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Recent spin-Seebeck experiments on thin ferromagnetic films apply atemperature difference $\Delta T_{x}$ along the length $x$ and measure a(transverse) voltage difference $\Delta V_{y}$ along the width $y$. Theconnection between these effects is complex, involving: (1) thermalequilibration between sample and substrate; (2) spin currents along the height(or thickness) $z$; and (3) the measured voltage difference. The present workstudies in detail the first of these steps, and outlines the other two steps.Thermal equilibration processes between the magnons and phonons in the sample,as well as between the sample and the substrate leads to two surface modes,with surface lengths $\lambda$, to provide for thermal equilibration.Increasing the coupling between the two modes increases the longer mode lengthand decreases the shorter mode length. The applied thermal gradient along $x$leads to a thermal gradient along $z$ that varies as $\sinh{(x/\lambda)}$,which can in turn produce fluxes of the carriers of up- and down- spins along$z$, and gradients of their associated \textit{magnetoelectrochemicalpotentials} $\bar{\mu}_{\uparrow,\downarrow}$, which vary as$\sinh{(x/\lambda)}$. By the inverse spin Hall effect, this spin current along$z$ can produce a transverse (along $y$) voltage difference $\Delta V_y$, whichalso varies as $\sinh{(x/\lambda)}$.

Thermal Equilibration and Thermally-Induced Spin Currents in a Thin-Film Ferromagnet on a Substrate