title: Namangan Institute of Engineering and Technology
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  Namangan Institute of Engineering and Technology Pdf ko'rish
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  Bog'liq ТўпламNamangan Institute of Engineering and Technology
  nammti.uz
  10.25.2023
  Pg.209
  0
  1
  exp
  k m
  n
  r
  ik r
  z
  z
  S
  (3)
  This function describes localized motion in the YZ plane and the state of motion of a free
  electron along the X axis. In equation (3), the
  0
  n
  z
  z
  function is responsible for localized
  motion. Then the solution to equation (3) will be as follows:
  2
  2
  *
  2
  2
  0
  0
  0
  *
  2
  1
  ( )
  (
  )
  (
  )
  (
  )
  2
  2
  c
  n
  n
  n
  d
  V z
  m
  z
  z
  z
  z
  E
  z
  z
  m dz
  (4)
  Here,
  0
  *
  ,
  y
  c
  k
  eB
  z
  eB
  m
  . Equation (4) is called the equation of a quantum harmonic
  oscillator, the motion of which is additionally limited by a quantum well, and E
  n
  is a discrete level.
  In a quantizing magnetic field, if the width of the quantum well increases, the energy spectrum
  of free electrons will increase. That is,
  a
  eB
  . Here, a is the width of the quantum well,
  is
  the magnetic length, which is equal in magnitude to the radius of the characteristic orbit of an
  electron in a quantizing magnetic field. Hence, the discrete energy levels E
  n
  will be equal to the
  energies of the harmonic quantum oscillator:
  1
  ,
  0,1, 2, 3....
  2
  N
  c
  E
  N
  N
  (5)
  According to equation (2), the velocity and momentum of charge carriers in the direction of
  the quantizing magnetic field can take any values. In other words, the motion of free electrons and
  holes in the direction of the XY plane (i.e., along the X axis) is not quantized. Hence, the total energy
  of free electrons in two-dimensional electron gases in the presence of a magnetic field directed
  along the X axis is determined by the following expression:
  2
  2
  1
  2
  2
  X
  N
  c
  k
  E
  N
  m
  (6)
  Where,
  1
  2
  c
  N
  is the energy of motion of a free electron in the YZ plane, these energies
  are called discrete Landau levels.
  2
  2
  2
  X
  k
  m
  is the energy of continuous motion along the X axis. Thus,
  in the presence of a longitudinal magnetic field, due to the quantization of the orbital motion of
  charge carriers in the YZ plane, the allowed energy zone is split into one-dimensional magnetic
  subbands, that is, into discrete Landau levels.
  In three-dimensional and two-dimensional electron gases, a change in the energy spectrum
  of charge carriers leads to a change in the oscillations of the density of states in a quantizing
  magnetic field.
  Now, let us first calculate the oscillations of the density of energy states in two-dimensional
  electron gases in the presence of a longitudinal strong magnetic field. When the width of the
  quantum well becomes comparable to the de Broglie wavelength, in two-dimensional
  semiconductor materials, then quantization occurs. That is,
  Z
  D
  L
  and
  Y
  Z
  L
  L
  . Hence, in the
  YZ plane, the cyclotron mass is calculated by the expression:
  2
  Y
  c
  L
  m
  E
  (7)
  For a parabolic dispersion law, the effective cyclotron mass will be constant. The energy in the
  interval between the two Landau levels is
  c
  E
  . Hence, for a two-dimensional semiconductor
  material, we find the difference in the section length of two isoenergy surfaces:
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