Treating some solid state problems with the Dirac equation

TL;DR

This paper introduces a relativistic approach using the Dirac equation to resolve ambiguities in effective-mass Hamiltonians for solid state problems, enabling accurate modeling of heterostructures.

Contribution

It extends the multistep approximation to relativistic cases, allowing arbitrary potential and mass profiles without ordering issues, and compares results with the Schrödinger equation approach.

Findings

Relativistic approach resolves effective-mass Hamiltonian ambiguities.

Results coincide with Schrödinger equation when using BenDaniel and Duke prescription.

Application demonstrated on semiconductor heterostructures.

Abstract

The ambiguity involved in the definition of effective-mass Hamiltonians for nonrelativistic models is resolved using the Dirac equation. The multistep approximation is extended for relativistic cases allowing the treatment of arbitrary potential and effective-mass profiles without ordering problems. On the other hand, if the Schrodinger equation is supposed to be used, our relativistic approach demonstrate that both results are coincidents if the BenDaniel and Duke prescription for the kinetic-energy operator is implemented. Applications for semiconductor heterostructures are discussed.