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A fully relativistic description of the spin-orbit induced spin Hall effect is presented that is based on Kubo’s linear response formalism. Using an appropriate operator for the spin-current density a Kubo-Středa-like equation for the spin Hall conductivity (SHC) is obtained. An implementation using the Korringa-Kohn-Rostoker band structure method in combination with the coherent potential approximation allow detailed investigations on various alloy systems. A decomposition of the SHC into intrinsic and extrinsic contributions is suggested. Accompanying calculations for the skew-scattering contribution of the SHC using the Boltzmann equation demonstrate the equivalence to the Kubo formalism in the dilute alloy regime and support the suggested decomposition scheme

Extrinsic and Intrinsic Contributions to the Spin Hall Effect of Alloys