A relativistic electron in an anisotropic conduction band

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.