Current distribution in the scanning vacuum tunnel microscope: A free-electron model

Insight into the resolution of the recently developed technique of scanning tunnelling microscopy (STM) is achieved by considering the transmission of free electrons through a constant potential barrier with corrugated boundaries representing the sampled surface and probing tip, respectively. The amplitudes of the reflected and transmitted waves are calculated via an extension of the so-called GR-method developed to treat scattering from a corrugated hard wall. Results for the distribution of current density, for the dependence of the tunnelling current on the horizontal and vertical positions of the scanning tip and for the resulting equicurrent lines (STM images) are presented for a two-dimensional model. Simple analytical approximations are shown to reproduce computed trends versus tip-sample separation, tip curvature and average barrier height.