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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.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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