Dimensional quantization in InSb and GaAs in three-zone model

Various cases of size quantization in narrow-gap crystals of cubic or tetrahedral symmetry in the three-band Kane approximation are analyzed. Expressions for the energy spectrum depending on the two-dimensional wave vector are obtained for two different cases, differing in the range of current carrier energy values. The size-quantized energy spectrum of electrons in the conduction band and holes in the subband of light holes in InSb and GaAs semiconductors is analyzed in the three-band Kane model.


Introduction
Modern advances in the field of nanoelectronics and nanotechnology have made it possible to observe and study new unique phenomena occurring in zero-, one-and two-dimensional semiconductor structures [1][2][3].The creation of such structures contributes to the study of the mechanisms of photonickinetic effects occurring in them [4,5].
Although quite a lot of work has been devoted to the study of the physical properties of lowdimensional systems of current carriers [6,7,8,9 and references therein], the size quantization of energy spectra in narrow-gap semiconductors and the associated photonic-kinetic effects have been studied quite little.Therefore, in this communication, size quantization is examined quantum mechanically in specific cases.Calculations are carried out in the multi-zone Kane model [8,10,11].

Schrödenger equation and analysis of its solutions for various cases.
In the first part of this work, it is shown that in the approximation of the multiband Kane model, the matrix Schrödinger equation cannot be solved analytically for a potential well of arbitrary shape.Therefore, it examines the dependence of the energy spectrum of a two-dimensional wave vector perpendicular to the direction of size quantization (on the Oz axis) for various cases that differ in the range of current carrier energy values.As the first part, to simplify the calculations, we will limit ourselves to linear terms in the Kane Hamiltonian, -the wave vector of current carriers.Then the Schrödinger equation in the Kane model, in contrast to equations (2,12,18) and at  0 = 0 ( )

2( )
If in expression (35) we take into account the condition of continuity of wave functions at the interface of the heterostructure: , then the size quantization of the energy spectrum of current carriers can be analyzed in the following cases: 1-case.When a particle moves in the field of a potential well ( ) 0 0 U = , then the energy spectrum of the two-dimensional wave vector is given by the relation ( ) ) ,

2( )
Figure 1 shows the energy spectrum for InSb and GaAs, calculated according to (5) for three sizequantized subbands.In the calculations, the following values were chosen: for the InSb crystal  In these figures, the region of negative values of the squared wave vector corresponds to the region of forbidden energies, represented by dashed lines, which represent the bandgap width and the spinorbit splitting zone.Fig. 1 and 2 show that with increasing width of the potential well, the width of the band gap decreases (due to a shift in the energy spectrum due to size quantization) and the energy distance between close states of size quantization levels (since the energy spectrum of size quantization is inversely proportional to the width of the well).Let us note here that in the Kane model the effective mass of heavy holes is considered infinite [10].Therefore, the states of electrons in the conduction band (solid line in the region of positive energy values) and the subband of light holes in the valence band (solid line in the region of negative energy values) are dimensionally quantized, and the subband of heavy holes in the valence band corresponds to a vertical line, since it (in the Kane approximation depends on the wave vector of holes).In the calculations, the minimum value of the conduction band was chosen as the starting point for energy, so the energy of electrons is positive and the energy of holes is negative.2-case.When a particle moves in the direction of size quantization (in this case), then the energy spectrum corresponds to a set of size-quantized levels, depending on the band parameters and the size of the well and is described by the expression In both of the above cases, the size quantization expressions can be simplified as follows: a) the energy region of current carriers that satisfies the condition is defined as , from which we have an expression for the energy spectrum, which has two size-quantized branches, which correspond to the size quantization of electron subbands in the conduction band (sign "+") and light holes (sign "-") ( ) Here and below 1 1, 2,.... ) c) provided that 0 g EE +   the dependence of the current carrier energy on the two-dimensional wave vector is determined as ( ) from which we obtain expressions for the dimensionally quantized energy spectrum of electrons (sign "+") and light holes (sign "-") ( ) The energy spectrum of electron size quantization calculated for InSb using expression (11) is shown in fig.3, where a) for two-dimensional; b) for the three-dimensional case.In the calculations, the above values of physical quantities were chosen; d) under the condition

Conclusion
Thus, various cases of size quantization in narrow-gap crystals of cubic or tetrahedral symmetry in the three-band Kane approximation have been analyzed.
Expressions are obtained for the energy spectrum depending on the two-dimensional wave vector directed along the heterostructure interface for two different cases, differing in the range of current carrier energy values.The dimensionally quantized energy spectrum of electrons in the conduction band and holes in the subband of light holes in InSb and GaAs semiconductors was analyzed in the three-band Kane model, where the mass of heavy holes is assumed to be infinite [10].
eV = is spin-orbit splitting energy, in for GaAs:

Figure 1 .
Figure 1.Size-quantized energy spectrum of electrons in InSb.

Figure 2 .
Figure 2. Size-quantized energy spectrum of electrons in GaAs.
dimensionally -quantized levels, defined by the relation

Figure 3 .
Figure 3. Dimensional quantized energy spectrum of electrons in InSb: a) two-dimensional; b) three-dimensional case.