Spin Pumping into Anisotropic Dirac Electrons

We study spin pumping into an anisotropic Dirac electron system induced by microwave irradiation to an adjacent ferromagnetic insulator theoretically. We formulate the Gilbert damping enhancement due to the spin current flowing into the Dirac electron system using second-order perturbation with respect to the interfacial exchange coupling. As an illustration, we consider the anisotropic Dirac system realized in bismuth to show that the Gilbert damping varies according to the magnetization direction in the ferromagnetic insulator. Our results indicate that this setup can provide helpful information on the anisotropy of the Dirac electron system.