To the theory of dimensional quantization in crystals in the Kane approximation

The microscopic mechanism of size quantization of the conduction band and valence band of semiconductors of cubic and tetrahedral symmetry has been theoretically studied in the Kane approximation. It is shown that the matrix Schrödinger equation obtained on the basis of the Kane model cannot be solved analytically for a potential well of an arbitrary profile. Therefore, expressions for the energy spectrum are obtained depending on the two-dimensional wave vector directed along the interface of the heterostructure for various cases that differ in the range of current carrier energy values.


Introduction
The development of quantum electronics has made it possible to study a number of mechanisms of nonlinear optical phenomena in solids, in particular, to determine the morphological, electronic and other properties of crystals [1].On the other hand, advances in the field of nanoelectronics and nanotechnology have made it possible to observe and study new natural phenomena occurring in lowdimensional structures, for example, in heterostructures [2.],where their symmetry changes [3].The creation of such structures also contributes to the study of nonlinear optical phenomena in them [4,5].
In general, although there is enough literature on the study of the physical properties of nanostructures [6,7,8,9], the phenomenon of size quantization in narrow-gap crystals and related nonlinear optical phenomena have been almost not studied.Therefore, this communication theoretically studies the microscopic mechanism of size quantization of the conduction band and valence band of semiconductors of cubic and tetrahedral symmetry in the Kane approximation [8,10,11] based on the Schrödinger matrix equation.

Schrödenger equations and their solutions.
In the approximation of the multiband Kane model, we will study size quantization in nanostructures grown on the basis of semiconductors of the A3B5 type in the Kane approximation [8,10,11].To simplify the calculations, we neglect the quadratic terms of k , i.e.Let us restrict ourselves to linear terms of k in the Kane Hamiltonian, k -where is the wave vector of current carriers.Then the Schrödinger equation in the Kane model has the form perpendicular to the dimensional quantization axis Oz) section, P is Kane parameter,  is spin-orbital splitting energy gap, calculated from the extremum of the valence band [12], ( ) Uzis potential, with which the physical nature of the quantum well in the Oz direction.
From the last relations it is easy to obtain equations for functions in the form of the following system of equations: Here ( ) , We are looking for a solution in the form Where , С  are constants of integration, the analytical form of which is determined by the boundary conditions of the problem.
If we write the degree of the exponential function as (4), then from the condition of the finiteness of the wave function we have that the quantity has two values: ( ) ( ) , where ( ) . From the last relations it is clear that the problem of dimensional quantization cannot be solved analytically within the framework of Kane's approach mentioned above.As a result, case studies are required, which we will undertake next.
It should be noted that in the general case, the energy spectra of current carriers in a homogeneous (satisfying ( ) 0 Uz= ) bulk crystal are determined using the relation  () E  is determined by the relation ( ) 2-case.When a particle moves in the direction of size quantization, i.e. 0, 0 xy kk == , then the energy spectrum is defined as 3-case.If the condition is satisfied , then the dependence of the energy spectrum on the two-dimensional wave vector is defined as ) 4-case.If the condition is satisfied In the case 0 xy kk == of the Schrödinger equation in the direction of size quantization is expressed as follows ( ) ( ) where From equation ( 9) it is clear that its solution depends on the type of potential and for an arbitrary type () Uzdoes not have an analytical solution, even if it has a linear or quadratic dependence on z.
Therefore, when where Then (13) can be rewritten for various special cases.For example, when and provided that ) . In particular, when electrons are in a quantum well ( then under the condition Thus, solution ( 12) can be represented as where CCare determined from the boundary conditions 2 ( ) 0 za  == for a particle moving in a potential well.
From (2) it is easy to obtain the Schrödenger equation in the Luttinger-Kohn approximation in the form ( ) ( ) where ( ) ( ) Then at ,0 kk → in it, where Here + −   the energy spectrum of electrons is determined from the following relation the dependence of the energy spectrum on the wave vector is described by the expression ( )( ) . Also, when To determine the analytical form of solutions to the Schrödenger equation s analyze a number of cases given below.
1-case.When a particle moves in a potential well, i.e. , 0 0 V = then we obtain an expression for the energy spectrum depending on k ⊥ in the form ( ) () 3 .

2( )
2-case.When current carriers move along the potential well interface, i.e. , then ( ) 3-case.Given the condition , the dependence ( ) Ek ⊥ will be written as 4-case.When the condition is satisfied , then the expression for the energy spectrum depending on the two-dimensional wave vector of the current carriers has the form: .

Conclusion
Thus, it is shown that the size quantization of the energy spectrum of current carriers in crystals of cubic or tetrahedral symmetry in the three-band Kane approximation is described by the matrix Schrödenger equation, the solution of which has eight components, two of which are analyzed in more detail.It is shown that the matrix Schrödinger equation obtained on the basis of the Kane model cannot be solved for a potential well of an arbitrary profile.Therefore, expressions for the energy spectrum are obtained depending on the two-dimensional wave vector directed along the heterostructure interface for nine different cases, differing in the range of current carrier energy values.The dimensionally quantized energy spectrum for a specific semiconductor is discussed in the second part of this article.
const == , i.e. when 1 0 k = , then solution (9) takes the form APITECH-V-2023 Journal of Physics: Conference Series 2697 , the solution of which can be sought in the form of (16) by replacing 3 4 )