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We study the effects of the coupling between magnetization dynamics and theelectronic degrees of freedom in a heterostructure of a metallic nanomagnetwith dynamic magnetization coupled with a superconductor containing a steadyspin-splitting field. We predict how this system exhibits a non-linear spintorque, which can be driven either with a temperature difference or a voltageacross the interface. We generalize this notion to arbitrary magnetizationprecession by deriving a Keldysh action for the interface, describing thecoupled charge, heat and spin transport in the presence of a precessingmagnetization. We characterize the effect of superconductivity on theprecession damping and the anti-damping torques. We also predict the fullnon-linear characteristic of the Onsager counterparts of the torque, showing upvia pumped charge and heat currents. For the latter, we predict a spin-pumpingcooling effect, where the magnetization dynamics can cool either the nanomagnetor the superconductor.

Non-linear spin torque, pumping and cooling in superconductor/ferromagnet systems