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 Greens 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.
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