Size effects of critical temperatures and order parameter distributions under four-fold symmetry of nano-structured superconductors

We investigate size and shape dependences of critical temperatures T_c's and order parameter distributions for nano-structured superconductors. We use the finite element method to solve the Gor'kov equations. We find that order parameter distributions hold four-fold symmetry for square superconductors. It is different from order parameter distributions of rectangular superconductors, which have two-fold symmetry. We show that the difference comes from symmetry of the superconductor and degeneracy of states of Cooper pairs, by investigating size dependences of order parameter distributions and those of eigen-energies of quasi-particles.