Electronic theory for itinerant in-plane magnetic fluctuations in $\mathrm{Na_xCoO_2}$

Starting from ab-initio band structure for $\mathrm{Na_xCoO_2}$, we derive the single-electron energies and the effective tight-binding description for the $t_{2g}$ bands using a projection procedure. We find that due to the presence of the next-nearest-neighbor hoppings a local minimum in the electronic dispersion close to the $\Gamma$ point of the first Brillouin zone forms. Therefore, in addition to a large Fermi surface an electron pocket close to the $\Gamma$ point emerges at high doping concentrations. The latter yields the new scattering channel resulting in a peak structure of the itinerant magnetic susceptibility at small momenta. This indicates itinerant in-plane ferromagnetic state above certain critical concentration $x_m$, in agreement with neutron scattering data. Below $x_m$ the magnetic susceptibility shows a tendency towards the antiferromagnetic fluctuations. We estimate the value of $0.56<x_m<0.68$ within the rigid band model and within the Hubbard model with infinite on-site Coulomb repulsion consistent with the experimental phase diagram.