We found a strong influence of the composition of the magnetic material on the temperature dependence of the tunneling magneto-Seebeck effect in $MgO$ based tunnel junctions. We use textit{ab initio} alloy theory to consider different $Fe_xCo_{1-x}$ alloys for the ferromagnetic layer. Even a small change of the composition leads to strong changes in the magnitude or even in the sign of the tunneling magneto-Seebeck effect. This can explain differences between recent experimental results. In addition, changing the barrier thickness from six to ten monolayers of $MgO$ leads also to a non-trivial change of the temperature dependence. Our results emphasize that the tunneling magneto-Seebeck effect depends very crucially and is very sensitive to material parameters and show that further experimental and theoretical investigations are necessary.
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