Enhanced orbital magnetic field effects in Ge hole nanowires

Hole semiconductor nanowires (NW) are promising platforms to host spin qubits and Majorana bound states for topological qubits because of their strong spin-orbit interactions (SOI). The properties of these systems depend strongly on the design of the cross section and on strain, as well as on external electric and magnetic fields. In this paper, we analyze in detail the dependence of the SOI and $g$ factors on the orbital magnetic field. We focus on magnetic fields aligned along the axis of the NW, where orbital effects are enhanced and result in a renormalization of the effective $g$ factor up to $400%$, even at small values of magnetic field. We provide an exact analytical solution for holes in Ge NWs and we derive an effective low-energy model that enables us to investigate the effect of electric fields applied perpendicular to the NW. We also discuss in detail the role of strain, growth direction, and high-energy valence bands in different architectures, including Ge/Si core/shell NWs, gate-defined one-dimensional channels in planar Ge, and curved Ge quantum wells. By comparing NWs with different growth directions, we find that the isotropic approximation is well justified. Curved Ge quantum wells feature large effective $g$ factors and SOI at low electric field, ideal for hosting Majorana bound states. In contrast, at strong electric field, these quantities are independent of the field, making hole spin qubits encoded in curved quantum wells to good approximation not susceptible to charge noise, and significantly boosting their coherence time.