Quantum-ohmic resistance fluctuation in disordered conductorsa - an invariant imbedding approach

Abstract

It is now well known that in the extreme quantum limit, dominated by the elastic impurity scattering and the concomitant quantum interference, the zero-temperature d.c. resistance of a strictly one-dimensional disordered system is non-additive and non-self-averaging. While these statistical fluctuations may persist in the case of a physically thin wire, they are implicitly and questionably ignored in higher dimensions. In this work, we have re-examined this question. Following an invariant imbedding formulation, we first derive a stochastic differential equation for the complex amplitude reflection coefficient and hence obtain a Fokker-Planck equation for the full probability distribution of resistance for a one-dimensional continuum with a gaussian white-noise random potential. We then employ the Migdal-Kadanoff type bond moving procedure and derive thed-dimensional generalization of the above probability distribution, or rather the associated cumulant function-'the free energy'. Ford=3, our analysis shows that the dispersion dominates the mobility edge phenomena in that (i) a one-parameter β-function depending on the mean conductance only does not exist, (ii) one has a line of fixed-points in the space of the first two cumulants of conductance, (iii) an approximate treatment gives a diffusion-correction involving the second cumulant. It is, however, not clear whether the fluctuations can render the transition at the mobility edge 'first-order'. We also report some analytical results for the case of the one-dimensional system in the presence of a finite electric field. We find a cross-over from the exponential to the power-law length dependence of resistance as the field increases from zero. Also, the distribution of resistance saturates asymptotically to a Poissonian form. Most of our analytical results are supported by the recent numerical simulation work reported by some authors.