Phase transitions in the mesoscopic superconducting square

In this paper we investigate phase transitions in a superconducting film of square shape as a function of temperature T and external magnetic fieldH in the limit a;j!l eff .W e solve the GLE for thin superconducting films with thickness d!l. Our results are also valid for a type-two superconducting cylinder of square cross section, if the Ginzburg-Landau parameter k5l/j@1. We show that a configuration with one antivortex in the center and four vortices on the diagonals of the square is unstable when we move away from the Hc2 line and the nonlinear term in the GLE is considered. On the contrary, at higher magnetic field, the configuration with four vortices on diagonals of the square remains stable. Similar to the superconducting disk 6‐8 we find a sequence of phase transitions of the first order between giant vortex states as well as between multivortex states with different vorticity. Second-order phase transition takes place when a giant vortex state splits into a multivortex state while simultaneously breaking the C4 symmetry. Such transitions are discussed in terms of the phenomenological theory of Landau. The GLE for dimensionless order parameter c has the form