We implement the Bogoliubov-de Gennes (BdG) equation in real-space using the screened Korringa-Kohn-Rostoker (KKR) method. This allows us to solve, self-consistently, the superconducting state for 3D crystals including substitutional impurities with a full normal-state DFT band structure. We apply the theoretical framework to bulk Nb with impurities. Without impurities, Nb has an anisotropic gap structure with two distinct peaks around the Fermi level. In the presence of nonmagnetic impurities, those peaks are broadened due to the scattering between the two bulk superconducting gaps, however the peaks remain separated. As a second example of self-consistent real-space solutions of the BdG equations, we examine superconducting clusters embedded within a nonsuperconducting bulk metallic host. This allows us to estimate the coherence length of the superconductor and we show that, within our framework, the coherence length of the superconductor is related to the inverse of the gap size, just as in bulk BCS theory.

中文翻译：

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含杂质超导体的实空间多重散射理论

我们使用筛选的Korringa-Kohn-Rostoker（KKR）方法在真实空间中实现Bogoliubov-de Gennes（BdG）方程。这使我们能够自洽地解决3D晶体的超导状态，其中包括具有完全正常态DFT能带结构的替代杂质。我们将理论框架应用于具有杂质的大量Nb。没有杂质，Nb具有各向异性的间隙结构，在费米能级附近有两个不同的峰。在存在非磁性杂质的情况下，由于两个体超导间隙之间的散射，这些峰变宽，但是这些峰保持分离。作为BdG方程的自洽实空间解的第二个示例，我们研究了嵌入非超导块状金属主体中的超导簇。