Variable-range hopping in the critical regime

Abstract

Mott's theory of variable range hopping (VRH) is extended to the critical regime approaching the metal-insulator transition where the hopping energy ${\ensuremath{\Delta}}_{h}(T)l\mathrm{kT}.$ The Miller-Abrahams impedance between a pair of sites is minimized for an arbitrary value of ${\ensuremath{\Delta}}_{h}/kT.$ The theory features spatial dispersion of the dielectric response function $\ensuremath{\epsilon}(R,n),$ which introduces a new length scale ${r}_{s}$ into the Miller-Abrahams resonance energy. In the regime ${\ensuremath{\Delta}}_{h}\ensuremath{\ll}\mathrm{kT}$ 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 $\ensuremath{\epsilon}(R,n)$ and the ${T}_{0}^{\ensuremath{'}}$ depends critically on the spatial dispersion of $\ensuremath{\epsilon}(R,n)$ and this feature allows an explanation of the Ge:Ga data of Watanabe et al., which exhibits ${T}_{0}^{\ensuremath{'}}\ensuremath{\propto}(1\ensuremath{-}{n/n}_{c}{)}^{1.0}.$ 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 ${T}_{0}$ values encompassing the crossover from high-temperature Mott VRH to conventional low-temperature Mott VRH. The reduced density dependence of the Mott VRH prefactor ${\ensuremath{\sigma}}_{0}(n)$ is also satisfactorily explained. In the critical regime for $1\ensuremath{-}{n/n}_{c}l0.05,$ Si:As and Si:P yield the localization length exponent $\ensuremath{\nu}\ensuremath{\sim}1.$