Creation of vortices in a superconducting film by ferromagnetic dots

We study the interaction between superconducting vortices and ferromagnetic dots with in-plane magnetization assuming that the proximity effect is suppressed. If the size of a ferromagnetic dot is large enough, the spontaneous vortices appear easily at the boundary of the dot. For a circular ferromagnetic dot with the diameter close to the lattice constant, the number of appearing vortices is necessarily even. We argue that the system of vortices has the same symmetry as the original array of ferromagnetic dots, whereas the symmetry may be broken spontaneously if the magnetization of the dots is perpendicular to the superconducting film. We calculate the phase diagrams of different vortex phases for the square array of circular and square dots.