Andreev States in long shallow SNS constrictions

We study Andreev bound states in a long shallow normal constriction, which is open to a superconductor at both ends. The interesting features of such setup include the absence of electron-hole symmetry and the interference of electron waves along the constriction. We compare results of a numerical approach based on the Bogoliubov equations with those of a refined semiclassical description. Three types of Andreev bound states occur in the constriction: {\it i}) one where both electron and hole wave part of the bound state propagate through the constriction, {\it ii}) one where neither electron nor hole wave part propagate, and {\it iii}) one where only the electron wave propagates. We show that in a wide energy region the spacing between the Andreev states is strongly modulated by the interference of electron waves passing the constriction.

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