Superconducting critical temperature and singlet and triplet pair functions of superconductor/normal-metal/ferromagnet trilayers

We calculate the superconducting critical temperature T_c, the singlet pair function Ψ⁺(x), and triplet pair function Ψ⁻(x) of superconductor/normal-metal/ferromagnet (S∕N∕F) trilayers using the linearized Usadel equation near T_c. The Green’s function method developed by Fominov et al. for the S∕F bilayers is extended to the S∕N∕F trilayer systems. The S of the trilayers is taken to be an s-wave singlet pairing superconductor, and the S∕N and N∕F interfaces are modeled in terms of the interface resistances parametrized, respectively, by γ_b^SN and γ_b^NF. We present the T_c, Ψ⁺(x), and Ψ⁻(x) for typical γ_b^SN, γ_b^NF, and the exchange energy E_ex: (a) For a small (large) γ_b^NF, T_c of S∕N∕F trilayers, as d_N is increased, increases (decreases) on the length scale of N coherence length ξ_N with a discontinuity at d_N=0 due to a boundary condition mismatch. (b) T_c(d_F) shows a nonmonotonic behavior like S∕F bilayers with a weakened shallow dip. (c) The odd frequency triplet component Ψ⁻(x), induced by E_ex and proximity effects, has a maximum near the N∕F interface and decreases on the length scale ξ_ex in F. It also penetrates into N and S regions on the length scale ξ_N and ξ_S, respectively. Based on these results we make comments on the experimental observation of the odd triplet components and the recent T_c measurements in Nb∕Au∕CoFe trilayer systems.