Absolute surface energies of polar and non-polar planes in GaN

Growth of high quality single crystals and epitaxial layers of GaN is very important for producing optoelectronic devices. {\it Ab initio} calculations can help in determining absolute surface energies, which are key quantities that control crystal-growth rates and fracture toughnesses. By means of hybrid functional calculations, we have determined absolute surface energies for the non-polar $\{11{\bar 2}0\}$ and $\{10{\bar 1}0\}$ and polar $(0001)$ and $(000{\bar 1})$ planes in wurtzite GaN. Low energy reconstructions of the bare and hydrogenated surfaces were considered under various conditions chosen to correspond to growth by molecular beam epitaxy (MBE) or metal-organic chemical vapor deposition (MOCVD). We find that the non-polar planes are close in energy, and lower in energy than the reconstructed $(000{\bar 1})$ polar plane under all conditions considered. The reconstructed $(0001)$ plane is lower in energy than the $(000{\bar 1})$ plane over the whole range of conditions, and lower in energy than the non-polar reconstructions for Ga-rich chemical potential conditions. From these surface energies, lower bounds on the anisotropic fracture toughness of GaN are determined. Surface energies of polar planes for other III-nitrides will be compared to those of GaN.