  The electronic transport and spincaloric properties of epitaxial magnetic
tunnel junctions with half-metallic Co$_2$MnSi Heusler electrodes, MgO
tunneling barriers, and different interface terminations are investigated by
using first-principles calculations. A new approach to spincaloric properties
is presented that circumvents the linear response approximation inherent in the
Seebeck coefficient and compared to the method of Sivan and Imry. This approach
supports two different temperatures in the two electrodes and provides the
exact current and/or voltage response of the system. Moreover, it accounts for
temperature-dependent chemical potentials in the electrodes and finite-bias
effects. We find that especially the former are important for obtaining
qualitatively correct results, even if the variations of the chemical
potentials are small. It is shown how the spincaloric properties can be
tailored by the choice of the growth conditions. We find a large effective and
spin-dependent Seebeck coefficient of $-65$ $\mu$V/K at room temperature for
the purely Co-terminated interface. We suggest to use such interfaces in
thermally operated magnetoresistive random access memory modules, which exploit
the magneto-Seebeck effect, to maximize the thermally induced readout voltage.
