Density functional Bogoliubov-de Gennes analysis of superconducting Nb and Nb(110) surfaces

We report on the implementation of the Bogoliubov-de Gennes method into the JuKKR code [https: //jukkr.fz-juelich.de], an implementation of the relativistic all-electron, full-potential Korringa-Kohn-Rostoker Green function method, which allows a material-specific description of inhomogeneous super-conductors and heterostructures on the basis of density functional theory. We describe the formalism and report on calculations for the s-wave superconductor Nb, a potential component of the materials platform enabling the realization of the Majorana zero modes in the field of topological quantum computing. We compare the properties of the superconducting state both in the bulk and for (110) surfaces. We compare slab calculations for different thicknesses and comment on the importance of spin-orbit coupling, the effect of surface relaxations and the influence of a softening of phonon modes on the surface for the resulting superconducting gap.