Role of nitrogen vacancies and related complexes in compensation and luminescence of Mg-doped GaN

Using first-principles calculations with the hybrid functional method (HSE), we investigate the effects of nitrogen vacancies and related complexes on the electrical and optical properties of Mg-doped GaN. We obtain information about the expected defect concentration, stable charge states, and defect levels by calculating the formation energies of vacancies and Mg$-$vacancy complexes. The $3+$ state of the nitrogen vacancy and the $2+$ state of the complex are found to be most stable when the Fermi level is near the valence-band maximum (VBM). Our calculations also enable us to study the role of these defects in luminescence. Vacancy-dopant complexes (including Mg$_{\rm Ga}$$-$$V_{\rm N}$) have been proposed as the origin of a deep level involved in the red (1.8 eV) photoluminescence (PL) band often observed in Mg$-$doped GaN. We investigate the optical absorption and emission energies by calculating the configuration coordinate diagram for the vacancy and for the Mg$_{\rm Ga}$$-$$V_{\rm N}$ complex. The emission, in which an electron in the conduction band is transferred to (Mg$_{\rm Ga}$$-$$V_{\rm N})^{2+}$, resulting in (Mg$_{\rm Ga}$$-$$V_{\rm N})^+$, peaks at 1.81 eV. Our calculated emission lines thus indicate that Mg$_{\rm Ga}$$-$$V_{\rm N}$ is a likely source for the red luminescence observed in Mg-doped GaN.