The interface between a ferro- or ferrimagnetic insulator and a normal metal can support spin currents polarized collinear with and perpendicular to the magnetization direction. The flow of angular momentum perpendicular to the magnetization direction (“transverse” spin current) takes place via spin torque and spin pumping. The flow of angular momentum collinear with the magnetization (“longitudinal” spin current) requires the excitation of magnons. In this article we extend the existing theory of longitudinal spin transport [Bender and Tserkovnyak, Phys. Rev. B 91, 140402(R) (2015)] in the zero-frequency weak-coupling limit in two directions: We calculate the longitudinal spin conductance nonperturbatively (but in the low-frequency limit) and at finite frequency (but in the limit of low interface transparency). For the paradigmatic spintronic material system YIG|Pt, we find that nonperturbative effects lead to a longitudinal spin conductance that is ca. 40% smaller than the perturbative limit, whereas finite-frequency corrections are relevant at low temperatures ≲100K only, when only few magnon modes are thermally occupied.
