Different phase leads to different transport behavior in Pb$_9$Cu(PO$_4$)$_6$O compounds

The recent claimed room-temperature superconductivity in Cu-doped lead apatite at ambient pressure are under highly debate. To identify its physical origin, we studied the crystal structures, energy band structures, lattice dynamics and magnetic properties of the parent Pb$_{10}$(PO$_4$)$_6$O compound, in which two different phases of the LK-99 compound are analyzed in detail. Our results show that the Pb$_{10}$(PO$_4$)$_6$O compound is an indirect band gap semiconductor, where Cu doping at the 4$f$ site of Pb leads to a semiconducting to half-metallic transition. Two half-filled flat bands spanning the Fermi energy levels are present in the 4$f$-phase of LK-99, which are mainly formed by hybridization of the $d_{x^2-y^2}$ and $d_{zy}$ orbitals of Cu with the 2$p$ orbitals of O. In addition, 6$h$-phase of LK-99 always has spin polarity at the bottom of the conduction band and at the top of the valence band, making the material a bipolar magnetic semiconductor. Our results are basically consistent with the recent experimental transport properties of LK-99 posted on arXiv:2308.05778.