Structure and Magnetization of Two-Dimensional Vortex Arrays in the   Presence of Periodic Pinning
  Ground-state properties of a two-dimensional system of superconducting vortices in the presence of a periodic array of strong pinning centers are studied analytically and numerically. The ground states of the vortex system at different filling ratios are found using a simple geometric argument under the assumption that the penetration depth is much smaller than the spacing of the pin lattice. The results of this calculation are confirmed by numerical studies in which simulated annealing is used to locate the ground states of the vortex system. The zero-temperature equilibrium magnetization as a function of the applied field is obtained by numerically calculating the energy of the ground state for a large number of closely spaced filling ratios. The results show interesting commensurability effects such as plateaus in the B-H diagram at simple fractional filling ratios. 
Toby Joseph and Chandan Dasgupta