We develop a method to solve the Bogoliubov–de Gennes equation for superconductors self-consistently, using the recursion method. The method allows the pairing interaction to be either local or nonlocal, corresponding to $s$- and $d$-wave superconductivity, respectively. Using this method we examine the properties of various $S-N$ and $S-S$ interfaces. In particular, we calculate the spatially varying density of states and order parameter for the following geometries: (i) $s$-wave superconductor to normal metal, (ii) $d$-wave superconductor to normal metal, (iii) $d$-wave superconductor to $s$-wave superconductor. We show that the density of states at the interface has a complex structure including the effects of normal surface Friedel oscillations, the spatially varying gap and Andeev states within the gap, and the subtle effects associated with the interplay of the gap and the normal van Hove peaks in the density of states. In the case of bulk $d$-wave superconductors, the surface leads to mixing of different order-parameter symmetries near the interface and substantial local filling in of the gap.

• Received 21 August 1997

DOI:https://doi.org/10.1103/PhysRevB.57.8709