Superconductivity induced by spin-orbit coupling in a two-valley ferromagnet

We analyze superconductivity in a ferromagnetically ordered state of multi-layer graphene system in proximity to WSe₂. We model this material by a two-valley system of interacting fermions with small pockets and Ising spin-orbit coupling. The magnetically-ordered state is a half-metal with a canted ferromagnetic order. We obtain the magnon spectrum and derive two types of magnon-mediated 4-fermion interactions: spin-flip interactions mediated by a single magnon and spin-preserving interactions mediated by two magnons. We argue that both have to be included into the pairing interaction between fermions from the filled bands. The full magnon-mediated interaction satisfies Adler criterion and is attractive in a spin-triplet, valley-odd and spatially-even pairing channel. The attraction is induced by spin-orbit coupling and is confined to energies which are parametrically smaller than the Fermi energy. We argue that near the onset of a canted ferromagnetic order this attraction wins over a repulsive intra-valley density-density interaction.