Finite transverse conductance in topological insulators under an applied in-plane magnetic field

Recently, in topological insulators (TIs) the phenomenon of planar Hall effect (PHE) wherein a current driven in presence an in-plane magnetic field generates a transverse voltage has been experimentally witnessed. There have been a couple of theoretical explanations of this phenomenon. We investigate this phenomenon based on scattering theory on a normal metal-TI-normal metal hybrid structure and calculate the conductances in longitudinal and transverse directions to the applied bias. The transverse conductance depends on the spatial location between the two NM-TI junctions where it is calculated. It is zero in the drain electrode when the chemical potentials of the top and the bottom TI surfaces ($\mu_t$ and $\mu_b$ respectively) are equal. The longitudinal conductance is $\pi$-periodic in $\phi$-the angle between the bias direction and the direction of the in-plane magnetic field. The transverse conductance is $\pi$-periodic in $\phi$ when $\mu_t=\mu_b$ whereas it is $2\pi$-periodic in $\phi$ when $\mu_t\neq\mu_b$. As a function of the magnetic field, the magnitude of transverse conductance increases initially and peaks. At higher magnetic fields, it decays for angles $\phi$ closer to $0,\pi$ whereas oscillates for angles $\phi$ close to $\pi/2$. The conductances oscillate with the length of the TI region. A finite width of the system makes the transport separate into finitely many channels. The features of the conductances are similar to those in the limit of infinitely wide system except when the width is so small that only one channel participates in the transport. When only one channel participates in transport, the transverse conductance in the region $0<x<L$ is zero for $\mu_t=\mu_b$ and the transverse conductance in the region $x>L$ is zero even for the case $\mu_t\neq\mu_b$. We understand the features in the obtained results.