  The influence of an asymmetric in-plane magnetic anisotropy on the thermally
activated spin current is studied theoretically for two different systems; (i)
the system consisting of a ferromagnetic insulator in a direct contact with a
nonmagnetic metal, and the sandwich structure consisting of a ferromagnetic
insulating part sandwiched between two nonmagnetic metals. It is shown that
when the difference between the temperatures of the two nonmagnetic metals in a
structure is not large, the spin pumping currents from the magnetic part to the
nonmagnetic ones are equal in amplitude and have opposite directions, so only
the spin torque current contributes to the total spin current. The spin current
flows then from the nonmagnetic metal with the higher temperature to the
nonmagnetic metal having a lower temperature. Its amplitude varies linearly
with the difference in temperatures. In addition, we have found that if the
magnetic anisotropy is in the layer plane, then the spin current increases with
the magnon temperature, while in the case of an out-of-plane magnetic
anisotropy 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 expressions
inferred from the Fokker-Planck approach and by the results obtained upon a
full numerical integration of the stochastic Landau-Lifshitz-Gilbert equation.
