The Cu induced ultraflat band in the room-temperature superconductor Pb$_{10-x}$Cu$_x$(PO$_4$)$_6$O$_4$ ($x=0,0.5$)

Based on the first principle calculations, we investigate the geometry and the electronic structures of the room temperature superconductor lead apatite with and without the Cu doped. Our calculations found that without the Cu doping the lead apatite crystal is an insulator with a flat band above the Fermi lever. Furthermore, our results indicate that by introducing the O1 vacancies would results in the disappear of the flat bands in the undopped crystal. While the Cu is doped forming the LK-99, there exist the ultra-flat bands crossing the Fermi level induced by the doped Cu atoms, which is due to the hybridization between the d states of the Cu atoms and the p states of the fully occupied 1/4 occupied O1 atoms. Moreover, we point out that the hybridization maybe the key for the realization of the room temperature superconductivity for the LK-99, and a possible solutions method to improve the quality of the superconductivity is proposed. Our work provides a foundation for the future studies on the unique electronic and the superconductivity properties of the LK-99 crystal.