Consistent hopping criterion in the Efros-Shklovskii regime

Abstract

We address the variable-range hopping regime in the domain where the measuring temperature $T$ is of the order of the characteristic Efros-Shklovskii temperature ${T}_{\mathit{ES}}$. In such a range, current theories imply ${r}_{\mathit{hop}}∕\ensuremath{\xi}<1$, where ${r}_{\mathit{hop}}$ is the hopping length and $\ensuremath{\xi}$ is the localization length, clearly in contradiction with the standard criterion for hopping conduction. We consider impurity overlap wave functions of the form $\ensuremath{\psi}(r)\ensuremath{\propto}{r}^{\ensuremath{-}n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}r∕\ensuremath{\xi})$ and include the preexponential factor of the hopping probability as a logarithmic correction in the Mott optimization procedure. From the general expressions derived, the standard Efros-Shklovskii law is recovered for $T⪡{T}_{\mathit{ES}}$, whereas an extended preexponential sensitive regime, consistent with ${r}_{\mathit{hop}}∕\ensuremath{\xi}>1$, is found for ${T}_{\mathit{ES}}\ensuremath{\gtrsim}T$. We argue that the expression resulting from an interplay between preexponential and exponential factors is a consistent extension of the classical Efros-Shklovskii argument. An additional parameter in the theory is directly related to the decay of the impurity wave functions and could be seen as a probe into their behavior. A fit of reference experimental data to the proposed theory yields consistent results.