Unfolding the Berry curvature of supercell calculations

Unfolding band structures of supercell calculations has become a valuable tool for visualizing the influence of point impurities on the electronic states in crystals. In the same spirit, we introduce a procedure which maps the $k$-space Berry curvature of the occupied states from the small BZ of a supercell onto the normal BZ of the perfect (or virtual) crystal. As an application, we analyze the $k$-space distribution of the unfolded curvature of bcc Fe$_{1-x}$Co$_x$ ordered alloys, to better understand the influence of alloying on the anomalous Hall conductivity. Comparing with the ordinary curvature calculated in the virtual-crystal approximation, we find that the lowering of translational symmetry by the Co ``impurities'' introduces ``extrinsic'' contributions, which correlate with changes in the spectral function near the Fermi surface. In particular, the unfolded curvature displays additional sharp peaks associated with low-energy \textit{pseudovertical} transitions. These occur in regions of $k$-space where two unfolded bands, which in the virtual crystal would be separated in $k$-space (and therefore would not jointly contribute to its Berry curvature), lie on either side of the Fermi level and are coupled by the impurity potential.