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

Abstract

Extraordinary Hall effect (EHE) in bilayer F/S(N) was investigated theoretically. The conductivity tensor $σ_{αβ}$ 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 $E^H$ 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 $R_s$ for F/S case. The value of the Hall constant is $R_H^{F/S} = R_H^{bulk} (σ^{\uparrow} + σ^{\downarrow})^2 / 4 σ^{\uparrow}σ^{\downarrow}$, where $σ^{\uparrow}$ and $σ^{\downarrow}$ are conductivities of electrons with up and down spins, and $R_H^{bulk}$ is the Hall constant in the bulk ferromagnetic metal. In fact, $R_H^{F/S}$ 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.