Theoretical study of extrinsic spin current generation in ferromagnets induced by anisotropic spin-flip scattering

The spin Hall effect (SHE) is responsible for electrical spin current generation, which is a key concept of modern spintronics. We present a theoretical study of an extrinsic mechanism of SHE arising from a spin-dependent s-d scattering in ferromagnets. In order to investigate the spin conductivity in a ferromagnetic 3d alloy model, we employ a microscopic transport theory based on the Kubo formula and the averaged T-matrix approximation. From the model, we derived an extrinsic mechanism that contributes to both the SHE and the time-reversal odd SHE known as the magnetic SHE. This mechanism can be understood as the contribution from anisotropic (spatial-dependent) spin-flip scattering due to the combination of the orbital-dependent anisotropic shape of s-d hybridization and spin flipping, with the orbital shift caused by spin-orbit interaction with the d orbitals. We also show that this mechanism is valid under crystal-field splitting among the d orbitals in either the cubic or tetragonal symmetry.