Coherent-medium approximation in the stochastic transport theory of random media

Abstract

An admittance for localized physical quantities is generally related to a random walk on the basis of linear-response theory. A coherent-medium approximation is introduced to solve a master equation which is assumed to govern the random walk. The general formalism is specialized to ac hopping conduction and applied to the bond-percolation model in one- and three-dimensional systems and to a lattice model for impurity conduction in doped semiconductors. For the one-dimensional bond-percolation model, the ac conductivity obtained by the coherent-medium approximation is in good agreement with exact results for both the frequency dependence and the critical behavior. The present method predicts a percolation transition and several critical behaviors of the ac conductivity at the transition point for the bond-percolation model in a simple cubic lattice. In particular, ${\lim }_{\omega \to 0}\frac{\mathrm{Re}\left[\sigma \right(\omega )-\sigma \mathrm{}(0\left)\right]}{{\omega }^{\frac{3}{2}}}$ diverges as ${(p-p_c)}^{-\frac{3}{2}}$ when $p=p_c+0\mathrm{}$ and ${\lim }_{\omega \to 0}\frac{\mathrm{Re}\sigma \left(\omega \right)}{{\omega }^2}$ diverges as ${(p_c-p)}^{-3\mathrm{}}$ when $p=p_c-0\mathrm{}$ and in the static limit $\mathrm{Re}\sigma \left(\omega \right)$ vanishes as ${\omega }^2$ for $p<\mathrm{}{p}_c$ and as ${\omega }^{\frac{1}{2}}$ at $p=p_c$. The present approximation also succeeds in reproducing the typical frequency dependence of the ac conductivity of the hopping conduction in doped semiconductors, namely, the transition from the dc behavior through an ${\omega }^s$ dependence to a plateau as the frequency $\omega$ is increased. The theoretical results are shown to be in good agreement with experiments.