Electronic Branched Flow in Graphene and Dirac Materials*

Electronic flow in graphene and other Dirac materials is ultra-relativistic with limiting velocity the Fermi velocity. When the two-dimensional material is placed on a substrate an additional potential is generated due to charged impurities; this weak random potential affects the Dirac dynamics. The resulting electronic flow coalesces into branches that merge and produce random local focusing and caustics. We investigate the onset of the electron caustic regime both analytically and numerically. We find that in the additional presence of a bias potential, electron caustics are generated at surface locations that scale with both the random and deterministic potential properties. In particular we find a scaling relationship that connects the caustic location to the external potentials that may be tested experimentally.