Ratchet effects for moving elastic vortex matter interacting with periodic pinning

Driven vortex matter interacting with periodic pinning potentials lacking spatial inversion symmetry is studied using an elastic continuum model. The center of mass velocity is calculated as a function of the driving force, for time-independent forces (dc), using perturbation theory, valid for weak pinning, and for strong pinning at large force magnitudes. The results are applied to rectification of the velocity when the driving force oscillates in time (ac) at low frequencies. Rectification (ratchet effect) occurs when the dc velocity vs. force relationship is not antisymmetric under inversion of the force. It is found that, to first order in perturbation, the velocity is antisymmetric under force inversion, independent of the spatial symmetry of the potential. To second order in perturbation, the velocity is antisymmetric under force inversion for spatially symmetric potentials, and symmetric for spatially antisymmetric ones. The rectified velocity is calculated for typical spatially antisymmetric potentials, and for sinus or square waves ac forces. The rectified velocity is found to be in the “easy” direction of motion of the potential, and to fall off as the inverse ac force amplitude square at large amplitudes.