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, a new imaging method is proposed where the scanning tip can be used to measure the local chemical potential. The numerical technique is illustrated with some examples.