a Masters project by Muhammad Asad-uz-Zaman
Summary: In a number of recent experiments, experimentalists have imaged the electron flow through a Quantum Point Contact in a Two-Dimensional Electron Gas at low temperature by a special technique of imaging called Scanning Probe Microscopy[M.A.Topinka,R.M.Westervelt and E.J.Heller, Physics Today 56, 12(2003)]. Some fascinating and novel features of their experiments are conductance quantization in units of 2e^2/hbar,where e is the electronic charge and hbar is Planck’s constant, branching and coherence behaviors of electron flow over the entire scanned region. But no analytical connection was made between the image and the electron flow. In this work, our focus is to develop a theoretical understanding of electron flow through a Quantum Point Contact in a Two-Dimensional Electron Gas and reconstruct the figure that model the data generated in the experiment. In order to accomplish that we model a Quantum Point Contact as a dipole source at the origin in the role of a Scanning Probe Microscope tip, at temperature 0 K. In an ideal case with no disorder, we have found that the variation of total current due to a two-dimensional attractive delta potential is proportional to the unperturbed charge density at the delta potential position. Then we include the disorder in the analysis, which alters the proportionality from charge density to charge amplitude.