  Sodium cobaltate Na_xCoO_2 as dopable strongly correlated layered material
with a triangular sublattice still poses a challenging problem in condensed
matter. The intriguing interplay between lattice, charge, spin and orbital
degrees of freedom leads to a complex phase diagram bounded by a nominal Mott
(x=0) regime and a band-insulating (x=1) phase. By means of the charge
self-consistent density functional theory (DFT) plus dynamical mean-field
theory (DMFT) scheme, built on a pseudopotential framework combined with a
continuous-time quantum Monte-Carlo solver, we here study the one-particle
spectral function A(k,\omega) as well as the thermopower S(T). The computations
may account for the suppression of the e_g' pockets in A(k,\omega) at lower
doping in line with photoemission experiments. Enhancement of the thermopower
is verified within the present elaborate multi-orbital method to treat
correlated materials. In addition, the two-particle dynamic spin susceptibility
\chi_s(\omega,q) is investigated based on a simplified tight-binding approach,
yet by including vertex contributions in the DMFT linear response. Besides the
identification of paramagnon branches at higher doping, a prominent high-energy
antiferromagnetic mode close to x=0.67 is therewith identified in
\chi_s(\omega,q), which can be linked to extended hopping terms on the CoO_2
sublattice.
