Dimensional effect in the AC vortex response in thin superconducting films: exact solution in a weak DC magnetic field

The surface impedance of thin superconducting films placed in a DC magnetic field was calculated by using a vortex dynamics model, which includes linear pinning restoring force (due to interaction with the film surface) and viscous drag force. Special account was made for dimensional effects in arrangement and AC dynamics of Abrikosov vortices, which are not considered in the well known Coffey-Clem and Brandt models. For the case of the DC magnetic field parallel to the film surface, the vortex ensemble inside the film exists in the form of separate vortex rows. The number of such rows changes discretely with the external DC magnetic field. The pinning constant (Labusch parameter) for vortex oscillations in the applied AC field is determined mainly by interaction of vortex rows with their images and strongly depends on the film thickness and on the magnetic field value. The "staircase" structure of the dependence of the surface impedance on the DC magnetic field is obtained. It corresponds to a discrete change of the number of the vortex rows inside the film. It was also shown that character of the dependence changes drastically with the frequency in the vicinity of the "depinning frequency".