Electron correlations in narrow energy bands: modified polar model approach
The electron correlations in narrow energy bands are examined within the framework of the modi ed form of
 polar model. This model permits to analyze the effect of strong Coulomb correlation, inter-atomic exchange
 and correlated hopping of electrons and explain some peculiarities of...
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| Zitieren: | Electron correlations in narrow energy bands: modified polar model approach / L. Didukh, Yu. Skorenkyy, O. Kramar // Condensed Matter Physics. — 2008. — Т. 11, № 3(55). — С. 443-454. — Бібліогр.: 50 назв. — англ. |
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| author | Didukh, L. Skorenkyy, Yu. Kramar, O. |
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| citation_txt | Electron correlations in narrow energy bands: modified polar model approach / L. Didukh, Yu. Skorenkyy, O. Kramar // Condensed Matter Physics. — 2008. — Т. 11, № 3(55). — С. 443-454. — Бібліогр.: 50 назв. — англ. |
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| description | The electron correlations in narrow energy bands are examined within the framework of the modi ed form of
polar model. This model permits to analyze the effect of strong Coulomb correlation, inter-atomic exchange
and correlated hopping of electrons and explain some peculiarities of the properties of narrow-band materials,
namely the metal-insulator transition with an increase of temperature, nonlinear concentration dependence of
Curie temperature and peculiarities of transport properties of electronic subsystem. Using a variant of generalized
Hartree-Fock approximation, the single-electron Green's function and quasi-particle energy spectrum of
the model are calculated. Metal-insulator transition with the change of temperature is investigated in a system
with correlated hopping. Processes of ferromagnetic ordering stabilization in the system with various forms of
electronic DOS are studied. The static conductivity and effective spin-dependent masses of current carriers
are calculated as a function of electron concentration at various DOS forms. The correlated hopping is shown
to cause the electron-hole asymmetry of transport and ferromagnetic properties of narrow band materials.
Електроннi взаємодiї у вузькозонних матерiалах дослiджено в рамках модифiкованої форми полярної моделi. Ця модель дозволяє врахувати вплив сильної кулонiвської кореляцiї, мiжатомного обмiну та корельованого переносу електронiв i пояснити деякi особливостi властивостей вузькозонних матерiалiв, зокрема перехiд метал-дiелектрик при зростаннi температури, нелiнiйну концентрацiйну залежнiсть температури Кюрi та особливостi транспортних властивостей електронної пiдсистеми. З використанням варiанту узагальненого наближення Гартрi-Фока отримано одноелектронну функцiю Грiна та квазiчастинковий енергетичний спектр моделi. В роботi дослiджено перехiд метал-дiелектрик при змiнi температури в системi з корельованим переносом. Також дослiджено процеси стабiлiзацiї феромагнiтного впорядкування в системi при рiзних формах електронної густини станiв. Розраховано статичну провiднiсть та спiн-залежнi ефективнi маси носiїв як функцiї електронної концентрацiї при рiзних густинах станiв. Показано, що корельований перенос зумовлює асиметрiю транспортних та феромагнiтних властивостей вузькозонних матерiалiв.
|
| first_indexed | 2025-12-07T17:46:37Z |
| format | Article |
| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 3(55), pp. 443–454
Electron correlations in narrow energy bands: modified
polar model approach∗
L.Didukh†, Yu.Skorenkyy, O.Kramar
Ternopil State Technical University, Department of Physics, 56 Rus’ka Str., 46001 Ternopil, Ukraine
Received April 29, 2008, in final form July 2, 2008
The electron correlations in narrow energy bands are examined within the framework of the modified form of
polar model. This model permits to analyze the effect of strong Coulomb correlation, inter-atomic exchange
and correlated hopping of electrons and explain some peculiarities of the properties of narrow-band materials,
namely the metal-insulator transition with an increase of temperature, nonlinear concentration dependence of
Curie temperature and peculiarities of transport properties of electronic subsystem. Using a variant of gener-
alized Hartree-Fock approximation, the single-electron Green’s function and quasi-particle energy spectrum of
the model are calculated. Metal-insulator transition with the change of temperature is investigated in a system
with correlated hopping. Processes of ferromagnetic ordering stabilization in the system with various forms of
electronic DOS are studied. The static conductivity and effective spin-dependent masses of current carriers
are calculated as a function of electron concentration at various DOS forms. The correlated hopping is shown
to cause the electron-hole asymmetry of transport and ferromagnetic properties of narrow band materials.
Key words: polar model of crystal, Mott-Hubbard material, ferromagnetic ordering, conductivity
PACS: 71.10.Fd, 71.27.+a, 72.20.-i, 75.10.-b
1. Introduction
Polar model of crystal, developed on the basis of pioneering work of Schubin and Wonsowsky [1],
proved to have rich physical content. Within the framework of the polar model the existence of
systems with charge ordering and gapless semiconductors was predicted, the criterium for metal-
insulator transition was formulated for the first time, the explanation of fractional magnetic mo-
ment of transition 3d-metals was proposed, the possibility of indirect (through polar states) ex-
change interaction was shown. In fact, Hubbard model [2], which today is a standard model for
describing strong electron correlations in crystals, is a partial case of the polar model. An idea
of the configurational representation [3] (in a polar model, site “configurations” are homeopolar
states, holes and doublons) appeared to be very fruitful in investigations of a wide class of ma-
terials with unique electrical and magnetic properties, namely oxides, sulphides and selenides of
transition and rare-earth elements. However, direct application of the method proposed in work [1]
in many cases was found to be inefficient. It was due to, firstly, the difficulties of constructing
Hamiltonians of each particular system, because the algorithm of transition from electron rep-
resentation to configurational one had not been devised, such a transition even in the simplest
model of s-band was too cumbersome. The second reason of inefficiency of early forms of a polar
model is uncontrolled character of approximations used, especially the postulating of commutation
relations (of bose-type) for operators of site excitations and replacement of some operators with
c-numbers (this way the Hamiltonian of “gas limit” was constructed). The other deficiency was
the constraint of equality of the mean electron number to unity, what restricted the use of the
model to the doped Mott-Hubbard insulators. Thus, in spite of the extraordinary heuristic value
of a polar model, the mathematical foundation of configurational representation was not devel-
oped. It caused also the absence of effective mathematical methods of model Hamiltonians (in
∗This article is dedicated to Prof. I.V. Stasyuk on the occasion of his 70th birthday.
†E-mail: didukh@tu.edu.te.ua
c© L.Didukh, Yu.Skorenkyy, O.Kramar 443
L.Didukh, Yu.Skorenkyy, O.Kramar
configurational representation) treatment within the framework of the traditional form of a polar
model.
The disadvantages mentioned above were partially removed in a “new form of polar model”
of Glauberman, Vladimirov and Stasyuk [4–6], where a rigorous algorithm of the transition from
electron operators to the so-called “site elementary excitation” operators was elaborated. For the
latter, exact commutation relations of two types, namely quasifermi and quasibose, were establi-
shed. However, the form of a polar model, presented in the works [4–6], is disadvantageous, namely,
operators of site excitation were built over some postulated magnetic “background” and, as a result,
the approach proposed in the mentioned papers is applicable only to ferromagnetic insulators.
The papers [7] and [8,9] were devoted to clarification of mathematical essence of configurational
representation of a polar model. In the first paper, transition operators Xkl
i (Hubbard operators)
were introduced and in papers [8,9] the relation between electron and configuration representations:
ais = X0s
i − ηsX
s̄2
i , (1)
was introduced. In the above equation, ais is an operator of annihilation of electron with spin
s =↑ or ↓ at site i, Xkl
i is an operator of site i transition from state |l〉 to state |k〉, possible
states are hole |0〉, doublon |2〉 and single occupied states with electron | ↑〉 or | ↓〉, ηs = 1 for
s =↑ and ηs = −1 otherwise (in paper [9] other notations were used, here we use standard ones).
The transition operator Xkl
i is nothing but a product of respective Shubin-Wonsowsky operators:
Xγδ
i = α+
iγαiδ (see [10] in this connection). Commutation relations for a pair of Shubin-Wonsowsky
operators are identical to the corresponding Hubbard operators. This fact closes the discussion
on the statistics of Shubin-Wonsowsky operators. It should be noted that the algebra for Shubin-
Wonsowsky operators as well as for operators introduced later in the papers [11,12], is “richer”
than Hubbard operators’ algebra, and the former operators can have both fermi and bose character
(see, for example, work [13]).
Despite its bulky appearance, configurational representation of the polar model (or Hubbard
model) for systems with strong intra-atomic Coulomb interaction essentially simplifies the analysis
due to a diagonal form of the principal term, namely intra-atomic Coulomb interaction.
A series of works were devoted to mathematical treatment of polar model Hamiltonian by dif-
ferent methods (see work [14] for a review). Here we note the papers [10,15–22] in the context of
further investigations concerning correlation effects in narrow-band materials. Within the frame-
work of this approach, the efficient form of perturbation theory [8,9] (generally adopted today) was
developed, an effective Hamiltonian was formulated, and generalized t − J model was proposed.
In the present paper specific electric and magnetic properties of some narrow-band materials
are investigated within the framework of modified form of a polar model. In section 2, the model
Hamiltonian is formulated. In section 3 the metal-insulator transition in a half-filled band with an
increase of temperature is studied. In section 4 ferromagnetic ordering in a partially filled band is
considered and nonlinear concentration dependence of Curie temperature is explained. In section 5
the peculiarities of transport properties of an electronic subsystem are analyzed.
2. Modified form of polar model
In electronic operator notations, the modified form of the polar model is represented by Hamil-
tonian [10]
H = − µ
∑
is
a+
isais +
∑
ijs
′
tij(n)a+
isajs +
∑
ijs
′ (
T (ij)a+
isajsnis̄ + h.c.
)
+ U
∑
i
ni↑ni↓ +
1
2
∑
ijss′
′
J(ij)a+
isa
+
js′ais′ajs +
1
2
∑
ijss′
′
V (ij)nisnjs′ , (2)
where µ is chemical potential,
tij(n) = t(ij) + n
∑
k 6=i
k 6=j
J(ikjk) (3)
444
Electron correlations in narrow energy bands: modified polar model approach
is hopping integral between the nearest neighbors dependent on electron concentration n,
J(ijkl) =
∫∫
φ∗(r −Ri)φ(r −Rk)
e2
|r − r′|φ
∗(r′ −Rj)φ(r′ −Rl)drdr
′, (4)
are the matrix elements which describe electron-electron interactions, U = J(iiii) is energy of
intra-atomic Coulomb repulsion, V (ij) = J(ijij) describes inter-atomic Coulomb interaction (i
and j are the nearest neighbors), T (ij) = J(iiij) describes the effect of site occupation on the
hopping processes (correlated hopping), J(ij) = J(ijji) is inter-atomic ferromagnetic exchange
interaction. Having neglected all matrix elements in (2) except t(ij) and U we obtain the Hubbard
Hamiltonian.
We rewrite the Hamiltonian of a correlated electron system in representation of Xkl
i Hubbard
operators using relations (1):
H = H0 + Htr + H ′
tr + Hex , (5)
H0 = −µ
∑
is
(
Xs
i + X2
i
)
+ U
∑
i
X2
i +
1
2
∑
ij
′
V (ij)
(
1 − X0
i + X2
i
) (
1 − X0
j + X2
j
)
,
Htr =
∑
ijs
′
tij(n)Xs0
i X0s
j +
∑
ijs
′
t̃ij(n)X2s
i Xs2
j ,
H ′
tr =
∑
ijs
′ (
t′ij(n)ηsX
s0
i X s̄2
j + h.c.
)
,
Hex = −1
2
∑
ijs
′
J(ij)
((
Xs
i +X2
i
) (
Xs
j +X2
j
)
+ Xss̄
i X s̄s
j
)
.
Here translation processes of holes and doublons are characterized by different hopping integrals,
tij(n) = (1 − τ1n)tij and t̃ij(n) = (1 − τ1n − 2τ2)tij , respectively; t′ij(n) = (1 − τ1n − τ2)tij is
a hopping parameter, which describes processes of creation and destruction of doublon-hole pair;
correlated hopping parameters τ2 and τ1 describe the effect of the sites involved the hopping process
and neighboring sites, respectively.
Configurational representation is especially useful in investigating a narrow-band system in
which the condition U � w (2w = 2z|t(ij)| is bare bandwidth, z is a number of the nearest
neighbors to a site) is satisfied. In this case, the system can be both Mott-Hubbard insulator at
n = 1 and doped Mott-Hubbard insulator at n 6= 1. Then, general Hamiltonian, using a suitable
form of perturbation theory [8], can be written in the form of effective Hamiltonian (EH), which is
suitable for the mathematical treatment. In this way the transition to the well-known t− J model
was done (see the review [15] as well as the papers [8,23], where the modern form of t − J model
was formulated for the first time). The canonical transformation is performed (see [10] for details)
and the Hamiltonian (5) is written in the form
H̃ = H0 + Htr + Hex + H̃ex + H̃tr−ex , (6)
where
H̃ex = −1
2
∑
ijs
′
J̃(ij)(Xs
i X s̄
j − Xss̄
i X s̄s
j − X2
i X0
j ),
H̃tr−ex = −1
2
∑
ijks
′
J(ijk)
(
Xs0
i X s̄
j X0s
k − Xs0
i X s̄s
j X0s̄
k + X2s
i Xss̄
j X s̄2
k − X2s
i X s̄
j Xs2
k
)
.
Here J̃(ij) = 2t′ij(n)t′ij(n)/∆ is integral of indirect exchange (through polar states), J(ijk) =
2t′ij(n)t′jk(n)/∆ is integral of indirect charge transfer, ∆ = U − V + zV
(
〈X0
i 〉 + 〈X2
i 〉
)
. The di-
stinctions of EH (6) from other forms of t − J models [24,25] are caused by the concentration
dependence of hopping integrals in lower and upper subbands, the difference of the noted hopping
integrals (the absence of electron-hole symmetry) and unusual form of the concentration depen-
dent superexchange and superhopping integrals. The peculiarities of the model EH listed above
are useful in interpreting the physical properties of narrow-band materials.
445
L.Didukh, Yu.Skorenkyy, O.Kramar
3. Metal-insulator transition at increase of temperature
Among the metal-insulator transitions (MIT) observed in narrow-bands materials the transiti-
ons from paramagnetic metal state to paramagnetic insulator state at increase of temperature are of
special interest. Such transitions are realized in systems NiS2−xSex [26,27] and (V1−xCrx)2O3 [26,
28]. In these systems, the paramagnetic insulator – paramagnetic metal transitions under external
pressure are observed as well. In this section temperature-induced MIT is considered and explained
based on the modified form of polar model.
Let us consider an electron system described by Hamiltonian (5) in the case of intermediate
correlation strength U ' w(n) and introduce a single-particle Green’s function
〈〈ap↑|a+
p′↑〉〉 = 〈〈X0↑
p |X↑0
p′ 〉〉 − 〈〈X↓2
p |X↑0
p′ 〉〉 − 〈〈X0↑
p |X2↓
p′ 〉〉 + 〈〈X↓2
p |X2↓
p′ 〉〉. (7)
The functions 〈〈X0↑
p |X↑0
p′ 〉〉 and 〈〈X↓2
p |X↑0
p′ 〉〉 satisfy the equations of motion
(E + µ)〈〈X0↑
p |X↑0
p′ 〉〉 =
δpp′
2π
〈X↑
p + X0
p〉 + 〈〈
[
X0↑
p , Htr
]
|X↑0
p′ 〉〉 + 〈〈
[
X0↑
p , H ′
tr
]
|X↑0
p′ 〉〉,
(E + µ − U)〈〈X↓2
p |X↑0
p′ 〉〉 = 〈〈
[
X↓2
p , Htr
]
|X↑0
p′ 〉〉 + 〈〈
[
X↓2
p , H ′
tr
]
|X↑0
p′ 〉〉, (8)
with [A, B] = AB−BA. To obtain the closed system of equations we take advantage of the two-pole
approximation [16,19]. Suppose in equation (8)
[
X0↑
p , Htr
]
=
∑
j
ε(pj)X0↑
j ,
[
X↓2
p , Htr
]
=
∑
j
ε̃(pj)X↓2
j , (9)
where ε(pj) and ε̃(pj) are non-operator quantities, which can be calculated using the method of
work [10]. For half band filling (electron concentration n=1) in a paramagnetic state we have
ε(pj) = (1 − 2d + 2d2)tpj − 2d2t̃pj , (10)
ε̃(pj) = (1 − 2d + 2d2)t̃pj − 2d2tpj , (11)
with d = 〈X2
p〉 being the concentration of the doubly occupied sites.
Terms of equations (8) describing processes of doublon-hole pair creation or destruction may be
taken into account in the mean-field approximation. This way we obtain a closed system of equa-
tions. Applying the analogous procedure to equations for functions 〈〈X0↑
p |X2↓
s 〉〉 and 〈〈X↓2
p |X2↓
s 〉〉,
after the Fourier transformation, we obtain Green’s function
Gk =
1
2π
(
Ak
E − E1(k)
+
Bk
E − E2(k)
)
, Ak =
1
2
(
1 − t′(k)
E2(k) − E1(k)
)
, Bk = 1 − Ak ,
(12)
where
E1,2(k) = −µ +
U
2
+ (1 − 2d)
t(k) + t̃(k)
2
∓ 1
2
√
[U − (t(k) − t̃(k))(1 − 2d + 4d2)]2 + (t′(k))2 (13)
is the energy spectrum of the system, t(k), t′(k) and t̃(k) are Fourier transforms of hopping
integrals tij , t′ij and t̃ij , respectively. Single-particle Green’s function (12) and energy spectrum
(13) are exact in the band and atomic limits and have peculiar dependence on concentration of
polar states, which explains the temperature-induced transition to insulator state in the systems
NiS2−xSex and (V1−xCrx)2O3. Equation (13) allows us to calculate the energy gap width as the
energy difference between the bottom of the upper and the top of the lower Hubbard bands:
∆E = −2w(1 − 2d)(1 − τ1 − τ2) +
Q1 + Q2
2
, (14)
446
Electron correlations in narrow energy bands: modified polar model approach
where
Q1,2 =
(
(U ∓ 2τ2Cw)2 + (1 − τ1 − 2τ2)
2w2
)
1
2 , C = 1 − 2d + 4d2.
Expression (14) describes the closure of the energy gap in the spectrum of paramagnetic insulator
at critical value (U/w)c, when the half-bandwidth w increases (under pressure or doping). This
expression also reproduces the exact result for a partial case of the model [29].
Figure 1. Phase diagram of paramagnetic
metal – paramagnetic insulator transition.
Figure 2. Polar states concentration as a func-
tion of the correlation strength parameter. Up-
per curve corresponds to τ2 = 0, middle curve:
τ2 = 0.1, lower curve: τ2 = 0.2.
At increasing temperature in metallic state the overlapping of subbands decreases and temper-
ature induced transition from metallic to insulating state can occur at some values of parameter
w/U (see figure 1). The energy gap depends on temperature through the concentration of polar
states which can be derived from function 〈〈X↓2
p |X2↓
s 〉〉. The dependence d(U/w) is plotted in fi-
gure 2. One can see that at critical value (U/w)c the slope of d(U/w)-dependence changes; the
concentration of doublons remains nonzero at any finite U/w. Our result for d(U/w) at τ1 = τ2 = 0
in the region of MIT is in good agreement with the results of the paper [30] obtained in the limit of
infinite dimensions and composite operator method [31]. We have found that the correlated hopping
processes reduce the polar states concentration and shift the transition point towards lower values
of U/w. To the best of our knowledge, a systematic study of MIT in Hubbard model with corre-
lated hopping, using the dynamical mean field theory, has not been performed so far, though an
approach for the description of correlated hopping in infinite dimensions has been developed in the
work [32]. Numerical data, obtained in the work [33] for a two-dimensional system with correlated
hopping by Monte-Carlo calculations do not make it possible to study the temperature-induced
MIT. Thus, the approximation developed in works [10,16], is at the moment the most conveni-
ent tool for describing peculiar electric properties of transition-metal compounds with correlated
hopping within the framework of the modified form of polar model.
4. Ferromagnetic ordering in the model
The regime of strong Coulomb interaction (U � w) is more beneficial for ferromagnetic or-
dering realization in the considered model. In this case one may apply an effective Hamiltonian (6)
obtained by the method of canonical transformation [10]. Energy spectrum of the system at n < 1
is obtained in the form [34]:
Es(k) = −µ + αstk(n) + βs − zJeffns , (15)
447
L.Didukh, Yu.Skorenkyy, O.Kramar
here
αs = 1 − ns̄ +
ns̄ns
1 − ns̄
=
2 − n + ηsm
2
+
n2 − m2
2(2 − n + ηsm)
(16)
is correlation narrowing factor,
βs = − 1
1 − ns̄
∑
k
tk(n)〈X s̄0
i X0s̄
j 〉k (17)
is spin-dependent shift of the subband center, Jeff is effective exchange parameter, determined
by competition of direct and indirect exchange interactions, ns = (n + ηsm)/2, m is the system
magnetization. For n > 1 we obtain Ẽs in a similar form; expressions for correlation narrowing
factor and shift of subband center are obtained from equations (16) and (17) replacing n → 2− n,
t(n) → t̃(n). The analytical calculation of spin-dependent shift of the subband center βs for model
rectangular density of states (DOS) has been done in the work [35]. In the present paper βs is
calculated numerically and its value is shown to depend drastically on non-perturbed DOS form.
Let us calculate the ground state energy of the system described by effective Hamiltonian (6)
in the case of n < 1. We use the expression for the ground state energy (per lattice site) in the
form [36]
E0 =
1
2N
∑
ks
∞
∫
−∞
[tk(n) + E] f(E)Ss
k(E)dE, (18)
and concentration of electrons with spin s
ns =
1
N
∑
k
∞
∫
−∞
f(E)Ss
k
(E)dE, (19)
here f(E) is the Fermi distribution function, Ss
k
(E) = (1− ns̄)δ(E −Es
k
) is spectral density of the
Green’s function.
We argue that the form of non-interacting DOS (which corresponds to some lattice structure)
substantially effects the critical electron concentration n1 at which ferromagnetic ordering occurs
as well as electron concentration n2 at which magnetic moment becomes saturated. By numerical
calculations of the ground state energy based on the expression (18) and subsequent minimization
we have investigated the condition of ferromagnetism stabilization for various DOS. In particular,
the numerical analysis has been done for DOS which corresponds to a simple cubic lattice [37], for
DOS that corresponds to the body-centered cubic lattice [38], as well as for the DOS with the peak
near the band-edge [39] ρ(ε) = c
√
w2 − ε2/(w + aε), with free parameter c = (1 +
√
1 − a2)/(πw).
Varying the asymmetry parameter a, from the latter DOS one can obtain both the semi-elliptical
DOS (at a = 0) and the DOS with peak near the band-edge (at a → 1).
Our results for sc-lattice n1 = 0.36 and n2 = 0.62 agree with the well-known results obtained by
Roth [36]: at n1 = 0.36 the ferromagnetic ordering occurs, and at n2 = 0.63 the magnetic moment
saturates. In the case of sc-lattice, spectral density approximation (SDA) [40,41] yields the following
results: spontaneous magnetization occurs at n1 = 0.34 and at n2 = 0.68 ferromagnetic ordering
reaches the saturation; in the case of bcc-lattice: the critical concentrations are n1 = 0.52 and
n2 = 0.68, respectively. Our results for bcc-lattice are n1 = 0.55 and n2 = 0.64. The Gutzwiller
variational approach [42] gives only the critical concentration of saturated ferromagnetic state
n2 = 0.68 for sc- and bcc-lattices. Our results also agree with the results obtained by the expansion
of one-particle Green’s functions over the inverse coordination number [43]. In the case of “tunable”
DOS with the peak near the band-edge we have n1 = 0.20 and n2 = 0.31 at a = 0.3, n1 = 0.09 and
n2 = 0.15 at a = 0.15. With an increase of parameter a, the critical concentration of ferromagnetism
onset decreases. Therefore, the existence of a peak near the band-edge in electronic DOS favors
the ferromagnetic ordering, in accordance with [39]. It should be noted, that in the case of strong
448
Electron correlations in narrow energy bands: modified polar model approach
electron correlation and half-filling of the band (when the shifts of subband center vanish) the
ferromagnetic ordering is stabilized only by the interatomic exchange interaction (independently
of DOS used).
Let us investigate the effect of non-interacting DOS form on the magnetization at non-zero
temperature. In the case n < 1, the concentration of electrons with spin projections ↑ and ↓ at
arbitrary DOS ρ(ε) and temperature Θ is as follows:
n↑ = (1 − n↓)
w(n)
∫
−w(n)
ρ(t)dt
exp(
E↑(t)
Θ ) + 1
, n↓ = (1 − n↑)
w(n)
∫
−w(n)
ρ(t)dt
exp(
E↓(t)
Θ ) + 1
. (20)
Taking into account Es(t) = µs + αst, where µs = −µ + βs − zJeffns, we write (20) as
n + m
2 − n + m
=
w(n)
∫
−w(n)
ρ(t)dt
exp(
µ↑+α↑t
Θ ) + 1
,
n − m
2 − n − m
=
w(n)
∫
−w(n)
ρ(t)dt
exp(
µ↓+α↓t
Θ ) + 1
. (21)
The difference between the values of the shifted chemical potentials for the electrons with different
spin projection is:
µ↑ − µ↓ = β↑ − β↓ − zJeffm. (22)
The numerical self-consistent analysis of the system of equations (21) and condition (22) has
been carried out. We have obtained the magnetization as a function of temperature, electron
concentration and energy parameters of the model, exchange interaction and correlated hopping,
in particular.
Figure 3. Temperature dependencies of mag-
netization for the DOS of bcc-lattice at
zJeff/w = 0.08. The curves (from down to up)
correspond to the case of electron concentrati-
ons n = 0.58, 0.62, 0.66, 0.70, 0.74, 0.78, 0.82,
0.84, 0.88, 0.92.
Figure 4. Concentration dependencies of
Curie temperature for the DOS of bcc-lattice
at τ1 = 0. The curves (from up to down) cor-
respond to the case of zJeff/w = 0, 0.04 and
0.08.
In the case of semi-elliptical DOS, the temperature dependence of magnetization is step-wise
and Curie temperature has rather low values. This fact is in accord with concentration dependence
of energy difference of paramagnetic and ferromagnetic states ∆EFM
0 /w for the semi-elliptic DOS.
With an increase of effective exchange interaction, the Curie temperature increases. In the case of
449
L.Didukh, Yu.Skorenkyy, O.Kramar
DOS with a peak near the band-edge (a 6= 0), the region of ferromagnetic ordering is shifted to low
electron concentration values and the difference ∆EFM
0 /w increases, which leads to a substantial
increase of Curie temperature.
In the case of DOS, which corresponds to the simple cubic lattice, the dependence m(Θ/w)
is also step-wise; the concentration dependence of Curie temperature has a maximum. Such non-
linear behavior of TC(n) is also obtained for the rectangular DOS, and Curie temperature is in
agreement with experimental values for sulphides of transition metals [44], that have cubic lattice
symmetry. Our results for the case of DOS of bcc-lattice are in qualitative agreement with the
papers [40,41,45,46], where spectral density approximation (SDA) is used. In figure 3 the tempera-
ture dependencies of magnetization at various electron concentrations are plotted (we assume that
half-bandwidth w = 1 eV, like in the cited works). In the absence of effective exchange interaction
we have obtained the values of Curie temperature, which are lower than the results of SDA. We
argue that this fact is due to the peculiarities of spin-dependent shift of band centers, which is
responsible for the translational mechanism of ferromagnetism stabilization. As one can see from
figure 3, when temperature increases at some fixed electron concentration, the continuous (or di-
scontinuous) transition to paramagnetic state is realized. The concentration dependence of Curie
temperature (see figure 4) is of nonlinear shape: ferromagnetic ordering occurs at some critical elec-
tron concentration (which has been determined based on the ground state energy minimization),
TC increases sharply and at n ' 0.82 reaches the maximum value. With an increase of effective
exchange interaction parameter, the maximum value increases and ferromagnetic concentration
region broadens. Since the translational mechanism of ferromagnetism is destabilized by the in-
crease of the correlated hopping parameter τ1, the Curie temperature decreases and the maximum
of ΘC/w is shifted to the lower electron concentration.
5. Peculiarities of the electronic conductivity
Using the method of the papers [47,48], we calculate the xx-component of static electronic
conductivity in the regime of strong intra-atomic correlation
σ = −
∑
s
σ0(1 − ns̄)
w(n)
∫
−w(n)
ρ(t)tdt
exp(Es(t)
Θ ) + 1
+
w̃(n)
∫
−w̃(n)
ρ(t)tdt
exp( Ẽs(t)
Θ ) + 1
, (23)
where the first sum is the conductivity of the lower subband, the second sum is the conductivity of
upper subband. Here the magnetization of the system is an important parameter which is calculated
using the method of the previous section separately for n < 1 and n > 1. In the above equation,
σ0 is a normalizing constant dependent on the mechanism of scattering and lattice parameter.
It should be noted here that one should perform a self-consistent calculation of spin-dependent
shifts of subband centers which are dependent on the DOS form as well as on band filling and
magnetization of the system.
Figure 5 shows the concentration dependencies in Hubbard model (both correlated hopping
parameters are taken to be zero) in a strong intraatomic correlation regime. This plot allows
us to analyze the effect of translation processes and DOS form on the conductivity of narrow-
band materials. The dependence of conductivity value on the chosen DOS form is related to the
corresponding dependence of kinetic energy. In particular, the maximum value of kinetic energy is
reached at rectangular DOS, and the minimum value corresponds to the body-centered cubic DOS.
The concentration dependence of conductivity is determined by the kinetic energy of electrons and
lattice magnetization. Both these factors, in their turn, depend on the peculiarities of bare density
of states and the interatomic interactions, which result in the appearance of correlation narrowing
of Hubbard subbands and spin-dependent shifts of subband centers.
The absence of sharp change of conductivity in the vicinity of half subband filling is a character-
istic feature of the curve corresponding to the model rectangular DOS. It is caused by the absence
of a transition to magnetically ordered state at this form of DOS. The maximum of concentration
450
Electron correlations in narrow energy bands: modified polar model approach
Figure 5. Dependencies of static conduc-
tivity on electron concentration at different
band DOS forms. Curve 1 corresponds to
bcc-lattice, curve 2 corresponds to sc-lattice,
curves 3 and 4 are for semi-elliptic and rect-
angular DOS, respectively.
Figure 6. Concentration dependence of con-
ductivity in the system with correlated hop-
ping and bcc-lattice. The correlated hopping
parameters for the upper curve are τ1 = τ2 =
0, for middle curve τ1 = τ2 = 0.1, for the lower
curve τ1 = τ2 = 0.2.
dependence of σ in this case is reached at n > 0.5, which is typical of paramagnetic state [49]. Step-
wise transition to saturated ferromagnetic state which is realized in a system with semi-elliptical
DOS, causes a sharp decrease of the conductivity at n = 0.59. Stabilization of magnetic ordering
(with partial spin polarization, see section 4), in the systems with body-centered cubic and sim-
ple cubic lattices, makes σ(n) dependencies more smooth. Note, that in the case of simple cubic
lattice, the concentration interval, in which partial magnetic polarization of the system is realized,
is much wider than that for the body-centered cubic, which has two important consequences. The
first, the changes of conductivity for sc-lattice are more smooth than for bcc-lattice. The second,
the difference between maximum values of the conductivity is small, because the maximum of the
curve for sc-lattice is in FM region and the respective maximum for bcc-lattice is in PM-region.
In the papers [48,49] the effect of correlated hopping of electrons on the conductivity in the case
of rectangular density of states was studied. Equation (23) permits to calculate the concentration
dependencies for conductivity of the system with correlated hopping at arbitrary DOS form. As
one can see from figures 5 and 6, both the DOS form and the correlated hopping substantially
effect the concentration dependencies of conductivity. Due to correlated hopping, conductivity in
more than half-filled band is substantially lower than that in less than half-filled band. An essential
peculiarity of the systems with bcc and sc lattices is the decrease of absolute value of conductivity
change if the transition takes place in more than half-filled band.
Within the framework of the considered model one can naturally introduce the notions of
“wide” (lower) and “narrow” (upper) energy subbands and, correspondingly, “light” and “heavy”
current carriers. Using the formula (16) for a coefficient of correlation narrowing of the subband
we obtain
ms
eff =
m0
(1 − τ1n)αs
(24)
for the effective mass of a current carrier in the lower subband, where m0 stands for effective mass
of current carriers in the absence of intra-atomic Coulomb correlation and correlated hopping. In
a similar way one obtains
m̃s
eff =
m0
(1 − τ1n − 2τ2)α̃s
(25)
451
L.Didukh, Yu.Skorenkyy, O.Kramar
for carriers in the upper subband. Renormalization of the current carrier mass in the system
under consideration is determined by two factors: the correlated hopping of electrons and the
correlation narrowing of a band. From the obtained formulae one can see that the effective masses
appear to be spin-dependent. Realization of magnetic ordering essentially modifies the behavior
of effective masses of current carriers. Conditions for magnetic ordering realization are determined
mainly by the form of unperturbed density of electronic states and exchange interaction. Numerical
calculations of the magnetization by minimization of the ground state energy permit to study the
effect of bare DOS form, model parameters (especially correlated hopping) and external effects
(doping, pressure, application of magnetic field) on the behavior of effective masses of current
carriers in the lower and upper quasiparticle subbands.
Figure 7. Concentration dependencies of effec-
tive mass in paramagnetic state of a system
with model rectangular DOS. Curves (from
down to up) correspond to values τ1 = τ2 = 0,
0.1 and 0.2.
Figure 8. Concentration dependencies of ef-
fective masses of carriers with spin ↓ (upper
curve) and ↑ (lower curve) in a saturated fer-
romagnetic state (zJeff/w = 0.02, τ1 = 0) for
the system with rectangular DOS.
It is worth emphasizing that here the term “effective mass of a current carrier” has a conditional
sense [49], which differs to some extent from that used in the standard band theory. Expressions (24)
and (25) are related to expressions for the band spectrum, which describe the transitions between
|s〉- and |0〉-states and transitions between |2〉- i |s〉-states. For this reason ms
eff and m̃s
eff are, in fact,
effective masses of the corresponding transitions. In the cases, when subbands are almost empty
or almost filled, ms
eff and m̃s
eff can be treated as effective masses of the electron and hole states,
respectively (in terms of band theory). In the transition from the regime, in which conduction
occurs due to the carriers from the lower band to the state when s ↔ 2-transitions make the main
contribution to the current, the effective mass increases stepwise at n = 1 (qualitatively, behavior
of the effective mass on the subband filling is shown in figure 7).
In the considered case, at rectangular density of states, in the absence of an effective exchange
interaction, a paramagnetic state is realized in the system [48]. Substantially different behavior is
the characteristic feature of the effective mass of carriers at Jeff 6= 0 caused by a strong dependence
of the energy spectrum on the system magnetization. In figure 8, dependencies of the effective
masses of the carriers with different spin directions on the band filling are shown for the case when
the magnetization reaches its saturation value. At decreasing n, the effective mass of a carrier
with the majority spin remains nearly constant, while for the carriers with the minority spin it
increases substantially. The localization of carriers with spin s =↓ is caused by the peculiarities
of the kinetic energy dependence on the system magnetization which makes the motion of the
majority spin carriers energetically more favorable. The increase of magnetization leads to the rise
of the difference in effective masses of spin-up and spin-down current carriers. This leads to a
decrease of overall transport in ferromagnetic state, although effective masses of the carrier with
452
Electron correlations in narrow energy bands: modified polar model approach
the majority spin become lower. Such a dependence of the current carrier mass on projection of
spin has been observed recently [50].
6. Conclusions
Peculiarities of the modified form of polar model in configurational representation, especially
the ability to treat the effects of interatomic Coulomb correlation accurately, permit to illus-
trate the formation of Hubbard subbands, to incorporate the effects of correlated hopping and to
introduce an effective exchange interaction in a natural way. The generalized mean field approx-
imation proposed in the papers [10,16,19], applied to this model, yields physically sound results,
temperature and concentration dependent energy spectrum, in particular. It enables us to inter-
pret non-typical temperature transition from metallic state to insulating state observed in the
systems NiS2−xSex and (V1−xCrx)2O3. Using the procedure of numerical self-consistent analysis,
temperature dependencies of magnetization have been calculated and the peculiarities of Curie
temperature concentration dependence in Fe1−xCoxS2 and Co1−xNixS2, electronic conductivity
and spin-dependent effective mass of current carriers as a function of energy parameters of the
model and electron concentration at various non-perturbed DOS have been investigated. It is
worth noting that the application of this approach has allowed to obtain proper estimations for
the values of ferromagnetic characteristics for real narrow-band materials.
Thus, the suitability of the model for reasonable description of the peculiarities of electric
and magnetic properties of narrow-band material, metal-insulator transition at an increase of
temperature, non-typical concentration dependence of Curie temperature and transport properties
of Mott-Hubbard materials, in particular, makes the modified form of polar model a convenient
and efficient tool for investigation of narrow-band systems.
Acknowledgements
The authors gratefully acknowledge numerous enlightening discussions with Prof. I.V. Stasyuk.
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Мiжелектроннi взаємодiї у вузьких енергетичних зонах:
метод полярної моделi
Л.Д.Дiдух, Ю.Л.Скоренький, О.I.Крамар
Тернопiльський державний технiчний унiверситет, Україна, Тернопiль 46001, вул. Руська, 56
Отримано 29 квiтня 2008 р., в остаточному виглядi – 2 липня 2008 р.
Електроннi взаємодiї у вузькозонних матерiалах дослiджено в рамках модифiкованої форми поляр-
ної моделi. Ця модель дозволяє врахувати вплив сильної кулонiвської кореляцiї, мiжатомного обмiну
та корельованого переносу електронiв i пояснити деякi особливостi властивостей вузькозонних ма-
терiалiв, зокрема перехiд метал-дiелектрик при зростаннi температури, нелiнiйну концентрацiйну
залежнiсть температури Кюрi та особливостi транспортних властивостей електронної пiдсистеми.
З використанням варiанту узагальненого наближення Гартрi-Фока отримано одноелектронну фун-
кцiю Грiна та квазiчастинковий енергетичний спектр моделi. В роботi дослiджено перехiд метал-
дiелектрик при змiнi температури в системi з корельованим переносом. Також дослiджено процеси
стабiлiзацiї феромагнiтного впорядкування в системi при рiзних формах електронної густини ста-
нiв. Розраховано статичну провiднiсть та спiн-залежнi ефективнi маси носiїв як функцiї електронної
концентрацiї при рiзних густинах станiв. Показано, що корельований перенос зумовлює асиметрiю
транспортних та феромагнiтних властивостей вузькозонних матерiалiв.
Ключовi слова: полярна модель кристалу, мотт-габбардiвський матерiал, феромагнiтне
впорядкування, провiднiсть
PACS: 71.10.Fd, 71.27.+a, 72.20.-i, 75.10.-b
454
|
| id | nasplib_isofts_kiev_ua-123456789-119340 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T17:46:37Z |
| publishDate | 2008 |
| publisher | Інститут фізики конденсованих систем НАН України |
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| spelling | Didukh, L. Skorenkyy, Yu. Kramar, O. 2017-06-06T13:43:05Z 2017-06-06T13:43:05Z 2008 Electron correlations in narrow energy bands: modified polar model approach / L. Didukh, Yu. Skorenkyy, O. Kramar // Condensed Matter Physics. — 2008. — Т. 11, № 3(55). — С. 443-454. — Бібліогр.: 50 назв. — англ. 1607-324X PACS: 71.10.Fd, 71.27.+a, 72.20.-i, 75.10.-b DOI:10.5488/CMP.11.3.443 https://nasplib.isofts.kiev.ua/handle/123456789/119340 The electron correlations in narrow energy bands are examined within the framework of the modi ed form of
 polar model. This model permits to analyze the effect of strong Coulomb correlation, inter-atomic exchange
 and correlated hopping of electrons and explain some peculiarities of the properties of narrow-band materials,
 namely the metal-insulator transition with an increase of temperature, nonlinear concentration dependence of
 Curie temperature and peculiarities of transport properties of electronic subsystem. Using a variant of generalized
 Hartree-Fock approximation, the single-electron Green's function and quasi-particle energy spectrum of
 the model are calculated. Metal-insulator transition with the change of temperature is investigated in a system
 with correlated hopping. Processes of ferromagnetic ordering stabilization in the system with various forms of
 electronic DOS are studied. The static conductivity and effective spin-dependent masses of current carriers
 are calculated as a function of electron concentration at various DOS forms. The correlated hopping is shown
 to cause the electron-hole asymmetry of transport and ferromagnetic properties of narrow band materials. Електроннi взаємодiї у вузькозонних матерiалах дослiджено в рамках модифiкованої форми полярної моделi. Ця модель дозволяє врахувати вплив сильної кулонiвської кореляцiї, мiжатомного обмiну та корельованого переносу електронiв i пояснити деякi особливостi властивостей вузькозонних матерiалiв, зокрема перехiд метал-дiелектрик при зростаннi температури, нелiнiйну концентрацiйну залежнiсть температури Кюрi та особливостi транспортних властивостей електронної пiдсистеми. З використанням варiанту узагальненого наближення Гартрi-Фока отримано одноелектронну функцiю Грiна та квазiчастинковий енергетичний спектр моделi. В роботi дослiджено перехiд метал-дiелектрик при змiнi температури в системi з корельованим переносом. Також дослiджено процеси стабiлiзацiї феромагнiтного впорядкування в системi при рiзних формах електронної густини станiв. Розраховано статичну провiднiсть та спiн-залежнi ефективнi маси носiїв як функцiї електронної концентрацiї при рiзних густинах станiв. Показано, що корельований перенос зумовлює асиметрiю транспортних та феромагнiтних властивостей вузькозонних матерiалiв. The authors gratefully acknowledge numerous enlightening discussions with Prof. I.V. Stasyuk. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Electron correlations in narrow energy bands: modified polar model approach Мiжелектроннi взаємодiї у вузьких енергетичних зонах: метод полярної моделi Article published earlier |
| spellingShingle | Electron correlations in narrow energy bands: modified polar model approach Didukh, L. Skorenkyy, Yu. Kramar, O. |
| title | Electron correlations in narrow energy bands: modified polar model approach |
| title_alt | Мiжелектроннi взаємодiї у вузьких енергетичних зонах: метод полярної моделi |
| title_full | Electron correlations in narrow energy bands: modified polar model approach |
| title_fullStr | Electron correlations in narrow energy bands: modified polar model approach |
| title_full_unstemmed | Electron correlations in narrow energy bands: modified polar model approach |
| title_short | Electron correlations in narrow energy bands: modified polar model approach |
| title_sort | electron correlations in narrow energy bands: modified polar model approach |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/119340 |
| work_keys_str_mv | AT didukhl electroncorrelationsinnarrowenergybandsmodifiedpolarmodelapproach AT skorenkyyyu electroncorrelationsinnarrowenergybandsmodifiedpolarmodelapproach AT kramaro electroncorrelationsinnarrowenergybandsmodifiedpolarmodelapproach AT didukhl miželektronnivzaêmodiíuvuzʹkihenergetičnihzonahmetodpolârnoímodeli AT skorenkyyyu miželektronnivzaêmodiíuvuzʹkihenergetičnihzonahmetodpolârnoímodeli AT kramaro miželektronnivzaêmodiíuvuzʹkihenergetičnihzonahmetodpolârnoímodeli |