Energy spectrum of a doubly orbitally degenerate model with non-equivalent subbands
In the present paper we investigate a doubly orbitally degenerate narrowband model with correlated hopping. The model peculiarity takes into account the matrix element of electron-electron interaction which describes intersite hoppings of electrons. In particular, this leads to the concentration...
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irk-123456789-1204722017-06-13T03:04:15Z Energy spectrum of a doubly orbitally degenerate model with non-equivalent subbands Didukh, L. Skorenkyy, Yu. Dovhopyaty, Yu. In the present paper we investigate a doubly orbitally degenerate narrowband model with correlated hopping. The model peculiarity takes into account the matrix element of electron-electron interaction which describes intersite hoppings of electrons. In particular, this leads to the concentration dependence of the effective hopping integral. The cases of the strong and weak Hund’s coupling are considered. By means of a generalized mean- field approximation the single-particle Green function and quasiparticle energy spectrum are calculated. Metal-insulator transition is studied in the model at different integer values of the electron concentration. Using the obtained energy spectrum we find criteria of metal-insulator transition. У роботі ми вивчаємо двократно орбітально вироджену вузькозонну модель з корельованим переносом електронів. Особливістю моделі є врахування матричного елемента електрон-електронної взаємодії, який описує міжвузлові переходи електронів. Це приводить, зокрема, до концентраційної залежності ефективного інтеграла пере- носу. Розглянуті випадки сильного та слабкого гундівського зв’язку. За допомогою узагальненого наближення середнього поля розраховані одночастинкова функція Гріна та енергетичний спектр. Перехід метал-діелектрик у моделі досліджений при різних цілих значеннях електронної концентрації. За допомогою отриманого енергетичного спектра знайдено критерії переходу метал-діелектрик 2001 Article Energy spectrum of a doubly orbitally degenerate model with non-equivalent subbands / L. Didukh, Yu. Skorenkyy, Yu. Dovhopyaty // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 491-498. — Бібліогр.: 22 назв. — англ. 1607-324X PACS: 71.28.+d, 71.27.+a, 71.10.Fd, 71.30.+h DOI:10.5488/CMP.4.3.491 http://dspace.nbuv.gov.ua/handle/123456789/120472 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
In the present paper we investigate a doubly orbitally degenerate narrowband
model with correlated hopping. The model peculiarity takes into account
the matrix element of electron-electron interaction which describes
intersite hoppings of electrons. In particular, this leads to the concentration
dependence of the effective hopping integral. The cases of the strong and
weak Hund’s coupling are considered. By means of a generalized mean-
field approximation the single-particle Green function and quasiparticle energy
spectrum are calculated. Metal-insulator transition is studied in the
model at different integer values of the electron concentration. Using the
obtained energy spectrum we find criteria of metal-insulator transition. |
| format |
Article |
| author |
Didukh, L. Skorenkyy, Yu. Dovhopyaty, Yu. |
| spellingShingle |
Didukh, L. Skorenkyy, Yu. Dovhopyaty, Yu. Energy spectrum of a doubly orbitally degenerate model with non-equivalent subbands Condensed Matter Physics |
| author_facet |
Didukh, L. Skorenkyy, Yu. Dovhopyaty, Yu. |
| author_sort |
Didukh, L. |
| title |
Energy spectrum of a doubly orbitally degenerate model with non-equivalent subbands |
| title_short |
Energy spectrum of a doubly orbitally degenerate model with non-equivalent subbands |
| title_full |
Energy spectrum of a doubly orbitally degenerate model with non-equivalent subbands |
| title_fullStr |
Energy spectrum of a doubly orbitally degenerate model with non-equivalent subbands |
| title_full_unstemmed |
Energy spectrum of a doubly orbitally degenerate model with non-equivalent subbands |
| title_sort |
energy spectrum of a doubly orbitally degenerate model with non-equivalent subbands |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120472 |
| citation_txt |
Energy spectrum of a doubly orbitally degenerate model with non-equivalent subbands / L. Didukh, Yu. Skorenkyy, Yu. Dovhopyaty // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 491-498. — Бібліогр.: 22 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT didukhl energyspectrumofadoublyorbitallydegeneratemodelwithnonequivalentsubbands AT skorenkyyyu energyspectrumofadoublyorbitallydegeneratemodelwithnonequivalentsubbands AT dovhopyatyyu energyspectrumofadoublyorbitallydegeneratemodelwithnonequivalentsubbands |
| first_indexed |
2025-07-08T17:56:29Z |
| last_indexed |
2025-07-08T17:56:29Z |
| _version_ |
1837102415517581312 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 3(27), pp. 491–498
Energy spectrum of a doubly orbitally
degenerate model with non-equivalent
subbands
L.Didukh∗, Yu.Skorenkyy, Yu.Dovhopyaty
Ternopil State Technical University, Department of Physics,
56 Rus’ka Str., 46001 Ternopil, Ukraine
Received August 14, 2000, in final form November 21, 2000
In the present paper we investigate a doubly orbitally degenerate narrow-
band model with correlated hopping. The model peculiarity takes into ac-
count the matrix element of electron-electron interaction which describes
intersite hoppings of electrons. In particular, this leads to the concentration
dependence of the effective hopping integral. The cases of the strong and
weak Hund’s coupling are considered. By means of a generalized mean-
field approximation the single-particle Green function and quasiparticle en-
ergy spectrum are calculated. Metal-insulator transition is studied in the
model at different integer values of the electron concentration. Using the
obtained energy spectrum we find criteria of metal-insulator transition.
Key words: narrow energy bands, orbital degeneracy, metal-insulator
transition, correlated hopping
PACS: 71.28.+d, 71.27.+a, 71.10.Fd, 71.30.+h
Both theoretical analysis [1–3] and available experimental data [4] point out that
the Hubbard model [5] should be generalized by taking into account orbital degener-
ation and correlated hopping. In the present paper we study a metal-insulator tran-
sition in the recently proposed [6] doubly orbitally degenerate narrow-band model
with correlated hopping. The peculiarity of the model is the electron-hole asymme-
try and the dependence of hopping integral on the average number of electrons per
site, thus the model shows much better properties than, for example, the Hubbard
model with doubly orbital degeneration. The model Hamiltonian is
H = −µ
∑
iγσ
a+iγσaiγσ +
∑
ijγσ
′
tij(n)a
+
iγσajγσ +
∑
ijγσ
′
(t′ija
+
iγσajγσniγ̄ + h.c.)
+
∑
ijγσ
′
(t′′ija
+
iγσajγσniγσ̄ + h.c.) + U
∑
iγ
niγ↑niγ↓ + U ′
∑
iσ
niασniβσ̄
∗E-mail: didukh@tu.edu.te.ua
c© L.Didukh,Yu.Skorenkyy, Yu.Dovhopyaty 491
L.Didukh,Yu.Skorenkyy, Yu.Dovhopyaty
+(U ′ − J)
∑
iσ
niασniβσ, (1)
where µ is the chemical potential, a+
iγσ, aiγσ are the creation and destruction oper-
ators of an electron of spin σ (σ =↑, ↓; σ̄ denotes spin projection which is opposite
to σ) on i-site and in orbital γ (γ = α, β denotes two possible values of orbital
states), niγσ = a+iγσaiγσ is the number operator of electrons of spin σ and in orbital
γ on i-site, niγ = niγ↑+niγ↓; tij is the hopping integral of an electron from γ-orbital
of j-site to γ-orbital of i-site (we neglect the electron hoppings between α- and β-
orbitals), t′ij (t
′′
ij) includes the influence of an electron on γ̄ (γ)-orbital of i- or j-site
on hopping process, the prime at the second sum in equation (1) signifies that i 6= j,
U is the intra-atomic Coulomb repulsion of two electrons of the opposite spins at
the same orbital (we assume that it has the same value at α- and β-orbitals), U ′
is the intra-atomic Coulomb repulsion of two electrons of the opposite spins at the
different orbitals, J is the intra-atomic exchange interaction energy which stabilizes
the Hund’s states forming the atomic magnetic moments, and the effective hopping
integral tij(n) = tij +nT1(ij) is concentration-dependent due to taking into account
the correlated hopping T1(ij).
The Hamiltonian (1) describes the model with non-equivalent subbands (the
analogues of Hubbard subbands). The non-equivalence of the subbands leads to dif-
ferent width of the subbands and different values of the density of states within the
subbands. At the same time, the density of states within each subband is symmet-
rical. As a consequence, the chemical potential is placed between the subbands at
integer values of the electron concentration n = 1, 2, 3. In these cases, in the model
described by the Hamiltonian (1), the metal-insulator transition (MIT) can occur.
1. Let us consider the case of the strong intra-atomic Coulomb interaction U ′ ≫
tij and the strong Hund’s coupling U ′ ≫ U ′ − J (values U ′ and J are of the same
order). These conditions allow us to neglect the states of site when there are more
than two electrons on the site and the “non-Hund’s” doubly occupied states (the
analogous conditions are used for an investigation of magnetic properties of the
Hubbard model with twofold orbital degeneration in [7–9]). Thus, lattice sites can
be in one of the seven possible states: a hole (a non-occupied by electron site); a
single occupied by electron site; the Hund’s doublon (a site with two electrons on
different orbitals with the same spins).
Using the method of works [10–13] we obtain the energy gap (here we neglect
the correlated hopping)
∆E = −2w(0.75− 1.5c) + (1/2)(F1 + F2),
F1,2 =
√
[(U ′ − J)∓ 0.5w)]2 + 16c2w2, (2)
where w = z|t|, z is the number of the nearest neighbours to a site, c is the hole
concentration. At T = 0 K MIT occurs when (U ′ − J)/(2w) = 0.75.
The energy gap width ∆E as a function of the parameters (U ′ − J)/(2w) and
(kT )/(2w) is presented in figure 1 and figure 2, respectively. With a change of the
parameter (U ′ − J)/(2w) the system undergoes the transition from an insulating to
492
Doubly orbitally degenerate model with non-equivalent subbands
Figure 1. The dependence of energy
gap width ∆E/(U ′ − J) on the param-
eter (U ′ − J)/(2w): the upper curve –
(kT )/(2w) = 0.1; the middle curve –
(kT )/(2w) = 0.05; the lower curve –
(kT )/(2w) = 0.
Figure 2. The dependence of energy
gap width ∆E/(U ′ − J) on the pa-
rameter (kT )/(2w): the upper curve –
(U ′ − J)/(2w) = 0.74; the lower curve
– (U ′ − J)/(2w) = 0.72.
a metallic state (negative values of the energy gap width correspond to the overlap-
ping of the Hubbard subbands). In the model under consideration at T = 0 K, an
insulator-metal transition at n = 1 occurs when (U ′ − J)/(2w) = 0.75 (figure 1, the
lower curve).
The transition from a metallic to an insulating state with the increase of tem-
perature at a given value of the parameter (U ′ − J)/(2w) is also possible (figure 2).
It can be explained by the fact that the energy gap width ∆E given by equation (2)
increases with the temperature T increase which is caused by the rise of the polar
states concentration at constant w, (U ′ − J).
2. The exchange interaction splits some of the bands. If the exchange interaction
is small comparative to the Coulomb interaction J ≪ U , then the splitting is small
and leads only to a weak broadening of the bands. Forasmuch we calculate the width
of the energy gap we can take into account the effect of J by an appropriate shift
of the band center resulting from the inclusion of J into the chemical potential by
means of mean-field approximation (see, e.g., [6,14]).
To describe MIT at the electron concentration n, we can take into account in
the Hamiltonian only the states of site with n − 1, n, n + 1 electrons (the analo-
gous simplification has been used in [15,16]). In the vicinity of the transition point
at the electron concentration n = 1, the concentrations of sites occupied by three
and four electrons are small. We can neglect the small amounts of these sites. For
calculation of single-particle Green functions we use the generalized mean-field ap-
proximation [10]. After transition to k-representation, we obtain the quasiparticle
493
L.Didukh,Yu.Skorenkyy, Yu.Dovhopyaty
energy spectrum:
E1,2(k) = −µ̃ +
U
2
+
ǫ(k) + ζ(k)
2
∓ 1
2
{
[U − ǫ(k) + ζ(k)]2 + 4ǫ̃(k)ζ̃(k)
}1/2
. (3)
By use of the mean-field approximation, in the case of t ′k = t′′k we obtain
ǫ(k), ǫ̃(k), ζ(k), ζ̃(k) as functions of t̃k = tk + 2t′
k
and c, b, d being the concentra-
tions of the holes and sites occupied by one, two electrons, respectively, connected
by the relations: c = 6d, b = 1
4
−3d. In the transition point, when the concentrations
of the holes and doublons are equal to zero, the energies of the electrons within the
subbands are
E1(k) = −µ̃ + tk,
E2(k) = −µ̃ + U + t̃k. (4)
From the equations (4) we obtain the criterion of MIT: U = w+ w̃, where w = z|tij |,
w̃ = z|t̃ij |. With the increase of the correlated hopping at the fixed value of parameter
U/2w, the energy gap width increases and the region of values of U/2w at which
the system is in a metallic state, decreases. In the partial case t ′
k
= t′′
k
= 0 (in this
case tk = t̃k) we have Uc/2w = 1.
Let us consider the MIT at electron concentration n = 2. In the vicinity of the
transition point in the case of two electrons per atom, the concentrations of holes and
sites occupied by four electrons are small. For the small values of the intra-atomic
exchange interaction (J ≪ U) we take J into account analogously to the case of
n = 1. To calculate single-particle Green functions we use the generalized mean-
field approximation. After transition to k-representation, we obtain the quasiparticle
energy spectrum:
E1,2(k) = −µ̃+
3U
2
+
ǫ(k) + ζ(k)
2
∓ 1
2
{
[U − ǫ(k) + ζ(k)]2 + 4ǫ̃(k)ζ̃(k)
}1/2
. (5)
By use of the mean-field approximation analogously to the above, in the case of t ′k =
t′′k we obtain ǫ(k), ǫ̃(k), ζ(k), ζ̃(k) as functions of t̃k = tk+2t′
k
, t∗
k
= tk+4t′
k
and b, d,
where b is the concentration of the sites occupied by one (or three) electrons, d is the
concentration of the doubly occupied sites, connected by the relation b = (1−8d)/6.
In the transition point, when the concentrations of the singly and triply occupied
sites are equal to zero, the quasiparticle energy spectrum is
E1,2(k) = −µ̃ +
3U
2
+
17
18
t∗
k
+ t̃k
2
∓ 1
2
{
[
U +
17
18
t∗
k
− t̃k
2
]2
+
[t∗
k
+ t̃k
18
]2
}1/2
. (6)
Using the quasiparticle energy spectrum (6), we find the energy gap width. In
the point of MIT the energy gap is equal to zero. From this condition we find the
criterion of MIT. With the increase of the correlated hopping at the fixed value of
parameter U/2w, the energy gap width increases faster than at n = 1 and the region
of values of U/2w at which the system is in the metallic state, decreases, analogously
494
Doubly orbitally degenerate model with non-equivalent subbands
to the case n = 1. In the partial case of t′
k
= t′′
k
= 0 (in this case t∗
k
= t̃k) we find
Uc/2ω = 2
√
2/3.
In a similar way, we consider the case of electron concentration n = 3. In the
vicinity of the transition point in the case of three electrons per atom, the concen-
trations of holes and sites occupied by one electron are small. Neglecting the small
amounts of these sites, we can calculate the single-particle Green functions analo-
gously to the above. We find the values of ǫ(k), ǫ̃(k), ζ(k), ζ̃(k) using the mean-field
approximation. They are functions of t∗
k
= tk+4t′
k
, t•
k
= tk+6t′
k
and d, t, f being the
concentrations of the sites occupied by two, three and four electrons, respectively,
connected by the relations: f = 6d, t = 1/4− 3d.
In the transition point, when the concentrations of the holes and single electrons
are equal to zero, the energies of the electrons within the subbands are
E1(k) = −µ̃+ 2U + t∗
k
,
E2(k) = −µ̃+ 3U + t•
k
. (7)
From the equation (7) we obtain the criterion of the MIT at the electron con-
centration n = 3: U = w∗ + w•, where w∗ = z|t∗ij |, w• = z|t•ij |. With the increase of
the correlated hopping at the fixed value of parameter U/2w, the energy gap width
increases faster than at n = 1, n = 2 and the region of values of U/2w at which the
system is in a metallic state, decreases. In the partial case t ′
k
= t′′
k
= 0 (in this case
tk = t̃k) we have Uc/2w = 1. This result coincides with the corresponding critical
value at the electron concentration n = 1 due to the electron-hole symmetry of the
model without the correlated hopping.
The peculiarities of the expressions for the quasiparticle energy spectrum are the
dependences on the concentration of polar states (holes, doublons at n = 1; single
electron and triple occupied sites at n = 2; doublons and sites occupied by four elec-
trons at n = 3) and on the hopping integrals (thus on external pressure). At given
values of U and hopping integrals (constant external pressure), the concentration
dependence of ∆E permits to study MIT under the action of external effects. In
particular, ∆E(T )-dependence can lead to the transition from a metallic state to
an insulating state with the increase of temperature (see figure 4). The described
transition is observed, in particular, in the (V1−xCrx)2O3 compound [4,17] and the
NiS2−xSex system [18,19]. The similar dependence of the energy gap width can be ob-
served at the change of the polar states concentration under the action of photoeffect
or magnetic field. The strong magnetic field can lead, for example, to the decrease
of the polar state concentration (see [20]) initiating the transition from a paramag-
netic insulator state to a paramagnetic metal state. The increase of the polar state
concentration under the action of light, stimulates the metal-insulator transition,
analogously to the influence of temperature change. At the increase of bandwidth
(for example, under the action of external pressure or composition changes) the
insulator-to-metal transition can occur.
If the correlated hopping is absent in the case n = 2, the MIT occurs at the
smaller value of U/2w than in the case n = 1 (figure 3). This result is in qualita-
tive accordance with the results of work [14], in distinction from [16,21]. Using the
495
L.Didukh,Yu.Skorenkyy, Yu.Dovhopyaty
Figure 3. The electron concentration
vs. interaction strength phase diagram
showing the paramagnetic metal (PM)
and paramagnetic insulator (PI) in the
absence of correlated hopping.
Figure 4. The dependence of critical
value (U/2w)c on the parameter of cor-
related hopping χ = t′ij/tij : the curve
1 – n = 1; the curve 2 – n = 2; the
curve 3 – n = 3.
critical values of the parameter U/(2w) at which MIT occurs for different integer
electron concentrations (see figure 3) we can interpret the fact that in the series of
disulphides MS2, the CoS2 (one electron within eg band corresponding to n = 1) and
CuS2 compounds (three electrons within eg-band corresponding n = 3) are metals,
and the NiS2 compound (two electrons within eg-band corresponding n = 2) is an
insulator. Really, for 0.94 6 U/2w 6 1 at the electron concentration n = 2 the
system described by the present model is an insulator, whereas for the same values
of the parameter U/2w at the electron concentrations n = 1, 3 the system is a metal
(according with the calculations of [22] the ratios U/2w in these compounds have
close values).
We have found that in the case of the strong Hund’s coupling at n = 1, the metal-
insulator transition occurs at a smaller value of the parameter ((U−J)/2w)c = 0.75
than in the case of the weak Hund’s coupling ((U − J)/2w)c = 1.
At nonzero values of correlated hopping, the point of MIT moves towards the
values of parameter U/2w at which the system is a metal (figure 4). The non-
equivalence of the cases n = 1 and n = 3 is a manifestation of the electron-hole
asymmetry which is a characteristic of the models with correlated hopping.
Thus, both orbital degeneracy and correlated hopping are the factors favouring
the transition of the system to an insulating state in the case of half-filling with
the increase of intra-atomic Coulomb repulsion in comparison with the single-band
Hubbard model.
496
Doubly orbitally degenerate model with non-equivalent subbands
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Перехід метал-діелектрик у двічі орбітально
виродженій моделі з нееквівалентними підзонами
Л.Дідух, Ю.Скоренький, Ю.Довгоп’ятий
Тернопільський державний технічний університет імені І.Пулюя,
кафедpа фізики
46001 Тернопіль, вул. Руська, 56
Отримано 14 серпня 2000 р., в остаточному вигляді –
21 листопада 2000 р.
У роботі ми вивчаємо двократно орбітально вироджену вузькозонну
модель з корельованим переносом електронів. Особливістю моде-
лі є врахування матричного елемента електрон-електронної взаємо-
дії, який описує міжвузлові переходи електронів. Це приводить, зо-
крема, до концентраційної залежності ефективного інтеграла пере-
носу. Розглянуті випадки сильного та слабкого гундівського зв’язку.
За допомогою узагальненого наближення середнього поля розрахо-
вані одночастинкова функція Гріна та енергетичний спектр. Перехід
метал-діелектрик у моделі досліджений при різних цілих значеннях
електронної концентрації. За допомогою отриманого енергетичного
спектра знайдено критерії переходу метал-діелектрик.
Ключові слова: вузькі зони провідності, орбітальне виродження,
перехід метал-діелектрик, корельований перенос
PACS: 71.28.+d, 71.27.+a, 71.10.Fd, 71.30.+h
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