Effective charging energy for a regular granular metal array

TL;DR

This paper analyzes the conductivity of a regular granular metal array using the AES model, revealing an Arrhenius law with an effective charging energy for conductance between 0 and 4, supported by analytical and numerical methods.

Contribution

It provides a detailed calculation of the effective charging energy in a granular array, combining analytical and Monte Carlo simulations, and discusses implications for experimental conductivity laws.

Findings

Conductivity follows an Arrhenius law with an effective charging energy for 0<g<4.

Numerical results confirm the analytical predictions.

Discusses disorder effects on the observed conductivity law.

Abstract

We study the Ambegaokar-Eckern-Sch\"{o}n (AES) model for a regular array of metallic grains coupled by tunnel junctions of conductance $g$ and calculate both paramagnetic and diamagnetic terms in the Kubo formula for the conductivity. We find analytically, and confirm by numerical path integral Monte Carlo methods, that for $0<g<4$ the conductivity obeys an Arrhenius law $\sigma(T)\sim\exp[-E^{*}(g)/T]$ with an effective charging energy $E^{*} (g)$ when the temperature is sufficiently low, due to a subtle cancellation between $T^2$ inelastic-cotunneling contributions in the paramagnetic and diamagnetic terms. We present numerical results for the effective charging energy and compare the results with recent theoretical analyses. We discuss the different ways in which the experimentally observed $\sigma(T)\sim\exp[-\sqrt{T_{0}/T}]$ law could be attributed to disorder.