Phenomenology of orbital torque, pumping and mixing conductance in metallic bilayers

Abstract: The conversion between spin and orbital currents is at the origin of the orbital torque and its Onsager reciprocal, the orbital pumping. Here, we propose a phenomenological model to describe the orbital torque in magnetic bilayers composed of an orbital source (i.e., a light metal such as Ti, Ru, CuOx...) and a spin-orbit coupled magnet (i.e., typically Ni, (Co/Pt)$_n$, etc.). This approach accounts for spin-to-orbit and orbit-to-spin conversion in the ferromagnet and at the interface. We show that the orbital torque arises from a compromise between orbital current injection from the orbital source to the ferromagnet and spin current backflow from the ferromagnet back to the orbital source. We also discuss the concept of orbital-mixing conductance and introduce the "orbit-spin-" and "spin-orbit-mixing" conductances that govern the orbital torque and orbital pumping, respectively.