Supercurrent transport in Josephson junctions with ferromagnetic trilayers

Abstract

We study the transport of the Josephson current in one-dimensional SF1F2F3S junctions, where S represents s-wave superconductors and F1, F2, and F3 denote ferromagnets with noncollinear magnetizations. The Josephson current displays distinct characteristics under various magnetization configurations as we vary the exchange fields (h1, h3) and thicknesses (d1, d3) of the F1 and F3 layers. Building on Eschrig's conversion mechanisms [Eschrig, Phys. Today 64(1), 43 (2011) and Eschrig, Rep. Prog. Phys. 78, 104501 (2015)], we rigorously derive the conversion and transport processes of spin-singlet and spin-triplet Cooper pairs as they traverse the ferromagnetic trilayers, which helps clarify the observed variations in the Josephson current. We demonstrate that opposite-spin Cooper pairs acquire an additional phase of Qjdj upon passing through each Fj layer (j = 1, 3). Here, Qj = 2hj/(ℏvF) represents the center-of-mass momentum of the Cooper pairs, where vF is the Fermi velocity. This result aligns with the description of these pairs provided in Eq. (73) of Eschrig [Rep. Prog. Phys. 78, 104501 (2015)]. The contributions of Qjdj to the Josephson current depend on the exchange field of the F2 layer and the magnetization directions of the F1 and F3 layers. When the F2 layer is a ferromagnet, the azimuthal angle difference (χ3 − χ1) between the F1 and F3 layers modulates the current amplitude. This angular difference has been previously discussed in Grein et al. [Phys. Rev. Lett. 102, 227005 (2009)] and Eschrig [Rep. Prog. Phys. 78, 104501 (2015)], and it has more recently been examined in detail in Schulz et al. [Phys. Rev. B 112, 104514 (2025) and Phys. Rev. B 112, 104515 (2025)]. In contrast, when the F2 layer is a half-metal, this angle difference induces a phase shift in the current without affecting its amplitude, consistent with the results presented in Eschrig and Löfwander [Nat. Phys. 4, 138 (2008)]. These findings open new possibilities for controlling long-range Josephson currents in practical applications.