Peculiarities of the density of states in SN junctions

TL;DR

This paper analytically investigates the density of states in diffusive SN junctions, revealing three distinct types of spectral peculiarities at the superconducting gap depending on the inverse proximity effect and material parameters.

Contribution

It provides a detailed analytical description of the density of states peculiarities in SN junctions, including the transition from check-mark to vertical features influenced by inverse proximity effects.

Findings

Identifies three types of DoS peculiarities at the gap energy.

Shows the transition from check-mark to vertical peculiarity with inverse proximity.

Highlights the role of material-matching parameters in spectral features.

Abstract

We study the density of states (DoS) $\nu(E)$ in a normal-metallic (N) film contacted by a bulk superconductor (S). We assume that the system is diffusive and the SN interface is transparent. In the limit of thin N layer (compared to the coherence length), we analytically find three different types of the DoS peculiarity at energy equal to the bulk superconducting order parameter $\Delta_0$. (i) In the absence of the inverse proximity effect, the peculiarity has the check-mark form with $\nu(\Delta_0)=0$ as long as the thickness of the N layer is smaller than a critical value. (ii) When the inverse proximity effect comes into play, the check-mark is immediately elevated so that $\nu(\Delta_0)>0$. (iii) Upon further increasing of the inverse proximity effect, $\nu(E)$ gradually evolves to the vertical peculiarity (with an infinite-derivative inflection point at $E=\Delta_0$). This crossover is controlled by a materials-matching parameter which depends on the relative degree of disorder in the S and N materials.