Injection of spin current at the superconductor/ferromagnetic insulator interface

We used the Linear Response Theory to determine the injection of spin current at the interface between a ferromagnetic insulator (FMI) and a superconductor (SC). As quasi-particles (the excited states on the superconductor ground state) cannot penetrate the FMI, there is no electric current across the interface. However, a spin-flip scattering allows quasi-particles to exchange spin with localized electrons in the FMI via magnons. Assuming an imbalance delta mu = mu(up arrow) - mu(down arrow) between the up and down-spin in the SC, quasi-particles in the most populated state are reflected with spin inversion while magnons are absorbing or emitting at the interface. Depending on the value of delta mu, we obtained two different behaviors of the spin current in the phase transition from a superconducting to a normal metal phase. For delta mu < 1.20 Delta(0), Delta(0) is the superconducting gap at zero temperature, the transition is of the second order while for delta mu > 1.20 Delta(0) the spin current presents a discontinuous jump at Tc. The dependence on delta mu is also determined as well as the spin conductance.