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Computational materials design of attractive Fermion system with large negative effective $U$ in the hole-doped Delafossite of CuAlO$_2$, AgAlO$_2$ and AuAlO$_2$

In order to realize super-high-critical temperature $(T_c)$ superconductors ($T_c$>1,000 K) based on general design rules for negative effective $U$ $(U_{eff})$ systems by controlling purely-electronic and attractive Fermion mechanisms, we perform computational materials design for the negative $U_{eff}$ system in hole-doped two-dimensional (2D) Delafossite CuAlO$_2$, AgAlO$_2$ and AuAlO$_2$ from ${\it ab\ initio}$ calculations. It is found that the large negative $U_{eff}$ in the hole-doped attractive Fermion systems for CuAlO$_2$ ($U_{eff}$ = -4.53 eV), AgAlO$_2$ ($U_{eff}$ = -4.88 eV), AuAlO$_2$ ($U_{eff}$ = -4.14 eV). These values are 10 times larger than that in hole-doped three-dimensional (3D) CuFeS$_2$ ($U_{eff}$ = -0.44 eV). For future calculations of the $T_c$ and phase diagram by quantum Monte Carlo simulations, we propose the negative $U_{eff}$ Hubbard model with the anti-bonding single $π$-band model for CuAlO$_2$, AgAlO$_2$ and AuAlO$_2$ by using the parameters obtained from ${\it ab\ initio}$ electronic structure calculations. The behavior of $T_c$ in the 2D Delafossite of CuAlO$_2$, AgAlO$_2$ and AuAlO$_2$ and 3D Chalcopyrite of CuFeS$_2$ shows the interesting chemical trend, ${\it i.e.,}$ $T_c$ increases exponentially in the weak coupling regime $|U_{eff}| < W$ ($\sim$ 2 eV) (where $W$ is the band width of Hubbard model) for the hole-doped CuFeS$_2$, and then $T_c$ goes through a maximum when $|U_{eff}| \sim W$ (2.8 eV, 3.5 eV) for the hole-doped AgAlO$_2$ and AuAlO$_2$, and finally $T_c$ decreases with increasing $|U_{eff}|$ in the strong coupling regime, where $|U_{eff}| > W$ (1.7 eV), for the hole-doped CuAlO$_2$. In this strong coupling regime, one can expect that $T_c$ = 1,000 $\sim$ 2,000 K by assuming the relation of the very strong coupling as $2Δ/ k_{\rm B}T_c$ = 50 $\sim$100 and the superconducting gap $Δ\sim |U_{eff}|$ = 4.53 eV $\sim$ 50,000 K.