Mott's law for the v.r.h. random resistor network and for Mott's random walk

TL;DR

This paper rigorously derives Mott's law for variable range hopping in disordered solids, confirming the law's form and determining the constant factor, for two effective models of electron transport.

Contribution

It provides the first rigorous derivation of Mott's law and calculates the constant factor in the exponential decay for two models of variable range hopping.

Findings

Confirmed the form of Mott's law for the models

Determined the explicit constant in the law's exponential

Validated the models' predictions with rigorous proofs

Abstract

Mott's variable range hopping (v.r.h.) is the phonon-induced hopping of electrons in disordered solids (such as doped semiconductors) within the regime of strong Anderson localization. It was introduced by N.~Mott to explain the anomalous low temperature conductivity decay in dimension $d\geq 2$, corresponding now to the so called Mott's law. We provide a rigorous derivation of this Physics law for two effective models of Mott v.r.h.: the random resistor network for v.r.h. of \cite[Section~IV]{AHL} and Mott's random walk. We also determine the constant multiplying the power of the inverse temperature in the exponent in Mott's law, which was an open problem also on a heuristic level.