Electron energy spectrum in core-shell elliptic quantum wire
The electron energy spectrum in core-shell elliptic quantum wire and elliptic semiconductor nanotubes are investigated within the effective mass approximation. The solution of Schrodinger equation based on the Mathieu functions is obtained in elliptic coordinates. The dependences of the electron siz...
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| Цитувати: | Electron energy spectrum in core-shell elliptic quantum wire / V. Holovatsky, V. Gutsul // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 61-67. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1180602017-05-29T03:04:18Z Electron energy spectrum in core-shell elliptic quantum wire Holovatsky, V. Gutsul, V. The electron energy spectrum in core-shell elliptic quantum wire and elliptic semiconductor nanotubes are investigated within the effective mass approximation. The solution of Schrodinger equation based on the Mathieu functions is obtained in elliptic coordinates. The dependences of the electron size quantization spectrum on the size and shape of the core-shell nanowire and nanotube are calculated. It is shown that the ellipticity of a quantum wire leads to the break of degeneration of quasiparticle energy spectrum. The dependences of the energy of odd and even electron states on the ratio between semiaxes are of a nonmonotonous character. The anticrosing effects are observed at the dependences of electron energy spectrum on the transversal size of the core-shell nanowire. В рамках наближення ефективних мас дослiджено енергетичний спектр електрона в складному елiптичному квантовому дротi та напiвпровiдниковiй елiптичнiй нанотрубцi. В елiптичних координатах на основi функцiй Мат’є отримано розв’язок рiвняння Шредiнгера. Розраховано залежностi спектру розмiрного квантування електрона вiд геометричних розмiрiв та форми складного квантового дроту та нанотрубки. Показано, що елiптичнiсть квантового дроту приводить до розщеплення енергетичних рiвнiв. Залежнiсть парних та непарних електронних енергетичних станiв вiд спiввiдношення пiвосей елiпса має немонотонний характер. На залежностi енергетичного спектру електрона від поперечних розмiрiв складного квантового дроту спостерiгається ефект розштовхування рiвнiв. 2007 Article Electron energy spectrum in core-shell elliptic quantum wire / V. Holovatsky, V. Gutsul // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 61-67. — Бібліогр.: 9 назв. — англ. 1607-324X PACS: 73.21.Fg, 73.21.Hb, 73.21.La DOI:10.5488/CMP.10.1.61 http://dspace.nbuv.gov.ua/handle/123456789/118060 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The electron energy spectrum in core-shell elliptic quantum wire and elliptic semiconductor nanotubes are investigated within the effective mass approximation. The solution of Schrodinger equation based on the Mathieu functions is obtained in elliptic coordinates. The dependences of the electron size quantization spectrum on the size and shape of the core-shell nanowire and nanotube are calculated. It is shown that the ellipticity of a quantum wire leads to the break of degeneration of quasiparticle energy spectrum. The dependences of the energy of odd and even electron states on the ratio between semiaxes are of a nonmonotonous character.
The anticrosing effects are observed at the dependences of electron energy spectrum on the transversal size of the core-shell nanowire. |
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Holovatsky, V. Gutsul, V. |
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Holovatsky, V. Gutsul, V. Electron energy spectrum in core-shell elliptic quantum wire Condensed Matter Physics |
| author_facet |
Holovatsky, V. Gutsul, V. |
| author_sort |
Holovatsky, V. |
| title |
Electron energy spectrum in core-shell elliptic quantum wire |
| title_short |
Electron energy spectrum in core-shell elliptic quantum wire |
| title_full |
Electron energy spectrum in core-shell elliptic quantum wire |
| title_fullStr |
Electron energy spectrum in core-shell elliptic quantum wire |
| title_full_unstemmed |
Electron energy spectrum in core-shell elliptic quantum wire |
| title_sort |
electron energy spectrum in core-shell elliptic quantum wire |
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Інститут фізики конденсованих систем НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/118060 |
| citation_txt |
Electron energy spectrum in core-shell elliptic quantum wire / V. Holovatsky, V. Gutsul // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 61-67. — Бібліогр.: 9 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT holovatskyv electronenergyspectrumincoreshellellipticquantumwire AT gutsulv electronenergyspectrumincoreshellellipticquantumwire |
| first_indexed |
2025-07-08T13:17:52Z |
| last_indexed |
2025-07-08T13:17:52Z |
| _version_ |
1837084891011874816 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 1(49), pp. 61–67
Electron energy spectrum in core-shell elliptic quantum
wire
V.Holovatsky, V.Gutsul
Fedkovych Chernivtsi National University, 2, Kotsyubinskoho St., Chernivtsi, 58012, Ukraine∗
Received September 4, 2006, in final form January 4, 2007
The electron energy spectrum in core-shell elliptic quantum wire and elliptic semiconductor nanotubes are in-
vestigated within the effective mass approximation. The solution of Schrodinger equation based on the Math-
ieu functions is obtained in elliptic coordinates. The dependences of the electron size quantization spectrum
on the size and shape of the core-shell nanowire and nanotube are calculated. It is shown that the ellipticity
of a quantum wire leads to the break of degeneration of quasiparticle energy spectrum. The dependences of
the energy of odd and even electron states on the ratio between semiaxes are of a nonmonotonous character.
The anticrosing effects are observed at the dependences of electron energy spectrum on the transversal size
of the core-shell nanowire.
Key words: elliptic quantum wire, elliptic nanotube, energy spectrum
PACS: 73.21.Fg, 73.21.Hb, 73.21.La
1. Introduction
Recently semiconductor nanowires have attracted great attention due to their small diameter
that leads to one-dimensional electron systems as well as due to the possibility of being used as the
building blocks in nanoscale electronics and photonics. The advantages of semiconductor nanowires
compared to carbon nanotubes lie in controllability of conduction types and in the formation
of heterojunctions and superlattices. These lead to a broad range of applications such as single
electron transistors, light emitting or laser diodes and chemical sensors. Such potential capabilities
can be greatly enhanced by introducing heterostructures within semiconductor nanowires. Modern
technologies permit to grow the nanowires of different shape and structure. Among them there are
the cylindrical, hexagonal and elliptical nanowires [1,2], multishell heterostructure nanowires [3]
and semiconductor nanotubes [4,5].
In order to apply the multishell heterostructure nanowires into different optical of the compo-
nents devices it is necessary to investigate the quasiparticles energy dependences on the shape and
geometrical size of the components of multishell nanowires.
The theory of quasiparticles spectra in complicated cylindrical semiconductor heterostructures
was developed much earlier than the growth technology of multishell heterostructure nanowires [6].
Within the framework of this theory it is impossible to describe the properties of one-dimensional
nanoheterostructures with the elliptic cross-section. These nanostructures can be grown using
selective-area metalorganic vapor phase epitaxy or under the action of one axis pressure. The
particular interest to the elliptic nanowires is also caused by the fact that the size quantization
problem for a carrier with anisotropic effective mass parameters in a cylindrical well (as it is shown
in [7]) is to be equivalent to that of a carrier with some isotropic effective mass in an elliptic well.
For the infinitely deep elliptic potential well model which can be used for the carriers in semicon-
ductor nanowires embedded into dielectric matrices, there exists an exact solution of Schrodinger
equation based on the Mathieu functions [8]. The approximated solutions can be obtained for the
case of potential elliptic well of a finite depth [8,9]. The comparison of the results of numerical
calculations within finite-difference method on a 400 × 400 uniform rectangular grid [9] with the
∗E-mail: theorphys@chnu.cv.ua
c© V.Holovatsky, V.Gutsul 61
V.Holovatsky, V.Gutsul
method of separation of radial and angular elliptic variables [8] proves that the second method can
be applied to the ratio between the ellipse semiaxes a/b < 3.
The size quantization problem for a carrier in core-shell elliptic semiconductor nanowire or el-
liptic nanotube has not been solved yet. Therefore, the peculiarities of the electron energy spectrum
in such nanostructures are investigated in this paper.
2. Hamiltonian of the system. Solutions of Schrodinger equation
2.1. Single elliptic nanowire
The elliptic quantum wire (EQW) GaAs (“0”) embedded into the semiconductor or dielectric
matrix (“1”) is under study. The coordinate system is chosen in such a way that Oz axis is directed
along the wire axis. Electron potential energy and effective mass in Cartesian coordinates have the
form
µ(x, y) =
{
µ0, x2/a2 + y2/b2 6 1,
µ1, x2/a2 + y2/b2 > 1 ,
(1)
U(x, y) =
{
0, x2/a2 + y2/b2 6 1,
V, x2/a2 + y2/b2 > 1 ,
(2)
where a and b are the ellipse semiaxes.
Electron can perform a free movement in the direction along the quantum wire. Due to the
tunnel effect, the wave function of quasiparticle can penetrate into the medium “1” and part of the
energy caused by the longitudinal movement of electron is Ez = ~
2k2
z/2µ∗, where µ∗ is averaged
effective mass, equal to µ0 in the case of infinite potential barrier. The energy caused by the
transversal movement of quasiparticle is found from the Schrodinger equation
−~
2
2
∇ 1
µ(x, y)
∇Ψ(x, y) + U(x, y)Ψ (x, y) = EΨ(x, y) . (3)
Setting (1), (2) into (3), the following equations are obtained for every medium
∆Ψ(i) (x, y) + k2
i Ψ(i) (x, y) = 0,(i = 0, 1), (4)
where k2
i = 2µi~
−2(E − Vi), V0 = 0, V1 = V .
Taking into account the elliptic symmetry, equation (4) is convenient to be solved in elliptic
coordinates (ξ, η, z) bound to the Cartesian by the relationships
x = f cosh ξ cos η, 0 6 ξ < ∞
y = f sinh ξ sin η, 0 6 η < 2π
z = z, −∞ < z < +∞
, (5)
where f =
√
a2 − b2 is the focus distance, ξ is the radial coordinate and η is the angular coordi-
nate. The radial coordinate (ξ) is defined by the ratio between ellipse semiwidths (tanh ξ = b/a).
Transiting in equation (4) from Cartesian coordinates to the elliptic, one can get
[
∂2
∂ξ2
+
∂2
∂η2
+
f2k2
i
2
(cosh 2ξ − cos 2η)
]
Ψ(i)(ξ, η) = 0, (i = 0, 1). (6)
The wave function permits the separation of the variables only for elliptic quantum well with
impenetrable walls. Then, the quantum states of quasiparticle are characterized by the definite
value of quantum number m
Ψm(ξ, η) = Rm(ξ)θm(η). (7)
Using the variable separation for elliptic quantum well of the finite depth with a/b < 3 we can
obtain the size quantization spectra of carriers with high accuracy. R
(i)
m (ξ) is the radial and θ
(i)
m (η)
is the angular part of electron wave functions in ith medium, satisfying Mathieu equations
∂2θ(i)
m (η)/∂η2 + (c − 2qi cos 2η)θ(i)
m (η) = 0, (8)
∂2R(i)
m (ξ)/∂ξ2 − (c − 2qi cosh 2ξ)R(i)
m (ξ) = 0, (9)
62
Electron energy spectrum in core-shell elliptic quantum wire
where qi = f2k2
i /4, c is the separating constant. The solution of radial equation (9) is the linear
combination of even (e) and odd (o) modified Mathieu functions of first and second type
Re
m(q, ξ) =
{
Ae
mJem(q, ξ) + Be
mNem(q, ξ), q > 0,
Ae
mIem(q, ξ) + Be
mKem(q, ξ), q < 0,
(10)
Ro
m(q, ξ) =
{
Ao
mJom(q, ξ) + Bo
mNom(q, ξ), q > 0,
Ao
mIom(q, ξ) + Bo
mKom(q, ξ), q < 0,
(11)
where Ae
m, Ao
m, Be
m, Bo
m – the coefficients determined by fitting and normalizing conditions.
The energy spectrum of quasiparticle is obtained within the fitting conditions for the wave
function. In case of impenetrable walls for the elliptic quantum wire, the even and odd wave func-
tions at the surface of elliptic cylinder (ξ = ξ0) are equal to zero. Their radial parts would contain
only the functions Jem(q, ξ) and Jom(q, ξ) (Be
m = Bo
m = 0) in analogy to the cylindrical functions.
Thus, the energies of even and odd states in EQW are characterized by the fixed magnitude of
quantum number m and are defined from the equations
Jem(q, ξ)|ξ=ξ0
= 0, m = 0, 1, 2, . . . ; Jom(q, ξ)|ξ=ξ0
= 0, m = 1, 2, . . . . (12)
The values q
e(o)
nm = f2E
e(o)
nm µ0
/
2~
2, satisfying equations (12), determine the quasiparticle di-
screte energy levels E
e(o)
nm , where n=1, 2,. . . is quantum number fixed by the order of the root of
respective equations (12).
In the case of elliptic quantum well with finite depth, the wave functions satisfy the boundary
conditions
1
µ0
∂R
e(0)
m (q0, ξ)/∂ξ
R
e(0)
m (q0, ξ)
∣
∣
∣
∣
∣
ξ=ξ0
=
1
µ1
∂R
e(1)
m (q1, ξ)/∂ξ
R
e(1)
m (q1, ξ)
∣
∣
∣
∣
∣
ξ=ξ0
, (13)
1
µ0
∂R
o(0)
m (q0, ξ)/∂ξ
R
o(0)
m (q0, ξ)
∣
∣
∣
∣
∣
ξ=ξ0
=
1
µ1
∂R
o(1)
m (q1, ξ)/∂ξ
R
o(1)
m (q1, ξ)
∣
∣
∣
∣
∣
ξ=ξ0
. (14)
The finiteness conditions of the radial wave functions determine the coefficients B
e(0)
m =
B
o(0)
m = 0 and A
e(1)
m = A
o(1)
m = 0. Setting (10), (11) into (13), (14) the dispersion equations
are obtained for electron spectrum in elliptic nanowire with finite potential barrier.
2.2. Complicated elliptic quantum wire
The elliptic quantum thread (“0”) covered with the elliptic shell (“1”) embedded into the
dielectric matrix (“2”) is under study. It is assumed that electron cannot penetrate into the matrix
and its potential energy in Cartesian coordinates is as follows
U(x, y) =
V0, x2/a2
0 + y2/b2
0 6 1,
V1, x2/a2
0 + y2/b2
0 > 1 ∩ x2/a2
1 + y2/b2
1 6 1,
∞, x2/a2
1 + y2/b2
1 > 1,
(15)
where a0, b0 and a1, b1 are the semiaxis of inner and outer ellipse, respectively.
The wave function at ξ = ξ1 = arctanh(b1/a1) is equal to zero and at the media interface
ξ = ξ0 = arctanh(b0/a0) it satisfies the boundary condition (13), (14), which determines the
dispersion equation for the electron energy spectrum. For the case V1 > V0 there is obtained
J ′
m
(q0,ξ0)
µ0Jm(q0,ξ0)
− J ′
m
(q1,ξ0)Nm(q1,ξ1)−N ′
m
(q1,ξ0)Jm(q1,ξ1)
µ1[Jm(q1,ξ0)Nm(q1,ξ1)−Nm(q1,ξ0)Jm(q1,ξ1)]
= 0, q1 > 0,
J ′
m
(q0,ξ0)
µ0 Jm(q0,ξ0)
− I′
m
(q1,ξ0)Km(q1,ξ1)−K′
m
(q1,ξ0)Im(q1,ξ1)
µ1[Im(q1,ξ0)Km(q1,ξ1)−Km(q1,ξ0)Im(q1,ξ1)]
= 0, q1 < 0,
(16)
63
V.Holovatsky, V.Gutsul
where qi = f2k2
i /4, k2
i = 2µi~
−2(E
e(o)
nm − Vi).
The numerical calculations performed over equations (16) make it possible to obtain even and
odd electron energies E
e(o)
nm , n=1, 2, 3, . . . is quantum number fixing the order of the corresponding
root of dispersion equation. Such solutions are possible only for the complicated elliptic quantum
wires with interfaces between ellipses which have equal focus distances since this is demanded by
the elliptic coordinate system (5).
2.3. Elliptic semiconductor nanotube
The elliptic semiconductor nanotube embedded into the dielectric or semiconductor matrix is
under study. Potential energy of the quasiparticle in Cartesian coordinates is as follows
U(x, y) =
{
V, x2/a2
0 + y2/b2
0 6 1 ∪ x2/a2
1 + y2/b2
1 > 1 ,
0, x2/a2
0 + y2/b2
0 > 1 ∩ x2/a2
1 + y2/b2
1 6 1.
(17)
In the case of elliptic quantum well with infinite depth (V = ∞) the even and odd radial wave
functions (10), (11) are equal to zero at the interfaces ξ = ξ0 and ξ = ξ1. These conditions make
it possible to obtaine the exact electron energy spectrum in elliptic nanotube with arbitrary ratio
between semiaxes of inner and outer ellipses but with equal focuses.
In the case of the elliptic quantum well with finite depth, using the conditions (13), (14) of
wave function and density of probability current continuity for the even and odd states at the
both interfaces there is obtained the electron energy spectrum in nanowires embedded into a
semiconductor matrix.
3. Results and discussion
3.1. Elliptic nanowire GaAs
The computer calculations over equations (12) were performed using the value of the electron
effective mass in the bulk crystal GaAs: µ0 = 0.067m0, where m0 is the pure electron mass.
5 10 15
0
1
2
e
(3,1)
(3,0)
e
e
e
o
o
e
e
o
o
a/b
e
(
o
)
2b
2a
(2,4)
(2,1)
(2,4)
(2,0)
(1,7)
(1,1)
(1,7)
(1,0)
E
n
m
,
e
V
Figure 1. Dependence of electron energy spectrum in elliptic quantum wires GaAs with im-
penetrable walls on the ratio between semiaxes. Solid curves – even states, dash curves – odd
states.
Figure 1 presents the results obtained for the energies of an electron transversal movement in
elliptic nanowire GaAs with impenetrable walls depending on the ratio between ellipse semiaxes at
its constant square S = πab, corresponding to the square of the circle with radius R = 10nm. The
even states are shown by solid curves and the odd ones by dashed. Figure 1 proves that even and
odd states create the sets of the levels. Herein the energies of even states are located lower than
64
Electron energy spectrum in core-shell elliptic quantum wire
20 30 40 50
100
200
300
e,o
e,o
e,o
o
E
11 e
E
41
E
31
E
21
E
11
e
(
o
)
a
0
/b
01.52 1.25 1.16 1.11 1.08 1.06
E
n
m
,
m
e
V
a
0
/a
GaAs
f=15a
GaAs
; a=20a
GaAs
; m=1
0 5 10 15 20 25
100
200
300
400 e,o
E
13
E
12
e,o
o
E
11
eE
11
e
E
10
a
1
/b
11.155 1.0141.0211.0321.061
e
(
o
)
E
1
m
,
m
e
V
a/a
GaAs
a
0
=10a
GaAs
; f=5a
GaAs
Figure 2. Dependences of even (Ee
nm
) and odd
(Eo
nm
) energy states of electron in EQW at
m=1 (solid curves) and energies (Enm) in the
respective CQW (dashed curves) on a0.
Figure 3. Dependences of even (solid curves)
and odd (dashed curves) states of electron en-
ergies in core-shell elliptic quantum well at
n=1 on the thickness of elliptic shell.
the respective energies of odd states. At the ellipse semiaxes ratio a/b = 1, the electron energy
spectrum is degenerated. The ground electron state (the even one) is non-degenerated just like in
cylindrical nanowire.
3.2. Elliptic core-shell nanowire GaAs/AlxGa1−xAs/dielectric
In figure 2 the electron energies of even and odd states (solid curves) with kz = 0, m=1 are pre-
sented for the complicated elliptic quantum wire GaAs/AlxGa1−xAs as functions of the magnitude
of semiaxis a0 of the inner thread (GaAs) of quantum wire at the constant thickness of the shell
(AlxGa1−xAs) in the direction of OX axis (∆a=20aGaAs) and constant focus distance (f=15aGaAs)
of both confining elliptic cylinders (aGaAs=5,65 Å is the lattice constant of bulk crystal GaAs). The
analogous dependences for the similar complicated cylindrical quantum wire (CQW) (dash curves)
are shown for comparison. Herein, the radii r0 and r1 for CQW were determined by the condition
of equality of cross–section square of the inner thread and shell for the elliptic and cylindrical
quantum wire: r0 =
√
a0b0, r1 =
√
a1b1 .
From figure 2 it is clear that the energies of electron even states in EQW are always smaller
while those of the odd ones are bigger than the corresponding energies in CQW. Such a splitting is
big for the lower energy levels and decreases for the states with bigger main quantum number. This
is explained by the fact that the ratio a0/b0 is always bigger than a1/b1 (since the focus distance
of both ellipses are equal). Since the skip of potential energy at the inner interface is V=109 meV,
then the electron in the states with bigger energy “feels” this interface less. At the increase of a0,
the EQW elliptic cross-sections approach the circle (this is obvious from a0/b0 magnitudes shown
at the upper horizontal axis of the figure and the pictures of the quantum wire cross-sections),
the energies of even and odd electron states coincide and tend to the respective values in CQW.
Toned region in figure 2 corresponds to the energies lower than the potential barrier. Here, the
energy spectrum is monotonously decreasing. As for the energies higher than the potential barrier,
one can see the anticrossing effect because the nanosystem under research consists of two potential
wells, each of which has its own system of energy levels, the crossings between which are forbidden
for the states with the same quantum number m. Consequently, the dependence of electron energy
spectrum on a0 magnitude in the region higher than the potential barrier is of a nonmonotonous
character. In the vicinity of the energy level anticrossings, the quasiparticle changes its location in
nanosystem.
65
V.Holovatsky, V.Gutsul
Figure 3 shows the energy dependences of even Ee
1 m(solid curves) and odd Eo
1 m (dashed curves)
of electron states on the thickness (∆a) for the elliptic shell (AlxGa1−xAs) at a0=10aGaAs and
f=5aGaAs. Figure proves that these energies are monotonously decreasing. In the limit case (∆a=0)
there is obtained the EQW with impenetrable potential wells. When the thickness of the shell (∆a)
increases, the size quantization becomes weaker and potential energy of electron in the shell is finite,
since the electron in the lowest energy state is bound by the potential well of the central part of
the wire. At ∆a > 8aGaAs, the energy dependence of the ground electron state on the thickness of
the shell is saturating. The latter corresponds to the energy of electron in elliptic wire, embedded
into the massive media AlxGa1−xAs, which may be obtained from roots of equations (6), (7). The
energies of the excited electron states becoming closer to each other, tend to the bottom of the
potential well created by the elliptic shell. The magnitude of the splitting of even and odd energy
states becomes smaller with the increase of the shell thickness because the ellipticity of the outer
interface is decreasing (magnitude of the ratio between semiaxes (a1/b1) is shown at the upper
axis of the graphics).
3.3. Elliptic nanotube InP/InAs/InP
The computer calculations were performed using the following parameters of semiconductor
bulk crystals: µ0 = 0, 0795m0, µ1 = 0, 026m0, are electron effective masses in InP and InAs,
respectively, V = 0, 33 eV in the height of the potential barrier for the electron at the interface
InP/InAs, aInP=5,87Å, aInAs=6,06Å are the lattice constants.
0 5 10 15
50
100
150
200
2.2941.0
a
0
/b
0
1.5121.1551.032
e
(
o
)
E
n
m
,
m
e
V
E
e
13
E
o
13
E
e
12 E
o
12
E
e
11
E
o
11
E
e
10
a
0
=20a
InP
; a=5a
InAs
f/a
InP
Figure 4. Dependences of even Ee
n m
(solid curves) and odd Eo
n m
(dashed curves) states of
electron energies on the focus distance f at the constant magnitudes a0=20aInP, ∆A=5aInAs .
Figure 4 presents the dependence of electron energy spectrum in nanotube on f at a0 = 20aInP
and ∆a = 5aInAs (values of the semiaxes ratio a0/b0 are noted on the top of figure). In the figure
one can see that at f = 0 (nanotube with circle cross-section) the energies of even and odd states
coincide. The increase of focus distance causes the splitting of the energies of even and odd states
due to the increase of the anisotropy of inner and outer elliptic interfaces.
4. Conclusions
In the elliptic coordinate system based on the Mathieu functions within the framework of the
effective mass approximation, the energy spectra of electron are calculated: for the bare quantum
wire GaAs, nanowire GaAs covered with the shell AlxGa1−xAs with impenetrable walls for the
quasiparticle and elliptic nanotube InP/InAs/InP. The electron energy spectrum consists of a series
of energy levels corresponding to the even and odd electron states. The results of investigation of
electron energy spectrum in core-shell elliptic quantum wire and elliptic nanotube prove that the
splitting between the energies of even and odd states depends on the ellipticity of both media
66
Electron energy spectrum in core-shell elliptic quantum wire
interfaces. Herein, even a small deformation of cylindrical quantum wire or nanotube essentially
effects the electron energy spectrum.
Acknowledgements
This work was financially supported by MPZ PP “Kolos”.
References
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Спектри електрона у складному елiптичному квантовому
дротi
В.А.Головацький, В.I.Гуцул
Чернiвецький нацiональний унiверситет iм. Юрiя Федьковича вул. Коцюбинського 2, 58012, Чернiвцi
Отримано 4 вересня 2006 р., в остаточному виглядi – 4 сiчня 2007 р.
В рамках наближення ефективних мас дослiджено енергетичний спектр електрона в складному елi-
птичному квантовому дротi та напiвпровiдниковiй елiптичнiй нанотрубцi. В елiптичних координатах
на основi функцiй Мат’є отримано розв’язок рiвняння Шредiнгера. Розраховано залежностi спектру
розмiрного квантування електрона вiд геометричних розмiрiв та форми складного квантового дроту
та нанотрубки. Показано, що елiптичнiсть квантового дроту приводить до розщеплення енергети-
чних рiвнiв. Залежнiсть парних та непарних електронних енергетичних станiв вiд спiввiдношення
пiвосей елiпса має немонотонний характер. На залежностi енергетичного спектру електрона вiд
поперечних розмiрiв складного квантового дроту спостерiгається ефект розштовхування рiвнiв.
Ключовi слова: елiптичний квантовий дрiт, елiптична нанотрубка, енергетичний спектр
PACS: 73.21.Fg, 73.21.Hb, 73.21.La
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