Effects of Band Nonparabolicity and Band Offset on the Electron Gas Properties in InAs/AlSb Quantum Well

One-band effective mass model is used to simulation of electron gas properties in quantum well. We calculate of dispersion curves for first three subbands. Calculation results of Fermi energy, effective mass at Fermi level as function of electron concentration are presented. The obtained results are good agreement with the experimental dates.


Introduction
In semiconductors, InAs and InSb of the conduction band are characterized by a strong nonparabolicity and recently intensively studied heterostructures based on them [1]- [3].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 [4]- [7].
In [8] has been investigated InAs/AlSb based QW with well width L = 15 nm, where two dimensional (2D) electron concentration ranges from 2.7 × 10 11 to 8 × 10 12 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.

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 ( ) ( ) and k − is in plane wave vector, ( ) 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 here ( ) Nonparabolicity of conduction band well takes into account by formulas ( ) ( ) where, 0 m -the free electron mass, P E -the Kane parameter, g E -the band gap, Δ - 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 where, E n -is bottom of n-th subbands.Now, approximation (8) is the best solution of (5).However, values of E n in (8) now are obtained from Equation ( 5) at k = 0 by use numeric method.
From Figure 1(a), Figure 1(b) follows that, the approximation ( 8) is sufficiently accurate and/or (8) is the best 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 n E is recalculated from Equation ( 5) at k = 0.

The Fermi Energy and Thermodynamic DOS
The total electron concentration is ( ) where According (8) we have ( ) where ( ) n s n -is n-th subband concentration.
In Equation (10), the terms in the sum should be positive.The negative terms in ( )     with the well width L = 15 nm can be found 1.52 10 cm 0 0 This estimation is close to experimental measured date E n 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

The Cyclotron Mass
According approximation (8), the electron effective mass at the Fermi level (cyclotron mass) ( ) The dependence

Conclusion
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 [8].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 3 5 A B .
Heaviside function.It establishes a link between the Fermi energy F E and full 2D electron concentration s n .They also determine the concentration of electrons in separate subbands ( ) n s n for a given s n .From (10), we can estimate the critical concentrations of 1 c n , in which the Fermi level comes to bottom of the second subband 2 F E E = .In the structure of InAs/AlSb QW

E n is shown in Figure 2 .
Similarly, the critical concentration of 2 c n , in which the Fermi level comes to bottom of the third subband 3 F E E = , is n c2 = 6.87 × 10 12 cm −2 .It is obtained from Equations (10) by changing the Fermi energy in the range of 0 level of the bottom of the next subband.The thermodynamically DOS of electron gas at Fermi level ( )

[ 8
shown in Figure 4.This dependence can be obtained from Equations (10) and (12) by changing the Fermi energy in the range 0 0.33 eV F E = − .This figure shows also the dependence of experimentally measured value of the effective masses (cyclotron mass) c m at the Fermi level of the total concentration of 2D s n

Figure 2 .
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 1 s c n n = , 2 F E E = and 2 s c n n = , 3 F E E = shown by the

Figure 4 .
Figure 4.The dependence of the effective mass on the total concentration for InAs/AlSb QW, L = 15 нм, V = 1.35 eV.

Table 1 .
Band parameters of InAs and AlSb.