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

Recently, in topological insulators (TIs) the phenomenon of planar Hall effect 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 (NM)-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 (μ_tandμ_brespectively) are equal. The longitudinal conductance isπ-periodic inϕ-the angle between the bias direction and the direction of the in-plane magnetic field. The transverse conductance isπ-periodic inϕwhenμ_t= μ_bwhereas it is 2π-periodic inϕwhenμ_t≠ μ_b. As a function of the magnetic field, the magnitude of transverse conductance increases initially and peaks. At higher magnetic fields, it decays for anglesϕcloser to 0,πwhereas oscillates for anglesϕclose toπ/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<Lis zero forμ_t= μ_band the transverse conductance in the regionx>Lis zero even for the caseμ_t≠ μ_b. We understand the features in the obtained results.