The effect of polarization charges on energy of univalent and bivalent donors in a spherical quantum dot
The energy of electrons of univalent and bivalent impurity of a spherical β -HgS/CdS nanoheterostructure is calculated as a function of quantum dot radius by the variation technique in the case of finite and infinite wells in the effective mass approximation. The effect of polarization charges whi...
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| Zitieren: | The effect of polarization charges on energy of univalent and bivalent donors in a spherical quantum dot / V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 653-661. — Бібліогр.: 18 назв. — англ. |
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| author | Boichuk, V.I. Bilynskyi, I.V. Leshko, R.Ya. |
| author_facet | Boichuk, V.I. Bilynskyi, I.V. Leshko, R.Ya. |
| citation_txt | The effect of polarization charges on energy of univalent and bivalent donors in a spherical quantum dot / V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 653-661. — Бібліогр.: 18 назв. — англ. |
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| container_title | Condensed Matter Physics |
| description | The energy of electrons of univalent and bivalent impurity of a spherical β -HgS/CdS nanoheterostructure is
calculated as a function of quantum dot radius by the variation technique in the case of finite and infinite wells
in the effective mass approximation. The effect of polarization charges which arise at the separation boundary
of the media is studied taking into account both the existence and the absence of an intermediate layer, where
dielectric permittivity depends on the coordinate.
Для сферичної наногетероструктури β-HgS/CdS у наближеннi ефективної маси варiацiйним способом обчислено енергiю електронiв одновалентної та двовалентної домiшки як функцiю радiуса квантової точки у випадку скiнченної та нескiнченної потенцiальної ями. Дослiджено вплив поляризацiйних зарядiв, що виникають на границi подiлу середовищ, з врахуванням iснування промiжного шару, де дiелектрична проникнiсть залежить вiд координат, i його вiдсутнiстю.
|
| first_indexed | 2025-12-07T15:48:15Z |
| format | Article |
| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 4(56), pp. 653–661
The effect of polarization charges on energy of univalent
and bivalent donors in a spherical quantum dot
V.I.Boichuk1, I.V.Bilynskyi1, R.Ya.Leshko1
1 Department of Theoretical Physics, Ivan Franko Drohobych State Pedagogical University,
24 Ivan Franko Str., 82100 Drohobych, Ukraine
Received March 4, 2008, in final form July 17, 2008
The energy of electrons of univalent and bivalent impurity of a spherical β-HgS/CdS nanoheterostructure is
calculated as a function of quantum dot radius by the variation technique in the case of finite and infinite wells
in the effective mass approximation. The effect of polarization charges which arise at the separation boundary
of the media is studied taking into account both the existence and the absence of an intermediate layer, where
dielectric permittivity depends on the coordinate.
Key words: energy, impurity, polarization charges, potential well
PACS: 71.55.-i, 73.21.La, 79.60.Jv
1. Introduction
Intensive investigations of impurity states in different nanoscale systems have been reported
in fundamental and applied publications for the recent 20 years. The use of effective mass ap-
proximation and of the dielectric continuum model yields good theoretical results that conform to
experimental data.
The properties of impurities in quantum dots (QDs) are discussed [1–11]. For a univalent
impurity in a spherical QD, exact solutions of the Schrödinger equation are obtained with extension
over the Coulomb potential interaction of particles. The potential was taken as a rectangular
spherical potential well due to a band mismatch at the separation boundary of the heterosystem.
The Calculated results reveal that the values of quantum levels of a confined electron in a spherical
QD can be quite different for the cases with finite and infinite well heights. The difference enhances
if a QD size decreases. The structure of donor quantum levels is similar to the confined electron
levels in a spherical QD if the well potential is stronger than that of a donor.
Based on exact solutions, quantum levels and energies of a donor located outside the center of
a spherical QD are obtained using a variation technique [5]. Thus, the results are obtained for the
ground and excited states of a donor in a spherical QD that is located inside or outside the QD.
The results show that the splitting and the change of quantum level order are dependent on QD
radius, impurity location, and well height. The splitting and the change of quantum level order are
caused by the violation of the system symmetry and by the competition of Coulomb interaction
and spatial confinement.
The dependence of energy of the system on the location of impurity was also studied [8]. It is
shown that energies of optical transitions depend on QD sizes and location of an admixture center.
By tuning QD sizes and the location of an admixture, it is possible for a heterosystem to emit light
of a certain wavelength.
The fine structure of energy levels of a hydrogenic impurity located in the center of a spherical
QD is calculated using a simple exact solution for a donor in a spherical QD [9]. It is mathematically
shown that there is no QD radius at which it would be impossible to find the energy of the
bound state of an impurity. The fine structure of an admixture is defined by perturbation theory.
c© V.I.Boichuk, I.V.Bilynskyi, R.Ya.Leshko 653
V.I.Boichuk, I.V.Bilynskyi, R.Ya.Leshko
The corrections are calculated for the terms that describe spin-orbit coupling, correction for the
relativistic kinetic energy and a Darwin term.
The effect of polarization charges of nanoheterostructure surface on the energy of a quasi-
particle is investigated [12–14]. In these papers for heterosystems of different nature a presence
of the bound surface charges at the separation boundary is taken into account with the help of
potential of forces of electrostatic images. The conditions of the origin of the electron boundary
states at the separation border are studied. The effect of the surface polarization charges of the
heterostructure on the exciton energy and oscillator strength is defined.
Modern technology of composing semiconductor nanoheterostructures makes it possible to get
nanosystems of sufficiently high quality. However, in the real conditions it is difficult to create a
heterogeneous system with a stepwise change of all physical parameters at the separation boundary.
Evidently, in the heterostructure there always exists a transitional layer, in which some physical
parameter changes from its value in one semiconductor (dielectric) to the corresponding value in
the other crystal. The effect of the surface transitional layer in which physical parameters are the
functions of coordinates on the quasi-particle energy is discussed [15–18].
In order to analyse the separately performed investigations regarding donors and polarization
charges, it is important to theoretically consider the effect of polarization charges on the univalent
donor energy in a spherical QD. In [1–11] the attention was focused on the research of a univalent
impurity. However, frequently the impurities in a QD have a valency greater than unity. Therefore,
a theoretical study of the effect of polarization charges arising at the interface of the media on
energy levels of a bivalent donor, as the simplest among polyvalent ones, that is dependent on QD
sizes will be important for further detailed research of donors.
The work is aimed at determining the ground donor state energy of a univalent and a bivalent
impurities in a spherical quantum dot with regard to polarization charges arising at the interface,
taking into account both the existence and the absence of a transitional layer. Specific calculations
are performed for the β-HgS/CdS nanoheterostructure.
2. Formulation of the problem and its solution
We consider a spherical nanoheterosystem. Its model is presented in figure 1. It is formed
by a nanocrystal of radius a with dielectric permittivity ε1, which is placed in a matrix with
dielectric permittivity ε2. At the separation boundary of the media there is a transitional layer
with the thickness L, where dielectric permittivity is a function of r coordinate. In the centre of
this spherical QD there is a univalent (bivalent) impurity. The univalent (bivalent) impurity stands
for an electro-neutral system with one (two) electrons.
Figure 1. Schematic diagram showing a model of quantum system.
654
Polarization charges and the energy of donors in a spherical QD
A point charge Ze placed at r0, induced the potential Φ(r, r0) at point r. The potential can be
found from the Poisson equation
∇2Φ (r, r0)+
d
dr
[ln ε (r)]
∂
∂r
[Φ (r, r0)] = −4πZe
ε (r)
δ (r, r0) , (1)
where δ(r, r0) is Dirac delta function. The solution of this equation allows us to find the interaction
energy of −e electron and the ion of a donor, interelectron interaction as well as the potential energy
of interaction of each particle with polarization charges at the interface.
By solving the Poisson equation (1) one can derive an expression that describes the interaction
of i-th electron with the impurity ion located in the quantum dot centre:
W (ri) = −e · Φ (ri) = −Ze2
{
(ε1 − ε2)/(ε1ε2a) + 1/(ε1ri), ri < a,
1/(ε2ri), ri > a.
(2)
The term −Ze2(ε1 − ε2)/(ε1ε2a) in (2) arises due to the existence of the interface, and all the
rest terms are purely Coulomb that describe the interaction among particles in the first ε1 or the
second ε2 medium. The expression of interelectronic interaction is given by:
W (r1, r2) = e2 ·
ε1−ε2
ε1
∞
∑
n=0
n+1
ε1n+ε2(n+1)
rn
1 rn
2
a2n+1 Pn (cos (θ))+
+ 1
ε1
∞
∑
n=0
rn
2
r
n+1
1
Pn (cos (θ)), r1 < a, r2 < a, r2 < r1,
ε1−ε2
ε1
∞
∑
n=0
n+1
ε1n+ε2(n+1)
rn
1 rn
2
a2n+1 Pn (cos (θ))+
+ 1
ε1
∞
∑
n=0
rn
1
r
n+1
2
Pn (cos (θ)), r1 < a, r2 < a, r1 < r2,
∞
∑
n=0
2n+1
ε1n+ε2(n+1)
rn
1
r
n+1
2
Pn (cos (θ)) , r1 < a, r2 > a,
∞
∑
n=0
2n+1
ε1n+ε2(n+1)
rn
2
r
n+1
1
Pn (cos (θ)) , r1 > a, r2 < a,
ε2−ε1
ε2
∞
∑
n=0
n
ε1n+ε2(n+1)
a2n+1
r
n+1
1
r
n+1
2
Pn (cos (θ))+
+ 1
ε2
∞
∑
n=0
rn
2
r
n+1
1
Pn (cos (θ)), r1 > a, r2 > a, r2 < r1,
ε2−ε1
ε2
∞
∑
n=0
n
ε1n+ε2(n+1)
a2n+1
r
n+1
1
r
n+1
2
Pn (cos (θ))+
+ 1
ε2
∞
∑
n=0
rn
1
r
n+1
2
Pn (cos (θ)), r1 > a, r2 > a, r1 < r2 ,
(3)
Pn(x) are Legendre polynomials.
The electron will induce polarization charges at the separation boundary of the media. The
corresponding energy of interaction of electron-polarization charges will be determined
V (ri) =
e2γ
4ε (ri)
∞
∫
0
dr
[
th
(
r − a
L
)
+
r
L
sech2
(
r − a
L
)]
1
r2 − r2
i
, (4)
ε (ri) =
ε1 + ε2
2
[
1 − γ · th
(
ri − a
L
)]
, (5)
γ =
ε1 − ε2
ε1 + ε2
, (6)
if a transitional layer of the thickness L is taken into account, in which dielectric permittivity ranges
between the value ε1 of one medium to ε2 of the other medium [17]. If it is neglected (L = 0), then
the energy, which is also called the potential of forces of electrostatic images, has a non-physical
divergence at r = a [16]:
V (ri) =
e2γ
2aε1
[
a2
a2 − r2
i
+
ε1
ε2
F
(
1,
ε2
ε1 + ε2
,
ε2
ε1 + ε2
+ 1,
(ri
a
)2
)]
, if ri < a, (7)
655
V.I.Boichuk, I.V.Bilynskyi, R.Ya.Leshko
V (ri) =
e2γ
2aε2
[
a2
a2 − r2
i
+
(
a
ri
)2
F
(
1,
ε2
ε1 + ε2
,
ε2
ε1 + ε2
+ 1,
(
a
ri
)2
)]
, if ri > a, (8)
where F (x, y, z, u) is confluent hypergeometric function.
The plots of the dependence of the potentials (4), (7)–(8) on the distance to the centre of
coordinates for nanoheterosystem β-HgS/CdS, when QD radius is 50 Å, are presented in figure 2.
Figure 2. The potential energy of electron-induced charges at the separation boundary of a
nanoheterosystem β-HgS/CdS with a transitional layer L = 5 Å(continuous curve) and L = 0 Å
(dotted curve).
Let there be a univalent impurity in the center of a spherical quantum dot. The Hamiltonian
of this system, using an effective mass approximation, will be written:
H = −~
2
2
∇ 1
m∗ (r)
∇ + U(r) + W (r) + V (r) = H0(r) + W (r) + V (r), (9)
where
Z = 1, m∗ (r) =
{
m∗
1, r < a,
m∗
2, r > a
(10)
is an effective mass of an electron in the corresponding crystal, V (r) will have the form of (4) or
(7)–(8). The potential energy due to the band mismatch (confinement potential) in a quantum dot
is expressed by the formula:
U(r) =
{
0, r < a,
U2, r > a,
U2 > 0. (11)
The Schrödinger equation with Hamiltonian (9) was solved using the variational method. In
order to choose a trial function, the problem of one particle (an electron) in both infinite and finite
spherical potential wells is used. The ground state of a charge in a spherical potential well with
infinitely high wells is described by the function
g1s(r) =
√
1/2πa
{
sin (πr/a)/r, r < a,
0, r > a,
(12)
and for the model of a finite potential at the interface it has the form
g1s(r) =
1√
4π
{
A1 · sin(kr)/r, r < a,
A2 · exp(−χr)/r, r > a,
=
{
g1(r), r < a,
g2(r), r > a,
(13)
656
Polarization charges and the energy of donors in a spherical QD
where
k =
√
2m∗
1
~2
E0, χ =
√
2m∗
2
~2
(U2 − E0). (14)
E0 is the energy of the particle ground state, constants A1 and A2 are defined from boundary
conditions and from a normalizing condition
g1(r)|r=a = g2(r)|r=a ,
1
m∗
1
d
dr
g1(r)
∣
∣
∣
∣
r=a
=
1
m∗
2
d
dr
g2(r)
∣
∣
∣
∣
r=a
,
∫
d~r · |g1s(r)|2 = 1. (15)
We consider QDs of small sizes, for which the kinetic energy of a particle is much greater than
the potential energy. That is why a trial variational function for the infinite potential model at the
separation boundary was chosen in the form
Ψ(r) = N1g1s(r) exp(−δr). (16)
The exponent in (16) describes the interaction of an electron and positively charged impurity ion,
δ is a variational parameter, N1 is a normalizing constant. For the finite potential model at the
interface, a trial function is written
Ψ(r) =
{
B1 · g1(r) exp [−αr] , r < a,
B2 · g2(r) exp [−βr] , r > a.
(17)
Values β, B2 are defined from the boundary conditions for function Ψ(r)
B1 (g1(r) exp(−αr))|r=a = B2 (g2(r) exp(−βr))|r=a ,
B1
m∗
1
d
dr
(g1(r) exp(−αr))
∣
∣
∣
∣
r=a
=
B2
m∗
2
d
dr
(g2(r) exp(−βr))
∣
∣
∣
∣
r=a
. (18)
It is found that
B2 = B1 exp [a (β − α)] , β =
m∗
2
m∗
1
α. (19)
According to the Ritz variational principle, the energy of the system is determined by mini-
mizing the functional E(α)
E (α) =
〈Ψ|H|Ψ〉
〈Ψ|Ψ〉 . (20)
Let there be a bivalent impurity in the center of a spherical QD. The Hamiltonian of this system
in the effective mass approximation will be written
H (r1, r2) = H0(r1) + H0(r2) + W (r1, r2) + V (r1) + V (r2) + W (r1) + W (r2), (21)
where H0(ri) is the Hamiltonian of a particle in a spherical potential well without impurity,
i = 1, 2, Z = 2. Here the interelectron interaction, interaction between each electron and ion of
an impurity, interaction between each electron and “intrinsic” polarization charges are taken into
account. The energy of the system is counted from the energy of interaction between an ion of
impurity and its “intrinsic” induced charges.
The energies and wave functions are the solutions of the stationary Schrödinger equation
H (r1, r2) Ψ (r1, r2) = EΨ (r1, r2) . (22)
It is not solved exactly. Therefore, the variational Ritz method was used to determine the ground
state. It is seen from the Hamiltonian that it is enough to consider the wave function which depends
657
V.I.Boichuk, I.V.Bilynskyi, R.Ya.Leshko
on r1, r2. Thus, the solutions of the problem of a particle in a spherical quantum well with finite
and infinite wells were used. The wave function for a model of an infinite potential at interface can
be written
Ψ (r1, r2) = N2 · g1s (r1) · g1s (r2) · exp [−δ (r1 + r2)] . (23)
For the model of a finite potential at the separation boundary of media the wave function will be
as follows:
Ψ (r1, r2) =
[{
B1 · g1(r1) exp [−αr1] , r1 < a,
B2 · g2(r1) exp [−βr1] , r1 > a.
]
·
[{
B1 · g1(r2) exp [−αr2] , r2 < a,
B2 · g2(r2) exp [−βr2] , r2 > a.
]
. (24)
In a similar manner, by minimizing a functional of form (20) for a presented system one can find
the energy of electrons.
3. The analysis of the obtained results
The results were obtained for a univalent and a bivalent donor located in the centre of a QD.
We assume the basic parameters of crystals that form the heterostructure to be equal to the volume
crystal parameters (table 1).
Table 1. Parameters of the nanoheterostructure β-HgS/CdS.
m1/m0 m2/m0 ε1 ε2 U2, meV Eg
1 , meV Eg
2 , meV
0.036 0.2 11.36 5.82 1200 500 2500
Figure 3 shows the dependencies of the ground state energy for univalent and bivalent donors in
the centre of the nanoheterostructure β-HgS/CdS taking into account the finite and infinite band
mismatch and the interaction of the particles and the interface of the heterostructure.
Figure 3. The energy of univalent (1, 4) and bivalent (2, 3) donors in a spherical QD with infinite
(1, 2) and finite (3, 4) potential at the separation boundary of the media.
In order to calculate the interaction of particles and the interface of the heterostructure for a
model of an infinite band mismatch equations (7)–(8) were used, and for a finite one equation (4)
was used. The magnitude of an intermediate layer was assumed L = 5 Å.
658
Polarization charges and the energy of donors in a spherical QD
It is seen from the plots that the energy of a QD almost does not depend on QD radius at large
radii. The smooth energy change is caused by the weak effect of the potential well and the surface
on the energy levels, and the energy is mainly defined by the interaction of electrons and an ion of
the impurity. A certain difference is observed between the energies obtained by using models of a
finite and an infinite well.
When the QD radius decreases, the confined potential (11) and polarization substantially effects
the energy of electrons. The leads to the increase of energy of the system. For the model of an
infinite confinement, the energies of a univalent and a bivalent donors grow faster than those for
a finite potential. Also, as a result of interelectronic interaction, the increase of the energy of a
bivalent impurity prevails over the rise of the energy of a univalent impurity within these models.
Figure 4. The energy caused by polarization charges at the separation boundary of the media
of a univalent (1, 4) and a bivalent (2, 3) donors in a spherical QD with finite (3, 4) and infinite
(1, 2) potentials at the interface.
To define a contribution given by polarization charges at the interface on the energy of a
nanoheterostructure, the calculation of the energy of univalent and bivalent donors was carried out
in a spherical QD without considering the interaction between electrons and polarization charges.
The difference between the total energy and the energy without induced charges at the interface of
heterostructure will show the effect of induced charges at the separation boundary of the media on
the energy of the system (figure 4). This quantity is denoted by Epol. It is worth mentioning that
in the considered nanoheterostructure, the dielectric permittivity of the matrix is considerably less
than that of a QD. Therefore, for charges in a QD the V (ri) is a positive function, which increases
while approaching to the surface. Thus, the contribution to the energies of particles caused by the
presence of the surface polarization is positive.
The energy of the system increases when a radius of a QD decreases, but if a QD radius is large,
the effect of polarization charges is small, because the rise of a QD radius reduces the effect of the
surface on electrons. From figure 4 it is also seen that Epol for a bivalent donor is greater than that
for a univalent donor. This is explained by the fact that a bivalent donor has two electrons, each of
them enhancing its energy due to the interaction with the induced charges, and a univalent donor
has only one electron. Therefore, the energy caused by polarization charges of a bivalent donor
is two times greater than that of a univalent. Since the potential energies of interaction between
electron and induced charges at the interface are different at small QD radii, it is seen from figure 4
that Epol for the models of finite and infinite potentials at small QD radii differ substantially. The
increase of the energy Epol with a decrease of a QD radius is caused by the growing effect of the
surface on the electrons.
659
V.I.Boichuk, I.V.Bilynskyi, R.Ya.Leshko
Figure 5. The dependence of a univalent (1) and a bivalent(2) donor energy on the value of a
nanoheterostructure intermediate layer with radius a = 50 Å.
In the present work, the energy of univalent and bivalent donors is calculated as a function of
the thickness of an intermediate layer. This dependence is shown in figure 5. It is seen from figure 5
that the reduction of an intermediate layer results in an increase of the energy. This is explained
by the fact that with reduction of L, the potential (4) approaches the potential (7)–(8) that is a
confinement of the growth of electrons, and thus the energy is enhanced.
Thus, in the present work, the electron energy dependencies of a univalent and a bivalent im-
purity are determinated as functions of a QD radius. The effect of a transitional layer L, where
dielectric permittivity is a function of coordinates on the energy of donors is studied. The results
showed that the reduction of a transitional layer enhances the energy of the system. The contri-
bution of polarization charges on the energy of a nanoheterosystem is defined. It is found that the
contribution of a bivalent donor is two times greater than that of a univalent donor. The calculation
showed that in the range of small QD radii the effect of the induced charges on the energy of a
nanostructure is essential.
660
Polarization charges and the energy of donors in a spherical QD
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18. Holovatsky V.A., Ukr. J. Phys., 2006, 51, 207.
Вплив поляризацiйних зарядiв на енергiю одновалентного та
двовалентного донора у сферичнiй квантовiй точцi
В.I.Бойчук1, I.В.Бiлинський1 , Р.Я.Лешко 1
1 Кафедра теоретичної фiзики, Дрогобицький державний педагогiчний унiверситет iм. I. Франка,
Дрогобич 82100, вул. Стрийська, 3
Отримано 4 березня 2008 р., в остаточному виглядi – 17 липня 2008 р.
Для сферичної наногетероструктури β-HgS/CdS у наближеннi ефективної маси варiацiйним спосо-
бом обчислено енергiю електронiв одновалентної та двовалентної домiшки як функцiю радiуса кван-
тової точки у випадку скiнченної та нескiнченної потенцiальної ями. Дослiджено вплив поляриза-
цiйних зарядiв, що виникають на границi подiлу середовищ, з врахуванням iснування промiжного
шару, де дiелектрична проникнiсть залежить вiд координат, i його вiдсутнiстю.
Ключовi слова: енергiя, домiшка, поляризацiйнi заряди, потенцiальна яма
PACS: 71.55.-i, 73.21.La, 79.60.Jv
661
662
|
| id | nasplib_isofts_kiev_ua-123456789-119577 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T15:48:15Z |
| publishDate | 2008 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Boichuk, V.I. Bilynskyi, I.V. Leshko, R.Ya. 2017-06-07T12:53:12Z 2017-06-07T12:53:12Z 2008 The effect of polarization charges on energy of univalent and bivalent donors in a spherical quantum dot / V.I. Boichuk, I.V. Bilynskyi, R.Ya. Leshko // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 653-661. — Бібліогр.: 18 назв. — англ. 1607-324X PACS: 71.55.-i, 73.21.La, 79.60.Jv DOI:10.5488/CMP.11.4.653 https://nasplib.isofts.kiev.ua/handle/123456789/119577 The energy of electrons of univalent and bivalent impurity of a spherical β -HgS/CdS nanoheterostructure is calculated as a function of quantum dot radius by the variation technique in the case of finite and infinite wells in the effective mass approximation. The effect of polarization charges which arise at the separation boundary of the media is studied taking into account both the existence and the absence of an intermediate layer, where dielectric permittivity depends on the coordinate. Для сферичної наногетероструктури β-HgS/CdS у наближеннi ефективної маси варiацiйним способом обчислено енергiю електронiв одновалентної та двовалентної домiшки як функцiю радiуса квантової точки у випадку скiнченної та нескiнченної потенцiальної ями. Дослiджено вплив поляризацiйних зарядiв, що виникають на границi подiлу середовищ, з врахуванням iснування промiжного шару, де дiелектрична проникнiсть залежить вiд координат, i його вiдсутнiстю. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics The effect of polarization charges on energy of univalent and bivalent donors in a spherical quantum dot Вплив поляризацiйних зарядiв на енергiю одновалентного та двовалентного донора у сферичнiй квантовiй точцi Article published earlier |
| spellingShingle | The effect of polarization charges on energy of univalent and bivalent donors in a spherical quantum dot Boichuk, V.I. Bilynskyi, I.V. Leshko, R.Ya. |
| title | The effect of polarization charges on energy of univalent and bivalent donors in a spherical quantum dot |
| title_alt | Вплив поляризацiйних зарядiв на енергiю одновалентного та двовалентного донора у сферичнiй квантовiй точцi |
| title_full | The effect of polarization charges on energy of univalent and bivalent donors in a spherical quantum dot |
| title_fullStr | The effect of polarization charges on energy of univalent and bivalent donors in a spherical quantum dot |
| title_full_unstemmed | The effect of polarization charges on energy of univalent and bivalent donors in a spherical quantum dot |
| title_short | The effect of polarization charges on energy of univalent and bivalent donors in a spherical quantum dot |
| title_sort | effect of polarization charges on energy of univalent and bivalent donors in a spherical quantum dot |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/119577 |
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