UC Berkeley

In a perfect crystal at zero temperature, the position of a single atom combined with the structural parameters of the crystal define the position of every atom in the crystal. In an amorphous solid, however, such long-range structural order is absent. Short-range (or local) order often persists thanks to atomic electrostatic fields and bonding preferences, manifesting as a narrow distribution of first-nearest-neighbor distances and, in some cases, a narrow distribution of bond angles; however, the distributions for second-nearest-neighbors are considerably broadened and for more distant neighbors are often indistinguishable. It is therefore remarkable that ferromagnetism and other forms of long-range magnetic order occur regularly in amorphous systems despite the absence of long-range structural order.

One such system is amorphous iron-silicon (a-FexSi1-x), which was previously found to show enhanced magnetization for a given composition x relative to its crystalline counterpart due to a reduction in the number of Si nearest neighbors for each Fe atom in the amorphous phase.More surprising was the discovery in a-FexSi1-x of an intrinsic anomalous Hall effect, usually attributed to the Berry curvature arising at band anti-crossing points near the Fermi level in the presence of spin-orbit coupling in crystalline materials.

The amorphous iron-germanium system (a-FexGe1-x, studied here for 0.38 ≤ x ≤ 0.61) also lacks long-range structural order and hence lacks a meaningful Brillouin zone. On top of the larger spin-orbit coupling from Ge compared to Si, the Fe-Ge system (in its crystalline states) has previously been shown to host a variety of long-range magnetic orderings, including ferromagnetism, helimagnetism, and a skyrmion lattice phase that has attracted intense research interest in recent years. This skyrmion lattice phase has allowed considerable experimental probing of a topologically nontrivial magnetic state, and promises energy-efficient adaptations to existing and theoretical data storage architectures. The possibility of enhancing this phase for applications, together with a search for the fundamental structure-independent origin of the intrinsic anomalous Hall effect, motivate our study of a-FexGe1-x and, later, its transition metal cousin a-CoxGe1-x.

The magnetization of a-FexGe1-x 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 k being a bad quantum number. Together with the shape of the low-temperature hysteresis loops, the temperature dependence of the magnetization for 2 ≤ T ≤ 300 K confirms that samples with x < 0.50 are weak ferromagnets, while those x > 0.50 are typical soft ferromagnets.

Magnetotransport measurements for 2 ≤ T ≤ 300 K yield a temperature-dependent ordinary Hall coefficient, indicating that a-FexGe1-x is a multi-band system. Two potential two-band models are explored, one abstract and one concretely based on the measured resistivity and Hall coefficient; each of these models assumes the existence of one electron band and one hole band. The concrete model yields real-valued solutions for electron and hole concentration only at T = 2 K, indicating that the two-band model remains an oversimplification and multiple energy bands of each carrier type are needed to understand the carrier density and its temperature dependence in a-FexGe1-x.

Hall measurements also reveal an enhanced anomalous Hall resistivity ϱxyAH (relative to crystalline FeGe) and no additional Hall effects; the low-temperature ϱxyAH 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 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-FexGe1-x. The density of curvature is the sum of spin-orbit correlations of local orbital states and can hence be calculated with no reference to 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.

Like a-FexGe1-x, the amorphous cobalt-germanium system (a-CoxGe1-x, studied here for 0.45 ≤ x ≤ 0.63) lacks long-range structural order. However, magnetic ordering appears in a-CoxGe1-x around x = 0.60 and manifests an extremely weak magnetic moment (MS < 0.1 μB/Co at x = 0.63) that increases with increasing x; this weak moment is predicted with remarkable quantitative accuracy by a simple Jaccarino-Walker model even as ab initio molecular dynamics and density functional theory calculations overestimate the magnetization by an order of magnitude.

The zero-field resistivity of a-CoxGe1-x, like a-FexGe1-x, is typical of an amorphous metal: weakly temperature-dependent but strongly composition-dependent, reflecting the composition dependence of the carrier density which increases with x. In fact, the residual resistivity measured at T = 4 K of a-CoxGe1-x agrees quantitatively with that of a-FexGe1-x for x in the range studied, which underscores the inconsistency of assuming a single-band model in calculating the carrier density from the Hall coefficient. The single-band assumption yields a carrier density in electron-dominated a-CoxGe1-x nearly double that in hole-dominated a-FexGe1-x for a given x; however, when the two-band model with one electron band and one hole band is applied to the data, the total carrier density of a-CoxGe1-x turns out to be very similar to that of a-FexGe1-x. Nevertheless, the two-band model only works in a-CoxGe1-x for x ≤ 0.50, indicating that this model likely remains an oversimplification of the true electronic energy band structure.

Magnetotransport measurements point to some of the other intricacies of this energy band structure; in particular, at low temperatures (T ≤ 20 K) and all x, the magnetoresistivity of a-CoxGe1-x shows a crossover from typical quadratic field dependence to a non-saturating linear field dependence extending up to 140 kOe in multiple sample-field orientations. This effect is believed to be a manifestation of Abrikosov's quantum magnetoresistance in an amorphous system, arising from a few very light carriers existing near the Fermi energy. This same feature of the energy band structure may be responsible for the enormous anomalous Hall effect (AHE), normalized by the magnetization, that develops at x = 0.60 (the AHE was absent from the Hall signal for x ≤ 0.55): the band structure of B20 CoGe shows a Dirac point near the Fermi energy, and if the disorder associated with the amorphous structure were to gap out a structure similar to this Dirac point, it would give rise to a very large density of Berry curvature and cause a large AHE.