Two Exponentials Associated with Temperature in Spin-Seebeck Effect Geometry

Recent experiments report the observation of a Spin-Seebeck effect, where an applied thermal gradient along (x) a very thin (z), narrow (y) ferromagnetic sample F is associated with a spin current.\footnote{K. Uchida et al, Nature 455, 778 (2008).} In present geometries this spin current is measured indirectly via a Pt bar above (z) the sample; a voltage difference $V$ along y is measured and interpreted as being due to a spin current $j_s$ into (z) the Pt, which then causes an inverse Spin Hall effect ($j_s$ causes transverse $V$). Measured voltages often show a $\sinh(x/s)$ dependence, where $s$ is long compared to any relevant spin-diffusion length.\footnote{C. M. Jaworski et al, Nature Materials 9, 898 (2010).} The spin current has been interpreted as accompanying a temperature disequilibrium between the phonons and magnons in F.\footnote{J. Xiao et al, Phys. Rev. B 81, 214418 (2010).} The present work uses irreversible thermodynamics to include magnon-phonon equilibration in F and the thermal properties of the (non-magnetic) substrate S. We find two exponentials describing the overall thermal response along x, the second one associated with equilibration between F and S. If the thermal coupling between F and S is poor, then the second length can be rather long.