Charging effects and the phase-ordering transition in granular superconductors

We examine the influence of the charging energy on the phase-ordering transition in a lattice of superconducting grains connected by Josephson coupling. We use a finite-temperature self-consistent phonon procedure which is a generalization of the $T=0\mathrm{}$ method of Simanek. The transition temperature $T_c$ is found to be monotonically reduced as the ratio of charging energy to Josephson coupling energy is increased. No evidence of reentrant normal behavior is seen. The specific heat for the phase degrees of freedom is seen to be phononlike ($\sim T^3$) at sufficiently low temperatures, diverging at $T_c$. The $T^3$ behavior persists to very small charging energies. Effects of site-dilution disorder are considered qualitatively using an estimate of "phase-phonon" softening derived from spin-wave-stiffness results of Harris and Kirkpatrick. For a given nonzero charging energy, the critical concentration for onset of phase coherence always exceeds the site-percolation threshold because of zero-point phase fluctuations. Criteria for possible experimental observation of "phase-phonon" effects are discussed, and a simple experiment is proposed.