  In this study, using first-principles calculations, we investigate the
behavior of electrons at the SiC/SiO$_2$ interface when nitrogen is introduced
as a dopant within a few nm of the SiC surface. When a highly doped nitrogen
layer (5$\times$10$^{19}$ cm$^{-3}$) is introduced within a few nm of the
SiC(11$\bar{2}$0) surface, the electronic state is not significantly affected
if the doping region is less than 4 nm. However, if the doping region exceeds 4
nm, the effect of quantum confinement decreases, which increases the electron
density induced in the inversion layer. As for the wave function, even when an
electric field is applied, the peak shifts toward the direction in which the
electrons are pulled away from the interface. This reduces the effect of
electron scattering at the interface and improves electron mobility.
