Charge-order on the triangular lattice: Effects of next-nearest-neighbor attraction in finite temperatures

The extended Hubbard model in the atomic limit, which is equivalent to lattice $S=1/2$ fermionic gas, is considered on the triangular lattice. The model includes onsite Hubbard $U$ interaction and both nearest-neighbor ($W_{1}$) and next-nearest-neighbor ($W_{2}$) density-density intersite interactions. The variational approach treating the $U$ term exactly and the $W_l$ terms in the mean-field approximation is used to investigate thermodynamics of the model and to find its finite temperature ($T>0$) phase diagrams (as a function of particle concentration) for $W_{1}>0$ and $W_{2}<0$. Two different types of charge-order (i.e., DCO and TCO phases) within $\sqrt{3} \times \sqrt{3}$ unit cells as well as the nonordered (NO) phase occur on the diagram. Moreover, several kinds of phase-separated (PS) states (NO/DCO, DCO/DCO, DCO/TCO, and TCO/TCO) are found to be stable for fixed concentration. Attractive $W_{2}<0$ stabilizes PS states at $T=0$ and it extends the regions of their occurrence at $T>0$. The evolution of the diagrams with increasing of $|W_{2}|/W_{1}$ is investigated. It is found that some of the PS states are stable only at $T>0$. Two different critical values of $|W_{2}|/W_{1}$ are determined for the PS states, in which two ordered phases of the same type (i.e., two domains of the DCO or TCO phase) coexist.