Thermal equilibration and thermally induced spin currents in a thin-film ferromagnet on a substrate

Abstract

Recent spin-Seebeck experiments on thin ferromagnetic films apply a temperature difference $\ensuremath{\Delta}{T}_{x}$ along the length $x$ and measure a (transverse) voltage difference $\ensuremath{\Delta}{V}_{y}$ along the width $y$. The connection between these involves: (1) thermal equilibration between sample and substrate, (2) spin currents along the height (or thickness) $z$, and (3) the measured voltage difference $\ensuremath{\Delta}{V}_{y}$. The present work models in detail the first of these steps, and outlines how to obtain the other two. In 1D, thermal equilibration between the magnons and phonons in the sample as well as additional equilibration between the sample and the substrate leads to two surface modes with lengths $\ensuremath{\lambda}$ to provide thermal equilibration. Increasing the coupling between the two modes increases the longer mode length and decreases the shorter mode length. In 2D, the applied thermal gradient along $x$ leads to a thermal gradient along $z$ that varies as $\mathrm{sinh}(x/\ensuremath{\lambda})$, which produces fluxes along $z$ of the up- and down-spin carriers, and gradients of their associated magnetoelectrochemical potentials ${\overline{\ensuremath{\mu}}}_{\ensuremath{\uparrow},\ensuremath{\downarrow}}$, which vary as $\mathrm{sinh}(x/\ensuremath{\lambda})$. There is also an infinite spectrum of shorter lengths $\ensuremath{\lambda}$ that are geometrically determined. By the inverse spin Hall effect, the spin current along $z$ can produce a transverse voltage difference $\ensuremath{\Delta}{V}_{y}$ that also varies as $\mathrm{sinh}(x/\ensuremath{\lambda})$. This is consistent with experiments if the longest $\ensuremath{\lambda}$ is comparable to or larger than the sample length $L$, and the shorter $\ensuremath{\lambda}$'s are smaller than the separation between the input or output lead and the nearest voltage probe. In this model, even seemingly linear voltage profiles are due to a surface mode.