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.234
  From here:
  1
  2
  2
  2
  1
  2
  F
  d
  d
  N
  k
  L
  (5)
  Now, we calculate the Fermi energy for a two-dimensional electron gas with parabolic law.
  Substituting (5) to (6), one can determine the Fermi energy in two-dimensional electron gases in the
  absence of a magnetic field:
  2
  2
  2
  2
  2
  2
  4
  d
  d
  p
  N
  m
  mL
  (6)
  Here, N
  2d
  is the concentration of electrons in a two-dimensional electron gas, L
  2
  is the surface
  of the plane of motion,
  p
  is the Fermi momentum.
  In the motion of a plane perpendicular to the magnetic field, the classical trajectories of
  electrons are circles. In quantum physics, such trajectories of electrons (periodic rotation of an
  electron) are equidistant discrete Landau levels:
  1
  2
  n
  c
  L
  E
  n
  (7)
  Where n
  L
  is the number of Landau levels.
  c
  eH
  mc
  - cyclotron frequency.
  In three-dimensional semiconductors, a continuous quadratic energy spectrum of the
  2
  2
  z
  p
  m
  is
  added to the energy spectrum of formula (7). However, in two-dimensional semiconductors, the
  movement of electrons along the Z-axis is quantized.
  Indeed, the thickness of the quantum well d is covered by the dimensional quantization
  condition, in other words, the thickness is relatively close to the de Broglie wavelength of the
  electron in the crystal. The movement of an electron along the Z axis is calculated from the potential
  V
  z
  :
  0, 0
  ,
  ( )
  ,
  0,
  z
  d
  V z
  z
  z
  d
  (8)
  In the absence of a magnetic field in two-dimensional electron gases, the normalized wave
  functions of particles have the following form:
  ,
  ,
  1
  2
  1
  1
  ( , , )
  exp(
  )
  exp(
  )
  ( )
  kfx kfy nfz
  fx
  fy
  nz
  f
  f
  x y z
  ik x
  ik y
  z
  L
  L
  (9)
  Where k
  fx
  , k
  fy
  are the wave numbers for the Fermi energy of electrons, n
  fz
  is the number of
  dimensional quantizers along the Z axis.
  In formula (9), the normalized functions in accordance with (8) are written in the following
  form:
  2
  ( )
  sin
  ,
  1, 2, 3...
  nz
  nz
  z
  n
  d
  d
  (10)
  The Fermi energy of electrons corresponding to state (9) will be
  2
  2
  2
  2
  2
  2
  2
  ,
  ,
  2
  2
  fx
  fy
  fz
  fx
  fy
  fz
  n
  E k
  k
  n
  k
  k
  m
  md
  (11)
  Substituting expressions (7), (11) into (6), we obtain the following formula in the presence of
  a magnetic field:
  2
  2
  2
  2
  2
  2
  2
  ( )
  4
  2
  d
  fz
  F
  n
  N
  H
  H
  mL
  md
  (12)
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