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

## TL;DR

This paper investigates how magnetization dynamics in superconductor/ferromagnet heterostructures induce non-linear spin torques, affect damping, and enable spin-pumping cooling effects, with a comprehensive theoretical framework.

## Contribution

It introduces a generalized Keldysh action for coupled charge, heat, and spin transport in precessing magnetization systems with superconductivity, revealing novel non-linear effects.

## Key findings

- Prediction of non-linear spin torque driven by temperature or voltage.
- Characterization of superconductivity's impact on damping and anti-damping torques.
- Discovery of spin-pumping cooling effect for nanomagnet and superconductor.

## Abstract

We study the effects of the coupling between magnetization dynamics and the electronic degrees of freedom in a heterostructure of a metallic nanomagnet with dynamic magnetization coupled with a superconductor containing a steady spin-splitting field. We predict how this system exhibits a non-linear spin torque, which can be driven either with a temperature difference or a voltage across the interface. We generalize this notion to arbitrary magnetization precession by deriving a Keldysh action for the interface, describing the coupled charge, heat and spin transport in the presence of a precessing magnetization. We characterize the effect of superconductivity on the precession damping and the anti-damping torques. We also predict the full non-linear characteristic of the Onsager counterparts of the torque, showing up via pumped charge and heat currents. For the latter, we predict a spin-pumping cooling effect, where the magnetization dynamics can cool either the nanomagnet or the superconductor.