Quantum-ohmic resistance fluctuation in disordered conductors—An invariant imbedding approach

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-parameterβ-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. The author felicitates Prof. D S Kothari on his eightieth birthday and dedicates this paper to him on this occasion.