Static and dynamic interaction of superconducting vortices with a periodic pinning potential

A model is developed which describes the static and dynamic interaction of the vortex lattice with the pinning potential due to harmonic one-dimensional variations of the thickness of superconducting films. Using a London-Pearl approach, an expression is derived for the free energy of the mixed state in thin modulated films from which static equilibrium configurations of the vortex lattice and the corresponding deformations due to the harmonic pinning force are deduced. Lattice configurations of lowest energy are found when the magnetic induction B corresponds to matching of the (undistorted) triangular vortex lattice to the periodic pinning structure. Particular attention is devoted to lattice configurations slightly deviating from a matching situation, for which long-wavelength transverse deformations are found predominant, and to Bragg configurations, where short-wavelength deformations of wave vector →k=(1/2)→g₁ (→g₁ is a nearest-neighbor reciprocal lattice vector) propagate along a high-symmetry direction of the triangular lattice. We estimate the critical current density, jc(B) for these lattice configurations. A matching peak in the jc(B)-curve is found to have approximately a resonantike shape whose width depends on the shear modulus of the lattice and on the strength of the elementary interaction of a single flux line with the pinning potential. The flux-flow regime is also investigated when steady-state vortex motion results from the equilibrium of the Lorentz driving force, the viscous damping force, the harmonic pinning force, and the pinning-induced lattice restoring force. A highly coherent flux-flow regime with rf properties similar to those of series arrays of resistively shunted Josephson junctions acting in phase and frequency coherence (super-radiant state) is found in the dynamic matching state. For nonmatching configurations we study the influence of dynamically excited lattice deformation modes on the current-voltage characteristics when vortex motion is driven by dc or by superimposed dc and rf transport currents. In particular, for nearly matching configurations the width of the rf-induced interference transitions is related to the shear modulus of the vortex lattice and to the flux-flow resistivity.