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The anomalous Hall effect (AHE) has been studied systematically in thelow-conductivity ferromagnetic oxide Fe$_{3-x}$Zn$_x$O$_4$ with $x = 0$, 0.1,and 0.5. We used (001), (110), and (111) oriented epitaxialFe$_{3-x}$Zn$_x$O$_4$ films grown on MgO and sapphire substrates in differentoxygen partial pressure to analyze the dependence of the AHE oncrystallographic orientation, Zn content, strain state, and oxygen deficiency.Despite substantial differences in the magnetic properties and magnitudes ofthe anomalous Hall conductivity $\sigma_{xy}^{\rm AHE}$ and the longitudinalconductivity $\sigma_{xx}$ over several orders of magnitude, a universalscaling relation $\sigma_{xy}^{\rm AHE} \propto \sigma_{xx}^{\alpha}$ with$\alpha = 1.69 \pm 0.08$ was found for all investigated samples. Our resultsare in agreement with recent theoretical and experimental findings forferromagnetic metals in the dirty limit, where transport is by metallicconduction. We find the same scaling relation for magnetite, where hoppingtransport prevails. The fact that this relation is independent ofcrystallographic orientation, Zn content, strain state, and oxygen deficiencysuggests that it is universal and particularly does not depend on the nature ofthe transport mechanism.

Anomalous Hall Effect in Magnetite: Universal Scaling Relation Between Hall and Longitudinal Conductivity in Low-Conductivity Ferromagnets