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Tunneling Conductance and Spin Transport in Clean Ferromagnet-Ferromagnet-Superconductor Heterostructures

We present a transfer matrix approach that combines the Blonder-Tinkham-Klapwijk (BTK) formalism and self-consistent solutions to the Bogolibuov-de Gennes (BdG) equations and use it to study the tunneling conductance and spin transport in ferromagnet (${\rm F}$)-superconductor (${\rm S}$) trilayers (${\rm F_1F_2 S}$) as functions of bias voltage. The self-consistency ensures that the spin and charge conservation laws are properly satisfied. We consider forward and angularly averaged conductances over a broad range of the strength of the exchange fields and ${\rm F}$ thicknesses, as the relative in-plane magnetization angle, $ϕ$, between the two ferromagnets varies. The $ϕ$-dependence of the self-consistent conductance curves in the trilayers can differ substantially from that obtained via a non-self-consistent approach. The zero bias forward conductance peak exhibits, as $ϕ$ varies, resonance effects intricately associated with particular combinations of the geometrical and material parameters. We find, when the magnetizations are non-collinear, signatures of the anomalous Andreev reflections in the subgap regions of the angularly averaged conductances. When ${\rm F_1}$ is half-metallic, the angularly averaged subgap conductance chiefly arises from anomalous Andreev reflection. The in-plane components of the spin current are strongly bias dependent, while the out-of-plane spin current component is only weakly dependent upon voltage. The components of the spin current aligned with the local exchange field of one of the F layers are conserved in that layer and in the S region, while they oscillate in the other layer. We compute the spin transfer torques, in connection with the oscillatory behavior of spin currents, and verify that the spin continuity equation is strictly obeyed in our method.