Stochastic treatment of AC hopping conduction in disordered systems

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Abstract
As was shown recently, the continuous-time random-walk model of hopping conduction introduced by Scher and Lax (1973) should give no frequency dependence of the conductivity. This stochastic model is generalized, taking into account the relatively slow polarization response of the charge carriers to the electric field produced by instantaneous hops of a test carrier. This modified stochastic model leads to AC conduction as a result of the polarization response, which is treated as a relaxation process. An exact expression is derived for the AC conductivity in terms of a distribution of waiting times between successive hops of a carrier. In the frequency range of interest, an omega v dependence (0<v<1) is found for the real and imaginary parts of the conductivity which satisfy the expected form of the Kramers-Kronig relation. The treatment suggests that v increases towards unity with decreasing temperature, in agreement with experimental findings.