  Various forms of carbon based complexes in GaN are studied with
first-principles calculations employing Heyd-Scuseria-Ernzerhof hybrid
functional within the framework of density functional theory. We consider
carbon complexes made of the combinations of single impurities, i.e.
$\mathrm{C_N-C_{Ga}}$, $\mathrm{C_I-C_N}$ and $\mathrm{C_I-C_{Ga}}$, where
$\mathrm{C_N}$, $\mathrm{C_{Ga}}$ and $\mathrm{C_I}$ denote C substituting
nitrogen, C substituting gallium and interstitial C, respectively, and of
neighboring gallium/nitrogen vacancies ($\mathrm{V_{Ga}}$/$\mathrm{V_N}$), i.e.
$\mathrm{C_N-V_{Ga}}$ and $\mathrm{C_{Ga}-V_N}$. Formation energies are
computed for all these configurations with different charge states after full
geometry optimizations. From our calculated formation energies, thermodynamic
transition levels are evaluated, which are related to the thermal activation
energies observed in experimental techniques such as deep level transient
spectroscopy. Furthermore, the lattice relaxation energies (Franck-Condon
shift) are computed to obtain optical activation energies, which are observed
in experimental techniques such as deep level optical spectroscopy. We compare
our calculated values of activation energies with the energies of
experimentally observed C-related trap levels and identify the physical origins
of these traps, which are unknown before.
