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In an intriguing recent experiment, it has been found that thetwo-dimensional (2D) surface states of a three-dimensional (3D) strongtopological insulator (TI) support a non-zero Hall voltage transverse to anapplied electric field even when the external magnetic field is in the plane(i.e., the in-plane Lorentz force vanishes). This so-called planar Hall effect(PHE) of TI surface states is found to be non-linear, i.e., the Hall voltagescales quadratically with the applied electric field and linearly with thein-plane magnetic field. In this paper, we derive the non-linear PHE for strongtopological insulator surface states and show that the derivations contained inthe previous literature are incomplete, which can lead to quantitative or evenin some cases qualitative errors in the estimates for the nonlinear planar Hallresistance. We derive the complete expressions for the non-linear planar Hallcurrents for TI surface states with broken particle-hole symmetry and provideresults for different regimes of the surface state Hamiltonian which can becompared with future experiments.

Magnetochiral anisotropy-induced nonlinear planar Hall effect in Topological Insulator surface states