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 $Psi^+(x)$, and triplet pair function $Psi^-(x)$ of superconductor/normal metal/ferromagnet (S/N/F) trilayers using the linearized Usadel equation near $T_c$. The Greens 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 parameterized, respectively, by $gamma_b^{SN}$ and $gamma_b^{NF}$. We present the $T_c$, $Psi^+(x)$, and $Psi^-(x)$ for typical $gamma_b^{SN}$, $gamma_b^{NF}$, and the exchange energy $ E_{ex}$: (a) For a small (large) $gamma_b^{NF}$, $T_c$ of S/N/F trilayers, as $d_N$ is increased, increases (decreases) on the length scale of N coherence length $xi_N$ with a discontinuity at $d_N=0$ due to a boundary condition mismatch. (b) $T_c(d_F)$ shows a non-monotonic behavior like S/F bilayers with a weakened shallow dip. (c) The odd frequency triplet component $Psi^-(x)$, induced by $E_{ex}$ and proximity effects, has a maximum near the N/F interface and decreases on the length scale $xi_{ex}$ in F. It also penetrates into N and S regions on the length scale $xi_N$ and $xi_S$, 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.