Transverse phase locking for vortex motion in square and triangular pinning arrays

We analyze transverse phase locking for vortex motion in a superconductor with a longitudinal dc drive and a transverse ac drive. For both square and triangular arrays we observe a variety of fractional phase-locking steps in the velocity versus dc drive which correspond to stable vortex orbits. The locking steps are more pronounced for the triangular arrays which is due to the fact that the vortex motion has a periodic transverse velocity component even for zero transverse ac drive. All the steps increase monotonically in width with ac amplitude. We confirm that the width of some fractional steps in the square arrays scales as the square of the ac driving amplitude. In addition we demonstrate scaling in the velocity versus applied dc driving curves at depinning and on the main step, similar to that seen for phase locking in charge-density-wave systems. The phase-locking steps are most prominent for commensurate vortex fillings where the interstitial vortices form symmetrical ground states. For increasing temperature, the fractional steps are washed out very quickly, while the main step gains a linear component and disappears at melting. For triangular pinning arrays we again observe transverse phase locking, with the main and several of the fractional step widths scaling linearly with ac amplitude.