In semiconductors, InAs and InSb of the conduction band are characterized by a strong nonparabolicity and recently intensively studied heterostructures based on them  -  . Nonparabolicity of the conduction band and the nature of the spin splitting of the electron in the quantum well (QW) are studied by the cyclotron resonance  -  .
In  has been investigated InAs/AlSb based QW with well width L = 15 nm, where two dimensional (2D) electron concentration ranges from 2.7 × 1011 to 8 × 1012 cm−2. In this work has been found increase of the effective mass of almost 2 times.
The purpose of this work―the calculation of: 1) subbands dispersion curves, 2) the density of states of 2D electron gas and 3) concentration dependence of effective mass in Fermi level for InAs/AlAs QW with width L = 15 nm.
It is shown that an abrupt change in the density of states leads to a peculiar change in the concentration dependence of effective mass.
2. The In-Plane Dispersion
Consider a single QW with width L (area A―InAs), concluded between barriers with height V (area B―AlAs). The energy is measured from the bottom of the band of the bulk InAs.
In the one band effective mass approximation, the solution of the three-dimensional Schrödinger equation can be represented as. Then for the area A and B, respectively, we can write the following one-dimensional equations
Here and is in plane wave vector,―energy-dependent effective mass of the electrons in the material A or B. Solving Equations (1) and (2), using the boundary condition
we find the dispersion equation
Nonparabolicity of conduction band well takes into account by formulas
where,―the free electron mass,―the Kane parameter,―the band gap,― the spin-orbital splitting of valence band, V―conduction band offset. Band parameters of InAs and AlSb are shown in Table 1.
To describe the statistics of electrons, Equation (5) is non convenient because it is not solvable with respect to E or k. Therefore, we replace Equation (5) is by simple approximation
Table 1. Band parameters of InAs and AlSb.
where, En―is bottom of n-th subbands. Now, approximation (8) is the best solution of (5). However, values of En in (8) now are obtained from Equation (5) at k = 0 by use numeric method.
For InAs/AlSb QW with L = 15 nm, we have: E1 = 0.0454 eV, E2 = 0.158 eV, E3 = 0.304 eV, and for case L = 6 nm we have: E1 = 0.163 eV, E2 = 0.509 eV, E3 = 0.903 eV.
From Figure 1(a), Figure 1(b) follows that, the approximation (8) is sufficiently accurate and/or (8) is the best solution of (5) in a wide range of width QW. It is convenient when studying the statistics of electrons, kinetic, optical, or other characteristics of the 2D electron gas. Inconveniences approximation (8) is such that depends on L, V and on other parameters of materials A,B. Therefore every time when changing these parameters, the value of is recalculated from Equation (5) at k = 0.
3. The Fermi Energy and Thermodynamic DOS
The total electron concentration is
According (8) we have
where―is n-th subband concentration.
In Equation (10), the terms in the sum should be positive. The negative terms in excluded by the Heaviside function. It establishes a link between the Fermi energy and full 2D electron concentration. They also determine the concentration of electrons in separate subbands for a given.
From (10), we can estimate the critical concentrations of, in which the Fermi level comes to bottom of the second subband. In the structure of InAs/AlSb QW
Figure 1. The dispersion curves of the first three subbands (n = 1, 2, 3) in InAs/AlSb QW: the full (red) line―according to Equation (5), the dotted (blue) line―approximation (8); (a) L = 15 nm, (b) L = 6 nm.
with the well width L = 15 nm can be found
This estimation is close to experimental measured date  . Similarly, the critical concentration of, in which the Fermi level comes to bottom of the third subband, is nc2 = 6.87 × 1012 cm−2.
The dependence is shown in Figure 2. It is obtained from Equations (10) by changing the Fermi energy in the range of. The graph shows that, depending on there exist a fractures―slowing of increase the Fermi’s energy. They are caused by abrupt changes (by jumps), the density of states at the critical points:, , ,.
These fractures occur at the critical concentrations of, where intersects the Fermi level of the bottom of the next subband.
The thermodynamically DOS of electron gas at Fermi level is shown in Figure 3.
4. The Cyclotron Mass
According approximation (8), the electron effective mass at the Fermi level (cyclotron mass) is
The dependence is shown in Figure 4.
This dependence can be obtained from Equations (10) and (12) by changing the Fermi energy in the range.
This figure shows also the dependence of experimentally measured value of the effective masses (cyclotron mass) at the Fermi level of the total concentration of 2D  . Fracture in the occur at critical points:, and,.
Figure 2. The dependence of the Fermi energy on the 2D concentration in InAs/AlSb QW with L = 15 nm, V = 1.35 eV. The critical points, and, shown by the shaded lines.
Figure 3. The thermodynamically DOS of electron gas at Fermi level in InAs/AlSb QW, L = 15 nm, V = 1.35 eV.
Figure 4. The dependence of the effective mass on the total concentration for InAs/AlSb QW, L = 15 нм, V = 1.35 eV.
In this study are provided useful approximation (8) of subband dispersions and simplified Equation (10) to calculate the statistics of a degenerate electron gas in heterostructured InAs/AlSb QW, which satisfactorily describes the experimental results  . They are also useful to study of calculation of the transport, optical and magnetic properties of electron gas in a Kane type 2D system. The above description of the algorithms can be applied to other QW heterostructures based on semiconductor group.
This work was supported by the Scientific and Technical program Republic of of Uzbekistan (Grant F2-OT-O-15494).
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