  GaAs-based two-dimensional electron gases (2DEGs) show a wealth of remarkable
electronic states, and serve as the basis for fast transistors, research on
electrons in nanostructures, and prototypes of quantum-computing schemes. All
these uses depend on the extremely low levels of disorder in GaAs 2DEGs, with
low-temperature mean free paths ranging from microns to hundreds of microns.
Here we study how disorder affects the spatial structure of electron transport
by imaging electron flow in three different GaAs/AlGaAs 2DEGs, whose mobilities
range over an order of magnitude. As expected, electrons flow along narrow
branches that we find remain straight over a distance roughly proportional to
the mean free path. We also observe two unanticipated phenomena in
high-mobility samples. In our highest-mobility sample we observe an almost
complete absence of sharp impurity or defect scattering, indicated by the
complete suppression of quantum coherent interference fringes. Also, branched
flow through the chaotic potential of a high-mobility sample remains stable to
significant changes to the initial conditions of injected electrons.
