Upper critical field and critical temperature in superconductor/ferromagnet sandwiches

Abstract

A theory of proximity effect for superconductor/ferromagnet sandwiches is developed. We show the equivalence between the Usadel transport-like differential equations and the Takahashi-Tachiki theory, which is based on the de Gennes correlation function method. Moreover the generalized Usadel equations, including the pair breaking effects such as orbital diamagnetism, Pauli spin paramagnetism, spin-orbit scattering and magnetic impurity scattering, are obtained for the first time. The method for solving the problem in the multimode method is proposed. The critical temperature and the perpendicular upper critical magnetic field of ferromagnet/ superconductor/ ferromagnet trilayers are calculated as a function of material parameters, such as the layer thickness, the coherence length and the normal state conductivity, as well as the mutual orientation of ferromagnetic moments. Therefore we treat the role of finite transparency at boundary interfaces and the arbitrary exchange energy, which account for both weak and strong ferromagnets. It is found that the dependence of the critical temperature, Tc , over the ferromagnetic layer thickness, df , exhibits various types of Tc(df) behavior; (i) the monotonic decay of Tc, superconductivity completely vanishes at a particular df, (ii) the nonmonotonic decay of Tc, superconductivity existing throughout the ferromagnet but Tc has a minimum value at a particular df, and (iii) the reentrant superconductivity occurs only in the parallel phase, Tc drops to zero at finite df and it restores again. Numerical results show that the antiparallel configuration enhances superconductivity than the parallel one. Investigations of the perpendicular upper critical magnetic field reveal that the low interface boundary resistance has influence only in the antiparallel phase due to the weak suppression character of the ferromagnetic exchange fields.