Electron-hole coherence in core-shell nanowires with partial proximity induced superconductivity

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.