Tersoff-Hamann STM Theory

In the perturbative transfer Hamiltonian approach developed by Bardeen 58), the tip and sample are treated as two non-interacting subsystems. Instead of trying to solve the problem of the combined system, each separate component is described by its wave function, i tip and i/zj, respectively. The tunneling current is then calculated by considering the overlap of these in the tunnel junction. This approach has the advantage that the solutions can be found, for many practical systems, at least approximately, by solution of the stationary Schrodinger equation. 

The purpose of doing STM is to learn about surface structures, and the tip as such is regarded as an uninteresting probe. In this sense, it is a problem that the electronic structure of the tip is contained in the formula for the tunnel current in the original work by Bardeen 58). Tersoff and Hamann 59,60), however, extended Bardeen s formalism and showed by simple, yet relevant approximations that the impact of the unwanted electronic structure of the tip is in many cases less pronounced for typical tunneling parameters. Fortunately, the Tersoff-Hamann model provides a simple conceptual framework for interpreting STM images, and therefore it is still the most widely used model. 

The difficulty of evaluating the effect on the tunneling current of the tip electronic structure was approached by Tersoff-Hamann by assuming a simple, s-wavc tip model with wave functions centered at a point Fq in the tip. In the limit of low-bias voltages, the total tunnel current can then be expressed as follows  

The strength of the Tersoff-Hamann theory is that it provides a simple interpretation of the tunnel current in terms of a physical quantity, representing the surface alone. However, it also explains the exponential dependence of the tunneling bias on the tip—sample separation. All electronic states decay exponentially into 

Consequently, one cannot in general assume that maxima expected from the surface topography coincide with maxima observed in STM images. This effect becomes especially evident on semiconductor surfaces (e.g., those of M0S2 or TiO2), for several reasons. In contrast to metals, semiconductors show a very strong 

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