Hopping conduction and the continuous-time random-walk model

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Abstract
The results of calculations of the short-time and infinite-time hopping conductivity of model disordered systems are used as the basis of a critical examination of the single-site approximation scheme (SSA) that has been employed extensively to study time-dependent transport phenomena in disordered insulators. The Scher and Montroll theory of dispersive transport, which has been used successfully to interpret transient photoconductivity experiments, invokes this approximation scheme to identify the hopping problem with a continuous-time random walk on an ordered lattice. Present results indicate that, while the SSA yields exact results in the short-time limit, it underestimates, sometimes seriously, the long-time or dc conductivity. In particular, the decay as a function of time of the mean drift velocity of photoexcited carriers hopping through a narrow band of localized states is considered here. Lower bounds are placed on the drift velocity which are often far larger than the smallest velocity predicted by the SSA. The implications of this failure of the SSA for the interpretation of experiment are also discussed.