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The recent claimed room-temperature superconductivity in Cu-doped leadapatite at ambient pressure are under highly debate. To identify its physicalorigin, we studied the crystal structures, energy band structures, latticedynamics 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. Ourresults show that the Pb$_{10}$(PO$_4$)$_6$O compound is an indirect band gapsemiconductor, where Cu doping at the 4$f$ site of Pb leads to a semiconductingto half-metallic transition. Two half-filled flat bands spanning the Fermienergy levels are present in the 4$f$-phase of LK-99, which are mainly formedby 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 thebottom of the conduction band and at the top of the valence band, making thematerial a bipolar magnetic semiconductor. Our results are basically consistentwith the recent experimental transport properties of LK-99 posted onarXiv:2308.05778.

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