Atomic and Electronic Structures of Oxygen on the $\beta-$Si$_3$N$_4$ $(10\overline{1}0)$ Surface*

The desirable mechanical and physical properties of Si$_3$N$_4$ ceramics in high temperature applications are hindered by their intrinsic brittleness. Doping Si$_3$N$_4$ with rare-earth oxides has long been known to overcome this limitation creating a tougher material. Precise information about the microscopic origin of this empirical observation has, however, been lacking for many years. In this study, we present {\em ab initio} calculations for the structural stability of $\beta-$Si$_3$N$_4$ $(10\overline{1}0)$ surfaces in the presence of different oxygen concentrations. Two different $(10 \overline{1}0)$ surface terminations, the ``open ring" and the ``half surface",\footnote{J. C. Idrobo {\em et al}., Phys. Rev. B {\bf 72}, 241301(R) (2005).} are investigated in detail using an asymmetric slab. We find that the Si-O bond plays the most important role in the structural stability and passivation of the surface. The theoretical results are analyzed in connection with recent electron microscopy studies on the interface.\footnote{A. Ziegler {\em et al.}, Science {\bf 306}, 1768 (2004); N. Shibata {\em et al.}, Nature {\bf 428}, 730 (2004); G. B. Winkelman {\em et al.}, Phil. Mag. Lett. {\bf 84}, 755 (2004).}