Spin pumping from a ferromagnetic insulator to an unconventional superconductor with interfacial Andreev bound states

Spin pumping from a ferromagnetic insulator into a high-${T}_c$ superconductor with a $d$-wave superconducting order parameter has recently been experimentally observed. Such unconventional superconducting order is known to produce interfacial bound states for certain crystallographic orientations. Here, we present a methodology which can be used to study spin pumping into unconventional superconductors, including the role of interfacial bound states. As an example, we determine how the crystallographic orientation of the $d$-wave order parameter relative to the interface changes the spin-pumping effect. We find that the spin-pumping effect is slightly enhanced at low temperatures for orientations hosting interfacial bound states compared to other superconducting states. However, the spin-pumping effect does not show a coherence peak close to ${T}_c$ for such orientations, and instead remains smaller than the normal-state value for all $T$. For orientations not hosting interfacial bound states, we find that the pumped spin current can be increased to several times the normal-state spin current at frequencies that are small compared to the superconducting gap. Our results show that the spin-pumping dependency on frequency and temperature changes qualitatively depending on the crystallographic orientation of unconventional superconducting order parameters relative to the interface.