Multi-vortex State Induced by Proximity Effects in a Small Superconducting Square

The influence of the different negative values of the deGennes parameter (b) in the thermodynamic properties of a superconducting infinitely long prism of square cross section area (S=9xi ^{2}(0)) in the presence of a magnetic field is investigated theoretically by solving numerically the nonlinear Ginzburg-Landau equations; (xi (0)) is the coherent length at zero temperature. We obtain the vorticity, magnetic induction, Cooper pair density, magnetization and phase of the order parameter as functions of the external applied magnetic field and the (b) parameter. Our results show that a multi-vortex state appear in the sample choosing a convenient value of (b<0) parameter, even for such small system. Also, we study a superconducting parallelepiped of volume (V=Sd) by means of true (3D) numerical simulations; (d) is the height of the parallelepiped. We focused our analysis on the way which the magnetization curves approximate from (d) finite to the characteristic curve of (drightarrow infty ) . This is the case for which the magnetic field and the order parameter are invariant along (z) -direction. For a superconductor of size (S=9xi ^2(0)) we find that the limit below which the system should be considered a real three-dimensional sample when is (d=8xi ) .