Motivated by recent electron microscopy studies on the ${\mathrm{Si}}_3{\mathrm{N}}_4∕\text{rare}\text{−}\text{earth}$ oxide interfaces, the atomic and electronic structures of bare $\mathrm{<ruby><rb>\beta </rb><rt>べーた</rt></ruby>}\text{−}{\mathrm{Si}}_3{\mathrm{N}}_4$ surfaces are investigated from first principles. The equilibrium shape of a ${\mathrm{Si}}_3{\mathrm{N}}_4$ crystal is found to have a hexagonal cross section and a faceted domelike base in agreement with experimental observations. The large atomic relaxations on the prismatic planes are driven by the tendency of Si to saturate its dangling bonds, which gives rise to resonant-bond configurations or planar $s{p}^2$-type bonding. We predict three bare surfaces with lower energies than the open-ring $\left(10\overline{1}0\right)$ surface observed at the interface, which indicate that nonstoichiometry and the presence of the rare-earth oxide play crucial roles in determining the termination of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ matrix grains.

• Received 28 October 2005

DOI:https://doi.org/10.1103/PhysRevB.72.241301