  In materials science, point defects play a crucial role in materials
properties. This is particularly well known for the wide band gap insulators
where the defect formation/compensation determines the equilibrium Fermi level
and generally the doping response of a given material. Similarly, the main
defect trends are also widely understood for regular metals (e.g., Cu and Zn
discussed herein). With the development of electronic structure theory, a
unique class of quantum materials - gapped metals (e.g., Ca6Al7O16, SrNbO3,
In15SnO24, and CaN2) that exhibit characteristics of both metals and insulators
- has been identified. While these materials have internal band gaps similar to
insulators, their Fermi level is within one of the main band edges, giving a
large intrinsic free carrier concentration. Such unique electronic structures
give rise to unique defect physics, where the formation of acceptor or donor
defect directly affects not only the electronic structure but also can
substantially shift the Fermi level. Motivated by this, herein, we develop a
fundamental first-principles theory of defect formation in gapped metal with a
primary focus on accurate calculations of defect formation energy. We
demonstrate that due to electron-hole recombination, the formation of acceptor
defects in n-type gapped metals results in significant dependence of defect
formation energy on supercell size, which we explain by the change of band
filling and its effect on the defect formation energetics. To accurately
describe defect formation energy, we revisit the phenomenology of band-filling
corrections and demonstrate the effect of this correction, accurate potential
alignment, and other factors that can affect defect energetics. Thus, this work
not only sheds light on the intrinsic properties of gapped metals but, in
general, establishes a theoretical foundation for analyzing defects in gapped
metals.
