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Th$_3$Te$_4$ materials are potential candidates for commercial thermoelectric(TE) materials at high-temperature due to their superior physical properties.We incorporate the multiband Boltzmann transport equations with firstprinciplescalculations to theoretically investigate the TE properties of Th$_3$Te$_4$materials. As a demonstration of our method, the TE properties of La$_3$Te$_4$are similar with that of Ce$_3$Te$_4$ at low-temperature, which is consistentwith the experiment. Then we systematically calculate the electricalconductivity, the Seebeck coefficient, and the power factor of the twomaterials above based on parameters obtained from first-principles calculationsas well as several other fitting parameters. Our results reveal that for theelectron--optical-phonon scattering at high temperatures, a linear dependenceof optical phonon energy on temperature explains better the experimentalresults than a constant optical phonon energy. Based on this, we predict thatthe TE properties of Ce$_3$Te$_4$ is better than La$_3$Te$_4$ at hightemperatures and the optimal carrier concentration corresponding toCe$_3$Te$_4$ shifts upward with increasing temperature. The optimal carrierconcentration of Ce$_3$Te$_4$ is around $1.6\times10^{21}$cm$^{-3}$ with thepeak power factor 13.07 $\mu$Wcm$^{-1}$K$^{-2}$ at $T=1200$K.

High-temperature thermoelectric properties with Th$_{3-x}$Te$_4$