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## Competitive 0 and π states in S/F/S trilayers: Multimode approach

We investigate the behavior of the critical temperature T_c in superconductor/ferromagnet/superconductor (S/F/S) trilayers in the dirty limit as a function of the ferromagnetic layer thickness d_f and the S/F interface transparency. We perform T_c calculations using the general self-consistent multimode approach based on the Usadel equations in Matsubara Green’s functions technique, and compare the results with the singlemode approximation, widely used in literature. Both methods produce similar results for sufficiently low interface transparency. For transparent interfaces, we obtain a qualitatively different T_c(d_f) behavior. Using the multimode approach, we observe multiple 0-π transitions in critical temperature, which cannot be resolved by the single-mode approximation. We also calculate the critical S layer thickness at given d_f when an S/F/S trilayer still has a nonzero critical temperature. Finally, we establish the limits of applicability of the single-mode approximation.