We present a microscopic theory of the proximity effect in the ferromagnet/superconductor/ferromagnet (F/S/F) nanostructures where S is an s-wave low-Tc superconductor and F's are layers of $3d$ transition ferromagnetic metal. Our approach is based on the direct analytical solution of Gor'kov equations for the normal and anomalous Green's functions together with a self-consistent evaluation of the superconducting order parameter. We take into account the elastic spin-conserving scattering of the electrons assuming s-wave scattering in the S layer and s-d scattering in the F layers. In accordance with previous quasiclassical theories, we found that due to exchange field in the ferromagnet the anomalous Green's function $F(z)$ exhibits the damping oscillations in the F layer as a function of distance z from the $S/F$ interface. In the given model, a half of the period of oscillations is determined by the length ξₘ⁰=πvF/εₑₓ, where vF is the Fermi velocity and εₑₓ is the exchange field, while damping is governed by the length l₀=(1/l_↑+1/l_↓)^-1, with l_↑ and l_↓ being spin-dependent mean free paths in the ferromagnet. The superconducting transition temperature Tc(dF) of the F/S/F trilayer shows the damping oscillations as a function of the F-layer thickness dF with period ξF=π/√mεₑₓ, where m is the effective electron mass. The oscillations of Tc(dF) are a consequence of the oscillatory behavior of the superconducting order parameter at the S/F interface vs thickness dF, which in turn is caused by the oscillations of $F(z)$ in the F region. We show that strong spin-conserving scattering either in the superconductor or in the ferromagnet significantly suppresses these oscillations. The calculated Tc(dF) dependences are compared with existing experimental data for Fe/Nb/Fe trilayers and Nb/Co multilayers.
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Bagrets et al. (2003) studied this question.