A MULTI-DIMENSIONAL TUNNELING THEORY WITH APPLICATION TO SCANNING TUNNELING MICROSCOPE

Abstract
We have developed a formal general approach for evaluating the WKB wave function in a forbidden region of a non-separable potential in a three-dimensional space. The wave function is described by two sets of orthogonal wave-fronts, the equi-phase and equi-amplitude surfaces, or equivalently by two sets of paths defined to be normal to these surfaces respectively. We have extended a Huygens type construction to the forbidden region to obtain the multi-dimensional wave function. The construction determines the wave-fronts and the paths. It is found that the equations for the paths obtained from the construction are coupled to each other and do not reduce to a set of ordinary differential equations. However, if the incident wave is normal to the turning surface, the equations satisfied by the paths can be de-coupled. These equations are found to be equivalent to Newton's equations of motion for the inverted potential and energy. This characteristics has been used to calculate the current density distribution for an STM. In this paper, we use the same approach to derive the expressions for the lateral resolutions and corrugations for a model STM.