Variable-range hopping in the critical regime

Mott's theory of variable range hopping (VRH) is extended to the critical regime approaching the metal-insulator transition where the hopping energy Delta(h)(T)<kT. The Miller-Abrahams impedance between a pair of sites is minimized for an arbitrary value of Delta(h)/kT. The theory features spatial dispersion of the dielectric response function epsilon(R,n), which introduces a new length scale r(s) into the Miller-Abrahams resonance energy. In the regime Delta(h)<< kT the Mott exponent 1/4 changes to 2/7 and the numerical factor [18.1] in the Mott T-0 is reduced to 1.51. Efros-Shklovskii VRH is also modified by the inclusion of epsilon(R,n) and the T-0(') depends critically on the spatial dispersion of epsilon(R,n) and this feature allows an explanation of the Ge:Ga data of Watanabe et al., which exhibits T-0('proportional to) (1 - n/n(c))(1.0). The new result for the Mott case allows the Si:As and Si:P results to be explained over a range of 10(6) in Mott To values encompassing the crossover from high-temperature Mott VRH to conventional low-temperature Mott VRH. The reduced density dependence of the Mott VRH prefactor sigma(0)(n) is also satisfactorily explained. In the critical regime for 1 - n/n(c) <0.05, Si:As and Si:P yield the localization length exponent v similar to 1.