Theory of magnetic 3d transition metal dopants in gallium nitride

Using first-principles density functional theory (DFT) methods and size-converged supercell models, we analyze the electronic and atomic structure of magnetic $3d$ transition metal dopants in cubic gallium nitride (c-GaN). All stable defect charge states for Fermi levels across the full experimental gap are computed using a method that correctly resolves the boundary condition problem (without a jellium approximation) and eliminates finite-size errors. The resulting computed defect levels are not impacted by the DFT band-gap problem, they span a width consistent with the experimental gap rather than being limited to the single-particle DFT gap. All defects with electronically degenerate (half-metal) ${T}_{d}$ ground states are found to have significant distortions, relaxing to ${D}_{2d}$ structures driven by the Jahn-Teller instability. This leads to insulating ground states for all substitutional $3d$ dopants, refuting claims in the literature that $+U$ or hybrid functional methods are required to avoid artificial half-metal results. Interpreting the ${d}^{n}$ atomic occupations within a crystal-field model and exchange splittings, we identify a systematic trend across the $3d$ transition metal series. Approaches to estimate excited-state energies as observed in photoluminescence from defect centers are assessed, ranging from a Koopmans-type single-particle energy interpretation to relaxed total energy differences in fully self-consistent DFT. The single-particle interpretations are found to be qualitatively predictive and the calculations are consistent with the limited available experimental data across the $3d$ dopant series. These results provide a baseline understanding to guide future studies and a conceptual framework within which to interpret new results.