Optical properties of GaAs/AlxGa1-xAs/GaAs quantum dot with off-central impurity driven by electric field
The effect of a constant electric field and donor impurity on the energies and oscillator strengths of electron intraband quantum transitions in double-well spherical quantum dot GaAs/AlxGa1−xAs/GaAs is researched. The problem is solved in the framework of the effective mass approximation and rect...
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irk-123456789-1570392019-06-20T01:26:47Z Optical properties of GaAs/AlxGa1-xAs/GaAs quantum dot with off-central impurity driven by electric field Holovatsky, V.A. Yakhnevych, M.Ya. Voitsekhivska, O.M. The effect of a constant electric field and donor impurity on the energies and oscillator strengths of electron intraband quantum transitions in double-well spherical quantum dot GaAs/AlxGa1−xAs/GaAs is researched. The problem is solved in the framework of the effective mass approximation and rectangular potential wells and barriers model using the method of wave function expansion over a complete set of electron wave functions in nanostructure without electric field. It is shown that under the effect of electric field, the electron in the ground state tunnels from the inner potential well into the outer one. It also influences on the oscillator strengths of intraband quantum transition. The binding energy of an electron with ion impurity is obtained as a function of electric field intensity at a different location of impurity. The effect of a constant electric field and donor impurity on the energies and oscillator strengths of electron intraband quantum transitions in double-well spherical quantum dot GaAs/AlxGa1−xAs/GaAs is researched. The problem is solved in the framework of the effective mass approximation and rectangular potential wells and barriers model using the method of wave function expansion over a complete set of electron wave functions in nanostructure without electric field. It is shown that under the effect of electric field, the electron in the ground state tunnels from the inner potential well into the outer one. It also influences on the oscillator strengths of intraband quantum transition. The binding energy of an electron with ion impurity is obtained as a function of electric field intensity at a different location of impurity. 2018 Article Optical properties of GaAs/AlxGa1-xAs/GaAs quantum dot with off-central impurity driven by electric field / V.A. Holovatsky, M.Ya. Yakhnevych, O.M. Voitsekhivska // Condensed Matter Physics. — 2018. — Т. 21, № 1. — С. 13703: 1–9. — Бібліогр.: 28 назв. — англ. 1607-324X PACS: 71.38.-k, 63.20.kd, 63.20.dk, 72.10.Di DOI:10.5488/CMP.21.13703 arXiv:1803.11425 http://dspace.nbuv.gov.ua/handle/123456789/157039 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
The effect of a constant electric field and donor impurity on the energies and oscillator strengths of electron
intraband quantum transitions in double-well spherical quantum dot GaAs/AlxGa1−xAs/GaAs is researched.
The problem is solved in the framework of the effective mass approximation and rectangular potential wells and
barriers model using the method of wave function expansion over a complete set of electron wave functions in
nanostructure without electric field. It is shown that under the effect of electric field, the electron in the ground
state tunnels from the inner potential well into the outer one. It also influences on the oscillator strengths of
intraband quantum transition. The binding energy of an electron with ion impurity is obtained as a function of
electric field intensity at a different location of impurity. |
| format |
Article |
| author |
Holovatsky, V.A. Yakhnevych, M.Ya. Voitsekhivska, O.M. |
| spellingShingle |
Holovatsky, V.A. Yakhnevych, M.Ya. Voitsekhivska, O.M. Optical properties of GaAs/AlxGa1-xAs/GaAs quantum dot with off-central impurity driven by electric field Condensed Matter Physics |
| author_facet |
Holovatsky, V.A. Yakhnevych, M.Ya. Voitsekhivska, O.M. |
| author_sort |
Holovatsky, V.A. |
| title |
Optical properties of GaAs/AlxGa1-xAs/GaAs quantum dot with off-central impurity driven by electric field |
| title_short |
Optical properties of GaAs/AlxGa1-xAs/GaAs quantum dot with off-central impurity driven by electric field |
| title_full |
Optical properties of GaAs/AlxGa1-xAs/GaAs quantum dot with off-central impurity driven by electric field |
| title_fullStr |
Optical properties of GaAs/AlxGa1-xAs/GaAs quantum dot with off-central impurity driven by electric field |
| title_full_unstemmed |
Optical properties of GaAs/AlxGa1-xAs/GaAs quantum dot with off-central impurity driven by electric field |
| title_sort |
optical properties of gaas/alxga1-xas/gaas quantum dot with off-central impurity driven by electric field |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157039 |
| citation_txt |
Optical properties of GaAs/AlxGa1-xAs/GaAs quantum dot with off-central impurity driven by electric field / V.A. Holovatsky, M.Ya. Yakhnevych, O.M. Voitsekhivska // Condensed Matter Physics. — 2018. — Т. 21, № 1. — С. 13703: 1–9. — Бібліогр.: 28 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT holovatskyva opticalpropertiesofgaasalxga1xasgaasquantumdotwithoffcentralimpuritydrivenbyelectricfield AT yakhnevychmya opticalpropertiesofgaasalxga1xasgaasquantumdotwithoffcentralimpuritydrivenbyelectricfield AT voitsekhivskaom opticalpropertiesofgaasalxga1xasgaasquantumdotwithoffcentralimpuritydrivenbyelectricfield |
| first_indexed |
2025-07-14T09:22:47Z |
| last_indexed |
2025-07-14T09:22:47Z |
| _version_ |
1837613678212415488 |
| fulltext |
Condensed Matter Physics, 2018, Vol. 21, No 1, 13703: 1–9
DOI: 10.5488/CMP.21.13703
http://www.icmp.lviv.ua/journal
Optical properties of GaAs/AlxGa1−xAs/GaAs
quantum dot with off-central impurity driven by
electric field
V.A. Holovatsky∗, M.Ya. Yakhnevych, O.M. Voitsekhivska
Yuriy Fedkovych Chernivtsi National University, 2 Kotsyubynsky St., 58012 Chernivtsi, Ukraine
Received November 24, 2017, in final form December 26, 2017
The effect of a constant electric field and donor impurity on the energies and oscillator strengths of electron
intraband quantum transitions in double-well spherical quantum dot GaAs/AlxGa1−xAs/GaAs is researched.The problem is solved in the framework of the effective mass approximation and rectangular potential wells and
barriers model using the method of wave function expansion over a complete set of electron wave functions in
nanostructure without electric field. It is shown that under the effect of electric field, the electron in the ground
state tunnels from the inner potential well into the outer one. It also influences on the oscillator strengths of
intraband quantum transition. The binding energy of an electron with ion impurity is obtained as a function of
electric field intensity at a different location of impurity.
Key words:multishell quantum dot, impurity, intraband transitions
PACS: 71.38.-k, 63.20.kd, 63.20.dk, 72.10.Di
1. Introduction
Multishell quantum dots (QDs) are new and very interesting objects of research in nanostructure
physics. The existence of several potential wells with sizes, which can be changed in the processes of
nanostructure growth, allows one to create multimodal sources and detectors of electromagnetic waves.
The relationships between the sizes of the potential wells determine the location of quasi-particles. The
external fields significantly change the energy spectra and location of quasi-particles too. For example,
in [1–4] it is shown that the electron location is changed under the effect of a magnetic field. The electron
in the ground state tunnels from the outer potential well into the inner one. This process is accompanied
by the varying oscillator strengths of intraband transitions.
The study of the impurity states in semiconductor nanostructures was initiated only in early 1980s
through the pioneering work of Bastard [5]. In spite of the growing interest to the topic of impurity
doping in nanocrystallites, the majority of theoretical works have been carried out on shallow donors in
spherical QDs employing perturbation methods [6–10] or variational approaches [11–16]. For example,
using the perturbation methods, Bose et al. [6–9] obtained the binding energy of a shallow hydrogenic
impurity in spherical QDs. Based on variational approaches, Zhu [11, 12] studied the energies of an
off-center hydrogenic donor confined by a spherical QD with a finite rectangular potential well. Using
the plane wave method, Li [17] calculated the electronic states of hydrogenic donor impurity in low-
dimensional semiconductor nanostructures in the framework of the effective mass envelope-function
theory. In [18], the influence of central impurity and external electric field on the energies of intraband
quantum transitions and absorption coefficient is studied in spherical quantum dot CdTe/ZnTe.
The effect of impurity on the energy spectra in multishell nanostructures is researched within different
methods in [19–25]. It is shown that electron-impurity binding energy depends on the sizes of nanos-
∗E-mail: ktf@chnu.edu.ua
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
13703-1
https://doi.org/10.5488/CMP.21.13703
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
V.A. Holovatsky, M.Ya. Yakhnevych, O.M. Voitsekhivska
tructure and impurity location. The external electric field changes the most probable electron location
and increases or decreases the binding energy depending on the impurity location. Besides, in multishell
nanostructures, the electron changes its location under the effect of an electric field. That is manifested
by the optical properties and can be used in new optoelectronic devices. The effect of electric field
on quasi-particles energy spectra in spherical nanostructures with one potential well is investigated in
[18, 26, 27].
In this paper we study the electron energy spectrum and the wave function of its ground state as
functions of the intensity of electric field taking into account the polarization effects, which arise due to
the presence of interfaces between nanostructure materials. The oscillator strengths of intraband quantum
transitions and binding energy of electron with donor impurity are calculated at its different location.
2. Theoretical model
A semiconductor spherical QD consisting of a core-well with the radius r0, a barrier of the thickness
∆ = r1 − r0 and a well of the width ρ = r2 − r1 placed into the semiconductor matrix-barrier is studied.
In order to investigate the effect of the electric field on the electron energy spectrum and wave functions
in the nanostructure with impurity, the Schrödinger equation with the Hamiltonian
H = −®∇
~2
2µ(r)
®∇ + VF (r, θ) + Vc(®r) + Vp(r, θ) +W (r) +U(r) (1)
is solved. The confining potential U(r) and the effective mass µ(r) are step-like functions:
U(r) =
0, r 6 r0 , r1 < r 6 r2 ,
V, r0 < r 6 r1 ,
∞, r > r2 ,
(2)
µ(r) =
{
m0 , r 6 r0 , r1 < r 6 r2 , wells,
m1 , r0 < r 6 r1 , barrier. (3)
The potential energy of interaction between the electron and the positive ion, which is located at z-axis
at the distance rimp from the center of the nanostructure, has the form
Vc(®r) = −
Ze2
ε |®r − ®rimp |
, (4)
where ε = √ε1 ε2 is an average dielectric constant, ε1, ε2 are the dielectric constants of semiconductor
materials of the wells and barriers, respectively.
Vp (r, θ) = −
Ze2(ε − ε3)
εr2
∞∑
k=0
rki rk
r2k
2
k + 1
kε + (k + 1)ε3
Pk(cos θ). (5)
Here, ε3 is the dielectric constant of semiconductor matrix-barrier and Pk(cos θ) is the Legendre poly-
nomial. A formula arises from the existence of the polarized surface charges at the dot boundary and
describes the interaction between the ion and electron [28].
W(r) term in formula (1) describes the electron self-polarization potential, which in the case of a
small difference between ε1 and ε2 is simplified to the following form [27, 28]
W(r) =
e2(ε − ε3)
8πεr2
∞∑
k=0
k + 1
kε + (k + 1)ε3
(r/r2)
2k . (6)
13703-2
Optical properties of GaAs/AlxGa1−xAs/GaAs
Here, VF (®r) is an electrostatic potential of the electron in external field ®F applied in z-direction
VF (r, θ) = −eF cos θ
a0r, r 6 r0 ,
a1r +
b1
r2 , r0 < r 6 r1 ,
a2r +
b2
r2 , r1 < r 6 r2 ,
r +
b3
r2 , r > r2.
(7)
The coefficients ai and bi (presented in the appendix A) are obtained as solutions of Poisson equation
with standard dielectric boundary conditions at the interfaces.
If F = 0, Z = 0, the Schrödinger equation with Hamiltonian (1) has exact solutions
Φn`m(r, θ, ϕ) = Rn`(r)Ỳ m(θ, ϕ), (8)
where Ỳ m(θ, ϕ) are spherical functions, and the radial wave functions are written as:
R(i)
n`
(r) = A(i)
n`
J(i)
`
(κn`r) + B(i)
n`
N (i)
`
(κn`r), i = 0, 1, 2, (9)
J(i)
`
(χn`r) =
{
j`(kn`r), i = 0, 2,
I`(χn`r), i = 1, (10)
N (i)
`
(χn`r) =
{
n`(kn`r), i = 0, 2,
K`(χn`r), i = 1. (11)
Here, j` , n` are spherical Bessel functions of the first and the second kind, I` , K` are modified spherical
Bessel functions of the first and the second kind, kn` = (2m0E0
n`
/~)1/2, χn` = [2m1(V − E0
n`
)/~]1/2.
The unknown coefficients A(i)
n`
, B(i)
n`
and the electron energies E0
n`
in QD are obtained using the
continuity conditions for the wave functions and their densities of currents at all interfaces:
R(i)
n`
(ri) = R(i+1)
n`
(ri)
1
mi
dR(i)
n`
(r)
dr
�����
r=ri
=
1
mi+1
dR(i+1)
n`
(r)
dr
�����
r=ri
i = 0, 1, 2 (12)
and the normalization condition for the radial wave function
∞∫
0
|Rn`(r)|2 r2dr = 1. (13)
The coefficients B(0)
n`
= 0, A(3)
n`
= 0 from the condition that the wave function is finite at r = 0 and
r → ∞, respectively. In order to study the electron properties in a nanostructure driven by the electric
field, we are going to use the method of expansion of the quasi-particle wave function using a complete
set of eigenfunctions of the electron in a spherical nanostructure without the external fields obtained as
the exact solutions of Schrödinger equation. When the external fields are applied, the spherical symmetry
of the problem is broken and the orbital quantum number ` becomes not a valid one. Now, the quantum
number j determines the number of the energy level with a fixed magnetic quantum number m. The new
states characterized by a magnetic quantum number m are presented as a linear combination of the states
characterized by the functions Φn`m(®r)
ψjm(®r) =
∑
n
∑̀
Cn`Φn`m(®r). (14)
Substituting (9) into the Schrödinger equation with Hamiltonian (1), we obtain a secular equation��Hn`,n′`′ − Ejmδn,n′δ`,`′
�� = 0. (15)
13703-3
V.A. Holovatsky, M.Ya. Yakhnevych, O.M. Voitsekhivska
In order to obtain the electron energy spectrum and wave functions under the effect of external fields and
impurity, one should calculate the eigenvalues and eigenvectors of the matrix. Herein,
Hn`,n′`′ = E0
n`δn′,n + ξ(α`,mδ`′,`+1 + β`,mδ`′,`−1)Un′`′,n` + In′`′,n`δ`′,` , (16)
In′`′,n` =
∫
V
Φ
∗
n′`′(®r)Vc(®r)Φn`(®r)d®r, Un′`′,n` =
∞∫
0
r2R∗n′`′(r)Rn`(r)dr,
α`,m =
√
`2 − (m + 1)2
(2` + 3)(2` + 1)
, β`,m =
√
`2 − m2
4`2 − 1
,
where ξ = eFε and ®r = (r, θ, ϕ).
In the nanostructure driven by an electric field, the intensity of intraband quantum transitions is given
by the formula
Ii− f ∼ (Ei − E f )
����� ∫
V
ψ∗i (®r)r cos θψf (®r)dV
�����2. (17)
In this paper, the energies and the intensities of quantum transition are found as functions of the electric
field strength.
The probability of intraband quantum transitions is determined by the oscillator strength
Fi− f = 2mωi f |di f |2/~e2
(ωi f is the frequency and di f is the dipole momentum of the transition). Using the atomic energy units, the
radial coordinate and taking into account the coordinate-dependent electron effective mass, the formula
for oscillator strength of the transitions from the ground state ( j = 1, m = 0) takes the form [2]:
Fi− f = (Ei − E f )
���〈i |√µ(r) r cos θ | f 〉
���2 . (18)
The Thomas-Reiche-Kuhn sum rule must be fulfilled∑
f
Fi− f = 1, (19)
where i denotes the initial state and f is the final state of the transition. It gives an opportunity to find all
quantum transitions and controls the accuracy of numerical calculations.
3. Results and discussion
The computer calculations were performed using the physical parameters of AlxGa1−xAs semicon-
ductor with Al concentration x = 0 for the potential wells and x = 0.4 for the barriers, m0 = 0.067me,
m1 = 0.1me, V = 297 meV, ε1 = 13.2, ε2 = 11.9, ε3 = 1, where me is the mass of a pure electron. The
sizes of potential wells are selected in such a way that the electron should be located in the core of a
nanostructure without an electric field and impurity (r0 = 6 nm, ρ = 5 nm, ∆ = 2 nm).
In figure 1, the radial distributions of probability densities for the electron in the ground and three
excited states are presented.We took into account not less than 36 terms in the expansion (9) (n = 1, . . . , 6
and ` = 0, . . . , 5). When the electric field intensity increases and the impurity shifts from QD center, the
number of states |n`m〉, which make a significant contribution into ψjm(®r), increases.
The results of |ψ10(®r)|
2calculations show, figure 2, that the electron, at certain geometric parameters, is
located in the core of a nanostructure. When the electron is driven by an electric field, it tunnels through
the barrier into the outer potential well. Moreover, when a central impurity is present, this transition
happens at a bigger value of the electric field intensity, which compensates the Coulomb attraction
between the impurity and electron.
13703-4
Optical properties of GaAs/AlxGa1−xAs/GaAs
Figure 1. (Colour online) The radial distribution of probability density for the electron in the states |10〉,
|11〉, |20〉 and |21〉.
without impurity with central impurity
F=200*10 V/m
5 F=400*10 V/m
F=400*10 V/m
F=300*10 V/m F=500*10 V/m
F=600*10 V/m5
5
5
5
5
Figure 2. Distribution of electron density in the ground state in a nanostructure with and without impurity
at different values of the electric field intensity (F).
The electron energies as functions of the electric field intensity (F) are presented in figure 3. It is clear
that the change of the electron location is accompanied by the effect of anti-crossing energy levels. If the
impurity is located at the distance rimp = −z0, the anti-crossing of energies happens at the same intensity
of the electric field the same as without impurity. The ground state energy of an electron, located in the
outer potential well, linearly decreases at a bigger intensity.
13703-5
V.A. Holovatsky, M.Ya. Yakhnevych, O.M. Voitsekhivska
Figure 3. Electron energies as functions of the electric field intensity in QD with on-center impurity
(solid curves), at rimp = −z0 (dashed curves) and without impurity (dotted curves).
Figure 4. Electron-impurity binding energy as a function of the electric field intensity at rimp = −z1,
−z0, −z0/2, 0, z0/2, z0, z1.
Under the influence of the electric field, the electron changes its most probable location generating,
in its turn, a varying binding energy with the ion impurity which is determined by the formula
Eb = E10 − E imp
10 , (20)
where E10 is the energy of electron ground state without impurity, E imp
10 — that with the impurity.
In figure 4, the electron-impurity binding energy as a function of the electric field intensity at
rimp = −z1, −z0, −z0/2, 0, z0/2, z0, z1 is shown. The figure proves that at z0 > 0, the binding energy
monotonously decreases because the electron moves from the impurity, but at z0 < 0, the function has a
non-monotonous character, because the electron at first moves to the impurity and then from it.
In figure 5, the dependences of the binding energy of an electron with impurity on its location are
shown at different values of the electric field intensity. An increasing electric field intensity shifts the
maximum of the binding energy and weakly changes its magnitude.
The results of calculations of oscillator strengths of intraband quantum transitions are presented in
figure 6. These non-monotonous dependences are explained by a varying symmetry of wave functions,
and their overlapping due to the electron changes its location.
In figure 6, the oscillator strengths of intraband quantum transitions are presented as functions of the
electric field intensity. The numeric calculations are performed at a magnetic quantum number m = 0,
13703-6
Optical properties of GaAs/AlxGa1−xAs/GaAs
F= 0 V/m
5
F=200 10 V/m
5
F=800 10 V/m
5
F=600 10 V/m
5
F=400 10 V/m
Figure 5. Electron-impurity binding energy as a function of the electron location.
Figure 6. (Colour online) Oscillator strengths of intraband quantum transitions as functions of the electric
field intensity. Solid curves — with impurity, dashed curves — without impurity.
thus we denote the states only by a quantum number j = 1, 2, 3, . . . . From the figure one can see that the
quantum transition between the ground and the first excited state (|1〉 → |2〉) is the most intensive in the
whole range of the electric field intensity when there is no impurity. When the central impurity appears,
besides the transition |1〉 → |2〉, the oscillator strengths of quantum transitions into the states with bigger
energies (|1〉 → |3〉, |1〉 → |4〉, |1〉 → |5〉) essentially increase. These optical properties of multishell
nanostructures can be used in semiconductor devices.
4. Summary
The effect of an electric field and a shallow charged impurity on the intraband quantum transition
of an electron in the spherical multishell QD with two potential wells is investigated. The Schrödinger
equation is solved using the matrix method and the effective mass approximation, taking into account
the polarization effects. We studied how the electron changes its location in a nanostructure driven by an
electric field. The sizes of potential wells are selected such that the electron, in the ground state, is to be
located in the core of nanostructure without electric field and impurity. Under the effect of an electric
field the depth of the outer potential wells increases, because the electron tunnels from the core-well into
the outer one. The electron changes its location which manifests in the complicated behavior of oscillator
strengths of intraband quantum transitions depending on the electric field intensity.
13703-7
V.A. Holovatsky, M.Ya. Yakhnevych, O.M. Voitsekhivska
A. Solutions of Poisson equation
a0 = 27r3
1 r3
2ε1ε2ε3
{
2r3
0 (ε1 − ε2)
[
r3
1 (2ε1 + ε2)(ε1 − ε3) + r3
2 (ε2 − ε1)(2ε3 + ε1)
]
+ r3
1 (2ε1 + ε2)
[
2r3
1 (ε2 − ε1)(ε1 − ε3) + r3
2 (ε2 + 2ε1)(2ε3 + ε1)
]}−1
,
a1 =
a0(2ε2 + ε1)
3ε2
, b1 =
r3
0 (ε2 − ε1)
ε1 + 2ε2
, a2 =
a1
[
2b1(ε1 − ε2) + r3
1 (ε2 + 2ε1)
]
3r3
1ε1
,
b2 =
r3
1
[
r3
1 (ε1 − ε2) + b1(ε1 + 2ε2)
]
2b1(ε1 − ε2) + r3
1 (2ε1 + ε2)
, b3 =
2a2b2ε1 − a2r3
2ε2 + r3
2ε3
2ε3
.
If ε1 ≈ ε2 and ε =
√
ε1 ε2, the coefficients ai and bi have the form
a0 = a1 = a2 =
3ε3
ε + ε3
, b1 = b2 = 0, b3 =
r3
2 (ε3 − ε)
ε + 2ε3
.
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13703-8
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https://doi.org/10.1016/j.physleta.2013.03.012
Optical properties of GaAs/AlxGa1−xAs/GaAs
Оптичнi властивостi квантової точки GaAs/AlxGa1−xAs/GaAs
з нецентральною домiшкою пiд впливом електричного поля
В.А. Головацький,М.Я. Яхневич, О.М. Войцехiвська
Чернiвецький нацiональний унiверситет iм.Ю. Федьковича,
вул. Коцюбинського, 2, 58012 Чернiвцi, Україна
Дослiджено вплив постiйного електричного поля та донорної домiшки на енергiї та сили осци-
лятора внутрiшньозонних квантових переходiв електрона в двоямнiй сферичнiй квантовiй точцi
GaAs/AlxGa1−xAs/GaAs. Задача розв’язана в рамках наближення ефективних мас та моделi прямокутнихпотенцiальних ям i бар’єрiв методом розкладу хвильової функцiї за повним набором хвильових функцiй
електрона у наносистемi без електричного поля. Показано,що пiд впливом електричного поля електрон
в основному станi може тунелювати з внутрiшньої потенцiальної ями в зовнiшню. Це супроводжується
змiною сил осциляторiв внутрiшньозонних квантових переходiв. Отримано залежнiсть енергiї зв’язку
електрона iоном домiшки вiд напруженостi електричного поля при рiзних положеннях домiшки.
Ключовi слова: багатошарова квантова точка, домiшка, внутрiшньозоннi квантовi переходи
13703-9
Introduction
Theoretical model
Results and discussion
Summary
Solutions of Poisson equation
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