Semiconductor quantum wells with BenDaniel - Duke boundary conditions: approximate analytical results

TL;DR

This paper derives approximate analytical formulas for bound state energies in semiconductor quantum wells with BenDaniel-Duke boundary conditions, simplifying the transcendental equations into algebraic forms for practical use.

Contribution

It introduces algebraic approximations for solving the Schrödinger equation in quantum wells with BenDaniel-Duke boundary conditions, providing simple formulas for bound state energies.

Findings

Derived cubic equations for ground and first excited states.

Provided algebraic formulas for bound state energies.

Analyzed higher excited states with approximate solutions.

Abstract

The Schrodinger equation for a particle moving in a square well potential with BenDaniel - Duke boundary conditions is solved. Using algebraic approximations for trigonometric functions, the transcendental equations of the bound states energy are transformed into tractable, algebraic equations. For the ground state and the first excited state, they are cubic equations; we obtain simple formulas for their physically interesting roots. The case of higher excited states is also analyzed. Our results have direct applications in the physics of type I and type II semiconductor heterostructures.