Quaternary alloy ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x{\mathrm{As}}_{1-y}{\mathrm{P}}_y$ hosts magnetic and electronic properties that can be tuned by varying the P concentration “$y$”, Mn concentration “$x$” and by annealing. In this work we make use of this tunability to probe the origin of the anomalous Hall effect (AHE) in ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x{\mathrm{As}}_{1-y}{\mathrm{P}}_y$ thin films grown on GaAs that host perpendicular magnetic anisotropy. Specifically, we find that AHE in this class of materials is determined primarily by two contributions: an intrinsic band component arising from the Berry curvature, and a component determined by hopping conduction. As we vary the properties of ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x{\mathrm{As}}_{1-y}{\mathrm{P}}_y$ from the metallic to the semi-insulating regime by changing the value of $y$ and by postgrowth annealing, we observe a clear crossover from a Berry-curvature-induced AHE to one that is caused by hopping. The transition occurs approximately at the point where the numbers of localized and itinerant holes become comparable. In this hopping regime, the conductivity follows the Efros-Shklovskii scaling law versus temperature indicating the presence of a Coulomb gap, but the AHE remains robustly present. These results indicate that ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x{\mathrm{As}}_{1-y}{\mathrm{P}}_y$ can host an interesting interplay between magnetism and Coulomb interactions.

• Received 15 April 2021
• Accepted 2 June 2021

DOI:https://doi.org/10.1103/PhysRevB.103.214437