Dirac and Klein-Gordon particles in one-dimensional periodic potentials

TL;DR

This paper investigates the energy dispersion and transmission properties of massless Dirac fermions and Klein-Gordon bosons in one-dimensional periodic potentials, with applications to graphene and superlattice structures.

Contribution

It provides a detailed analysis of dispersion relations and transmission characteristics for relativistic particles in periodic potentials, including numerical results and their relation to superlattice behavior.

Findings

Zero energy gap for zero-momentum carriers in Dirac particles

Numerical energy spectrum and density of states for fermions

Transmission properties related to superlattice configurations

Abstract

We evaluate the dispersion relation for massless fermions, described by the Dirac equation, and for zero-spin bosons, described by the Klein-Gordon equation, moving in two dimensions and in the presence of a one-dimensional periodic potential. For massless fermions the dispersion relation shows a zero gap for carriers with zero momentum in the direction parallel to the barriers in agreement with the well-known "Klein paradox". Numerical results for the energy spectrum and the density of states are presented. Those for fermions are appropriate to graphene in which carriers behave relativistically with the "light speed" replaced by the Fermi velocity. In addition, we evaluate the transmission through a finite number of barriers for fermions and zero-spin bosons and relate it with that through a superlattice.