  In a number of current experiments in the field of spin-caloritronics a
temperature gradient across a nanostructured interface is applied and
spin-dependent transport phenomena are observed. However, a lack in the
interpretation and knowledge let it unclear how the temperature drop across a
magnetic nanostructured interface looks like where both phonons and electrons
may contribute to thermal transport. We answer this question for the case of a
magnetic tunnel junction (MTJ) where the tunneling magneto Seebeck effect
occurs. Nevertheless, our results can be extended to other nanostructured
interfaces as well. Using an \textit{ab initio} method we explicitly calculate
phonon and electron thermal conductance across the Fe/MgO/Fe-MTJs by using
Green's function method. Further, by estimating the electron-phonon interaction
in the Fe leads we are able to calculate the electron and phonon temperature
profile across the Fe/MgO/Fe-MTJ. Our results show that there is an
electron-phonon temperature imbalance at the Fe-MgO interfaces. In consequence,
a revision of the interpretation of current experimental measurements might be
necessary.
