  Graphene as well as more generally Dirac solids constitute two dimensional
materials where the electronic flow is ultra relativistic. When a Dirac solid
is deposited on a different substrate surface with roughness, a local random
potential develops through an inhomogeneous charge impurity distribution. This
external potential affects profoundly the charge flow and induces a chaotic
pattern of current branches that develops through focusing and defocusing
effects produced by the randomness of the surface. An additional bias voltage
may be used to tune the branching pattern of the charge carrier currents. We
employ analytical and numerical techniques in order to investigate the onset
and the statistical properties of carrier branches in Dirac solids. We find a
specific scaling-type relationship that connects the physical scale for the
occurrence of branches with the characteristic medium properties, such as
disorder and bias field. We use numerics to test and verify the theoretical
prediction as well as a perturbative approach that gives a clear indication of
the regime of validity of the approach. This work is relevant to device
applications and may be tested experimentally.
