ON THE 3-DIMENSIONAL SCANNING TUNNELING MICROSCOPY FORMALISM

The tunneling current using Bardeen formalism is calculated in a scanning tunneling microscope (STM). We take into account that the process is nonspecular tunneling, so the no conservation of the transverse wave number is considered. The tip is a three-dimensional box with free electrons inside and with the transversal dimensions in the atomic range. Note that with this description the allowed states inside the tip are quantified and that it is necessary for the electrons to have at least a minimum value for the energy in the parallel direction. The wave functions inside the sample are described as extended plane waves. Using a perturbative potential, the wave functions of the electrons inside and outside the tip are calculated. The matrix element to apply Bardeen formalism is calculated using the second Green's identity which makes unnecessary the tip's wave function outside it. Thus, instead of doing the integration on a tunnel surface, we have done the integration over the tip volume. Finally a discussion of the lateral resolution of the STM in terms of the transverse wave number is made.