Nonalgebraic length dependence of transmission through a chain of barriers with a Levy spacing distribution

TL;DR

This paper analyzes how transmission through a one-dimensional chain of barriers with Levy-distributed spacings scales with length, revealing a nonalgebraic dependence and specific noise characteristics.

Contribution

It provides the first detailed calculation of conductance moments and noise behavior for a Levy-distributed barrier chain, highlighting nonalgebraic length dependence.

Findings

Average transmission scales as L^(-alpha) ln L for 0<alpha<1.

Fano factor approaches 1/3 with 1/ln L corrections.

Transmission exhibits nonalgebraic, Levy-like length dependence.

Abstract

The recent realization of a "Levy glass" (a three-dimensional optical material with a Levy distribution of scattering lengths) has motivated us to analyze its one-dimensional analogue: A linear chain of barriers with independent spacings s that are Levy distributed: p(s)~1/s^(1+alpha) for s to infinity. The average spacing diverges for 0<alpha<1. A random walk along such a sparse chain is not a Levy walk because of the strong correlations of subsequent step sizes. We calculate all moments of conductance (or transmission), in the regime of incoherent sequential tunneling through the barriers. The average transmission from one barrier to a point at a distance L scales as L^(-alpha) ln L for 0<alpha<1. The corresponding electronic shot noise has a Fano factor (average noise power / average conductance) that approaches 1/3 very slowly, with 1/ln L corrections.