Variable Range Hopping Conduction in Complex Systems and a Percolation Model with Tunneling

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For the low-temperature electrical conductance of a disordered quantum insulator in d-dimensions, Mott mott had proposed his Variable Range Hopping (VRH) formula, G(T) = G₀ exp[-(T₀/T)γ], where G₀ is a material constant and T₀ is a characteristic temperature scale. For disordered but non-interacting carrier charges, Mott had found that γ= 1/(d+1) in d-dimensions. Later on, Efros and Shkolvskii esh found that for a pure (i.e., disorder-free) quantum insulator with interacting charges, γ=1/2, independent of d. Recent experiments indicate that γ is either (i) larger than any of the above predictions; and, (ii) more intriguingly, it seems to be a function of p, the dopant concentration. We investigate this issue with a semi-classical or semi-quantum RRTN (Random Resistor cum Tunneling-bond Network) model, developed by us in the 1990's. These macroscopic granular/ percolative composites are built up from randomly placed meso- or nanoscopic coarse-grained clusters, with two phenomenological functions for the temperature-dependence of the metallic and the semi-conducting bonds. We find that our RRTN model (in 2D, for simplicity) also captures this continuous change of γ with p, satisfactorily.