Electron Transfer through a One-Dimensional Fractal Structure

An adequate model of electron tunneling through a self-similar fractal potential (SFP) defined on a Cantor set is extended to a generalized Cantor set. It is demonstrated that, as in a specific case, the Schrödinger equation for the SFP is reduced to a functional equation for the transfer matrix which admits solutions of three types. Two of them are single-parameter solutions corresponding to SFP barriers and lacunas with arbitrary powers. In both cases, the transfer matrices are nonanalytic in the long-wavelength region and have fractal dimensionalities there. The third solution type includes a unique solution corresponding to the SFP barrier with fixed power for a given barrier width. The corresponding transfer matrix is analytic at the point k = 0. It is shown that generally the SFP possesses only the property of approximate scale invariance on the generalized Cantor set in the long- and short-wavelength regions. Only the limiting SFP, whose fractal dimensionality is equal to unity, possesses the property of rigorous scale invariance irrespective of its power. It is shown that SFPs with identical fractal dimensionalities but different lacunas are described by different transfer matrices.