The Zeeman, Spin-Orbit, and Quantum Spin-Hall Interactions in Anisotropic and Low-Dimensional Conductors

TL;DR

This paper extends the Dirac equation to anisotropic low-dimensional conductors, revealing how magnetic interactions like Zeeman and spin-orbit effects depend on crystal anisotropy and dimensionality, with implications for superconductivity and quantum spin Hall states.

Contribution

It provides a relativistic framework for understanding magnetic interactions in anisotropic and low-dimensional conductors, including explicit formulas for g-factors and quantum spin Hall Hamiltonians.

Findings

g-factors depend on effective mass anisotropy

g_{||} is much less than 2 in 2D systems

Quantum spin Hall Hamiltonian derived for 2D metals

Abstract

When an electron or hole is in a conduction band of a crystal, it can be very different from 2, depending upon the crystalline anisotropy and the direction of the applied magnetic induction ${\bf B}$. In fact, it can even be 0! To demonstrate this quantitatively, the Dirac equation is extended for a relativistic electron or hole in an orthorhombically-anisotropic conduction band with effective masses $m_j$ for $j=1,2,3$ with geometric mean $m_g=(m_1m_2m_3)^{1/3}$. The appropriate Foldy-Wouthuysen transformations are extended to evaluate the non-relativistic Hamiltonian to $O({\rm m}c^2)^{-4}$, where ${\rm m}c^2$ is the particle's Einstein rest energy. For ${\bf B}||\hat{\bf e}_{\mu}$, the Zeeman $g_{\mu}$ factor is $2{\rm m}\sqrt{m_{\mu}}/m_g^{3/2} + O({\rm m}c^2)^{-2}$. While propagating in a two-dimensional (2D) conduction band with $m_3\gg m_1,m_2$, $g_{||}<<2$, consistent with recent measurements of the temperature $T$ dependence of the parallel upper critical induction $B_{c2,||}(T)$ in superconducting monolayer NbSe$_2$ and in twisted bilayer graphene. While a particle is in its conduction band of an atomically thin one-dimensional metallic chain along $\hat{\bf e}_{\mu}$, $g<<2$ for all ${\bf B}={\bf\nabla}\times{\bf A}$ directions and vanishingly small for ${\bf B}||\hat{\bf e}_{\mu}$. The quantum spin Hall Hamiltonian for 2D metals with $m_1=m_2=m_{||}$ is $K[{\bf E}\times({\bf p}-q{\bf A})]_{\perp}\sigma_{\perp}+O({\rm m}c^2)^{-4}$, where ${\bf E}$ and ${\bf p}-q{\bf A}$ are the planar electric field and gauge-invariant momentum, $q=\mp|e|$ is the particle's charge, $\sigma_{\perp}$ is the Pauli matrix normal to the layer, $K=\pm\mu_B/(2m_{||}c^2)$, and $\mu_B$ is the Bohr magneton.