Coulomb blockade and quantum tunnelling in the low-conductivity phase of granular metals

Abstract

We study the effects of Coulomb interaction and inter-grain quantum tunnelling in an array of metallic grains using the phase-functional approach for temperatures $T$ well below the charging energy $E_c$ of individual grains yet large compared to the level spacing in the grains. When the inter-grain tunnelling conductance $g\gg 1$, the conductivity $\sigma$ in $d$ dimensions decreases logarithmically with temperature ($\sigma /{\sigma }_0\sim 1-\frac{1}{2\pi gd}\ln (g{E}_c/T)$), while for $g\to 0$, the conductivity shows simple activated behaviour ($\sigma \sim \exp (-E_c/T)$). We show, for bare tunnelling conductance $g\gtrsim 1$, that the parameter $\gamma \equiv g(1-2/(g\pi )\ln (g{E}_c/T))$ determines the competition between charging and tunnelling effects. At low enough temperatures in the regime $1\gtrsim \gamma \gg 1/\sqrt{\beta E_c}$, a charge is shared among a finite number $N=\sqrt{(E_c/T)/\ln (\pi /2\gamma z)}$ of grains, and we find a soft activation behaviour of the conductivity, $\sigma \sim z^{-1}\exp (-2\sqrt{(E_c/T)\ln (\pi /2\gamma z)})$, where $z$ is the effective coordination number of a grain.

11 pages REVTeX, 3 Figures. Appendix added, replaced with published version

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