Advanced percolation solution for hopping conductivity

Hopping of carriers between localized states dominates charge transport in amorphous organic and inorganic semiconductors. We suggest a comprehensive description of this transport regime based on the percolation approach that allows one to determine not only very pronounced exponential dependencies of the hopping conductivity on material parameters, but also the more weakly dependent pre-exponential factors. The problem of the variable-range hopping (VRH) via sites with exponential energy distribution is mapped onto a universal geometrical problem of percolation via spheres with distributed sizes. An exact solution of the latter problem provides accurate results for the VRH in systems with exponential density-of-states (DOS). Our analytical results are confirmed by straightforward computer simulations and compared to former results present in the literature. We also discuss the case of nearest-neighbor hopping on a lattice, where the pre-exponential factors are provided by the percolation approach for any shape of the DOS.