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.
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