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

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 $k \rightarrow \infty$.