Linear response Kubo-Bastin formalism with application to the anomalous and spin Hall effects: A first-principles approach

We present a general first-principles approach to treat various linear response phenomena relevant for spintronics. It is based on a Kubo-Bastin formalism and implemented within the multiple-scattering Korringa-Kohn-Rostoker (KKR) Green's function method with the underlying electronic structure determined by density functional theory. The symmetric (e.g., longitudinal electronic transport) as well as the antisymmetric (e.g., transverse transport) parts of the response tensor are determined, including both the so-called Fermi-sea and the Fermi-surface contributions. To describe spin-orbit-induced phenomena, such as the anomalous and spin Hall effects, a fully relativistic description is employed. Exploiting the adopted Green's function method substitutional disorder in the full concentration range of alloys is treated within the coherent potential approximation, taking full account of occurring vertex corrections in the averaging procedure for the linear response quantities. Extrinsic (scattering related, e.g., side-jump and skew scattering) and intrinsic (band structure–related) contributions to the transport tensors are treated on equal footing. Other phenomena, such as Gilbert damping and spin-orbit torques, are particular cases of the general framework and their determination is briefly addressed. The versatility of the method is demonstrated by presenting results for the anomalous and spin Hall conductivities for elemental transition metals and their alloys.

Figure 1

The AHC of fcc-${\mathrm{Fe}}_x{\mathrm{Pd}}_{1-x}$ as a function of concentration $x$ determined within the Kubo-Bastin formalism. The total AHC (${\sigma }_{\mathrm{xy}}^{\mathrm{vc}}$) and different contributions to it are shown: the on-site term (${\sigma }_{\mathrm{xy}}^0$), the off-site Fermi-surface term, including vertex corrections (${\sigma }_{\mathrm{xy}}^{1,I,\mathrm{vc}}$) and the off-site Ferm-sea contribution (${\sigma }_{\mathrm{xy}}^{1,II}$). Additionally, the off-site term omitting the vc (${\sigma }_{\mathrm{xy}}^{1,I,\mathrm{nvc}}$) is shown for comparison. Experimental data [71] for ${\sigma }_{\mathrm{xy}}$ (full circles) determined at $T=4.2$ K is also displayed.

Figure 2

The SHC of ${\mathrm{Cu}}_x{\mathrm{Au}}_{1-x}$ as a function of concentration $x$ determined within the Kubo-Bastin formalism. The total SHC and different contributions (notation: see text and Fig. 1 caption) to it are displayed.