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In this chapter we have shown how theorists actually proceed from a given SFM experimental result to arrive at a realistic simulation of the imaging process. It turned out that the key to successful modeling lies in the ability to successively refine the theoretical model, especially with regard to allowing flexibility in tip selection. This process is inherently iterative: it is usually not possible to arrive at a consistent model that agrees with experimental data without several iteration cycles to fine-tune the model. Contrary to what one might believe, theoretical modelling of SFM experiments is therefore no black box, at least not at the present stage. A general approach for real understanding in SFM simulations must include the following components: Justification for the interaction simulation method itself: empirical potentials can be useful, but must be carefully tested, and are usually inflexible.An attempt to model the real experimental tip if enough data exists, or at least several plausible models must be considered.For high-resolution imaging, tip and surface relaxations must be included since they have a significant influence on the interactions.The dynamics of the cantilever and experimental electronics must be treated at a level appropriate for the phenomenon being simulated. [ABSTRACT FROM AUTHOR]

In reflecting on the simulations and the gradual resolution of the puzzle, it becomes clear that only a very limited part of the whole physical situation is actually accessible in the experiments. The change of the position of the SPM tip is a result of measurements of constant-current/height contours. But the change of the position of surface atoms under given experimental conditions cannot be determined. This makes simulations the only source of information on both the stability of a system under specific conditions determined from interaction energies, the elastic limit of a surface and tip system, and the relation between true surface properties (properties of its ground state) and virtual properties that are due to the measurement itself. It might seem that the last distinction is far-fetched. But one only has to consider that atomic positions on a surface can be determined by a number of different methods, e.g., electron diffraction, photon diffraction, and electron tunneling, to understand that different experimental methods might lead to different results. And in this case, the possibility in theory to switch on or off a particular effect makes it quite adaptable to a whole range of experimental data. This becomes even more important in the case of SFM. [ABSTRACT FROM AUTHOR]

In this chapter we have presented an overview over the most common methods used in tunneling problems, which are, in increasing order of complexity: the Tersoff-Hamann model, the Bardeen model, the Landauer-Büttiker model, and the Keldysh model. The treatment of the tunneling junction in these models is described by one of the following: restricted to the surface only (Tersoff-Hamann); includes both sides of the junction, without considering interference effects (Bardeen); is based on elastic tunneling conditions (Landauer-Büuttiker); includes the full nonequilibrium formulation of the problem (Keldysh). Readers interested in a general formulation of transport theory are referred to the previous chapter, where the whole framework is treated in some detail. [ABSTRACT FROM AUTHOR]

We have given in this chapter an introduction to the current state of electron transport theory, in view of applications to tunneling problems. The theoretical framework, based on Green's functions of open systems, was shown to be adaptable, via its perturbative extension into nonequilibrium environments, to treat all relevant physical processes at the atomic scale, essentially from first principles. The present implementations rely on tight-binding schemes or local orbital geometry; within these limits the theory can cope with finite-bias potentials and inelastic effects due to electron-electron or electron-phonon interactions. It can be foreseen that the framework, once it is extended to cover also plane-wave and full-potential methods, will provide the backbone of transport simulations on the atomic scale, whenever high accuracy is the decisive issue. [ABSTRACT FROM AUTHOR]