# Robustness of Yu-Shiba-Rusinov resonances in presence of a complex superconducting order parameter

## TL;DR

This study investigates how complex, energy-dependent superconducting order parameters affect Yu-Shiba-Rusinov (YSR) states, revealing they can appear outside the gap and have increased broadening, challenging previous assumptions about their protection.

## Contribution

It demonstrates that nontrivial order parameters in superconductors cause YSR states to appear outside the gap with increased broadening, supported by experimental data and theoretical calculations.

## Key findings

- YSR states can appear outside the superconducting gap.
- Intrinsic broadening of YSR states relates to the imaginary part of order parameters.
- Non-thermal relaxation mechanisms affect YSR state lifetime.

## Abstract

Robust quantum systems rely on having a protective environment with minimized relaxation channels. Superconducting gaps play an important role in the design of such environments. The interaction of localized single spins with a conventional superconductor generally leads to intrinsically extremely narrow Yu-Shiba-Rusinov (YSR) resonances protected inside the superconducting gap. However, this may not apply to superconductors with nontrivial, energy dependent order parameters. Exploiting the Fe-doped two-band superconductor NbSe$_2$, we show that due to the nontrivial relation between its complex valued and energy dependent order parameters, YSR states are no longer restricted to be inside the gap. They can appear outside the gap (i. e. inside the coherence peaks), where they can also acquire a substantial intrinsic lifetime broadening. T-matrix scattering calculations show excellent agreement with the experimental data and relate the intrinsic YSR state broadening to the imaginary part of the host's order parameters. Our results suggest that non-thermal relaxation mechanisms contribute to the finite lifetime of the YSR states, even within the superconducting gap, making them less protected against residual interactions than previously assumed. YSR states may serve as valuable probes for nontrivial order parameters promoting a judicious selection of protective superconductors.