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Pg.221
papers did not consider the influence of microwave radiation absorption on quantum oscillation
phenomena.
The purpose of this section is to mathematical model the temperature dependence of
microwave magnetoabsorption oscillations in semiconductors under the action of an
electromagnetic wave and to study the effect of sample temperature on the results of experimental
data processing.
As is known, oscillations of the spectral density of states play an important role in determining
the Shubnikov-de Haas, de Haas-van Alphen oscillations and the quantum Hall effect in bulk and
nanoscale semiconductors.
Let us consider the Shubnikov-de Haas oscillations in narrow-gap semiconductors under the
action of temperature and a strong electromagnetic field. The power of absorbed microwave
radiation per unit volume is determined by the following expression:
2
E
P
E
(1)
Here,
is the conductivity of the semiconductor,
E
E
is the electric field strength of the wave.
In a quantizing magnetic field, the distribution function satisfies the kinetic equation:
0
N
N
N
N
z
z
N
f
f
f
f
eE
t
k
.
(2)
From (2), one can determine the current density of
zN
j
and the longitudinal electrical
conductivity of
zz
for each N-th quantum level:
2
0
N
z
zN
z
N
z
z
f
e E
j
k
dk
m
k
.
(3)
1
2
2
0
2
3
/ 2
( )
(2 )
( )
(
,
)
c
zz
c
N
s
N
N
f
E
m
e
E N E
H
dE
E
(4)
As seen from (4), in a quantizing magnetic field, the longitudinal conductivity of
zz
strongly
depends on the oscillations of the spectral density of states and the
(
)
E
relaxation time.
For a unit volume of a semiconductor, the following condition is satisfied:
1
zz
zz
zz
R
.
Here,
zz
is the longitudinal specific magnetoresistance.
In the general case, the relaxation time is determined by the following expression:
0
( )
r
E
E
(5)
where,
0
and
r
- have different values for different semiconductors.
Substituting expressions (3), (4) and (5) into (2), we obtain the following expression:
2
,
,
,
,
,
,
,
( )
k
i
E
i
E
P Gauss E E T
H E
N
Gauss E E T
H
E
E
(6)
2
0
0
(
, , )
(
, , )
,
,
,
( )
k
i
i
E
E
f E
T
f E
T
P
H E
N
H
E
E
E
E
(7)
2
,
,
,
,
,
,
,
( )
k
i
E
i
E
P Lorentz E E T
H E
N
Lorentz E E T
H
E
E
(8)
We differentiate expressions (6), (7) and (8) with respect to H, that is,
, , ,
k
E
dP
H T E E
dH
.
Namangan Institute of Engineering and Technology
nammti.uz
10.25.2023
Pg.222
From here, we obtain the expression for the dependence of the Shubnikov-de Haas
oscillations on the absorption of microwave radiation and temperature in narrow-gap
semiconductors:
2
2
3
2
3
2
3
2
1
,
,
,
,
2
,
,
,
1
( )
1
2
(
)
2
( )
;
(2)
i
E
k
i
E
g
AH N
dP Gauss E E T
H E
C
N
Gauss E E T
H
E E
dH
E
E
N
AH
E
e
m
A
C
mc
(9)
0
2
0
2
(
, , )
1
,
,
(
, , )
2
,
1
( )
1
2
(
)
2
i
E
k
i
E
g
f E
T
dP
H E
AH N
f E
T
E
C
N
H
E
E
dH
E
E
E
N
AH
E
(10)
2
2
1
,
,
,
,
2
,
,
,
1
( )
1
2
(
)
2
i
E
k
i
E
g
AH N
dP Lorentz E E T
H E
C
N
Lorentz E E T
H
E E
dH
E
E
N
AH
E
(11)
Here,
k
P
is the microwave absorption power for the Kane model.
Using (8), (9) and (10) one can determine the
, , ,
E
dP H T E E
dH
oscillations for wide-gap
semiconductors. The working formula for the Shubnikov-de Haas oscillations in the absorption of
microwave radiation and temperature for wide-gap materials is as follows:
max
2
0
1
1
, , ,
1
2
1
,
,
( )
1
1
(
)
2
2
k
m
E
i
E
N
i
AH N
dP H T E E
C
Gauss E E T
E E
dH
E
N
AH
E
N
AH
(12)
From here, we have the opportunity, using formulas (9), (10), (11) and (12), to calculate the
, , ,
E
dP H T E E
dH
oscillations in narrow-gap and wide-gap semiconductors in a strong
electromagnetic field and at various temperatures.
Thus, a new mathematical model has been created to determine the oscillations of the
absorption of microwave radiation in narrow-gap semiconductors. On the basis of the proposed
model, it is possible to investigate to explain the experimental oscillations at different temperatures.
Let us consider Shubnikov-de Haas oscillations in the presence of absorption of microwave
radiation in narrow-gap semiconductors. In particular, we will obtain the
, , ,
k
E
dP
H T E E
dH
oscillation
plot for InSb using formula (11).
Fig.6 shows the dependence of
dP
dH
on the magnetic field strength H in InSb
(0)
0.234
Eg
at T=3 K and
3
10
E
V
E
sm
. In this case,
E
E
const
.
Namangan Institute of Engineering and Technology
nammti.uz
10.25.2023
Pg.223
Fig.1. Oscillations
dP
dH
in InSb at temperature T=3 K and electromagnetic wave strength
3
10
E
V
E
sm
, calculated using formula (11).
References:
1.
Erkaboev, U.I., Rakhimov, R.G. Determination of the dependence of the oscillation of
transverse electrical conductivity and magnetoresistance on temperature in heterostructures based
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2.
Erkaboev, U.I., Rakhimov, R.G. Simulation of temperature dependence of oscillations of
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Electrical Engineering, Electronics and Energy, 2023, 5, 100236.
3.
Gulyamov, G., Erkaboev, U.I., Rakhimov, R.G., Mirzaev, J.I., Sayidov, N.A. Determination
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4.
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5.
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x
Ga
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