Separation of the individual contributions to the spin Hall effect in dilute alloys within the first-principles Kubo-Středa approach

We present a procedure for the separation of the intrinsic, side-jump, and skew-scattering contributions to the spin Hall conductivity within the ab initio Kubo-Středa approach. Furthermore, two distinct contributions to the side-jump mechanism, either independent of the vertex corrections or solely caused by them, are quantified as well. This allows for a detailed analysis of individual microscopic contributions to the spin Hall effect. The efficiency of the proposed method is demonstrated by a first-principles study of dilute metallic alloys based on Cu, Au, and Pt hosts.