Nonmonotonic critical temperature in superconductor/ferromagnet bilayers

The critical temperature T_c 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, the "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 T_c as a function of the bilayer's parameters in the dirty case. Comparing our theory with experiment, we obtain good agreement. In the general case, we observe three characteristic types of T_c(d_f) behavior: 1) nonmonotonic decay of T_c to a finite value exhibiting a minimum at particular d_f, 2) reentrant behavior, characterized by vanishing of T_c in a certain interval of d_f and finite values otherwise, 3) monotonic decay of T_c and vanishing at finite d_f. Qualitatively, the nonmonotonic behavior of T_c(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.