OSCILLATIONS OF THE COMBINED DENSITY OF STATES IN SEMICONDUCTORS WITH A NON-
QUADRATIC DISPERSION LAW
R.G.Rakhimov
Namangan Institute of Engineering and Technology
Annotation. The development of technology for the growth of semiconductor nanostructures
has led to the creation of highly efficient electronic and optoelectronic devices. It became possible
to design multilayer structures with the necessary profile of localization, distribution of charge
carriers and electronic spectrum. This progress would not have been possible without the
development of methods for calculating the band structure and intensive experimental studies of
the electrical, magnetic, and optical properties of single-crystal semiconductors. In this article,
oscillations of interband magneto-optical absorption in semiconductors with the Kane dispersion
law are considered. The change in the oscillations of the combined density of states with respect to
the photon energy is compared for different Landau levels in parabolic and non-parabolic bands.
Keywords: semiconductor, electron gas, oscillation, microwave, Landau levels, electric field,
mathematical model, Shubnikov-de Haas oscillations.
Namangan Institute of Engineering and Technology
nammti.uz
10.25.2023
Pg.218
One of the most effective methods for studying the band structure of semiconductors is the
study of magnetoabsorption, which can be caused by both intraband and interband transitions. In
weak magnetic fields, this method gives the value of the effective mass of carriers at the Fermi level,
and in quantizing fields, it makes it possible to determine the distance between Landau levels
between which optical transitions occur [1–4]. It can be used to reveal the non-parabolicity of the
dispersion law and thus obtain information about the features of the band structure of new
materials. In addition, this method makes it possible to reveal "subtle" effects, for example, the
influence of electron-electron interaction [5] and the absence of an inversion center in the unit cell
[6]. To obtain detailed information in structures with a strong nonparabolicity, it is necessary to
study magnetoabsorption in a wide energy range, which, in turn, requires the use of high magnetic
fields.
The purpose of this study is to calculate the oscillations of interband magnetoabsorption in
semiconductors with a nonparabolic dispersion law.
The nonparabolicity of the conduction band for electrons can be written as the following
expression [7]:
2
2
2
1
1
4
(1)
2
2
2
2
g
z
c
g
g
c
n
E
k
E
E
E
N
m
The bottom of the conduction band was chosen as the energy reference point. Hence, the
non-quadratic dispersion law for holes is written in a similar way:
2
2
2
1
1
4
(2)
2
2
2
2
p z
g
v
g
g
g
c
p
k
E
E
Ec
E
E
E
N
m
Here, E
v
is the energy of the valence band ceiling.
The quasi-momentum conservation law will look like this: k
n
=k
c
=k. Where, k
n
, k
c
are the wave
vectors of holes and electrons, respectively [8].
For such energy bands, expressions (1) and (2) imply:
2
2
2
1
( , )
( , )
4
(3)
2
2
z
c
z
v
z
g
g
c
k
E
h
E k B
E k B
E
E
N
m
Here,
h
is the energy of the absorbed photon.
Let us now find the combined number of states with energies in the interval between two
Landau levels. Using the expression for the cyclotron mass, from equation (3), we determine k
z
:
1
2
2
(2 )
(
)
1
(
4(
)
)
(4)
2
z
g
c
g
m
h
k
E
N
E
and
2
Z
Z
Z
k
n
L
(5)
According to expressions (4) and (5), the number of states in the energy range from
1
(
)
2
c
N
to E
1
2
2
1
2
(
)
1
(
4(
)
)
(6)
2
2
z
Z
g
c
g
m
L
h
n
E
N
E
The combined density of states (CDS) with energies less than E is equal to
max
3
2
2
3
0
2
3
2
(
)
1
( , )
(
4(
)
)
(7)
2
2
N
x
y
z
c
g
c
N
g
L L L m
h
N E B
E
N
E
As a result, we will determine the CDS per unit volume with the Kane dispersion law:
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