Hopping transport on a randomly substituted lattice for long range and nearest neighbor interactions

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Abstract
A theoretical study of hopping transport of excitations or charge carriers among particles randomly distributed on a lattice is presented. The method used is an extension of the diagrammatic technique applied by Gochanour, Andersen, and Fayer to hopping transport in a continuum. We present an exact diagrammatic analysis of the configuration averaged Green function of the Pauli master equation. We obtain a self-consistent approximation to the Green function from which transport properties such as the mean squared displacement may be calculated for any transfer rate, any lattice type and any concentration. For a three dimensional lattice, the results are shown to be accurate in the low concentration limit and for the filled lattice, and are expected to be accurate at intermediate concentration. This is the first theory of hopping transport on a randomly substituted lattice, which is not restricted to low concentration, that can be applied in the case of a long range transfer rate. Results are presented for a Förster dipole–dipole transfer rate and for a transfer rate limited to nearest neighbors for a simple cubic lattice. The latter has a percolation threshold that is described in a qualitatively correct manner by our approximation.