Title: Meta-GGAs vs. Hybrid Functionals for Point Defects: The Best of Both Worlds Applied to Layered MnO$_2$, NiO$_2$ and KCoO$_2$

Abstract: Defects in a material can significantly tune its properties and enhance its utility. Hybrid functionals like HSE06 are often used to describe solids with defects. However, geometry optimization using hybrid functionals (e.g., HSE06), often used to describe solids with defects, is challenging for a large supercell, as needed for defect study. The proposed r$^2$SCAN+rVV10+U+U$_d$ method, which is computationally much cheaper and faster than hybrid functionals, can successfully describe defects in materials with the proper choice of U (for the d orbitals of the host atom) and U$_d$ (for those of the defect atom), as shown here for small polarons in layered transition-metal oxides. We use a literature value of U or U$_d$ appropriate to a given transition-metal ion and its oxidation state. The materials MnO$_2$ and NiO$_2$, with one K atom intercalated between layers in a supercell, are found to have one localized occupied e$_g$ state on the transition metal ion that takes an electron from the K atom, when the geometry is calculated as above, for standard U values but not for U=U$_d$=0. K-intercalated KCoO$_2$ is surprisingly different, due to a dramatic change of electronic configuration of the defected Co$^{+2}$ ion.