Nonmonotonic critical temperature in superconductorÕferromagnet bilayers

The critical temperature Tc of a superconductor/ferromagnet ~SF! bilayer can exhibit nonmonotonic dependence on the thickness d f of the F layer. SF systems have been studied for a long time; according to the experimental situation, a ‘‘dirty’’ limit is often considered which implies that the mean free path in the layers is the second smallest spatial scale after the Fermi wavelength. However, all calculations reported for the dirty limit were done with some additional assumptions, which can be violated in actual experiments. Therefore, we develop a general method ~to be exact, two independent methods! for investigating Tc as a function of the bilayer parameters in the dirty case. Comparing our theory with experiment, we obtain good agreement. In the general case, we observe three characteristic types of Tc(d f) behavior: ~1! a nonmonotonic decay of Tc to a finite value exhibiting a minimum at particular d f ; ~2! a reentrant behavior, characterized by a vanishing of Tc in a certain interval of d f and finite values otherwise; and ~3! a monotonic decay of Tc and a vanishing at finite d f . Qualitatively, the nonmonotonic behavior of Tc(d f) is explained by the interference of quasiparticles in the F layer, which can be either constructive or destructive depending on the value of d f .