{\it Ab initio} calculations of BaTiO$_3$, PbTiO$_3$ and SrTiO$_3$ (001) and (011) surfaces

We present results of calculations of surface relaxations and rumplings\footnote{R. I. Eglitis and D. Vanderbilt, Phys. Rev. B {\bf 76}, 155439 (2007).} for the (001) and (011) surfaces of ATiO$_3$ perovskites (A=Ba, Pb, and Sr) using a hybrid B3PW description of exchange and correlation. On the (001) surfaces, we consider both AO and TiO$_2$ terminations. In the former case, the surface AO layer is found to relax inward for all three materials, while outward relaxations of all atoms in the second layer are found for both kinds of (001) terminations. The surface relaxation energies of AO and TiO$_2$ terminations are found to be comparable with each other for all three materials. For (011) surfaces, we consider terminations on a TiO layer, an A layer, or an O layer. The surface relaxation energies for BaTiO$_3$, PbTiO$_3$ and SrTiO$_3$ (011) surfaces for all terminations are considerable larger than for (001) surfaces. Among the (011) surfaces, the relaxation energy is much larger for the TiO-terminated surface than for the Ba- or Pb-terminated surfaces for the BaTiO$_3$ and PbTiO$_3$ perovskites. We predict a considerable increase in the Ti-O chemical bond covalency near the (011) surfaces as compared to both the bulk and the (001) surfaces.