The Scaling of the Anomalous Hall Effect in the Insulating Regime

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We develop a theoretical approach to study the scaling of anomalous Hall effect (AHE) in the insulating regime, which is observed to be σxyAH∝σxx¹.⁴⁰ ¹.⁷⁵ 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 σxyAH σxxγ with γ predicted to be 1.38≤γ≤1.76, 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.