Vortex-lattice melting in two-dimensional superconducting networks and films

We carry out an extensive Monte Carlo study of phase transitions in two-dimensional (2D) superconducting networks, in an applied magnetic field, for square and honeycomb geometries. We consider both systems with a dilute vortex density 1/q, and dense systems near full frustration with vortex density 1/2–1/q. The dilute case gives the continuum limit as q→∞, and serves as a model for a uniform superconducting film. For this dilute case, we find a transition temperature ${\mathit{T}}_{\mathit{c}}$∼1/q, at which the vortex lattice unpins from the network and forms a ‘‘floating solid’’ phase. At a higher temperature ${\mathit{T}}_{\mathit{m}}$, this floating solid melts into a vortex liquid. We analyze the transition at ${\mathit{T}}_{\mathit{m}}$ according to the Kosterlitz-Thouless theory of dislocation mediated melting in 2D. While we find a discontinuous jump in the vortex shear modulus at ${\mathit{T}}_{\mathit{m}}$ which is consistent with this theoy, we find (in opposition to this theory) that the transition is weakly first order, and we find no evidence for a hexatic liquid phase. For the case near full frustration, we find that the system can be described in terms of the density of defects in an otherwise fully frustrated vortex pattern. These dilute defects result in similar behavior as that found in the dilute vortex system, with pinned, floating, and liquid defect phases.