Electron energy spectrum in a spherical quantum dot with smooth confinement
The electron energy spectrum in a quantum dot (QD) with smooth dependences of the quasiparticle potential energy and the effective mass at the interface between semiconductor media is calculated in the effective mass approximation. It is shown that the electron energy corrections due to the taili...
Збережено в:
| Дата: | 2007 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2007
|
| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/117995 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Electron energy spectrum in a spherical quantum dot with smooth confinement / V. Holovatsky, O. Voitsekhivska, V. Gutsul // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 3. — С. 10-16. — Бібліогр.: 9 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-117995 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1179952017-05-28T03:05:33Z Electron energy spectrum in a spherical quantum dot with smooth confinement Holovatsky, V. Voitsekhivska, O. Gutsul, V. The electron energy spectrum in a quantum dot (QD) with smooth dependences of the quasiparticle potential energy and the effective mass at the interface between semiconductor media is calculated in the effective mass approximation. It is shown that the electron energy corrections due to the tailing of the interface are nonmonotonous functions of the QD radius, the increasing of which brings to the rapid increasing of shifts, reaching their maxima, and slowly decreasing for the QDs of big sizes. The calculations prove that the relative corrections for the different electron energy levels in a spherical QD are placed closer to each other with increase in the radius. The growth of the parameter of interface tailing leads to the proportional increase in the corrections to electron energy spectra. Numerical calculations are performed for HgS/CdS and GaAs/AlxGa₁₋xAs QDs, all dependences being qualitatively similar. 2007 Article Electron energy spectrum in a spherical quantum dot with smooth confinement / V. Holovatsky, O. Voitsekhivska, V. Gutsul // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 3. — С. 10-16. — Бібліогр.: 9 назв. — англ. 1560-8034 PACS 68.35.-p, 71.18.+y, 73.21.La http://dspace.nbuv.gov.ua/handle/123456789/117995 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The electron energy spectrum in a quantum dot (QD) with smooth
dependences of the quasiparticle potential energy and the effective mass at the interface
between semiconductor media is calculated in the effective mass approximation. It is
shown that the electron energy corrections due to the tailing of the interface are
nonmonotonous functions of the QD radius, the increasing of which brings to the rapid
increasing of shifts, reaching their maxima, and slowly decreasing for the QDs of big
sizes. The calculations prove that the relative corrections for the different electron energy
levels in a spherical QD are placed closer to each other with increase in the radius. The
growth of the parameter of interface tailing leads to the proportional increase in the
corrections to electron energy spectra. Numerical calculations are performed for
HgS/CdS and GaAs/AlxGa₁₋xAs QDs, all dependences being qualitatively similar. |
| format |
Article |
| author |
Holovatsky, V. Voitsekhivska, O. Gutsul, V. |
| spellingShingle |
Holovatsky, V. Voitsekhivska, O. Gutsul, V. Electron energy spectrum in a spherical quantum dot with smooth confinement Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Holovatsky, V. Voitsekhivska, O. Gutsul, V. |
| author_sort |
Holovatsky, V. |
| title |
Electron energy spectrum in a spherical quantum dot with smooth confinement |
| title_short |
Electron energy spectrum in a spherical quantum dot with smooth confinement |
| title_full |
Electron energy spectrum in a spherical quantum dot with smooth confinement |
| title_fullStr |
Electron energy spectrum in a spherical quantum dot with smooth confinement |
| title_full_unstemmed |
Electron energy spectrum in a spherical quantum dot with smooth confinement |
| title_sort |
electron energy spectrum in a spherical quantum dot with smooth confinement |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117995 |
| citation_txt |
Electron energy spectrum in a spherical quantum dot with smooth confinement / V. Holovatsky, O. Voitsekhivska, V. Gutsul // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2007. — Т. 10, № 3. — С. 10-16. — Бібліогр.: 9 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT holovatskyv electronenergyspectruminasphericalquantumdotwithsmoothconfinement AT voitsekhivskao electronenergyspectruminasphericalquantumdotwithsmoothconfinement AT gutsulv electronenergyspectruminasphericalquantumdotwithsmoothconfinement |
| first_indexed |
2025-07-08T13:08:04Z |
| last_indexed |
2025-07-08T13:08:04Z |
| _version_ |
1837084272663461888 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 10-16.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
10
PACS 68.35.-p, 71.18.+y, 73.21.La
Electron energy spectrum in a spherical quantum dot
with smooth confinement
V. Holovatsky, O. Voitsekhivska, V. Gutsul
Chernivtsi National University
2, Kotsyubynsky str., 58012 Chernivtsi, Ukraine; e-mail: theorphys@chnu.cv.ua
Abstract. The electron energy spectrum in a quantum dot (QD) with smooth
dependences of the quasiparticle potential energy and the effective mass at the interface
between semiconductor media is calculated in the effective mass approximation. It is
shown that the electron energy corrections due to the tailing of the interface are
nonmonotonous functions of the QD radius, the increasing of which brings to the rapid
increasing of shifts, reaching their maxima, and slowly decreasing for the QDs of big
sizes. The calculations prove that the relative corrections for the different electron energy
levels in a spherical QD are placed closer to each other with increase in the radius. The
growth of the parameter of interface tailing leads to the proportional increase in the
corrections to electron energy spectra. Numerical calculations are performed for
HgS/CdS and GaAs/AlxGa1-xAs QDs, all dependences being qualitatively similar.
Keywords: quantum dot, smooth confinement, interface, electron energy spectrum.
Manuscript received 07.02.07; accepted for publication 27.09.07; published online 31.10.07.
1. Introduction
The wide perspectives of the utilization of semi-
conductor nanoheterosystems in laser and nano-
scheme computer technique determine the intensive
development of the physics of quantum wells, wires,
and dots, though the physical phenomena in such
objects are often essentially different from those
having place in bulk crystals. Therefore, the
investigation of nanoheterosystems is of great interest
for the scientists.
The great number of theoretical and experimental
researches has been performed in this branch of physics
till now. The modern technological methods allow one
to produce the high-quality nanoheterostructures of
various shapes created of different semiconductor
materials. The experimental investigations stimulate the
intensive theoretical study of physical phenomena in
such nanosystems.
Problems of the establishment of a theory
describing the physical phenomena in nanohetero-
systems are related to the fact that these objects are so
small that the size quantization becomes essential.
Nevertheless, the microscopic approach taking into
account the atomic-molecular structure of a
heterosystem is very complicated for such systems; it is
used only for the nanocrystals containing several
hundreds of atoms. At the same time, the effective mass
approximation and the dielectric continuum model
introduced for the description of physical processes
running in bulk semiconductor crystals are often used in
nanophysics after some modifications.
The skiplike rectangular potential or the parabolic
one of a harmonic oscillator (see Fig. 1) (for which there
are the exact solutions of the Schrödinger equation) is
usually used in the calculations of quasiparticle energy
spectra in spherical quantum dots (QDs). Nevertheless,
the simplicity of the solutions has some disadvantages in
the description of potentials.
The main disadvantage of parabolic potential is the
neglect of the difference between the quasiparticle
effective mass inside a QD and in the external medium.
Moreover, it is impossible to obtain the continuum
energy spectrum of quasiparticles with high energies due
to the fact that this potential is not confined at infinity.
The authors of [1], taking into account the above-
mentioned disadvantages of the classical parabolic
potential, used a model potential. It is equal to the
parabolic one inside the QD, and it is constant in the
external medium. The calculation of the energy spectrum
is rather difficult and performed within some
approximations.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 10-16.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
11
Fig. 1. Geometrical scheme of spherical quantum dots and
schemes of the skiplike rectangular potential and the potential
of a harmonic oscillator.
The other disadvantage of the rectangular
potential model is the unphysical skiplike behavior of
the potential energy and the effective mass at the
interface of semiconductor media. The same unphysical
features are observed in the research of electrostatic
image forces due to the skiplike varying of the
dielectric constant at the media interface. In works [2,
3], the potential of image forces and corrections to the
electron binding energies were calculated by using the
smooth functions for the dielectric constants such as
ε+ε
ε−ε
+
ε+ε
=ε
L
z
z tanh1
2
)(
21
1221 . (1)
In work [2], it was shown that the chosen form of
the diffuse tailing of the interface of media is proved by
experiment. In the approach proposed in [3-5] for the
interface between a spherical quantum dot and the
external medium, the smooth functions
−
=
L
Rr
rf 0tanh)( , (2)
−
π
=
L
Rr
rf 0arctan
2
)( , (3)
and the linear approximating function
<+
+<<−
−
−<−
=
r
L
R
L
Rr
L
R
L
Rr
L
Rr
rf
2
,1
22
,
2
,1
)(
0
00
0
0
, (4)
where L is the parameter characterizing the interface
tailing width, were used.
In works [4, 5], it was shown that the results of
calculations of energy spectra are less sensitive to the
form of the approximating function than to the width of
the transition layer L.
In this paper, the corrections to the energies of
spherically symmetric stationary states of an electron
which arise due to accounting the tailing of the potential
energy and the effective mass functions at the interface
of the QD are calculated. The functional dependences of
the potential energy of an electron and its effective mass
on the distance from the center of a spherical QD are
assumed to be as follows:
−
+=
L
RrV
rU 0tanh1
2
)( , (5)
−
+
−
+
+
=
L
Rr
mm
mmmm
rm 0
21
1221 tanh1
2
)( . (6)
Here, m1 and m2 are, respectively, the electron
effective masses in a QD and in the external medium,
and V is some parameter.
Using the smooth functions m(r) and U(r) is more
physical because, in addition to the presence of the
transition layer at the interface, the macroscopic
parameters – effective mass (m) and potential energy (U)
– at a fixed point of the nanosystem are the results of the
averaging which cannot be skiplike.
The energy spectrum of a quasiparticle with the
effective mass (6) in the potential well (5) is found from
the solution of the Schrödinger equation
( ) ( ) ( )rErrU
rm
Ψ=Ψ
+∇∇−
)(
1
2
2
. (7)
It is obvious that this equation cannot be solved
exactly; therefore, we have to use some approximating
methods. For example, the ground state energy can be
calculated in the framework of the variational method,
but the problem becomes quite difficult for the energies
of excited states.
We propose to solve Eq. (7) by using the
approximation of the smooth functions (5) and (6) by the
skiplike functions shown in the Fig. 1. These functions
give the opportunity to obtain the exact solution of the
respective Schrödinger equation. When the number of
skips is big enough, the obtained solutions will be
sufficiently close to the solutions of Eq. (7) with the
smooth functions (5) and (6). Thus, the smooth interface
is substituted by several imaginary intermediate layers
with effective masses mi and potentials Vi. Then the
calculations are carried out on the base of the conditions
of equality of the areas under the smooth functions (5)
and (6) and the corresponding skiplike functions which
are shown in Fig. 2. The problem of solving the
Schrödinger equation (7) is the same as the problem to
RQD
U(r)=mω2r2/2
V
0
mCdS
mHgS
CdSHgS
RQD
U(r)
r
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 10-16.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
12
find electron energy spectra in complicated spherical
semiconductor nanoheterosystems [6, 7].
2. Solution of the Schrödinger equation
Taking into account the spherical symmetry of the
problem and the skiplike functions for the effective mass
m(r) and the potential U(r) chosen as
)()(
0
rVrU i
N
i
iσ=∑
=
, )()(
1
rmrm i
N
i
iσ=∑
=
(8)
with
≤≤
=σ −
regionother,0
,1
)( 1 ii
i
rrr
r , r-1 = 0,
it is convenient to write the radial part of the wave
function as
)()(
0
rRrR i
N
i
iσ=∑
=
. (9)
From the Schrödinger equation (7), we obtain the
system of N + 1 equations for the radial wave functions
)(rRi
0)()(
)(1
2
2
2
2
=−+
∂
∂
∂
∂
− rREV
r
rR
r
rrm ii
i
i
,
i=0, 1, …N.
(10)
Every of them contains the constant potential
energy Vi and the constant effective mass mi. The
solutions of Eqs. (10) are the linear combinations of
Bessel spherical functions of the first and second orders
NirknBrkjArR iiiii ,...,1,0),()()( 00 =+= , (11)
or the respective Hankel functions of the first and second
orders
NirkhSrkhArR iiiii ,...,1,0),()(()( 00 =+= +− , (12)
where )(21
iii VEmk −= − . For E < Vi, the Bessel
functions of the first and second orders are transformed
into the correspondent modified spherical Bessel
functions )(0 rkI i and )(0 rkK i .
The conditions of continuity of the wave function
and its probability current density at all points ri bring to
the system of 2(N + 1) equations
.,...,1,0
)(1)(1
)()(
1
1
1
Ni
dr
rdR
mdr
rdR
m
rRrR
ii
ii
rr
i
irr
i
i
rrirri
=
=
=
=
+
+=
=+=
(13)
Inserting representation (11) into the system of
equations (13), we obtain the system of linear equations
for the determination of the coefficients ii BA , ,
[ ]
[ ]
=+−
−+
=−
−−+
++++
+
++
++
0)()(
1
)()(
1
0)(
)()()(
11
'
011
'
0
1
'
0
'
0
110
0
1100
iiiiii
i
iiiiii
i
iii
iiiiiiiii
BrknArkj
m
BrknArkj
m
Brkn
ArkjBrknArkj
, (14)
where
)(
)(
)( 1
0'
0 iii
rr
i
ii rkjk
dr
rkjd
rkj
i
−==
=
, (15)
)(
)(
)( 1
0'
0 iii
rr
i
ii rknk
dr
rkdn
rkn
i
−==
=
. (16)
Using the condition that the wave function should
be finite at ∞→→ rr and0 , two unknown
coefficients are fixed as:
.0
,0
1
0
=
=
+NB
B
(17)
The other unknown 2(N+1) coefficients are
determined from the system of equations (14) and the
normalizing condition
( ) 12
0
2
=∫
∞
drrrR . (18)
Herein, the system of linear equations for the
coefficients Ai and Bi has the nontrivial solutions when
the determinant built from Bessel functions of the first
and second orders and their derivatives is equal to zero.
Thus, we arrive at the dispersion equation for the
determination of the electron energy spectrum in a
quantum dot with smooth interface.
The same electron energy spectrum can be found
using the system of equations (13) with a radial wave
function written in the form (12). Using the relation
S0 = 1, we obtain the expression for the S-matrix (SN+1),
whose poles define the energy of stationary states in the
nanosystem [8, 9].
3. Analysis of the results
The greater the number of skips, the closer are the
approximating functions to functions (5) and (6), and the
solution of Eq. (7) becomes more exact. In order to
define the optimal number of skips (N) in the
approximating functions U(r) and m(r), we performed
the calculation of the electron energy in spherically
symmetric states for various values of N.
In Fig. 2, we present the dependences of the
electron effective mass and the potential energy in
quantum dots HgS/CdS and GaAs/AlxGa1-xAs on the
distance along the radial coordinate at the fixed number
of skips N = 10 and the tailing parameter L = 1.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 10-16.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
13
12 14 16 18 20
0.05
0.10
0.15
0.20
0
500
1000
1500
RQD
V
V(r), meV
0.036
m, m0
mCdS
mHgS
CdSHgS
r, aHgS
a)
12 14 16 18 20
V(r), meV
RQD
0.1
0.067
m, m0
mAl
0.4
Ga
0.6
As
mGaAs
Al0.4Ga0.6AsGaAs
r, aGaAs
300
100
200
387
0
V
b)
Fig. 2. Dependence of the electron potential energy and the
effective mass on the radius r in the HgS/CdS (a) and
GaAs/AlxGa1-xAs (b) quantum dots. The solid curves are the
model skiplike function with N = 10, and the dashed ones are
m(r) and U(r) by formulas (5) and (6).
Parameters of the materials used in the numerical
calculations are shown in Table 1.
Table. Parameters of the materials.
Material ( )0mme *) )A(
o
a ( )eVeV
HgS 0.036 5.851 0
CdS 0.2 5.818 1.35
GaAs 0.067 5.65 0
Al0.4Ga0.6As 0.1 0.387
* ) 0m – bare mass of an electron.
4 5 6 7 8 9 10 11
0
10
20
30
∆ n
0=
E
N n0
- E
1 n0
, m
eV
n=3
n=2
n=1
N
a)
3 4 5 6 7 8 9 10 11
0
20
40
60
80
n=1
n=2
n=3
N
dE
n0
=E
N n0
-E
1 n0
, m
еV
b)
Fig. 3. Dependences of the absolute corrections to the energy
levels E10, E20, E30 on N in HgS/CdS (a) and GaAs/AlxGa1-xAs
(b) QDs with the radius r0 = 15.
In Fig. 3, we show the results of calculations of the
absolute corrections to the electron energy in spherically
symmetric states,
1
000 n
N
nn EE −=∆ , (19)
where N
nE 0 – the energies of electron stationary states in a
model potential with N skips, and 1
0nE – the energies of
electron stationary states in a rectangular potential (1 skip).
One can see from Fig. 3 that, as the number N
increases, the magnitudes of the absolute corrections are
saturated. Herein, for the higher excited states, the
saturation is observed at great numbers of skips. The sa-
turation is observed at N = 7-8 for a HgS/CdS QD and at
N = 10-11 for a GaAs/AlxGa1-xAs QD. The satisfactory
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 10-16.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
14
exactness (2-3 %) for all states for both types of
quantum dots and for the tailing parameter L = 1aHgS (a
HgS/CdS QD) and L = 1aGaAs (a GaAs/Al0.4Ga0.6As QD)
is reached at N = 10. For the greater value of L, the
corresponding bigger values of corrections and the
electron energy are observed. Therefore, the relative
corrections stay the same. Since, the further calculations
were performed with the model skiplike functions of the
potential and the effective mass at N = 10.
In Fig. 4, we display the dependence of the electron
spectrum on the size of a quantum dot at the different
magnitudes of the tailing parameter of the interface. It is
seen that the tailing of the interface does not influence
the qualitative shape of the dependence of the energy
spectrum on the size of a QD.
5 10 15 20 25 30 35
0
200
400
600
800
1000
1200
n=3
n=2
n=1
RQ D, aHgS
En, meV
L=0
L=1 aHgS
L=2 aHgS
a)
10 15 20 25 30 35 40 45 50
0
50
100
150
200
250
En, meV
L=0 aGaAs
L=1 aGaAs
L=2 aGaAs
n=3
n=2
n=1
RQD, aGaAs b)
Fig. 4. Dependence of the electron energy in spherically
symmetric states (l = 0) with n = 1, 2, 3 on the radius of
HgS/CdS (a) and GaAs/AlxGa1-xAs (b) QDs. Solid curves –
without the interface tailing, dashed curves – at L = 1 aHgS, and
doted curves – at L = 2 aHgS .
In order to evaluate the quantitative influence of the
smooth interface between the media on three lowest
electron energy levels, we present the results of
calculations of the relative corrections to the energies of
spherically symmetric states at various values of the
parameter L in Fig. 5:
1
0
1
00
1
0
0
0
n
n
N
n
n
n
n
E
EE
E
−
=
∆
=δ . (20)
0 10 20 30
0.0
0.1
0.2
0.3
n=3
n=2
n=1
L=2aHgS
L=1aHgS
RQD, aHgS
δ
n0
a)
0 10 20 30 40 50
0.00
0.04
0.08
0.12
δn0
n=3
n=2
n=1
L=1aGaAs
L=2aGaAs
RQD, aGaAs
b)
Fig. 5. Dependences of the relative corrections to the energies
1
0nE at n = 1, 2, 3 on the size of HgS/CdS (a) and GaAs/
AlxGa1-xAs (b) QDs for different values of the tailing
parameter L.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 10-16.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
15
Figures 4 and 5 prove that the absolute corrections
to the ground energy level are the smallest but the
relative correction is the biggest. The calculations show
that the relative correction to the energies of spherically
symmetric states increase with the QD size, and then
they decrease and become closer to one another. Such a
dependence is understandable because the tailed
potential well is thinner near the bottom, and it is wider
at the great energies than the respective rectangular well.
Therefore, at the small sizes of QDs, the corrections to
the energy levels located in the region of energies, where
the tailed potential well is wider than that for the
rectangular potential, can be negative. The increase in
the QD size leads to a decrease of the energies of
discrete levels shifted into the region of the potential
well where it is thinner than the respective rectangular
one, which is accompanied by the increase in the
respective energy corrections. The further increase of the
QD radius causes a decrease of the influence of the
interface tailing, because the probability of the
quasiparticle location at the heterointerface decreases
rapidly. As a result, the relative corrections for great-size
QDs are not sensitive to their radius.
The double increase of the tailing parameter causes
the increase of the relative correction by two times. For
big QDs, it is somewhat bigger than two.
The account for the smooth behavior of the
potential energy and the effective mass at the interface
between a QD and the external medium brings to a shift
of the energy levels. Its magnitude has the nonmo-
notonous dependence on the QD size. For rather big
QDs, the energy levels are shifted into the region of high
energies. Herein, the magnitudes of relative corrections
for the different energies of spherically symmetric states
are equal. The increase of the tailing parameter causes
the proportional increase of relative corrections to the
energy levels. All dependences for HgS/CdS and
GaAs/AlxGa1-xAs quantum dots are qualitatively similar,
the quantitative differences are caused by the different
values of the effective masses and the different depths of
potential wells.
0.5 1.0 1.5 2.0 2.5 3.0
50
100
150
E10, meV
L, aGaAs
R=10
R=15
R=20
Fig. 6. Dependences of the electron ground state energy on the
tailing parameter L in an AlxGa1-xAs QD at a fixed radius
R = (10, 15, 20) aGaAs.
In order to study the influence of the interface
tailing on the electron energy spectrum, we calculated
the electron ground state energy as a function of the
tailing parameter L at various values of the quantum dot
radius. The results of calculations are presented in Fig. 6.
As seen, with increase in the tailing parameter, the
location of the electron ground energy level is shifted
into the region of great energies. Herein, the interface
tailing influence is stronger for the quantum dots of
small radii. For quantum dots with R > 15 aGaAs, the
energies of the ground state show almost a linear
dependence on the tailing parameter L.
4. Conclusions
The results of studies allow us to draw the following
conclusions.
The skiplike potential at the interface of media used
instead of the smooth one gives the opportunity to solve
the Schrödinger equation and to calculate the corrections
to the energy spectrum obtained with the rectangular
potential and to take the diffuse tailing of the interface
into the account.
The introduction of the smooth functions m(r) and
U(r) for HgS/CdS quantum dots with R > 15 aHgS and
GaAs/Al0.4Ga0.6 QDs with the radii R > 20 aGaAs causes a
shift of all energy levels into the region of higher
energies.
It is established that the corrections to three lowest
spherically-symmetric energy levels become closer to
one another with increase in the size of QDs and weakly
depend on the radii of big quantum dots.
The increase of the tailing parameter causes the
increase of the corrections to the energy spectrum of an
electron in a quantum dot.
References
1. M. Ciurla, J. Adamowski., B. Szafran, S. Bednarek,
Modelling of confinement potentials in quantum
dots // Physica E: Low-Dimens. Syst. Nanostr. 15,
p. 162-168 (2002).
2. T. Nakamura, Modification of the image potential
for the interfacial zone with varying dielectric
constant // J. Phys. Soc. Jpn 52, p. 973-980 (1983).
3. V.I. Boichuk, R.Yu. Kubay, The potential energy
of a charge near the surface of the spherical
semiconductor microcrystal at the presence of
intermediate layer of varying dielectric constant //
Zhurn. Fiz. Doslidzhen’ 3, p. 492-497 (1999) (in
Ukrainian).
4. V.I. Boichuk, R.Yu. Kubay, I.V. Bilynsky, The
influence of the image potential on the electron
energy spectrum in complicated spherical
microcrystal CdS/HgS/H2O // Zhurn. Fiz.
Doslidzhen’ 3, p. 187-191 (1999) (in Ukrainian).
5. V.I. Boichuk, I.V. Bilynsky, R.Yu. Kubay, Energy
spectrum of a charged near-surface complicated
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2007. V. 10, N 3. P. 10-16.
© 2007, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
16
spherical heterostructure with taking into account
noninertial polarization // Ukr. J. Phys. 45, p. 236-
241 (2000).
6. M. Tkach, V. Holovatsky, O. Voitsekhivska,
M. Mikhalyova, Spectrum and interactions of
quasiparticles in complicated spherical
nanoheterosystems CdS/HgS/ZnO/H2O // Electro-
chemical Society Proc. 25, p. 316-321 (1998).
7. N.V. Tkach, О.N. Voitsehivskaya, V.А. Golo-
vatskii, M.Ya. Mikhal'ova, Spectrum of electron
and holes in a complicated spherical semiconductor
nanoheterosystem // Izv. Vuzov. Fiz., No 12, p. 58-
65 (1998).
8. M. Tkach, V. Holovatsky, O. Voitsekhivska,
Ya. Berezovsky, Electron resonance states in
opened spherical two-barrier nanoheterostructures
// Ukr. J. Phys. 46, p. 859-864 (2001).
9. M. Tkach, V. Holovatsky, Lifetime of electron and
hole quasistationary states in a spherical
nanoheterosystem β-HgS/ β-CdS/ β-HgS // Ukr. J.
Phys. 45, p. 972-975 (2000).
|