Nucleation of superconductivity in a mesoscopic rectangle

We have studied the nucleation of superconductivity in a mesoscopic rectangle. We used an analytical gauge transformation for the vector potential A which gives A(n) = 0 for the normal component along the boundary of the rectangle. Consequently, the linearized Ginzburg-Landau equation is reduced to an eigenvalue problem in the basis set of functions obeying the Neumann boundary condition. Through the application of this technique we are able to accurately determine the field-temperature superconducting phase boundary together with the corresponding vortex patterns. A range of aspect ratios for the rectangle has been investigated and compared with a superconducting square (aspect ratio = 1) and with a superconducting line (aspect ratio = infinity). This also allows us to determine the stability of the vortex patterns with an anti-vortex in the centre, which have been predicted for a superconducting square, with respect to the deformation of the square.