Numerical calculations of a.c. hopping conductivity

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Abstract
The first direct computer calculations of a.c. conductivity <[sgrave](ω)> are presented for electrons in three-dimensional, r-percolation hopping systems. Kirchhoff's equations are solved for 1600 randomly distributed sites using constant capacitances and conductances in which the dependence on the intersite separation r has the form r 3/2 exp (– 2αLr). The computer data are compared with the predictions of the pair approximation, corrected by the addition of the known d.c. limit <[sgrave](0)>, and with the predictions of the continuous-time random-walk (CTRW) model proposed by Scher and Lax (1973). Good agreement is obtained with the predictions of the corrected pair approximation over the entire frequency range. The CTRW predictions are qualitatively incorrect both at zero frequency and at high frequencies. At intermediate frequencies, however, the CTRW predictions are in excellent numerical agreement with the computer results, for which <[sgrave](ω)>∝ωs but with s significantly less than 0.8.

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