Temperature and impurity effects of the polaron in an asymmetric quantum dot
We study the temperature and impurity effects of the ground state energy and the ground state binding energy in an asymmetric quantum dot by using the liner combination operator method. It is found that the ground state energy and the ground state binding energy will increase with increasing the tem...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | Temperature and impurity effects of the polaron in an asymmetric quantum dot / Shu-Ping Shan, Ya-Min Liu, Jin-Lin Xiao // Физика низких температур. — 2013. — Т. 39, № 7. — С. 786–789. — Бібліогр.: 22 назв. — англ. |
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irk-123456789-1186622017-05-31T03:03:09Z Temperature and impurity effects of the polaron in an asymmetric quantum dot Shan, Shu-Ping Liu, Ya-Min Xiao, Jin-Lin Низкоразмерные и неупорядоченные системы We study the temperature and impurity effects of the ground state energy and the ground state binding energy in an asymmetric quantum dot by using the liner combination operator method. It is found that the ground state energy and the ground state binding energy will increase with increasing the temperature. The ground state ener-gy is a decreasing function of the Coulomb bound potential, whereas the ground state binding energy is an in-creasing one of it. 2013 Article Temperature and impurity effects of the polaron in an asymmetric quantum dot / Shu-Ping Shan, Ya-Min Liu, Jin-Lin Xiao // Физика низких температур. — 2013. — Т. 39, № 7. — С. 786–789. — Бібліогр.: 22 назв. — англ. 0132-6414 PACS: 71.38.–k, 73.63.Kv http://dspace.nbuv.gov.ua/handle/123456789/118662 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы Shan, Shu-Ping Liu, Ya-Min Xiao, Jin-Lin Temperature and impurity effects of the polaron in an asymmetric quantum dot Физика низких температур |
| description |
We study the temperature and impurity effects of the ground state energy and the ground state binding energy in an asymmetric quantum dot by using the liner combination operator method. It is found that the ground state energy and the ground state binding energy will increase with increasing the temperature. The ground state ener-gy is a decreasing function of the Coulomb bound potential, whereas the ground state binding energy is an in-creasing one of it. |
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Article |
| author |
Shan, Shu-Ping Liu, Ya-Min Xiao, Jin-Lin |
| author_facet |
Shan, Shu-Ping Liu, Ya-Min Xiao, Jin-Lin |
| author_sort |
Shan, Shu-Ping |
| title |
Temperature and impurity effects of the polaron in an asymmetric quantum dot |
| title_short |
Temperature and impurity effects of the polaron in an asymmetric quantum dot |
| title_full |
Temperature and impurity effects of the polaron in an asymmetric quantum dot |
| title_fullStr |
Temperature and impurity effects of the polaron in an asymmetric quantum dot |
| title_full_unstemmed |
Temperature and impurity effects of the polaron in an asymmetric quantum dot |
| title_sort |
temperature and impurity effects of the polaron in an asymmetric quantum dot |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2013 |
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Низкоразмерные и неупорядоченные системы |
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http://dspace.nbuv.gov.ua/handle/123456789/118662 |
| citation_txt |
Temperature and impurity effects of the polaron in an asymmetric quantum dot / Shu-Ping Shan, Ya-Min Liu, Jin-Lin Xiao // Физика низких температур. — 2013. — Т. 39, № 7. — С. 786–789. — Бібліогр.: 22 назв. — англ. |
| series |
Физика низких температур |
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2025-07-08T14:24:37Z |
| last_indexed |
2025-07-08T14:24:37Z |
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1837089086569971712 |
| fulltext |
© Shu-Ping Shan, Ya-Min Liu, and Jin-Lin Xiao, 2013
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 7, pp. 786–789
Temperature and impurity effects of the polaron
in an asymmetric quantum dot
Shu-Ping Shan
College of Physics and Electromechanics, Fujian Longyan College, Longyan 364012, P.R. China
E-mail: ssping04@126.com
Ya-Min Liu
College of Physics and Electronic Information, Inner Mongolia Hulunbei’er College, Hulunbei’er 021008, P.R. China
Jin-Lin Xiao
College of Physics and Electronic Information, Inner Mongolia National University, Tongliao 028043, P.R. China
Received January 14, 2013, revised April 4, 2013
We study the temperature and impurity effects of the ground state energy and the ground state binding energy
in an asymmetric quantum dot by using the liner combination operator method. It is found that the ground state
energy and the ground state binding energy will increase with increasing the temperature. The ground state ener-
gy is a decreasing function of the Coulomb bound potential, whereas the ground state binding energy is an in-
creasing one of it.
PACS: 71.38.–k Polarons and electron–phonon interactions;
73.63.Kv Quantum dots.
Keywords: asymmetric quantum dot, impurity, polaron, linear combination operator.
1. Introduction
With the rapid development of nanofabrication technolo-
gy, the investigations of quasi-zero-dimensional quantum
dots (QDs) have aroused great interest. Due to the small
structures of QDs, some physical properties such as optical
and electron transport characteristics are quite different from
those of the bulk material. Consequently, there has been a
large amount of experimental work [1–4] on QDs. Many
investigators have studied the properties of the QDs in many
aspects by a variety of theoretical methods [5–8]. Since the
electron–phonons interaction is enhanced by the geometric
confinement, research on the polaron effect has become a
main subject in QD system. For example, a quasi-analytical
approach to study energy levels of two-electron QD was
studied by Hassanabadi [9]. Bondar [10] calculated the per-
colation and excitonic luminescence in SiO2/ZnO two-phase
structure with a density of QDs randomly distributed over a
spherical surface by using the effective mass method. Em-
ploying Landau–Pekar variational theory, Yao et al. [11].
calculated the ground state energy and effective mass in a
two- and three-dimensional QD. Based on the rotational
wave approximation and the effective mass approximation,
Guo et al. [12] investigated the parameter-dependent optical
nutation in PbSe/CdSe/ZnS QD. The energy levels structure
of two interacting electrons in a parabolic QD under an ex-
ternal magnetic field of arbitrary strength was studied via the
asymptotic iteration method by Al-Dossary [13]. With the
effective mass approximation and finite barrier potential,
Sadegh and Rezaie [14] studied the effect of magnetic field
on the impurity binding energy of the excited states in spher-
ical QD. Yu et al. [15] investigated the geometric phase of
QDs in the time-dependent isotropic magnetic field using
the invariant theory. Using the Green’s function technique,
the shot noise in the mesoscopic system composed of a QD
coupled to ferromagnetic terminals under the perturbation of
ac fields was studied by Zhao et al. [16]. Under the effective
mass approximation and modified by the field theory, Liu
et al. [17] theoretically described the nonlinear optical
properties of the CdSe/ZnS QD quantum well in the vi-
cinity of a spherical metal nanoparticle. Within the spin-
density-functional theory, Zhang et al. [18] investigated the
electronic properties of a QD formed by the potentials asso-
ciated with the surface acoustic wave and constrictions.
Temperature and impurity effects of the polaron in an asymmetric quantum dot
Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 7 787
Chen et al. [19] studied the temperature dependence of the
binding energy of an impurity bound magnetopolaron in a
GaAs parabolic QD by using the second-order Rayleigh–
Schrödinger perturbation theory. However, the temperature
and impurity effects on the ground state energy and the
ground state binding energy of strong-coupling polaron in an
asymmetry QD has never been studied yet.
Comparing with the methods mentioned above, the li-
near combination operator method is simple and concise to
study the electron–phonon interaction with arbitrary coupl-
ing strength (strong- and weak-coupling) [20,21] in the
low-dimensional quantum system. Moreover, the proper-
ties of the polaron’s vibrational frequency are easier to
study by this method [22]. In the present paper, we investi-
gated the temperature and impurity effects on the ground
state energy and the ground state binding energy of the
polaron in an asymmetry QD using the linear combination
operator method.
2. Theoretical model
We consider a system in which an electron is moving in
a polar crystal and interacting with bulk longitudinal opti-
cal (LO) phonons. On the basis of the effective mass ap-
proximation, the Hamiltonian of the electron–LO phonon
interaction system with a hydrogenic impurity at the center
can be written as
2 2
2 2 2 2
1 2
1 1
2 2 2 2
z
LO
p
H a a m m z
m m
q q
q
p
2
0
exp(i ) ,q
e
V a hc
r
q
q
q r (1)
where ( , ),zpp p ,( )zr , ( , ),m, qq q , 1 and 2
stand for the momentum, the coordinate vector, the mass of
the electron, the wave vector, the electron–phonons coupl-
ing strength, the parallel confinement strength and the per-
pendicular confinement strength, respectively. aq and aq
denote the creation and annihilation operators of the LO
phonon with the frequency LO . 2
0/( ) e r denotes the
Coulomb potential between the electron and the hydrogen-
like impurity. qV and in Eq. (1) are
1/4 1/2
4
,
2
LO
q
LO
V i
q m V
(2)
1/22
0
2 1 1
,
2
LO
LO
me
(3)
where V refers to the volume of the crystal. Using the
Fourier expansion to the Coulomb bound potential as follow:
2 2
2
0 0
4 1
exp( i )
e e
r V qq
q r . (4)
Then, we carry out the unitary transformation to Eq. (1):
*exp [ ( )]q qU a f a fq q
q
, (5)
where qf and *
qf are the variational functions, we intro-
duce the famous linear combination operator:
1/2
1/2
( )
2
, , , ,
( )
2
j j j
j j j
m
p b b
j x y z
r i b b
m
(6)
where is the variational parameter. The ground state
wave function of the system is chosen as
0| | 0 | 0a b , (7)
| 0 a refers to the unperturbed zero phonon state and | 0 b
denotes the vacuum state of the b operator. To make the
calculation simpler, we choose the usual polaron units
( 2 1).LOm The ground state energy is obtained
in the following form:
0 4 4
1 2
3 1 2 1
2
4
E
l l
, (8)
where 1 1/l m and 2 2/l m are the parallel and
the perpendicular confinement lengths and
2
0( / ) /( )e m is the Coulomb bound potential. Per-
forming the variational of 0E with respect to , we obtain
2 3/2
4 4
1 2
2 4 8 4
0
33 3 3l l
, (9)
where is the vibrational frequency of the polaron. If eE
and pE denote the energies of the uncoupled electron and
phonon, respectively, then the ground state binding energy
of the strong-coupling impurity polaron is given by
0 4 4
1 2
1 2 1
2 2b e pE E E E
l l
. (10)
The mean number of optical phonons around the ground
state is
1
0 0| |N U a a Uq q
q
. (11)
3. Temperature effects
At finite temperature, electron–phonon system is no
longer entirely in the ground state. The lattice vibrations
excite not only the real phonon but also electron in a para-
bolic potential. According to the quantum statistics theory,
the mean number of phonons is
Shu-Ping Shan, Ya-Min Liu, and Jin-Lin Xiao
788 Low Temperature Physics/Fizika Nizkikh Temperatur, 2013, v. 39, No. 7
1
exp 1LO
B
N
k T
, (12)
where Bk is the Boltzmann constant. The value of re-
lates not only to the N but also to .N Through the above
analysis, we can obtain the relationship between 0 ,E ,bE
N and .T
4. Numerical results and discussion
In this section, to show more obviously the influence of
the temperature and impurity on the properties of the strong-
coupling polaron in an asymmetric QD, we perform the nu-
merical calculation. The results are presented in Fig. 1.
Figure 1(a) shows the relationship between the ground
state energy 0E of the strong-coupling impurity polaron
varying with the temperature T and the Coulomb bound
potential for fixed 1 21.2, 1.6l l and 6.0. The
lines correspond to the cases of the Coulomb bound poten-
tial 0.2 and 0.3. Figure 1(b) illustrates the ground
state binding energy bE as functions of the temperature T
and the Coulomb bound potential for fixed 1 1.2,l
2 1.6l and 5.5. The lines correspond 1.0 and
0.2. From this figures we can see that the ground state
energy and the ground state binding energy increased with
increasing temperature. Due to the thermal movement
speed of the electron and phonons will be enhanced with
rising temperature, so that the electron will interact with
more phonons, and the energy of the polaron will be in-
creased. For this reason, the ground state energy and the
ground state binding energy increased with the increase of
the temperature. From Fig. 1 one can also find that the
ground state binding energy is an increasing function of the
Coulomb bound potential, whereas the ground state energy
is a decreasing one of it. There is Coulomb bound potential
between the electron and the hydrogen-like impurity be-
cause of the existence of a hydrogen-like impurity in the
center of an asymmetric QD. Since the presence of the
Coulomb potential is equivalent of introducing another
new confinement on the electrons, which leads to greater
electron wavefunction overlapping with each other, the
electron–phonon interaction will be enhanced. Therefore,
the ground state binding energy increases with the rising of
the Coulomb bound potential. In Eqs. (8) and (10), we
know that the Coulomb bound potential influences the
ground state energy with a negative value, but the contri-
bution of the Coulomb bound potential to the ground state
binding energy is a positive effect. Hence, we can obtain
the results mentioned above. Thus the ground state energy
and the ground state binding energy of the asymmetric
QDs can be tuned by changing temperature and Coulomb
bound potential (impurity).
4. Conclusion
In conclusion, based on the linear combination operator
method, we have investigated the temperature and impurity
effects of the polaron in an asymmetric quantum dot. It is
found that the ground state energy and the ground state
binding energy will increase with increasing temperature.
The ground state energy is a decreasing function of the
Coulomb bound potential, whereas the ground state bind-
ing energy is an increasing functions of it.
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Temperature and impurity effects of the polaron in an asymmetric quantum dot
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