How disorder affects the Berry-phase anomalous Hall conductivity: A reciprocal-space analysis

The anomalous Hall conductivity of “dirty” ferromagnetic metals is dominated by a Berry-phase contribution which is usually interpreted as an intrinsic property of the Bloch electrons in the pristine crystal. In this work we evaluate the geometric Hall current directly from the electronic ground state with disorder and then recast it as an integral over the crystalline Brillouin zone. The integrand is an effective $k$-space Berry curvature, obtained by unfolding the Berry curvature from the small Brillouin zone of a large supercell. Therein, disorder yields a net extrinsic Hall contribution, which we argue is related to the elusive side-jump effect. As an example, we unfold the first-principles Berry curvature of an ordered ${\mathrm{Fe}}_3\mathrm{Co}$ alloy from the original fcc-lattice Brillouin zone onto a bcc-lattice zone with four times the volume. Comparison with the virtual-crystal Berry curvature clearly reveals the symmetry-breaking effects of the substitutional Co atoms.

Figure 1

(Top) Energy bands of bcc Fe. (Bottom) Energy bands of the fcc ${\mathrm{Fe}}_3\mathrm{Co}$ alloy unfolded onto the BZ of bcc Fe. Colors indicate the spin polarization $\langle S_z\rangle$ of the states. The points labeled P all have coordinates $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$. The bands of the bcc $\langle {\mathrm{Fe}}_3\mathrm{Co}\rangle$ virtual crystal (not shown) are almost indistinguishable from those of bcc Fe, except for a shift in the Fermi level.

Figure 2

Energy bands color coded by the spin polarization $\langle S_z\rangle$ and Berry curvature summed over the occupied states, plotted along the path $\Gamma$-H-P. (Top) bcc Fe. (Middle) bcc $\langle {\mathrm{Fe}}_3\mathrm{Co}\rangle$ virtual crystal. (Bottom) fcc ${\mathrm{Fe}}_3\mathrm{Co}$ alloy, using BZ unfolding.

Figure 3

Heat map of the Berry curvature in the plane $k_y=0$, in atomic units (note the log scale). The lines of intersection between the Fermi surface and the plane are also shown. The top and middle panels show $-{\Omega }_z^{\mathrm{occ}}$ for bcc Fe and bcc $\langle {\mathrm{Fe}}_3\mathrm{Co}\rangle$ respectively, and the bottom panel shows $-{\Omega }_z^{\mathrm{unf}}$ for fcc ${\mathrm{F}}_3\mathrm{Co}$.