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 $g\gg1$, the conductivity $σ$ in $d$ dimensions decreases logarithmically with temperature ($σ/σ_{0}\sim1-\frac{1}{2πgd}\ln(gE_{c}/T)$), while for $g\to0$, the conductivity shows simple activated behaviour ($σ\sim \exp(-E_c/T)$). We show, for bare tunnelling conductance $g \gtrsim 1$, that the parameter $γ\equiv g(1-2/(gπ)\ln(gE_{c}/T))$ determines the competition between charging and tunnelling effects. At low enough temperatures in the regime $1\gtrsim γ\gg 1/\sqrt{βE_{c}}$, a charge is shared among a finite number $N=\sqrt{(E_{c}/T)/\ln(π/2γz)}$ of grains, and we find a soft activation behaviour of the conductivity, $σ\sim z^{-1}\exp(-2\sqrt{(E_{c}/T)\ln(π/2γz)})$, where $z$ is the effective coordination number of a grain.

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