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We implement the Bogoliubov-de Gennes (BdG) equation in real-space using thescreened Korringa-Kohn-Rostoker (KKR) method. This allows us to solve,self-consistently, the superconducting state for 3d crystals includingsubstitutional impurities with a full normal-state DFT band structure. We applythe theoretical framework to bulk Nb with impurities. Without impurities, Nbhas an anisotropic gap structure with two distinct peaks around the Fermilevel. In the presence of non-magnetic impurities those peaks are broadened dueto the scattering between the two bulk superconducting gaps, however the peaksremain separated. As a second example of self-consistent real-space solutionsof the BdG equations we examine superconducting clusters embedded within anon-superconducting bulk metallic host. This allows us to estimate thecoherence length of the superconductor and we show that, within our framework,the coherence length of the superconductor is related to the inverse of the gapsize, just as in bulk BCS theory.

Real-space multiple scattering theory for superconductors with impurities