2 2
(0)
,
( )
n iA A
n i n i n i n i n ih T v
r r . Off-diagonal (by site
number) matrix elements can be presented in the following form:
1 1 2 2
1 1 1 1 2 2 2 2 1 1 1 1 1 1 2 2 2 2 2 2
(0)
,
( ) ( )
n i n iA A
n i n i n i n i n i n ih T v v
r r r r .
Matrix elements in the right side of the last formula are calculated
by the Slater–Koster method [18, 19].
Potential energy of an electron in the ion core field in the expression
(11) is determined by formulas (76)–(78), in which, instead of the total
electron density, the density of electrons, ( )
i
n
r (78), in states of the A
kind ion core should put.
Matrix elements
1 1 1 2 2 2
3 3 3 4 4 4
(2) ,
,
n i n i
n i n i
v
of Hamiltonian (1) can be calculated by
integrating the angular variables. Integral of the product of three
spherical functions (Gaunt integral) can be represented using the
Clebsch–Gordan coefficients [34]. As a result, for matrix elements
1 1 2 2
3 3 4 4
(2) ,
,
l m l m
l m l m
v , we obtain
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 217
1 1 2 2
3 3 4 4
3
2 1 3 2 1
3
2 1 3 1 1
1 2
(2) , 2 3 3 1
,
0 3 2
2 , 0,1,2,3,...
2 , 0,1,2,3,...
(2 1)(2 1)(2 1)(2 1)1
2 1 (2 1)(2 1)
l m l m
l m l m
l
l l l l l
l l l l l
l l l n n
l l l n n
l l l l
v e
l l l
3 3 3 1 2 3 1 2 2 1 1
( ;0,0) ( ; , ) ( ;0,0) ( ; , )c l l l c l l l m m m c l l l c l l l m m m
1 3
2 2 1 1 3
2 2 2
1 1 1 ' ' 1 2 2 2 2 1
10 0
( ) ( ) ( ) ( )
r l
nl n l n l n l l
r
dr r R r R r dr r R r R r
r
2 3
2 2 1 1 3
2 2 1
2 2 2 2 1 1 1 1 1
20 0
( ) ( ) ( ) ( )
r l
n l n l nl n l l
r
dr r R r R r dr r R r R r
r
, (81)
where l, m are orbital and magnetic quantum numbers, respectively,
c(lll; m, m)—the Clebsch–Gordan coefficients [34], Rnl(r)—the ra-
dial part of wave function, n—the main quantum number.
Matrix elements on the basis of real wave functions [18],
1 1 1 2 2 2
3 3 3 4 4 4
(2) ,
,
n i n i
n i n i
v
, for each site are expressed by linear combinations of ma-
trix elements
1 1 2 2
3 3 4 4
(2) ,
,
l m l m
l m l m
v . This calculation procedure for matrix ele-
ments can be easily programmed.
Matrix elements on the basis of real wave functions,
1 1 1 2 2 2
3 3 3 4 4 4
(2) ,
,
n i n i
n i n i
v
, for
different sites can be approximately represented in the form similar to
formula (81), if we describe radial part of the wave function by Gaus-
sian function (Gaussian orbital) as this is done in the method of mo-
lecular orbitals—linear combinations of atomic orbitals [35]. In this
approximation, the multicentre integrals
1 1 1 2 2 2
3 3 3 4 4 4
(2) ,
,
n i n i
n i n i
v
have the form of
one-centre integrals, as the product of two Gaussian orbitals, which
are localized at different centres, can be reduced to the product of orbi-
tals, which are localized at the joint centre.
To investigate the influence of an external magnetic field on the en-
ergy spectrum of electrons and electrical conductivity member describ-
ing the additive energy of the crystal in an external magnetic field, it
is need to be added to Hamiltonian of the system H (1):
1 1 1 1 1 1 1 1
1 1 1 1
1
2 H
B n i n i
n i
H a a
, (82)
where B, H, 1 are the Bohr magneton, intensity of external magnetic
field, and electron spin projection on the direction of the magnetic
field, respectively.
218 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
Spin-dependent transport in systems with strong electron correla-
tions is described by partial constituent of diagonal conductivity ten-
sor element (79), which corresponds to a specific value of the electron
spin projection.
Thus, in the obtained expressions for the Green’s functions of elec-
trons and phonons, free energy and electrical conductivity of the crys-
tal, the processes of electron scattering on the potential of ion core,
charge and spin fluctuations of density and lattice vibrations are taken
into account. The accuracy of calculating the electronic structure, vi-
brational spectra and electrical conductivity of the crystal is deter-
mined by precision of vertex renormalization parts of the mass opera-
tors of electron–electron and electron–phonon interactions and the
small parameter of the cluster expansion for the Green’s functions of
electrons and phonons.
6. RESULTS OF CALCULATIONS AND CONCLUSIONS
Here are the results of calculation of the energy spectrum of electrons
and phonons and conductivity of carbon nanotubes doped with chro-
mium. In calculation, renormalization of vertex parts of mass operator
of electron–electron and electron–phonon interactions (21)–(23) was
neglected. Real wave functions of 2s-, 2p-, 3s-, 3d-states of neutral non-
interacting atoms of carbon were chosen. In the above-mentioned self-
consistent iterative procedure of calculation of electronic structure and
free energy, in the first step of the iteration formula (44),
1i
Z
were put
equal to occupation numbers of the corresponding electron states of
neutral noninteracting atoms of carbon and Cr. The off-diagonal (by
site index (ni)) matrix elements of Hamiltonian (1) were calculated by
taking into account the first three coordination spheres. Contribution
to the static displacements of atoms was neglected in the calculations.
Calculations were performed for the temperature T 300 K.
We performed geometry optimization of the crystal structure of
carbon nanotube of chirality (3,0) with Cr impurity. Geometric optimi-
zation of the crystal structure was achieved by minimizing the free en-
ergy (32). Carbon nanotube doped with Cr has a one-dimensional crys-
tal structure. Primitive cell contains 18 non-equivalent atom posi-
tions. Carbon atoms are located in 12 positions on the surface of the
inner cylinder. The distance between the carbon atoms is 0.142 nm.
The Cr atoms are randomly located in six positions on the outer surface
of the cylinder opposite the centre of a hexagon, the vertices of which
are carbon atoms. Through the study of free energy minimum, it is
found that Cr atoms are randomly located on the surface of nanotubes.
This indicates that the probability of Cr atoms arrangement in the ex-
pression (51) 0lj i lj
P P ñ
, where ñ
—the ratio of Cr atoms with re-
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 219
spect to the six possible positions of the atoms Cr within the primitive
cell. The distance between carbon atoms and Cr is 0.22 nm. The relative
position of carbon atoms and Cr is similar to the location of atoms of
transition metals on the surface of carbon nanotubes of large diameter,
which are described in [35] by ultrasoft pseudopotential method.
The value of localized magnetic moment projection of the Cr atom and
induced localized magnetic moment of the C atom in the direction of the
magnetic field increases with the intensity of field. For carbon nano-
tubes with 5 Cr atoms in primitive cell, value of projection magnetic
moment of the Cr atom varies as Cr
(1.02;2.24)
B
m , and the magnetic
moment of the C atom varies as C
(0.0036;0.02)
B
m with increasing
values of the magnetic field intensity from zero to H 200 A/m.
As calculated by means of the formula (39), Figure 7 shows the par-
tial ( 0 0
, ,
1
( ) ( )
e i i
i
g P g
v
) and total ( ( ) ( )
e e
g g
) electron densi-
ties of states of carbon nanotube with an admixture of Cr in the ab-
sence of external magnetic field. Vertical line shows the Fermi level F.
In Figure 8, points show the dependence of free energy F (32) on the
parameter of pair correlations in arrangement of Cr impurities on lat-
tice sites, 0
BB BB
lj i
(54) for the first coordination sphere (atom of Cr is
denoted as the B kind atom). The dependence F(BB) is shown in the re-
gion of free energy minimum. The free energy is measured from aver-
age electrostatic interaction energy of the A kind ions, 0 (61).
Figure 9 shows the partial (ge()) and total (ge()) electron densities
of states of carbon nanotube with five atoms of Cr per primitive cell in
external magnetic field of H 100 A/m. In Figure 9, the part of the en-
ergy spectrum that is close to the Fermi level is shown. As shown in
Fig. 7. Electron densities of states of carbon nanotube with an admixture of Cr.
220 S. P. REPETSKY, O. V. TRETYAK, I. G. VYSHIVANAYA, and V. V. SHASTUN
Figure 9, for a given sign of the projection of magnetic moment local-
ized on sites of lattice, energies of the electron with spin
1
2
are
shifted relative to values of the energy of the electron with spin
1
2
. This leads to the formation of the Coulomb gap in the energy
spectrum of electrons, which is visible on 1
2
( )
e
g and on g() of Fig. 9.
The results presented in Fig. 9 are qualitatively consistent with re-
sults obtained by another method in Ref. [37].
In Figure 10, the dependence of the spin polarized electric current,
Fig. 8. Dependence of free energy F for carbon nanotubes with five atoms of
Cr per primitive cell on the parameter of pair correlations in arrangement of
Cr impurities on lattice sites,
BB.
Fig. 9. Electron densities of states of carbon nanotube with five atoms of Cr
per primitive cell in external magnetic field of H 100 A/m.
MODEL OF STRONG COUPLING THEORY OF ELECTRON CORRELATIONS 221
1 2 1 2
, in carbon nanotube with chirality (3,0) on the
magnitude of the external magnetic field as calculated by the formula
(37) for temperature of 300 K is shown.
Thus, the phenomenon of spin-dependent electron transport occurs
in systems with strong electron correlations and is associated with the
location of the Fermi level relative to the Coulomb gap in the energy
spectrum of electrons.
In an external magnetic field, there is relative displacement of elec-
trons’ energy levels (Coulomb gap) for different spin projections
(Fig. 9).
Value of spin polarization of electron transport depends on the dif-
ference of electron density states with different spin projection at the
Fermi level.
In Ref. [38], the electronic structure and properties of carbon nano-
tubes with transition metal chains adsorbed on the surface is calcu-
lated, based on the density functional method using ultrasoft pseudo-
potential. Our results are qualitatively consistent with the results of
[38], in which ab initio electron density functional method shows that
the chains of transition metals adsorbed on the surface of carbon nano-
tubes open a gap in the electrons’ states with a certain spin value.
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| id | nasplib_isofts_kiev_ua-123456789-98332 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1608-1021 |
| language | English |
| last_indexed | 2025-12-07T17:09:32Z |
| publishDate | 2012 |
| publisher | Інститут металофізики ім. Г.В. Курдюмова НАН України |
| record_format | dspace |
| spelling | Repetsky, S.P. Tretyak, O.V. Vyshivanaya, I.G. Shastun, V.V. 2016-04-11T17:52:39Z 2016-04-11T17:52:39Z 2012 Self-consistent model of strong coupling theory of electron correlations in disordered crystals / S.P. Repetsky, O.V. Tretyak, I.G. Vyshivanaya, V.V. Shastun // Успехи физики металлов. — 2012. — Т. 13, № 2. — С. 189-223. — Бібліогр.: 38 назв. — англ. 1608-1021 PACS numbers: 63.20.K-, 71.15.Ap, 71.20.Tx, 71.27.+a, 72.10.-d, 72.25.-b, 73.63.Fg https://nasplib.isofts.kiev.ua/handle/123456789/98332 The method of the description of electron correlations in the disordered crys-tals based on a Hamiltonian of many-particle system as well as the diagram technique for calculation of Green’s functions are developed. Розвинено методу опису електронних кореляцій у невпорядкованих кри-сталах, що базується на Гамільтоніяні багаточастинкової системи та діяг-рамній техніці для розрахунку Ґрінових функцій. Развит метод описания электронных корреляций в неупорядоченных кристаллах, основанный на гамильтониане многочастичной системы и диаграммной технике для расчёта функций Грина. en Інститут металофізики ім. Г.В. Курдюмова НАН України Успехи физики металлов Self-consistent model of strong coupling theory of electron correlations in disordered crystals Самоузгоджений модель теорії сильного зв’язку еле-ктронних кореляцій у невпорядкованих кристалах Article published earlier |
| spellingShingle | Self-consistent model of strong coupling theory of electron correlations in disordered crystals Repetsky, S.P. Tretyak, O.V. Vyshivanaya, I.G. Shastun, V.V. |
| title | Self-consistent model of strong coupling theory of electron correlations in disordered crystals |
| title_alt | Самоузгоджений модель теорії сильного зв’язку еле-ктронних кореляцій у невпорядкованих кристалах |
| title_full | Self-consistent model of strong coupling theory of electron correlations in disordered crystals |
| title_fullStr | Self-consistent model of strong coupling theory of electron correlations in disordered crystals |
| title_full_unstemmed | Self-consistent model of strong coupling theory of electron correlations in disordered crystals |
| title_short | Self-consistent model of strong coupling theory of electron correlations in disordered crystals |
| title_sort | self-consistent model of strong coupling theory of electron correlations in disordered crystals |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/98332 |
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