  The proximity effect refers to the phenomenon whereby superconducting
properties are induced in a normal conductor that is in contact with an
intrinsically superconducting material. In particular, the combination of
nano-structured semiconductors with bulk superconductors is of interest because
these systems can host unconventional electronic excitations such as Majorana
fermions when the semiconductor's charge carriers are subject to a large
spin-orbit coupling. The latter requirement generally favors the use of
hole-doped semiconductors. On the other hand, basic symmetry considerations
imply that states from typical simple-metal superconductors will predominantly
couple to a semiconductor's conduction-band states and, therefore, in the first
instance generate a proximity effect for band electrons rather than holes. In
this article, we show how the superconducting correlations in the conduction
band are transferred also to hole states in the valence band by virtue of
inter-band coupling. A general theory of the superconducting proximity effect
for bulk and low-dimensional hole systems is presented. The interplay of
inter-band coupling and quantum confinement is found to result in unusual
wave-vector dependencies of the induced superconducting gap parameters. One
particularly appealing consequence is the density tunability of the proximity
effect in hole quantum wells and nanowires, which creates new possibilities for
manipulating the transition to nontrivial topological phases in these systems.
