Tempered Fractional Equations for Quantum Transport in Mesoscopic One-Dimensional Systems with Fractal Disorder

Abstract

New aspects of electron transport in quantum wires with Lévy-type disorder are described. We study the weak scattering and the incoherent sequential tunneling in one-dimensional quantum systems characterized by a tempered Lévy stable distribution of spacing between scatterers or tunneling barriers. The generalized Dorokhov−Mello−Pereyra−Kumar equation contains the tempered fractional derivative on wire length. The solution describes the evolution from the anomalous conductance distribution to the Dorokhov function for a long wire. For sequential tunneling, average values and relative fluctuations of conductance and resistance are calculated for different parameters of spatial distributions. A tempered Lévy stable distribution of spacing between barriers leads to a transition in conductance scaling.

Keywords

• fractal
• quantum wire
• anomalous diffusion
• tempered fractional equation
• stable law
• weak scattering
• tunneling
• conductance