Computational materials design of negative effective $U$ system in the hole-doped Delafossite of CuAlO$_{2}$, AgAlO$_{2}$ and AuAlO$_{2}$

In order to realize the super-high-$T_{\mathrm{C}}$ superconductors ($T_{\mathrm{C}}$\textgreater 1,000K) based on the general design rules [1] for the negative $U_{eff}$ system, we have performed computational materials design for the$ U_{eff} $\textless 0 system in the hole-doped two-dimensional (2D) Delafossite CuAlO$_{2}$, AgAlO$_{2}$ and AuAlO$_{2}$ from the first principles. We find the interesting chemical trend of $T_{\mathrm{C}}$ in 2D and 3D systems; where the $T_{\mathrm{C}}$ increases exponentially in the weak coupling regime (\textbar $U_{eff}$ (-0.44eV)\textbar \textless $W$(2eV), $W$ is the band width) for hole-doped CuFeS$_{\mathrm{2}}$ [2], then the $T_{\mathrm{C}}$ goes through a maximum when \textbar $U_{eff}$ (-4.88eV, -4.14eV)\textbar $\approx \quad W$(2.8eV, 3.5eV) for hole-doped AgAlO$_{\mathrm{2}}$ and AuAlO$_{\mathrm{2}}$, and the $T_{\mathrm{C}}$ decreases with increasing \textbar $U_{eff}$\textbar in strong coupling regime, where \textbar $U_{eff}$ (-4.53eV)\textbar \textgreater $W$(1.7eV) for hole-doped CuAlO$_{2}$ \\[4pt] [1] H. Katayama-Yoshida et al., \textit{Appl. Phys. Express, }\textbf{1 }081703, 2008.\\[0pt] [2] T. Fukushima et al., \textit{J. Phys. Condens. Matter, }\textbf{26 }355502, 2014.