Fermi-Level Pinning of Yu-Shiba-Rusinov States in a Superconductor with Weakly Broken Spin-Rotational Invariance

TL;DR

This paper demonstrates that self-consistent modeling reveals a critical impurity coupling window where Yu-Shiba-Rusinov states are pinned to the Fermi level, enabling continuous control of impurity orientation in superconductors with weak spin symmetry breaking.

Contribution

It shows that self-consistency is essential to accurately describe YSR state pinning and impurity orientation control, contrasting with non-self-consistent approaches that miss this critical behavior.

Findings

YSR resonance pinning occurs within a finite coupling window proportional to magnetic field strength.

Self-consistent treatment predicts a finite critical window, unlike zero-width in non-self-consistent models.

Impurity moments rotate continuously when tuning through the critical window.

Abstract

As is well known, magnetic impurities adsorbed on superconductors, e.g. of the s-wave type, can introduce a bound gap-state (Yu-Shiba-Rusinov resonance). We here investigate within a minimal model how the impurity moment arranges with respect to a weak homogeneous internal magnetic field employing a fully self-consistent mean-field treatment. Our investigation reveals a critical window of impurity-to-substrate coupling constants, $J$. The width of the critical region, $\delta J$, scales like $\delta J \sim B/\Delta$, where $B$ is the magnitude of the internal field, that breaks the spin-rotation symmetry, and $\Delta$ is the bulk order parameter. While tuning $J$ through the window, the energy of the Yu-Shiba-Rusinov (YSR) resonance is pinned to the Fermi energy $\varepsilon_{\rm F}$ and the impurity moment rotates in a continuous fashion. We emphasize the pivotal role of self-consistency: In treatments ignoring self-consistency, the critical window adopts zero width, $\delta J=0$; consequently, there is no pinning of the YSR-resonance to $\varepsilon_{\rm F}$, the impurity orientation jumps and therefore this orientation cannot be controlled continuously by fine-tuning the coupling $J$. In this sense, our study highlights the significance of self-consistency for understanding intricate magnetic interactions between superconductive materials and Shiba chains.