Exact Solutions of Two-Band Models of Graded-Gap Superlattices

TL;DR

This paper derives exact solutions for two-band models of graded-gap superlattices with linearly varying gaps, providing explicit dispersion relations using hypergeometric functions within the envelope-function approximation.

Contribution

It presents a novel analytical approach to solve Dirac-like equations in graded-gap superlattices with exact expressions for dispersion relations.

Findings

Exact solutions for Dirac-like equations in graded-gap superlattices

Closed-form dispersion relations using confluent hypergeometric functions

Analytical framework for understanding miniband structures

Abstract

We have theoretically investigated two-band models of graded-gap superlattices within the envelope-function approximation. Assuming that the gap varies linearly with spatial coordinate, we are able to find exact solutions of the corresponding Dirac-like equation describing the conduction- and valence-band envelope-functions. The dispersion relation inside allowed miniband of the superlattice may be expressed in terms of confluent hypergeometric functions in a closed form.