Model for the onset of transport in systems with distributed thresholds for conduction

Abstract

We present a model supported by simulation to explain the effect of temperature on the conduction threshold in disordered systems. Arrays with randomly distributed local thresholds for conduction occur in systems ranging from superconductors to metal nanocrystal arrays. Thermal fluctuations provide the energy to overcome some of the local thresholds, effectively erasing them as far as the global conduction threshold for the array is concerned. We augment this thermal energy reasoning with percolation theory to predict the temperature at which the global threshold reaches zero. We also study the effect of capacitive nearest-neighbor interactions on the effective charging energy. Finally, we present results from Monte Carlo simulations that find the lowest-cost path across an array as a function of temperature. The main result of the paper is the linear decrease of conduction threshold with increasing temperature: ${V}_{t}(T)={V}_{t}(0)[1\ensuremath{-}4.8{k}_{B}TP(0)∕{p}_{c}]$, where $1∕P(0)$ is an effective charging energy that depends on the particle radius and interparticle distance, and ${p}_{c}$ is the percolation threshold of the underlying lattice. The predictions of this theory compare well to experiments in one- and two-dimensional systems.