# Effects of nonmagnetic disorder on the energy of Yu-Shiba-Rusinov states

## TL;DR

This paper investigates how nonmagnetic disorder affects Yu-Shiba-Rusinov states in superconductors, finding that the energy remains relatively stable even with moderate disorder, especially in three dimensions.

## Contribution

The study introduces a scattering approach to quantify the impact of nonmagnetic disorder on Yu-Shiba-Rusinov states, revealing their robustness in various dimensions.

## Key findings

- Disorder-induced effects on YSR states are small even at moderate disorder levels.
- Energy variance of YSR states remains small in the dirty limit, especially in 3D.
- Impurities within a few Fermi wavelengths significantly influence YSR states in 3D, while in 2D the effect is logarithmically larger.

## Abstract

We study the sensitivity of Yu-Shiba-Rusinov states, bound states that form around magnetic scatterers in superconductors, to the presence of nonmagnetic disorder in both two and three dimensional systems. We formulate a scattering approach to this problem and reduce the effects of disorder to two contributions: disorder-induced normal reflection and a random phase of the amplitude for Andreev reflection. We find that both of these are small even for moderate amounts of disorder. In the dirty limit in which the disorder-induced mean free path is smaller than the superconducting coherence length, the variance of the energy of the Yu-Shiba-Rusinov state remains small in the ratio of the Fermi wavelength and the mean free path. This effect is more pronounced in three dimensions, where only impurities within a few Fermi wavelengths of the magnetic scatterer contribute. In two dimensions the energy variance is larger by a logarithmic factor because impurities contribute up to a distance of the order of the superconducting coherence length.