  We report the density-functional calculations that systematically clarify the
stable forms of carbon-related defects and their energy levels in amorphous
SiO$_2$ using the melt-quench technique in molecular dynamics. Considering the
position dependence of the O chemical potential near and far from the
SiC/SiO$_2$ interface, we determine the most abundant forms of carbon-related
defects: Far from the interface, the CO$_2$ or CO in the internal space in
SiO$_2$ is abundant and they are electronically inactive; near the interface,
the carbon clustering is likely and a particular mono-carbon defect and a
di-carbon defect induce energy levels near the SiC conduction-band bottom, thus
being candidates for the carrier traps.
