Oxygen-Induced Reconstructions on the $\beta-$Si$_3$N$_4$ $(10\overline{1}0)$ Surfaces

Motivated by recent electron microscopy studies at the Si$_3 $N$_4$/rare-earth oxide (REO) interfaces, we present first principles calculations for the preferred bonding sites and configurations of oxygen on various terminations of the $\beta- $Si$_3$N$_4$ $(10\overline{1}0)$ surface as a function of coverage and surface stoichiometry. We predict that oxygen induces various surface reconstructions, and it has a strong tendency to replace N on the surface. The structural stability of most low-energy surfaces is driven by the tendency of Si to saturate its dangling bonds and of oxygen to bridge two Si atoms similar to the bonding in SiO$_2$. The present {\em ab initio} results resolve the discrepancy between the experimental observations at the Si$_3$N$_4$/REO interfaces and previous theoretical studies\footnote{Juan C. Idrobo {\em et al.}, Phys. Rev. B {\bf 72}, 241301(R) (2005).} for bare surfaces regarding the lowest energy surface termination.