Effective nonlinear susceptibility of a grained superconducting film

The effective nonlinearity of a thin grained superconducting film is calculated theoretically. It is assumed that the film has the form of a continuous superconducting matrix with superconducting grained inclusions of cylindrical structure with a generatrix perpendicular to the film surface. The current and field distributions are determined to calculate the effective nonlinearity; both the matrix and the grains are considered in the framework of the Ginzburg-Landau theory. The spacing between the grains is assumed to be much larger than their diameter so that the Maxwell-Garnet approximation can be used for calculating fields. The analytic formula derived for effective nonlinear susceptibility is analyzed for various values of physical parameters. It is shown that the effective nonlinearity exceeds the nonlinearity of the matrix and grains by a factor of several units when the grain size is much larger than the magnetic field penetration depth in the substance of the matrix and the conductivity of the grains is lower than that of the matrix.