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Thermal spin pumping mediated by magnon in the semiclassical regime

We microscopically analyze thermal spin pumping mediated by magnons, at the interface between a ferromagnetic insulator and a non-magnetic metal, in the semiclassical regime. The generation of a spin current is discussed by calculating the thermal spin transfer torque, which breaks the spin conservation law for conduction electrons and operates the coherent magnon state. Inhomogeneous thermal fluctuations between conduction electrons and magnons induce a net spin current, which is pumped into the adjacent non-magnetic metal. The pumped spin current is proportional to the temperature difference. When the effective temperature of magnons is lower than that of conduction electrons, localized spins lose spin angular momentum by emitting magnons and conduction electrons flip from down to up by absorbing all the emitted momentum, and vice versa. Magnons at the zero mode cannot contribute to thermal spin pumping because they are eliminated by the spin-flip condition. Consequently thermal spin pumping does not cost any kinds of applied magnetic fields. We have discussed the distinction from the theory proposed by Xiao et al. [Phys. Rev. B, 81 (2010) 214418], Adachi et al. [Phys. Rev. B, 83 (2011) 094410], and Bender et al. [arXiv:1111.2382].