Cluster approximation in the theory of the AC hopping conductivity in disordered systems. I. One-dimensional systems

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A theory is developed for the low-frequency AC hopping conductivity in disordered systems which accounts for multiple hops. The ideas of percolation theory are used to reduce the problem to a study of sigma ( omega ) (the frequency-dependent conductivity) in a cluster model. It is found that the size of the clusters increases with decreasing frequency and that sigma ( omega ) approximately omega s,s<1. An expression is obtained for the exponents. At low temperatures sigma depends weakly on temperature, but at higher temperatures it exhibits activated behaviour. Results are obtained for a one-dimensional system and compared with numerical results given in the literature.