Quasi-extended electron states in strongly disordered systems

Abstract

Intuitive arguments based on the concept of a typical state conclude that transport in localised systems is dominated by the states near the centre of the material. They maximise the transmission coefficient, but have narrow linewidths depending on length, L, as w_min approximately=exp(- delta ^1/2L/4). Rigorous 1D theory shows that these arguments are in error. A more circumspect application of intuition shows that transport is dominated by a new sort of band of states, delocalised over a necklace of square root L sites spaced evenly across the material. There are very few of these states; they have fractal dimension ¹/₂; and they dominate the conductivity because of their large bandwidth: w_1/2 approximately=exp(- delta ' square root L). In a typical situation w_min approximately=10^-8Hz, w_1/2 approximately=10⁶ Hz. Whereas the rigorous results are derived only in the 1D case, the intuitive arguments are valid for strongly localised systems irrespective of dimension.