  Thermal-bias-induced spin angular momentum transfer between a paramagnetic
metal and ferromagnetic insulator is studied theoretically based on the
stochastic Landau-Lifshitz-Gilbert (LLG) phenomenology. Magnons in the
ferromagnet establish a nonequilibrium steady state by equilibrating with
phonons via bulk Gilbert damping and electrons in the paramagnet via spin
pumping, according to the fluctuation-dissipation theorem. Subthermal magnons
and the associated spin currents are treated classically, while the appropriate
quantum crossover is imposed on high-frequency magnetic fluctuations. We
identify several length scales in the ferromagnet, which govern qualitative
changes in the dependence of the thermally-induced spin current on the magnetic
film thickness.
