Planar Hall effect in topological Weyl and nodal-line semimetals

Using symmetry analysis and a semiclassical Boltzmann equation, we theoretically explore the planar Hall effect (PHE) in three-dimensional materials. We demonstrate that PHE is a general phenomenon that can occur in various systems regardless of band topology. Both the Lorentz force and Berry curvature effects can induce significant PHE, and the leading contributions of both effects linearly depend on the electric and magnetic fields. The Lorentz force and Berry curvature PHE coefficient possess only antisymmetric and symmetric parts, respectively. Both contributions respect the same crystalline symmetry constraints but differ under time-reversal symmetry. Remarkably, for topological Weyl semimetals, the Berry curvature PHE coefficient is a constant that does not depend on the Fermi energy, while the Lorentz force contribution linearly increases with the Fermi energy, resulting from the linear dispersion of the Weyl point. Furthermore, we find that the PHE in topological nodal line semimetals is mainly induced by the Lorentz force, as the Berry curvature in these systems vanishes near the nodal line. Our study not only highlights the significance of the Lorentz force in PHE, but also reveals its unique characteristics, which will be beneficial for determining the Lorentz force contribution experimentally.