Static and Dynamic Properties of Commensurate and Incommensurate Phases of a Two-Dimensional Lattice of Superconducting Vortices

Phase transitions in two-dimensional (2D) systems have received considerable attention recently. In several experiments the 2D crystal under consideration is exposed to the force field created by a periodic substrate. Among other situations this is the case of a 2D lattice of superconducting vortices interacting with a periodic pinning potential. As pointed out by Martinoli and coworkers1-3 some years ago, thin superconducting films, whose thickness is periodically modulated in one dimension, provide such a system. In this lecture we discuss the static and dynamic behaviour of this model system in which the 2D vortex lattice can be driven through a variety of phases4, 5 simply by changing the conditions of flux-line density and/or temperature. In particular, we show how measurements of the critical currents and of the complex rf impedance of thickness-modulated layers can be used to probe the transition of the 2D vortex lattice from a “locked” commensurate (C) phase in registry with the substrate periodicity to a “floating” incommensurate (I) solid phase exhibiting 2D topological order6, 7 or to a fluid-like phase.