Enhanced orbital magnetic field effects in Ge hole nanowires

TL;DR

This paper investigates how orbital magnetic fields influence the spin-orbit interaction and g-factors in Ge hole nanowires, revealing enhanced effects and potential advantages for quantum computing applications like Majorana bound states.

Contribution

It provides an exact analytical solution and an effective low-energy model for Ge NWs, highlighting growth directions and architectures optimal for Majorana states and qubit coherence.

Findings

Orbital magnetic fields can renormalize the g factor up to 400%.

Core/shell NWs along [110] have twice the g factor of other directions.

Curved Ge quantum wells exhibit large g factors and SOI, beneficial for Majorana states.

Abstract

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 work, 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. In core/shell NWs grown along the $[110]$ direction the $g$ factor can be twice larger than for other growth directions which makes this growth direction advantageous for Majorana bound states. Also curved Ge quantum wells feature large effective $g$ factors and SOI, again ideal for hosting Majorana bound states. Strikingly, because these quantities are independent of the electric field, hole spin qubits encoded in curved quantum wells are to good approximation not susceptible to charge noise, significantly boosting their coherence time.