Critical currents and electric fields of two-dimensional systems

We calculate the critical current of a thin, superconducting film with periodically modulated thickness placed in a perpendicular magnetic field. This problem is shown to be closely connected with the problem of the commensurate-incommensurate phase transition in two dimensions. The main mechanism responsible for the vortex motion appears to be the creation of a soliton in a vortex lattice on the boundary of a sample. The critical current is zero in the incommensurate phase. An analogous phenomenon is the critical longitudinal field for the Wigner crystal of electrons on the surface of liquid helium. The shear modulus for an Abrikosov vortex lattice in a thin film is found for a wide range of parameters.