The London Study of Vortex States in a Superconducting Film Due to a   Magnetic Dot
  Here we report a study of vortex states in a thin superconducting film with a magnetic dot grown upon it by means of a method based on London-Maxwell equations. Vortices with single quantum flux ($\Phi_0 = h c / 2 e$), giant vortices (vortices with multiple flux) as well as antivortices (vortices with negative vorticity) are taken into consideration. It turns out that giant vortices occur, when the dot's size is sufficiently smaller ($R \leq 4.5 \xi $) than the effective penetration depth $\Lambda$. In the case of a dot with sufficiently large size ($R \geq 6 \xi $), the vortices with single quantum flux dominate the vortex states. Their geometrical patterns are predicted up to seven vortices. Our calculations do not show the spontaneous appearance of antivortices. 
Serkan Erdin