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Namangan Institute of Engineering and Technology
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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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