We introduce energy-projected equations of motion to treat the diffusive transport of charge carriers that undergo long-range (i.e., greater than nearest-neighbor) hopping among the sites of an energetically disordered lattice. This approach leads naturally to an energy-projected effective-medium theory for treating such systems. Exact expressions for the diffusion constant associated with the energy-projected effective medium theory are obtained. Using the formalism in conjunction with what is normally a rather poor approximation, i.e., the virtual-crystal approximation, we are able to obtain the exact diffusion constant for the long-range symmetric-random-well problem. Effective-medium calculations and numerical simulations are presented for nearest-neighbor and long-range hopping on a disordered binary lattice.
P. E. Parris and B. D. Bookout, "Energy-Projected Effective-Medium Theory of Long-Range Hopping on Energetically Disordered Lattices," Physical Review B (Condensed Matter), vol. 48, no. 13, pp. 9354-9358, American Physical Society (APS), Oct 1993.
The definitive version is available at https://doi.org/10.1103/PhysRevB.48.9354
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