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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.233
INFLUENCE OF A QUANTIZING MAGNETIC FIELD ON THE FERMI ENERGY OSCILLATIONS IN TWO-
DIMENSIONAL SEMICONDUCTORS
R.G.Rakhimov
Namangan Institute of Engineering and Technology
Annotation. In this article we investigated the effects of quantizing magnetic field and
temperature on Fermi energy oscillations in nanoscale semiconductor materials. It is shown that
the Fermi energy of a nanoscale semiconductor material in a quantized magnetic field is quantized.
The distribution of the Fermi-Dirac function is calculated in low-dimensional semiconductors at
weak magnetic fields and high temperatures. The proposed theory explains the experimental results
in two-dimensional semiconductor structures with a parabolic dispersion law.
Keywords: semiconductor, Fermi energy, quantizing magnetic field, dispersion law, two-
dimensional semiconductor structure, 2D electron gas.
In k-space isoenergic surfaces E(k)=const are closed and are represented in the form of a
sphere. The allowed energy states have a constant density V/8π
3
and are distributed in k-space.
Here, V is the volume of the crystal. Since two opposite orientations of the spin of the electron state
are responsible for each value of k, then the wave numbers of all states that will be filled have values
no more than k
F
in the volume of the crystal V, according to the Pauli principle and k
F
is determined:
3
3
3
4
2
3
8
d
F
V
k
N
(1)
From here
1
2
3
3
3
d
F
N
k
V
(2)
Here, N
3d
is the electron concentration for a three-dimensional electron gas.
If the system of electrons is due to the Fermi-Dirac statistics, then the energy in the ground
state, i.e., at absolute temperature, is called maximum:
2
2
2
F
F
k
E
m
(3)
E
F
- called Fermi energy for 3D electron gas. The Fermi surface will have a spherical shape with
a radius of k
F
for the isotopic dispersion law. The expressions given above were obtained only for
bulk materials and do not consider changes in the oscillations of the Fermi energy in two-
dimensional electron gases.
Now, consider the dependence of the Fermi energy on the quantizing magnetic field in two-
dimensional electron gases. In the absence of a magnetic field in two-dimensional electron gases,
the electron energy is quantized along the Z-axis, so the electron moves freely only in the XY plane.
These quantizations are called dimensional quantization. However, if the magnetic induction B is
directed perpendicular to the XY plane, then the free energy of the electron is also quantized along
the XY plane.
The question arises: how will the Fermi energy change in two-dimensional electron gases in
the presence of a quantizing magnetic field.
For a 2D electron gas, the allowed energy states have a constant density S/4π
2
and are
distributed in the XY plane. Here, S is the surface area of the crystal. Then, using formulas (1) and
(2), we determine the electron concentration for a two-dimensional electron gas:
2
2
2
2
2
4
4
d
F
L
N
k
(4)
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