Systematic band-filling dependence in the strongly-correlated triangular lattice

Strongly correlated triangular lattices are of interest because of their applicability to real systems such as superconducting organic charge-transfer solids (CTS) and the hydrated sodium cobaltate. Experimental work on nonhydrated Na$_x$CoO$_2$ have found (a) a charge-ordered semiconductor at $x=0.5$, (b) Curie-Weiss metal for $x>0.5$, and (c) paramgnetic metal for $x<0.5$. The strong $x$-dependence is reminescent of the band-filling dependence in CTS conductors \footnote{S. Mazumdar and A. N. Bloch, Phys. Rev. Lett. \textbf{50}, 207 (1983.} We have performed numerical calculations based on the extended Hubbard Hamiltonian to understand the $x$-dependence. We show that for {\it finite} Hubbard $U$, and moderate nearest neighbor interaction $V$, the normalized probability of double occupancy, which is a measure of the strength of the electron correlation, varies strongly as a function of the density of carriers. We are able to explain (i) the different behavior of $x<0.5$ and $x>0.5$, (ii) the absence of the $\sqrt{3} \times \sqrt{3}$ charge-ordering at $x=1/3$, and (iii) why $x=0.5$ is unique. Cobalt valence of 3.5 in the superconducting hydrated cobaltate is in agreement with our proposed mechanism of superconductivity in the CTS. \footnote {S. Mazumdar and R. T. Clay, Phys. Rev. B 77, 180515 (R), (2008).}