Effects of magnetic fluctuation on 0-$\pi$ transition in a superconductor-ferromagnet-superconductor junction

There has been growing interest in a superconductor- ferromagnetic metal-superconductor (SFS) junction, in which the Josephson critical current, $I_{\rm c}$, shows a cusp as a function of thickness of ferromagnetic-layer, $d$, and/or temperature, $T$. Such a non-monotonous behavior, which is in marked contrast to $I_{\rm c}$ in a conventional Josephson junction, originates from the fact that the current-phase relation is shifted by $\pi$. This is called $\pi$-state. We study the influence of magnetic fluctuation on $I_{\rm c}$ in the SFS junction by a tunneling Hamiltonian approach. An analytical formula of $I_{\rm c}$ is given in the fourth order perturbation theory as regards the tunneling matrix element. Electrons propagate diffusively in the FM due to non-magnetic- and magnetic scatterings. The $I_{\rm c}$ exhibits the damped oscillatory dependence on $d$, and shows the transition between ${\it 0}$- and $\pi$-{\it states}. When the superconducting transition temperature is comparable to the ferromagnetic Curie temperature, the period of oscillation is elongated by increasing $T$ due to the magnetic fluctuation, which plays an important role in the $0$-$\pi$ transition, in particular, with $T$. Our results present an appropriate combination of a superconductor and a ferromagnetic metal to control the ${\it 0} $- and the $\pi$-{\it states}.