  The spin Hall effect (SHE) and the magnetic spin Hall effect (MSHE) are
responsible for electrical spin current generation, which is a key concept of
modern spintronics. We theoretically investigated the spin conductivity induced
by spin-dependent s-d scattering in a ferromagnetic 3d alloy model by employing
microscopic transport theory based on the Kubo formula. We derived a novel
extrinsic mechanism that contributes to both the SHE and MSHE. 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.
