Enhanced Antiferromagnetic Exchange between Magnetic Impurities in a Superconducting Host

It is generally believed that superconductivity only weakly affects the indirect exchange between magnetic impurities. If the distance $r$ between impurities is smaller than the superconducting coherence length ($r\lesssim \xi$), this exchange is thought to be dominated by Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions, identical to the those in a normal metallic host. This perception is based on a perturbative treatment of the exchange interaction. Here, we provide a nonperturbative analysis and demonstrate that the presence of Yu-Shiba-Rusinov bound states induces a strong $1/r^2$ antiferromagnetic interaction that can dominate over conventional RKKY even at distances significantly smaller than the coherence length ($r\ll \xi$). Experimental signatures, implications, and applications are discussed.

Figure 1

(a) Schematic illustration of a magnetic impurity which binds a localized electronic YSR state. The associated spectrum is shown below, with the BCS ground state ${\psi }_{\mathrm{BCS}}$ separated from excited states by $\mathrm{\Delta }$. There exists a single midgap YSR state of energy $E_b$. (b) When two impurities are separated by distances $r<\xi$, their YSR states overlap and hybridize. This hybridization causes both an overall energy shift $\eta$ and a splitting ${\eta }^{\prime }$.

Figure 2

For concreteness, all plots are calculated using actual parameters for superconducting aluminum, with $E_f=11.7\mathrm{eV}$, $k_f=20.1{\mathrm{nm}}^{-1}$, $N_0=35\mathrm{eV}/{\mathrm{nm}}^3$, and $\xi =1.6\mu \mathrm{m}$ [38]. Comparison between bare RKKY and the $J_{\mathrm{YSR}}$ for $E_b\sim 10^{-2}\mathrm{\Delta }$. Resonant enhancement enables $J_{\mathrm{YSR}}$ to dominate at distances $r\ll \xi$.