Properties of a charge in the nanoheterosystems with quantum wells and barriers
In the paper a quasi-one-dimensional three-layer nanowire (NW) with an intermediate layer at the separation boundaries is described by the Kronig-Penney model with δ-function potentials. The distance between the last or supreme atoms of the intermediate layer is a parameter of the problem and ranges...
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irk-123456789-320452012-04-07T12:21:14Z Properties of a charge in the nanoheterosystems with quantum wells and barriers Boichuk, V.I. Bilynskyi, I.V. Shakleina, I.O. Holskyi, V.B. In the paper a quasi-one-dimensional three-layer nanowire (NW) with an intermediate layer at the separation boundaries is described by the Kronig-Penney model with δ-function potentials. The distance between the last or supreme atoms of the intermediate layer is a parameter of the problem and ranges from zero to two lattice parameters of crystals. A precise solution is obtained for the quantum-mechanical reflection coefficient R which makes it possible to determine the dependence of the factor on a wave vector, widths of an intermediate layer and a monolayer of the nanoheterostructure. The specific calculations are performed for the GaAs/AlAs/GaAs and AlAs/GaAs/AlAS nanowires. The comparison of the reflection coefficients in the envelope-function approximation and in the effective mass method is performed at small values of a wave vector. It is shown that similar results can be received using the refined procedure, originally proposed by Harrison, with appropriately taken parameters. For the cosinusoidal dependence of the energy on a wave vector which arises in the Kronig-Penney model within the framework of the S-matrix scattering method, we determined the binding states energy for a AlAs/GaAs/AlAs quantum well wire. 2010 Article Properties of a charge in the nanoheterosystems with quantum wells and barriers / V.I. Boichuk, I.V. Bilynskyi, I.O. Shakleina, V.B. Holskyi // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13701: 1-11. — Бібліогр.: 24 назв. — англ. 1607-324X PACS: 71.55.-i, 73.21.La, 79.60.Jv http://dspace.nbuv.gov.ua/handle/123456789/32045 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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English |
| description |
In the paper a quasi-one-dimensional three-layer nanowire (NW) with an intermediate layer at the separation boundaries is described by the Kronig-Penney model with δ-function potentials. The distance between the last or supreme atoms of the intermediate layer is a parameter of the problem and ranges from zero to two lattice parameters of crystals. A precise solution is obtained for the quantum-mechanical reflection coefficient R which makes it possible to determine the dependence of the factor on a wave vector, widths of an intermediate layer and a monolayer of the nanoheterostructure. The specific calculations are performed for the GaAs/AlAs/GaAs and AlAs/GaAs/AlAS nanowires. The comparison of the reflection coefficients in the envelope-function approximation and in the effective mass method is performed at small values of a wave vector. It is shown that similar results can be received using the refined procedure, originally proposed by Harrison, with appropriately taken parameters. For the cosinusoidal dependence of the energy on a wave vector which arises in the Kronig-Penney model within the framework of the S-matrix scattering method, we determined the binding states energy for a AlAs/GaAs/AlAs quantum well wire. |
| format |
Article |
| author |
Boichuk, V.I. Bilynskyi, I.V. Shakleina, I.O. Holskyi, V.B. |
| spellingShingle |
Boichuk, V.I. Bilynskyi, I.V. Shakleina, I.O. Holskyi, V.B. Properties of a charge in the nanoheterosystems with quantum wells and barriers Condensed Matter Physics |
| author_facet |
Boichuk, V.I. Bilynskyi, I.V. Shakleina, I.O. Holskyi, V.B. |
| author_sort |
Boichuk, V.I. |
| title |
Properties of a charge in the nanoheterosystems with quantum wells and barriers |
| title_short |
Properties of a charge in the nanoheterosystems with quantum wells and barriers |
| title_full |
Properties of a charge in the nanoheterosystems with quantum wells and barriers |
| title_fullStr |
Properties of a charge in the nanoheterosystems with quantum wells and barriers |
| title_full_unstemmed |
Properties of a charge in the nanoheterosystems with quantum wells and barriers |
| title_sort |
properties of a charge in the nanoheterosystems with quantum wells and barriers |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/32045 |
| citation_txt |
Properties of a charge in the nanoheterosystems with quantum wells and barriers / V.I. Boichuk, I.V. Bilynskyi, I.O. Shakleina, V.B. Holskyi // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13701: 1-11. — Бібліогр.: 24 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT boichukvi propertiesofachargeinthenanoheterosystemswithquantumwellsandbarriers AT bilynskyiiv propertiesofachargeinthenanoheterosystemswithquantumwellsandbarriers AT shakleinaio propertiesofachargeinthenanoheterosystemswithquantumwellsandbarriers AT holskyivb propertiesofachargeinthenanoheterosystemswithquantumwellsandbarriers |
| first_indexed |
2025-07-03T12:31:43Z |
| last_indexed |
2025-07-03T12:31:43Z |
| _version_ |
1836628998440878080 |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 1, 13701: 1–11
http://www.icmp.lviv.ua/journal
Properties of a charge in the nanoheterosystems with
quantum wells and barriers
V.I. Boichuk, I.V. Bilynskyi, I.O.Shakleina∗, V.B. Holskyi
Department of Theoretical Physics, Ivan Franko Drohobych State Pedagogical University,
24 Ivan Franko Str., 82100 Drohobych, Ukraine
Received December 11, 2008, in final form August 28, 2009
In the paper a quasi-one-dimensional three-layer nanowire (NW) with an intermediate layer at the separa-
tion boundaries is described by the Kronig-Penney model with δ–function potentials. The distance between
the last or supreme atoms of the intermediate layer is a parameter of the problem and ranges from zero to
two lattice parameters of crystals. A precise solution is obtained for the quantum-mechanical reflection co-
efficient R which makes it possible to determine the dependence of the factor on a wave vector, widths of
an intermediate layer and a monolayer of the nanoheterostructure. The specific calculations are performed
for the GaAs/AlAs/GaAs and AlAs/GaAs/AlAS nanowires. The comparison of the reflection coefficients in the
envelope-function approximation and in the effective mass method is performed at small values of a wave
vector. It is shown that similar results can be received using the refined procedure, originally proposed by
Harrison, with appropriately taken parameters. For the cosinusoidal dependence of the energy on a wave
vector which arises in the Kronig-Penney model within the framework of the S-matrix scattering method, we
determined the binding states energy for a AlAs/GaAs/AlAs quantum well wire.
Key words: heterostructure, intermediate layer, Kronig-Penney model, reflection coefficient, S-matrix
scattering method
PACS: 71.55.-i, 73.21.La, 79.60.Jv
1. Introduction
Recently various nanostructures have been intensively studied both experimentally and the-
oretically due to their interesting electric and optical properties. A number of works is devoted
to the heterostructures with interface planes GaAs/AlAs in which the layers of AlAs are placed
in crystals GaAs or vice versa [1–4]. Electronic properties of such heterostructures are associated
with electrons of the Γ-valley of the band spectrum. The problem of electron transmission through
a boundary of different crystals arises in heterojunction electronics and in transport properties of
polycrystalline materials [5–10]. In order to solve the problem precisely, it is necessary to know
the exact Bloch functions of constituent materials and the evanescent states near the interface. In
many works the Bloch functions are replaced with the envelope ones and the effective mass approx-
imation is used [5–9]. It is usual to assume that the envelope function and its derivatives need to
be continuous across the interface [10–13]. In principle, these conditions would follow from a valid
expression for the current operator in terms of the envelope functions. Such an expression would
depend on the proper form of the effective kinetic energy operator acting on the envelope functions
– as well as be correct in the vicinity of the interfaces where the band structure changes abruptly.
Although the expression of this operator was discussed in [5, 14–16], it has never been generally
derived from the first principles. Instructive results, however, have been obtained in theories based
on the Kane ~k ·~p model and tight binding approximation [14–16]. Harrison in [5] used the continuity
conditions of ξ Φ and (m ξ)−1 ∇Φ, where Φ is an envelope function, m(x) is an effective mass, ξ(x)
is a scalar parameter which is hopefully not too strongly dependent on the energy near the band
edge in each crystal. For one-dimensional crystals, the Harrison conditions in their most general
form (β Φ and α (∂Φ/∂x) are continuous) can be justified by the fact that the effective Schrödinger
∗
E-mail: ioshak@mail.ru
c© V.I. Boichuk, I.V. Bilynskyi, I.O. Shakleina, V.B. Holskyi 13701-1
http://www.icmp.lviv.ua/journal
V.I. Boichuk et al.
equations, defining the envelope functions in each crystal, are second-order differential equations.
Hence, their solutions can be matched with two parameters, α and β. The continuity of current
density requires that α β m be continuous across the interface. Thus, the matching problem reduces
to one unknown parameter (ξ). In most publications on the problem of choice and substantiation
of boundary conditions, the authors studied the heterostructures containing one interface at which
all the values, and the distance between atoms in particular, change abruptly.
Recent developments in nanotechnologies have led to the publication of a number of experimen-
tal studies of physical properties of quantum wires. Significant progress in electronic microscopy
makes it possible to experimentally investigate the structure of grown quantum wires, observe the
size and form dependences of these quasi-one-dimensional nanoscale heterosystems on their growth
conditions. In particular, for the InGaAs, GaAs, ZnO quantum wires carrier concentrations and
mobilities with different external physical parameters are defined, and luminescence, electro- and
photoconductivity measurements are done [19, 20].
The fabrication of axial heterostructure in the direction of quantum wire axis by means of
alternating semiconductor materials with various energy gaps substantially widens the potential
application of nanowires. In this way resonant diodes and quantum transistors used in novel nano-
electronic devices are made. Hence, in plenty of experimental as well as theoretical works, electron,
hole,and exciton energy states are studied in composite quantum wires [10–13, 17–20].
The purpose of the present work is to develop the theory of reflection coefficient in an infinite
quasi-one-dimensional three-layer wire (with quantum well or quantum barrier) with account of an
intermediate layer at the interfaces in the Kronig-Penney model with δ -function potentials peaks.
In the intermediate layer the distance between adjacent atoms can range from zero to two lattice
parameters. We also investigate the dependence of reflection coefficient on the wave vector, the
thickness of an intermediate layer, width of a quantum well (a barrier) of the heterostructure and
define the bound state energies by the S-matrix scattering method.
2. A heterostructure model. General theory
A model of the quantum wire which consists of three regions is presented in figure 1. The left
“crystal” (labeled I) and the right one (labeled III) are identical half-infinite quasi-one-dimensional
crystals. The internal regions (labeled II) correspond to the quantum well or barrier of the het-
erosystem. All three crystalline regions represent arrays of δ-functions of strength Pi and periodicity
di (i = 1, 2). The size of δ-function peaks is different in the first (third) and second regions. In ad-
dition, one can have a potential discontinuity at the interface plane. This permits us to isolate the
band-structure effects from those associated with a potential step at the interface. An important
characteristic of the boundaries is the size b = b1+b2 of the intermediate layers containing “atoms”
of both crystals. This is a source of uncertainty even in our model problem because the reflection
coefficient and particle energy will be a strong function of b since the intermediate layers break the
translation symmetry. We shall not restrict ourselves to any specific value of b. For simplicity of
presentation we shall assume that the gap (the intermediate layer) is not too large: b < d1 + d2.
Figure 1. A heterosrtucture model.
Then we can always choose the crystal unit cells at the left interface so that they abut at
the point x = 0 without a gap or overlap. Then the Schrödinger equation of an electron for
13701-2
Properties of a charge in the nanoheterosystems with quantum wells and barriers
−b1 − d1 < x < b2 + d2 is written as
{
−~
2
2m0
d2
dx2
− Piδ(x ± bi) + Vi
}
Ψ
(i)
ki
= E Ψ
(i)
ki
, i = 1, 2, (1)
where m0 is a free electron mass. In equation (1) i = 1 is used in the first region, while i = 2
corresponds to the second region, thus V1 = 0, V2 = −V . Similar equations can be written for an
electron at the interface x = L.
The first and the third region are half-infinite, that is why the functions Ψ
(1)
k1
and Ψ
(3)
k1
are the
Bloch functions which satisfies the relation
Ψ
(1)
k1
(0) = eik1d1Ψ
(1)
k1
(−d1), Ψ
(3)
k1
(L) = e−ik1d1Ψ
(3)
k1
(L + d1). (2)
Similar relations can be written for the wave-function derivatives. Note that at the pole points
of the Dirac δ- function the wave-function derivative changes stepwise [17]. In particular
Ψ
(1)′
k1
(−b1 + 0) − Ψ
(1)′
k1
(−b1 − 0) = −
2m0P1
~2
Ψ
(1)
k1
(−b1) ,
Ψ
(3)′
k1
(L + b1 + 0) − Ψ
(3)′
k1
(L + b1 − 0) = −
2m0P1
~2
Ψ
(3)
k1
(L + b1). (3)
The use of translational properties of the wave functions makes it possible to obtain the dis-
persion equation of an electron in the first and third regions
cos(k1d1) = cos(q1d1) − U1
sin(q1d1)
q1d1
. (4)
where U1 = m0P1d1
~2 , q1 =
√
2m0E
~2 , k1 is an electron wave vector.
The total wave functions in the first and third regions are as follows:
Ψ1(x) = Ψ
(1)
k1
(x) + r Ψ
(1)
−k1
(x),
Ψ3(x) = t Ψ
(3)
k1
(x). (5)
In the vicinity of the interfaces the electron wave functions with a certain wave vector can be
written
Ψ
(1)
k1
(x) = Ak1
[
1 − e−i (q1+k1)d1
1 − e−i (q1−k1)d1
eiq1(x+b1) + e−iq1(x+b1) − (1 − e−i (q1+k1)d1)eiq1|x+b1|
]
,
Ψ
(3)
k1
(x) = Ak1
[
1 − e−i(q1+k1)d1
1 − e−i(q1−k1)d1
eiq1(x−L−b1) + e−iq1(x−L−b1) − (1 − e−i(q1+k1)d1)eiq1|x−L−b1|
]
. (6)
For other points of space the functions Ψ
(1)
k1
(x) and Ψ
(3)
k1
(x) can be found from the transcendent
conditions imposed on the wave function.
In the second region of the heterosystem it is also possible to find the wave function Ψ2(x)
which describes a particle. Its form is rather complicated and depends on the number of atoms N
of the region (see Appendix).
From the matching conditions of functions Ψ1(x), Ψ2(x) and Ψ3(x) at x = 0, x = L
Ψ1(0) = Ψ2(0), Ψ2(L) = Ψ3(L),
Ψ
′
1(x)
∣
∣
∣
x=0
= Ψ
′
2(x)
∣
∣
∣
x=0
, Ψ
′
2(x)
∣
∣
∣
x=L
= Ψ
′
3(x)
∣
∣
∣
x=L
(7)
one can define the reflection amplitude r. Once r is known, we can determine the coefficients of
reflection (R) and transmission (T ) of a charge in the heterostructure
R = |r|
2
, T = 1 − R. (8)
13701-3
V.I. Boichuk et al.
If the 2nd region of the heterostructure contains a great number of atoms (N � 1), it is
possible to neglect the absence of translation symmetry and use the Kronig-Penney model. Then
the wave function of the first cell in the second layer can be written in the form of (5) provided q1
is substituted with q2 =
√
2m0(E+V )
~2 , and k1 with k2 respectively.
The wave function of the last cell can be found using the translative property of the Bloch
function. The total wave function in this layer is represented by
Ψ2(x) = pΨ
(2)
k2
(x) + f Ψ
(2)
−k2
(x). (9)
Using the matching conditions of functions (7), one can find the expression for the reflection
coefficient amplitude r
r =
Ψ
(1)
k1
(0)
[
µΨ
(2)′
k2
(0) + Ψ
(2)′
−k2
(0)
]
− Ψ
(1)′
k1
(0)
[
µΨ
(2)
k2
(0) + Ψ
(2)
−k2
(0)
]
Ψ
(1)′
−k1
(0)
[
µΨ
(2)
k2
(0) + Ψ
(2)
−k2
(0)
]
− Ψ
(1)
−k1
(0)
[
µΨ
(2)′
k2
(0) + Ψ
(2)′
−k2
(0)
] , (10)
where the following notation is used:
µ =
Ψ
(2)′
−k2
(0)Ψ
(3)
k1
(L) − Ψ
(2)
−k2
(0)(d2)Ψ
(3)′
k1
(L)
Ψ
(2)
k2
(0)Ψ
(3)′
k1
(L) − Ψ
(2)′
k2
(0)Ψ
(3)
k1
(L)
e−2ik2L.
The reflection coefficient obtained due to (10) can be compared to its value from equation (7)
and expressions (D.5)–(D.8). Then we can establish the criteria of the use of the infinite crystal
approach in the 2nd layer of the heterostructure.
The Bloch wave functions of the crystals which form real heterostructures are unknown. There-
fore, as a rule, the envelope-function model is used. In our case the envelope function looks like
Φ(x) =
Φ1(x) = eik1x + r e−ik1x, x < 0,
Φ2(x) = A eik2x + B e−ik2x, 0 < x < L,
Φ3(x) = t eik1(x−L), x > L.
(11)
Functions Φi(x) (i = 1, 2, 3) represent linear combinations of plane-wave solutions of the
effective-mass Hamiltonians,
H(i) = E(i)
n −
~
2
2mi
∂2
∂x2
, i = 1, 2, 3 (12)
describing electronic motion near the band edges E
(i)
n in each crystal. Boundary conditions of
envelope functions are frequently based on the continuity condition of Φ and 1
m
∂ Φ
∂ x
[10–14, 19–21].
More general conditions which are imposed on envelope functions are used by Harrison [5]
β1 Φ1(0) = β2 Φ2(0),
β2 Φ2(L) = β1 Φ3(L),
α1
∂ Φ1
∂x
∣
∣
x=0
= α2
∂ Φ1
∂x
∣
∣
x=0
,
α2
∂ Φ2
∂x
∣
∣
x=L
= α1
∂ Φ3
∂x
∣
∣
x=L
. (13)
Parameters αi and βi (i = 1, 2) define physical conditions at the interface. They (or part of
them) are considered matching parameters of the problem.
Continuity of the flux (in the effective-mass approximation) in this case requires that
α1 β1 m1 = α2 β2 m2 . (14)
13701-4
Properties of a charge in the nanoheterosystems with quantum wells and barriers
Using (13) and (14), we determine the reflection coefficient for the heterostructure
R =
ξ2
1
ξ2
2
[
m2
2
k2
1
ξ2
2
m2
1
k2
2
ξ2
1
− 1
]2
sin2 k2L
4
m2
2
k2
1
m2
1
k2
2
+
ξ2
1
ξ2
2
[
m2
2
k2
1
ξ2
2
m2
1
k2
2
ξ2
1
− 1
]2
sin2 k2L
,
β2
1
β2
2
=
ξ1
ξ2
. (15)
The expressions of the wave function allow us to obtain the energies of the quantum wire. We
consider the composite nanowire which contains a quantum well in the second region. In this case
there should exist particle binding states. The energy of these states is obtained by the S-matrix
scattering method [22].
Using the S-matrix scattering method, we can express the particle wave function in the first
layer in the form
Ψ1(x) = A(Ψ
(1)
−k1
(x) − S(E)Ψ
(1)
k1
(x)). (16)
Using the matching conditions of functions (7), S(E) is written
S(E) = −r−1(E). (17)
In order to find discrete particle energy values in the heterostructure, it is necessary to find the
S-matrix poles.
3. Specific calculations. The analysis of the obtained results
We apply the present model to the study of GaAs/AlAs/GaAs and AlAs/GaAs/AlAS quantum
wire. For this purpose, once we know the effective masses, lattice parameters and a potential step,
it is necessary to determine parameters of the problem: P1, P2, d1, d2.
From dispersion relation (4) one defines extreme energies in the limit ki → 0. If in the het-
erosystem at the interface there are no additional fields, and the electron potential energy does not
change (V2 = 0), then for both crystals (i = 1, 2) the extreme energies are found from the following
equations:
tg
√
m0d2
i
2~2
En = −
Ui
√
m0d2
i
2~2 En
, n = 1, 3 , 5, . . . , (18)
sin
√
m0d2
i
2~2
En = 0, n = 2, 4, 6, . . . . (19)
One can also derive effective masses of odd and even energy bands
mn
m0
= −
2~
2 Ui
m0d2
i En
[
1 −
2Ui
m0d2
i
2~2 En + U2
i
]
, n = 1, 3, 5, . . . .
mn
m0
= −
Ui
π2n2
, n = 2, 4 , 6, . . . . (20)
From equation (4) we obtained the dispersion relation for crystals GaAs and AlAs (d1 = 5.653Ȧ,
m = 0.066m0−GaAs; d2 = 5.66Ȧ, m = 0.124m0−AlAs, V0 = 570 meV is a misalignment between
third bands). The spectrum under consideration is realized providing that
U1 =
~
2
m0P1d1
= −3.01, U2 =
~
2
m0P2d2
= − 4.79.
Having the necessary parameters, it is possible to determine the coefficient of reflection R or
transmission T .
Reflection and transmission of electrons in such NW’s essentially depend on the size of the
intermediate layer (boundary), that is b = b1 + b2, which results from the distortion of lattice
13701-5
V.I. Boichuk et al.
Figure 2. The electron reflection coefficient of a GaAs/AlAs/GaAs (a) and AlAs/GaA/AlAs
(b) three-layer system. U1 = −3.01, U2 = −4.79. Curve 1 corresponds to b = b1 + b2 = d2
(b1 = 0.1d2, b2 = 0.9d2). Curve 2 corresponds to b = b1 + b2 = 1.8d2 (b1 = 0.1d2, b2 = 1.7d2).
parameters near the interface of the heterostructure. The change of these values leads to the
change of local maxima in the dependence R = R(k1) as well as to their shift.
In figure 2 the dependence of the reflection coefficient R on the dimensionless wave vector of an
electron incident on a quantum well (barrier) at a varying size of the intermediate layer is shown
for the GaAs/AlAs/GaAs and AlAs/GaAs/AlAs quasi-one-dimensional three-layer heterosystems.
For the first heterostructure (figure 2 (a)), in contrast to the second system the dependence
R = R(k1) consists of two regions – scattering under the barrier and above it (for k1d1
π
< 0, 2
electron transmission is not practically observed). However, in general, the wave-vector dependence
of the reflection coefficient R = R(k1) for different sizes of the intermediate layer has identical
behavior for both heterostructure types. At small k1 the reflection coefficient is close to unity
and decreases as the wave vector rises. Each structure exhibits specific points at which perfect
transmission (R = 0) is observed. It is noteworthy that the number of maxima (and accordingly,
perfect transmission points) is larger for the GaAs/AlAs/GaAs nanowire in spite of the fact, that
a transmission interval of this system is much smaller than that of AlAs/GaAs/AlAs. The sizes of
the corresponding maxima differ little for both crystal types. At k1d1
π
= 1 the reflection coefficient
reaches its maximum possible value (R = 1).
For both heterostructures, the results R = R(k1) obtained by exact and approximated solutions
are compared. At large sizes of the second region R = R(k1) L (L > 4 d2) both expressions yield
qualitatively similar results. However, a better agreement is obtained for the GaAs/AlAs/GaAs
system.
It can be concluded that for composite NW’s in the limit of quantum well or barrier widths
larger than 4d2 an electron can be described by the Bloch function.
Let us analyze the reflection coefficient received within the envelope-function approach and the
effective mass approximation refined by the Harrison boundary conditions. We consider a special
case of the surface when the intermediate region containing atoms of both crystals is of the size
b = d1+d2
2 . In this case one can estimate the exact values of matching parameters ξi ≡ β2
i , i = 1, 2
through the crystal parameters and the electron energy at the bottom of the band [8]
ξ1,2 =
U1,2
(qn d1,2)2
m0
m1,2
, qn =
√
2m0En
~2
.
If in (21) one takes into account the expression for the effective mass (20), one can see that
for the higher-lying bands, when qn d1,2 � |U1,2|, values of ξi tend to unity, and formula (18)
is transferred into formula (15) of the reflection coefficient. It can be obtained by the envelope-
function approach
(
ξ1
ξ2
= 1
)
. For the heterosystem models under consideration the following exact
13701-6
Properties of a charge in the nanoheterosystems with quantum wells and barriers
values were obtained: ξ1 = 0, 91, ξ2 = 0, 768, which, in its turn, makes it possible to compare the
results received by the effective mass approximation with the exact solution.
In figure 3 (a) the reflection coefficient R = R(k1) for the AlAs/GaAs/AlAs heterosystem,
calculated according to the exact solution and effective mass approximation refined by Harrison
boundary conditions (15), is shown for different barrier sizes. One can see that in the limit of small
wave-vector values k1 → 0, where the effective-mass approach is applicable, a good qualitative
agreement of the results is observed. A further wave-vector increase leads to the shift of points of
perfect electron transmission (R = 0) in the direction of bigger k1 and a certain disagreement of
maxima.
Figure 3. The electron reflection coefficient of a AlAs/GaAs/AlAs (a) and AlAs/GaA/AlAs (b)
three-layer system. A solid curve stands for an accurate solution, dashed curve stands for the
effective-mass method specified by Harrison conditions. Curves 1 and 1′ correspond to L=8d2
(45.224 A); curves 2 and 2′ correspond to L=2d2 (11.306 A).
The calculations performed for the GaAs/AlAs/GaAs heterostructure (figure 3 (b)) have also
shown a quite good qualitative agreement of the results obtained by the effective-mass approxi-
mation with the Harrison matching conditions and the exact solution. In the limit k1 → 0, both
methods yield almost identical results. With a further wave vector increase, the divergence becomes
more essential. This can be assigned to the fact that the effective mass approximation works only
in the interval of small wave-vector values. Note that for the heterostructure under consideration,
in contrast to the structure AlAs/GaAs/AlAs, with an increase of the wave vector one can observe
the shift of perfect electron transmission points (R = 0) in the direction of smaller k1 received
in the effective mass approximation. It is accompanied by a more essential disagreement of max-
ima. However, it is necessary to mention that this divergence is much smaller than that when we
compare the exact solution and the results obtained in the effective-mass approximation in case of
ξ1,2 = 1.
As is seen from the plots, with the reduction of a quantum well (barrier) size the mean value
of the electron transmission coefficient increases which, in its turn, is expected due to physical
reasons.
Therefore, one can come to the conclusion that for the considered surface type, which corre-
sponds to the heterostructures with close lattice parameters, the Harrison matching conditions are
applicable and yield good results.
Using the S-matrix scattering method, the energy of binding states is determined for the
AlAs/GaAs/AlAs quantum well nanowire. According to this procedure, the state energy is set
by those energy values which are the S-matrix scattering poles. This problem is far from being
simple because of complex numerical calculations. An analytical expression of the S-matrix (espe-
cially of an exact expression of the wave function in the second layer) is rather complicated. Besides,
13701-7
V.I. Boichuk et al.
the values of the functions ki = ki(E) (i = 1, 2) are found from the transcendental equations for
wave vectors.
The results of calculations of the binding state energy in a quantum well which is formed by
third bands for L = 4d2 are presented in figure 4. One can see that at a fixed value of b2 = d2/2 the
energy depends on b1 and therefore, on the distance between the abutting atoms at the interfaces.
An increase of b1 results in the reduction of the state energy which can be explained by an effective
increase of the quantum well size. Besides, one can see that the exact solution (solid line) makes
it possible to receive slightly smaller energies than those in the case of the approximate solution
(dashed lines), though the difference between the energies is small.
Figure 4. The dependence E(b1) for the AlAs/GaAs/AlA heterostructure (a third band, L=4d2).
Figure 5. The dependence E = E(L) for the AlAs/GaAs/AlAs heterostructure. Curves 1,1′, 2,2′
are obtained by S-matrix scattering method, curves 3,3′, 4,4′ – by effective-mass approximation:
1,1′, 3,3′ – E=E(L) for a third band 2,2′, 4,4′ – E = E(L) for a second band.
The dependence of the binding state energy on the quantum well size in the second and third
bands is shown in figure 5. The third bands are characterized by a larger misalignment at the
13701-8
Properties of a charge in the nanoheterosystems with quantum wells and barriers
interface than the second ones. The energy of the binding states calculated from the bottom or top
of the corresponding bands in the 2nd layer is greater for the 3rd band. The increase of quantum
well size gives rise to new binding states and a particle monotonous energy reduction obtained
both by the S-matrix scattering method (solid lines) and the effective-mass approximation (dashed
lines). The derived results are in a good agreement with those in publication [13].
Within the quantum well, the ground state (lines 1,2,3,4) appears at smaller values of L than
the excited state (lines 1′, 2′, 3′, 4′) both for the third and second energy bands. The comparison of
particle energies obtained by different procedures shows that the energies obtained by the S-matrix
scattering method are greater than the corresponding energies in the effective-mass approximation.
In particular, for L = 6d3, the difference of ground state energies obtained by different methods
makes 200 meV for third bands and 90 meV for second ones.
Thus, in the paper a quasi-one-dimensional three-layer nanoheterostructure with an interme-
diate layer at the interfaces is described by the Kronig-Penney model with δ–function potentials.
The distance between the last or supreme atoms of the intermediate layer is a parameter of the
problem and ranges from zero to two lattice parameters of crystals. The exact solution is obtained
for the reflection coefficient R which allows us to determine the wave-vector dependence of the
coefficient, intermediate layer size and quantum well size of the nanowire. The specific calcula-
tions performed for the GaAs/AlAs/GaAs and AlAs/GaAs/AlAs heterosystems showed that the
envelope-functions and effective-mass approximations for small wave-vector values yield quite good
results provided the Harrison matching conditions are used with appropriately taken parameters.
Using the S-matrix scattering method, the energy of binding states is determined for the
AlAs/GaAs/AlAs quantum well nanowire. The dependence of the particle energy on the inter-
mediate layer and second layer sizes (quantum well width) is obtained.
Appendix
The Schrödinger equation for a boundary cell at x = 0 has the form
{
−~
2
2m0
d2
dx2
− Piδ(x ± b1)
}
Ψ
(i)
ki
= E Ψ
(i)
ki
. (D.1)
A general form of the particle wave function which is described by the wave vector k1, for the
layer containing an atom with the coordinate x = −b1, is as follows:
Ψ
(1)
k1
(x) = A e−iq1(x+b1) + B eiq1(x+b1) + Ceiq1|x+b1| . (D.2)
Using the well-known condition applied on the wave function at x = −b1, we get
Ψ
(1)′
k1
(−b1 + 0) − Ψ
(1)′
k1
(−b1 − 0) = −
2m0P1
~2
Ψ
(1)
k1
(−b1), (D.3)
C = −
A1 + B1
1 + iq1d1
U1
, where U1 =
m0P1d1
~2
.
From
Ψ
(1)
k1
(−d1) = e−ik1d1Ψ
(1)
k1
(0) ,
dΨ
(1)
k1
dx
∣
∣
∣
∣
∣
x=−d
= e−ik1d1
dΨ
(1)
k1
dx
∣
∣
∣
∣
∣
x=0
we obtain the following relation between the coefficients:
C = −A1
(
1 − e−i (q1+k1)d1
)
, B = A1
1 − e−i (q1+k1)d1
1 − e−i (q1−k1)d1
.
Hence, the normalizing wave function of a particle in the extreme cells of the first and third
layers is written
Ψ
(1)
k1
(x) = Ak1
[
1−e−i (q1+k1)d1
1−e−i (q1−k1)d1
eiq1(x+b1) + e−iq1(x+b1)−(1−e−i (q1+k1)d1)eiq1|x+b1|
]
,
Ψ
(3)
k1
(x) = Ak1
[
1−e−i (q1+k1)d1
1−e−i (q1−k1)d1
eiq1(x−L−b1) + e−iq1(x−L−b1)−(1−e−i (q1+k1)d1)eiq1|x−L−b1|
]
. (D.4)
13701-9
V.I. Boichuk et al.
Let us consider the second layer. For the layer near a first atom we have
f1 = A1e
−iq2(x−b2) + A2e
iq2(x−b2) + B1e
iq2|x−b2|, 0 6 x 6 b2 +
1
2
d2 . (D.5)
From the condition of discontinuity of the derivative at x = b2 we obtain the following relation
between the coefficients: B1 = − U2
U2+iq2d2
(A1 + A2).
The electron wave function in the region of the Nth cell is written as follows:
fN = A2N−1e
−iq2(x−b2−(N−1)d2) + A2N eiq2(x−b2−(N−1)d2) + BNeiq2|x−b2−(N−1)d2|,
b2 + 2N−3
2 d2 6 x 6 L , (D.6)
where BN = −
U2
U2 + iq2d2
(A2N−1 + A2N ).
To find the wave function of the whole second layer, we use the following matching conditions:
f1
(
x = b2 +
d2
2
)
= f2
(
x = b2 +
d2
2
)
,
f ′
1
(
x = b2 +
d2
2
)
= f ′
2
(
x = b2 +
d2
2
)
,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
fN−1
(
x = b2 +
2N − 3
2
d2
)
= fN
(
x = b2 +
2N − 3
2
d2
)
,
f ′
N−1
(
x = b2 +
2N − 3
2
d2
)
= f ′
N
(
x = b2 +
2N − 3
2
d2
)
, (D.7)
where N is the number of atoms in the second layer. The afore-mentioned matching conditions
permit us to determine the following recurrent relations between the coefficients:
A2N−1 =
d1
d5
A2N−3 +
d2
d5
A2N−2 ; A2N =
d3
d5
A2N−3 +
d4
d5
A2N−2 , (D.8)
where
d1 = ϕ1(d2/2)ϕ′
1(−d2/2)− ϕ2(−d2/2)ϕ′
2(d2/2),
d2 = ϕ2(d2/2)ϕ′
2(−d2/2)− ϕ2(d2/2)ϕ′
2(−d2/2),
d3 = ϕ1(−d2/2)ϕ′
1(d2/2) − ϕ1(d2/2)ϕ′
2(−d2/2),
d4 = ϕ1(−d2/2)ϕ′
2(d2/2) − ϕ2(d2/2)ϕ′
1(−d2/2),
d5 = ϕ1(−d2/2)ϕ′
1(d2/2) − ϕ1(d2/2)ϕ′
1(−d2/2),
ϕ1(x) = e−iq2(x−b2) −
U2
U2 + iq2d2
eiq2|x−b2|, ϕ2(x) = eiq2(x−b2) −
U2
U2 + iq2d2
eiq2|x−b2|.
In order to define the reflection coefficient amplitude r in matching conditions (7) it is necessary
to take into account that fi(x)(i = 1, 2, . . . , N) are the pieces of the function Ψ2(x), hence Ψ2(0) =
f1(0), and Ψ2(L) = fN (L).
13701-10
Properties of a charge in the nanoheterosystems with quantum wells and barriers
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Властивостi заряду в наногетеросистемах з квантовими
ямами та бар’єрами
В.I. Бойчук, I.В. Бiлинський, I.О. Шаклеїна, В.Б. Гольський
Кафедра теоретичної фiзики, Дрогобицький державний педагогiчний унiверситет iм. I. Франка,
вул. Франка, 24, Дрогобич, 82100
На основi моделi Кронiга-Пеннi з δ-подiбними потенцiалами дослiджено тришарову наногетеро-
структуру, в якiй бiля меж подiлу iснує перехiдний шар, вiдстань мiж крайнiми атомами якого є
параметром задачi i може змiнюватись вiд нуля до двох сталих ґраток кристалiв. Отримана точна
формула для коефiцiєнта вiдбивання R, з якої можна визначити залежнiсть коефiцiєнта вiд хви-
льового вектора, величини перехiдного шару та величини квантової ями (бар’єру) гетероструктури.
Конкретнi обчислення проведенi для гетеросистем GaAs/AlAs/GaAs та AlAs/GaAs/AlAs. Проведено
порiвняння коефiцiєнтiв вiдбивання у наближеннi огинаючих та у методi ефективної маси при малих
значеннях хвильового вектора; показано, що близькi результати можна отримати, якщо використа-
ти граничнi умови Гарiсона при вiдповiдним чином пiдiбраних параметрах. Для косинусоїдальної
залежностi енергiї вiд хвильового вектора, яка виникає в моделi Кронiга-Пеннi, в рамках методу S-
матрицi розсiяння визначено енергiю зв’язаних станiв для гетероструктури з квантовою ямою (типу
AlAs/GaAs/AlAs).
Ключовi слова: гетероструктура, перехiдний шар, модель Кронiга-Пеннi, коефiцiєнт вiдбивання,
S-матриця розсiяння
13701-11
Introduction
A heterostructure model. General theory
Specific calculations. The analysis of the obtained results
|