Theory of proximity effect in superconductor/ferromagnet heterostructures

Abstract

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/ɛex, where vF is the Fermi velocity and ɛex 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/mɛex, 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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