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

Abstract

We calculate the superconducting critical temperature Tc, 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 Tc. 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 γbSN and γbNF. We present the Tc, Ψ+(x), and Ψ−(x) for typical γbSN, γbNF, and the exchange energy Eex: (a) For a small (large) γbNF, Tc of S∕N∕F trilayers, as dN is increased, increases (decreases) on the length scale of N coherence length ξN with a discontinuity at dN=0 due to a boundary condition mismatch. (b) Tc(dF) shows a nonmonotonic behavior like S∕F bilayers with a weakened shallow dip. (c) The odd frequency triplet component Ψ−(x), induced by Eex and proximity effects, has a maximum near the N∕F interface and decreases on the length scale ξex in F. It also penetrates into N and S regions on the length scale ξN and ξS, respectively. Based on these results we make comments on the experimental observation of the odd triplet components and the recent Tc measurements in Nb∕Au∕CoFe trilayer systems.

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