Transparent boundary conditions for the Bogoliubov–de Gennes equation in (1+1)D

Abstract

Discrete transparent boundary conditions (TBCs) are presented for a Crank–Nicolson implementation of the (1+1)D Bogoliubov–de Gennes equation for s- and p-wave superconductors. The whole-space dynamics is reformulated as a finite-space problem augmented by transparent boundary conditions. The latter are derived for the underlying space–time grid under the assumption of constant coefficients (mass, electric potential, and gap function) in the exterior regions. The translation matrix for the spinor on the spatial grid is expressed in Z space. The condition for out-projection of (exponentially) growing wave contributions establishes the TBCs. Their inverse Z transform then are fed into the Crank–Nicolson scheme. Next to basic numerical tests, the scheme is illustrated for the example of Andreev reflection at normal-metal–superconductor interfaces and compared to analytic results, where available. Suppression factors for reflected waves at superconducting boundaries are determined for selected numerical examples.