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

Abstract

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 Green's function method developed by Fominov 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 gamma(SN)(b) and gamma(NF)(b). We present the T-c, Psi(+)(x), and Psi(-)(x) for typical gamma(SN)(b), gamma(NF)(b), and the exchange energy E-ex: (a) For a small (large) gamma(NF)(b), 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 nonmonotonic 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.