Screening effects caused by electron-electron interactions in crystaline structures with intermediate dimensionality
In the paper studied are polarization properties of an electron subsystem in the structures of intermediate dimension, particularly, in the quasi-two-dimensional structures depending on the degree of quasi-two-dimensionality. These investigations were founded on the model dispersion law that most co...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2005
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| Цитувати: | Screening effects caused by electron-electron interactions in crystaline structures with intermediate dimensionality / N.K. Tovstyuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 1. — С. 32-36. — Бібліогр.: 13 назв. — англ. |
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Tovstyuk, N.K. 2017-06-13T17:49:19Z 2017-06-13T17:49:19Z 2005 Screening effects caused by electron-electron interactions in crystaline structures with intermediate dimensionality / N.K. Tovstyuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 1. — С. 32-36. — Бібліогр.: 13 назв. — англ. 1560-8034 PACS: 73.20 Dx, 21.60 Jz https://nasplib.isofts.kiev.ua/handle/123456789/121251 In the paper studied are polarization properties of an electron subsystem in the structures of intermediate dimension, particularly, in the quasi-two-dimensional structures depending on the degree of quasi-two-dimensionality. These investigations were founded on the model dispersion law that most completely governs the peculiarities of the structure. Here, we studied the character of alternating oscillations of the potential screened by interelectron interaction, namely, by its anisotropy depending on microscopic parameters that determine the shape of isoenergetic surface and the degree of quasi-two-dimensionality. It is shown that the obtained potential, in comparison with similar results for isotropic Fermi gas, oscillates at smaller distances with amplitude of an order of magnitude higher and with a period of an order of that smaller. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Screening effects caused by electron-electron interactions in crystaline structures with intermediate dimensionality Article published earlier |
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Screening effects caused by electron-electron interactions in crystaline structures with intermediate dimensionality |
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Screening effects caused by electron-electron interactions in crystaline structures with intermediate dimensionality Tovstyuk, N.K. |
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Screening effects caused by electron-electron interactions in crystaline structures with intermediate dimensionality |
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Screening effects caused by electron-electron interactions in crystaline structures with intermediate dimensionality |
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Screening effects caused by electron-electron interactions in crystaline structures with intermediate dimensionality |
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screening effects caused by electron-electron interactions in crystaline structures with intermediate dimensionality |
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Tovstyuk, N.K. |
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Tovstyuk, N.K. |
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2005 |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Article |
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In the paper studied are polarization properties of an electron subsystem in the structures of intermediate dimension, particularly, in the quasi-two-dimensional structures depending on the degree of quasi-two-dimensionality. These investigations were founded on the model dispersion law that most completely governs the peculiarities of the structure. Here, we studied the character of alternating oscillations of the potential screened by interelectron interaction, namely, by its anisotropy depending on microscopic parameters that determine the shape of isoenergetic surface and the degree of quasi-two-dimensionality. It is shown that the obtained potential, in comparison with similar results for isotropic Fermi gas, oscillates at smaller distances with amplitude of an order of magnitude higher and with a period of an order of that smaller.
|
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1560-8034 |
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https://nasplib.isofts.kiev.ua/handle/123456789/121251 |
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Screening effects caused by electron-electron interactions in crystaline structures with intermediate dimensionality / N.K. Tovstyuk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 1. — С. 32-36. — Бібліогр.: 13 назв. — англ. |
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AT tovstyuknk screeningeffectscausedbyelectronelectroninteractionsincrystalinestructureswithintermediatedimensionality |
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2025-11-26T19:06:46Z |
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2025-11-26T19:06:46Z |
| _version_ |
1850769241988399104 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 2. P. 32-36.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
32
PACS: 73.20 Dx, 21.60 Jz
Screening effects through electron-electron interaction
in the crystal structures of an intermediate dimension
N.K. Tovstyuk
Ivan Franko Lviv National University, Department of Electronics
50, Dragomanova str., 29005 Lviv, Ukraine
E-mail: tose@polynet.lviv.ua
Abstract. In the paper studied are polarization properties of an electron subsystem in the
structures of intermediate dimension, particularly, in the quasi-two-dimensional structures
depending on the degree of quasi-two-dimensionality. These investigations were founded
on the model dispersion law that most completely governs the peculiarities of the structure.
Here, we studied the character of alternating oscillations of the potential screened by
interelectron interaction, namely, by its anisotropy depending on microscopic parameters
that determine the shape of isoenergetic surface and the degree of quasi-two-
dimensionality. It is shown that the obtained potential, in comparison with similar results
for isotropic Fermi gas, oscillates at smaller distances with amplitude of an order of
magnitude higher and with a period of an order of that smaller.
Keywords: quasi-two-dimensional structure, interelectron interaction, polarization
properties, singuliarity points.
Manuscript received: 18.03.05; accepted for publication 18.05.05.
1. Introduction
The recently subject of intensive studies are low-
dimensional structures, i.e., quantum dots, one-
dimensional (1D) wire and two-dimensional (2D)
structures [1, 2]. Such investigations are not only of
fundamental interest, but also are more important for the
applications of these objects. A number of precise results
of the studies of such objects are well known [3, 4].
However, the structures which can be considered as
intermediate structures between zero- and 1D, or 1D and
2D, 2D and 3D ones have been realized [5], the latter
including the layer crystals. The layer crystal can be
represented as a set of packed "sandwiches" bonded by
weak van der Waals forces. Each sandwich is a set of
monoatomic layers with covalent or ion-covalent bond.
Such discrimination in chemical bonds causes a number
of phenomena typical for the layer crystals. The degree
of 2D or 3D can be significantly changed by different
ways, such as a pressure hydrostatic or axial (along the
normal to the layers) or intercalation [6]. In other words,
the layer crystals can be considered as a quasi 2D
structure (Q2D), or, in the case of decreased interlayer
binding, we can consider it as the system of 2D
structures weakly bonded to one another. It is known
that electron mixing at the sites of a crystal determines
the width of the conduction band. Hence, the difference
of the value of mixing inside the layer and between the
layers is the reason why the bandwidth in the layer is
much higher than that between the layers in the reciprocal
space. It demands to refuse from the widely used
representation of the dispersion law of carriers by the
parabolic dependence, which is possible for a wide band
(i.e., within the layer), but is not valid for a narrow band. In
the latter case, even at a very low degree of band
occupancy, the deviation from the parabolic dispersion law
can be significant. It means that the peculiarities of a layer
crystal manifest themselves at the stage of its one particle
studies, which should be all the more so taken into
consideration in the case of many-body effects.
The aim of this work is
• to analyze some many-body characteristics depending
on the degree of “quasi-two-dimensionality” of the
system on the assumption of a model dispersion law,
that most completely governs the structure peculiarities
of the crystal of an intermediate dimension;
• to study how the dimension of crystal structure
including the nonparabolic band structure (in those
cases when the effective mass approximation is
forbidden) affects the polarization properties of the
electron subsystem.
2. Model dispersion law and its application to the
description of many-particle effects
Let us consider the following model dispersion law (in the
case when OZ axis coincides with the normal to the
layers) [7]
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 2. P. 33-37.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
33
( ) ( ) )cos1(,, 22
zyxzyx ktkkkkkE −++=α (1)
where *
,2
1
yxm
=α ( *
, yxm is the isotropic effective mass
of an electron inside the layer), t is an integral of
electron mixing between the nearest layers along the
normal to them, which determines the conduction band-
width, 2t, in the OZ direction of the reciprocal space.
The lattice constants and h were chosen to be equal to
unity.
The discontinuity of the ion structure is considered as
the discontinuity of the value of wavevectors kx ky, kz,
that is caused by the Born – Karman conditions. The
choice of the effective mass approximation for the
description of electrons inside the XOY layer and the
tight binding across the layers is imposed by the above-
mentioned significant difference of chemical bondings in
various OX, OY, and OZ crystallographical directions of
the layer crystal, which causes the condition 2t << π2α,
i.e., the case of the intermediate dimension.
The model dispersion law (1) is convenient for the
mentioned problem solution, inasmuch as the change of t
parameter makes it possible to realize the 2D structure in
the t = 0 limit. When t increases (2t→π2α) the
nonparabolic effects along kz can be neglected, and then
we take this 3D anisotropic parabolic dispersion law. The
limit value of the parameter α = 0 describes the 1D
structure. The model dispersion law (1) causes the phase
2
12 Lifshitz transition, i.e., topological changes of
isoenergetic surfaces according to the energy or chemical
potential changes, which should manifest themselves in
polarization properties of layer crystals.
It is clear that interpretating such crystals as the
system with an intermediate dimension, their physical
properties, in particular, polarization ones should be
compared to the properties of pure 2D (when t = 0) or
3D structures (when 2t → π2α). However, the smooth
transition from 2D to 3D case for t < α may not be
observed.
Many-body effects in the electron subsystem of the
crystal with intermediate dimension can have a specific
character due to the peculiarities of the band structure.
For example, 1D system studies [8, 9] using the
dispersion law (1) in the case α = 0 showed an additive
singularity of the dielectric function when qz = 2(π − kF).
The Fourier form of the screened Coulomb potential
related with the core potential V0(q) through the well-
known expression [10] that, in the Ω = 0 case, is as
follows:
( ) ( )
( )
0V
V
ε
=
q
q
q
% , (2)
where ( )q0V for 3D structures has the form [10]
( )
2
0 2
0
4 eV π
ε
=q
q
(3)
and ε(q) is the dielectric function. We will be interested in
the behaviour of the potential screened by electron-electron
interaction for systems with an intermediate dimension
which is inspected for anisotropy of the conduction band.
That is why both in the t < α case and in the limit cases
t→0, t → π 2α the Coulomb potential will be given as (3).
Statistic description of many-body effects, particularly
within the framework of the model of the interacting
electron gas in the solid state, foresees the introduction of
electron basis for functions. In our case, the Bloch
functions instead of the plane waves are taken as the basis
of functions, i.e., wave functions of carriers are described
by the dispersion law (1). The effects of electron-electron
interaction causes the screening the Coulomb potential,
which is found by summing the loop diagrams (random
phase approximation (RPA)). In the paper [11], q-depen-
dence of static dielectric function of the Bloch electrons
with the parabolic dispersion law is described by the
following model expression
( )
( ) ( )
2
22
, 1
0
q
q
p
g i
ω
ε
ω ω
Ω = +
− +
, (4)
where ωg(q) is the effective q-dependent gap frequency and
ωp is the plasma frequency of valence electrons.
3. Polarization properties of the intermediate dimension
structures
In this paper, the dielectric function is calculated from the
Bloch functions, considering the environment anisotropy
(the dispersion law (1)). Since we deal with an electron
subsystem, dielectric function should be studied depending
on the direction being vector rather than a tensor.
In the RPA, ε(q,Ω) components, namely, real ε1(q,Ω)
and imaginary ε2(q,Ω) ones are connected with the
polarization loop ),( ΩΠ q [13] as follows:
( ) ( ) ( )1 01 Req, q, qVε Ω = + Π Ω , (5)
( ) ( ) ( )2 0Imq, q, qVε Ω = Π Ω . (5′)
Here,
.2,22,2
),(
00 Ω⎟
⎠
⎞⎜
⎝
⎛ Ω−−⎟
⎠
⎞⎜
⎝
⎛ Ω++=
=ΩΠ
∫ ddGGi kqkqk
q
ωω
(6)
),( ΩΠ q is the polarization loop calculated in terms of
Green’s functions when T = 0 [12]:
[ ]{ } .)(sign)(
),(
1
0
−−++−=
=
FF EE
G
ειδεω
ω
kk
k
(7)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 2. P. 32-36.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
34
For 1D case such polarization loop was analyzed in
[8, 9]. Integrating over the frequency by residue method,
we get
,
2
1
),(
}{
1
}{
2
33
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−=
=ΩΠ
∫∫
A
zyx
B
zyx dkdkdkRdkdkdkR
π
q
(8)
where
).2,1(
)1()()( 22
1
=
−+Ω++−−=−
α
δα
α iEER qq kk
(9)
Then in the case 0=Ω , the real and imaginary
components of ),( ΩΠ q become
( )
{ } { }
( ) ( )
{ } { }
3 3
3 3
11Re 2 2 sin ;
2
1Im 2 2 sin
2
q
q
x x y y z x y z
B A
x x y y z x y z
B A
q k q k b k dk dk dk
q k q k b k dk dk dk
α α
π
δ α α
−
−
+
⎡ ⎤Π = + +⎣ ⎦
Π = − − −
∫
∫
(10)
Eqs (9) and (10) are taken in the sense of their
principal values and b = 2tsin(qz/2). In the 3D case, the
regions {A3} і {B3} are as follows:
⎪
⎪
⎩
⎪
⎪
⎨
⎧
>⎟
⎠
⎞
⎜
⎝
⎛ −
<⎟
⎠
⎞
⎜
⎝
⎛ −
∈
⎪
⎪
⎩
⎪
⎪
⎨
⎧
<⎟
⎠
⎞
⎜
⎝
⎛ −
>⎟
⎠
⎞
⎜
⎝
⎛ −
∈
F
F
F
F
E
E
B
E
E
A
ε
ε
ε
ε
2
2
}{and/or
2
2
}{ 33 qk
qk
qk
qk
.
(11)
Varying the free carriers concentration one can
change the value of the Fermi energy. It is well known
that the carriers located in the vicinity of the Fermi level
play the main role in kinetics effects. The Fermi level
transition to higher than the bandwidth along kz, 2t,
makes it possible to consider at least Q2D problems in
the vicinity εF→2t. We can realize the similar Q2D one
by changing t parameter. Hereinafter, we will consider
the above-mentioned transition by changing εF that is,
for εF < 2t, we deal with the anisotropical nonparabolic
case, and when εF > 2t, this is a Q2D case with
isoenergetic surfaces open along OZ axis, and, for εF >> 2t,
this is a 2D case. All the parameters such as α, t, εF are
given in electron-volts.
4. Calculations of polarization loop
Firstly, let us consider 1D case (α = 0), for which the
dispersion law (1) is as follows:
( ) )cos1( zz ktkE −= . (12)
In this case, the disperse dependence of real and
imaginary components ( )ΩΠ ,zq are written in the follo-
wing manner:
( ) ( )
{ } { }
∫
−
−−Ω−=ΩΠ
AB
zzz dkkbq 1sin
2
1,Re
π
, (13)
( ) ( )
{ } { }
∫
+
−Ωδ=ΩΠ
AB
zzz dkkbq sin
2
1
,Im . (14)
The condition (11) gives us the nonzero values of the
polarization loop (13) and (14). The regions of integration
are given in Fig. 1.
1. Region 1>Q where
t
tQ Fε−= , then { } { } { }0, ∈BA
or according to (5) ( ) 0=Π zq .
2. Region 1≤≤QR : then { } [ ] { } [ ]1403 ,,, zzBzzA ∈∈
(Fig. 1а), where 43, zz are the roots of the equation
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+=
2
cos zq
zQ , and 10 , zz are the roots of the
equation ⎟
⎠
⎞
⎜
⎝
⎛ −=
2
cos zqzQ .
3. Region RQR ≤≤− : { } [ ] { } [ ]3401 ,,, zzBzzA ′′∈′′∈
(Fig. 1b).
4. Region RQ −≤≤−1 : { } [ ] { } [ ]3041 ,,, zzBzzA ′′′′∈′′′′∈
(Fig. 1c).
The integrals (13) and (14) can be found analytically. In
the static case ( 0=Ω ), it is
z4
z1z3
z0 z4
z1 z3
z0
z4
z1 z3
z0
+– +– +–
kz
a b c
Fig. 1. The regions of integration {A}, {B} according to (11) in the 1D case.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 2. P. 33-37.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
35
( )
2
~
tg
2
~
tg
ln
2
sin2
1Re
0
1
z
z
qt
q
z
z
π
=Π . (15)
The limits of integration are determined from the
condition (6) (Fig. 1), and they are different depending
on Q value [8]. They can be compared with ( )zqΠRe
for the 1D case with the parabolic dispersion law [12]
( ) ( )
Fz
Fz
z
F
Dz kq
kq
q
k
Nq
2
2
ln
2
0Re 1 −
+
=Π . (16)
As seen from (16), the polarization loop has
singuliarity in one point F2kqz = while, for our case, it
has an additional point ( )F2 kqz −= π (see Fig. 1 in [8]).
In the case of 3D anisotropy (α ≠ 0, t ≠ 0), the similar
calculations of { } { }BA , regions and ( ) ( )qq ΠΠ Im,Re
were performed numerically. They showed that the
behaviour of the polarization loop does not change
qualitively at the transition from 1D-system to 3D-
anisotropical one for given band filling (Fermi level),
i.e., the quantity of singuliarity points does not decrease,
while the ( ),Re zqΠ ( )zqΠIm values increase. The
further increase of band filling, i.e., the transition from
3D-case to Q2D-one, causes the decrease of the quantity
of singuliarity points and an increase of
( ) ( )zz qq ΠΠ Im,Re values (Figs 2, 3).
5. The potential screened by interelectron interaction
The obtained disperse dependences for the components
of dielectric function were used for the )(~ qV
r calculations
according to (2). Here, the Coulomb potential )(0 qV
r is
taken from (3), i.e., the potential used for 3D-case. Such
a choice is caused by the above-mentioned purpose of
the paper which lies in studying the screening effects in the
structures of intermediate dimension. These effects are
caused by strongly anisotropic but still 3D-structure. That
is why the coordinate dependence of ( )rV
~
( ) ( ) zyx dqdqdqVrV qrcosq~~ ∫
−
=
π
π
(17)
is considered as 3D integral.
Let us analyze the ( )rV
~
dependence on the coordinates
in the directions (001) and (111). In the (001) case, ( )rV
~
dependence for different εF is represented in Fig. 4. As
seen from Fig. 4, the amplitude oscillations (curve 1) take
place at smaller distances than that for isotropic Fermi-gas
(insertion in Fig. 4). This difference is evidently caused by
the nonparabolic dispersion law. The decrease of ( )rV
~
(curves 1-3), i.e., at the transition from quasi 2D to 3D
anisotropic case, causes a decrease of the first minimum in
( )rV
~
dependence just to negative values and shifts them in
εF = 0.01eV
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.00 0.79 1.57 2.36 3.14 3.93 4.71 5.50 6.28
qz
П
( q
z)
ReП
ImП
Fig. 2. Dependence of the polarization loop ( ),0,0,Re zqΠ
( )0,0,Im zqΠ for the 3D anisotropic case (εF = 0.01 eV).
εF = 0.21eV
qz
П
( q
z)
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0.00 0.79 1.57 2.36 3.14 3.93 4.71 5.50 6.28
ReП
ImП
Fig. 3. Dependence of the polarization loop ( ),0,0,Re zqΠ
( )0,0,Im zqΠ for the Q2D case (εF = 0.21 eV).
x y = = 0
–2
0
2
4
6
8
10
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
z
V
( z
)
0.01
0.1
0.21
εF =
Fig. 4. Dependence of ( )zV
~
in the direction (001) for different
εF values.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2005. V. 8, N 2. P. 33-37.
© 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
36
the direction of smaller z-distances. In the vicinity of
εF = 2t, electrons at distances z ∼ 1.5a0 (a0 is the lattice
constant) are most attracted, while when ( )rV~ amplitude
oscillations practically do not depend on εF .
( )rV~ dependence along (111) direction looks
somewhat different: the first minimum in the z ∼ 1.5a0
vicinity (Fig. 5a) disappears and oscillations in the
coordinate dependence are noticeable only after 15-fold
extension.
The coordinate ( )rV
~
dependence in (111) direction
compared with that in the direction (001) changes as
follows: i) when εF = 0.01, the oscillation period
significantly increases, while the first local minimum
(typical for (001) direction) vanishes; ii) when εF = 0.21,
the oscillation period also increases significantly, and the
depth of the first minimum decreases.
6. Conclusions
Polarization properties of the electron subsystem in the
structures of intermediate dimension, in particular, in the
Q2D structures depending on the degree of quasi-two-
dimensionality are studied. The peculiarities of their
electron band structure are characterized by the model
dispersion law.
The analysis of the polarization properties of electron
subsystem in the Q2D structures within the framework
of the RPA shows:
1. When changing the degree of Q2D, the number of
singuliarity points in the disperse behaviour of the
polarization loop )( zqΠ and dielectric function in the kz
direction do not change while its number changes inside
the XOY plane.
2. The character of alternating oscillations of the
potential screened by interelectron interaction is stronger
pronounced as compared to the isotropic crystal structures.
This is caused by nonparabolic effects.
Thus, changing the degree of quasi-two-dimensionality
of the layer crystal by intercalation, stage ordering,
hydrostatic or axial pressure makes it possible to
significantly affect the properties of the electron subsystem.
That is why we should take into consideration the
screening effects analyzing the phenomena that take place
in layer crystals.
References
1. L. Gordstein, F. Glas, J.Y. Marzin et al. // Appl. Phys.
Lett. 47, p.1099 (1985).
2. V.M. Ustinov // Physics and Technique of
Semiconductors 38, N8, p. 963 (2004) (in Russian).
3. G.B. Ibragimov // J. Phys.: Condens Matter 14, p. 8145
(2002).
4. B. Vinter and L. Thibaudean, Bound to free state
infrared absorption and selection rules in quantum
wells. Intersubband transition in quantum wells, Ed. by
E. Rosencher, B. Vinter, B. Levine, New York (1992).
5. H.N. Spector // Phys. Rev. B 28, p. 971 (1983).
6. I.I. Grigorchak // Phys. Chem. Solid State 2, N1, p. 7
(2003).
7. R.F. Fivaz // J. Phys. Chem. Solids 28, N4, p. 839
(1967).
8. B. Lukiyanets, N. Tovstyuk // Condensed Matter Phys.
8, p. 99 (1996).
9. B.A. Lukiyanets, N.K. Tovstyuk // Zhurn. Fiz.
Doslidgen’ 1, p. 251 (1997).
10. N. Ashcroft, N. Mermin, Solid State Physics, Mir,
Moscow (1979)(in Russian).
11. F. Bechstedt et al. // Phys. status solidi (b) 126, p. 575
(1984).
12. D. Pines, The many-body problem, W.A. Benjamin Inc.,
New York (1961).
13. H.I. Starnberg, H.E. Brauer, H.P. Hughes // Phys. and
Chem. of Materials with Low-Dimensional Structures
24, p. 41 (2000).
x y = = 0
z
V
(z
)
—0.6
—0.4
—0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
= 0.001
= 0.021
= 0.03
F
F
F
ε
ε
ε
Fig. 5. Dependence of ( )zV
~
in the direction (111) for different
εF values.
|