Fast algorithms for Josephson-junction arrays: Busbars and defects.

We critically review the fast algorithms for the numerical study of two–dimensional Josephson junction arrays and develop the analogy of such systems with electrostatics. We extend these procedures to arrays with bus–bars and defects in the form of missing bonds. The role of boundaries and of the guage choice in determing the Green’s function of the system is clarified. The extension of the Green’s function approach to other situations is also discussed. The dynamical properties of Josephson junction arrays (JJAs) are currently the foci of several experimental and theoretical investigations [1]. These arrays can now be routinely fabricated in several sizes and geometries, and the characteristics of their junctions can be varied at will over a wide range of values [2]. A large body of high–precision experimental data has consequently become available for JJAs in the presence of external magnetic fields [3–5]. On the theoretical front, several insights into the behaviour of JJAs have come from numerical studies of the underlying equations of motion as given by the resistively- and capacitively-shunted junction (RCSJ) model using input current drives and defects, both controlled [6] and random [7]. With the size of experimental arrays increasing continuously, and with the number of interesting effects best seen only in large arrays going up in equal measure, it has become imperative to find ever more efficient algorithms for implementing the corresponding simulations, inclusive of all the experimental conditions. An example of the latter for current–driven arrays is the presence of bus-bars, through which the external current can be conveniently injected or withdrawn. To understand the problem which these algorithms must address, we recall that in the RCSJ model, the total current, iij, (inclusive of external drives where applicable) flowing through the junction between sites i and j, is viewed as consisting of three ‘channels’ in parallel: superconductive, resistive ( or ohmic) and capacitive. The currents in each of these channels can be expressed in terms of the phase difference, θij = θi − θj, across the junction. This leads to the following equation for the evolution of the latter in time: C¯