Classical transport theory for the planar Hall effect with threefold symmetry

TL;DR

This paper demonstrates that the threefold-symmetric planar Hall effect can be explained purely by classical transport theory, specifically through a third-order expansion of the Boltzmann equation, without invoking quantum mechanisms.

Contribution

It introduces a classical explanation for the threefold-symmetric PHE based on crystal symmetry and a third-order Boltzmann equation expansion, challenging quantum-centric interpretations.

Findings

The threefold component arises from crystal mirror plane orientations.

The classical effect is widespread across various crystal symmetries.

The amplitude of the classical threefold PHE is comparable to the anomalous Hall effect.

Abstract

In recent years, the planar Hall effect (PHE) has become a key probe of Berry curvature and the anomalous Hall effect (AHE). Threefold-symmetric signals under in-plane fields are often attributed to such quantum mechanisms. Here, we establish a purely classical origin for a three-fold-symmetric PHE. The idea is simple yet decisive: a third-order expansion of the Boltzmann equation in the magnetic field reveals that the threefold component originates from the relative positions of the mirror planes in the crystals with respect to the measurement setups. Remarkably, the threefold contribution should be ubiquitous because this symmetry condition can be realized across a broad range of crystals. Numerical estimates based on concrete models further show that its amplitude is comparable to that expected from the AHE.