Physics Department, University of Wisconsin, Madison, WI, USA (1999)
Since the appearance of the STM most tunneling theories are based on Bardeen's tunneling current formalism^{8}, which has been adapted for the STM by Tersoff and Hamann^{9}:
where f(E) is the Fermi function, V - the bias voltage, M_{μν}—the tunneling matrix element between the states ψ_{μ} of the tip and ψ_{ν} of the surface. In special cases M_{μν} and the whole above expression can be simplified. In a particular case of interest to STS of semiconductors and insulators, we are dealing with the applied bias of the order of 2 eV, which is not small in comparison to kT (0.026 eV at room temperature). For tip-sample bias smaller than a typical work function (4–5 eV) the tunneling current at a fixed location on the surface can be approximated^{6} as:
where ρ_{T}(E+eV) is the density of states of the tip, ρ_{S}(E) is the density of states of the sample and T(E,V) is the transmission probability of the electron of energy E through the tunneling barrier. Even in this simplified model, a number of assumptions about all the three functions in the integrand can be made to construct more or less realistic models. Several such models will be considered in the following sections.