Based on the solution of the stochastic Landau-Lifshitz-Gilbert equation discretized for a ferromagnetic chain subject to a uniform temperature gradient, we present a detailed numerical study of the spin dynamics with a focus particularly on finite-size effects. We calculate and analyze the net longitudinal spin current for various temperature gradients, chain lengths, and external static magnetic fields. In addition, we model an interface formed by a nonuniformly magnetized finite-size ferromagnetic insulator and a normal metal and inspect the effects of enhanced Gilbert damping on the formation of the space-dependent spin current within the chain. A particular aim of this study is the inspection of the spin Seebeck effect beyond the linear response regime. We find that within our model the microscopic mechanism of the spin Seebeck current is the magnon accumulation effect quantified in terms of the exchange spin torque. According to our results, this effect drives the spin Seebeck current even in the absence of a deviation between the magnon and phonon temperature profiles. Our theoretical findings are in line with the recently observed experimental results by M. Agrawal et al., Phys. Rev. Lett. 111, 107204 (2013).
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