Extended Holstein Polaron Mass
The renormalization of the effective mass of an electron due to the small polaron formation is studied within an extended Holstein model. It is assumed that an electron moves along a onedimensional chain of ions and interacts with ions vibrations of the neighboring chain via a long-range density-dis...
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| citation_txt | Extended Holstein Polaron Mass / B.Ya. Yavidov // Укр. фіз. журн. — 2010. — Т. 55, № 3. — С. 335-341. — Бібліогр.: 21 назв. — англ. |
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| description | The renormalization of the effective mass of an electron due to the small polaron formation is studied within an extended Holstein model. It is assumed that an electron moves along a onedimensional chain of ions and interacts with ions vibrations of the neighboring chain via a long-range density-displacement type force. By means of exact calculations, the renormalized mass of a nonadiabatic small polaron is obtained in the strong coupling limit. The obtained results are compared with analogous ones within the ordinary Holstein model. The effect of the polarization of vibrations of ions on the small polaron mass is discussed.
Вивчено перенормування маси електрона в результатi утворення малого полярону в межах розширеної моделi Холстейна. Передбачається, що електрон рухається по одномiрному ланцюжку iонiв i взаємодiє з коливаннями iонiв сусiднього ланцюжка внаслiдок далекодiйних сил. Шляхом прямих обчислень отримано перенормовану масу недiабатичного малого полярону в межах сильного зв’язку. Отриманi результати порiвняно з аналiтичними результатами Холстейна. Обговорено вплив коливань iонiв з рiзними поляризацiями на масу малого полярону.
|
| first_indexed | 2025-12-07T17:26:23Z |
| format | Article |
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GENERAL PROBLEMS OF THEORETICAL PHYSICS
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 335
EXTENDED HOLSTEIN POLARON MASS
B.YA. YAVIDOV1, 2
1Institute of Nuclear Physics, Academy of Science of Uzbekistan
(100214 Ulughbek, Tashkent, Uzbekistan; e-mail: yavidov@ inp. uz )
2Ajiniyaz State Pedagogical Institute of Nukus
(104, A. Dosnazarov Str., 230105 Nukus, Karakalpakstan, Uzbekistan)
PACS 72.80
c©2010
The renormalization of the effective mass of an electron due to
the small polaron formation is studied within an extended Hol-
stein model. It is assumed that an electron moves along a one-
dimensional chain of ions and interacts with ions vibrations of
the neighboring chain via a long-range density-displacement type
force. By means of exact calculations, the renormalized mass of
a nonadiabatic small polaron is obtained in the strong coupling
limit. The obtained results are compared with analogous ones
within the ordinary Holstein model. The effect of the polarization
of vibrations of ions on the small polaron mass is discussed.
1. Introduction
A model of a polaron with a long-range “density-
displacement” type interaction was introduced in Ref.
[1] by Alexandrov and Kornilovitch. The model by it-
self represents an extension of the large Fröhlich polaron
(LFP) model [2] to a discrete ionic crystal lattice or an
extension of the Holstein polaron model (HM) [3] to a
case where an electron interacts with many ions of the
lattice due to the long-range electron-phonon interac-
tion. Subsequently, the model was named an extended
Holstein model (EHM) [4]. In the model, a polaron has
an internal structure different from those of both HM
polaron and LFP. In the HM, a carrier is coupled to in-
tramolecular vibrations and self-trapped on a single site.
The size of a Holstein polaron is the same as the size of
the phonon cloud, both are about the lattice constant. In
the case of large Fröhlich polarons, the size of a polaron
is also the same as the size of the phonon cloud, but
the polaron extends over many lattice constants. The
size of a polaron in EHM is about the lattice constant,
but its phonon cloud spreads over the whole crystal. As
was shown in Ref. [1] in the strong-coupling limit of
EHM, a polaron consists of an electron localized on a
site n and the phonon cloud spread over other lattice
sites m. The mass enhancement of such a quasiparticle
increases exponentially with coupling as in the standard
small polaron theory. In EHM, one has to work with a
new situation where the electron wave function size in a
polaron and the size of a lattice deformation surrounding
an electron are different. The former is the atomic size,
while the latter is spread over the whole crystal. Accord-
ing to Ref. [1], we use the term small polaron for such
a quasiparticle (for an alternative viewpoint, see Ref.
[4]). Within the model, a renormalized mass appears to
be much smaller as compared with that in the ordinary
Holstein model. Conclusions of [1] were confirmed later
on by other authors [4–6]. In addition, Fehske, Loos,
and Wellein [4] investigated electron-lattice correlations,
a single-particle spectral function, and the optical con-
ductivity of a polaron in EHM in strong and the weak
coupling regimes by means of the exact Lancroz diag-
onalization method. Other properties of EHM such as
the ground-state spectral weight, the average kinetic en-
ergy, and the mean number of phonons by means of the
variational and Quantum Monte Carlo simulation ap-
proaches were studied in [7, 8]. All numerical and ana-
lytical results in [1] were obtained in the nonadiabatic
or near-nonadibatic regime. In work [9], we extended
this model to the adiabatic limit and found that the
mass of a polaron in EHM is much less renormalized
than the mass of a small Holstein polaron in this limit
as well. Work [1, 9] considered an electron interacting
with vibrations of ions of an upper chain which are po-
larized perpendicularly to the chain. This case mimics
high − Tc cuprates, where the in-plane (CuO2) carriers
B.YA. YAVIDOV
Fig. 1. Electron hops on a lower chain and interacts with vibrations
of ions of the upper infinite chain via a density-displacement type
force fm,α(n). The distances between chains and between ions are
assumed equal to 1
are strongly coupled with the c-axis polarized vibrations
of apical oxygen ions [10]. A more realistic case where
apical ions vibrate in all directions and their effect on
the mass of a small polaron in EHM were studied in [11].
At the same time, polarons were experimentally recog-
nized as quasiparticles in novel materials, in particular,
in superconducting cuprates and manganites with colos-
sal magnetoresistance [12–20]. In the previous papers
[9, 11], the mass renormalization of an electron due to
the formation of a small polaron in EHM was restricted
only to a simple two-site model. Here, we extend these
studies for a many-site system and derive an analytical
expression for the mass of a nonadiabatic small polaron
in EHM in the strong coupling regime and compare it
with that in the ordinary Holstein model. In addition,
the effect of polarized vibrations and their contributions
to the mass of a polaron are discussed within EHM.
2. The Model
We consider an electron performing the hopping motion
on a lower chain consisting of static sites, but interacting
with all ions of an upper chain via a long-range density-
displacement type force, as shown in Fig. 1, similar to a
case considered in [5, 6]. So, the motion of an electron is
always one-dimensional, but vibrations of upper chain’s
ions are isotropic and two-dimensional.
The Hamiltonian of the model is
H = He +Hph +He−ph, (1)
where
He = −t
∑
n
(c†ncn+a + H.c.) (2)
is the electron hopping energy,
Hph =
∑
m,α
(
− ~2∂2
2M∂u2
m,α
+
Mω2u2
m,α
2
)
(3)
is the Hamiltonian of vibrating ions, and
He−ph =
∑
n,m,α
fm,α(n) · um,αc
†
ncn (4)
describes the interaction between an electron that be-
longs to lower chain and ions of the upper chain. Here,
c†n(cn) is the creation (annihilation) operator of an elec-
tron on the site n, um,α is the α = y, z-polarized dis-
placement of the m-th ion, fm,α(n) is an interaction
density-displacement type force between the electron on
site n and the α polarized vibration of the m-th ion, M
is the mass of vibrating ions, and ω is their frequency.
The explicit dependence of the interaction force on the
y and z coordinates is
fm,y(n) =
κy|n−m|
(|n−m|2 + b2)3/2
(5)
and
fm,z(n) =
κzb
(|n−m|2 + b2)3/2
, (6)
where κy and κz are some coefficients. The distance
along a chain |n−m| is measured in units of the lattice
constant |a| = 1. The distance between the chains is b.
3. Strong Coupling and Nonadiabatic Limit
In the strong coupling limit (λ = Ep/D > 1) and the
nonadiabatic approximation, the wave function of the
system is presented as a superposition of normalized
Wannier functions W (r− n) localized on the site n,
Ψ =
∑
n
An(um,α)W (r− n). (7)
For a convenience, we consider 2N + 1 ions in the lower
chain. Then the Schrödinger equation HΨ = EΨ is re-
duced to a system of coupled second-order differential
equations with respect to the infinite number of vibra-
tional coordinates um,α(
E −Hph −
∑
m,α
fm,α(ni) · um,α
)
Ani
(um,α) =
= t
∑
n6=ni
An(um,α) (8)
336 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3
EXTENDED HOLSTEIN POLARON MASS
with i = 0,±1,±2, . . . ,±(N − 1),±N . Further, we omit
the argument um,α of An, but keep in mind that it de-
pends on them. The common tool to investigate (8) is a
perturbation approach with respect to the hopping inte-
gral. In the zero order (t = 0), the system is (2N+1)-fold
degenerate with the electron localized, for example, on
site ni, so that An = Ãni if n = ni, where
Ãni
= exp
[
−Mω
2~
∑
m,α
(
um,α +
fm,α(ni)
Mω2
)2
]
(9)
and zero otherwise. In the first order in the hopping inte-
gral t, we are looking for a solution of system (8) as a sin-
gle column matrix
(
An−N
, An(−N+1) , . . . , An(N−1) , AnN
)T
(T stands for a transposed matrix) which is a linear com-
bination of Ãni
:(
An−N
, An(−N+1) , . . . , An(N−1) , AnN
)T =
= α−N
(
Ãn−N
, 0, . . . , 0
)T
+ αi
(
0, . . . , Ãni
, . . . , 0
)T
+
+α−N
(
0, · · · , 0, ÃnN
)T
. (10)
Substituting (10) into the system of equations (8), we
get a system of linear equations for the coefficients
α−N , α−N+1, · · · , αN−1, αN ,
E(ni)Ãniαi − t
∑
k 6=i
Ãni+k
αi+k = 0, (11)
where
E(ni) =
(
E −Hph −
∑
m,α
fm,α(ni) · um,α
)
. (12)
The system of equations (11) have a square (2N + 1)×
(2N + 1) matrix. Diagonal elements of the matrix are
products of (9) and (12). Then we introduce the Born–
von Karman boundary condition Ãn−N
= ÃnN
which
ensures the translation invariance of the system and en-
ables us rewrite the system of equations (11) as
Ẽαi −
∑
k 6=i
ti,kαk = 0. (13)
Here, Ẽ = E −N ′~ω/2 − Ep, N ′ is the number of ions
in the upper chain,
t̃k,k′ = t
∫
Ãnk
Ãnk′dum,α∫
|Ãnk
|2dum,α
(14)
are the renormalized hopping integrals and
Ep = Ep(ni) =
∑
m,α
f2
m,α(ni)
2Mω2
(15)
is the polaronic shift which is independent of ni. Ex-
pressing all nondiagonal elements t̃k,k′ of the matrix
through t̃ = t̃1,2, we find
Ẽαi − t̃
∑
k 6=i
g̃2
i,kαk = 0 (16)
and
g̃2
i,k = −(1/2M~ω3)×
×
∑
m,α
fm,α(ni)(fm,α(ni+1)− fm,α(n′k)). (17)
It appears that the matrix of the system of equations
(16) is symmetric. Then the system of equations is sep-
arated into block 2 × 2 diagonal matrix equations that
couple only αi and αi+1 as(
Ẽ −t̃e−g̃
2
i,i+1
−t̃e−g̃
2
i,i+1 Ẽ
)(
αi
αi+1
)
= 0. (18)
From (18), we obtain a secular equation for the energy∣∣∣∣ E −N ′~ω/2 + Ep −t̃
−t̃ E −N ′~ω/2 + Ep
∣∣∣∣ = 0. (19)
The energy levels of the system are found as
E± = N ′~ω/2− Ep ± t̃ (20)
The evaluation of (14) with regard for (9) results in
t̃ = te−g
2
, (21)
where
g2 =
1
2M~ω3
∑
m,α
(
f2
m,α(n)− fm,α(n)fm,α(n + a).
)
.
(22)
Formulas (15) and (21) are the main analytical results
of the present work.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 337
B.YA. YAVIDOV
Fig. 2. Schematic representation of the extended Holstein model
and the ordinary Holstein model for a two-site system. The elec-
tron on site 1 interacts (a) with sites m = −1 and m = 0 and (b)
with only site m = 1 of the ion chain, in the extended Holstein
model and the Holstein model, respectively
4. Results
Analytical expressions for the polaronic shift (15) and
the renormalized hopping integral (21) were obtained
early in Ref. [1] by using the canonical Lang–Firsov
transformation. Work [1] studied the renormalization of
the effective mass of an electron due to only z-polarized
vibrations of the upper chain. However, a role of y-
polarized vibrations of the upper chain and their influ-
ence on the mass renormalization in EHM was not dis-
cussed, and no quantitative results were presented. In
this section, we calculate the small polaron mass in EHM
for both density-displacement type interactions (5) and
(6). Moreover, we calculate the mass of a small polaron
in EHM with two-dimensional isotopic vibrations of ions
of the upper chain as well. In our model, the electron-
phonon coupling constant λ = Ep/2t, and the polaron
mass
mp =
~2
2t̃a2
= m∗ exp[g2], (23)
where m∗ = ~2/2ta2 is the bare electron band mass.
One can express the polaron mass in terms of λ and t/ω
(parameter of nonadiabaticity) as
mp/m
∗ = e2γλt/~ω, (24)
where
γEHM = 1−
∑
m,α fm,α(n) · fm,α(n + a)∑
m,α f2
m,α(n)
. (25)
The dimensionless parameter γ in the exponent of (24)
depends on the type of interacting force and the geom-
etry of a lattice. For the ordinary Holstein model, it
Fig. 3. Ratio of the masses of nonadiabatic small polarons in EHM
and HM with the only z− polarized vibrations of ions as a function
of the electron-phonon coupling constant λ at different values of
t/~ω. Open (filled) circles and squares show γ calculated within
two models (for a whole chain)
is always equal to 1 (γHM = 1). So the ratio of small
polaron masses in EHM and ordinary HM is given by
mp,EHM
mp,HM
= exp
[
2λ(γEHM − γHM)
t
~ω
]
. (26)
We would like to stress that the model yields a less renor-
malization of the effective mass than the Holstein model.
This is true not only for c-axis polarized vibrations of
apical oxygen ions, but for their isotropic vibrations as
well [11]. For simplicity, let us first consider z-polarized
vibrations of ions of the upper chain and only nearest-
neighbors interactions, as in Fig. 2,a. In this case, our
model yields Ep = f2
0z(1)/Mω2 and the mass renor-
malization mp/m
∗ = exp(Ep/2~ω), while the Holstein
model with the local interaction, Fig. 2,b, for the same
Ep yields mp/m
∗ = exp(Ep/~ω).
The factor 1/2 in the exponent provides much lighter
small polarons in EHM as compared with those within
the Holstein model. If one considers the Coulomb-like
interaction with the whole upper chain, one gets the fac-
tor γz = 0.28 [6] instead of 0.5 in the exponent, which
means an even less renormalized effective mass. The
results for the mass of a nonadiabatic small polaron in
EHM with the only z-polarized vibrations of ions in com-
parison with those of HM are presented in Fig. 3.
Now we discuss the influence of y-polarized vibrations
of the upper chain on the small polaron mass in EHM.
In this case, the density-displacement type interaction
force (5) is longer ranged than (6), as it decays as r−2,
338 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3
EXTENDED HOLSTEIN POLARON MASS
Fig. 4. The mass ratio for nonadiabatic small polarons in EHM
and HM with only y− polarized vibrations of ions as a function
of the electron-phonon coupling constant λ at different values of
t/~ω
while force (6) decays as r−3. For a lattice in Fig. 1, one
finds γy = 0.652657. The results for the mass of a small
polaron in EHM with only y− polarized vibrations of
the upper chain and their comparison with those of HM
are given in Fig. 4.
The comparison of curves in Figs. 3 and 4 shows that
the effective mass of an electron is more renormalized
with y-polarized vibrations of ions than with z-polarized
vibrations. For example, at λ = 1 and t/~ω = 1,
mp,EHM,z ' 1.77m∗, while mp,EHM,y ' 3.68m∗. If
we switch-on both z- and y-polarized vibrations, then
T a b l e 1. Calculated masses of polarons for the same
polaron shift with z- and y-polarized two-dimensional vec-
tor vibrations of ions in EHM. The fifth column presents
the polaron mass in HM
λ t/~ω = 0.50
mp,z mp,y mp,EHM mp,HM
1 1.33213 1.92064 1.47995 2.71828
2 1.77457 3.68885 2.19025 7.38906
3 2.36396 7.08494 3.24146 20.0855
4 3.14910 13.6076 4.79720 54.5982
5 4.19501 26.1353 7.09961 148.413
λ t/~ω = 0.75
mp,z mp,y mp,EHM mp,HM
1 1.53752 2.66175 1.80041 4.48169
2 2.36396 7.08494 3.24146 20.0855
3 3.63462 18.8584 5.83594 90.0171
4 5.58829 50.1963 10.5071 403.429
5 8.59209 133.610 18.9130 1808.04
Fig. 5. The mass ratio for nonadiabatic small polarons in EHM
and HM with vector vibrations of ions as a function of the electron-
phonon coupling constant λ at different values of t/~ω
each of them contributes to the mass renormalization.
The overall effect of both contributions of vibrations
of ions to a small polaron mass in EHM depends on
the ratio κz/κy. In the case where κz = κy, ions are
isotropic oscillators. The calculations of (25) with a vec-
tor fm(n) force (taking both (5) and (6) into account)
yields γ = 0.392008. This result shows that a nonadia-
batic small polaron in EHM remains lighter than a small
polaron of HM. The comparison of the results of EHM
with two-dimensional vector vibrations of ions and ordi-
nary HM is given in Fig. 5. For the illustrative purpose,
the calculated masses of small polarons (i) for each type
of ions vibrations in EHM and (ii) in HM are presented
in Table 1 at the same polaron shift. As many exper-
iments with cuprates show, the polaron mass is of the
order of ∼ (2 ÷ 5)m∗ [21]. In our model, such values of
mp can be explained in the κz ≥ κy limit.
It should be emphasized than the full polaron massmp
can be presented as a product of mp,z and mp,y: mp =
mp,z×mp,y (see Table 2). However, the full polaron shift
Ep is given as the sum of Ep,z and Ep,y: Ep = Ep,z +
Ep,y. The same is true for the electron-phonon coupling
constant λ: λ = λz + λy, λz = Ep,z/2t = δE,zλ and
λy = Ep,y/2t = δE,yλ. Here, λz and λy are the electron-
phonon coupling constants due to only z- and y-polarized
vibrations of ions of the upper chain, respectively, and
δE,z =
Ep,z
Ep
=
∑
m f2
m,z(n)∑
m(f2
m,z(n) + f2
m,y(n))
, (27)
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 339
B.YA. YAVIDOV
δE,y =
Ep,y
Ep
=
∑
m f2
m,y(n)∑
m(f2
m,z(n) + f2
m,y(n))
(28)
are the relative contributions of z- and y-polarized vi-
brations to a full polaron shift. In the case of isotropic
vibrations of the upper chain ions and κz = κy, we
find δE,z = 0.712393 and δE,y = 0.287603 (within HM,
δE,z = δE,y = 0.5). As one can see, the main contri-
bution to a full polaron shift comes from z polarized
vibrations. In general, δE,α (α = z, y) depends (i) on
the ratio κz/κy, (ii) on the type (range) of interaction
forces fm,α(n), and (iii) on a lattice geometry. For the
ordinary Holstein model, δE,α depends only on the ratio
κz/κy. As far as the effects of polarized vibrations on a
small polaron in HM and EHM are concerned, there are
qualitative and quantitative differences that can be seen
in the following:
– in the Holstein model: polarized vibrations of both
types contribute to the mass renormalization and a full
polaron shift mp,z = mp,y, Ep,z = Ep,y if κz = κy.
– in the extended Holstein model: as in HM, both types
of polarized vibrations contribute to the mass renormal-
ization and to a full polaron shift but now with a different
weights. mp,z 6= mp,y and Ep,z 6= Ep,y even if κz = κy.
z-polarized vibrations of ions give rise to a mobile po-
laron, while y-polarized vibrations give rise to a heavy
polaron. When both types of polarization are switched
on, the full polaron shift is mainly determined by the z
contribution ' 71%, and the value of mp,z exceeds that
of mp,y (Table 2). So, the anisotropic properties of a po-
laron due to polarized vibrations are more pronounced
in EHM.
T a b l e 2. Calculated masses of polarons in EHM with
regard for z-, y-polarized, and two-dimensional vector vi-
brations of ions
λ t/~ω = 0.50
mp,z mp,y
mp,z−mp,y
mp,y
mp = mp,z ×mp,y
1 1.22667 1.20648 0.01673 1.47995
2 1.50471 1.45560 0.03374 2.19025
3 1.84577 1.75615 0.05103 3.24146
4 2.26415 2.11877 0.06861 4.79720
5 2.77735 2.55626 0.08649 7.09961
λ t/~ω = 0.75
mp,z mp,y
mp,z−mp,y
mp,y
mp = mp,z ×mp,y
1 1.35859 1.32520 0.02520 1.80041
2 1.84577 1.75615 0.05103 3.24146
3 2.50765 2.32725 0.07752 5.83594
4 3.40688 3.08408 0.10467 10.5071
5 4.62856 4.08702 0.13250 18.9130
5. Conclusion
We have solved the extended Holstein model with a
long-range density-displacement type interaction in the
strong coupling limit and in the nonadiabatic regime.
We have found the mass of a small polaron in the ex-
tended Holstein model and compared it with that in
the ordinary Holstein model. It is established that y-
polarized vibrations of ions give a more renormalization
of the polaron mass than z-polarized vibrations. In gen-
eral, both y- and z-polarized vibrations contribute to the
mass renormalization. The overall effect of both types
of polarized vibrations depends (i) on the ratio κz/κy,
(ii) on the type (range) of interaction forces fm,α(n),
and (iii) on a lattice geometry. In the limit κz ≥ κy, it
is found that a small polaron in EHM is lighter than a
small Holstein polaron in the nonadiabatic regime.
This work is supported by the Uzbek Academy of Sci-
ence (Grant No. FA-F2-070) and the Ministry of Public
Education of Uzbekistan.
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Received 21.04.09
МАСА ПОЛЯРОНУ В РОЗШИРЕНIЙ МОДЕЛI
ХОЛСТЕЙНА
Б.Я. Явидов
Р е з ю м е
Вивчено перенормування маси електрона в результатi утворен-
ня малого полярону в межах розширеної моделi Холстейна. Пе-
редбачається, що електрон рухається по одномiрному ланцюж-
ку iонiв i взаємодiє з коливаннями iонiв сусiднього ланцюжка
внаслiдок далекодiйних сил. Шляхом прямих обчислень отри-
мано перенормовану масу недiабатичного малого полярону в
межах сильного зв’язку. Отриманi результати порiвняно з ана-
лiтичними результатами Холстейна. Обговорено вплив коли-
вань iонiв з рiзними поляризацiями на масу малого полярону.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 3 341
|
| id | nasplib_isofts_kiev_ua-123456789-13407 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 2071-0194 |
| language | English |
| last_indexed | 2025-12-07T17:26:23Z |
| publishDate | 2010 |
| publisher | Відділення фізики і астрономії НАН України |
| record_format | dspace |
| spelling | Yavidov, B.Ya. 2010-11-08T14:43:42Z 2010-11-08T14:43:42Z 2010 Extended Holstein Polaron Mass / B.Ya. Yavidov // Укр. фіз. журн. — 2010. — Т. 55, № 3. — С. 335-341. — Бібліогр.: 21 назв. — англ. 2071-0194 PACS 72.80 https://nasplib.isofts.kiev.ua/handle/123456789/13407 The renormalization of the effective mass of an electron due to the small polaron formation is studied within an extended Holstein model. It is assumed that an electron moves along a onedimensional chain of ions and interacts with ions vibrations of the neighboring chain via a long-range density-displacement type force. By means of exact calculations, the renormalized mass of a nonadiabatic small polaron is obtained in the strong coupling limit. The obtained results are compared with analogous ones within the ordinary Holstein model. The effect of the polarization of vibrations of ions on the small polaron mass is discussed. Вивчено перенормування маси електрона в результатi утворення малого полярону в межах розширеної моделi Холстейна. Передбачається, що електрон рухається по одномiрному ланцюжку iонiв i взаємодiє з коливаннями iонiв сусiднього ланцюжка внаслiдок далекодiйних сил. Шляхом прямих обчислень отримано перенормовану масу недiабатичного малого полярону в межах сильного зв’язку. Отриманi результати порiвняно з аналiтичними результатами Холстейна. Обговорено вплив коливань iонiв з рiзними поляризацiями на масу малого полярону. This work is supported by the Uzbek Academy of Science (Grant No. FA-F2-070) and the Ministry of Public Education of Uzbekistan. en Відділення фізики і астрономії НАН України Загальні питання теоретичної фізики Extended Holstein Polaron Mass Маса полярону в розширеній моделі Холстейна Article published earlier |
| spellingShingle | Extended Holstein Polaron Mass Yavidov, B.Ya. Загальні питання теоретичної фізики |
| title | Extended Holstein Polaron Mass |
| title_alt | Маса полярону в розширеній моделі Холстейна |
| title_full | Extended Holstein Polaron Mass |
| title_fullStr | Extended Holstein Polaron Mass |
| title_full_unstemmed | Extended Holstein Polaron Mass |
| title_short | Extended Holstein Polaron Mass |
| title_sort | extended holstein polaron mass |
| topic | Загальні питання теоретичної фізики |
| topic_facet | Загальні питання теоретичної фізики |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/13407 |
| work_keys_str_mv | AT yavidovbya extendedholsteinpolaronmass AT yavidovbya masapolâronuvrozšireníimodelíholsteina |