Mixed state and magnetization of a thin type II superconducting film in the applied parallel magnetic field: Variational technique of the inclusion of the contribution from the core of a vortex

The equilibrium vortex structure and magnetization for a thin superconducting film (with the Ginzburg-Landau parameter $\kappa\gg 1$) in the magnetic field applied parallel to the film surface have been calculated with the inclusion of the spatial inhomogeneity of the order parameter within the core of a vortex by using a variational function. The variational approach allows obtaining more accurate values of the core self-energy in comparison to those found in the London approximation (including its changes with the growth of the vortex lattice density) and gives a more adequate picture of the interactions between vortices and of their interaction with the surface. The formula determining a correction to the London expression for the magnetic flux related to vortices in the film is derived. The comparison of the calculations performed using these two models demonstrates that the vortex core structure provides an important contribution if the film thickness is $d<50\xi$. For such thin films, the novel size effect manifesting itself in a significant difference between the values of magnetization for the film and a bulk sample within the magnetic field range $0.08H_{c2}<H<0.5H_{c2}$ has been found. It has also been demonstrated that if $d$ decreases to $d=10\xi$, the splitting of the central vortex row occurs as a second order phase transition.