The Temperature Dependence of the Density of States in Semiconductors

The temperature dependence of the density of energy states in semiconductors is considered. With the help of mathematical modeling of the thermal broadening of the energy levels, the temperature dependence of the band gap of semiconductors 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.


Introduction
Density of states determines thermal, optical, magnetic, electric and other physical properties of semiconductors.The change of the energy gaps with increasing temperature can be explained by the influence of lattice vibrations 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][4][5][6][7][8] the temperature dependence of the density of states is determined by relaxation spectroscopy of energy levels in semiconductors.It is shown that the density of surface states varies according to temperature.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 conduction 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 dependence 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 semiconductor, 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 comparison between theory and experiment is necessary to consider the specific form of the band structure of the semiconductor and the dependence of the effective mass of the charge carriers of the temperature.
The aim of this work is to study the temperature dependence of the band gap semiconductor with the band structure and temperature dependence of the effective mass of carriers and comparison of theory with experiment.

The Dependence of the Energy Gap with Temperature
As was shown in [3][4][5][6][7][8], the density of states can be decomposed 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 discrete states by GN-function takes into account the thermal broadening of each discrete level.We assume that the density of states at absolute zero parabolic N s (E).We define a specific form of the electron dispersion.For example, the electron dispersion parabola or according to the Kane model.According to the procedure [3][4][5][6][7][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: where N s (E)-the density of states at zero temperature [12,13]   where 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 condition given in [8].We assume that the density of states corresponding to the energy band gap edges E c and E v is N k .Energy region where the density of states N s (T) is less than the critical N k assume band gap.The energy range where N s (T) > N k permission from the zones.The values of the edges of the band gap position of the bottom of the conduction band E c (T) and valence bands E v (T) is deter-mined by the solution of the equation transcendent 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 temperature dependence of the resistance of the semiconductor.One of the reasons is that different values for N k optical and electrical measuring techniques.Figure 2 shows plots of the density of states at a temperature 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.

The Influence of the
Figure 3 shows the temperature dependence of the graphics of the band gap for the solid solutions p-Bi 2−x Sb x Te 3−y Se y to changing the effective mass density of states taken from Figure 3 [9].For example, for a solid solution of p-Bi . 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   This shows that the reduction of the band gap by changing the effective mass with increasing temperature from

Comparison of Theory with Experiment
The temperature dependence of the width of the energy gap depends on the density of states at the absolute temperature.In the model used in [3][4][5][6][7][8] the temperature dependence of the density of states is determined by the temperature dependence of GN-functions and band structure of the allowed bands at the bottom of the conduction  [14] and Si [15].Using mathematical modeling of the temperature dependence of E g (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 experimental data.It follows that the temperature dependence of the band gap is satisfactorily described by a mathematical 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 energy levels in the zones and in the forbidden zone.

Conclusion
The temperature dependence of the energy spectrum of the density of states of solid solutions of  Experiment for InAs [14] calculations for paraboliс zony calculation for the model of Kane 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 broadening of the energy levels with the GN function satisfactorily describes the process of the temperature dependence of the band gap of Si and InAs.
transcendent solution of Equation (3) with respect to E at the specified temperature T, and the critical value of the density of states N k determines the position of the edges of the gap E c (N k , T) and E v (N k , T).In Equation (3) N k included as a parameter.N k value is determined by the condition of the experiment and by the accuracy of measurement techniques.Then the band gap is defined as the difference between the values of E c (T, N k ) and E v (T, N k ): Effective Mass of the Density of States at the Temperature Dependence of the Band Gap in Solid p-Bi 2−x Sb x Te 3−y Se yIn[9] found that in solid p-Bi 2−x Sb x Te 3−y Se y effective density of states in the valence band is strongly dependent on temperature.Figure1shows the temperature dependence of the effective mass of the density of states in solid p-Bi 2−x Sb x Te 3−y Se y from[9].Using the data of Figure 1 calculated by the model bandgap variation with temperature.
2−x Sb x Te 3−y Se y change in the band gap by changing the effective mass at T = 100 K is

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

Figure 3 .
Figure 3. Graphic E g (T)-temperature dependence of the band gap.и * p m соnst 

Figure 4 .
Figure 4.The temperature dependence of the band gap of Si.

Figure 5 .
Figure 5.The temperature dependence of the band gap of InAs.p-Bi 2−x Sb x Te 3−y Se y 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 effective mass of the density of states by increasing T has significant 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