Electron energy spectrum and oscillator strengths of quantum transitions in double quantum ring nanostructure driven by electric field
The effect of homogeneous electric field on the energy spectrum, wave functions of electron and oscillator strengths of intra-band quantum transitions in a double cylindrical quantum ring (GaAs/AlxGa1−xAs) is studied within the approximations of effective mass and rectangular potentials. The calcu...
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Інститут фізики конденсованих систем НАН України
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irk-123456789-1574672019-06-22T01:27:00Z Electron energy spectrum and oscillator strengths of quantum transitions in double quantum ring nanostructure driven by electric field Makhanets, O.M. Gutsul, V.I. Kuchak, A.I. The effect of homogeneous electric field on the energy spectrum, wave functions of electron and oscillator strengths of intra-band quantum transitions in a double cylindrical quantum ring (GaAs/AlxGa1−xAs) is studied within the approximations of effective mass and rectangular potentials. The calculations are performed using the method of expansion of quasiparticle wave function over a complete set of cylindrical wave functions obtained as exact solutions of Schrödinger equation for an electron in a nanostructure without electric field. It is shown that the electric field essentially affects the electron localization in the rings of a nanostructure. Herein, the electron energies and oscillator strengths of intra-band quantum transitions non-monotonously depend on the intensity of electric field. У моделi ефективних мас та прямокутних потенцiалiв дослiджено вплив однорiдного електричного поля на енергетичний спектр, хвильовi функцiї електрона та сили осциляторiв внутрiшньозонних квантових переходiв у подвiйних напiвпровiдникових (GaAs/AlxGa1−xAs) цилiндричних квантових кiльцях. Розрахунки виконанi методом розкладу хвильових функцiй квазiчастинки за повним набором цилiндричних хвильових функцiй, отриманих як точний розв’язок рiвняння Шредiнгера для електрона в наноструктурi за вiдсутностi електричного поля. Показано, що електричне поле суттєво впливає на локалiзацiю електрона у системi нанокiлець. При цьому як енергiї електрона, так i сили осциляторiв внутрiшньозонних квантових переходiв немонотонно залежать вiд величини напруженостi електричного поля. 2018 Article Electron energy spectrum and oscillator strengths of quantum transitions in double quantum ring nanostructure driven by electric field / O.M. Makhanets, V.I. Gutsul, A.I. Kuchak // Condensed Matter Physics. — 2018. — Т. 21, № 4. — С. 43704: 1–9. — Бібліогр.: 16 назв. — англ. 1607-324X PACS: 73.21.La, 78.67.H DOI:10.5488/CMP.21.43704 arXiv:1812.08551 http://dspace.nbuv.gov.ua/handle/123456789/157467 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 homogeneous electric field on the energy spectrum, wave functions of electron and oscillator
strengths of intra-band quantum transitions in a double cylindrical quantum ring (GaAs/AlxGa1−xAs) is studied
within the approximations of effective mass and rectangular potentials. The calculations are performed using
the method of expansion of quasiparticle wave function over a complete set of cylindrical wave functions obtained as exact solutions of Schrödinger equation for an electron in a nanostructure without electric field. It is
shown that the electric field essentially affects the electron localization in the rings of a nanostructure. Herein,
the electron energies and oscillator strengths of intra-band quantum transitions non-monotonously depend on
the intensity of electric field. |
| format |
Article |
| author |
Makhanets, O.M. Gutsul, V.I. Kuchak, A.I. |
| spellingShingle |
Makhanets, O.M. Gutsul, V.I. Kuchak, A.I. Electron energy spectrum and oscillator strengths of quantum transitions in double quantum ring nanostructure driven by electric field Condensed Matter Physics |
| author_facet |
Makhanets, O.M. Gutsul, V.I. Kuchak, A.I. |
| author_sort |
Makhanets, O.M. |
| title |
Electron energy spectrum and oscillator strengths of quantum transitions in double quantum ring nanostructure driven by electric field |
| title_short |
Electron energy spectrum and oscillator strengths of quantum transitions in double quantum ring nanostructure driven by electric field |
| title_full |
Electron energy spectrum and oscillator strengths of quantum transitions in double quantum ring nanostructure driven by electric field |
| title_fullStr |
Electron energy spectrum and oscillator strengths of quantum transitions in double quantum ring nanostructure driven by electric field |
| title_full_unstemmed |
Electron energy spectrum and oscillator strengths of quantum transitions in double quantum ring nanostructure driven by electric field |
| title_sort |
electron energy spectrum and oscillator strengths of quantum transitions in double quantum ring nanostructure driven by electric field |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157467 |
| citation_txt |
Electron energy spectrum and oscillator strengths of quantum transitions in double quantum ring nanostructure driven by electric field / O.M. Makhanets, V.I. Gutsul, A.I. Kuchak // Condensed Matter Physics. — 2018. — Т. 21, № 4. — С. 43704: 1–9. — Бібліогр.: 16 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT makhanetsom electronenergyspectrumandoscillatorstrengthsofquantumtransitionsindoublequantumringnanostructuredrivenbyelectricfield AT gutsulvi electronenergyspectrumandoscillatorstrengthsofquantumtransitionsindoublequantumringnanostructuredrivenbyelectricfield AT kuchakai electronenergyspectrumandoscillatorstrengthsofquantumtransitionsindoublequantumringnanostructuredrivenbyelectricfield |
| first_indexed |
2025-07-14T09:53:41Z |
| last_indexed |
2025-07-14T09:53:41Z |
| _version_ |
1837615621961940992 |
| fulltext |
Condensed Matter Physics, 2018, Vol. 21, No 4, 43704: 1–9
DOI: 10.5488/CMP.21.43704
http://www.icmp.lviv.ua/journal
Electron energy spectrum and oscillator strengths of
quantum transitions in double quantum ring
nanostructure driven by electric field
O.M. Makhanets∗, V.I. Gutsul, A.I. Kuchak
Yuriy Fedkovych Chernivtsi National University, 2 Kotsyubinsky St., 58012 Chernivtsi, Ukraine
Received July 31, 2018, in final form September 26, 2018
The effect of homogeneous electric field on the energy spectrum, wave functions of electron and oscillator
strengths of intra-band quantum transitions in a double cylindrical quantum ring (GaAs/AlxGa1−xAs) is studiedwithin the approximations of effective mass and rectangular potentials. The calculations are performed using
the method of expansion of quasiparticle wave function over a complete set of cylindrical wave functions ob-
tained as exact solutions of Schrödinger equation for an electron in a nanostructure without electric field. It is
shown that the electric field essentially affects the electron localization in the rings of a nanostructure. Herein,
the electron energies and oscillator strengths of intra-band quantum transitions non-monotonously depend on
the intensity of electric field.
Key words: nanoring, electron, energy spectrum, oscillator strength, electric field
PACS: 73.21.La, 78.67.Hc
1. Introduction
Multilayered semiconductor nanostructures are studied both theoretically and experimentally for
quite a long time. Unique properties of quasiparticles in such structures allow us to use them as the basic
elements of modern nanoelectronic devices, such as tunnel diodes, lasers and detectors [1–3].
Semiconductor quantum rings occupy a separate place among various types of nanostructures. As a
rule, they have cylindrical symmetry as well as quantum wires [4]. However, unlike the latter, their height
is finite and can be of several nanometers. Therefore, the current of charge carriers in such nanostructures
will be confined in all three dimensions and, in this respect, they are similar to quantum dots. Modern
experimental possibilities allow one to grow nanoheterostructures with double cylindrical quantum rings
on the basis of GaAs/AlxGa1−xAs semiconductors [5–7].
Cylindrical semiconductor quantum rings are intensively investigated theoretically [8–14]. Changing
the geometric sizes of the rings one can affect the energy spectra of quasiparticles and obtain the necessary
optical properties. The external fields also essentially influence the spectra. In [8, 9], the influence of a
magnetic field on the energy spectrum of the electron and on the oscillator strengths of its intra-band
quantum transitions in GaAs/AlxGa1−xAs rings was studied. It was shown that the electron energies and
the oscillator strength of intra-band quantum transitions non-monotonously depend on the induction of
the magnetic field. Besides, there was observed an anti-crossing of energy levels of the same symmetry
over the magnetic quantum number (the Aaronov-Bohm effect) and brightly expressed maxima and
minima in the dependences of oscillator strengths on induction.
In the papers [10–12], the authors investigated the effect of a homogeneous electric field on the
optical properties of quantum nanorings using the model of a parabolic potential. The wave function of
∗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.
43704-1
https://doi.org/10.5488/CMP.21.43704
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
O.M. Makhanets, V.I. Gutsul, A.I. Kuchak
an electron in an electric field was written as an expansion over a complete set of its wave functions in
an infinitely deep potential well with a further solution of the corresponding secular equation.
In the proposed paper, we are going to study the structure similar to that of [10–12]. However, an
optimal model of confined potential is to be used with an orthonormal basis of cylindrical wave functions
obtained in the model of finite potential. The oscillator strengths of intra-band quantum transitions as
functions of the electric field intensity in a double quantum ring GaAs/AlxGa1−xAs nanostructure are
analyzed.
2. Theory of electron energy spectrum and oscillator strengths of intra-
band quantum transitions in double quantum ring nanostructure
driven by electric field
The nanostructure consisting of two concentric rings (quantumwells GaAs) separated by a concentric
and tunnel transparent ring AlxGa1−xAs of the width ∆ is studied. The heights of three constructing parts
are L. The inner and outer radii of the first ring are ρ0 and ρ1, its width is h1, while those of the second
ring are ρ2 and ρ3, respectively, its width is h2. The cross-section by the plane z = 0 and potential energy
scheme of the structure is shown in figure 1. A vector of electric field intensity ®F is directed along Ox
axis.
According to the symmetry considerations, the further calculations are performed in a cylindrical
coordinate system (ρ, ϕ, z) with Oz axis directed along the axial axis of the rings. The electron effective
masses are fixed in all parts of the structure
µ(®r) =
{
µ0 , |z | > L/2 or |z | 6 L/2 and 0 6 ρ 6 ρ0 , ρ1 6 ρ 6 ρ2 , ρ > ρ3 ,
µ1 , |z | 6 L/2 and ρ0 < ρ < ρ1 , ρ2 < ρ 6 ρ3.
(1)
0
1
2
“0” “1” “2” “3”
GaAs
GaAs
U
0
U0
AlGaAs
AlGaAs
3
“4”
F
0
ρ ρ ρ ρ
1 2 3
h h1 2∆
0
e
x
y
ϕ
Figure 1. Cross-section of nanostructure, at height L = 5 nm, by plane z = 0 and potential energy profile.
43704-2
Electron energy spectrum and oscillator strengths
In order to study the electron energy spectrum, there is solved the Schrödinger equation
ĤΨ(ρ, ϕ, z) = EΨ(ρ, ϕ, z) (2)
with Hamiltonian
Ĥ =
1
2µ(®r)
[
−~2
(
∂2
∂ρ2 +
1
ρ
∂
∂ρ
+
1
ρ2
∂2
∂ϕ2 +
∂2
∂z2
)]
−
~2
2µ(®r)
∂2
∂z2 +U(®r) − |e|Fρ cos ϕ, (3)
where e is the electron charge, F is the magnitude of the electric field intensity, U(®r) is the potential of
size quantization.
Taking into account that the electric field does not influence the energy spectrum of the electron
moving along Oz axis and that the electron is mostly located in the quantum wells weakly penetrating
into the barriers, the potential energy U(®r) is conveniently written as a sum
U(®r) = U(z) +U(ρ), (4)
where
U(z) =
{
U0 , |z | > L/2,
0, |z | 6 L/2, U(ρ) =
{
U0 , 0 6 ρ 6 ρ0 , ρ1 6 ρ 6 ρ2 , ρ > ρ3 ,
0, ρ0 < ρ < ρ1 , ρ2 < ρ 6 ρ3.
(5)
In this case, z variable is separated in a Schrödinger equation with Hamiltonian (3), and the wave function
can be written as follows:
Ψ (®r) = Φ(ρ, ϕ) f (z). (6)
The Schrödinger equation for the electron moving along Oz axis is easily solved [15]. The wave
functions f (z) are obtained in the form
f (z) =
{
A(+) cos(k0z),
A(−) sin(k0z),
0 6 z 6 L/2,
B exp(−k1z), z > L/2.
(7)
Using the condition of continuity of the wave function f (z) and its density of current at the interface
z = L/2 together with the normality condition, the unknown coefficients (A(±), B) are found and the
dispersion equations too:
k0
µ0
tan
(
k0
L
2
)
=
k1
µ1
,
k0
µ0
cot
(
k0
L
2
)
= −
k1
µ1
, (8)
where k0 =
√
2µ0Enz /~
2, k1 =
√
2µ1(U0 − Enz )/~
2. The energy spectrum (Enz ) of the electron moving
along Oz axis is obtained from equations (8) with quantum number nz numerating their solutions.
If there is no electric field (F = 0), the Schrödinger equation with Hamiltonian (3) is solved exactly
Φ
0
nρm
(ρ, ϕ) =
1
√
2π
Rnρm(ρ)e
imϕ . (9)
Here, nρ and m are the radial and magnetic quantum number, respectively, and the radial wave functions
are written as follows:
R(i)nρm(ρ) = A(i)nρm j(i)m (χρ) + B(i)nρmn(i)m (χρ), i = 0, 1, 2, 3, 4, (10)
j(i)m (χρ) =
{
Im(χ0ρ), i = 0, 2, 4,
Jm(χ1ρ), i = 1, 3, (11)
n(i)m (χρ) =
{
Km(χ0ρ), i = 0, 2, 4,
Nm(χ1ρ), i = 1, 3, (12)
43704-3
O.M. Makhanets, V.I. Gutsul, A.I. Kuchak
where Jm, Nm are cylindrical Bessel functions of the first and second kind, Im,Km aremodified cylindrical
Bessel functions of the first and second kind, χ0 =
√
2µ0(U0 − E0
nρm)/~
2, χ1 =
√
2µ1E0
nρm/~
2.
All unknown coefficients A(i)nρm, B(i)nρm (hence, the wave functions) and electron energies E0
nρm
are
obtained from the conditions of continuity of wave functions (10)–(12) and their densities of currents at
the interfaces of nanostructure and normality condition for the radial wave function. As far as the wave
function should be finite at ρ = 0 and ρ→∞, it means that the coefficients B(0)nρm = 0, A(4)nρm = 0.
If the structure is driven by an outer electric field, the Schrödinger equation with Hamiltonian (3)
cannot be solved analytically. In order to find the electron spectrum at F , 0, the unknownwave functions
are written as an expansion over a complete set of wave functions (9)
Φn(ρ, ϕ) =
1
√
2π
∑
nρ
∑
m
cnnρmRnρm(ρ)e
imϕ . (13)
Setting the expansion (13) into Schrödinger equation, the secular equation is obtained��Hnρm,n
′
ρm
′ − Enδnρ,n′ρδm,m′
�� = 0, (14)
where the matrix elements Hnρm,n
′
ρm
′ are of the form:
Hnρm,n
′
ρm
′ = Enρmδnρ,n′ρδm,m′ +
(
δm′,m+1 + δm′,m−1
) eF
2
∞∫
0
Rnρm(ρ)Rn′ρm
′(ρ)ρ2dρ. (15)
We should note that, as it is clear from (13) and (14), the new states of electron at its transversal
movement are characterized by only one quantum number n.
Thus, the problem of the energy spectrum En andwave functionsΦn(ρ, ϕ) is reduced to the calculation
of eigenvalues and eigenvectors of the obtained matrix. Hence, the complete wave functions Ψnnz (®r) (6)
of an electron and its energy Ennz = En + Enz become known. They make it possible to evaluate the
oscillator strengths of intra-band optical quantum transitions using the formula from [16]
Fn′n′z
nnz ∼ (En′n′z − Ennz )
��Mn′n′z
nnz
��2, (16)
where
Mn′n′z
nnz =
〈
n′n′z
��√µ(ρ)eρ cos ϕ
��nnz
〉
(17)
is the dipole momentum of the transition.
3. Analysis of the results
The electron energies and its oscillator strengths of intra-band quantum transitions are studied as
functions of the electric field intensity (F) for the double quantum ring GaAs/Al0.4Ga0.6As nanostructure
with the physical parameters µ0 = 0.063m0, µ1 = 0.096m0,U0 = 297meV (m0 is mass of pure electron in
vacuum); aGaAs = 5.65 Å is GaAs lattice constant. All spectral parameters were calculated at a quantum
number nz = 1, that is why it is omitted further.
In figure 2 the distribution of probability of electron (in ground state) location in nanostructure
|Φ1(ρ, ϕ)|
2 ρ is shown at L = 5 nm, ρ0 = 5aGaAs, h1 = 18aGaS, ∆ = 3aGaS, h2 = 17aGaS and at different
intensities of electric field: F = 0, 1, 1.5, 2.5 MV/m. It is clear that an increasing intensity changes the
location of electron in nanostructure. If F = 0, it is located in the inner ring with the width h1, while the
angular distribution of probability is uniform. When intensity increases, the electron, in the ground state,
tunnels from the inner ring into the outer ring in such a way that at F = 2.5 MV/m it completely locates
into the outer ring of the width h2. Herein, its angular distribution essentially changes. The obtained
result is in good qualitative agreement with the results of paper [14]. If the nanostructure studied is driven
by a homogeneous magnetic field with the induction B directed along Oz axis, then, an increasing B (at
43704-4
Electron energy spectrum and oscillator strengths
Figure 2. Probability density of electron (in ground state) location in a nanostructure |Φ1(ρ, ϕ)|
2 ρ at
L = 5 nm, ρ0 = 5aGaAs, h1 = 18aGaS, ∆ = 3aGaS, h2 = 17aGaS and different electric field intensity:
F = 0, 1, 1.5, 2.5 MV/m.
F = 0) causes an increasing localization of the electron (in ground state) in the inner ring. Herein, the
angular distribution of probability remains uniform [9].
In figure 3 (a) the electron energy (En) as a function of electric field intensity (F) is shown at
L = 5 nm, ρ0 = 5aGaAs, h1 = 18aGaS, ∆ = 3aGaS, h2 = 17aGaS. The figure proves that the ground
state (|1〉) energy only decreases when F increases. However, the energies of excited states demonstrate
a different behaviour. In particular, the energy of the state |5〉 increases at first and then decreases.
The energies of the states |6〉 and |8〉 increase in the whole interval of the intensity studied. Generally,
an increase or a decrease of electron energies is determined by the location of the electron, in the
corresponding state, in the particular ring and by the character of angular distribution of probability with
respect to the direction of the electric field (for the ground state, figure 2).
Since a potential barrier, which separates the rings, is of a finite height and width, then, the electron
can tunnel from one quantum well into the other. This leads to a complicated and non-monotonous
dependence of the electron energy spectrum on the intensity of the electric field. In particular, there
43704-5
O.M. Makhanets, V.I. Gutsul, A.I. Kuchak
0,0 0,5 1,0 1,5 2,0 2,5
40
50
60
70
80
90
100
110
120
130
140
15
22
14
13
12
21
20
11
n =1, m=0
zn =1
2
3
4
5
7
9
8
6
n=1
F, MV/m
E
n
,
m
e
V
0,0 0,5 1,0 1,5 2,0 2,5
0,0
0,2
0,4
0,6
0,8
1,0
F, MV/m
W
4
W
n
W
3
a) b)
Figure 3. Electron energy En (a) and complete probability Wn of its location in the inner ring in quantum
states |3〉 and |4〉 (b) as functions of the electric field intensity (F) at L = 5 nm, ρ0 = 5aGaAs, h1 = 18aGaS,
∆ = 3aGaS, h2 = 17aGaS.
are observed anti-crossings of energy levels [for example, |1〉 and |2〉 at F ∼ 1.5 MV/m; |3〉 and |4〉
at F ∼ 1.4 MV/m; |5〉 and |6〉 at F ∼ 0.8 MV/m in figure 3 (a)]. Anti-crossings appear depending on
whether the electron, being in the neighbouring quantum states, is located in the outer or in the inner
ring. This is well illustrated in figure 3 (b), which shows the dependence of the complete probability
(Wn =
∫ρ1
ρ0
∫2π
0 |Φn(ρ, ϕ)|
2 ρdρdϕ) of electron location in the states |3〉 and |4〉 in the inner ring on the
electric field intensity at the same geometrical parameters of the structure.
Figure proves that if F = 0, the electron in both states |3〉 and |4〉 is located in the outer ring with a big
probability. When the intensity increases, the probability of electron location in the inner ring increases
0,0 0,5 1,0 1,5 2,0 2,5
0,0
0,2
0,4
0,6
0,8
1,0
F
n
'
n
,
a
.u
.
F
4
1
F, MV/m
F
4
3
F
4
2
F
3
2
F
3
1
F
2
1
Figure 4. Oscillator strengths of intra-band quantum transitions as functions of the electric field inten-
sity (F) at L = 5 nm, ρ0 = 5aGaAs, h1 = 18aGaS, ∆ = 3aGaS, h2 = 17aGaS.
43704-6
Electron energy spectrum and oscillator strengths
for the state |3〉 and, at first, almost does not change for the state |4〉. At F ∼ 1.1 MV/m, W3 approaches
the maximum and rapidly decreases while W4 rapidly increases. In the vicinity of F ∼ 1.4 MV/m, these
probabilities become equal and an exchange of localization of the electron between the rings occurs for
these states. Thus, the anti-crossing of energies E3 and E4 as functions of F is observed [figure 3 (a)].
We should note that such an effect is absent in a single nano-ring with one potential well. Moreover, it
should be mentioned that quite similar series of levels, with respect to n quantum number, will occur at
nz = 2, 3, . . . , but they will be located in the high-energy region of the spectrum.
The capability of an electron, in different states, to be located in the inner (h1) or the outer (h2) ring
causes a complicated and non-monotonous dependence of oscillator strengths of intra-band quantum
transitions on the intensity F with brightly expressed maxima and minima (figure 4). Herein, it turns out
that such a non-monotonous behaviour of Fn′
n is, mainly, determined by overlapping wave functions of
the electron in the corresponding quantum states.
Let us observe, for example, the transition of an electron from the ground state |1〉 to the state |2〉
(curve F2
1 in figure 4) and the dependence of the probability density of the electron location in a
nanostructure in these states (|Φn(ρ, ϕ)|
2 ρ) at F = 1.5 MV/m, where the oscillator strength is maximal,
and at F = 2.5 MV/m, where it is minimal (figure 5). Figure shows that at F = 1.5 MV/m, the electron
E1=22.2 meV, F=1.5 MV/m E1=9.8 meV, F=2.5 MV/m
E2=29.3 meV, F=1.5 MV/m E2=22.8 meV, F=2.5 MV/m
Figure 5. Contour of probability distribution of the electron location in a nanostructure in the states |1〉
and |2〉 at F = 1.5 MV/m and F = 2.5 MV/m, L = 5 nm, ρ0 = 5aGaAs, h1 = 18aGaS, ∆ = 3aGaS,
h2 = 17aGaS.
43704-7
O.M. Makhanets, V.I. Gutsul, A.I. Kuchak
in the states |1〉 and |2〉 is mainly localized in the outer ring. Herein, the overlapping of the respective
wave functions in formula (17) is essential and the oscillator strength is maximal though the difference of
energies (E2−E1) is not big. When the electric field intensity increases, the difference of energies E2−E1
becomes bigger [figure 3 (a)]. However, the electron in the state |2〉, under the influence of an electric
field, begins to tunnel into the inner ring, which causes a smaller overlapping of the wave functions in
(17) and, hence, a smaller oscillator strength of the respective transition. At F = 2.5 MV/m, the electron
in the state |1〉 is completely localized in the outer ring, while in the state |2〉—mainly in the inner ring
(figure 5). The wave functions of the corresponding states weakly overlap and the oscillator strength of
the respective transition is small.
Quite similarly, due to the changes of location of the electron in the space of tunnel-connected quantum
rings driven by an electric field, one can explain the non-monotonous behaviour of the oscillator strengths
of quantum transitions between the other states.
Finally, we should note that an increasing nanostructure height L causes a decrease of the electron
energy Enz at nz = 1. It tends to zero in the limit case (limL→∞ Enz=1 = 0) and the spectrum Enρm
completely corresponds to that in the structure consisting of two cylindrical nanotubes with an axial
quasimomentum kz = 0.
4. Conclusions
1. The electron energy spectrum and oscillator strengths of intra-band quantum transitions in a
double quantum ring GaAs/AlxGa1−xAs nanostructure are studied as functions of the electric field
intensity (F) within the approximations of effective mass and rectangular potentials.
2. To calculate the energy spectrum and probability densities of the electron location in nano-rings
driven by electric field, the stationary Schrödinger equation is solved using the method of expansion
of a quasiparticle wave function over a complete set of wave functions in a nanostructure without
the electric field.
3. It is shown that the electric field essentially changes the distribution of probability of the electron
location in a nanostructure. Thus, if the electron, in ground state, is located in the inner ring, then
at an increasing electric field intensity, the quasiparticle tunnels into the outer ring.
4. The electron energies and oscillator strengths of intra-band quantum transitions non-monotonously
depend on the intensity of the electric field. One can observe anti-crossings of energy levels and
brightly expressed minima and maxima in oscillator strengths as functions of F. Such a behaviour
is caused by the change of the location of an electron, in different quantum states, in the space of
two rings due to the varying electric field intensity.
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Енергетичний спектр електрона та сили осциляторiв
внутрiшньозонних квантових переходiв у подвiйному
нанокiльцi в електричному полi
О.М.Маханець, В.I. Гуцул, А.I. Кучак
Чернiвецький нацiональний унiверситет iменiЮрiя Федьковича,
вул. Коцюбинського, 2, 58012 Чернiвцi, Україна
У моделi ефективних мас та прямокутних потенцiалiв дослiджено вплив однорiдного електричного поля
на енергетичний спектр, хвильовi функцiї електрона та сили осциляторiв внутрiшньозонних квантових
переходiв у подвiйних напiвпровiдникових (GaAs/AlxGa1−xAs) цилiндричних квантових кiльцях. Розра-хунки виконанi методом розкладу хвильових функцiй квазiчастинки за повним набором цилiндричних
хвильових функцiй, отриманих як точний розв’язок рiвняння Шредiнгера для електрона в наноструктурi
за вiдсутностi електричного поля. Показано, що електричне поле суттєво впливає на локалiзацiю еле-
ктрона у системi нанокiлець. При цьому як енергiї електрона, так i сили осциляторiв внутрiшньозонних
квантових переходiв немонотонно залежать вiд величини напруженостi електричного поля.
Ключовi слова: нанокiльце, електрон, енергетичний спектр, сила осцилятора, електричне поле
43704-9
https://doi.org/10.21272/jnep.9(6).06015
https://doi.org/10.1016/j.optmat.2016.07.024
https://doi.org/10.1016/j.physe.2015.11.016
https://doi.org/10.1016/j.physe.2015.01.044
https://doi.org/10.1103/PhysRevB.97.041304
https://doi.org/10.1016/j.spmi.2013.01.011
https://doi.org/10.1016/S0921-4526(00)00471-3
Introduction
Theory of electron energy spectrum and oscillator strengths of intra-band quantum transitions in double quantum ring nanostructure driven by electric field
Analysis of the results
Conclusions
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