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The influence of an asymmetric in-plane magnetic anisotropy on the thermallyactivated spin current is studied theoretically for two different systems; (i)the system consisting of a ferromagnetic insulator in a direct contact with anonmagnetic metal, and the sandwich structure consisting of a ferromagneticinsulating part sandwiched between two nonmagnetic metals. It is shown thatwhen the difference between the temperatures of the two nonmagnetic metals in astructure is not large, the spin pumping currents from the magnetic part to thenonmagnetic ones are equal in amplitude and have opposite directions, so onlythe spin torque current contributes to the total spin current. The spin currentflows then from the nonmagnetic metal with the higher temperature to thenonmagnetic metal having a lower temperature. Its amplitude varies linearlywith the difference in temperatures. In addition, we have found that if themagnetic anisotropy is in the layer plane, then the spin current increases withthe magnon temperature, while in the case of an out-of-plane magneticanisotropy the spin current decreases when the magnon temperature enhances.Enlarging the difference between the temperatures of the nonmagnetic metals,the linear response becomes important, as confirmed by analytical expressionsinferred from the Fokker-Planck approach and by the results obtained upon afull numerical integration of the stochastic Landau-Lifshitz-Gilbert equation.

Magnon-driven longitudinal spin Seebeck effect in F|N and N|F|N structures: role of asymmetric in-plane magnetic anisotropy