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

## TL;DR

This study investigates the magnetic and anomalous Hall effects in amorphous iron-germanium, demonstrating that intrinsic Berry curvature mechanisms dominate the anomalous Hall conductivity despite the lack of long-range order.

## Contribution

It provides a unified framework for understanding the intrinsic anomalous Hall effect in amorphous materials using a density of curvature approach that does not rely on crystal momentum.

## Key findings

- Intrinsic mechanism dominates anomalous Hall conductivity in amorphous FeGe.
- Density of curvature can be calculated without Brillouin zone or crystal momentum.
- Enhanced anomalous Hall resistivity observed compared to crystalline FeGe.

## Abstract

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.