  Low-dimensional germanium hole devices are promising systems with many
potential applications such as hole spin qubits, Andreev spin qubits, and
Josephson junctions, and can serve as a basis for the realization of
topological superconductivity. This vast array of potential uses for Ge largely
stems from the exceptionally strong and controllable spin-orbit interaction
(SOI), ultralong mean free paths, long coherence times, and compatibility with
complementary metal-oxide-semiconductor (CMOS) technology. However, when
brought into proximity with a superconductor (SC), metallization normally
diminishes many useful properties of a semiconductor, for instance, typically
reducing the g factor and SOI energy, as well as renormalizing the effective
mass. In this paper, we consider metallization of a Ge nanowire (NW) in
proximity to a SC, explicitly taking into account the three-dimensional (3D)
geometry of the NW. We find that proximitized Ge exhibits a unique
phenomenology of metallization effects, where the 3D cross section plays a
crucial role. For instance, in contrast to expectations, we find that SOI can
be enhanced by strong coupling to the superconductor. We also show that the
thickness of the NW plays a critical role in determining both the size of the
proximity-induced pairing potential and metallization effects, since the
coupling between the NW and SC strongly depends on the distance of the NW wave
function from the interface with the SC. In the absence of electrostatic
effects, 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 by
electrostatic effects in order to achieve a sizable proximity gap such that the
required electrostatic field strength can simultaneously induce a strong SOI.
