Magnons versus electrons in thermal spin transport through metallic interfaces

We develop a theory for spin transport in magnetic metals that treats the contribution of magnons and electrons on equal footing. As an application we consider thermally-driven spin injection across an interface between a magnetic metal and a normal metal, i.e., the spin-dependent Seebeck effect. We show that the ratio between magnonic and electronic contribution scales as $\sqrt{T/T_C}T_F/T_C$, with the Fermi temperature $T_F$ and the Curie temperature $T_C$. Since, typically, $T_C \ll T_F$, the magnonic contribution may dominate the thermal spin injection, even though the interface is more transparent for electronic spin current.