Semiclassical transport in two-dimensional Dirac materials with spatially variable tilt

We use Boltzmann theory to study the semi-classical dynamics of electrons in a two-dimensional (2D) tilted Dirac material in which the tilt varies in space. The spatial variation of the tilt parameter induces a non-trivial spacetime geometry on the background of which the electrons roam about. As the first manifestation of graivto-electric phenomena, we find a geometric planar Hall effect according to which a current flows in a direction transverse to the chemical potential gradient and is proportional to $g^{xy}$ component of the emergent spacetime structure. The longitudinal conductivity contains information about the gravitational red-shift factors. Furthermore, in the absence of externally applied electric field there can be "free-fall" or zero-bias currents that can be used as detectors of terahertz radiation.