The Scaling of the Anomalous Hall Effect in the Insulating Regime

TL;DR

This paper presents a theoretical model explaining the scaling behavior of the anomalous Hall effect in insulators, aligning well with experimental data and enhancing understanding of AHE in insulating materials.

Contribution

The paper introduces a phonon-assisted hopping and percolation theory-based model to predict the scaling of AHE in insulators, matching experimental exponents.

Findings

Derived a formula for anomalous Hall conductivity in insulators.

Predicted the scaling exponent range as 1.38 to 1.76.

Confirmed the theory's predictions with experimental data.

Abstract

We develop a theoretical approach to study the scaling of anomalous Hall effect (AHE) in the insulating regime, which is observed to be $\sigma_{xy}^{AH}\propto\sigma_{xx}^{1.40\sim1.75}$ in experiments over a large range of materials. This scaling is qualitatively different from the ones observed in metals. Basing our theory on the phonon-assisted hopping mechanism and percolation theory, we derive a general formula for the anomalous Hall conductivity, and show that it scales with the longitudinal conductivity as $\sigma_{xy}^{AH}\sim\sigma_{xx}^{\gamma}$ with $\gamma$ predicted to be $1.38\leq\gamma\leq1.76$, quantitatively in agreement with the experimental observations. Our result provides a clearer understanding of the AHE in the insulating regime and completes the scaling phase diagram of the AHE.