title: Namangan Institute of Engineering and Technology
text: Namangan Institute of Engineering and Technology
  nammti.uz
  10.25.2023
  Pg.221
  papers did not consider the influence of microwave radiation absorption on quantum oscillation
  phenomena.
  The purpose of this section is to mathematical model the temperature dependence of
  microwave magnetoabsorption oscillations in semiconductors under the action of an
  electromagnetic wave and to study the effect of sample temperature on the results of experimental
  data processing.
  As is known, oscillations of the spectral density of states play an important role in determining
  the Shubnikov-de Haas, de Haas-van Alphen oscillations and the quantum Hall effect in bulk and
  nanoscale semiconductors.
  Let us consider the Shubnikov-de Haas oscillations in narrow-gap semiconductors under the
  action of temperature and a strong electromagnetic field. The power of absorbed microwave
  radiation per unit volume is determined by the following expression:
  2
  E
  P
  E
  (1)
  Here,
  is the conductivity of the semiconductor,
  E
  E
  is the electric field strength of the wave.
  In a quantizing magnetic field, the distribution function satisfies the kinetic equation:
  0
  N
  N
  N
  N
  z
  z
  N
  f
  f
  f
  f
  eE
  t
  k
  .
  (2)
  From (2), one can determine the current density of
  zN
  j
  and the longitudinal electrical
  conductivity of
  zz
  for each N-th quantum level:
  2
  0
  N
  z
  zN
  z
  N
  z
  z
  f
  e E
  j
  k
  dk
  m
  k
  .
  (3)
  1
  2
  2
  0
  2
  3
  / 2
  ( )
  (2 )
  ( )
  (
  ,
  )
  c
  zz
  c
  N
  s
  N
  N
  f
  E
  m
  e
  E N E
  H
  dE
  E
  (4)
  As seen from (4), in a quantizing magnetic field, the longitudinal conductivity of
  zz
  strongly
  depends on the oscillations of the spectral density of states and the
  (
  )
  E
  relaxation time.
  For a unit volume of a semiconductor, the following condition is satisfied:
  1
  zz
  zz
  zz
  R
  .
  Here,
  zz
  is the longitudinal specific magnetoresistance.
  In the general case, the relaxation time is determined by the following expression:
  0
  ( )
  r
  E
  E
  (5)
  where,
  0
  and
  r
  - have different values for different semiconductors.
  Substituting expressions (3), (4) and (5) into (2), we obtain the following expression:
  2
  ,
  ,
  ,
  ,
  ,
  ,
  ,
  ( )
  k
  i
  E
  i
  E
  P Gauss E E T
  H E
  N
  Gauss E E T
  H
  E
  E
  (6)
  2
  0
  0
  (
  , , )
  (
  , , )
  ,
  ,
  ,
  ( )
  k
  i
  i
  E
  E
  f E
  T
  f E
  T
  P
  H E
  N
  H
  E
  E
  E
  E
  (7)
  2
  ,
  ,
  ,
  ,
  ,
  ,
  ,
  ( )
  k
  i
  E
  i
  E
  P Lorentz E E T
  H E
  N
  Lorentz E E T
  H
  E
  E
  (8)
  We differentiate expressions (6), (7) and (8) with respect to H, that is,
  , , ,
  k
  E
  dP
  H T E E
  dH
  .
  Namangan Institute of Engineering and Technology
  nammti.uz
  10.25.2023
  Pg.222
  From here, we obtain the expression for the dependence of the Shubnikov-de Haas
  oscillations on the absorption of microwave radiation and temperature in narrow-gap
  semiconductors:
  2
  2
  3
  2
  3
  2
  3
  2
  1
  ,
  ,
  ,
  ,
  2
  ,
  ,
  ,
  1
  ( )
  1
  2
  (
  )
  2
  ( )
  ;
  (2)
  i
  E
  k
  i
  E
  g
  AH N
  dP Gauss E E T
  H E
  C
  N
  Gauss E E T
  H
  E E
  dH
  E
  E
  N
  AH
  E
  e
  m
  A
  C
  mc
  (9)
  0
  2
  0
  2
  (
  , , )
  1
  ,
  ,
  (
  , , )
  2
  ,
  1
  ( )
  1
  2
  (
  )
  2
  i
  E
  k
  i
  E
  g
  f E
  T
  dP
  H E
  AH N
  f E
  T
  E
  C
  N
  H
  E
  E
  dH
  E
  E
  E
  N
  AH
  E
  (10)
  2
  2
  1
  ,
  ,
  ,
  ,
  2
  ,
  ,
  ,
  1
  ( )
  1
  2
  (
  )
  2
  i
  E
  k
  i
  E
  g
  AH N
  dP Lorentz E E T
  H E
  C
  N
  Lorentz E E T
  H
  E E
  dH
  E
  E
  N
  AH
  E
  (11)
  Here,
  k
  P
  is the microwave absorption power for the Kane model.
  Using (8), (9) and (10) one can determine the
  , , ,
  E
  dP H T E E
  dH
  oscillations for wide-gap
  semiconductors. The working formula for the Shubnikov-de Haas oscillations in the absorption of
  microwave radiation and temperature for wide-gap materials is as follows:
  max
  2
  0
  1
  1
  , , ,
  1
  2
  1
  ,
  ,
  ( )
  1
  1
  (
  )
  2
  2
  k
  m
  E
  i
  E
  N
  i
  AH N
  dP H T E E
  C
  Gauss E E T
  E E
  dH
  E
  N
  AH
  E
  N
  AH
  (12)
  From here, we have the opportunity, using formulas (9), (10), (11) and (12), to calculate the
  , , ,
  E
  dP H T E E
  dH
  oscillations in narrow-gap and wide-gap semiconductors in a strong
  electromagnetic field and at various temperatures.
  Thus, a new mathematical model has been created to determine the oscillations of the
  absorption of microwave radiation in narrow-gap semiconductors. On the basis of the proposed
  model, it is possible to investigate to explain the experimental oscillations at different temperatures.
  Let us consider Shubnikov-de Haas oscillations in the presence of absorption of microwave
  radiation in narrow-gap semiconductors. In particular, we will obtain the
  , , ,
  k
  E
  dP
  H T E E
  dH
  oscillation
  plot for InSb using formula (11).
  Fig.6 shows the dependence of
  dP
  dH
  on the magnetic field strength H in InSb
  (0)
  0.234
  Eg
  at T=3 K and
  3
  10
  E
  V
  E
  sm
  . In this case,
  E
  E
  const
  .
  Namangan Institute of Engineering and Technology
  nammti.uz
  10.25.2023
  Pg.223
  Fig.1. Oscillations
  dP
  dH
  in InSb at temperature T=3 K and electromagnetic wave strength
  3
  10
  E
  V
  E
  sm
  , calculated using formula (11).
  References:
  1.
  Erkaboev, U.I., Rakhimov, R.G. Determination of the dependence of the oscillation of
  transverse electrical conductivity and magnetoresistance on temperature in heterostructures based
  on quantum wells. East European Journal of Physics, 2023(3), pp.133–145.
  2.
  Erkaboev, U.I., Rakhimov, R.G. Simulation of temperature dependence of oscillations of
  longitudinal magnetoresistance in nanoelectronic semiconductor materials. e-Prime - Advances in
  Electrical Engineering, Electronics and Energy, 2023, 5, 100236.
  3.
  Gulyamov, G., Erkaboev, U.I., Rakhimov, R.G., Mirzaev, J.I., Sayidov, N.A. Determination
  of the dependence of the two-dimensional combined density of states on external factors in
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  4.
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  5.
  Erkaboev, U.I., Sayidov, N.A., Negmatov, U.M., Mirzaev, J.I., Rakhimov, R.G. Influence
  temperature and strong magnetic field on oscillations of density of energy states in heterostructures
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  6.
  Erkaboev, U.I., Sayidov, N.A., Negmatov, U.M., Rakhimov, R.G., Mirzaev, J.I. Temperature
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  x
  Ga
  1-x
  As quantum well in presence of transverse strong
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  7.
  Erkaboev, U., Rakhimov, R., Mirzaev, J., Sayidov N., Negmatov, U., Abduxalimov, M.
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  8.
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