Superconducting order parameter $\pi$-phase shift in magnetic impurity wires

TL;DR

This paper investigates how magnetic impurity wires induce a $$-phase shift in the superconducting order parameter, linking it to quasiparticle resonance conditions and showing its persistence with spin-orbit coupling in topological phases.

Contribution

It reveals the microscopic origin of the $$-phase shift in magnetic impurity wires and demonstrates its stability in topologically non-trivial superconducting phases with spin-orbit coupling.

Findings

Resonance condition causes $$-phase shift in impurity wires.

Superconducting order parameter decreases in steps with increasing magnetic strength.

$$-phase shift persists with finite spin-orbit coupling in topological phases.

Abstract

It has previously been found that a magnetic impurity in a conventional $s$-wave superconductor can give rise to a local $\pi$-phase shift of the superconducting order parameter. By studying a finite wire of ferromagnetic impurities, we are able to trace the origin of the $\pi$-phase shift to a resonance condition for the Bogoliubov-de Gennes quasiparticle states. When non-resonating states localized at the impurity sites are pulled into the condensate for increasing magnetic strength, the superconducting order parameter is reduced in discrete steps, eventually resulting in a $\pi$-phase shift. We also show that for a finite spin-orbit coupling, the $\pi$-phase shift is preserved and occurs in a large portion of the topologically non-trivial phase.