  Based on DFT calculations, we present the basic electronic structure of
CuPb9(PO4)6O (Cu-doped lead apatite, LK-99), in two scenarios: (1) where the
structure is constrained to the P3 symmetry and (2) where no symmetry is
imposed. At the DFT level, the former is predicted to be metallic while the
latter is found to be a charge-transfer insulator. In both cases the filling of
these states is nominally d9, consistent with the Cu2+ valence state, and Cu
with a local magnetic moment ~0.7mB. In the metallic case we find these states
to be unusually flat (0.2 eV dispersion), giving high DOS at EF that we argue
can be a host for novel electronic physics, including potentially high
temperature superconductivity. The flatness of the bands is the likely origin
of symmetry-lowering gapping possibilities that would remove the spectral
weight from EF. Since some experimental observations show
metallic/semiconducting behavior, we propose that disorder is responsible for
closing the gap. We consider a variety of possibilities that could possibly
close the gap, but limit consideration to kinds of disorder that preserve
electron count. For all possibilities we considered (spin disorder, O on
vacancy sites, Cu on different Pb sites), the local Cu moment, and consequently
the gap remains robust. We conclude that disorder responsible for metallic
behavior entails some kind of doping where the electron count changes. We claim
that the emergence of the flat bands should be due to weak wave function
overlap between the Cu and O orbitals, owing to the directional character of
the constituent orbitals. So, finding an appropriate host structure for
minimizing hybridization between Cu and O while allowing them to still weakly
interact should be a promising route for generating flat bands at EF which can
lead to interesting electronic phenomena, regardless of whether LK-99 is a
room-temperature superconductor.
