Chiral Anomaly as the Origin of the Planar Hall Effect in Weyl Semimetals

In condensed matter physics, the term “chiral anomaly” implies the violation of the separate number conservation laws of Weyl fermions of different chiralities in the presence of parallel electric and magnetic fields. One effect of the chiral anomaly in the recently discovered Dirac and Weyl semimetals is a positive longitudinal magnetoconductance. Here we show that chiral anomaly and nontrivial Berry curvature effects engender another striking effect in Weyl semimetals, the planar Hall effect (PHE). Remarkably, the PHE manifests itself when the applied current, magnetic field, and the induced transverse “Hall” voltage all lie in the same plane, precisely in a configuration in which the conventional Hall effect vanishes. In this work we treat the PHE quasiclassically, and predict specific experimental signatures for type-I and type-II Weyl semimetals that can be directly checked in experiments.

Figure 1

3D band dispersion of the lattice model of a Weyl semimetal ($k_z$ is suppressed) obtained by the diagonalizing Hamiltonian in Eq. (15) for (a) $\gamma =0$ (type-I Weyl nodes), (b) $\gamma =0.05$ (type-I Weyl nodes), and (c) $\gamma =0.15$ (type-II Weyl nodes), respectively. The chemical potential is set at zero energy (indicated by the dashed line). The Weyl cones are at $(k_0,0,0)$ and $(\text{−}{k}_0,0,0)$. The parameters used are $t=-0.005\mathrm{eV}$, $m=0.15\mathrm{eV}$, and $k_0=(\pi /2)$.

Figure 2

(a) Illustration for the planar Hall effect measurement setup ($V_H$ is the in-plane Hall voltage). (b) The normalized amplitude of planar Hall conductivity ($\mathrm{\Delta }\sigma$) as a function of magnetic field for the lattice model of the Weyl semimetal given by Eq. (15) for $\gamma =0$. (Inset shows longitudinal magnetoconductivity as a function of magnetic field). Both PHC and LMC amplitudes show $B^2$ dependence on the magnetic field. (c)–(d) The angular dependence of longitudinal magnetoconductivity and planar Hall conductivity for $B=5\mathrm{T}$. [We have normalized the $y$ axes of (c) and (d) by ${\sigma }_{xx}(\theta =0)$ and ${\sigma }_{yx}^{\mathrm{ph}}(\theta =\pi /4)$ respectively.]

Figure 3

(a) The normalized longitudinal magnetoconductivity computed numerically for the lattice model of the Weyl semimetal given by Eq. (15) with tilt parameter $\gamma =0.15$ as a function of the magnetic field $\mathbf{B}$ applied along the tilt direction ($x$ axis). (b) The angular dependence of the longitudinal magnetoconductivity for $B=5\mathrm{T}$ (applied parallel to the tilt direction) and $\gamma =0.15$. (c)–(d) The same for planar Hall conductivity for the parameters mentioned above.