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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.210
2
c
Y
c
m
L
(8)
The number of states for quantization, in the presence of a longitudinal quantizing magnetic
field, in the YZ plane, due to the cyclic conditions, is equal to
2
Y
L
. In expression (8), the number of
states between two quantum orbits is:
2
2
2
c
Y
Y
Y
c
c
Y
m
L
L
L
m
L
(9)
From formula (6) we find k
x
1/ 2
1/ 2
2
1
2
X
N
c
m
k
E
N
(10)
In the presence of a longitudinal strong magnetic field, the movement of charge carriers along
the X axis is not quantized in k
x
and takes the following form:
2
X
X
X
k
n
L
(11)
With the help formulas (10) and (11), in the energy range from
1
2
c
N
to E, it is possible
to determine the number of states along the X axis:
1/ 2
1/ 2
2
1
2
2
X
X
N
c
L
m
n
E
N
(12)
Using formulas (9) and (12), in the presence of a longitudinal magnetic field and for a
rectangular quantum well, we obtain the total number of quantum states by the following
expression:
max
3 / 2
1/ 2
2
1/ 2
2
0
1
( ,
)
2
2
N
X
Y
d
X
c
N
c
N
L L
m
N
E H
E
N
(13)
Differentiate expression (13) with respect to energy E per unit area (L
X
L
Y
=1) and define
2
( ,
)
d
S X
N
E H
:
max
3 / 2
2
1/ 2
2
1/ 2
0
1
( ,
)
2
1
2
N
d
S X
c
N
N
c
m
N
E H
E
N
(14)
This formula is called the density of energy states, in two-dimensional electron gases (that is,
in a rectangular quantum well), in the presence of a longitudinal quantizing magnetic field. This
formula is analogous to the quantum thread equation (Fig.1). Obviously, with a longitudinal
quantizing magnetic field, in a two-dimensional electron gas, the energies of free electrons in the
YZ plane can take only some fixed values, but the electron energy along the X axis remains free (not
quantized). Formula (14), at Н
0, turns into work:
2
2
( )
d
S
m
N
E
(15)
This formula describes the density of energy states in two-dimensional electron gases in the
absence of a magnetic field (Fig.1). In conclusion, we note that the main feature of the oscillation of
the density of energy states for a two-dimensional electron gas, in the presence of a longitudinal
strong magnetic field, is that it does not depend on the width of the quantum well or the size of the
size quantization and is determined only by the magnitude of the magnetic field induction and
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