Individual and Multi Vortex Pinning in Systems with Periodic Pinning Arrays

We examine multi and individual vortex pinning in thin superconductors with periodic pinning arrays. For multi-vortex pinning we observe peaks in the critical current of equal magnitude at every matching field, while for individual vortex pinning we observe a sharp drop in the critical current after the first matching field in agreement with experiments. We examine the scaling of the critical current at commensurate and incommensurate fields for varied pinning strength and show that the depinning force at incommensurate fields decreases faster than at the commensurate fields.