Green-function theory of scanning tunneling microscopy: Tunnel current and current density for clean metal surfaces

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Abstract
A theory of scanning tunneling microscopy (STM) is presented that accounts for a realistic treatment of the electronic structure of the sample surface. The sample is represented by a semi-infinite crystal built from muffin-tin potentials describing the atomic structure and surface electronic wave functions of s, p, d, etc., electrons. The other electrode carrying the tip atom is a planar free-electron metal surface. The potential of the tip atom is expanded in a localized basis set. Within the single-particle approach, the exact equation for the scattering wave for the combined system (tip plus sample) is derived. It is evaluated using a Green-function technique. From the scattering wave function the spatial distribution of the current density is obtained. The method is applied to study the tunnel current to clean Al(111), Pd(111), and Pd(100) surfaces. At typical tip-sample separations (≥3 Å) the substrate atoms appear as protrusions. Quite remarkable, the contrast is found to be larger in the Al(111) images than in the Pd(111) and Pd(100) images. This is a consequence of the tip-sample interaction. As a further consequence of tip-sample interactions we find that at close distances between the tip and the Pd surfaces the interstitial regions appear as maxima in the variation of the tunnel current. The theory covers a broad range of tip-sample separations, including those where perturbation treatments of STM theory (as, e.g., developed by Tersoff and Hamann) break down. A systematic analysis of the different aspects that may affect the tunnel current is presented.