Semiclassical analysis of the quantum interference corrections to the conductance of mesoscopic systems

TL;DR

This paper uses semiclassical analysis to derive simple expressions for quantum interference effects on conductance in mesoscopic systems, extending previous work to longer times and applying results to chains of ergodic scatterers.

Contribution

It provides a semiclassical framework for analyzing long-time quantum interference corrections to conductance, bridging results between random matrix theory and diffusive wire models.

Findings

Weak localization correction approaches -1/3 as the number of scatterers increases.

Variance of conductance fluctuations approaches 2/15 for many scatterers.

Results interpolate between known limits of random matrices and diffusive wires.

Abstract

The Kubo formula for the conductance of a mesoscopic system is analyzed semiclassically, yielding simple expressions for both weak localization and universal conductance fluctuations. In contrast to earlier work which dealt with times shorter than $O(\log \hbar^{-1})$, here longer times are taken to give the dominant contributions. For such long times, many distinct classical orbits may obey essentially the same initial and final conditions on positions and momenta, and the interference between pairs of such orbits is analyzed. Application to a chain of $k$ classically ergodic scatterers connected in series gives the following results: $-{1 \over 3} [ 1 - (k+1)^{-2} ]$ for the weak localization correction to the zero--temperature dimensionless conductance, and ${2 \over 15} [ 1 - (k+1)^{-4} ]$ for the variance of its fluctuations. These results interpolate between the well known ones of random scattering matrices for $k=1$, and those of the one--dimensional diffusive wire for $k \rightarrow \infty$.