Itinerant ferromagnetism and intrinsic anomalous Hall effect in amorphous iron-germanium

The amorphous iron-germanium system ($a$-Fe$_x$Ge$_{1-x}$) lacks long-range structural order and hence lacks a meaningful Brillouin zone. The magnetization of aFeGe is well explained by the Stoner model for Fe concentrations $x$ above the onset of magnetic order around $x=0.4$, indicating that the local order of the amorphous structure preserves the spin-split density of states of the Fe-$3d$ states sufficiently to polarize the electronic structure despite $mathbf{k}$ being a bad quantum number. Measurements reveal an enhanced anomalous Hall resistivity $rho_{xy}^{mathrm{AH}}$ relative to crystalline FeGe; this $rho_{xy}^{mathrm{AH}}$ is compared to density functional theory calculations of the anomalous Hall conductivity to resolve its underlying mechanisms. The intrinsic mechanism, typically understood as the Berry curvature integrated over occupied $mathbf{k}$-states but shown here to be equivalent to the density of curvature integrated over occupied energies in aperiodic materials, dominates the anomalous Hall conductivity of $a$-Fe$_x$Ge$_{1-x}$ ($0.38 leq x leq 0.61$). The density of curvature is the sum of spin-orbit correlations of local orbital states and can hence be calculated with no reference to $mathbf{k}$-space. This result and the accompanying Stoner-like model for the intrinsic anomalous Hall conductivity establish a unified understanding of the underlying physics of the anomalous Hall effect in both crystalline and disordered systems.