Within a Supersymmetric Quantum Mechanics (SUSY-QM) framework, the (3+1)
Dirac equation describing a Dirac material in the presence of external parallel
electric and magnetic fields is solved. Considering static but non-uniform
electric and magnetic profiles with translational symmetry along the
y-direction, the Dirac equation is transformed into two decoupled pairs of
Schr\"odinger equations, one for each chirality of the fermion fields. Taking
trigonometric and hyperbolic profiles for the vector and scalar potentials,
respectively, we arrive at SUSY partner P\"oschl-Teller-like quantum
potentials. Restricting to the conditions of the potentials that support an
analytic zero-mode solution, we obtain a nontrivial current density in the same
plane where the electric and magnetic fields lie, but perpendicular to both of
them, indicating the possibility of realizing the Planar Hall Effect.
Furthermore, this non-vanishing current density is the sum of current densities
for the left- and right-chiralities, suggesting that the net current is a
consequence of chiral symmetry.Comment: 10 pages, 6 figure