Interplay of paramagnetic, orbital, and impurity effects on the phase transition of a normal metal to the superconducting state

We derive the generalized Ginzburg-Landau free-energy functional for conventional and unconventional singlet superconductors in the presence of paramagnetic, orbital, and impurity effects. Within the mean-field theory, we determine the criterion for the appearance of the nonuniform (Fulde-Ferrell-Larkin-Ovchinnikov) superconducting state, with vortex lattice structure and additional modulation along the magnetic field. We also discuss the possible change of the order of transition from a normal to a superconducting state. We find that the superconducting phase diagram is very sensitive to geometrical effects such as the nature of the order parameter and the shape of the Fermi surface. In particular, we obtain the qualitative phase diagrams for three-dimensional isotropic s-wave superconductors and in quasi-two-dimensional d-wave superconductors under magnetic field perpendicular to the conducting layers. In addition, we determine the criterion for instability toward a nonuniform superconducting state in s-wave superconductors in the dirty limit.