Spin Seebeck effect and phonon energy transfer in heterostructures containing layers of a normal metal and a ferromagnetic insulator

In the framework of the kinetic approach based on the Boltzmann equation for the phonon distribution function, we analyze phonon heat transfer in a heterostructure containing a layer of a normal metal ($ N $) and a layer of a ferromagnetic insulator ($ F $). Two realistic methods for creating a temperature gradient in such a heterostructure are considered: by heating of the $N$-layer by an electric current and by placing the $N/F$-bilayer between massive dielectrics with different temperatures. The electron temperature $ T_e $ in the $ N $-layer and the magnon temperature $ T_m $ in the $ F $-layer are calculated. The difference in these temperatures determines the voltage $ V_{ISHE} $ on the $ N $-layer in the Seebeck spin effect regime. The dependence of $ V_{ISHE} $ on the bath temperature and on the thickness of the $ N $ and $ F $ layers is compared with the available experimental data.