The Temperature Dependence of the Density of States in Semiconductors

doi:10.4236/wjcmp.2013.34036

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World Journal of Condensed Matter Physics, 2013, 3, 216-220

Published Online November 2013 (http://www.scirp.org/journal/wjcmp)

http://dx.doi.org/10.4236/wjcmp.2013.34036

Open Access WJCMP

The Temperature Dependence of the Density of States in

Semiconductors

Gafur Gulyamov1, Nosir Yusupjanovich Sharibaev1,2, Ulugbek Inoyatillaevich Erkaboev1

1Namangan Engineering Pedagogical Institute, Namangan, Uzbekistan; 2Namangan Engineering Institute of Technology, Namangan,

Uzbekistan.

Email: [email protected]

Received September 17th, 2013; revised October 21st, 2013; accepted November 10th, 2013

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

The temperature dependence of the density of energy states in semiconductors is considered. With the help of mathe-

matical modeling of the thermal broadening of the energy levels, the temperature dependence of the band gap of semi-

conductors is studied. In view of the non-parabolic and the temperature dependence of the effective mass of the density

of states in the allowed bands, graphs of temperature dependence of the band gap are obtained. The theoretical results of

mathematical modeling are compared with experimental data for Si, InAs and solid solutions of p-Bi2−xSbxTe3−ySey. The

theoretical results satisfactorily explain the experimental results for Si and InAs. The new approach is investigated by

the temperature dependence of the band gap of semiconductors.

Keywords: Band Gap; The Effective Mass Density of States; The Energy Spectrum; The Numerical Simulation and

Experiment

1. Introduction

Density of states determines thermal, optical, magnetic,

electric and other physical properties of semiconductors.

The change of the energy gaps with increasing tempera-

ture can be explained by the influence of lattice vibra-

tions on the energy levels in crystals. Statistical analysis

of the problem is carried out by analyzing the free energy

of the crystal as the sum of the energies of the electron

gas, lattice vibrations and electron-phonon interactions [1,

2]. On the other hand, one should take into account the

thermal broadening of the energy levels of the radiative

transitions [1,2]. In [3-8] the temperature dependence of

the density of states is determined by relaxation spec-

troscopy of energy levels in semiconductors. It is shown

that the density of surface states varies according to tem-

perature. Due to the thermal broadening of the levels, the

discrete spectrum with hanging temperature becomes a

continuous energy spectrum. With the expansion of the

energy spectrum of the density of states in the energy

derived from the probability required energy level, it was

shown that the amount of energy slits is dependent on

temperature. The temperature dependence of the band

gap is determined by the density of states of the conduc-

tion band and valence band of the semiconductor. Due to

the thermal broadening of the density of states near the

bottom of the conduction band, valence band reduces

band gap. In the calculation of the temperature depend-

ence of the forbidden assumed for simplicity, the density

of states in the areas of constant edge of the conduction

band and valence band is sharp and has a stepped shape.

In these works, the effective mass of the density of states

does not depend on the temperature. However, as shown

in experiments [9], the effective mass of the density of

states depends on the temperature. This change in the

effective mass changes the temperature dependence of

the band gap. However, in the real state of the semicon-

ductor, density is a function of speed and energy band

structure of the sample is determined. Moreover, the

density of states is so general that it can be used even

when there is no Brillouin zone and sharp boundaries of

permitted and prohibited zones [10,11].

Thus, the analysis of experimental results for compari-

son between theory and experiment is necessary to con-

sider the specific form of the band structure of the semi-

conductor and the dependence of the effective mass of

the charge carriers of the temperature.

The aim of this work is to study the temperature de-

pendence of the band gap semiconductor with the band

The Temperature Dependence of the Density of States in Semiconductors 217

structure and temperature dependence of the effective

mass of carriers and comparison of theory with experi-

ment.

2. The Dependence of the Energy Gap with

Temperature

As was shown in [3-8], the density of states can be de-

composed into a series of GN functions. The temperature

dependence of GN function will determine the statistical

thermal broadening of the discrete levels. The resulting

density of states is determined by the expansion of dis-

crete states by GN-function takes into account the ther-

mal broadening of each discrete level. We assume that

the density of states at absolute zero parabolic Ns(E). We

define a specific form of the electron dispersion. For ex-

ample, the electron dispersion parabola or according to

the Kane model. According to the procedure [3-8] we

expand the density of states in a series of GN-functions.

We will take into account nonparabolicity zones via the

temperature dependence of the effective mass of the

temperature:



 

ssiii

NEТNEGNEET



,,

(1)

where



 

11 1

exp exp

GNEE T

EE EE

kT kTkT























Ns(E)—the density of states at zero temperature [12,

13]



0at

scv

pv v

NEEEE

NEEE E

NEE EE





























(2)

where



023

2π

N,



023

2π

N and Ec, Ev—

The value of the energy of the conduction band and

valence band at T = 0.

Substituting (2) into (1) we obtain the density of states

at temperature T. In determining the width of the gap, we

use the density of states, which depends on temperature.

When modeling the process of measuring the width of

the band gap with increasing temperature, use the condi-

tion given in [8]. We assume that the density of states

corresponding to the energy band gap edges Ec and Ev is

Nk. Energy region where the density of states Ns(T) is less

than the critical Nk assume band gap. The energy range

where Ns(T) > Nk permission from the zones. The values

of the edges of the band gap position of the bottom of the

conduction band Ec(T) and valence bands Ev(T) is deter-

mined by the solution of the equation transcendent

 

ii ik

NEGNЕЕТ N



 (3)

transcendent solution of Equation (3) with respect to E at

the specified temperature T, and the critical value of the

density of states Nk determines the position of the edges

of the gap Ec(Nk, T) and Ev(Nk, T). In Equation (3) Nk

included as a parameter. Nk value is determined by the

condition of the experiment and by the accuracy of meas-

urement techniques. Then the band gap is defined as the

difference between the values of Ec(T, Nk) and Ev(T, Nk):











kc kv k

ETNETN ETN (4)

It follows that the method of determining the accuracy of

the experiment and the important factors in determining

the width of the gap. Indeed the band gap, determined by

optical methods, “optical width” of the band gap can not

match the value of the band gap, determined by the tem-

perature dependence of the resistance of the semicon-

ductor. One of the reasons is that different values for Nk

optical and electrical measuring techniques.

3. The Influence of the Effective Mass of the

Density of States at the Temperature

Dependence of the Band Gap in Solid

p-Bi2−xSbxTe3−ySey

In [9] found that in solid p-Bi2−xSbxTe3−ySey effective

density of states in the valence band is strongly depend-

ent on temperature. Figure 1 shows the temperature de-

pendence of the effective mass of the density of states in

solid p-Bi2−xSbxTe3−ySey from [9]. Using the data of Fig-

ure 1 calculated by the model bandgap variation with

temperature.

Figure 2 shows plots of the density of states at a tem-

perature T = 100 K and T = 300 K. As can be seen from

Figure 2 into account the change of the effective mass

density of states significantly affects the density of states

near the valence band.

Figure 3 shows the temperature dependence of the

graphics of the band gap for the solid solutions p-

Bi2−xSbxTe3−ySey to changing the effective mass density

of states taken from Figure 3 [9]. For example, for a

solid solution of p-Bi2−xSbxTe3−ySey change in the band

gap by changing the effective mass at T = 100 K is









100100, 0.93100, const

ggp gp

EEm Em



0.001эВ



. By increasing the temperature to T = 300 K,

changing the width of the band gap due to change in the

effective mass of the density of states is





300

E









300,1.35300,const 0.0165

gp gp

Em EmэВ

This shows that the reduction of the band gap by chang-

ing the effective mass with increasing temperature from

Open Access WJCMP

The Temperature Dependence of the Density of States in Semiconductors

218

Figure 1. The temperature dependence of the effective mass

of the density of states m/m0 in solid solutions

p-Bi2−xSbxTe3−ySey [14]. 1→ x = 1, y = 0.06; 2→ x = 1.1, y =

0.06; 3→ x = 1.2, y = 0.06; 4→ x = 1.2, y = 0.09; 5→ x = 1.3,

y = 0.09, 6→ x = 1.3, y = 0.07; 7→ x = 1.5, y = 0.09.

Figure 2. Graphic density of states at T = 100 K and 300 K.





mmT and





mmT. ______ ------

mсоnst*





mT.

100 K to 300 K can be increased more than tenfold. Fig-

ure 3 shows plots of the temperature dependence of the

band gap of the solid solutions p-Bi2−xSbxTe3−ySey for the

changes in the effective mass of the density of states

taken from Figure 1. Thus, changes in the effective mass

of the density of states with temperature can greatly af-

fect the temperature dependence of the band gap.

4. Comparison of Theory with Experiment

The temperature dependence of the width of the energy

gap depends on the density of states at the absolute tem-

perature. In the model used in [3-8] the temperature de-

pendence of the density of states is determined by the

temperature dependence of GN-functions and band struc-

ture of the allowed bands at the bottom of the conduction

Figure 3. Graphic Eg(T)—temperature dependence of the

band gap. и

mсоnst





mmT. _____ ; ----

mсоnst





mmT, p-Bi0.7Sb1.3Te2.93Se0.07; ......





mmT,

p-Bi0.6Sb1.2Te2.91Se0.09; ····





mmT

p-Bi0.5Sb1.5Te2.91Se0.09.

band and at the top of the valence band. Analysis of the

results of numerical modeling of changes in the density

of states at the temperature showed that the value of the

density of states near the band edges is determined by the

number of states of the band edges, a few tens of kT or

about 0.1 meV, the density of states in the depths of the

allowed zones does not affect the width of the gap. Since

the GN-function of deep area of the zone does not pene-

trate into the region band gap semiconductor, the main

contribution to the shift of the band edges give the states

lying close to the edges of the allowed bands. According

to this law of dispersion near the top of the valence band

and the conduction band edge is crucial in determining

the temperature dependence of the band gap Eg(T). Fig-

ures 4 and 5 are graphs of temperature dependence of the

band gap of InAs [14] and Si [15].

Using mathematical modeling of the temperature de-

pendence of Eg(T) for a parabolic band and Kane model

obtained plots of the band gap of the temperature. As can

be seen in the investigated temperature range of parabolic

dispersion and a model for the use of Kane’s model is in

good agreement with experimental data for InAs [14] and

Si [15]. Theoretical calculations of the theoretical given

for these materials is in good agreement with the experi-

mental data. It follows that the temperature dependence

of the band gap is satisfactorily described by a mathe-

matical model of expansion of the density of states in a

series of GN-functions, which describes the temperature

dependence of the thermal broadening of individual en-

ergy levels in the zones and in the forbidden zone.

5. Conclusion

The temperature dependence of the energy spectrum of

the density of states of solid solutions of

Open Access WJCMP

The Temperature Dependence of the Density of States in Semiconductors 219

1.050

1.070

1.090

1.110

1.130

1.150

1.170

1.190

100

150

200

250

300

350

400 450 500

Т, К

, eV

Experiment for Si [13]

calculations for the parabolic zone

calculation for the of Kane

Figure 4. The temperature dependence of the band gap of

Si.

0.3400

0.3500

0.3600

0.3700

0.3800

0.3900

0.4000

0.4100

100

150 200250300

T, K

, eV

0.4200

Experiment for InAs [14]

calculations for paraboliс zony

calculation for the model of Kane

Figure 5. The temperature dependence of the band gap of

InAs.

p-Bi2−xSbxTe3−ySey takes into account of the temperature

dependence of the effective mass of the density of states

in the valence band. The temperature dependence of the

band gap for the changes in the effective mass of the

density of states is obtained. The numerical experiments

show that at temperatures T > 120 K, change in the effec-

tive mass of the density of states by increasing T has sig-

nificant effect on the temperature dependence of the band

gap.

In this temperature range ([0, 300 K], [0, 500 K]),

mathematical modeling of the temperature dependence of

the band gap is satisfactorily described by a parabolic

dispersion law and the Kane model. The experimental

results of changing the band gap of silicon [13] and InAs

[14] within the accuracy of measurement is consistent

with the results of theoretical calculations. Comparison

of theory and experiment shows that the thermal broad-

ening of the energy levels with the GN function satisfac-

torily describes the process of the temperature depend-

ence of the band gap of Si and InAs.

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