Numerical calculations of d.c. hopping conductivity

Abstract

Abstract Computer calculations of d.c. hopping conductivity ⟨σ⟨ are presented for electrons in both narrow and wide energy bands in both two and three dimensions. Excellent agreement with recently proposed analytical formulae is obtained in all cases. Kirchoff's equations are solved for up to 2500 randomly distributed sites using the effective conductances introduced by .Miller and Abrahams (1960) in which the dependence on intersite separation r has the form r 1.5 exp (−2αr) where α is the decay constant of the localized wavefunctions. For narrow energy bands the fluctuations in the computed values of ⟨σ⟨ are less than 7%. A reduction in the number of connected neighbours from 24 to 12 has significant effects when αa < 6 where a is the average intersite spacing. Similarly, for wide energy bands more than 12 connected neighbours are necessary to obtain macroscopic conductivities at temperatures above a few degrees Kelvin. It is suggested that the anomalous computer results for ⟨σ⟩ reported by previous authors are a consequence of their computational procedures.