Andreev reflection and geometrical resonance effects for a gradual variation of the pair potential near the normal-metal–superconductor interface

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Abstract
The probability of Andreev reflection is calculated as a function of the energy for quasiparticles that are incident on a normal-metal–superconductor (N-S) interface with a gradual variation of the pair potential. These calculations are an extension of the work of Blonder, Tinkham, and Klapwijk [Phys. Rev. B 25, 4515 (1982)], who assumed a step function for the position dependence of the pair potential. We integrate the Bogoliubov equations numerically in the region in which the pair potential varies with position and apply boundary conditions to find the quantities of interest. This approach is also used to calculate the geometrical resonances in the transmission of a tunnel junction on an N-S bilayer. For a steplike variation of the pair potential, the same expression for the transmission is found as with the usual density-of-states approach. Also results are given for a gradual variation of the pair potential at the interface. Both the probability of Andreev reflection and the geometrical resonance effects begin to change if the region in which the pair potential varies with position becomes of the order of the coherence length of the superconductor.