densities of states of the sample surface and the tip, . How the tunneling current depends on distance between the two electrodes is contained in the tunneling matrix element . This formula can be transformed so that no explicit dependence on the potential remains. First, the part is taken out from the Schrödinger equation for the tip, and the elastic tunneling condition is used so that . Now is present in the Schrödinger equation for the sample, and equals the kinetic plus the potential operator acting on . However, the potential part containing US is on the tip side of the barrier nearly zero. What remains, can be integrated over z because the integrand in the parentheses equals . Bardeen's tunneling matrix element is an integral of the wave functions and their gradients over a surface separating the two planar electrodes, . The exponential dependence of the tunneling current on the separation of the electrodes comes from the very wave functions that leak through the potential step at the surface and exhibit exponential decay into the classically forbidden region outside of the material. The tunneling matrix elements show appreciable energy dependence, which is such that tunneling from the upper end of the eV interval is nearly an order of magnitude more likely than tunneling from the states at its bottom. When the sample is biased positively, its unoccupied levels are probed as if the density of states of the tip is concentrated at its Fermi level. Conversely, when the sample is biased negatively, its occupied electronic states are probed but the spectrum of the electronic states of the tip dominates. In this case it is important that the density of states of the tip is as flat as possible. The results identical to Bardeen's can be obtained by considering adiabatic approach of the two electrodes and using the standard time-dependent perturbation theory. This leads to Fermi's golden rule for the transition probability in the form given above. Bardeen's model is for tunneling between two planar electrodes and does not explain scanning tunneling microscope's lateral resolution. Tersoff and Hamann used Bardeen's theory and modeled the tip as a structureless geometric point. This helped them disentangle the properties of the tip—which are hard to model—from the properties of the sample surface. The main result was that the tunneling current is proportional to the local density of states of the sample at the