  By solving the Bogoliubov-de Gennes Hamiltonian, the electron-hole coherence
within a partially proximitized n-doped semiconductor shell of a core-shell
nanowire heterostructure is investigated numerically and compared with the
Andreev reflection interpretation of proximity induced superconductivity.
Partial proximitization is considered to quantify the effects of a reduced
coherence length. Three cases of partial proximitization of the shell are
explored: radial, angular and longitudinal. For the radial case, it is found
that the boundary conditions impose localization probability maxima in the
center of the shell in spite of off-center radial proximitization. The induced
superconductivity gap is calculated as a function of the ratio between the
proximitized shell thickness and the total shell thickness. In the angular
case, the lowest energy state of a hexagonal wire with a single proximitized
side is found to display the essence of Andreev reflection, only by lengthwise
summation of the localization probability. In the longitudinal case, a clear
correspondence with Andreev reflection is seen in the localization probability
as a function of length along a half-proximitized wire. The effect of an
external magnetic field oriented along the wire is explored.
