The effect of midgap states on the magnetic exchange interaction mediated by a $d$-wave superconductor

TL;DR

This paper investigates how midgap states in a $d$-wave superconductor influence the magnetic exchange interaction between ferromagnetic contacts, revealing edge-dependent effects that alter the typical RKKY interaction.

Contribution

It provides a theoretical analysis of the impact of midgap states on magnetic interactions at superconductor edges, highlighting a qualitative change in behavior at the $ ext{110}$ edge.

Findings

Oscillating RKKY interaction at the $ ext{010}$ edge.

Midgap states favor magnet alignment at the $ ext{110}$ edge.

Magnet alignment suppresses midgap states, increasing condensation energy.

Abstract

We theoretically study the indirect interaction between two ferromagnetic contacts located on the surface of a $d$-wave superconductor. When the magnets are connected to a $\{010\}$ edge of the superconductor we find an oscillating RKKY interaction that varies in sign as the distance between the magnetic contacts is varied. However, when coupling the magnets to an $\{110\}$ edge of the superconductor, we find that the presence of midgap states qualitatively changes the results. The ground state of the system is then found to always favor alignment of the magnets as this configuration most strongly suppresses the midgap states, leading to a larger condensation energy which dominates over the intrinsic RKKY interaction.