Spectral parameters of electron in multi-shell open semiconductor nanotube
The spectral parameters (resonance energies and resonance widths) of electron in multi-shell open cylindrical semiconductor nanotube are theoretically investigated within the effective mass and rectangular potentials model by using the S-matrix approach. These parameters as functions of the nanotube...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2015
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irk-123456789-1211572017-06-14T03:05:56Z Spectral parameters of electron in multi-shell open semiconductor nanotube Makhanets, O.M. Kuchak, А.І. Gutsul, V.I. Voitsekhivska, O.M. The spectral parameters (resonance energies and resonance widths) of electron in multi-shell open cylindrical semiconductor nanotube are theoretically investigated within the effective mass and rectangular potentials model by using the S-matrix approach. These parameters as functions of the nanotube thickness and axial quasimomentum are analyzed for the nanostructure composed of GaAs and AlxGa₁₋xAs semiconductors 2015 Article Spectral parameters of electron in multi-shell open semiconductor nanotube / O.M. Makhanets, А.І. Kuchak, V.I. Gutsul, O.M. Voitsekhivska // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 1. — С. 57-62. — Бібліогр.: 15 назв. — англ. 1560-8034 DOI: 10.15407/spqeo18.01.057 PACS 73.21-Hb, 73.21.-b, 78.67.Ch http://dspace.nbuv.gov.ua/handle/123456789/121157 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The spectral parameters (resonance energies and resonance widths) of electron in multi-shell open cylindrical semiconductor nanotube are theoretically investigated within the effective mass and rectangular potentials model by using the S-matrix approach. These parameters as functions of the nanotube thickness and axial quasimomentum are analyzed for the nanostructure composed of GaAs and AlxGa₁₋xAs semiconductors |
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Makhanets, O.M. Kuchak, А.І. Gutsul, V.I. Voitsekhivska, O.M. |
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Makhanets, O.M. Kuchak, А.І. Gutsul, V.I. Voitsekhivska, O.M. Spectral parameters of electron in multi-shell open semiconductor nanotube Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Makhanets, O.M. Kuchak, А.І. Gutsul, V.I. Voitsekhivska, O.M. |
| author_sort |
Makhanets, O.M. |
| title |
Spectral parameters of electron in multi-shell open semiconductor nanotube |
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Spectral parameters of electron in multi-shell open semiconductor nanotube |
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Spectral parameters of electron in multi-shell open semiconductor nanotube |
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Spectral parameters of electron in multi-shell open semiconductor nanotube |
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Spectral parameters of electron in multi-shell open semiconductor nanotube |
| title_sort |
spectral parameters of electron in multi-shell open semiconductor nanotube |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/121157 |
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Spectral parameters of electron in multi-shell open semiconductor nanotube / O.M. Makhanets, А.І. Kuchak, V.I. Gutsul, O.M. Voitsekhivska // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2015. — Т. 18, № 1. — С. 57-62. — Бібліогр.: 15 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT makhanetsom spectralparametersofelectroninmultishellopensemiconductornanotube AT kuchakaí spectralparametersofelectroninmultishellopensemiconductornanotube AT gutsulvi spectralparametersofelectroninmultishellopensemiconductornanotube AT voitsekhivskaom spectralparametersofelectroninmultishellopensemiconductornanotube |
| first_indexed |
2025-07-08T19:18:17Z |
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2025-07-08T19:18:17Z |
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1837107595511332864 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 57-62.
doi: 10.15407/ spqeo18.01.057
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
57
PACS 73.21-Hb, 73.21.-b, 78.67.Ch
Spectral parameters of electron
in multi-shell open semiconductor nanotube
O.M. Makhanets, А.І. Kuchak, V.I. Gutsul, O.M. Voitsekhivska
Chernivtsi National University,
2, Kotsiubynsky str., 58012 Chernivtsi, Ukraine
Phone: (0372)244-816, e-mail: ktf@chnu.edu.ua
Abstract. The spectral parameters (resonance energies and resonance widths) of electron
in multi-shell open cylindrical semiconductor nanotube are theoretically investigated
within the effective mass and rectangular potentials model by using the S-matrix
approach. These parameters as functions of the nanotube thickness and axial quasi-
momentum are analyzed for the nanostructure composed of GaAs and AlxGa1–xAs
semiconductors.
Keywords: nanotube, quasi-stationary state, resonance energy, resonance width.
Manuscript received 23.12.14; revised version received 12.01.15; accepted for
publication 00.00.15; published online 00.00.15.
1. Introduction
The multi-shell semiconductor nanotubes have been
recently studied both theoretically and experimentally
[1-7]. The unique properties of quasi-particles (electrons,
excitons and so on) in these nanostructures allow using
them as basic elements for the devices of modern
nanoelectronics [8-10].
The authors of ref. [3] have been grown the arrays
of semiconductor nanotubes consisting of the sequence
of GaAs and AlxGa1–xAs nanoshells by using the method
of molecular beam epitaxy. This nanostructure was
covered by rather thick shell of GaAs in order to avoid
AlxGa1–xAs oxidizing.
The multi-shell nanotube under study is an open
one, because the potential energy of electron in GaAs is
smaller than that in AlxGa1–xAs. In open nanotubes, on
the contrary to the closed ones, the quasi-particles can
tunnel through the potential barrier into the outer
medium, creating an additional channel of energy
relaxation for the quasi-particles excited in the quantum
well. It is clear that the quasi-particles energy spectra in
these nanosystems are quasi-stationary and characterized
by the resonance energies and resonance widths.
The theory of exciton and phonon stationary spectra
together with the theory of electron- and exciton-phonon
interaction well correlating to the experimental data and
general physical considerations is already developed for
the closed cylindrical and hexagonal nanotubes [5-7].
The quasi-stationary spectra of electrons, holes and
excitons were theoretically studied for the spherically-
symmetric quantum dots and single cylindrical quantum
wires [11-15]. In this paper, we present the theoretical
study of electron quasi-stationary spectrum in multi-shell
open cylindrical semiconductor nanotube. The
dependences of resonance energies and resonance widths
on the nanotube thickness and axial quasi-momentum of
GaAs and Al0.4Ga0.6As are obtained and analyzed.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 57-62.
doi: 10.15407/ spqeo18.01.057
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
58
2. Electron energy spectrum and wave functions
in multi-shell cylindrical nanotube
The multi-shell open cylindrical semiconductor
nanotube consisting of inner wire with the radius 0
(“0” – GaAs), barrier-shell with the thickness Δ1 (“1” –
AlxGa1–xAs), nanotube with the thickness h (“2” – GaAs)
and one more barrier-shell with the thickness Δ2 (“3” –
AlxGa1–xAs) embedded into the outer structure (“4” –
GaAs) is studied. The cross-section and energy scheme
of this nanostructure is presented in Fig. 1. The potential
energy of electron in outer medium is smaller than that
in barrier-shells, thus the system is an open one, and the
electron energy spectrum is quasi-stationary.
Considering the symmetry of the system, all further
calculations were performed in the cylindrical coordinate
system (, , z) with Oz axis directed along the axial
axis of nanotube. The effective masses and potential
energies of electron are fixed as
.,,
,,,0,0
)(
,)(
32100
3210
1
0
U
U
(1)
The stationary Schroedinger equation
),,(),,(),,( zEzzH
(2)
Fig. 1. Cross-section and energy scheme of multi-shell
nanotube.
with Hamiltonian
)(
)(2
1
)(
1
)(2
,,
2
22
2
22
U
z
zH
(3)
was solved in order to obtain the electron energy
spectrum and wave functions. Considering the
symmetry, the latter ( z,, ) is written as
imikz
mkmk eeR
L
r )(
2
1
)(
. (4)
Here, k – the axial quasi-momentum; m = 0, ±1,
±2, …– magnetic quantum number; L – the effective
length of electron movement along the axial axis of
nanotube. Substituting (4) and (3) into the equation (2),
the variables (, , z) are separated and the equation for
the radial wave functions )(mkR is obtained
.0)(
)()()(
1
2
2
2
22
kmREU
km
(5)
This equation is exactly solved for each part of
nanostructure. The solutions are written as
,
,)(
,)(
,)(
,)(
,)(
)(
3
00
)4()4(
32
1
)3(
1
)3()3(
21
0
)2(
0
)2()2(
10
1
)1(
1
)1()1(
0
00
)0()0(
kHESkHΑR
ikHESikHΑR
kHESkHΑR
ikHESikHΑR
kHkHΑR
R
mmkmmmk
mmkmmmk
mmkmmmk
mmkmmmk
mmmmk
mk
(6)
where
2
2
0
0
2
kEk
, 2
02
1
1 )(
2
kEUk
, (7)
mm HH , are the Hankel functions of the whole order,
Smk (E) is the scattering matrix.
Using the condition of wave function and its
density of current continuity at all nanotube interfaces
pp
p
km
p
p
km
p
p
p
kmp
p
km
RR
RR
)(1)(1
)()(
)1(
1
)(
)1()(
(p = 0, 1, 2, 3)
(8)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 57-62.
doi: 10.15407/ spqeo18.01.057
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
59
together with the normality condition
)()()( 00
0
*
00
kkdRR
kmkkmk
, (9)
we obtained all unknown coefficients ( )(i
mΑ , )(i
kmS ) and
analytical expression for the S-matrix
,
)()()()(
)()()()(
3031
)3(
01
01
3031
)3(
3031
)3(
3031
)3(
10
01
kHki
k
ki
kHki
kHkikHki
k
ki
S
mkmmkm
mkmmkm
km
(10)
where
)()()( 31
)3(
3131
)3( kiHSkiHki mmkmkm ,
,
)()()()(
)()()()(
2131
)2(
01
10
2120
)2(
2120
)2(
2120
)2(
01
10
)3(
ikHki
ik
k
ikHk
ikHkkiHk
ik
k
S
mkmmkm
mkmmkm
mk
)()()( 20
)2(
2020
)2( kHSkHk mmkmkm ,
,
)()()()(
)()()()(
1011
)1(
10
01
1011
)1(
1011
)1(
1011
)1(
01
01
)2(
kHki
k
ki
kHki
kHkikHki
k
ki
S
mkmmkm
mkmmkm
km
(11)
)()()( 11
)1(
1111
)1( ikHSikHik mmkmkm ,
.
)()()()(
)()()()(
0100
01
10
0100
01000100
01
10
)1(
ikHkJ
ik
k
ikHkJ
ikHkJkiHkJ
ik
k
S
mmmm
mmmm
mk
According to the general theory, the poles of S-
matrix in the complex energy plane define the electron
resonance energies )(kE
mn
and resonance widths
)(k
mn
or its life times )(/)( kk mnmn
in quasi-
stationary states
2/)()()(
~
kikEkE mnmnmn
. (12)
The quantum number nρ denotes S-matrix poles
at fixed m.
Finally, the formulas (10)-(12) definitely determine
the spectral parameters (resonance energies and widths)
of electron in multi-shell open cylindrical semiconductor
nanotube.
3. Analysis of results
The numeric calculation of electron spectral parameters
was performed for the multi-shell nanotube composed of
GaAs/AlxGa1−xAs semiconductors with the parameters:
μ0 = 0.063m0, μ1 = (0.063+0.083x)
m0 (m0 is the mass of
free electron in vacuum), U0 = 0.57(1.155x + 0.37x
2
)
and
the lattice parameter aGaAs = 5.65Å.
In Fig. 2, the resonance energies mnE
(а) and
widths mn
(b) are presented as functions of the
nanotube thickness h at k = 0, m = 0, a fixed radius of the
inner wire ρ0 = 10aGaAs and barrier-shells thicknesses Δ1 =
Δ2 = 4aGaAs. Figure proves that both the resonance
energies and widths non-monotonously depend on the
nanotube thickness. Herein, the resonance energies as
functions of h are the sequence of horizontal and
decaying plots, while at the functions of resonance
widths the clearly visible maxima are observed for small
Γ. Horizontal plots in Fig. 2а correspond to the states
where electron is located in the inner wire with a higher
probability. In the states corresponding to the decaying
plots, electron is mainly located in nanotube. Its
increasing thickness causes the decrease in the resonance
energy.
The dependences of resonance widths on h
(Fig. 2b) are explained in the following way. Let us
observe for example the ground electron state (nρ = 1,
m = 0): at h = 0 the nanotube is absent and electron is
localized in the inner wire, in order to transit into the
outer medium it has to tunnel through the rather strong
potential barrier with the thickness Δ1 + Δ2. Thus, the
resonance width Γ10 of energy level is small. Electron
penetrates into nanotube more and more when h
increases. Now, it has to tunnel through the only one
barrier-shell with the thickness Δ2 to transit into the
outer medium. Consequently, the resonance width of
energy level increases, approaching its maximum.
Further, it only decays because the resonance energy
becomes smaller and the height of the potential barrier
effectively increases.
The function Γ20 does not look the same as Γ10
(Fig. 2b). One can see that for small h, the electron
energy of the second quasi-stationary state rapidly
decreases when h increases (Fig. 2а). Electron is
localized in nanotube, and its resonance width is rather
large but rapidly decays due to the bigger effective
height of potential barrier (Fig. 2b). It approaches the
minimal magnitude at the nanotube thickness changing
from h ≈ 15aGaAs up to h ≈ 25aGaAs , when electron is
located in the inner wire and its energy almost does not
depend on h (Fig. 2а). The electron energy E20 decreases
when h increases further. The quasi-particle is localized
in nanotube and function Γ20 is similar to Γ10: at first it
increases and then decreases only.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 57-62.
doi: 10.15407/ spqeo18.01.057
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
60
10 20 30 40 50 60
0
50
100
150
200
250
a
60
50
40
30
20
10
h, a
GaAs
E
n
m
,
m
eV
10 20 30 40 50 60
0
1
2
3
b
6050403020
10
n
m
,m
eV
h, a
GaAs
Fig. 2. Resonance energies mnE
(a) and resonance widths mn
(b) as functions of the nanotube thickness h at k = 0, m = 0,
fixed radius of the inner wire ρ0 = 10aGaAs and barrier-shells thicknesses Δ1 = Δ2 = 4aGaAs.
The non-monotonous behavior of resonance widths
of other energy states is also explained by different
location of electron in the space of multi-shell
nanostructure.
It should be noted that, contrary to the single open
quantum wires where the higher energy level (over nρ
quantum number) at fixed m has a wider resonance
width [13], in the case of the studied nanotube one can
see that, for example, when the nanotube thickness
varies from h ≈ 12aGaAs up to h ≈ 26aGaAs , E20 > E10,
however Γ20 < Γ10. This peculiarity of spectral
parameters gives opportunity to produce multi-shell
nanotubes with the inverse occupied levels, which can
be used as active elements of semiconductor lasers.
In Fig. 3, the electron resonance energies (Figs. 3а
and 3c) and widths (Figs. 3b and 3d) are presented as
functions of the axial quasi-momentum k at m = 0, fixed
radius of the inner wire 0 = 10aGaAs , thicknesses of
barrier-shells Δ1 = Δ2 = 4aGaAs , nanotube thickness h =
25aGaAs and two different concentrations x = 0.2 and
x = 0.4. In Figs. 3а and 3c, one can also see the
dispersion laws of electron energy in bulk
semiconductor crystals GaAs and AlxGa1−xAs, being the
composition materials of studied nanotube,
0
22
GaAs
2
k
E
,
1
22
0AsGaAl
2x1x
k
UE
.
Figs. 3а and 3c show that these curves separate the
plane (E, k) into three regions: I – the region where the
S-matrix has not any poles and, respectively, there are
not any states here; II – the region where the S-matrix
has several poles in complex energy plane defining the
resonance energy and width as functions of the quasi-
momentum k; III – the region with )(AsGaAl x1x
kEE
where electron is in stationary states of continuum
energy spectrum.
Dependences of resonance energies in all the states
(Figs. 3а and 3c, region ІІ) are well approximated by the
law
mn
mnmn
k
EkE
2
)0()(
22
,
and electron effective masses
0
2
2
2
)(11
k
mn
mn k
kE
,
as it is clear from Table, are close to its effective mass in
bulk GaAs. For the higher n, the effective mass weakly
increases due to the higher energy of electron, more and
more tunneling into the barrier-shell where the effective
mass is higher.
All the bands of resonance energies are
characterized by the maximal longitudinal quasi-
momentum
mn
k
defined from the expression
1
22
0
2
)(
mn
mn
k
UkE
. (13)
The higher magnitudes of maximal quasi-
momentum correspond to the lower energy levels.
Besides,
mn
k
increases for the higher concentrations x
(Figs. 3а and 3b).
Table. Effective mass of electron moving along the
nanotube axis.
x 0.2 0.4
mn 10 20 30 10 20 30 40
mn
0.0642 0.0649 0.0656 0.0638 0.0646 0.052 0.067
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 57-62.
doi: 10.15407/ spqeo18.01.057
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
61
0,00 0,05 0,10 0,15 0,20
100
200
300
400
500
600
700
800
a
0.60.4Al Ga AsE
E
GaAs
co
nt
in
uu
m
III
II
I
30
20
10
x=0.2
k (/a)
E
n
m
,
m
eV
0,00 0,05 0,10 0,15 0,20
5
10
15
b
x=0.2
30
20
10
k(/a)
n
m
,m
eV
0,00 0,05 0,10 0,15 0,20
100
200
300
400
500
600
700
800
k
40
c
0.60.4Al Ga AsE
n
m
III
II
I
co
nt
in
uu
m
E
GaAs
40
30
20
10
k (/a)
E
,
m
eV x=0.4
0,00 0,05 0,10 0,15 0,20
5
10
15
d
x=0.4
40
30
20
10
n
m
,m
eV
k(/a)
Fig. 3. Electron resonance energies (а, c) and resonance widths (b, d) as functions of axial quasi-momentum k at m = 0, fixed
radius of the inner wire ρ0 = 10aGaAs, thicknesses of barrier-shells Δ1 = Δ2 = 4aGaAs , nanotube thickness h = 25aGaAs and
concentrations x = 0.2 and x = 0.4.
Figs. 3b and 3d prove that the resonance widths of
all the quasi-stationary states rapidly increase for the
higher longitudinal quasi-momentum. It is clear, because
the increasing quasi-momentum causes the decrease of
the effective height of potential barrier for the electron
tunneling into the out of the barrier space with a higher
probability. This feature of nanotubes can be used in
devices – separators of electrons over their velocities,
eliminating the rapid ones through the barrier and
leaving the slow ones.
4. Conclusions
The spectral parameters of electron in multi-shell open
cylindrical semiconductor nanotube were investigated
within the effective mass and rectangular potentials
model by using the S-matrix approach. These
parameters as functions of the nanotube thickness and
axial quasi-momentum have been analyzed for the
nanostructure composed of GaAs and AlxGa1−xAs
semiconductors.
It has been shown that both the resonance energies
and widths non-monotonously depend on the nanotube
thickness. For the resonance energies, the non-
monotonous character is pronounced as alternating
horizontal and decaying plots, while the functions of
resonance widths display clearly observed maxima and
minima. This behavior of spectral parameters is caused
by the complicated character of probability density of
electron location in the space of nanostructure.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2015. V. 18, N 1. P. 57-62.
doi: 10.15407/ spqeo18.01.057
© 2015, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
62
It has been ascertained that all minibands of
resonance energies are characterized by the effective
masses that are close to the electron effective mass in
bulk GaAs and by the maximal longitudinal quasi-
momentum. The electron resonance energies and width
monotonously increase when the axial quasi-momentum
increases.
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