Reentrant Fulde-Ferrell-Larkin-Ovchinnikov state in small-sized superconductors

We study the effect of a parallel magnetic field in a thin and small superconductor. The field suppresses superconductivity through Zeeman coupling while stabilizing the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state at high fields before superconductivity is destroyed. When the spatial period of FFLO state is comparable to the size of the superconductor, there is a strong commensuration effect, which modifies the superconducting phase diagram. We investigate the FFLO state and the phase diagram in the presence of strong commensuration effect both for the $s$- and $d$-wave superconductors using the Bogoliubov--de Gennes equation, Green function approach, and Ginzburg-Landau theory. We found that the superconducting phase diagram is strongly modulated. Interestingly, there is reentrance of superconductivity upon increasing the magnetic field. The commensuration effect of the FFLO state can be used to detect the FFLO state in experiments.