On Temperature Dependence of Longitudinal Electrical Conductivity Oscillations in Narrow-gap Electronic Semiconductors

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JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ
Vol. 12 No 3, 03012(5pp) (2020) Том 12 № 3, 03012(5cc) (2020)
2077-6772/2020/12(3)03012(5) 03012-1  2020 Sumy State University
On Temperature Dependence of Longitudinal Electrical Conductivity Oscillations
in Narrow-gap Electronic Semiconductors
G. Gulyamov2, U.I. Erkaboev1,
*
, R.G. Rakhimov1, J.I. Mirzaev1
1 Namangan Institute of Engineering and Technology, 160115 Namangan, Uzbekistan
2 Namangan Engineering – Construction Institute, 160103 Namangan, Uzbekistan
(Received 08 November 2019; revised manuscript received 15 June 2020; published online 25 June 2020)
Oscillations of longitudinal electrical conductivity, oscillations of magnetic susceptibility and oscilla-
tions of electronic heat capacity for narrow-gap electronic semiconductors are considered. A theory is con-
structed of the temperature dependence of quantum oscillation phenomena in narrow-gap electronic semi-
conductors, taking into account the thermal smearing of Landau levels. Oscillations of longitudinal electri-
cal conductivity in narrow-gap electronic semiconductors at various temperatures are studied. An integral
expression is obtained for the longitudinal conductivity in narrow-gap electronic semiconductors, taking in-
to account the diffuse broadening of the Landau levels. A formula is obtained for the dependence of the o s-
cillations of longitudinal electrical conductivity on the band gap of narrow-gap semiconductors. The theory
is compared with the experimental results of Bi2Se3. A theory is constructed of the temperature depend-
ence of the magnetic susceptibility oscillations for narrow-gap electronic semiconductors. Using these oscil-
lations of magnetic susceptibility, the cyclotron effective masses of electrons are determined. The calcula-
tion results are compared with experimental data. The proposed model explains the experimental results
in p-Bi2 – xFexTe3 at different temperatures.
Keywords: Oscillations of electronic heat capacity, Oscillations of magnetic susceptibility and oscillations
of electrical conductivity, Electronic narrow-gap semiconductors, Cyclotron effective mass.
DOI: 10.21272/jnep.12(3).03012
PACS numbers: 71.20. – b, 71.28. + d
*
Erkaboev1983@mail.ru
1. INTRODUCTION
It is known that, using quantum oscillation phe-
nomena it is possible to determine the basic physical
quantities (longitudinal conductivity, magnetic suscep-
tibility, thermoelectric power and other transport phe-
nomena) in electronic and nanoscale semiconductors. In
particular, oscillations of longitudinal electrical conduc-
tivity and oscillations of magnetic susceptibility provide
valuable information on the energy spectra of free elec-
trons in electronic semiconductor structures. In a
strong magnetic field, the longitudinal conductivity is
determined using the following expression [1]:
11/2
22
2
200
2 2 3
/2 / 2
( ) ( )
(2 ) 1
( ) ( )
22
cc
zz c Z N Z c c N
NN
f E f E
e m e
k E dk E N E dE
m E E

     





     







(1)
Here, N is the number of Landau levels,

c is the cyclo-
tron frequency,

N(E) is the relaxation time, E is the
energy of a free electron in a quantizing magnetic field,
f0(E)/E is the energy derivative of the Fermi-Dirac
function, which takes on the character of a delta func-
tion at low temperatures. From formula (1) it is seen
that the effective mass is a constant, that is, this ex-
pression is applicable only for the parabolic dispersion
law. But, if the dispersion law is nonparabolic (Kane’s
dispersion law), then the effective mass is strongly
dependent on energy (m(E)). It is known that, just in
narrow-gap electronic semiconductors, the effective
mass depends on the energy (m(E)) [2-4]. Recently,
many experiments have been performed on oscillations
of longitudinal electrical conductivity and oscillations of
magnetic susceptibility in narrow-gap electronic semi-
conductors [5-8]. In these works, quantum oscillation
phenomena at a constant temperature were studied.
However, until now, the theory of temperature depend-
ence has not been developed for these processes in nar-
row-gap electronic semiconductors. The study of quan-
tum oscillation phenomena associated with equilibrium
and nonequilibrium quantities allows us to identify
new properties of massive, low-dimensional, and elec-
tronic semiconductors. Such values include longitudinal
magnetic susceptibility, electronic heat capacity, ther-
modynamic potential, electrical conductivity, and oth-
ers. In a quantizing magnetic field and at low tempera-
tures, such quantities oscillate. All quantum oscillation
phenomena depend on the spectral density of energy
states in semiconductors. The spectral density of states
in semiconductors is determined by the energy spec-
trum of electrons and holes. As experiments show, the
density of states depends on temperature. The temper-
ature dependence is explained by thermal broadening
of discrete levels in the sample. As shown in [9, 10], the
density of states at low temperatures from a continuous
spectrum turns into a discrete one. This is because, at
low temperatures the thermal broadening of the dis-
crete levels decreases, disappears, and the continuous
spectrum turns into discrete levels. The temperature
dependence of the spectral density of states in a quan-
tizing magnetic field was considered in [9, 10]. It is
shown that with increasing temperature, the density of

G. GULYAMOV, U.I. ERKABOEV, R.G. RAKHIMOV, J.I. MIRZAEV J. NANO- ELECTRON. PHYS. 12, 03012 (2020)
03012-2
states in a strong field turns into a continuous spec-
trum of the density of states of electrons in the absence
of a magnetic field. In this case, with increasing tem-
perature, in the collision of electrons, the thermal mo-
tion smears the discrete Landau levels and turns them
into a continuous spectrum of density of states. The
discontinuous nature of the function, the spectral den-
sity of states near the points E  (N + 1/2)ћ
c
leads to
significant features of the phenomena of transport and
magnetic susceptibility with the parabolic dispersion
law. In works [11-13], oscillations of the longitudinal
magnetic susceptibility were observed in wide-gap and
narrow-gap semiconductors at constant temperatures.
And also, in these works the temperature dependence
of the oscillation amplitude of the longitudinal magnet-
ic susceptibility was considered in a strong magnetic
field. However, in the above works, a concrete theory of
oscillations of the longitudinal magnetic susceptibility
in narrow-gap semiconductors, taking into account the
temperature dependence of the spectral density of
states, was not constructed.
The aim of this work is to construct a theory of the
temperature dependence of the oscillations of longitu-
dinal electric conductivity and oscillations of the mag-
netic susceptibility in narrow-gap electronic semicon-
ductors, taking into account the thermal broadening of
the Landau levels.
2. THEORY
2.1 Dependence of Oscillations of Longitudinal
Electrical Conductivity on the Band Gap in
Narrow-gap Electronic Semiconductors
Let us consider oscillations of longitudinal electrical
conductivity in narrow-gap electronic semiconductors.
In a quantizing magnetic field, the electron energy of
the conduction band is determined by the following
expression [1]:
22
20
11
4
2 2 2 2 2
gzB
N g g c
n
Ek g H
E E E N m





      




(2)
where EN is the electron energy of the conduction band
in a quantizing magnetic field with a nonparabolic
dispersion law, Eg is the band gap of narrow-gap semi-
conductors.
We define kz from formula (2) excluding spin. From
here, we find kz2 and determine the wave function
along the Z-axis with the nonparabolic dispersion law:
2
21
2
N
z N c
g
E
m
k E N
E


   


(3)
Differentiating formula (3), we obtain the following
expression and we determine the expression for the
longitudinal conductivity in narrow-gap electronic sem-
iconductors:
11/2
2
2
2
0
23
/2
()
2
(2 ) 1
1 ( )
2
c
NN
zz c N c N
Ngg
fE
EE
me E N E dE
E E E

   

    

      



 


   


(4)
Now, let us analyze the longitudinal conductivity
oscillations for various narrow-gap electronic semicon-
ductors with a nonparabolic dispersion law. Formula
(4) allows to graphically analyze the dependence of

zz(E, H, T, Eg(T)). Fig. 1 shows the dependence of the
longitudinal conductivity oscillations on a strong mag-
netic field in InSb. Here, T  1 K, Eg  0.234 eV and the
number of Landau levels in the conduction band is
N  10 [14, 15]. As can be seen from Fig. 1, with in-
creasing magnetic field induction, the amplitudes of
oscillations of the longitudinal conductivity increase. It
can also be seen from the figure that the amplitude of
the conductivity oscillation is 10. Each oscillation of the
amplitude of the longitudinal conductivity corresponds
to one discrete Landau level.
With the help of formula (4), we compare the oscil-
lations of the longitudinal electrical conductivity for
various values of the band gap. In Fig. 1, oscillation
phenomena are presented for InSb and InAs at a con-
stant temperature. Here, Т  4 K, Eg  0.234 eV [15] for
InSb, Eg  0.414 eV [15] for InAs and the number of
Landau levels in the conduction band is equal to
N  12. As can be seen from Fig. 1, with an increase in
the band gap, one can observe a downward movement
of the oscillation graph. For example, longitudinal elec-
trical conductivity at Eg  0.234 eV, B  0.5 T, T  1 K is
equal to

zz  0.266 (Ohmcm) – 1. Longitudinal conduc-
tivity at Eg  0.414 eV, B  0.5 T, T  1 K is equal to

zz  0.246 (Ohmcm) – 1. It follows that with the help of
the band gap of narrow-gap semiconductors at constant
temperatures, it is possible to control the oscillations of
longitudinal electrical conductivity. Thus, from Fig. 1, a
strong dependence of the longitudinal electrical conduc-
tivity on the band gap in narrow-gap semiconductors is
seen. But, as can be seen from formula (1), for a spec-
trum with a parabolic dispersion law, the longitudinal
electrical conductivity oscillations do not depend on the
band gap.
Fig. 1 – Longitudinal electrical conductivity oscillations in
narrow-gap semiconductors at T  1 K calculated by formula
(4): 1 – for InSb; 2 – for InAs
2.2 Temperature Dependence of Longitudinal
Conductivity Oscillations in Narrow-gap
Electronic Semiconductors
Let us consider the temperature dependence of the
longitudinal electric conductivity oscillations in nar-

ON TEMPERATURE DEPENDENCE OF LONGITUDINAL… J. NANO- ELECTRON. PHYS. 12, 03012 (2020)
03012-3
row-gap electronic semiconductors. The graphs in Fig. 1
are obtained at low temperatures and strong magnetic
fields. In this case, the Landau levels are manifested
sharply and the thermal broadening is very weak. The
broadening of the discrete levels is described by the
derivative of the Fermi-Dirac energy distribution func-
tion f(E,

, T)/E. To take into account the tempera-
ture dependence of the longitudinal conductivity oscil-
lations, we expand

zz(E, H, T, Eg(T)) in the derivative
of the Fermi-Dirac distribution function f(E,

, T)/E.
Then the longitudinal conductivity oscillations will
depend on the temperature. As known, the band gap of
semiconductors is highly dependent on temperature
(Eg(T)) [15, 16]. The temperature dependence of the
band gap of semiconductors can be determined using
the empirical relation of Varshni [15, 16] or the analyt-
ical expression of Feng [16] and other relations. Hence,
we obtain the temperature dependence of the longitu-
dinal conductivity oscillations in narrow-gap semicon-
ductors in the presence of a strong magnetic field:
1/ 2
1
2
2
2
0
0
2 3 2 2
11
/2
22
( , , )
2
(2 ) 1
( , , , ( )) 1 2
(0) (0)
c
rNN
zz g c N c
N
gg
fE Т
EE
me
E H T E T E E N dE
T T E
EE
TT


   
  


  






       









  


(5)
Thus, it becomes possible to calculate the longitudi-
nal conductivity oscillations in narrow-gap semiconduc-
tors at various temperatures.
We plot the graph of the

zz(E, H, T, Eg(T)) depend-
ences with the help of formula (5). Fig. 2 shows the
oscillations of the longitudinal conductivity in InSb at
temperatures T  1 K, 25 K, and 77 K. It can be seen
from Fig. 2 that at a temperature of 77 K, the ampli-
tudes of the longitudinal electrical conductivity oscilla-
tions are practically unnoticeable and coincide with

zz(E, H, T, Eg(T)) in the absence of a magnetic field.
2.3 Investigation of Magnetic Susceptibility
Oscillations in Narrow-gap Semiconductors
at Various Temperatures
Let us consider the temperature dependence of the
longitudinal magnetic susceptibility oscillations in
narrow-gap semiconductors taking into account the
temperature dependence of the density of states. For
narrow-gap semiconductors, the spectral density of
states is determined by the following expression [10]:
 
max
32
12
23
20
21
()
,21
(2) ()
2
N
g
c
S
N
c
g
E
E
m
N E H
EEN
E





  

, (6)
where NS(E, H) is the spectral density of energy states
with nonparabolic dispersion law. Integrating formula
(6), we obtain the total number of quantum states per
unit volume. In quantizing magnetic fields, the free
energy of electrons without taking into account spin is
expressed in terms of the total number of quantum
states in the following form [1, 14]:
max
11
2
2
2
0
1
( , , ) ( ) 1 exp
( ) 2
N
Ng
m eH E eH E
F E H T n E N dE
c E mc kT






   
        

   
   



(6)
where, n is the concentration of charge carriers,

is the
Fermi level. Differentiating (7) with respect to H we
find dF(E, H, T)/dH, and differentiating again with
respect to H we obtain d2F(E, H, T)/dH2:
 
max
22
1
22
2
22 3
02
1 3 1
4
22
( , , ) 1
,, 4 ( ) 11 exp
2
N
g
N
g
eH E
N N E
mc E
d F E H T e m
E H T dE
dH c E
E eH
EN kT
E mc




   
   

   
   
     




 

  

 






(7)
Here,

(E, H, T) are magnetic susceptibility oscilla-
tions in narrow-gap electronic semiconductors. Thus,
using formula (8), one can calculate the temperature
dependences of the magnetic susceptibility oscillations
in narrow-gap electronic semiconductors. Now consider
the numerical calculations using the computer program
Maple.
Using formula (8), we construct a graph of the de-
pendence of the longitudinal magnetic susceptibility
oscillations on the strong magnetic field strength in n-
Bi2Te2.85Se0.15 (Fig. 3). Here, Eg(0)  0.18 eV [17], mag-
netic field strength B  0.1  3 T (or H  1  30 kOe) at
T  2 K. From Fig. 3 it follows that with an increase in
the magnetic field induction, the oscillation amplitude
of the longitudinal magnetic susceptibility increases
Fig. 2 – Temperature dependence of the longitudinal electrical
conductivity oscillations in various narrow-gap electronic
semiconductors calculated by the formula (5): 1 – for InSb and
2 – for InAs

G. GULYAMOV, U.I. ERKABOEV, R.G. RAKHIMOV, J.I. MIRZAEV J. NANO- ELECTRON. PHYS. 12, 03012 (2020)
03012-4
Fig. 3 – Dependence of the magnetic susceptibility oscillations
on temperature and magnetic field in n-Bi2Te2.85Se0.15 calculat-
ed by formula (8)
significantly. At low temperatures, the discrete Landau
levels manifest themselves sharply and the thermal
broadening of the discrete levels is not felt. Thermal
broadening of the levels in a strong magnetic field leads
to smoothing of discrete levels.
2.3.1 Influence of Temperature on Electronic
Heat Capacity Oscillations
One of the methods for the determination of the
spectral density of energy states oscillations of semi-
conductors in a strong magnetic field is based on meas-
urements of oscillations of the electronic heat capacity. In
the works [12, 13], oscillations of the electron specific heat
in semiconductors at low temperatures were studied.
However, in these works, the temperature dependence of
the oscillations of the electronic heat capacity for narrow-
gap semiconductors was not considered.
Now, we consider the oscillations of the electronic
heat capacity in narrow-gap semiconductors at various
temperatures. For a degenerate electron gas, the deriv-
ative of the Fermi-Dirac temperature distribution func-
tion has the following form:
   
0
2
2
exp
,, 1
1 exp
E
E
f E T kT
T kT E
kT






 










(9)
Using formulas (6) and (9), we obtain the following
analytical expression for the temperature dependence
of oscillations of the electronic heat capacity in a quan-
tizing magnetic field:
   
max
32
12
22
23
20
21exp
()
,, 21
(2) () 1 exp
2
N
g
c
N
c
g
EE
E
E
mkT
C E H T E dE
kT EE
EN
EkT








 

    


   






(10)
Here, C(E, H, T) are electronic specific heat oscilla-
tions for narrow-gap electronic semiconductors. Thus,
with the help of the formula (10), it is possible to calcu-
late the oscillations of the electronic heat capacity in
narrow-gap semiconductors at various temperatures.
3. COMPARISON OF THEORY WITH
EXPERIMENTAL RESULTS
In the work [18], the de Haas-van Alphen effect in
magnetic semiconductors was observed. The magnetic
susceptibility oscillations in p-Bi2 – xFexTe3 were obtained
at T  2 K, x  0 [18] and Eg(0)  0.2 eV [17]. Fig. 4 pre-
sents the theoretical and experimental graphs for
p-Bi2 – xFexTe3 (x  0) at T  2 K. Using formula (8), a theo-
retical graph is obtained. As can be seen in this figure, the
amplitude of the Landau levels on the theoretical curve is
observed much higher than on the experimental graph.
With the help of formulas (8), one can plot the graph of the
magnetic susceptibility oscillations for p-Bi2 – xFexTe3 at
various temperatures. It can be seen from Fig. 4 that at
high temperatures, the oscillation amplitudes erode, and a
strong magnetic field is not felt. This is due to the fact that
the thermal broadening of the Landau levels is enhanced
at high temperatures.
Let us analyze the longitudinal conductivity oscilla-
tions of specific narrow-gap electronic materials in a
quantizing magnetic field. For a unit volume of semi-
conductors, the following condition is satisfied:
   
( , , , ( )) ( , , , ( ))
1
( , , , ( ))
zz g zz g
zz g
R E H T E T E H T E T
E H T E T




(11)
Fig. 4 – De Haas-van Alphen oscillations in p-Bi2 – xFexTe3 at
T  2 K and x  0: 1 – experiment [18], 2 – theory calculated by
formula (8)
Here, Rzz is the longitudinal magnetoresistance.
In Fig. 5, the results of theoretical calculations are
compared with experimental data for Bi2Se3 [8] at a
measurement temperature of T  4.2 K, Eg(T)  0.15 eV
and in the magnetic field induction range B  0 ÷ 32 T.
The theoretical curve for Rzz(E, H, T, Eg(T)) is obtained
with the help formula (11). As can be seen from this
figure, discrete Landau levels are not observed in the
range of the magnetic field induction B  5 ÷ 10 T in the
experimental graph. But, oscillations of the longitudi-
nal magnetoresistance in the theoretical curve are
manifested precisely in this interval of the magnetic
field induction. Using formula (11), we can calculate
the oscillations of the longitudinal magnetoresistance
in Bi2Se3 at various temperatures. As can be seen from
Fig. 5, the theoretical curve and experimental data are
in good agreement.

ON TEMPERATURE DEPENDENCE OF LONGITUDINAL… J. NANO- ELECTRON. PHYS. 12, 03012 (2020)
03012-5
Fig. 5 – Magnetoresistance oscillations in Bi2Se3 at T  4.2 K:
1 – theory calculated by formula (11); 2 – experiment [8]
4. CONCLUSIONS
Based on the study, the following conclusion can be
made: for the first time, the theory of the temperature
dependence on the longitudinal electrical conductivity
and magnetic susceptibility oscillations in narrow-gap
semiconductors was constructed taking into account
the thermal smearing of Landau levels. Generalized
mathematical expressions were obtained for the mag-
netic susceptibility, longitudinal electrical conductivity
and electronic specific heat oscillations for narrow-gap
electronic semiconductors in quantizing magnetic
fields. The theory is compared with the experimental
results of Bi2Se3 and p-Bi2 – xFexTe3. Using these oscilla-
tions of magnetic susceptibility, the cyclotron effective
masses of electrons are determined.
ACKNOWLEDGEMENTS
This research has been supported by the Ministry of
innovative development of the republic of Uzbekistan
(Grant No. OT-F2-70).
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Про температурну залежність поздовжніх коливань електричної провідності
в вузькозонних електронних напівпровідниках
G. Gulyamov2, U.I. Erkaboev1, R.G. Rakhimov1, J.I. Mirzaev1
1 Namangan Institute of Engineering and Technology, 160115 Namangan, Uzbekistan
2 Namangan Engineering – Construction Institute, 160103 Namangan, Uzbekistan
Розглянуто коливання поздовжньої електричної провідності, коливання магнітної сприйнятливо-
сті та коливання електронної теплоємності для вузькозонних електронних напівпровідників. Побудо-
вана теорія температурної залежності явищ квантових коливань у вузькозонних електронних напівп-
ровідниках з урахуванням термічного розмивання рівнів Ландау. Досліджено коливання поздовжньої
електричної провідності у вузькозонних електронних напівпровідниках при різних температурах.
Отримано інтегральний вираз для поздовжньої електропровідності у вузькозонних електронних напі-
впровідниках з урахуванням дифузного розширення рівнів Ландау. Знайдена формула залежності
коливань поздовжньої електричної провідності від ширини забороненої зони вузькозонних напівпро-
відників. Порівняно теорію з експериментальними результатами для Bi2Se3. Побудована теорія тем-
пературної залежності коливань магнітної сприйнятливості для вузькозонних електронних напівпро-
відників. За допомогою цих коливань магнітної сприйнятливості визначають ефективні циклотронні
маси електронів. Результати розрахунків порівнюються з експериментальними даними. Запропоно-
вана модель пояснює результати експериментів у p-Bi2 – xFexTe3 при різних температурах.
Ключові слова: Коливання електронної теплоємності, Коливання магнітної сприйнятливості та ко-
ливання електропровідності, Електронні вузькозонні напівпровідники, Ефективна циклотронна маса.