Effect of long-range Coulomb interactions on the superconducting transition in Josephson-junction arrays

In this paper we treat the effect of Coulomb interactions between the grains of a Josephson-junction array or a granular superconductor, in the presence of a magnetic field. By using the duality transformation developed by Chui and Weeks, we are able to study both the low-$T_c$ and the high self-capacitance limits for d-dimensional cubic arrays. We examine two models for the Coulomb interaction matrix U. In the first, only the diagonal and nearest-neighbor interactions are nonzero. In the other, only the diagonal and nearest-neighbor components of the capacitance matrix C=$U^{\mathrm{-}1}$ are nonzero. Our results for the nearest-neighbor $U_{\mathrm{ij}}$ model agree with the results of Fazekas, who found no sign of normal state reentrance in two or three dimensions. In the second model, reentrance occurs when the dimensionless parameter λ, related to the inverse screening length of the Coulomb interactions, becomes smaller than a critical value ${\lambda }^{\mathrm{*}}$. For both models, the off-diagonal interactions enhance the transition temperature by reducing the cost of Cooper-pair tunneling. We compare the results of the second model with experimental observations of charging effects and reentrance in three dimensions.