Surface density of states in superconductors with inhomogeneous pairing constant: Analytical results

TL;DR

This paper analytically investigates how surface suppression of the pairing constant in superconductors affects the local density of states, revealing a surface gap, a distinctive vertical feature at the bulk gap energy, and higher-energy corrections.

Contribution

It provides analytical expressions for the surface density of states in superconductors with inhomogeneous pairing, highlighting differences from the order parameter and describing a unique vertical peculiarity.

Findings

Surface gap in DOS differs from the order parameter due to spatial scale differences.

A vertical peculiarity at the bulk gap energy exhibits asymmetric square-root behavior.

Perturbative corrections to DOS are derived at higher energies.

Abstract

We consider a superconductor with surface suppression of the BCS pairing constant $\lambda(x)$. We analytically find the gap in the surface density of states (DOS), behavior of the DOS $\nu(E)$ above the gap, a "vertical" peculiarity of the DOS around an energy equal to the bulk order parameter $\Delta_0$, and a perturbative correction to the DOS at higher energies. The surface gap in the DOS is parametrically different from the surface value of the order parameter due to a difference between the spatial scale $r_c$, at which $\lambda(x)$ is suppressed, and the coherence length. The vertical peculiarity implies an infinite-derivative inflection point of the DOS curve at $E=\Delta_0$ with square-root behavior as $E$ deviates from $\Delta_0$. The coefficients of this dependence are different at $E<\Delta_0$ and $E>\Delta_0$, so the peculiarity is asymmetric.