The theoretical investigation of the dissipation due to a pure spin current generated by spin pumping in a ferromagnetic/nonmagnetic bilayer is studied. The analytical solution of the dissipation is specifically derived. We show that the dissipation becomes zero when the spin diffusion length of the nonmagnet becomes sufficiently longer than its thickness. We also show that the second law of the thermodynamics is guaranteed.
Taniguchi, T., & Saslow, W. M. (2016). Heat production by diffusion of pure spin current. Journal of Magnetism and Magnetic Materials, 400, 168–170. https://doi.org/10.1016/j.jmmm.2015.07.041