Extraordinary Hall effect in hybrid ferromagnetic/superconductor (F/S) bilayer

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Extraordinary Hall effect (EHE) in bilayer F/S(N) was investigated theoretically. The conductivity tensor σαᵦ is calculated in the Kubo formalism with Green functions found as the solutions of the Gorkov equations. We considered diffuse transport in the ferromagnetic layer, taking into account as a main mechanism of electron resistivity s-d scattering. In this model Gorkov equations for s-electrons in the ferromagnetic layer remain linear and are solved easily. It is shown that Hall field Eʰ for both F/S and F/N contacts are step-functions of the coordinate perpendicular to the planes of the layers and have zero value in S(N) layer. The Andreev reflection increases the value of Hall constant Rₛ for F/S case. The value of the Hall constant is Rₕᶠ/ˢ = Rₕbulk (σ + σ)² / 4 σσ, where σ and σ are conductivities of electrons with up and down spins, and Rₕbulk is the Hall constant in the bulk ferromagnetic metal. In fact, Rₕᶠ/ˢ coincides with EHE constant of the bilayer of two ferromagnetic metals with equal thickness and opposite directions of their magnetizations. So we can make a conclusion, that the ideal interface between ferromagnetic metal and superconductor may be considered like a mirror with inversion in spin space.