Received 25 November 2015; accepted 22 February 2016; published 25 February 2016
Recently, investigations of narrow band gap semiconductors InAs, InSb  - based heterostructures have been paid much attention because they are essential to developing semiconductor devices. In these structures, the conduction band is strongly nonparabolicity. Properties of subband nonparabolicity of a two dimensional electron gas (2DEG) can be studied by the cyclotron resonance experiments. In heterostructures InAs/AlSb, a quantum well (QW) depth for electrons reaches up to ~1.3 eV. Modern technology makes it possible to obtain a structure
with a 2DEG with a concentration of up to  . Depending on the width of the QW, one or more subbands may be formed in the system.
It is known that with an increase electron concentration when the Fermi level crosses the bottom of the next series subbands, the density of state (DOS) of 2DEG drastically changes. In case of parabolic band energy dependence of DOS is abruptly changing rectangular steps  . Nonparabolicity can affect the structure of the resulting subbands in the QW, and can shift the bottom of the subbands  . Therefore, it should be manifested in the energy dependence of the DOS of 2DEG. With increasing temperature, due to thermal broadening of the levels, as well as the temperature dependence of the band gap of (the degree of nonparabolicity directly depend on), the energy dependence of the DOS of 2DEG in general is complex.
The purpose of this work is to obtain expressions for the DOS of 2DEG with allowance conduction band nonparabolicity and graphical analysis of the expressions. To study the effect of temperature on the DOS, the calculations will be carried out numerically for heterostructures of InAs and InSb based QW.
2. Electron Energy Spectrum in QW
In the simplest approximation, the nonparabolic energy spectrum of the 2DEG can be represented as
Here, , , and the effective mass at the bottom of the conduction band. In approaching an infinitely high barrier
In real situations, the energy levels and form of the function can determine the 8-zone hamiltonian   considering the effects of the finite height of the barrier, the deformation caused by the mismatch of the lattice constant, etc.
The dependence of is a complex in the Kane model  . However, the majority of cases it is possible to apply various approximations. For simplicity, below we use the following approximation 
where, -free electron mass, -band gap of the bulk semiconductor. The energy is measured from the bottom of the conduction band of the bulk semiconductor. The parameters used in the calculations for InAs, InSb are shown in Table 1.
According to (1-3), bottom n-th subband is determined from the equations
Solving Equation (4), it is possible to determine of interband transition energy, for example. Figure 1 shows the results of calculations L dependence of for single InAs QW. The same figure also shows the results of experimental studies on the energy of intersubband transitions  -  .
The figure shows that a simple model (4) gives qualitatively correct results only for wide quantum wells. For example, in QW width value obtained from (4) to the bottom of the first and second subband respectively equal to 0.0636 and 0.2 eV. Hence, estimated intersubband transition energy is, an experimental measured value is 0.134 eV  . With a decrease in L results obtained by the model (4) strongly deviates from the experiment. This deviation may be due to the finiteness of the barrier height (~1.36 eV), the
Table 1. The parameters used in the calculations for InAs, InSb.
Figure 1. Intersubband energy E12 for the InAs single QW. The crosses represent the experimental data  , squares  , circles  and the solid line-Equation (4).
deformation of the structure due to mismatch of the lattice constant, and the effect of depolarizing shift unaccounted for in the model (4). Therefore, following a study of DOS carried out for the QW width on the example of InAs and InSb.
3. DOS at Zero Temperatures
Knowing the energy spectrum of the electron gas can determine its DOS. To determine the DOS of 2DEG, we use the equation of the total number of particles. After summation over the spin degree it has the form
where -Fermi-Dirac distribution function.
Writing to polar coordinates, expressing according to (1) through a total energy E and taking into account, that is positive, we have
where, -Heaviside function. From (6) we obtain the equation for the two dimensional concentration
Hence, we obtain the formula for DOS of 2DEG, where band nonparabolicity (3) is taking into account
In the particular case when, from (9) follows known formula for the DOS corresponding parabolic dispersion 
Formula (9) can be rewritten as
is energy-dependent electron effective mass. It is also obtained from the definitions of the transport mass
taking into account the spectrum (1) and (3).
From these graphs follows that the band nonparabolicity can be lead to the following results:
-The bottom of each subbands moves down as compared to the parabolic case.
-This is clear from the fact that the bottom of the subbands is inversely proportional to the effective mass of.
-The height of the jumps increases with energy, since it is proportional to.
-Within each subbands the DOS increases linearly.
According to (9), the change of the DOS within each subbands must be square.
However, this amendment to the plot is weak, as for the structures of InAs, InSb the approximation is also satisfactory.
4. DOS at Finite Temperatures
In letter  it is shown that the temperature dependence of the DOS can be described decomposition to the number of GN-function (defined as energy derived probability of filling the energy levels). In  , study using the DOS expansion in GN-functions able to explain the temperature dependence of the DOS in quantizing magnetic fields. This expansion can be written as
Figure 2. DOS for the InAs single QW, L = 150 Å, T = 0 K. The solid line represents the nonparabolic and dotted line―the parabolic approximation.
Figure 3. DOS for the InSb single QW, L = 150 Å, T = 0 K. The solid line represents the nonparabolic and dotted line―the parabolic approximation.
Using (9) (14) and (15) can be plotted for specific QW. Figure 4 and Figure 5 shows the energy dependence of must for the QW width appropriately InAs, InSb, and calculated according to the Formula (14) for any temperatures.
As seen from Figure 4 and Figure 5 the temperature greatly affects the shape depending. When low temperature the thermal generated electrons from the energy states is weak and it does not lead to a strong broadening of the energy levels and the DOS has sharply stepped shape. At the same angles of steps slightly smoothed. There dependency nonparabolicity dispersion law remains the same as Figure 2 and Figure 3. The low temperature has small effect on the energy form of DOS. With further increase of temperature the thermal generated electrons from the energy levels increased. This leads to a strong broadening thermodynamic DOS. It significantly changes its shape. In this case, the thermal broadening almost washed away a step change in the DOS and rectangular steps turned into a smooth blend smooth curve.
At low energies steps completely disappear. With increasing energy, appear slight deviations to the provisions of the former steps of the respective discrete levels. With further increase of the energy―the thermodynamic DOS become a smooth curve, but nonparabolicity dispersion law manifests itself in a wide range of temperatures.
In this paper, the analysis of the density of states of the 2DEG in a single quantum is well. The calculations are performed for narrow-gap semiconductors InAs and InSb.
The nonparabolicity of conduction band leads to the following features of DOS:
-The height of the jumps increases with energy, since it is proportional to;
-Within each subband DOS increases linearly;
Figure 4. DOS for InAs single QW, L = 150 Å at nonzero temperatures.
Figure 5. DOS for InSb single QW, L = 150 Å at nonzero temperatures.
This work was supported by the Scientific and Technical program Republic of Uzbekistan (Grant F2-OT- O-15494).
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