Scattering approach to scanning gate microscopy

We present a perturbative approach to the conductance change caused by a weakly invasive scattering potential in a two-dimensional electron gas. The resulting expressions are used to investigate the relationship between the conductance change measured in scanning gate microscopy as a function of the position of a scattering tip and local electronic quantities like the current density. We use a semiclassical approach to treat the case of a strong hard-wall scatterer in a half-plane facing a reflectionless channel. The resulting conductance change is consistent with the numerically calculated quantum conductance.