  In an intriguing recent experiment, it has been found that the
two-dimensional (2D) surface states of a three-dimensional (3D) strong
topological insulator (TI) support a non-zero Hall voltage transverse to an
applied 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 voltage
scales quadratically with the applied electric field and linearly with the
in-plane magnetic field. In this paper, we derive the non-linear PHE for strong
topological insulator surface states and show that the derivations contained in
the previous literature are incomplete, which can lead to quantitative or even
in some cases qualitative errors in the estimates for the nonlinear planar Hall
resistance. We derive the complete expressions for the non-linear planar Hall
currents for TI surface states with broken particle-hole symmetry and provide
results for different regimes of the surface state Hamiltonian which can be
compared with future experiments.
