Effect of magnetic and electric fields on optical properties of semiconductor spherical layer
Theoretical investigation of the influence of magnetic and electric fields on the energy spectrum and wave functions of electron in semiconductor spherical layer has been performed. The case of co-directed electric and magnetic fields has been considered. The Schrödinger equation has been solved...
Збережено в:
| Дата: | 2014 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2014
|
| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/118351 |
| 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: | Effect of magnetic and electric V.A. Holovatsky, I.B. Bernikfields on optical properties of semiconductor spherical layer / V.A. Holovatsky, I.B. Bernik // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 1. — С. 7-13. — Бібліогр.: 28 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-118351 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1183512020-11-18T10:04:26Z Effect of magnetic and electric fields on optical properties of semiconductor spherical layer Holovatsky, V.A. Bernik, I.B. Theoretical investigation of the influence of magnetic and electric fields on the energy spectrum and wave functions of electron in semiconductor spherical layer has been performed. The case of co-directed electric and magnetic fields has been considered. The Schrödinger equation has been solved using the method of expansion for the wave function of electron in the spherical layer under external fields by applying the complete set of wave functions of a quasi-particle in a spherical nanostructure without the external fields. It has been shown that electric and magnetic fields take off the spectrum degeneration with respect to the magnetic quantum number. The external fields rebuild the energy spectrum and deform wave functions of electron. Moreover, their influence on the spherically symmetric state is the largest one. Increasing the magnetic field induction entails a monotonous dependence of the electron energy for the states with m > 0 and non-monotonous one for the states with m < 0. The ground state of electron is successively formed by the states with m = 0, –1, –2, … with increasing the induction of magnetic field. The enhancement of the electric field mainly diminishes the electron energy. The influence of field on the energy and intensities of the 1p-1s intraband transition has been studied. It has been shown that there exists a certain value of the electric field, at which the energy of quantum transition doesn’t depend on the magnetic field induction. 2014 Article Effect of magnetic and electric V.A. Holovatsky, I.B. Bernikfields on optical properties of semiconductor spherical layer / V.A. Holovatsky, I.B. Bernik // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 1. — С. 7-13. — Бібліогр.: 28 назв. — англ. 1560-8034 PACS 73.21.La, 78.67.Hc http://dspace.nbuv.gov.ua/handle/123456789/118351 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 |
Theoretical investigation of the influence of magnetic and electric fields on the
energy spectrum and wave functions of electron in semiconductor spherical layer has
been performed. The case of co-directed electric and magnetic fields has been
considered. The Schrödinger equation has been solved using the method of expansion for
the wave function of electron in the spherical layer under external fields by applying the
complete set of wave functions of a quasi-particle in a spherical nanostructure without
the external fields. It has been shown that electric and magnetic fields take off the
spectrum degeneration with respect to the magnetic quantum number. The external fields
rebuild the energy spectrum and deform wave functions of electron. Moreover, their
influence on the spherically symmetric state is the largest one. Increasing the magnetic
field induction entails a monotonous dependence of the electron energy for the states
with m > 0 and non-monotonous one for the states with m < 0. The ground state of
electron is successively formed by the states with m = 0, –1, –2, … with increasing the
induction of magnetic field. The enhancement of the electric field mainly diminishes the
electron energy. The influence of field on the energy and intensities of the 1p-1s
intraband transition has been studied. It has been shown that there exists a certain value
of the electric field, at which the energy of quantum transition doesn’t depend on the
magnetic field induction. |
| format |
Article |
| author |
Holovatsky, V.A. Bernik, I.B. |
| spellingShingle |
Holovatsky, V.A. Bernik, I.B. Effect of magnetic and electric fields on optical properties of semiconductor spherical layer Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Holovatsky, V.A. Bernik, I.B. |
| author_sort |
Holovatsky, V.A. |
| title |
Effect of magnetic and electric fields on optical properties of semiconductor spherical layer |
| title_short |
Effect of magnetic and electric fields on optical properties of semiconductor spherical layer |
| title_full |
Effect of magnetic and electric fields on optical properties of semiconductor spherical layer |
| title_fullStr |
Effect of magnetic and electric fields on optical properties of semiconductor spherical layer |
| title_full_unstemmed |
Effect of magnetic and electric fields on optical properties of semiconductor spherical layer |
| title_sort |
effect of magnetic and electric fields on optical properties of semiconductor spherical layer |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118351 |
| citation_txt |
Effect of magnetic and electric V.A. Holovatsky, I.B. Bernikfields on optical properties
of semiconductor spherical layer / V.A. Holovatsky, I.B. Bernik // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 1. — С. 7-13. — Бібліогр.: 28 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT holovatskyva effectofmagneticandelectricfieldsonopticalpropertiesofsemiconductorsphericallayer AT bernikib effectofmagneticandelectricfieldsonopticalpropertiesofsemiconductorsphericallayer |
| first_indexed |
2025-07-08T13:45:54Z |
| last_indexed |
2025-07-08T13:45:54Z |
| _version_ |
1837086680448761856 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 7-13.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
7
PACS 73.21.La, 78.67.Hc
Effect of magnetic and electric fields on optical properties
of semiconductor spherical layer
V.A. Holovatsky, I.B. Bernik
Chernivtsi National University,
2, Kotsiubynsky str., 58012 Chernivtsi;
Phone: +38(037)2244816, e-mail: ktf@chnu.edu.ua
Abstract. Theoretical investigation of the influence of magnetic and electric fields on the
energy spectrum and wave functions of electron in semiconductor spherical layer has
been performed. The case of co-directed electric and magnetic fields has been
considered. The Schrödinger equation has been solved using the method of expansion for
the wave function of electron in the spherical layer under external fields by applying the
complete set of wave functions of a quasi-particle in a spherical nanostructure without
the external fields. It has been shown that electric and magnetic fields take off the
spectrum degeneration with respect to the magnetic quantum number. The external fields
rebuild the energy spectrum and deform wave functions of electron. Moreover, their
influence on the spherically symmetric state is the largest one. Increasing the magnetic
field induction entails a monotonous dependence of the electron energy for the states
with m 0 and non-monotonous one for the states with m < 0. The ground state of
electron is successively formed by the states with m = 0, –1, –2, … with increasing the
induction of magnetic field. The enhancement of the electric field mainly diminishes the
electron energy. The influence of field on the energy and intensities of the 1p-1s
intraband transition has been studied. It has been shown that there exists a certain value
of the electric field, at which the energy of quantum transition doesn’t depend on the
magnetic field induction.
Keywords: spherical layer, intraband transitions, electric field, magnetic field.
Manuscript received 29.10.13; revised version received 02.12.13; accepted for
publication 20.03.14; published online 31.03.14.
1. Introduction
Semiconductor nanostructures have attracted a particular
attention of scientists for about two decades. Recent
advances in semiconductor technology allow growing
more complex structures than the simple quantum wells,
wires or dots. Among them: multiple quantum rings [1],
complex quantum wires [2] and multilayered quantum
dots [3-5]. The latter are grown using the chemical
colloidal method on the basis of CdS, CdSe, ZnS, ZnSe,
HgS and other semiconductor materials and are
promising for many technological applications such as
infrared photodetectors [6], lasers [7, 8], biolabels and
biosensors [9, 10], etc.
Multilayered spherical quantum dots awake a
particular interest of researchers because these structures
can be implemented for production of light emitting
diodes as they give high-efficient luminescence [11-13].
The simplest multilayered spherical structures are
quantum dots composed of the semiconductor core and
shell embedded into the matrix [14-16]. If the core
serves as a potential barrier and the shell – potential
well, the system is called a spherical layer.
External electrical and magnetic fields influence on
localization of quasi-particles in quantum wells and
affect optical properties of the structures. When the
external fields are present, the spherical symmetry is
violated and, therefore, calculation of the energy
spectrum becomes complicated due to the fitting
conditions at the interfaces. The influence of electric
field on properties of nanosystems is described by the
linear term with respect to the electric field in the
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 7-13.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
8
Schrödinger equation for electron, so the problem can be
solved using the variational method or perturbation
theory for a simple spherical quantum dot [17, 18] and
spherical layer [19, 20]. In the article [17], the Stark shift
is calculated using the variational method within the
framework of the infinite confining potential well model
for electron in the spherical quantum dot. The authors of
[18] investigated intersubband absorption spectra in the
simple spherical quantum dot with definite walls with or
without hydrogen on-center impurity. The article [19] is
dedicated to the study of the electron energy shift in the
spherical layer with impenetrable walls within the model
of the perturbation theory. The analytical calculation of
the influence of strong electric field on electron states
was performed in [20]. But the proposed theory is
applicable only for limit case of a “thin enough” layer
laying fairly far from the center of the system, which not
always corresponds to real nanosystems. The interband
transitions in the spherical layer driven by electric field
are investigated in [21, 22] using the perturbation theory
approach.
The study of the magnetic field effect on the
electron spectrum and wave functions is much more
complicated, as the Hamiltonian of electron comprises
not only a linear term, but the quadratic one of the
magnetic field induction. Therefore, in the majority of
theoretical studies, these investigations are performed
for simple quantum dots [23, 24]. The analogous
problem for the spherical layer is solved only using the
numerical calculations [25]. The proposed studies of
spherical layer don’t give any information about optical
properties of the structure. Therefore, this problem of the
influence of electric and magnetic fields on spherical
layer properties needs more detailed investigation.
In this work, we investigate the effect of external
electric and magnetic fields on the electron energy
spectrum and its probability density of localization in the
spherical layer (SL) with infinite confining potentials.
Also, the intensities of intraband 1p – 1s quantum
transitions as functions of the electric field and magnetic
field induction are obtained.
2. Schrödinger equation for electron in spherical
layer driven by magnetic and electric fields
We consider a spherical nanostructure consisting of a
core-barrier (r < r1), a shell-well (r1 < r < r2) embedded
into semiconductor matrix (r > r2), the so-called
spherical layer (SL). The directions of the magnetic field
induction and electric field are chosen along the Oz axis.
To obtain the energies and wave functions of electron in
SL driven by external fields, the Schrödinger equation
with the Hamiltonian (1) is to be solved
,)()(cos
)(2
1
2
rUrV
r
D
CreF
A
c
e
p
r
A
c
e
pH
p
(1)
where A
is the vector potential, F is the magnitude of
the electric field, )(rV p is the self-polarization potential
with account of the influence of induced charges on the
interfaces, )(rU is the confining potential and
2101
2100
,,
,,
)(
rrrrrm
rrrrrm
r , (2)
211
210
,,
,
)(
rrrr
rrr
r , (3)
21
21
,,
,0
)(
rrrr
rrr
rU . (4)
The potential of the field inside the spherical shell
in uniform electrostatic field E has the form:
cos)(
2r
D
CreFrV E . (5)
The coefficients C and D are obtained as the
solutions of the Poisson equation [26]:
)(
)2(
10
3
1
01
r
D
C , (6)
2
01
3
11001
3
2
3
2
3
1101
)(2)2)(2(
)(3
rr
rr
D . (7)
The potential )(rV p was calculated in [27] in the
form:
n
n
nn
p
Znn
n
r
r
nn
n
r
r
r
e
r
r
F
rr
r
r
e
r
r
F
r
r
rr
r
r
e
rV
1
2
)1(
1
)1(2
,1,,1
2
,1,,1
2
)(
10
10
0
12
201
01
12
1
0
2
2
2
22
1
0
22
2
2
2
10
10
02
2
2
1
11
2
1
22
1
2
1
01
01
01
2
(8)
2
0
2
1
1001
12
1
2 11
1
1
1
nnnn
nnr
r
Z
n
n ,
(9)
10
1
2
01
0
1 ,
. (10)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 7-13.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
9
It is taken into account that the relationship
between A
and magnetic induction B
: 2/][ BrA
,
and introduced are the new dimensionless magnitudes:
Ry* = e2 / (2a*) – effective Rydberg energy,
)/( 2
0
2* ema – effective Bohr radius, and parameters
)Ry2/( *cmeB , ** Ry/eFa . As a result, the
Hamiltonian (1) transforms to the form:
,)()(cos
4/)sin(
2
2
rVrU
r
D
Cr
rLH
p
z
(11)
where Lz is the z component of the angular momentum
operator of electron.
If B = 0, F = 0, the Schrödinger equation with the
Hamiltonian (11) has the exact solutions:
),()(),,( lmnlnlm YrRr , (12)
where )()()( rknBrkjArR nllnlnllnlnl , nlB
= )(/)( 11 rknrkjA nllnllnl , )(),( znzj ll – Bessel
spherical functions of the first and the second kind,
respectively; the values knl are fixed by the condition
Rnl(r2) = 0, and the coefficients Anl – by the normality
condition.
In order to study the electron properties in SL
driven by electric and magnetic fields, we are going to
use the method of expansion of the quasi-particle wave
function by applying the complete set of eigen functions
of electron in the spherical nanostructure without the
external fields, obtained as the exact solutions of the
Schrödinger equation [23]. When the external fields are
applied, the spherical symmetry of the problem is
lowered to the axial one and the orbital quantum number
l becomes inconvenient. The new states characterized by
a magnetic quantum number m are represented as a
linear combination of the states )(rnlm
:
n l
nlmlnjm rcr )()(
. (13)
Substituting (13) into the Schrödinger equation
with the Hamiltonian (11), we obtain the secular
equation for the electron energy spectrum:
0',','', llnnjmlnln EH , (14)
where the matrix elements '', lnlnH have the form:
,
4
1
,''2,',,',2,',
2
,''1,',1,',
,',',''
0
,''
B
nllnllmlllmlllml
F
nllnllmlllml
llnn
p
nllnnllnln
I
I
ImEH
(15)
2
1
)()(''
4
,''
r
r
nlln
B
nlln drrRrRrI ,
2
1
)()(
2''
2
,''
r
r
nlln
F
nlln drrR
r
D
CrrRrI ,
2
1
)()()(''
2
,''
r
r
nl
p
ln
p
nlln drrRrVrRrI ,
)12)(32(
)1( 22
,
ll
ml
ml ,
14 2
22
,
l
ml
ml ,
)12()32)(52(
])1][()2[(
2
2222
,
lll
mlml
ml ,
14)32)(12(
)1(
1
2
2222
,
l
ml
ll
ml
ml ,
)32()12)(12(
)]()1[(
2
2222
,
lll
mlml
ml .
Calculating the energy spectrum and wave
functions of electron in SL driven by electric and
magnetic fields, the intensity of 1s1p intraband
quantum transitions can be obtained as follows:
2
* )(cos)(~ dVrrrEEI
V
fififi
. (16)
In this work, the energies and intensities of
quantum transition are obtained as functions of the
magnetic field induction and electric field.
3. Analysis of the results
The computer calculations were performed using the
physical parameters: electron effective mass
emm 13.00 (me – the mass of pure electron), dielectric
constant 6.100 of CdSe for shell-well and dielectric
constant 25.81 of ZnS semiconductor material for
potential barriers.
Expanding the wave functions (13), we took into
account enough quantity of terms, providing the
condition that the sum of squares of the expansion
coefficients was equal to unity with the accuracy not less
than 0.01%.
The dependences of the electron energy spectrum
on the magnetic field induction (a) and electric field (b)
for the cases of applying one or another external field are
presented in Fig. 1. The spherical symmetry of the
problem is violated by the applied fields, so only two
quantum numbers characterizing the electron states
should be used, but for convenience, we use the same
quantum numbers characterizing the states of electron in
a nanostructure driven by electric and magnetic fields as
the ones without the external fields.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 7-13.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
10
Fig. 1. Energy spectrum of electron in SL with r1 = 4 nm, r2 =
12 nm as a function of the magnetic field induction (F = 0) (a),
electric field (B = 0) (b).
From Fig. 1a, one can see that the energy spectrum
of the electron becomes non-degenerate due to the
influence of magnetic field. The energies of the states
with m 0 become higher under the influence of the
magnetic field. For the states with m < 0, the non-
monotonous dependence of the energy on magnetic field
is caused by the linear and quadratic terms contributed
by the magnetic field in the Hamiltonian (11). When the
magnetic field increases, the states with m = ...,2,1,0
successively play the role of the ground state. The
ground state of electron with the certain value of the
magnetic quantum number transforms into the state with
another m when the magnetic field intensity increases at
the equal magnitude. Theoretical investigations show
that the distance between the points of these transitions
increases when the SL radius becomes smaller. Similar
behavior of the electron ground state energy was
theoretically obtained and experimentally confirmed for
the semiconductor quantum rings [28]. Any analogous
effects aren’t observed for simple quantum dots, for
which the ground state is formed by the electron energy
state with m = 0 independently on the magnetic field
induction.
Fig. 1b shows that external electrical field partially
takes off degeneration of the energy spectrum with
respect to the magnetic quantum number, mainly
decreasing the electron energy.
Fig. 2 displays the dependences of the electron
energy states with m = 0, ±1 on the electric field at the
applied external magnetic field with different values of
induction. Magnetic field shifts up the energy spectrum.
From Fig. 2b, one can see that the ground state is formed
by the lowest energy level with m = 1 at B = 40 T in
the case of the electric field absence. But the increase of
the electric field rebuilds the energy spectrum, to higher
extent pulling down the energies of the states with m = 0.
So, at a certain value of F, the ground state is formed by
the state with m = 0 as in the case of the external fields
absence. In Fig. 2, one can observe the anti-crossing
effect for the states with the same value of the magnetic
quantum number and the crossing one for the states with
different m (the insert of Fig. 2a).
The distributions of probability densities for
electron localization in SL in different quantum states
are presented in Fig. 3.
Fig. 2. Energy spectrum of electron in SL with r1 = 4 nm, r2 =
12 nm as a function of the electric field at B = 20 T (a), B =
40 T (b).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 7-13.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
11
F = 0 F = 20 kV/cm F = 50 kV/cm
B = 0, 100
B = 0, 110
B = 40 T, 100
B = 40 T, 110
Fig. 3. Distributions of probability densities for electron localization in SL with r1 = 4 nm, r2 = 12 nm for the quantum states
with 100 , 110 at B = 0, 40 T and F = 0, 20, 50 kV/cm.
Fig. 3 proves that the electron wave functions are
deformed due to the influence of external fields. The
turned on electric and magnetic fields deform the
distributions of electron densities in SL. In the case of
the electric field absence, the angular probability
increases near = 0, and decreases near = /2, while
the magnetic field induction increases. The electron 1s
state is characterized by the most obvious deformation of
its wave function. Its view approaches to the form of the
excited 1p state under the influence of magnetic field.
The energies of the mentioned electron states become
closer with enhancement of B, too (Fig. 1a). Electric
field localizes electron in the potential well made by it
near = . Magnetic field mainly promotes this effect.
The depicted distributions together with the
energies of quantum transitions determine the intensities
of intraband transitions. We calculate the intensities of
the quantum transitions between the first excited and
ground states (m = 0). The obtained energies and
intensities of the transitions are shown in Fig. 4.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 7-13.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
12
0.2 0.4 0.6 0.8
0.00
0.25
0.50
0.75
a
I 1p
-1
s, a
rb
. u
.
F, kV/cm
0.2 0.4 0.6 0.8
0
20
40
E
0
F
0
b
B = 0
B = 20 T
B = 30 T
B = 40 T
E
1p
-1
s, m
eV
F, kV/cm
Fig. 4. Dependences of the energies (a) and intensities (b) of
the 1sp1 intraband quantum transition on the electric field
at B = 0, 20, 30, 40 T.
From Fig. 4a, one can see that the intensities of the
studied transitions decrease with increasing the magnetic
field induction, obtaining non-monotonous character of
its dependence on the electric field. This behavior is
entailed by complicated evolution of energies and
distributions of electron probability densities in the
ground and first excited states in SL driven by external
fields. From Fig. 4b, it is clear that the energy of the
1p – 1s intraband transition increases with electric field
strength growth at all the values of the magnetic field
induction. At F0 17 kV/cm, this energy obtains the
characteristic value E0 that is constant at arbitrary B.
The intensity of the quantum transition at F0 can be
controlled by fitting the magnetic field induction.
4. Conclusions
We have studied the electron energy spectrum in
semiconductor spherical layer under the influence of
electric and magnetic fields. The problem has been
solved using the method of electron wave function
expansion over the set of eigen functions being the exact
solutions of the Schrödinger equation for the same
structures without the external fields.
Turning on the magnetic field takes off degeneracy
of the energy spectrum with respect to the magnetic
quantum number. The electron energies for the states
with positive and negative values of the magnetic
quantum number differently depend on the magnetic
field induction: for the states with m 0, the energy
monotonously increases, and for m < 0, the energy
decreases at first and then, only when the magnetic field
becomes strong enough, enhances. The electric field
takes off degeneracy of the spectrum partially. Due to
applying the electric field, additional lowering the
potential well is introduced, so the energy levels are
pulled down.
The distributions of electron probability densities in
SL are deformed by both electric and magnetic fields.
Electric field localizes electron near , magnetic
field with large induction promotes this effect.
The complicated dependence of electron
localization on the magnitudes of external fields defines
non-monotonous character of the intensities of intraband
quantum transitions. The specific value of the electric
field exists, at which the intensity of the 1s1p
transition can be changed by applying the magnetic field
with different induction at the constant transition energy.
The results of the investigation can be applied for
constructing new devices on the basis of multilayered
quantum dots. Nevertheless the problem is solved for
CdSe/ZnS SL, the proposed method can be used for
calculating electrical and optical properties of SLs for
other materials and sizes. The introduced theory can be
applied for further investigation of the features of
multilayered quantum dots.
References
1. T. Mano, T. Kuroda, S. Sanguinetti, T. Ochiai,
T. Tateno, J. Kim, T. Noda,. M. Kawabe,
K. Sakoda, G. Kido, N. Koguchi, Self-assambly of
concentric cuantum double rings // Nano Lett. 5(3),
p. 425-428 (2005).
2. M. Heigoldt, J. Arbiol, D. Spirkoska et al., Long
range epitaxial growth of prismatic heterostructures
on the facets of catalyst-free GaAs nanowires // J.
Mater. Chem. 19(7), p. 840-848 (2009).
3. D. Battaglia, B. Blackman, X. Peng, Coupled and
decoupled dual quantum systems in one
semiconductor nanocrystal // J. Am. Chem. Soc.
127(31), p. 10889-10897 (2005).
4. M. Braun, C. Burda, M. El-Sayed, Variation of the
thickness and number of wells in CdS/HgS/CdS
quantum dot quantum well system // J. Phys.
Chem. A, 105(23), p. 5548-5551 (2001).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 7-13.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
13
5. M. Uehara, H. Nakamura, H. Maeda, Preparation
of ZnS/CdSe/ZnS quantum dot quantum well by
using a microfluidic reactor // J Nanosci.
Nanotechnol. 9(1), p. 577-83 (2009).
6. S. Krishna, Quantum dot-in-a-well infrared
photodetectors // Infrared Phys. & Technol. 47,
p. 151-163 (2005).
7. S. Ivanov, J. Nanda, A. Piryanitski, M. Achermann,
L. Balet, I. Bezel, P. Anikeeva, S. Tretiak,
V. Klimov, Light amplification using inverted core-
shell nanocrystals: towards lasing in the single-
exciton regime // J. Phys. Chem. B, 108(30),
p. 10625-16030 (2004).
8. Y. Chan, J. Steckel, P. Snee, J-M. Caruge,
J. Hodgkiss, D. Nocera, M. Bawendi, Blue
semiconductor nanocrystal laser // Appl. Phys. Lett.
86(7), 073102 (2005).
9. U. Resch-Genger, M. Grabolle, S. Cavaliere-
Jaricot, R. Nitschke, T. Nann, Quantum dots versus
organic dyes as fluorescent label // Nature
Methods, 5(9), p. 763-775 (2008).
10. K. Sapsford, T. Pons, I. Medintz, H. Mattoussi,
Biosensing with luminescent semiconductor
quantum dots // Sensors, 6(8), p. 925-953 (2006).
11. S. Sapra, S. Mayilo, T. Klar, A. Rogach,
J. Feldmann, Bright white-light emission from
semiconductor nanocrystals: by chance and by
design // Adv. Mater. 19(4), p. 569-572 (2007).
12. S. Nizamoglu, E. Mutlugun, T. Ozel, H. Demir,
S. Sapra, N. Gaponik, A. Eychmuller, Dual-color
emitting quantum-dot-quantum-well CdSe/ZnS
heteronanocrystals hybridized on InGaN/GaN light
emitting diodes for high-quality white light generation
// Appl. Phys. Lett. 92(11), 113110 (2008).
13. S. Kim, T. Kim, M. Kang, S. Kwak, T. Yoo, L. Park,
I. Yang, S. Hwang, J. Lee, S. Kim, S.-W. Kim,
Highly luminescent InP/GaP/ZnS nanocrystals and
their application to white light-emitting diodes // J.
Am. Chem. Soc. 134(8), p. 3804-3809 (2012).
14. B. Dabbousi, J. Rodrigues-Viejo, F. Micules,
J. Heine, H. Mattousi, R. Ober, K. Jensen,
M. Bawendi, (CdSe)ZnS Core-shell quantum dots:
synthesis and characterization of a size series of
highly luminescent nanocrystallites // J. Chem.
Phys. B, 101(46), p. 9463-9475 (1997).
15. R. Little, M. El-Sayed, G. Bryant, S. Burke,
Formation of quantum-dot quantum-well
heteronanostructures with large lattice mismatch:
ZnS/CdS/ZnS // J. Chem. Phys. 114(4), p. 1813-
1822 (2001).
16. G. Murugadoss, Luminescense properties of
multilayer coated single structure ZnS/CdS/ZnS
nanocomposites // Spectrochimia Acta, Part A, 93,
p. 53-57 (2012).
17. G. Vazquez, M. Castillo-Mussot, C. Mendoza,
H. Spector, Spherical quantum dot under the
electric field // Phys. Status Solidi (c), 1(S1),
p. S54-S57 (2004).
18. R. Kostic, D. Stojanovic, Electric field effect on the
nonlinear and linear intersubband absorption
spectra in CdTe/ZnTe spherical quantum dot // J.
Nanophotonics, 6(1), 061606 (2012).
19. M. Zoheir, A. Manaselyan, H. Sarkisyan,
Electronic states and the Stark shift in narrow band
InSb quantum spherical layer // Phys. E, 40(9),
p. 2945-2949 (2008).
20. V. Harutyunyan, Nanospherical heterolayer in
strong electrostatic field // Appl. Nanosci. 2, p. 399-
344 (2012).
21. E. Kazaryan, A. Kostanyan, H. Sarkisyan,
Interband transitions in a spherical quantum layer
in the presence of an electric field: spherical rotator
model // Proc. Nat. Academy of Sci. of Armenia,
Phys. 42(4), p. 218-226 (2007).
22. M. Zuhair, A. Manaselyan, H. Sarkisyan, Interband
transitions in narrow gap InSb spherical layer
quantum dot in the presence of electric field // J.
Phys.: Conf. Series, 129, 012017 (2008).
23. S. Wu, L. Wan, Electronic structures in a CdSe
spherical quantum dot in a magnetic field:
Diagonalization method and variational method //
J. Appl. Phys. 111(6), 063711 (2012).
24. Z. Xiao, Exciton binding energy in spherical
quantum dots in a magnetic field // J. Appl. Phys.
86(8), p. 4509 (1999).
25. J. Planelles, J. Diaz, J. Climente, W. Jaskolski,
Semiconductor nanocrystals in a magnetic field //
Phys. Rev. B, 65, 245302 (2002).
26. W. Smythe, Static and Dynamic Electricity.
McGraw-Hill Book Company, Inc. New York-
Toronto-London, p. 140-141, 1950.
27. V. Boichuk, R. Kubay, I. Bilynskii, Influence of the
image potential on the energy spectrum of electron
in complex spherical microcrystal CdS/β-HgS/H2O
// J. Phys. Studies, 3(2), p. 187-191 (1999).
28. A. Lorke, R. Luyken, A. Govorov, J. Kotthaus,
J. Garsia, P. Petroff, Spectroscopy of nanoscopic
semiconductor rings // Phys. Rev. Lett. 84(10),
p. 2223-2226 (2000).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 1. P. 7-13.
PACS 73.21.La, 78.67.Hc
Effect of magnetic and electric fields on optical properties
of semiconductor spherical layer
V.A. Holovatsky, I.B. Bernik
Chernivtsi National University,
2, Kotsiubynsky str., 58012 Chernivtsi;
Phone: +38(037)2244816, e-mail: ktf@chnu.edu.ua
Abstract. Theoretical investigation of the influence of magnetic and electric fields on the energy spectrum and wave functions of electron in semiconductor spherical layer has been performed. The case of co-directed electric and magnetic fields has been considered. The Schrödinger equation has been solved using the method of expansion for the wave function of electron in the spherical layer under external fields by applying the complete set of wave functions of a quasi-particle in a spherical nanostructure without the external fields. It has been shown that electric and magnetic fields take off the spectrum degeneration with respect to the magnetic quantum number. The external fields rebuild the energy spectrum and deform wave functions of electron. Moreover, their influence on the spherically symmetric state is the largest one. Increasing the magnetic field induction entails a monotonous dependence of the electron energy for the states with m ( 0 and non-monotonous one for the states with m < 0. The ground state of electron is successively formed by the states with m = 0, –1, –2, … with increasing the induction of magnetic field. The enhancement of the electric field mainly diminishes the electron energy. The influence of field on the energy and intensities of the 1p-1s intraband transition has been studied. It has been shown that there exists a certain value of the electric field, at which the energy of quantum transition doesn’t depend on the magnetic field induction.
Keywords: spherical layer, intraband transitions, electric field, magnetic field.
Manuscript received 29.10.13; revised version received 02.12.13; accepted for publication 20.03.14; published online 31.03.14.
1. Introduction
Semiconductor nanostructures have attracted a particular attention of scientists for about two decades. Recent advances in semiconductor technology allow growing more complex structures than the simple quantum wells, wires or dots. Among them: multiple quantum rings [1], complex quantum wires [2] and multilayered quantum dots [3-5]. The latter are grown using the chemical colloidal method on the basis of CdS, CdSe, ZnS, ZnSe, HgS and other semiconductor materials and are promising for many technological applications such as infrared photodetectors [6], lasers [7, 8], biolabels and biosensors [9, 10], etc.
Multilayered spherical quantum dots awake a particular interest of researchers because these structures can be implemented for production of light emitting diodes as they give high-efficient luminescence [11-13]. The simplest multilayered spherical structures are quantum dots composed of the semiconductor core and shell embedded into the matrix [14-16]. If the core serves as a potential barrier and the shell – potential well, the system is called a spherical layer.
External electrical and magnetic fields influence on localization of quasi-particles in quantum wells and affect optical properties of the structures. When the external fields are present, the spherical symmetry is violated and, therefore, calculation of the energy spectrum becomes complicated due to the fitting conditions at the interfaces. The influence of electric field on properties of nanosystems is described by the linear term with respect to the electric field in the Schrödinger equation for electron, so the problem can be solved using the variational method or perturbation theory for a simple spherical quantum dot [17, 18] and spherical layer [19, 20]. In the article [17], the Stark shift is calculated using the variational method within the framework of the infinite confining potential well model for electron in the spherical quantum dot. The authors of [18] investigated intersubband absorption spectra in the simple spherical quantum dot with definite walls with or without hydrogen on-center impurity. The article [19] is dedicated to the study of the electron energy shift in the spherical layer with impenetrable walls within the model of the perturbation theory. The analytical calculation of the influence of strong electric field on electron states was performed in [20]. But the proposed theory is applicable only for limit case of a “thin enough” layer laying fairly far from the center of the system, which not always corresponds to real nanosystems. The interband transitions in the spherical layer driven by electric field are investigated in [21, 22] using the perturbation theory approach.
The study of the magnetic field effect on the electron spectrum and wave functions is much more complicated, as the Hamiltonian of electron comprises not only a linear term, but the quadratic one of the magnetic field induction. Therefore, in the majority of theoretical studies, these investigations are performed for simple quantum dots [23, 24]. The analogous problem for the spherical layer is solved only using the numerical calculations [25]. The proposed studies of spherical layer don’t give any information about optical properties of the structure. Therefore, this problem of the influence of electric and magnetic fields on spherical layer properties needs more detailed investigation.
In this work, we investigate the effect of external electric and magnetic fields on the electron energy spectrum and its probability density of localization in the spherical layer (SL) with infinite confining potentials. Also, the intensities of intraband 1p – 1s quantum transitions as functions of the electric field and magnetic field induction are obtained.
2. Schrödinger equation for electron in spherical layer driven by magnetic and electric fields
We consider a spherical nanostructure consisting of a core-barrier (r < r1), a shell-well (r1 < r < r2) embedded into semiconductor matrix (r > r2), the so-called spherical layer (SL). The directions of the magnetic field induction and electric field are chosen along the Oz axis. To obtain the energies and wave functions of electron in SL driven by external fields, the Schrödinger equation with the Hamiltonian (1) is to be solved
,
)
(
)
(
cos
)
(
2
1
2
r
U
r
V
r
D
Cr
eF
A
c
e
p
r
A
c
e
p
H
p
+
+
q
÷
ø
ö
ç
è
æ
+
-
-
÷
ø
ö
ç
è
æ
-
m
÷
ø
ö
ç
è
æ
-
=
r
r
r
r
(1)
where
A
r
is the vector potential, F is the magnitude of the electric field,
)
(
r
V
p
is the self-polarization potential with account of the influence of induced charges on the interfaces,
)
(
r
U
is the confining potential and
î
í
ì
>
£
<
£
<
£
=
m
2
1
0
1
2
1
0
0
,
,
,
,
)
(
r
r
r
r
r
m
r
r
r
r
r
m
r
,
(2)
î
í
ì
>
<
e
£
£
e
=
e
2
1
1
2
1
0
,
,
,
)
(
r
r
r
r
r
r
r
r
,
(3)
î
í
ì
>
<
¥
£
£
=
2
1
2
1
,
,
,
0
)
(
r
r
r
r
r
r
r
r
U
.
(4)
The potential of the field inside the spherical shell in uniform electrostatic field E has the form:
q
÷
ø
ö
ç
è
æ
+
-
=
cos
)
(
2
r
D
Cr
eF
r
V
E
.
(5)
The coefficients C and D are obtained as the solutions of the Poisson equation [26]:
)
(
)
2
(
1
0
3
1
0
1
e
-
e
e
+
e
=
r
D
C
,
(6)
2
0
1
3
1
1
0
0
1
3
2
3
2
3
1
1
0
1
)
(
2
)
2
)(
2
(
)
(
3
e
-
e
-
e
+
e
e
+
e
e
-
e
e
=
r
r
r
r
D
.
(7)
The potential
)
(
r
V
p
was calculated in [27] in the form:
(
)
(
)
(
)
(
)
(
)
n
n
n
n
p
Z
n
n
n
r
r
n
n
n
r
r
r
e
r
r
F
r
r
r
r
e
r
r
F
r
r
r
r
r
r
e
r
V
1
2
)
1
(
1
)
1
(
2
,
1
,
,
1
2
,
1
,
,
1
2
)
(
1
0
1
0
0
1
2
2
0
1
0
1
1
2
1
0
2
2
2
2
2
1
0
2
2
2
2
2
1
0
1
0
0
2
2
2
1
1
1
2
1
2
2
1
2
1
0
1
0
1
0
1
2
þ
ý
ü
-
+
e
-
e
+
e
-
e
´
ï
î
ï
í
ì
´
÷
÷
ø
ö
ç
ç
è
æ
-
+
e
-
e
e
-
e
÷
ø
ö
ç
è
æ
e
+
+
ï
þ
ï
ý
ü
ï
î
ï
í
ì
÷
÷
÷
ø
ö
ç
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
+
a
a
e
e
+
-
÷
÷
ø
ö
ç
ç
è
æ
e
+
e
e
-
e
e
+
+
ï
þ
ï
ý
ü
ï
î
ï
í
ì
÷
÷
ø
ö
ç
ç
è
æ
÷
ø
ö
ç
è
æ
+
a
a
÷
ø
ö
ç
è
æ
+
-
´
´
÷
÷
ø
ö
ç
ç
è
æ
e
+
e
e
-
e
e
=
å
¥
=
+
+
(8)
(
)
(
)
(
)
(
)
(
)
2
0
2
1
1
0
0
1
1
2
1
2
1
1
1
1
1
e
-
e
+
e
+
e
+
e
+
e
+
÷
÷
ø
ö
ç
ç
è
æ
+
=
+
n
n
n
n
n
n
r
r
Z
n
n
,
(9)
1
0
1
2
0
1
0
1
,
e
+
e
e
=
a
e
+
e
e
=
a
.
(10)
It is taken into account that the relationship between
A
r
and magnetic induction
B
r
:
2
/
]
[
B
r
A
r
r
r
´
=
, and introduced are the new dimensionless magnitudes: Ry* = e2 / (2(a*) – effective Rydberg energy,
)
/(
2
0
2
*
e
m
a
e
=
h
– effective Bohr radius, and parameters
)
Ry
2
/(
*
c
m
e
B
h
=
h
,
*
*
Ry
/
eFa
=
x
. As a result, the Hamiltonian (1) transforms to the form:
,
)
(
)
(
cos
4
/
)
sin
(
2
2
r
V
r
U
r
D
Cr
r
L
H
p
z
+
+
q
÷
ø
ö
ç
è
æ
+
x
-
-
q
h
+
h
+
D
-
=
(11)
where Lz is the z component of the angular momentum operator of electron.
If B = 0, F = 0, the Schrödinger equation with the Hamiltonian (11) has the exact solutions:
)
,
(
)
(
)
,
,
(
j
q
=
j
q
F
lm
nl
nlm
Y
r
R
r
,
(12)
where
)
(
)
(
)
(
r
k
n
B
r
k
j
A
r
R
nl
l
nl
nl
l
nl
nl
+
=
,
=
nl
B
=
)
(
/
)
(
1
1
r
k
n
r
k
j
A
nl
l
nl
l
nl
-
,
)
(
),
(
z
n
z
j
l
l
– Bessel spherical functions of the first and the second kind, respectively; the values knl are fixed by the condition Rnl(r2) = 0, and the coefficients Anl – by the normality condition.
In order to study the electron properties in SL driven by electric and magnetic fields, we are going to use the method of expansion of the quasi-particle wave function by applying the complete set of eigen functions of electron in the spherical nanostructure without the external fields, obtained as the exact solutions of the Schrödinger equation [23]. When the external fields are applied, the spherical symmetry of the problem is lowered to the axial one and the orbital quantum number l becomes inconvenient. The new states characterized by a magnetic quantum number m are represented as a linear combination of the states
)
(
r
nlm
r
F
:
å
å
F
=
y
n
l
nlm
l
n
jm
r
c
r
)
(
)
(
r
r
.
(13)
Substituting (13) into the Schrödinger equation with the Hamiltonian (11), we obtain the secular equation for the electron energy spectrum:
0
'
,
'
,
'
'
,
=
d
d
-
l
l
n
n
jm
l
n
l
n
E
H
,
(14)
where the matrix elements
'
'
,
l
n
l
n
H
have the form:
(
)
{
}
{
}
,
4
1
,
'
'
2
,
'
,
,
'
,
2
,
'
,
2
,
'
'
1
,
'
,
1
,
'
,
,
'
,
'
,
'
'
0
,
'
'
B
nl
l
n
l
l
m
l
l
l
m
l
l
l
m
l
F
nl
l
n
l
l
m
l
l
l
m
l
l
l
n
n
p
nl
l
n
nl
l
n
l
n
I
I
I
m
E
H
-
+
-
+
d
l
+
d
c
+
d
g
h
+
+
d
b
+
d
a
x
-
-
d
d
+
h
+
=
(15)
ò
=
2
1
)
(
)
(
'
'
4
,
'
'
r
r
nl
l
n
B
nl
l
n
dr
r
R
r
R
r
I
,
ò
÷
ø
ö
ç
è
æ
+
=
2
1
)
(
)
(
2
'
'
2
,
'
'
r
r
nl
l
n
F
nl
l
n
dr
r
R
r
D
Cr
r
R
r
I
,
ò
=
2
1
)
(
)
(
)
(
'
'
2
,
'
'
r
r
nl
p
l
n
p
nl
l
n
dr
r
R
r
V
r
R
r
I
,
)
1
2
)(
3
2
(
)
1
(
2
2
,
+
+
-
+
-
=
a
l
l
m
l
m
l
,
1
4
2
2
2
,
-
-
-
=
b
l
m
l
m
l
,
)
1
2
(
)
3
2
)(
5
2
(
]
)
1
][(
)
2
[(
2
2
2
2
2
,
+
+
+
-
+
-
+
-
=
g
l
l
l
m
l
m
l
m
l
,
1
4
)
3
2
)(
1
2
(
)
1
(
1
2
2
2
2
2
,
-
-
-
+
+
-
+
-
=
c
l
m
l
l
l
m
l
m
l
,
)
3
2
(
)
1
2
)(
1
2
(
)
](
)
1
[(
2
2
2
2
2
,
-
-
+
-
-
-
-
=
l
l
l
l
m
l
m
l
m
l
.
Calculating the energy spectrum and wave functions of electron in SL driven by electric and magnetic fields, the intensity of
1s
1p
-
intraband quantum transitions can be obtained as follows:
(
)
2
*
)
(
cos
)
(
~
dV
r
r
r
E
E
I
V
f
i
f
i
f
i
ò
y
q
y
-
-
r
r
.
(16)
In this work, the energies and intensities of quantum transition are obtained as functions of the magnetic field induction and electric field.
3. Analysis of the results
The computer calculations were performed using the physical parameters: electron effective mass
e
m
m
13
.
0
0
=
(me – the mass of pure electron), dielectric constant
6
.
10
0
=
e
of CdSe for shell-well and dielectric constant
25
.
8
1
=
e
of ZnS semiconductor material for potential barriers.
Expanding the wave functions (13), we took into account enough quantity of terms, providing the condition that the sum of squares of the expansion coefficients was equal to unity with the accuracy not less than 0.01%.
The dependences of the electron energy spectrum on the magnetic field induction (a) and electric field (b) for the cases of applying one or another external field are presented in Fig. 1. The spherical symmetry of the problem is violated by the applied fields, so only two quantum numbers characterizing the electron states should be used, but for convenience, we use the same quantum numbers characterizing the states of electron in a nanostructure driven by electric and magnetic fields as the ones without the external fields.
Fig. 1. Energy spectrum of electron in SL with r1 = 4 nm, r2 = 12 nm as a function of the magnetic field induction (F = 0) (a), electric field (B = 0) (b).
From Fig. 1a, one can see that the energy spectrum of the electron becomes non-degenerate due to the influence of magnetic field. The energies of the states with m ( 0 become higher under the influence of the magnetic field. For the states with m < 0, the non-monotonous dependence of the energy on magnetic field is caused by the linear and quadratic terms contributed by the magnetic field in the Hamiltonian (11). When the magnetic field increases, the states with m =
...
,
2
,
1
,
0
-
-
successively play the role of the ground state. The ground state of electron with the certain value of the magnetic quantum number transforms into the state with another m when the magnetic field intensity increases at the equal magnitude. Theoretical investigations show that the distance between the points of these transitions increases when the SL radius becomes smaller. Similar behavior of the electron ground state energy was theoretically obtained and experimentally confirmed for the semiconductor quantum rings [28]. Any analogous effects aren’t observed for simple quantum dots, for which the ground state is formed by the electron energy state with m = 0 independently on the magnetic field induction.
Fig. 1b shows that external electrical field partially takes off degeneration of the energy spectrum with respect to the magnetic quantum number, mainly decreasing the electron energy.
Fig. 2 displays the dependences of the electron energy states with m = 0, ±1 on the electric field at the applied external magnetic field with different values of induction. Magnetic field shifts up the energy spectrum. From Fig. 2b, one can see that the ground state is formed by the lowest energy level with m =
1
-
at B = 40 T in the case of the electric field absence. But the increase of the electric field rebuilds the energy spectrum, to higher extent pulling down the energies of the states with m = 0. So, at a certain value of F, the ground state is formed by the state with m = 0 as in the case of the external fields absence. In Fig. 2, one can observe the anti-crossing effect for the states with the same value of the magnetic quantum number and the crossing one for the states with different m (the insert of Fig. 2a).
The distributions of probability densities for electron localization in SL in different quantum states are presented in Fig. 3.
Fig. 2. Energy spectrum of electron in SL with r1 = 4 nm, r2 = 12 nm as a function of the electric field at B = 20 T (a), B = 40 T (b).
Fig. 3 proves that the electron wave functions are deformed due to the influence of external fields. The turned on electric and magnetic fields deform the distributions of electron densities in SL. In the case of the electric field absence, the angular probability increases near ( = 0, ( and decreases near ( = (/2, while the magnetic field induction increases. The electron 1s state is characterized by the most obvious deformation of its wave function. Its view approaches to the form of the excited 1p state under the influence of magnetic field. The energies of the mentioned electron states become closer with enhancement of B, too (Fig. 1a). Electric field localizes electron in the potential well made by it near ( = (. Magnetic field mainly promotes this effect.
100
y
The depicted distributions together with the energies of quantum transitions determine the intensities of intraband transitions. We calculate the intensities of the quantum transitions between the first excited and ground states (m = 0). The obtained energies and intensities of the transitions are shown in Fig. 4.
0.2
0.4
0.6
0.8
0.00
0.25
0.50
0.75
a
I
1p-1s
, arb. u.
F, kV/cm
0.2
0.4
0.6
0.8
0
20
40
D
E
0
F
0
b
B = 0
B = 20 T
B = 30 T
B = 40 T
D
E
1p-1s
, meV
F, kV/cm
Fig. 4. Dependences of the energies (a) and intensities (b) of the
1s
p
1
-
intraband quantum transition on the electric field at B = 0, 20, 30, 40 T.
From Fig. 4a, one can see that the intensities of the studied transitions decrease with increasing the magnetic field induction, obtaining non-monotonous character of its dependence on the electric field. This behavior is entailed by complicated evolution of energies and distributions of electron probability densities in the ground and first excited states in SL driven by external fields. From Fig. 4b, it is clear that the energy of the 1p – 1s intraband transition increases with electric field strength growth at all the values of the magnetic field induction. At F0 ( 17 kV/cm, this energy obtains the characteristic value (E0 that is constant at arbitrary B. The intensity of the quantum transition at F0 can be controlled by fitting the magnetic field induction.
4. Conclusions
We have studied the electron energy spectrum in semiconductor spherical layer under the influence of electric and magnetic fields. The problem has been solved using the method of electron wave function expansion over the set of eigen functions being the exact solutions of the Schrödinger equation for the same structures without the external fields.
Turning on the magnetic field takes off degeneracy of the energy spectrum with respect to the magnetic quantum number. The electron energies for the states with positive and negative values of the magnetic quantum number differently depend on the magnetic field induction: for the states with m ( 0, the energy monotonously increases, and for m < 0, the energy decreases at first and then, only when the magnetic field becomes strong enough, enhances. The electric field takes off degeneracy of the spectrum partially. Due to applying the electric field, additional lowering the potential well is introduced, so the energy levels are pulled down.
The distributions of electron probability densities in SL are deformed by both electric and magnetic fields. Electric field localizes electron near
p
=
q
, magnetic field with large induction promotes this effect.
The complicated dependence of electron localization on the magnitudes of external fields defines non-monotonous character of the intensities of intraband quantum transitions. The specific value of the electric field exists, at which the intensity of the
1s
1p
-
transition can be changed by applying the magnetic field with different induction at the constant transition energy.
The results of the investigation can be applied for constructing new devices on the basis of multilayered quantum dots. Nevertheless the problem is solved for CdSe/ZnS SL, the proposed method can be used for calculating electrical and optical properties of SLs for other materials and sizes. The introduced theory can be applied for further investigation of the features of multilayered quantum dots.
References
1. T. Mano, T. Kuroda, S. Sanguinetti, T. Ochiai, T. Tateno, J. Kim, T. Noda,. M. Kawabe, K. Sakoda, G. Kido, N. Koguchi, Self-assambly of concentric cuantum double rings // Nano Lett. 5(3), p. 425-428 (2005).
2. M. Heigoldt, J. Arbiol, D. Spirkoska et al., Long range epitaxial growth of prismatic heterostructures on the facets of catalyst-free GaAs nanowires // J. Mater. Chem. 19(7), p. 840-848 (2009).
3. D. Battaglia, B. Blackman, X. Peng, Coupled and decoupled dual quantum systems in one semiconductor nanocrystal // J. Am. Chem. Soc. 127(31), p. 10889-10897 (2005).
4. M. Braun, C. Burda, M. El-Sayed, Variation of the thickness and number of wells in CdS/HgS/CdS quantum dot quantum well system // J. Phys. Chem. A, 105(23), p. 5548-5551 (2001).
5. M. Uehara, H. Nakamura, H. Maeda, Preparation of ZnS/CdSe/ZnS quantum dot quantum well by using a microfluidic reactor // J Nanosci. Nanotechnol. 9(1), p. 577-83 (2009).
6. S. Krishna, Quantum dot-in-a-well infrared photodetectors // Infrared Phys. & Technol. 47, p. 151-163 (2005).
7. S. Ivanov, J. Nanda, A. Piryanitski, M. Achermann, L. Balet, I. Bezel, P. Anikeeva, S. Tretiak, V. Klimov, Light amplification using inverted core-shell nanocrystals: towards lasing in the single-exciton regime // J. Phys. Chem. B, 108(30), p. 10625-16030 (2004).
8. Y. Chan, J. Steckel, P. Snee, J-M. Caruge, J. Hodgkiss, D. Nocera, M. Bawendi, Blue semiconductor nanocrystal laser // Appl. Phys. Lett. 86(7), 073102 (2005).
9. U. Resch-Genger, M. Grabolle, S. Cavaliere-Jaricot, R. Nitschke, T. Nann, Quantum dots versus organic dyes as fluorescent label // Nature Methods, 5(9), p. 763-775 (2008).
10. K. Sapsford, T. Pons, I. Medintz, H. Mattoussi, Biosensing with luminescent semiconductor quantum dots // Sensors, 6(8), p. 925-953 (2006).
11. S. Sapra, S. Mayilo, T. Klar, A. Rogach, J. Feldmann, Bright white-light emission from semiconductor nanocrystals: by chance and by design // Adv. Mater. 19(4), p. 569-572 (2007).
12. S. Nizamoglu, E. Mutlugun, T. Ozel, H. Demir, S. Sapra, N. Gaponik, A. Eychmuller, Dual-color emitting quantum-dot-quantum-well CdSe/ZnS heteronanocrystals hybridized on InGaN/GaN light emitting diodes for high-quality white light generation // Appl. Phys. Lett. 92(11), 113110 (2008).
13. S. Kim, T. Kim, M. Kang, S. Kwak, T. Yoo, L. Park, I. Yang, S. Hwang, J. Lee, S. Kim, S.-W. Kim, Highly luminescent InP/GaP/ZnS nanocrystals and their application to white light-emitting diodes // J. Am. Chem. Soc. 134(8), p. 3804-3809 (2012).
14. B. Dabbousi, J. Rodrigues-Viejo, F. Micules, J. Heine, H. Mattousi, R. Ober, K. Jensen, M. Bawendi, (CdSe)ZnS Core-shell quantum dots: synthesis and characterization of a size series of highly luminescent nanocrystallites // J. Chem. Phys. B, 101(46), p. 9463-9475 (1997).
15. R. Little, M. El-Sayed, G. Bryant, S. Burke, Formation of quantum-dot quantum-well heteronanostructures with large lattice mismatch: ZnS/CdS/ZnS // J. Chem. Phys. 114(4), p. 1813-1822 (2001).
16.
G. Murugadoss, Luminescense properties of multilayer coated single structure ZnS/CdS/ZnS nanocomposites // Spectrochimia Acta, Part A, 93, p. 53-57 (2012).
17. G. Vazquez, M. Castillo-Mussot, C. Mendoza, H. Spector, Spherical quantum dot under the electric field // Phys. Status Solidi (c), 1(S1), p. S54-S57 (2004).
18. R. Kostic, D. Stojanovic, Electric field effect on the nonlinear and linear intersubband absorption spectra in CdTe/ZnTe spherical quantum dot // J. Nanophotonics, 6(1), 061606 (2012).
19. M. Zoheir, A. Manaselyan, H. Sarkisyan, Electronic states and the Stark shift in narrow band InSb quantum spherical layer // Phys. E, 40(9), p. 2945-2949 (2008).
20. V. Harutyunyan, Nanospherical heterolayer in strong electrostatic field // Appl. Nanosci. 2, p. 399-344 (2012).
21. E. Kazaryan, A. Kostanyan, H. Sarkisyan, Interband transitions in a spherical quantum layer in the presence of an electric field: spherical rotator model // Proc. Nat. Academy of Sci. of Armenia, Phys. 42(4), p. 218-226 (2007).
22. M. Zuhair, A. Manaselyan, H. Sarkisyan, Interband transitions in narrow gap InSb spherical layer quantum dot in the presence of electric field // J. Phys.: Conf. Series, 129, 012017 (2008).
23. S. Wu, L. Wan, Electronic structures in a CdSe spherical quantum dot in a magnetic field: Diagonalization method and variational method // J. Appl. Phys. 111(6), 063711 (2012).
24. Z. Xiao, Exciton binding energy in spherical quantum dots in a magnetic field // J. Appl. Phys. 86(8), p. 4509 (1999).
25. J. Planelles, J. Diaz, J. Climente, W. Jaskolski, Semiconductor nanocrystals in a magnetic field // Phys. Rev. B, 65, 245302 (2002).
26. W. Smythe, Static and Dynamic Electricity. McGraw-Hill Book Company, Inc. New York-Toronto-London, p. 140-141, 1950.
27. V. Boichuk, R. Kubay, I. Bilynskii, Influence of the image potential on the energy spectrum of electron in complex spherical microcrystal CdS/β-HgS/H2O // J. Phys. Studies, 3(2), p. 187-191 (1999).
28. A. Lorke, R. Luyken, A. Govorov, J. Kotthaus, J. Garsia, P. Petroff, Spectroscopy of nanoscopic semiconductor rings // Phys. Rev. Lett. 84(10), p. 2223-2226 (2000).
F = 0 F = 20 kV/cm F = 50 kV/cm
B = 0, � EMBED Equation.3 ��� � � �
B = 0, � EMBED Equation.3 ��� � � �
B = 40 T, � EMBED Equation.3 ��� � � �
B = 40 T, � EMBED Equation.3 ��� � � �
Fig. 3. Distributions of probability densities for electron localization in SL with r1 = 4 nm, r2 = 12 nm for the quantum states with � EMBED Equation.3 ���, � EMBED Equation.3 ��� at B = 0, 40 T and F = 0, 20, 50 kV/cm.
© 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
7
110
y
100
y
110
y
100
y
110
y
_1451733039.unknown
_1453301640.unknown
_1453303062.unknown
_1459847215.unknown
_1459847238.unknown
_1453303068.unknown
_1453302064.unknown
_1453302491.unknown
_1453302841.unknown
_1453302787.unknown
_1453302480.unknown
_1453301667.unknown
_1453301758.unknown
_1451733209.unknown
_1451734005.unknown
_1451737558.unknown
_1451749139.unknown
_1451749408.unknown
_1451737605.unknown
_1451737618.unknown
_1451737564.unknown
_1451737434.unknown
_1451737549.unknown
_1451734187.unknown
_1451733280.unknown
_1451733292.unknown
_1451733532.bin
_1451733288.unknown
_1451733219.unknown
_1451733233.unknown
_1451733215.unknown
_1451733188.unknown
_1451733201.unknown
_1451733205.unknown
_1451733195.unknown
_1451733046.unknown
_1451733164.unknown
_1451733042.unknown
_1451732879.unknown
_1451732928.unknown
_1451732995.unknown
_1451733001.unknown
_1451732991.unknown
_1451732916.unknown
_1451732925.unknown
_1451732909.unknown
_1451732740.unknown
_1451732793.unknown
_1451732868.unknown
_1451732789.unknown
_1451732671.unknown
_1451732701.unknown
_1451732152.unknown
_1451732633.unknown
_1442305008.unknown
|