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    Hopping-transport regimes and dimensionality transition: A unified Monte Carlo random-resistor-network approach

    Alejandro Toral-Lopez* and Gianluca Fiori

    Damiano Marian

    • Dipartimento di Ingneneria della Informazione, University of Pisa, Via Caruso 16, 56126 Pisa, Italy

    • Dipartimento di Fisica “E. Fermi”, University of Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy

    • *Contact author: alejandro.lopez@ing.unipi.it

    Phys. Rev. Applied 25, 024040 – Published 12 February, 2026

    DOI: https://doi.org/10.1103/bx6m-635y

    Abstract

    Hopping transport, characterized by carrier tunneling between localized states, is a key mechanism in disordered materials such as organic semiconductors, perovskites, nitride alloys, and two-dimensional-material- (2DM) based inks. Two main regimes are typically observed—variable-range hopping and nearest neighbor hopping—with a transition between them upon temperature variation. Despite numerous experimental observations, the modeling of this transition remain insufficiently explored and not fully understood. In this work, we present an in-house Monte Carlo random-resistor-network-based simulator capable of capturing both hopping-transport regimes. We demonstrate how material properties that define the network, such as localization length and the spatial and energetic distribution of sites, determine the dominant transport regime. The simulator has been successfully validated against experimental data, showing excellent agreement, reproducing the transition from one regime to the other and accurately capturing one-dimensional, two-dimensional, and three-dimensional variable-range hopping behavior, providing both a theoretical framework for interpreting experiments and a powerful tool for studying transport mechanisms.

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