ENERGY STATES OF CURRENT CARRIERS IN MULTILAYER SEMICONDUCTOR STRUCTURES. WENTZEL-KRAMERS-BRILLOUIN APPROXIMATION

I. Introduction

The progress of modern microelectronics is largely determined by the study of the properties of systems with inhomogeneously distributed parameters, the development of methods for the effective theoretical analysis of such systems, the development and provision of objective methods for controlling technological processes that make it possible to create semiconductor layers with desired properties [1, 2]. In this regard, below we consider the general questions of the propagation of electron waves in a medium whose properties change only along a certain direction. The approach is based on the use of the one-electron stationary Schrödinger equation to describe the processes of elastic scattering and tunneling of noninteracting spinless particles under the condition that their total energy is conserved.

The study of the electronic properties of both symmetric and asymmetric with respect to the geometric dimensions of the layers of a semiconductor structure is relevant in connection with the use of these structures in micro- or nanoelectronics and in other areas of solid state physics [3–9].

At present, molecular beam epitaxy and other methods of modern technology make it possible to obtain semiconductor layers with an arbitrary profile of composition change (structure with a quantum well) to improve the characteristics of devices based on them [2]. In this case, the problem of electronic states is reduced to the problem of the behavior of a particle in potential wells of an arbitrary shape. In particular, to create a new generation of resonant tunneling diodes and heterolasers with separated electronic and optical confinement, structures with rectangular size-quantized wells are used, in the center of which there is an additional energy dip.

The study of the electronic states in the structures mentioned above leads to the calculation of the one-electron wave functions of the stationary Schrödinger equation in the semiclassical approximation in the presence of the potential U(x), which we will consider as a slowly varying function of the x coordinate.

Then the one-dimensional Schrödinger equation can be written as

,                                 (1)

where, making the substitution , we obtain the equation for the function  [19]

.                      (2)

Assuming that the system under consideration is close to the classical one in its properties, we will look for a solution in the form of a rows indegrees of the Planck constant, i.e.

                      (3)

Then the general solution of equation (1) has the form

,           (4)

where  ,  and  are the effective mass and energy of current carriers.

In classically inaccessible energy regions, i.e. at , the momentum of the current carriers becomes imaginary. Then in these regions (4) takes the form

       (5)

Note that the accuracy of the semiclassical approximation does not allow taking into account both terms simultaneously, and therefore, in some cases, we will not take into account the exponentially small term in (4) and (5).

II. Linear and quadratic approximation

Let us consider an isolated classical turning point at , far from which the semiclassical approximation is applicable for calculating the transparency coefficient of a potential barrier [20]. Therefore, the solutions of the Schrödenger equation in the allowed and forbidden areas can be found by formulas (4) - (5).

The wave function near the turning point can be found by solving the Schrödinger equation, where the potential energy  near the turning point  can be represented as

                 (6 a)

or

.                               (6 b)

Then the Schrödinger equation can be written as

or

                                      (7)

whose general solution is an arbitrary linear combination of hypergeometric functions, i.e.

, (8)

where , ,   , .

In the general case,   , which corresponds to an exponentially growing wave function. Therefore, to choose a wave function that satisfies the conditions of finiteness of the wave functions at infinity, i.e. satisfying this quantum mechanical approach, there are two alternative cases:

1. ,  and . In this case, the wave function takes the form

,            (9)

and the energy spectrum of current carriers is quantized and is defined as

.                                             (10)

From (10) we obtain an expression for the size-quantized energy spectrum in the form

or

.           (11)

2.  ,  and . In this case, the wave function takes the form

,       (12)

and the energy spectrum of current carriers is quantized and is determined by the relation:

.                     (13)

For a quantitative analysis of the size-quantized energy spectrum, we assume that  . Then we have an expression for the size-quantized energy spectrum in a form convenient for quantitative calculation

  ,                          (14)

 

where  ,

Similarly, it is easy to obtain the following expression

 .                                    (15)

It can be seen from formulas (11) and (13) that to fulfill the condition of finiteness of the wave functions at infinity, there are two types of energy spectrum, and both depend nonlinearly on the size quantization number, i.e. the dimensionally quantized energy spectrum is not equidistant.

Figure 1 a and b shows the dependences of the size-quantized energy spectra, characterized by the values of  and , on the parameter  . It can be seen from these figures that as  increases, the values of  and  decrease from . These energy values from  with the growth of  first decreases and is reached to a minimum, and then increases.

We note here that in quantitative calculations it is convenient to use the connection of the above hypergeometric functions with Hermite polynomials:

, .

 

Fig.1 Dependences of the energy quantities  and  on  for different values of the size quantization number, where  .

III. Cubic and biquadratic approximation

Next, consider the following cubic and biquadratic terms in (6), i.e.

                   (16)

where .  and  are the expansion coefficients of  in a series of . The solution of the Schrödinger equation can be done in a similar way. In this case, it passes into the Schrödenger equation for a harmonic oscillator at  and . Then it can be solved using perturbation theory [19]. In this case, the energy of particles in potential (16) in the zeroth approximation is equal to the energy of a harmonic oscillator:

   ,                                     (17)

and the wave function in the zero approximation has the form [19]

.                     (18)

Then the calculation of the energy spectrum of electrons according to the perturbation theory gives the following result

 ,                         (19)

Fig. 2. Dependence of  on parameter  for various  . From above, the first line corresponds to the value  , the second to  and the third to  for a different value of  .

where  is the effective electron mass, the  axis is chosen as the size quantization axis,  , in the spherical approximation in the  energy spectrum. .

Figure 2 shows the dependence of  on the parameter  for various  . It can be seen from Fig. 2 that the energy spectrum of electrons in potential (16) takes discrete values and the steepness of the energy spectrum is more noticeable, the greater , and it also decreases with increasing  for arbitrary values of  .

 

 

Literature

1.       Shchuka A. A.. Nanoelectronics. -M., Fizmatkniga. 2007. -465 p.[in Russian]

2.       Usanov, D. A., & AV, S. Physical foundations of nanoelectronics. Saratov: Saratov State University. 2013. -162 p.

3.       Dragunov, V. P., Neizvestnyi, I. G., & Gridchin, V. A. Fundamentals of nanoelectronics. Logos, Moscow. 2006. -496 p.

4.       Ivchenko E. L. Optical spectroscopy of semiconductor nanostructures // E.L. Ivchenko.-Harrow (UK) : Alpha Science, 2005.-350 p.

5.       Rasulov R.Ya., Rasulov V.R., Eshboltaev I., Mamadalieva N.S. Absorption of linearly polarized radiation in a dimensionally quantized well. Uzbek Physics Journal: 2018. -Vol. 20. -No. 3-P.361-365.

6.       Petrov A.G., Shik A. Absorption of light by holes in quantum wells. FTP. -1994. -Vol.28. -P.2185 -2191. (In Russian).

7.       Rasulov, V. R., Rasulov, P. Y., Eshboltaev, I. M., & Sultonov, R. R. Dimensionally Quantization in n-GaP. Semiconductors. 2020. -Vol. 54(4), -P.429–432. (https://doi.org/10.1134/S1063782620040132)

8.       Golub L.E., Ivchenko E.L., Rasulov R.Ya. Absorption of light by holes in quantum wells. FTP. 1995.  -Vol. 29. -P. 1093-1099. (In Russian).

9.       Rasulov, R. Y., Rasulov, V. R., Mamadalieva, N. Z., & Sultanov, R. R. Subbarrier and Overbarrier Electron Transfer through Multilayer Semiconductor Structures. Russian Physics Journal. 2020. -63(4), 537-546.