Hopping Conduction in One-Dimensional Random Chains

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Abstract
Hopping conduction in one-dimensional chains is studied for two types of random distribution of the nearest neighbor jump rate w. The exact frequency dependence of the ac conductivity is obtained for a bond-percolation model where the jump rate vanishes randomly. The real and imaginary part of the ac conductivity are shown to vanish quadratically and linearly with the frequency, respectively. Critical behaviors at the percolation threshold are discussed. A distribution ρwρ−1/wρ0 (0 ≤ w ≤ w 0, ρ > 0) for the jump rate is used to describe hopping conduction among randomly-located sites, where p is the dimensionless number density of the sites. Using the coherent medium approximation, it is shown that an insulator-to-metal transition takes place at ρ = 1. Five regimes, depending on the value of ρ, are possible for the low frequency behavior of the conductivity, the ac part of the conductivity at low frequencies is quasi-symmetric around ρ = 1 and for 0<ρ≤1 a carrier can disappear from its initial site even though the dc conductivity vanishes.