title: OSCILLATIONS OF THE COMBINED DENSITY OF STATES IN SEMICONDUCTORS WITH A NON-
text: 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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