Energy Levels in Nanowires and Nanorods with a Finite Potential Well

G. Gulyamov,Kh. N. Juraev,A. B. Davlatov,A. G. Gulyamov,Kamakhya Prasad Ghatak

doi:10.1155/2020/4945080

Abstract

The energy of electrons and holes in cylindrical quantum wires with a finite potential well was calculated by two methods. An analytical expression is approximately determined that allows one to calculate the energy of electrons and holes at the first discrete level in a cylindrical quantum wire. The electron energy was calculated by two methods for cylindrical layers of different radius. In the calculations, the nonparabolicity of the electron energy spectrum is taken into account. The dependence of the effective masses of electrons and holes on the radius of a quantum wires is determined. An analysis is made of the dependence of the energy of electrons and holes on the internal and external radii, and it is determined that the energy of electrons and holes in cylindrical layers with a constant thickness weakly depends on the internal radius. The results were obtained for the InP/InAs heterostructures.

Highlights

A lot of work has been done on calculating the energy of electrons and holes in quantum wells based on InP/InAs/InP heterostructures, due to the fact that today, for the creation of new-generation devices, it largely depends on semiconductor nanostructures
In type III-V semiconductors and in heterostructures based on them, electron dispersion is strongly nonparabolic; the Kane model was used to study the spectrum of charge carriers of these materials [9]
5 6 7 8 9 10 11 12 13 15 R1 that the wave function inside the barrier at a distance R1 is zero, and the error in this case is 0.01 meV. e results shown in Figure allow us to draw the following conclusion: for a constant thickness of cylindrical nanorod, there is a weak dependence of the particle energy on the internal radius R1, i.e., with an increase in R1, a slight increase in particle energy is observed

Summary

Introduction

A lot of work has been done on calculating the energy of electrons and holes in quantum wells based on InP/InAs/InP heterostructures, due to the fact that today, for the creation of new-generation devices, it largely depends on semiconductor nanostructures. In [23,24,25,26,27,28], to determine the energy spectrum and wave function of an electron in quantum wires with a rectangular cross-sectional shape, solutions of the Schrodinger equation were obtained using various mathematical methods. E authors of [30,31,32,33] obtained analytical solutions of the Schrodinger equation for cylindrical quantum wires with a finite potential and a parabolic dispersion law. E solution of equation (8) gives a linear combination of the imaginary argument Il(ζ) of the Bessel function and the Macdonald function Kl(ζ) of the l-th order [37]: ψ2(ρ) A2Kl cBρ􏼁 + B2Il cBρ􏼁, ρ > R. erefore, ψ(ρ) for a radial wave function is appropriate for the following: ψ(ρ) 􏼨 A1Jl kAρ􏼁 + B1Nl kAρ􏼁, 0 ≤ ρ ≤ R,. MA/mB􏼁Jl kAR1􏼁I′l cBR1􏼁 − J′l kAR1􏼁Il cBR1􏼁 N′l kAR1􏼁Il cBR1􏼁 − mA/mB􏼁Nl kAR1􏼁I′l cBR1􏼁 . mA/mB􏼁Jl kAR2􏼁K′l cBR2􏼁 − J′l kAR2􏼁Kl cBR2􏼁 N′l kAR2􏼁Kl cBR2􏼁 − mA/mB􏼁Nl kAR2􏼁K′l cBR2􏼁

Valence band

InAs InP

Nonparabolic Parabolic