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Spin Seebeck effect and magnon diffusion length in $\rm{\mathbf{Fe}}_{\mathbf{3}}\rm{\mathbf{O}}_{\mathbf{4}}$

The determination of the magnon diffusion length (MDL) is important for increasing the efficiency of spin Seebeck effect (SSE) based devices utilising non-metallic magnets. We extract the MDL at $50$ and $300\,\rm{K}$ in an $\rm{Fe}_{3}\rm{O}_{4}$ single crystal from the magnon dispersion obtained using inelastic neutron scattering (INS) and find them to be equal within error. We then measure the heat flux normalised SSE responses and in-plane magnetization of $\rm{Fe}_{3}\rm{O}_{4}$ thin films and normalise by the static magnetization contribution to the SSE before determining the MDLs from a fit of the thickness dependence. We find that the MDLs determined in this way are smaller than that measured from INS which maybe due to differences in magnon propagation between bulk and thin film $\rm{Fe}_{3}\rm{O}_{4}$.