Surface topography in scanning tunneling microscopy: A free-electron model

Abstract

The topographic image as given by scanning tunneling microscopy (STM) is deduced in analytic form in a free-electron, or Sommerfeld model. The method is non-numerical and employs perturbed wave functions for an arbitrarily modified plane metal surface to approximate the local density of states (LDOS), at the Fermi level. The curves of constant LDOS, hence also the contours followed by the probe in an s-wave tip model, are calculated in terms of h${(\mathrm{x}}_{?}$), a surface profile function. The image of an arbitrary periodic or nonperiodic surface structure is determined by contours of the form z${(\mathrm{x}}_{?}$)=z¯+Δ${(\mathrm{x}}_{?}$,z¯) where z¯ is the average probe-surface separation, and Δ${(\mathrm{x}}_{?}$,z¯) is a convolution over h${(\mathrm{x}}_{?}$). We also discuss the parallel and perpendicular resolution of surface structures such as a one- or two-dimensional Gaussian, a perfect step, and a cosine surface, as a function of distance and tip radius. We find there is considerable smoothing of the image in STM for finite surface defects for typical tip-surface separations and tip radii.