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<p>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.&#xa0;Mott to explain the anomalous low temperature conductivity decay in dimension <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(d\ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, 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 [<CitationRef CitationID="CR1">1</CitationRef>, Section&#xa0;IV] 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.</p>

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

  • 期刊论文

  • 发表日期：

    2026年1月5日

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  • Alessandra Faggionato

摘要

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 \(d\ge 2\) d ≥ 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 [1, Section IV] 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.