Effective charging energy for a regular granular metal array

We study the Ambegaokar-Eckern-Schön 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 σ(T)∼exp[−E_σ(g)∕T] with an effective charging energy E_σ(g) when the temperature is sufficiently low, due to a subtle cancellation between T² inelastic-cotunneling contributions in the paramagnetic and diamagnetic terms. We present numerical results for this effective charging energy. We discuss the different ways in which the experimentally observed σ(T)∼exp[−√T₀∕T] law could be attributed to disorder.