g-factor symmetry and topology in semiconductor band states.

ABSTRACT

The [Formula see text] tensor, which determines the reaction of Kramers-degenerate states to an applied magnetic field, is of increasing importance in the current design of spin qubits. It is affected by details of heterostructure composition, disorder, and electric fields, but it inherits much of its structure from the effect of the spin-orbit interaction working at the crystal-lattice level. Here, we uncover interesting symmetry and topological features of [Formula see text] for important valence and conduction bands in silicon, germanium, and gallium arsenide. For all crystals with high (cubic) symmetry, we show that large departures from the nonrelativistic value [Formula see text] are guaranteed by symmetry. In particular, considering the spin part [Formula see text], we prove that the scalar function [Formula see text] must go to zero on closed surfaces in the Brillouin zone, no matter how weak the spin-orbit coupling is. We also prove that for wave vectors [Formula see text] on these surfaces, the Bloch states [Formula see text] have maximal spin-orbital entanglement. Using tight-binding calculations, we observe that the surfaces [Formula see text] exhibit many interesting topological features, exhibiting Lifshitz critical points as understood in Fermi-surface theory.