Electron tunnelling through a self-similar fractal potential on the generalized Cantor set

Abstract

A proper formalism developed earlier to study electron tunnelling through a self-similar fractal potential (SSFP) posed on the Cantor set is extended here to describe the SSFP whose levels consist of N fractals of the next level. We have derived a functional equation for the transfer matrix of this potential and found three different solutions. Two of them correspond to SSFP barriers and SSFP wells whose power may be arbitrary. The third one relates to the only SSFP barrier whose power has a definite value. These solutions show that SSFPs, in the general case, are approximately scale invariant in the long- and short-wave regions, and only the limiting SSFP whose fractal dimension is equal to unity should be strictly scale invariant. We have shown that except for the limiting case the tunnelling parameters of SSFPs, with the same fractal dimension depend on N. In addition, we have established a link between the solutions of the functional equation and the power of SSFPs.