1-D Dirac Equation, Klein Paradox and Graphene

TL;DR

This paper provides exact solutions to the 1D Dirac equation with step potentials, clarifying misconceptions about the Klein Paradox and exploring implications for graphene's electronic properties.

Contribution

It demonstrates that the Klein Paradox does not exist when solutions are correctly matched and analyzes the distinct current and momentum behaviors in graphene.

Findings

Klein Paradox is a result of incorrect solution matching

Exact solutions show no paradox in 1D Dirac systems

Graphene exhibits unique current and momentum characteristics

Abstract

Solutions of the one dimensional Dirac equation with piece-wise constant potentials are presented using standard methods. These solutions show that the Klein Paradox is non-existent and represents a failure to correctly match solutions across a step potential. Consequences of this exact solution are studied for the step potential and a square barrier. Characteristics of massless Dirac states and the momentum linear band energies for Graphene are shown to have quite different current and momentum properties.