An investigation of coexistence between superconductivity and ferromagnetism in ferromagnetic superconductors

In this paper, we investigate the coexistence between superconductivity and ferromagnetism in a ferromagnetic superconductor (FS) where FS is in antiparallel cooper pairing state and parallel cooper pairing state. By using spin generalized Bogoliubov de Gennes (SBdG) equations, we derive the superconducting order parameter of FS as a function of temperature (T) and exchange field (h) of FS. We find that for some values of T and h the order parameter behaves as a multi-value function, so to determine the physical value of order parameter we calculate the difference in thermodynamic potential between superconducting and normal states, and we present a phase diagram of an FS in T-h plane which separates the superconducting region from the normal region. As a result, we deduce that the non-zero order parameter is not sufficient for having superconducting state and it is necessary to calculate the difference in thermodynamic potential of normal and superconducting states. In other word, in some points of T and h, although there are non-zero order parameters, since in these points the thermodynamic potential of the normal state is lower than the superconducting state, the system falls into the normal state.