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Low-dimensional germanium hole devices are promising systems with manypotential applications such as hole spin qubits, Andreev spin qubits, andJosephson junctions, and can serve as a basis for the realization oftopological superconductivity. This vast array of potential uses for Ge largelystems from the exceptionally strong and controllable spin-orbit interaction(SOI), ultralong mean free paths, long coherence times, and compatibility withcomplementary metal-oxide-semiconductor (CMOS) technology. However, whenbrought into proximity with a superconductor (SC), metallization normallydiminishes many useful properties of a semiconductor, for instance, typicallyreducing the g factor and SOI energy, as well as renormalizing the effectivemass. In this paper, we consider metallization of a Ge nanowire (NW) inproximity to a SC, explicitly taking into account the three-dimensional (3D)geometry of the NW. We find that proximitized Ge exhibits a uniquephenomenology of metallization effects, where the 3D cross section plays acrucial role. For instance, in contrast to expectations, we find that SOI canbe enhanced by strong coupling to the superconductor. We also show that thethickness of the NW plays a critical role in determining both the size of theproximity-induced pairing potential and metallization effects, since thecoupling between the NW and SC strongly depends on the distance of the NW wavefunction from the interface with the SC. In the absence of electrostaticeffects, we find that a sizable gap opens only in thin NWs ($d \lesssim 3$ nm).In thicker NWs, the wave function must be pushed closer to the SC byelectrostatic effects in order to achieve a sizable proximity gap such that therequired electrostatic field strength can simultaneously induce a strong SOI.

Microscopic analysis of proximity-induced superconductivity and metallization effects in superconductor-germanium hole nanowires