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 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 the interface.