Vortex configurations in a superconducting film with a square array of small antidots are studied within the Ginzburg-Landau (GL) theory. We find that in addition to the conventional vortex structures at the matching fields, a variety of vortex states can be stabilized by decreasing the pinning strength of the antidots, including (i) the triangular vortex lattice where some vortices are pinned by the antidots and others are located between them, (ii) vortex line structures, and (iii) a lattice of vortex cluster structures around the empty pinning centers. Although these partially pinned vortex structures are obtained more frequently in field cooled experiments than the square pinned vortex lattice, they are not the lowest energy states, i.e., the ground state, contrary to the results from a London approach. This result can be understood as due to the presence of a broad local minimum in the GL free energy which keeps the vortices away from the pinning centers. Our results can also be related to recent experiments on macroscopic metallic particles that move in a plane in the presence of a weak electrostatic pinning potential.
Berdiyorov, G. R., Milošević, M. V., & Peeters, F. M. (2007). Superconducting film with weak pinning centers: Incommensurate vortex lattices. Physical Review B, 76(13), 134508. https://doi.org/10.1103/PhysRevB.76.134508