  We report static and dynamic first-principles calculations that provide
atomistic pictures of the initial stage of the oxidation processes occurring at
the ($11\bar20$) surface of 4H-SiC. Our results unveil reaction pathways and
their associated free-energy barriers for the adsorption of oxygen and the
desorption of carbon atoms. We find that oxygen adsorption shows structural
multi-stability and that the surface-bridge sites are the most stable and
crucial sites for subsequent oxidation. We find that an approaching O$_2$
molecule is adsorbed, then dissociated and finally migrates toward these
surface-bridge sites with a free-energy barrier of 0.7 eV at the ($11\bar20$)
surface. We also find that a CO molecule is desorbed from the metastable
oxidized structure upon the overcoming of a free-energy barrier of 2.4$\sim$2.6
eV, thus constituting one of the annihilation process of C during the
oxidation. The results of the CO molecule desorption on the ($11\bar20$)
surface are compared with the ($000\bar1$) surface. A catalytic effect of
dangling bonds at the surface, causing a drastic reduction of the CO desorption
energy, is found on the ($000\bar1$) surface and the microscopic picture of the
effect is ascribed to an electron transfer from the Si to C dangling bonds. The
intrinsic ($11\bar20$) surface does not show this catalytic effect, and this is
because the surface consists of an equal amount of Si and C dangling bonds and
the electron transfer occurs before the desorption.
