Zero-energy bound state at the interface between an $s$-wave superconductor and a disordered normal metal with repulsive electron-electron interactions

TL;DR

This paper investigates how disorder and interfacial barriers affect zero-energy bound states at superconductor-normal metal interfaces with repulsive interactions, revealing the transition from bound states to minigap formation.

Contribution

It provides a detailed analysis of the impact of disorder and barriers on zero-energy bound states using quasiclassical theory, extending understanding of topological superconductivity interfaces.

Findings

Zero-energy peak broadens and diminishes with increased disorder.

Minigap is restored in the diffusive limit, eliminating bound state signatures.

Interfacial barriers shift the zero-energy peak without changing its shape.

Abstract

In recent years, there has been a renewed interest in the proximity effect due to its role in the realization of topological superconductivity. Here, we study a superconductor--normal metal proximity system with repulsive electron-electron interactions in the normal layer. It is known that in the absence of disorder or normal reflection at the superconductor--normal metal interface, a zero-energy bound state forms and is localized to the interface [Fauchere et al., Phys. Rev. Lett. 82, 3336 (1999)]. Using the quasiclassical theory of superconductivity, we investigate the low-energy behavior of the density of states in the presence of finite disorder and an interfacial barrier. We find that as the mean free path is decreased, the zero-energy peak in the density of states is broadened and reduced. In the quasiballistic limit, the bound state eliminates the minigap pertinent to a noninteracting normal layer and a distinct peak is observed. When the mean free path becomes comparable to the normal layer width, the zero-energy peak is strongly suppressed and the minigap begins to develop. In the diffusive limit, the minigap is fully restored and all signatures of the bound state are eliminated. We find that an interfacial potential barrier does not change the functional form of the density of states peak but does shift this peak away from zero energy.