Green's function technique for studying electron flow in 2D mesoscopic samples

In a recent series of scanning probe experiments, it became possible to visualize local electron flow in a two-dimensional electron gas. In this paper, a Green's function technique is presented that enables efficient calculation of the quantity measured in such experiments. Efficient means that the computational effort scales like $M^3 N$ ($M$ is the width of the tight-binding lattice used, and $N$ is its length), which is a factor $MN$ better than the standard recursive technique for the same problem. Moreover, within our numerical framework it is also possible to calculate (with the same computational effort $M^3 N$) the local density of states, the electron density, and the current distribution in the sample, which are not accessible with the standard recursive method. Furthermore, an imaging method is discussed where the scanning tip can be used to measure the local chemical potential. The numerical technique is used to study electron flow through a quantum point contact. All features seen in experiments on this system are reproduced and a new interference effect is observed resulting from the crossing of coherent beams of electron flow.