Frequency-dependent hopping conductivity in disordered networks in the presence of a biased electric field

The frequency-dependent hopping conductivity is calculated for a one-dimensional disordered chain as well as for a cubic percolating network in the presence of a biased electric field. In this biased case, there exists a drift-conducting region at low frequencies crossing over to the diffusive behavior at higher frequencies. Explicit expressions for the conductivity in various regions are presented. In the one-dimensional weak-disordered chain, one finds the leading term of the real (${\sigma }_R$) and imaginary (${\sigma }_I$) parts of the ac part of the conductivity behave as ${\omega }^2$ and ω, respectively, in the drift region, crossing over to the form in which both ${\sigma }_R$ and ${\sigma }_I$ behave as ${\omega }^{1/2}$ in the diffusion region. In the three-dimensional cubic percolation model, one obtains an additional anomalous diffusion region corresponding to short-time diffusion behavior on percolating clusters. One obtains ${\sigma }_R$=$A_1$ ${\omega }^2$ and ${\sigma }_I$=$B_1$ω in the lowest-frequency drift region, and ${\sigma }_R$=$A_2$ ${\omega }^{3/2}$ and ${\sigma }_I$=$B_2$ω in the normal diffusion region, while ${\sigma }_R$=$A_3$ ${\omega }^{1/2}$ and ${\sigma }_I$=$B_3$ ${\omega }^{1/2}$ in the anomalous diffusion region. These coefficients depend on the bias and reduce to the form of Odagaki and Lax in the absence of a bias.