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.

Figure 1

Schematic representation of the considered individual contributions to the spin Hall effect.

Figure 2

The spin Hall conductivity as a function of the longitudinal charge conductivity is shown for four dilute Cu(Au) alloys labeled by the corresponding impurity concentrations.

Figure 3

The skew-scattering contribution ${\sigma }_{xy}^{z\mathrm{skew}}$ (left) at the impurity concentration 1 at. % as well as the sum of the intrinsic and side-jump contributions ${\sigma }_{xy}^{z\mathrm{intr}}+{\sigma }_{xy}^{z\mathrm{sj}}$ (right) are shown as obtained for different dilute alloys based on Cu, Au, and Pt hosts.

Figure 4

The two parts of the side-jump contribution, ${\sigma }_{xy}^{z\mathrm{sj}\left(\mathrm{nvc}\right)}$ (left) and ${\sigma }_{xy}^{z\mathrm{sj}\left(\mathrm{vc}\right)}$ (right), are shown for different dilute alloys based on Cu, Au, and Pt hosts. The dashed curves in the left panels are plotted (in arbitrary units) according to Eq. (7) obtained within the Fert-Levy model [12].

Figure 5

The sum of the intrinsic contribution ${\sigma }_{xy}^{z\mathrm{intr}}$ and the first part of the side-jump contribution ${\sigma }_{xy}^{z\mathrm{sj}\left(\mathrm{nvc}\right)}$ calculated for Li, Be, Na, and Mg impurities in Cu (green circles), Au (blue triangles), and Pt (red squares) hosts is shown in comparison to the corresponding intrinsic contribution (dashed lines and numbers) obtained from its straightforward calculations.