Full capacitance-matrix effects in driven Josephson-junction arrays

We study the dynamic response to external currents of periodic arrays of Josephson junctions, in a resistively capacitively shunted junction model, including full capacitance-matrix effects. We define and study three different models of the capacitance matrix $C_{\overrightarrow{r},r^{\prime }}$: model A includes only mutual capacitances; model B includes mutual and self-capacitances, leading to exponential screening of the electrostatic fields; model C includes a dense matrix $C_{\overrightarrow{r},r^{\prime }}$ that is constructed approximately from superposition of an exact analytic solution for the capacitance between two disks of finite radius and thickness. In the latter case the electrostatic fields decay algebraically. For comparison, we have also evaluated the full capacitance matrix using the MIT FASTCAP algorithm, good for small lattices, as well as a corresponding continuum effective-medium analytic evaluation of a finite-voltage disk inside a zero-potential plane. In all cases the effective $C_{\overrightarrow{r},r^{\prime }}$ decays algebraically with distance, with different powers. We have then calculated current-voltage characteristics for dc+ac currents for all models. We find that there are giant capacitive fractional steps in the $I-V$’s for models B and C, strongly dependent on the amount of screening involved. We find that these fractional steps are quantized in units inversely proportional to the lattice sizes and depend on the properties of $C_{\overrightarrow{r},r^{\prime }}$. We also show that the capacitive steps are not related to vortex oscillations but to localized screened phase locking of a few rows in the lattice. The possible experimental relevance of these results is also discussed.