A Relativistic One Dimensional Band Model with Position Dependent Mass

TL;DR

This paper introduces a one-dimensional relativistic band model with position-dependent mass, analyzing how mass variation affects electron and positron band structures, revealing that non-locality has minimal impact at low energies but significant effects at high energies.

Contribution

It presents a novel relativistic band model incorporating a position-dependent mass function represented as a non-local operator, extending the standard Kronig-Penney model.

Findings

Constant mass case resembles standard relativistic Kronig-Penney bands.

At low wavenumber, particles behave as if mass is constant.

High wavenumber bands are distorted by periodic mass variations.

Abstract

In this note a one-dimensional band model is proposed based on a periodic Dirac comb having an identical mass distribution $m(x)$ . in each unit cell. The mass function is represented as a Hermitian, non-local separable operator. Two specific cases--a constant mass model and a sinusoidal mass model--are examined. The lowest electron and positron bands for the constant mass case are similar to those for the standard relativistic Kronig-Penney model, suggesting that non-locality has little influence. The results for the sinusoidal case are consistent with the expectation that at low wavenumber an electron "feels" it has am average constant mass, but at high wave number, the particle "sees" the periodic mass variation and the band is distorted.