Energy stability of branching in the scanning gate response of two-dimensional electron gases with smooth disorder

The branched pattern typically observed through the scanning gate microscopy (SGM) of two-dimensional electron gases in the presence of weak, smooth disorder has recently been found to be robust against a very large shift in the Fermi energy of the electron gas. We propose a toy model, where the potential landscape reduces to a single localized feature, which makes it possible to recast the understanding of branch formation through the effect of caustics in an appropriate set of classical trajectories, and it is simple enough to allow for a quantitative analysis of the energy and spatial dependence of the branches. We find the energy stability to be extremely generic, as it rests only upon the assumptions of weak disorder, weak scattering, and the proportionality of the SGM response to the density of classical electron trajectories. Therefore the robustness against changes of the electron's Fermi energy remains when adopting progressively realistic models of smooth disorder.