Revisiting the surface density of states of midgap Andreev edge states

Abstract

We revisit the effect of surface roughness on midgap Andreev edge states (MAES) in p- and d- wave superconductors. For a perfectly specular surface, MAES form a flat band at the Fermi energy, which manifests as a sharp midgap peak in the surface density of states (SDOS). Previous theoretical studies have shown that MAES in p- and d-wave superconductors respond markedly differently to surface roughness. In the d-wave state, diffuse surface scattering significantly broadens the midgap peak in the SDOS, accompanied by the emergence of a V-shaped structure centered at the Fermi energy. In contrast, the midgap peak in the p-wave state remains robust against diffuse scattering. In this work, we clarify the physical origin of this contrasting behavior. A key aspect of our analysis is that the flat band in the d-wave state consists of two distinct types of MAES modes. We show that inter-mode diffuse scattering leads to substantial broadening of the midgap peak and to the formation of the V-shaped structure. By contrast, the robustness of MAES in the p-wave state arises from the presence of a single MAES mode in the flat band. These results provide new insight into the response of MAES to surface roughness.