Fermi-level pinning of Yu-Shiba-Rusinov states in a superconductor with weakly broken spin-rotational invariance

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, SJ, scales like SJ B/A, where B is the magnitude of the internal field, that breaks the spin-rotation symmetry, and A 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 epsilon 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, SJ = 0; consequently, there is no pinning of the YSR-resonance to epsilon 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.