Extrinsic and Intrinsic Contributions to the Spin Hall Effect of Alloys

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.

Figure 1

Spin Hall conductivity ${\sigma }_{xy}^z$ for the alloy systems ${\mathrm{Au}}_x{\mathrm{Pt}}_{1-x}$ and ${\mathrm{Ag}}_x{\mathrm{Au}}_{1-x}$. The full squares correspond to calculations including vertex corrections while the open squares represent calculations without vertex corrections. Results from other ab initio investigations on the intrinsic SHE of pure Pt [10] and Au [8, 11] are included. Because of the different definitions for the spin-current operator the results from Refs. 10, 11 have been multiplied by a factor of 2 for the sake of consistency.

Figure 2

The extrinsic spin Hall conductivity ${\sigma }_{xy}^{z\mathrm{extr}}$ versus the longitudinal conductivity ${\sigma }_{xx}$ for ${\mathrm{Ag}}_x{\mathrm{Au}}_{1-x}$ and ${\mathrm{Au}}_x{\mathrm{Pt}}_{1-x}$ (black line/circles). The blue and orange lines are explained in the text.

Figure 3

The side-jump and intrinsic contributions to the spin Hall conductivity, ${\sigma }_{xy}^{z\mathrm{sj}}$ and ${\sigma }_{xy}^{z\mathrm{intr}}$, respectively, for various dilute transition metal alloys $A(B)$ with the concentration of 1 at. % for the dissolved component $B$.

Figure 4

The longitudinal conductivity ${\sigma }_{xx}$ (top) and the skew-scattering contribution to the spin Hall conductivity ${\sigma }_{xy}^{z\mathrm{skew}}$ (bottom) for various dilute alloys $A(B)$ obtained on the basis of the Kubo-Středa equation (left bar) and the Boltzmann formalism (right bar), respectively. The concentration for the dissolved component $B$ is 1 at. %.