The Ruderman-Kittel-Kasuya-Yosida interaction of magnetic impurities in a superconductor exponentially decreases when the distance $r$ between them is larger than the superconductor's coherence length because this interaction is mediated by quasiparticles, which have a gap in their energy spectra. At the same time, the spin-singlet superconducting condensate was always assumed to stay neutral to magnetic impurities. Due to a spin-orbit coupling (SOC), however, Cooper pairs gain an admixture of spin-triplet correlated states, which provide for a link between impurity spins and an $s$-wave condensate. It is shown that perturbations of its phase mediate the $1/r^2$ interaction of these spins in two-dimensional (2D) systems. This effect is considered within two models: of a clean 2D $s$-wave superconductor with the strong Rashba SOC and of a bilayer system which combines a 2D Rashba coupled electron gas and an $s$-wave superconducting film. The predicted long-range interaction can have a strong effect on spin orders in superconductor-magnetic impurity systems that are expected to host Majorana fermions.

• Received 8 May 2018
• Revised 24 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.054504