Electrodynamics of a vortex array in a periodic planar pinning potential

Abstract

Electrodynamics of a vortex array is investigated in a periodic planar pinning potential (called here the superlattice) in a perpendicular dc magnetic field B. In this system a maximum of ac losses is found at a certain field ${B}_{m},$ in contrast to point-defect pinning or collective pinning where the ac loss is a monotonic function of B. Dependences of ${B}_{m}$ on the distance between the pinning planes, l, and the ac frequency are presented and the effect of orientation of the ac field is discussed. In all the considered cases, ${B}_{m}$ is found to be proportional to the local part of the tilt modulus ${c}_{44}$ of the flux line lattice. In a superlattice with l much larger than the London penetration depth \ensuremath{\lambda}, a crossover from two-dimensional (2D) to three-dimensional (3D) behavior occurs when the dc magnetic field is increased. In the 3D regime only one long-wave mode is important for a description of the ac response while in the 2D regime the whole spectrum of the eigenmodes must be taken into account. Only the 3D regime is realized in a superlattice having $l<~\ensuremath{\lambda}.$ In the high-field 3D regime the field dependence of the superconducting electron density is weaker in the vortex array in the planar pinning potential $(\ensuremath{\propto}{1/B}^{1/2})$ than in the point-defect pinning potential $(\ensuremath{\propto}1/B).$ This is found to be responsible for the anomalous field dependence of the ac loss.