  This work examines the carbon defects associated with recently reported and
novel peaks of infrared (IR) absorption and Raman scattering appearing in GaN
crystals at carbon ($^{12}C$) doping in the range of concentrations from
$3.2*10^{17}$ to $3.5*10^{19} cm^{-3}$. 14 unique vibrational modes of defects
are observed in GaN samples grown by hydride vapor phase epitaxy (HVPE) and
then compared with defect properties predicted from first-principles
calculations. The vibrational frequency shift in two $^{13}C$ enriched samples
related to the effect of the isotope mass indicates six distinct configurations
of the carbon-containing point defects. The effect of the isotope replacement
is well reproduced by the density functional theory (DFT) calculations.
Specific attention is paid to the most pronounced defects, namely tri-carbon
complexes($C_N=C=C_N$) and carbon substituting for nitrogen $C_N$. The position
of the transition level (+/0) in the bandgap found for $C_N=C=C_N$ defects by
DFT at 1.1 eV above the valence band maximum, suggest that $(C_N=C=C_N)^+$
provides compensation of ${C_N}^-$. $C_N=C=C_N$ defects are observed to be
prominent, yet have high formation energies in DFT calculations. Regarding
${C_N}$ defects, it is shown that the host Ga and N atoms are involved in the
defect's delocalized vibrations and significantly affect the isotopic frequency
shift. Much more faint vibrational modes are found from di-atomic carbon-carbon
and carbon-hydrogen (C-H) complexes. Also, we note changes of vibrational mode
intensities of $C_N$, $C_N=C=C_N$, C-H, and $C_N-C_i$ defects in the IR
absorption spectra upon irradiation in the defect-related UV/visible absorption
range. Finally, it is demonstrated that the resonant enhancement of the Raman
process in the range of defect absorption above 2.5 eV enables the detection of
defects at carbon doping concentrations as low as $3.2*10^{17} cm^{-3}$.
