Classical transport theory for the planar Hall effect with threefold symmetry

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.

Figures (4)

• Figure 1: (a) Setups that show no $\sigma_{xy}^{(3)}$ (left) and finite $\sigma_{xy}^{(3)}$ (right) in cubic crystals. (b) Schematic images of sufficient condition where $\sigma_{xy}^{(3)}$ vanishes. $m_{1,2,3}$ denotes the mirror planes in the crystals, and $x,y,z$ axes corresponds to the directions of $\mathbf{j}$, $\mathbf{E}$, and $\mathbf{j}\times \mathbf{E}$ respectively.
• Figure 2: (a) Schematic images for ellipsoidal Fermi surfaces distributed around the $C_3$ axis (left). The mirror planes with $\theta=0$ and $\theta\neq0$ (right). (b) Calculated angular dependence of PHE with several values of $\theta$. (c) Extracted the three-fold component from the PHE. $n_{1,2,3}=10^{17}$ cm$^{-3}$, $B=0.3$ T, $\mu_0=14$ T$^{-1}$, and $\beta=0.005$.
• Figure 3: (a) Fermi surface in a sc, fcc, and bcc lattice. (b) Three-fold angular dependence of $\sigma_{yx}$ in cubic systems. $E_F/t=-0.5$ and $k_BT/t=2.8\times10^{-2}$ where $t$ is the hopping parameter set to $30$ meV. The carrier lifetime $\tau$ is $1$ ps.
• Figure 4: (a) Color map of the amplitude of the three-fold PHE in the three-ellipsoidal model as a function of the magnetic field and tilt angle. The carrir density is $3\times10^{17}$ cm$^{-3}$. $\mu_0=14$ T$^{-1}$ and $\beta=\mu_1/\mu_2=0.005$. (b) Maximum values of the three-fold PHE as a function of anisotropy in the mobility.