Self-consistent evaluation of proximity and inverse proximity effects with pair-breaking in diffusive SN junctions

We consider a planar superconducting-normal-metal (SN) junction with both inelastic and spin-flip scattering processes present. In the diffusive limit, we use a one-dimensional formulation of the Usadel equation to compute the self-consistent energy dependence of the single-particle density of states as a function of distance from the interface on both the superconducting and metallic sides for various spatial profiles of a pair-breaking spin-flip term. The pair-breaking processes fill in the superconducting gap at zero energy, which is reflected in the zero-bias tunneling conductance in scanning tunneling microscopy/spectroscopy experiments, in the vicinity of the junction. We also investigate the impact of having a partially transparent interface at the junction. We compare our findings with the observed exponential rise in the zero-bias conductance at the 1H step edge in recent experiments on 4Hb-TaS$_2$ [A. K. Nayak et al., Nat. Phys. 17, 1413 (2021)].