Theory of pinning in a Superconducting Thin Film Pierced by a   Ferromagnetic Columnar Defect
  This is an analytical study of pinning and spontaneous vortex phase is a system consisting of a superconducting thin film pierced by a long ferromagnetic columnar defect of finite radius $R$. The magnetic fields, screening currents, energy and pinning forces for this system are calculated. The interaction between the magnetic field of vortices and the magnetization outside the plane of the film and its close proximity enhances vortex pinning significantly. Spontaneous vortex phase appears when the magnetization of the columnar defect is increased above a critical value. Transitions between phases characterized by different number of flux quanta are also studied. These results are generalized to the case when the superconductor is pierced by an array of columnar defects. 
M. Amin Kayali