Peeping electrons going by is one of the most fascinating issues of Quantum Mechanics. Attempting such an experiment in a solid-state environment, like in a two-dimensional electron gas (2DEG), involves subtle fundamental ingredients such as decoherence and many-body effects, as well as cross-talks and other technical limitations. The Scanning Gate Microscopy (SGM), developed in recent years, is an attempt to measure the electron flow. This technique is based on the measurement of the conductance through the sample. A local gate electrode, typically a charged tip of a scanning tunnelling microscope, creates a local perturbation in the 2DEG, and leads to a change of the measured conductance that depends on the position of the tip. These conductance changes have often been interpreted as a mapping of the electron current density along a nanostructured device. In order to understand what is really measured by such conductance changes, we developed a systematic perturbation theory for the effects of a local perturbation on the conductance of a nanostructured 2DEG. In agreement with the dictates of Quantum Mechanics, our analysis shows that the SGM measurements in a phase-coherent nanostructure are not given by a local quantity. The resulting expressions for the conductance change in first and second order in the perturbation contain matrix elements of the perturbing potential with two scattering states impinging from opposite electrodes.
The case of a Quantum Point Contact (QPC) is of particular interest because a number of experimental SGM data is already available. A QPC is a narrow constriction in the 2DEG that can lead to conductance quantization in units of 2e²/h. When one varies the Fermi wavelength, the conductance of the QPC exhibits steps and plateaus (see the inset in the upper part of the figure). We find that the first-order contribution for weak tip voltages is significant only on the conductance steps (upper figure; dashed line of the inset). Therefore, the second-order correction (lower figure) is the dominant one on the plateaus. The latter contribution is always negative, exhibits fringes, and has a spatial decay consistent with SGM experiments on QPC.