  In this study, we perform a comparative theoretical study on the
thermoelectric performance of materials with Cu$Ch_4$ ($Ch=$ S, Se) tetrahedra,
including famous thermoelectric materials BiCuSeO and tetrahedrite
Cu$_{12}$Sb$_4$S$_{13}$, by means of first-principles calculations. By
comparing these electronic band structures, we find that many of these
materials possess a Cu-$t_{2g}$ band structure consisting of
quasi-one-dimensional band dispersions and the isotropic (two-dimensional for
layered compounds) band dispersion near the valence-band edge. Therefore, the
key factors for the thermoelectric performance are the anisotropy of the former
band dispersion and the degeneracy of these two kinds of band dispersions. We
also find that a large extension of the chalcogen orbitals often improves their
thermoelectric performance by improving these two factors or by going beyond
such a basic band structure through a large alternation of its shape. Such a
large extension of the chalcogen orbitals might partially originate from the
anisotropic Cu-$Ch$ bond geometry of a tetrahedron. Our study reveals
interesting similarities and differences of materials with Cu$Ch_4$, which
provides important knowledge for a future search of high-performance
thermoelectric materials.
