Since the appearance of the STM most tunneling theories are based on Bardeen's tunneling current formalism⁸, which has been adapted for the STM by Tersoff and Hamann⁹:

where f(E) is the Fermi function, V - the bias voltage, M_mn - the tunneling matrix element between the states y_m of the tip and y_n of the surface. In special cases M_mn and the whole above expression can be simplified. In 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 approximated⁶ as:

where r_T(E+eV) is the density of states of the tip, r_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.