Anomalous Hall Effect

Published

November 10, 2025

The Anomalous (Spin) Hall Effect (AHE) gets its name from anomalous transport.

In topological insulators, its intrinsic contribution is given by the integral of the Berry curvature over each occupied band (related to the Chern number). It describes non-dissipative transport. In a 2D FM material, it can be calculated by integrating over the first BZ

\[ \sigma_{xy}=\frac{-e^2}{(2\pi)^2 \hbar}\int_{BZ}\Omega(\mathbf{k})d^2\mathbf{k} = -\frac{e^2}{(2\pi)^2}C \]

AH resistance is \(\rho_{xy}\) and longitudinal resistance \(\rho_{xx}\) (\(\rho\) and \(\sigma\) are tensors, \(\rho = \sigma^{-1}\)) are related by the following mechanisms

Skew scattering \(\rho_{xy}^{-1}\propto \rho_{xx}\)
Intrinsic and side-jump \(\rho_{xy}^{-1}\propto \rho_{xx}^2\)

In metals, dissipation is always present along with the AHE.

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