  We present a theory of the spin Seebeck effect driven by subthermal non-local
phonon heat transfer and spectral non-uniform temperature distribution. The
theory explains the non-local behavior of the effect arising from the fact that
phonons that store the energy (thermal) and the phonons that transfer it
(subthermal) are located in different parts of the spectrum and have very
different kinetics. This gives rise to a spectral phonon distribution function
that deviates from local equilibrium along the substrate and is sensitive to
boundary conditions. The theory also predicts a non-magnon origin of the effect
in ferromagnetic metals in agreement with observations in recent experiments.
Equilibration of the heat flow out of the substrate to the Pt probe and
backwards leads to a measurable vertical spin-current produced by the spin
polarized electrons dragged by the local thermal phonons. We predict specific
sample length limits and other dependencies that can be probed experimentally,
and obtain the correct magnitude of the effect.
