Abstract
We have presented the coherent medium approach to hopping conduction problems where the motion of carriers obey the usual random walk equation and discussed the existence of the coherent medium which is defined through an average of the random walk propagator. We have introduced the coherent medium approximation (CMA) to obtain an approximate but easily tractable coherent medium. The CMA is a generalized use of the coherent potential approximation (CPA) in the master equation. The CPA is one of the most fruitful methods in treating random systems and is widely used in electron 19 and phonon problems. 18 We have applied the CMA to various cases of hopping conduction in one- and three-dimensions and compared the results with experiment. A simple comparison of the CMA with other methods has also been given.
We have derived the CMA condition Eq. (5.11) from the multiple scattering formalism due to Lax.12,13 An alternative derivation of Eq. (5.11) was given by Odagaki and Lax,l10 where a traditional idea of the effective medium approximation was used.36 Namely, a random unit is subjected to an as yet unknown effective medium and the effective medium is determined to be such that the resulting extra perturbation is required to vanish on the average over all possibilities of the random unit. A similar condition to Eq. (5.11) has also been used in the problem of random resistor networks.37
We hope that further improvements of the coherent medium approximation will be developed including traps, asymmetric jump rates, cluster effects and a similar method will be applied to other problems, for example, optical properties and magnetic properties of random systems.
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Lax, M., Odagaki, T. (1982). Coherent medium approach to hopping conduction. In: Burridge, R., Childress, S., Papanicolaou, G. (eds) Macroscopic Properties of Disordered Media. Lecture Notes in Physics, vol 154. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-11202-2_11
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