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CALCULATION OF OSCILLATIONS OF THE DENSITY OF ENERGY STATES IN TWO-DIMENSIONAL
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Bog'liq ТўпламCALCULATION OF OSCILLATIONS OF THE DENSITY OF ENERGY STATES IN TWO-DIMENSIONAL
MATERIALS IN THE PRESENCE OF A LONGITUDINAL AND TRANSVERSE STRONG MAGNETIC FIELD
U.I.Erkaboev, R.G.Rakhimov
Namangan Institute of Engineering and Technology
Annotation. In this article, we investigated the effect of temperature and a quantizing
magnetic field on oscillations of the density of energy states in the conduction band of nanoscale
semiconductor structures. A new mathematical model has been developed for calculating the
temperature dependence of the oscillations of the density of states in a rectangular quantum well
under the influence of a transverse quantizing magnetic field. Using the proposed model, the
experimental results were explained at different temperatures and magnetic fields.
Keywords: semiconductor, nanoscale semiconductor structures, quantizing magnetic field,
quantum well, oscillation, density of energy states
According to the band theory of a solid, the wave function of a free electron, in the presence
of an external field, is a solution of the stationary Schrödinger equation with a parabolic dispersion
law:
2
2
*
( )
( )
( )
2
V r
r
E
r
m
(1)
Here, V(r) is the energy of free electrons in the presence of an external field, E is the energy
of charge carriers in the absence of an external field,
( )
r
is the wave function. The dependence of
the quantizing magnetic field on the wave function of electrons and the energy spectra of charge
carriers in two-dimensional electron gases is determined using equation (1), in which the
momentum operator should be replaced by the generalized momentum operator in a quantizing
magnetic field:
2
*
1
( )
( )
( )
2
i
eA
V z
r
E
r
m
(2)
Here,
A
is the vector potential of the induction of a strong magnetic field,
( )
B
rot A
. To
solve equation (2), the direction of the vector B is chosen in two different ways. In the first case, this
vector will be directed along the plane of the two-dimensional layer (along the X-axis) and
perpendicular to the Z-axis. For a longitudinal quantizing magnetic field, vector potential A can be
chosen in the form of
0,
, 0
A
Bz
.
k m
from the Schrödinger equation (2), for a deep
rectangular quantum well, takes the following form:
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