The theory of electron states on the dynamically deformed adsorbed surface of a solid
Dispersion relations for the spectra of surface electron states on a dynamically deformed adsorbed surface of a monocrystal with the Zinc blende structure are received. It is established that the dependences of the band gap width and of the concentration of electrons on the concentration of adatom...
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Peleshchak, R.M. Seneta, M.Ya. 2019-06-19T13:44:58Z 2019-06-19T13:44:58Z 2018 The theory of electron states on the dynamically deformed adsorbed surface of a solid / R.M. Peleshchak, M.Ya. Seneta // Condensed Matter Physics. — 2018. — Т. 21, № 2. — С. 23701: 1–9. — Бібліогр.: 15 назв. — англ. 1607-324X PACS: 73.20.At, 73.0.Hb, 68.60.Bs DOI:10.5488/CMP.21.23701 arXiv:1806.09944 https://nasplib.isofts.kiev.ua/handle/123456789/157056 Dispersion relations for the spectra of surface electron states on a dynamically deformed adsorbed surface of a monocrystal with the Zinc blende structure are received. It is established that the dependences of the band gap width and of the concentration of electrons on the concentration of adatoms N0d upon the solid surface are of nonmonotonous character Отримано дисперсiйнi спiввiдношення для спектру поверхневих електронних станiв на динамiчно деформованiй адсорбованiй поверхнi монокристалу зi структурою цинкової обманки. Встановлено, що залежностi ширини забороненої зони та концентрацiї електронiв на поверхнi твердого тiла вiд концентрацiї адсорбованих атомiв N0d мають немонотонний характер en Інститут фізики конденсованих систем НАН України Condensed Matter Physics The theory of electron states on the dynamically deformed adsorbed surface of a solid Теорiя електронних станiв на динамiчно деформованiй адсорбованiй поверхнi твердого тiла Article published earlier |
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| title |
The theory of electron states on the dynamically deformed adsorbed surface of a solid |
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The theory of electron states on the dynamically deformed adsorbed surface of a solid Peleshchak, R.M. Seneta, M.Ya. |
| title_short |
The theory of electron states on the dynamically deformed adsorbed surface of a solid |
| title_full |
The theory of electron states on the dynamically deformed adsorbed surface of a solid |
| title_fullStr |
The theory of electron states on the dynamically deformed adsorbed surface of a solid |
| title_full_unstemmed |
The theory of electron states on the dynamically deformed adsorbed surface of a solid |
| title_sort |
theory of electron states on the dynamically deformed adsorbed surface of a solid |
| author |
Peleshchak, R.M. Seneta, M.Ya. |
| author_facet |
Peleshchak, R.M. Seneta, M.Ya. |
| publishDate |
2018 |
| language |
English |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Теорiя електронних станiв на динамiчно деформованiй адсорбованiй поверхнi твердого тiла |
| description |
Dispersion relations for the spectra of surface electron states on a dynamically deformed adsorbed surface of a
monocrystal with the Zinc blende structure are received. It is established that the dependences of the band gap
width and of the concentration of electrons on the concentration of adatoms N0d upon the solid surface are of
nonmonotonous character
Отримано дисперсiйнi спiввiдношення для спектру поверхневих електронних станiв на динамiчно деформованiй адсорбованiй поверхнi монокристалу зi структурою цинкової обманки. Встановлено, що залежностi ширини забороненої зони та концентрацiї електронiв на поверхнi твердого тiла вiд концентрацiї адсорбованих атомiв N0d мають немонотонний характер
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/157056 |
| citation_txt |
The theory of electron states on the dynamically deformed adsorbed surface of a solid / R.M. Peleshchak, M.Ya. Seneta // Condensed Matter Physics. — 2018. — Т. 21, № 2. — С. 23701: 1–9. — Бібліогр.: 15 назв. — англ. |
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AT peleshchakrm thetheoryofelectronstatesonthedynamicallydeformedadsorbedsurfaceofasolid AT senetamya thetheoryofelectronstatesonthedynamicallydeformedadsorbedsurfaceofasolid AT peleshchakrm teoriâelektronnihstanivnadinamičnodeformovaniiadsorbovaniipoverhnitverdogotila AT senetamya teoriâelektronnihstanivnadinamičnodeformovaniiadsorbovaniipoverhnitverdogotila AT peleshchakrm theoryofelectronstatesonthedynamicallydeformedadsorbedsurfaceofasolid AT senetamya theoryofelectronstatesonthedynamicallydeformedadsorbedsurfaceofasolid |
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2025-11-25T19:06:17Z |
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2025-11-25T19:06:17Z |
| _version_ |
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| fulltext |
Condensed Matter Physics, 2018, Vol. 21, No 2, 23701: 1–9
DOI: 10.5488/CMP.21.23701
http://www.icmp.lviv.ua/journal
The theory of electron states on the dynamically
deformed adsorbed surface of a solid
R.M. Peleshchak, M.Ya. Seneta∗
Drohobych Ivan Franko State Pedagogical University, 24 Franko St., 82100 Drohobych, Ukraine
Received January 12, 2018, in final form March 23, 2018
Dispersion relations for the spectra of surface electron states on a dynamically deformed adsorbed surface of a
monocrystal with the Zinc blende structure are received. It is established that the dependences of the band gap
width and of the concentration of electrons on the concentration of adatoms N0d upon the solid surface are ofnonmonotonous character.
Key words: surface electron states, acoustic quasi-Rayleigh wave, adsorbed atoms
PACS: 73.20.At, 73.0.Hb, 68.60.Bs
1. Introduction
The creation of a new class of microelectronic and nanoelectronic devices with controlled parame-
ters needs a research of the excitation mechanisms of electronic states upon the adsorbed surface of
semiconductors. One of these mechanisms of excitation is a dynamic deformation in the subsurface
layer of a solid. Such a dynamic deformation upon the adsorbed surface of a solid can be formed by
a quasi-Rayleigh acoustic wave [1]. Interaction between the quasi-Rayleigh acoustic wave and adatoms
renormalizes the spectrum of the surface electron states due to the deformation potential [2]. Technolo-
gically changing the concentration of adsorbed atoms, it is possible to change the frequency of a surface
acoustic wave (SAW) and the electronic structure of a subsurface layer. Such a correlation between the
concentration of adsorbed atoms, the frequency of a surface acoustic wave and the electron structure of a
subsurface layer can be used in practice for the change of coefficients of electromagnetic waves reflection
from the interface of the media and for the change of a dispersion law of plasma oscillations [3].
At present, there are many works [3–7] devoted to the investigation of surface quantum electronic
states. At the same time, the main focus was concentrated on the study of electron states upon the crystal
surface, due to the breakdown of periodic crystalline potential. It is known that if the surface is smooth,
then the surface electron states do not arise.
In particular, the authors of the work [3] investigated the surface electron states of a semiconductor
bounded by an uneven surface having an infinitely high potential barrier. The surface of a semiconductor
was considered without adsorbed atoms, and the surface roughness was formed by a quasi-Rayleigh
acoustic wave.
The influence of interaction between adatoms and the self-consistent acoustic quasi-Rayleigh wave
on its dispersion and on the width of the phonon mode at various values of concentration of adatoms was
investigated in the works [8, 9].
In the work [10] within the long-wave approximation, taking into account the nonlocal elastic inter-
action between the adsorbed atom and the matrix atoms and the mirror image forces, the deformation
potential of the acoustic quasi-Rayleigh wave is investigated, and the dependences of the deformation
potential amplitude and of the surface roughness height on the concentration of adsorbed atoms are
calculated.
∗E-mail: marsen18@i.ua
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
23701-1
https://doi.org/10.5488/CMP.21.23701
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
R.M. Peleshchak, M.Ya. Seneta
The conditions of the appearance of localized electron states upon the semiconductor surface with
inequalities formed by the adsorbed atoms and by the acoustic quasi-Rayleigh wave were investigated in
the work [11].
The purpose of this work is to investigate the influence of the concentration of adsorbed atoms on
the spectrum of the surface electron states and on the distribution of electron density upon a dynamically
deformed adsorbed surface of a monocrystal with the Zinc blende structure.
2. Formulation of the problem
Here, we are considering the subsurface layer of a cubic crystal in a molecular beam epitaxy or
implantation process [8]. Under the action of a flow of atoms, the atoms are adsorbed with an average
concentration N0d. Due to the deformation potential and the local renormalization of the surface energy,
the adatoms and the deformation field of the surface acoustic quasi-Rayleigh wave inhomogeneously
deform the subsurface layer. In its turn, byway of the deformation potential the self-consistent deformation
redistributes the adsorbed atoms along the surface. The influence of adsorbed atoms is reduced to
a change of boundary conditions for a stress tensor σi j on z = 0 surface. The layer, being defect-
enriched by adatoms, can be considered as a film of a thickness, ρ density and Young’s modulus E . It is
rigidly connected with a substrate, the other monocrystal part, having elastic parameters ρs and Es. The
connecting plane of the film and substrate z = a is parallel to the surface (100). The z-axis is directed into
the single crystal depth, the axes x and y are directed along two orthogonal crystallographic directions
of type [100]. A surface acoustic quasi-Rayleigh wave, extending in x-axis direction with ω′( ®q, N0d)
frequency, forms a dynamic deformation and interacts with the adatoms. The deformed surface form
along the x-axis, depending on time, can be described by the following function [11]:
z0(x, t) = ς̃(N0d) cos [qx − ω′(q, N0d) · t] , (2.1)
where q = 2π
Lx
, Lx is the period of roughnesses (the length of acoustic wave) along the x-axis. The
dispersion law ω′(q, N0d) = Reω of a quasi-Rayleigh wave and the width ω′′(q, N0d) = Imω of the
acoustic phonon mode are determined [12]:
ω′(q, N0d) = ctqξ0
©«1 −
1
f ′(ξ0)
2ξ0θdN0d
kBT ρc2
l
·
Dd
2q
[
1 − 2
3
1−2ν
K(1−ν)a
θd
2
kBT
N0d
]
q{
Dd
[
1 − 2
3
1−2ν
K(1−ν)a
θd
2
kBT
N0d
] }2
q2 + c2
t ξ0
2
×
(
1 − l2
dq2
) [
q
√
1 − ξ0
2 ∂F
∂N1d
+
(
2 − ξ0
2
) θd
2a
] )
, (2.2)
ω′′(q, N0d) = −ctq
1
f ′(ξ0)
2ξ2θdN0d
kBT ρc2
l
·
Ddq{
Dd
[
1 − 2
3
1−2ν
K(1−ν)a
θd
2
kBT
N0d
] }2
q2 + c2
t ξ0
2
×
(
1 − l2
dq2
) [
q
√
1 − ξ0
2 ∂F
∂Nd1
+
(
2 − ξ0
2
) θd
2a
]
, (2.3)
where N0d is a the spatially homogeneous component of the concentratoin of adatoms [9]; K is the
modulus of elasticity; l2
d is the average of the square of a characteristic distance of the interaction between
the adatom and the matrix atoms; ν is the Poisson coefficient; θd is the surface deformation potential [10];
Dd is the diffusion coefficient of the adatom; T is the temperature of a substrate; kB is the Boltzmann
constant.
The height of the roughness ς̃(N0d) is equal to the sum of the normal components of the displacement
vector ®ul(®r, t), ®ut(®r, t) of longitudinal and transverse waves, respectively, on the plane z = 0 [10]:
ς̃(N0d) =
��ulz(0)�� + ��utz(0)�� . (2.4)
23701-2
The theory of electron states on the dynamically deformed adsorbed surface of a solid
The components ulz(0), utz(0) of the displacement vector of the medium points are found from the
equation solution [13]:
∂2 ®u
∂t2 = c2
t ∆®r ®u + (c
2
l − c2
t )
−−−→
grad(div ®u). (2.5)
The solution of equation (2.5) for the Rayleigh surface wave, extending in the x-axis direction, is
represented as:
ux(x, z) = −iqAeiqx−iωt−klz − iktBeiqx−iωt−ktz, (2.6)
uz(x, z) = klAeiqx−iωt−klz + qBeiqx−iωt−ktz , (2.7)
where k2
l,t(N0d) = q2 − ω2
c2
l,t
; A, B are the amplitudes of SAW. The deformation potential created by the
surface acoustic quasi-Rayleigh wave and by the adsorbed atoms is determined by the following relation:
V(x, z, t) = λi j
∂ui(x, z, t)
∂xj
= V0e−klz cos [qx − ω′(q, N0d) · t] , (2.8)
where ui = uli+uti; λi j is a tensor of deformation potential;V0 = −
|λ |ς̃(N0d)q
2(2−ξ0)
kl
c2
t
c2
l
; 1
kl
is the depth of the
sound penetration into semiconductor, ξ0 is a quantity dependent on the ratio between the longitudinal cl
and transversal ct sound velocity [14], and the expression for the height of roughnesses ς̃(N0d) is defined
in the work [10].
3. Electron states upon the dynamically deformed adsorbed surface
The motion of electrons upon the dynamically deformed adsorbed surface (i.e., in the field of a quasi-
Rayleigh acoustic wave) essentially depends on the value of the ratio between the wavelength 2π/kx (kx
is the component of the electron wave vector) and the mean free path l of the electrons. In semiconductors
without the inversion center kx l � 1, i.e., at the distances of an order of a wavelength, there is observed
a multiple impact of electrons, during which the equilibrial distribution of electrons is established. The
surface electron states on the rough boundary of a monocrystal having a Zinc blende structure are found
from the non-stationary Schrödinger equation [11]:
i~
∂ψ(x, z, t)
∂t
= −
~2
2m∗
(
∂2ψ
∂x2 +
∂2ψ
∂z2
)
+ V0e−klz cos [qx − ω′(q, N0d) · t]ψ(x, z, t). (3.1)
The surface roughnesses are formed by the dynamically deformed (acoustic quasi-Rayleigh wave)
and adsorbed atoms having concentration N0d. Interaction between the surface acoustic wave and the
adsorbed atoms is attained through the deformation potential. We assume that the motion of conduction
electrons is bounded by an uneven wall which is an infinitely high potential barrier.
The solution of equation (3.1) is obtained as a sum of space-time harmonics [7]:
ψ(x, z, t) =
∞∑
n=−∞
ψn(z)eiΦn(x,t), (3.2)
where Φn(x, t) = (kx + nq)x − [ωe + nω′(q, N0d)] t, ~ωe = E is the electron energy, and ~kx is its
momentum.
Substituting (3.2) into equation (3.1) and using the condition of orthogonality of functions, we obtain:(
k2
n +
∂2
∂z2
)
ψn(z) = −2β2qς̃(N0d)e−klz [ψn+1(z) + ψn−1(z)] , (3.3)
where k2
n =
2m∗
~2
[E + n~ω′(q, N0d)] − (kx + nq)2, β2 = m∗ |V0 |
~2 ς̃(N0d)q
is the effective electron mass.
Equation (3.3) is solved using the method of successive approximations for a small parameter
qς̃(N0d) � 1. In the zero approximation, the wave function ψ0
n(z) has the form:
ψ0
n(z) = Aneiknz . (3.4)
23701-3
R.M. Peleshchak, M.Ya. Seneta
Substituting ψ0
n(z) in the right-hand part of equation (3.3), we find the solution of the inhomogeneous
equation:
ψ̃n(z) = −β2qς̃(N0d)
[
ei(kn−1+ikl)z
k2
n − (kn−1 + ikl)2
An−1 −
ei(kn+1+ikl)z
k2
n − (kn+1 + ikl)2
An+1
]
. (3.5)
The solution of Schrödinger equation (3.1) can be represented as the series:
ψn(x, z, t) =
∞∑
n=−∞
[
Aneiknz −
β2qς̃(N0d)ei(kn−1+ikl)z
k2
n − (kn−1 + ikl)2
An−1 −
β2qς̃(N0d)ei(kn+1+ikl)z
k2
n − (kn+1 + ikl)2
An+1
]
eiΦn(x,t). (3.6)
Finding the dispersion law of surface electron states E = E(kx) we use the boundary conditions
for the wave function at infinity and at the interface boundary. Taking into account the conditions for a
smooth roughness of the surface qς̃(N0d) � 1
( ∂z0
∂x � 1
)
, the boundary condition on the plane z = 0
[15] takes the form [9]:
∂ψ
∂z
+
∂2ψ
∂z2 z0 −
∂ψ
∂x
∂z0
∂x
����
z=0
= 0. (3.7)
At a distribution of a surface acoustic wave the electrons on the surface scatter on its deformation
potential V(x, z, t) [11] within the volume, as well as on the roughnesses which are formed by both the
surface acoustic wave and the heterogeneous distribution of adsorbed atoms. As a result, the harmonics
are actuated with wave numbers kn+1 and kn−1. Within the boundary condition (3.7) we substitute the
wave function in the form (3.6) [11]. In the following calculations, the energy of the Rayleigh wave
quantum is neglected, i.e.:
~ω′(q, N0d) �
~2q2
2m∗
or
ξ0ct �
~q
2m∗
. (3.8)
If the surface roughness is formed by a quasi-Rayleigh wave, then the electrons dissipate not only
on the roughness created by the inhomogeneous distribution of adsorbed atoms Nd(x) but also on the
deformation potential of the quasi-Rayleigh wave. This additional scattering prevents the formation of
surface electron states.
Now, we consider the existence of surface electron states for a zero harmonic. In this case, the
interaction with the harmonic n > 2 is neglected, because the connection with the zero harmonic is
proportional to [qς̃(N0d)]
n � 1. The spectrum of electron states on a surface with roughnesses, formed
by the surface acoustic wave and by adsorbed atoms, has the form [11]:
k0 = −
ς̃2(N0d)
4
[ (
k2
0 + kxq
)2
k−1
+
(
k2
0 − kxq
)2
k1
]
−
iβ2qς̃2(N0d)
2
×
{
k2
0 + kxq
k−1
[
k0 + ikl
k2
−1 − (k0 + ikl)2
+
k−1 + ikl
k2
0 − (k−1 + ikl)2
]
+
k2
0 − kxq
k1
×
[
k0 + ikl
k2
1 − (k0 + ikl)2
+
k1 + ikl
k2
0 − (k1 + ikl)2
]
− k0
(
Γ1
k1
+
Γ−1
k−1
)
+ Γ0
}
, (3.9)
where Γ0 =
q(kx−q)−(k0+ikl)2
k2
−1−(k0+ikl)2
−
q(kx+q)+(k0+ikl)2
k2
1−(k0+ikl)2
, Γ±1 =
(k±1+ikl)∓kxq
k2
0−(k±1+ikl)2
.
We find the solution of equation (3.9) using the method of successive approximations k0 (ς̃(N0d)) =
k0 (ς̃(0)) + δk0 (ς̃(N0d)) for a small parameter qς̃(N0d) � 1 for amplitudes with n = 1; 0;−1, where
ς̃(N0d) = ς̃(0) + δς̃(N0d), ς̃(0) is the height of the roughnesses of the surface formed only by a surface
acoustic wave. For the case ς̃(N0d) = ς̃0 + δς̃(N0d) = 0 (the absence of the adsorbed atoms and a surface
acoustic wave, i.e., the case of a smooth surface), the solution of equation (3.9) is k0 ≡ k(0)0 = 0. Then,
the dispersion law of electrons moving along a smooth surface has the form E0 =
~2k2
x
2m∗ . In this case, the
23701-4
The theory of electron states on the dynamically deformed adsorbed surface of a solid
region where the electrons localize Le(N0d) = 1/|δk0(N0d)| occupies a half of x ∈ (0;∞) space, since
|δk0(N0d)| → 0. At ς̃ , 0, the zero harmonic (n = 0) k2
0 =
2m∗
~2
E − k2
x . Then,
E =
~2
2m∗
(
k2
0 + k2
x
)
=
~2k2
x
2m∗
(
1 +
k2
0
k2
x
)
. (3.10)
Let k0 = k(0)0 + δk0. Then,
E =
~2k2
x
2m∗
[
1 +
(δk0)
2
k2
x
]
, k(0)0 = 0;
δk0 = −
ς̃2(N0d)k2
xq2
4
(
1
k1
+
1
k−1
)
−
iβ2qς̃2(N0d)
2
{(
q2 − k2
l
) (
1
k2
1 + kl2
+
1
k2
−1 + kl2
)
+2iklkxq
[
1
k1
(
k2
1 + kl2
) − 1
k−1
(
k2
−1 + kl2
) ] }
. (3.11)
Now, we find the quantity δk0 in the boundary cases: long-wave (kx � q/2 ), resonance (kx ∼ q/2 )
and short-wave (kx � q/2 ) cases.
In the long-wave approximation, the value δk0 has the form:
δk0 =
(
k2
x − 2β2) q2
4
(
1
k1
+
1
k−1
) [
ς̃2
0 + 2ς̃0δς̃(N0d)
]
, (3.12)
where k1 = i
√
q(q + 2kx), k−1 = i
√
q(q − 2kx).
In the resonance case, the dispersion equation has the form:
δk0δk−1 = −
ζ̃2
0 + 2ζ̃0δζ̃(N0d)
16
q4
(
1 −
8β2
qkl
)
,
where δk0 =
√
2m∗
~2
δE − qδkx , δk−1 =
√
2m∗
~2
δE + qδkx .
At the point kx =
q
2
(
kx = π
Lx
)
, the energy changes jump-like, i.e., there is a band gap whose value is
equal to
2δE = −
~2q4
16m∗
(
1 −
8β2
qkl
) [
ζ̃2
0 + 2ζ̃0δζ̃(N0d)
]
. (3.13)
As can be seen from formula (3.13), in the case of the absence of adsorbed atoms
[
δζ̃(N0d) = 0
]
, the
width of the band gap in the subsurface layer of the GaAs(100) semiconductor coincides with the results
of work [7].
In the short-wave approximation, the values δk0 and E are complex, i.e., the electron states are
quasi-stationary.
E = Re E + i Im E,
Re E =
~2k2
x
2m∗
[
1 +
(Re δk0)
2 − (Im δk0)
2
k2
x
]
,
Im E =
~2
2m∗
Re δk0 Im δk0 < 0,
where
Re δk0 = −
q2
k−1
[
ζ̃2
0 + 2ζ̃0δζ̃(N0d)
] (
k2
x + 2β2),
Im δk0 = −
q2
4 |k1 |
[
ζ̃2
0 + 2ζ̃0δζ̃(N0d)
] (
k2
x − 2β2),
k1 = i
√
q(q + 2kx) , k−1 =
√
q(2kx − q).
At the same time, the relaxation time is equal to τ = ~/|Im E |.
23701-5
R.M. Peleshchak, M.Ya. Seneta
4. Influence of the concentration of adsorbed atoms on the band gap
width and the electron density distribution on the surface of a mono-
crystal having a Zinc blende structure
Figure 1 shows a plot of the dependence of the band gap width on the concentration of adsorbed
atoms upon the plane of a monocrystal having a Zinc blende structure at the point of the Brillouin zone
kx = q/2 or kx = π/Lx . The upper and lower branches are determined according to the relations [11]:
E+
(
kx =
q
2
)
= −
~2q2
8m∗
{
1 +
q2
4
(
1 −
8β2
qkl
) [
ζ̃2
0 + 2ζ̃0δζ̃(N0d)
]}
, (4.1)
E−
(
kx =
q
2
)
= −
~2q2
8m∗
{
1 −
q2
4
(
1 −
8β2
qkl
) [
ζ̃2
0 + 2ζ̃0δζ̃(N0d)
]}
. (4.2)
The calculationwasmade for GaAs(100) semiconductor with the following parameter values [13, 15]:
ld = 2.9 nm; a = 0.565 nm; cl = 4400 m/s; ct = 2475 m/s; ρ = 5320 kg/m3; θd = 10 eV; N0d =
3 · 1013 cm−2; Dd = 5 · 10−2 cm2/s; ∂F
∂Nd
= 0.1 eV; m∗ = 6.1 · 10−32 kg; λ = 0.02 eV.
Figure 1 shows that the functional dependence of the band gap width on the concentration of adsorbed
atoms is nonmonotonous. There is observed an increase of a functional dependence δE = δE(N0d) on
the concentration interval of adatoms 0 < N0d 6 2.1 · 1012 cm−2. At this concentration interval, the
semiconductor band gap width increases by 8%. With a further increase of the concentration of adsorbed
atoms, the band gap width decreases. Moreover, at the interval 2.1 · 1012 < N0d 6 1013 cm−2, the
band gap width decreases by 56%. Such nonmonotonous dependence of δE = δE(N0d) is explained by
the nonmonotonous dependence of the roughness height ς̃ = ς̃(N0d) on the concentration of adsorbed
atoms [11].
The length of the spatial localization of an electron wave function Le = Le(N0d) in the long-wave
approximation is obtained by the relation:
L(N0d) =
4
q2
(
1
k1
+ 1
k−1
) [
ς̃2
0 + 2ς̃0δς̃(N0d)
] (
k2
x − 2β2) . (4.3)
We can see from formula (4.3), that the reduction of the roughness period Lx
(
q = 2π
Lx
)
along the
x-axis or an increase of the height ς̃(N0d) = ς̃0 + δς̃(N0d) of the adsorbed surface roughness leads to a
Figure 1. The band gap width on the surface of GaAs(100) semiconductor at the point kx = q/2 of the
Brillouin zone, depending on the concentration of adsorbed atoms.
23701-6
The theory of electron states on the dynamically deformed adsorbed surface of a solid
stronger localization of the electron wave function ψ0 = A0eikx x−z/L(N0d). A decrease of the length of an
electron de Broglie wave (an increase of kx) reduces to the same effect.
Figure 2 shows the dependence of the length of a spatial localization of the electron wave function
Le = Le(N0d) on the concentration of adsorbed atoms in the case of long-wave approximation.
Analyzing the graphic dependence in figure 2, we see that curve Le = Le(N0d) is nonmonotonous. At
small concentrations of adsorbed atoms (0 < N0d 6 2.1 ·1012 cm−2) with an increasing concentration, the
length of the spatial localization of the wave function of the electron decreases. With a further increase
of the concentration of adsorbed atoms, the function Le = Le(N0d) increases monotonously [12]. The
roughness of the two media division is caused by the appearance of surface electron states.
The wave function of the surface electronic states is determined by the following expression [5]
ψk = A0e−|δkz |z+i|kx x−ω ®k t | . From the normalization condition
∭
ψkψ
∗
k
dxdydz = 1, we determine the
amplitude where A0 =
√
2 |δkz |/S , where S = LxLy; Lx, Ly are the model sizes in x and y directions,
respectively.
The dependence of the wave function of surface electron states on the z coordinate leads to the
appearance of an inhomogeneous electron plasma in the z > z0 region. The basic parameters of this
plasma can be determined due to the characteristics of the surface roughness. We determine the change
of the concentration of electrons δn0(z) as:
δn0(z) =
∑
kx
ψkψ
∗
knkx =
2
S
∑
kx
nkz |δk0 |e−2 |δkz |z,
where nkx is the number of electrons with the wave vector kx ; the sum is carried out at all values of kx . In
this case, the minimum kx value is determined by the model sizes in the x-direction, that is, the value Lx ,
and the maximum is determined by the Fermi momentum ~kF. The total number of particles in the z > 0
region is equal to
∑
kx nkx .
The surface density nS =
∫∞
0 δn0(z)dz =
∑
kx nkx /S. For a degenerate electron gas at nkx = 0 and
nkx = 1, the values δn0(0) = 2ns |δk0 | are as follows:
- if kF < q/2 then δn0(0) = 2 |δk0 |
kF
2
4π ;
- if kF > q/2 then δn0(0) = 2 |δk0 |
kFq
2π2 .
A graphic representation of the dependence of the change of the concentration of electrons δn(0)
on the concentration of adsorbed atoms upon the z = 0 surface is given in figure 3. A nonmonotonous
character of the curves is determined by a nonmonotonous functional dependence of the height of the
roughnesses of semiconductor surfaces on the concentration of adsorbed atoms [10].
Figure 2.The length of the spatial localization of an electronwave function in the subsurface of GaAs(100)
semiconductor, depending on the concentration of adsorbed atoms at kx = q/3; q = 0.03 Å−1.
23701-7
R.M. Peleshchak, M.Ya. Seneta
Figure 3. The change of the concentration of electrons upon the z = 0 surface of the GaAs(100)
semiconductor, depending on the concentration of adsorbed atoms.
The maximum change of the concentration of electrons upon z = 0 surface of GaAs(100) semicon-
ductor is observed at N0d = 1.9 ·1012 cm−2. In particular, if kF > q/2 (kF = 0.06 Å−1; q/2 = 0.015 Å−1),
then δn(0) = 9.4 · 1017 cm−3; if kF < q/2 (kF = 0.006 Å−1) then δn(0) = 2.95 · 1018 cm−3. As the
concentrations of adsorbed atoms increases (N0d > 1.9 · 1012 cm−2), the change of the electron density
distribution upon the surface of GaAs(100) semiconductor monotonously decreases.
5. Conclusions
1. The dispersion relations for the spectra of surface electron states upon a dynamically deformed
adsorbed surface of GaAs (CdTe) semiconductor were derived in the long-wave (kx � q/2),
resonant (kx ∼ q/2) and short-wave (kx � q/2) approximations.
2. It was established that the energy band gap width upon a dynamically deformed adsorbed surface
of a semiconductor at the point kx = q/2 of the Brillouin zone, depending on the concentration of
adsorbed atoms, is of a nonmonotonous character.
3. It was shown that the dependence of the change of the concentration of electrons on the concentra-
tion of adsorbed atoms upon the surface is of a nonmonotonous character, determined by the height
of the inequality of the solid surface. In particular, the maximum change of the electron density
upon the surface of a GaAs(100) semiconductor is achieved at the concentration of the adsorbed
atoms N0d = 1.9 · 1012 cm−2.
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Теорiя електронних станiв на динамiчно деформованiй
адсорбованiй поверхнi твердого тiла
Р.М. Пелещак,М.Я. Сенета
Дрогобицький державний педагогiчний унiверситет iменi Iвана Франка,
вул. Iвана Франка, 24, 82100 Дрогобич, Україна
Отримано дисперсiйнi спiввiдношення для спектру поверхневих електронних станiв на динамiчно де-
формованiй адсорбованiй поверхнi монокристалу зi структурою цинкової обманки. Встановлено, що за-
лежностi ширини забороненої зони та концентрацiї електронiв на поверхнi твердого тiла вiд концентра-
цiї адсорбованих атомiв N0d мають немонотонний характер
Ключовi слова: електроннi стани, акустична квазiрелеєвська хвиля, адсорбованi атоми
23701-9
https://doi.org/10.5488/CMP.17.23601
https://doi.org/10.21272/jnep.9(3).03032
https://doi.org/10.21272/jnep.9(5).05023
https://doi.org/10.5488/CMP.19.43801
https://doi.org/10.5488/CMP.18.43801
https://doi.org/10.1134/1.1466485
Introduction
Formulation of the problem
Electron states upon the dynamically deformed adsorbed surface
Influence of the concentration of adsorbed atoms on the band gap width and the electron density distribution on the surface of a monocrystal having a Zinc blende structure
Conclusions
|