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 $\mathbf{B}$. 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 $m_g=(m_1{m}_2{m}_3{)}^{1/3}$. The appropriate Foldy-Wouthuysen transformations are extended to evaluate the non-relativistic Hamiltonian to $O(\mathrm{m}{c}^2{)}^{-4}$, where $\mathrm{m}{c}^2$ is the particle's Einstein rest energy. For $\mathbf{B}||{\hat{\mathbf{e}}}_{\mu }$, the Zeeman $g_{\mu }$ factor is $2\mathrm{m}\sqrt{m_{\mu }}/m_g^{3/2}+O(\mathrm{m}{c}^2{)}^{-2}$. 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 ${\hat{\mathbf{e}}}_{\mu }$, $g<<2$ for all $\mathbf{B}=\nabla \times \mathbf{A}$ directions and vanishingly small for $\mathbf{B}||{\hat{\mathbf{e}}}_{\mu }$. The quantum spin Hall Hamiltonian for 2D metals with $m_1=m_2=m_{||}$ is $K[\mathbf{E}\times (\mathbf{p}-q\mathbf{A}){]}_{\perp }{\sigma }_{\perp }+O(\mathrm{m}{c}^2{)}^{-4}$, where $\mathbf{E}$ and $\mathbf{p}-q\mathbf{A}$ 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.

arXiv admin note: text overlap with arXiv:1906.10164