Anisotropic g-tensor and strong spin-orbit interaction of holes in hexagonal germanium nanowires

Abstract

Hexagonal germanium combines direct-gap optical activity with the low-nuclear-spin environment of a group-IV material, making it a candidate platform for spin qubits and spin-photon interfaces. Using a ten-band $\mathbf{k}\cdot\mathbf{p}$ model, we study the response of confined hole states in hexagonal germanium nanowires to magnetic and electric fields. We show that, by symmetry, the top valence band of the bulk crystal has a vanishing $g$ factor in the plane perpendicular to the $c$ axis, in analogy with the in-plane $g$ factor of heavy holes in two-dimensional hole gases. In nanowires, confinement induces only small transverse $g$-tensor components, so the $g$ tensor remains strongly anisotropic. A transverse electric field strongly reduces the nanowire band gap, providing a gate-tunable optical signature and limiting the usable field range. The same field induces a Rashba spin-orbit interaction that is markedly stronger for fields perpendicular to the $c$ axis and depends nonlinearly on the field strength. In narrow wires at strong fields, the spin-orbit length reaches a few tens of nanometers, comparable to hole devices in cubic silicon. We identify the microscopic origin of the Rashba coupling as a second-order process mediated by a few intermediate subbands with distinct symmetry selection rules. Our results provide guidance for the design of electrically controlled hole-spin qubits in hexagonal germanium nanowires.