Length scale for magnon-polaron formation from nonlocal spin transport

We develop a theory for nonlocal spin transport through magnetic insulators that treats the coherent magnetoelastic interaction on equal footing with incoherent relaxation processes. In particular, our theory is able to describe the formation of magnon polarons, hybridized spin and elastic waves, near an interface where spin is injected into the magnetic insulator. Our theory is based on the stochastic Landau-Lifshitz-Gilbert equation coupled to stochastic equations of motion for the lattice displacement. By solving these equations, we obtain the charge voltage generated in a detector on one side of the magnetic insulator in response to spin biasing with an injector on the other side. We find that though magnon-polaron formation causes anomalous features in the spin transport, a length scale exists, however, below which magnetoelastic coupling does not affect the nonlocal spin current. This finding may motivate experiments to explore this aspect of magnon-phonon coupling in magnetic materials.