The Two-Dimensional Physics of Josephson Junction Arrays

This chapter discusses the two-dimensional physics of Josephson junction arrays. Josephson junction arrays consist of islands of superconductor, usually arranged on an ordered lattice, coupled by Josephson junctions. They may be divided into classical and quantum arrays depending on the ratio of the Josephson coupling energy to the charging energy. Josephson junction arrays may also be divided into overdamped and underdamped arrays, referring to the fact that the equation of motion for a single Josephson junction is identical to a damped pendulum. The presence of vortices is one of the natural consequences of arranging such junctions in a two-dimensional lattice. Large arrays have proven to be very useful model systems for studying a wide variety of other physical problems, for example, phase transitions in frustrated and random systems, the dynamics of coupled non-linear systems and macroscopic quantum effects. Classical two-dimensional arrays may be shown to be isomorphic to a two-dimensional XY spin system—they are physical representations of the XY model, which is a two-dimensional lattice of spins free to rotate in the XY plane. Classical two-dimensional arrays are used in zero magnetic fields for studying phase transitions such as the Kosterlitz–Thouless–Berezinskii transition, for studying the effects of disorder on phase transitions, and for studying dimensional crossover effects in phase transitions.