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Origin of coexisting large Seebeck coefficient and metallic conductivity in the electron doped SrTiO$_3$ and KTaO$_3$

We study the origin of the large Seebeck coefficient despite the metallic conductivity in the La-doped SrTiO$_3$ and Ba-doped KTaO$_3$. We calculate the band structure of SrTiO$_3$ and KTaO$_3$, from which the Seebeck coefficient is obtained using the Boltzmann's equation. We conclude that the multiplicity of the $t_{2g}$ bands in these materials is one major origin of the good thermoelectric property in that when compared at a fixed total number of doped electrons, the Seebeck coefficient and thus the power factor are larger in multiple band systems than in single band ones because the number of doped electron bands {\it per band} is smaller in the former. We also find that the second nearest neighbor hopping integral, which generally has negative values in these materials and works destructively against the Seebeck effect, is nearly similar between KTaO$_3$ and SrTiO$_3$ despite the larger band width in the former. This can be another factor favorable for thermopower in the Ba-doped KTaO$_3$.