Electronic structure of the $\rm Ca_3Co_4O_9$ compound from ab initio local interactions

We used fully correlated ab initio calculations to determine the effective parameters of Hubbard and t - J models for the thermoelectric misfit compound $\rm Ca_3Co_4O_9$. As for the $\rm Na_xCoO_2$ family the Fermi level orbitals are the $a_{1g}$ orbitals of the cobalt atoms ; the $e'_g$ being always lower in energy by more than 240\,meV. The electron correlation is found very large $U/t\sim 26$ as well as the parameters fluctuations as a function of the structural modulation. The main consequences are a partial $a_{1g}$ electrons localization and a fluctuation of the in-plane magnetic exchange from AFM to FM. The behavior of the Seebeck coefficient as a function of temperature is discussed in view of the ab initio results, as well as the 496\,K phase transition.