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Pudding-mold type band in a potential thermoelectric material CuAlO$_2$ : comparison with Na$_x$CoO$_2$

A potential thermoelectric material CuAlO$_2$ is theoretically studied. We first construct a model Hamiltonian of CuAlO$_2$ based on the first principles band calculation, and calculate the Seebeck coefficient. Then, we compare the model with that of a well-known thermoelectric material Na$_x$CoO$_2$, and discuss the similarities and the differences. It is found that the two materials are similar from an electronic structure viewpoint in that they have a peculiar pudding-mold type band shape, which is advantageous as thermoelectric materials. There are however some differences, and we analyze the origin of the difference from a microscopic viewpoint. The band shape of CuAlO$_2$ is found to be even more ideal than that of Na$_x$CoO$_2$, and we predict that once a significant amount of holes is doped in CuAlO$_2$, thermoelectric properties (especially the power factor) even better than that of Na$_x$CoO$_2$ can be expected.