RKKY Interaction On Surfaces of Topological Insulators With Superconducting Proximity Effect

We consider the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between magnetic impurities on the surface of a three-dimensional topological insulator with proximity induced superconductivity. A superconductor placed on the top of the topological insulator induces a gap in the surface electron states and gives rise to a long-ranged in-plane antiferromagnetic RKKY interaction. This interaction is frustrated due to strong spin-orbit coupling, decays as $1/r$ for $r<\xi$, where $r$ is the distance between two magnetic impurities and $\xi$ the superconducting coherence length, and dominates over the ferromagnetic and Dzyaloshinskii-Moriya type interactions for $r>\xi$. We find the condition for the Yu-Shiba-Rusinov intragap states that are bound to the magnetic impurities.