Normal modes and nonlinear effects in Josephson arrays

We consider Josephson tunneling in an array of superconducting grains which interact both by (long-range) capacitative effects and by nearest-neighbor Josephson tunneling; this model may apply to (disordered) granular superconductors. Both analytic and numerical methods were employed. The normal modes have dispersion relations which depend on the dimensionality; however, neglect of the long-range Coulomb interaction (e.g., ground plane screening) leads to a wave equation with a velocity on the order of a Josephson frequency times the distance between the grains. Numerical calculations verify that result when a small amount of charge is deposited on one site of a chain. However, an interesting localization effect occurs if one site is given a large amount of charge. This causes the phase at that site to increase rapidly with time, and thus the Josephson currents associated with that site oscillate rapidly; this prevents the charge, on average, from leaving the initial site. A one-site-dominant model appropriate to this nonlinear regime is developed and is shown to be in good agreement with the numerical calculations. A numerical study is also made of the onset of the nonlinear regime.