  The reports of room-temperature superconductivity in Cu-doped Pb-apatite draw
intense interests and debates. Herein, based on the density functional theory,
we show that the Cu-insertion in Pb-apatite is thermodynamically stable and
further carrier doping can convert the system CuPb$_{10}$(PO$_4$)$_6$O into
metal. The electric conduction is mainly along the one-dimensional (1D) Cu-O
chains in the c-axis (out-of-plane). The calculated conductance along the
c-axis is larger than the in-plane ones by 2 magnitude orders, indicating the
1D conduction behavior. Moreover, the 1D Cu-O chain is an anti-ferromagnetic
Mott insulator at zero-doping point due to the super-exchange. Further
electron/hole-doping will erase the anti-ferromagnetism. Therefore, the
Cu-inserted system CuPb$_{10}$(PO$_4$)$_6$O show transport and magnetic
features similar to the cuprate superconductors. On the other hand, the
O-insertion can also induce the metallic states, in which the conductance along
the out-of-plane direction is higher than the in-plane direction by 6-folds.
Our results display the metallization of Pb$_{10}$(PO$_4$)$_6$O via
Cu/O-insertions, and suggesting the electric conductions along the c-axis might
dominate the transport behaviors.
