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

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 ${\bf B}||\hat{\bf e}_μ$, the Zeeman $g_μ$ factor is $2{\rm m}\sqrt{m_μ}/m_g^{3/2} + O({\rm m}c^2)^{-2}$. 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 $\hat{\bf e}_μ$, $g<<2$ for all ${\bf B}={\bf\nabla}\times{\bf A}$ directions and vanishingly small for ${\bf B}||\hat{\bf e}_μ$. The quantum spin Hall Hamiltonian for 2D metals with $m_1=m_2=m_{||}$ is $K[{\bf E}\times({\bf p}-q{\bf A})]_{\perp}σ_{\perp}+O({\rm m}c^2)^{-4}$, 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, $σ_{\perp}$ is the Pauli matrix normal to the layer, $K=\pmμ_B/(2m_{||}c^2)$, and $μ_B$ is the Bohr magneton.