Competitive 0 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>π</mml:mi></mml:math> states in S/F/S trilayers: Multimode approach

Abstract

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 single-mode 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-$\ensuremath{\pi}$ 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.