Simulation of current in the scanning tunneling microscope

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Abstract
Considering simple models of the scanning tunneling microscope and metallic samples, we use a finite-element method to solve Schrödinger’s equation for the electrons tunneling from the tip to the sample. We plot current-density maps for various geometries of the electrodes: hemispherical or cylindrical tip facing a planar surface or a surface with a Gaussian boss or dip. It can be seen on the current-density maps that the electron flow passes preferentially through the thinnest region of the barrier. From the current density in the case of a planar sample, we investigate the width of the tunnel current beam when it penetrates into the sample. From the dependence of the current on the distance between a hemispherical tip and a Gaussian boss or dip, we show that the corrugation of the sample surface is attenuated by a factor of two in the constant-current image. The effective work function, determined from the logarithmic derivative of the current with respect to the distance, differs from the real work function of the sample and, as an effect of the image potential, decreases when the tip approaches the sample. A comparison between a numerical resolution of the exact Schrödinger equation and the transfer Hamiltonian approximation shows that the latter gives good results, even when the tip is close to the sample.