![]() Very similar results were obtained by applying effective medium theory for the ac behavior of random systems. 69 The crossover frequency shifts to lower values with decreasing dc conductivity. 52,67 The general theory of hopping in disordered systems elaborated by Movaghar et al., 68 as well as extended pair approximation of Summerfield and Butcher, predicts a nonzero dc conductivity and subsequent dependence of conductivity on frequency also as ω s. In fact all the models for hopping transport predict similar exponential frequency dependence of conductivity which can be expressed by σ ac( ω) ∼ ω s with s in the range 0.4 < s < 1.0 assuming a phonon frequency of the order of 10 12 Hz in most cases s ≈ 0.8. In general one can say that while the dc conduction requires a continuous network of hopping sites and the weakest links determine the value of σ dc, for the ac conductivity when the frequency increases these links are shunted and the conductivity increases. ![]() For ω ≪ ω ph the exponent s ≈ 0.8 and is a weak function of frequency.
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