Scattering of relativistic electrons and analogies with optical phenomena: A study of longitudinal and transverse shifts at step potentials

TL;DR

This paper explores how relativistic electrons behave at potential steps, drawing analogies with optical phenomena like the Goos-H"anchen and Imbert-Fedorov shifts, and discusses implications for Dirac and topological materials.

Contribution

It provides a novel analysis of electron shifts at potential steps using current and angular momentum conservation, linking quantum effects to optical phenomena.

Findings

Goos-H"anchen shift decomposed into two components from current and interference

Transverse Imbert-Fedorov shift related to spin Hall effect

Results align with stationary phase method findings

Abstract

We investigate the behavior of relativistic electrons encountering a potential step through analogies with optical phenomena. By accounting for the conservation of Dirac current, we elucidate that the Goos-H\"anchen shift can be understood as a combination of two components: one arising from the current entering the transmission region and the other originating from the interference between the incident and reflected beams. This result has been proven to be consistent with findings obtained utilizing the stationary phase method. Moreover, we explore the transverse Imbert-Fedorov shift, by applying both current conservation and total angular momentum conservation, revealing intriguing parallel to the spin Hall effect. Beyond enriching our comprehension of fundamental quantum phenomena, our findings have potential applications for designing and characterizing devices using Dirac and topological materials.