A relativistic electron in an anisotropic conduction band

TL;DR

This paper extends the Dirac equation to describe a relativistic electron in an anisotropic conduction band, revealing effects on magnetic properties and quantum spin Hall effect in various anisotropic materials.

Contribution

It introduces a covariant formulation of the Dirac equation for anisotropic conduction bands, linking relativistic quantum mechanics with anisotropic material properties.

Findings

Quantum spin Hall effect derived for anisotropic bands

Zeeman energy is negligible in certain low-dimensional materials

Covariance established under Lorentz transformations

Abstract

The Dirac equation is extended for a relativistic electron in an orthorhombically-anisotropic conduction band. Its covariance is established with general proper and improper Lorentz transformations. In the non-relativistic limit, the kinetic and Zeeman energy terms of the Hamiltonian are both determined by the same three effective masses, and the quantum spin Hall effect is derived. This has important consequences for magnetic measurements of many classes of clean anisotropic semiconductors, metals, and superconductors. The Zeeman energy is vanishingly small for magnetic fields parallel to clean monolayers and in all directions in quasi-one-dimensional materials.