We performed first-principles calculations of the spontaneous polarization and electronic band structures in Scx AlyGa1-x-yN alloys assuming their growth on freestanding GaN. We found an apparent deviation from the Vegard's law of the lattice constants of the Scx AlyGa 1-x-yN alloys lattice-matched to GaN. It was supposed that this deviation comes from the different bonding properties of IIIB and IIIA nitrides, resulting in different crystal structures, such as hexagonal and wurtzite structures. As was reported in our previous report on Yx AlyGa 1-x-yN [K. Shimada et al.: J. Appl. Phys. 110 (2011) 074114], we also found that in Scx AlyGa1-x-yN alloys, the superlattice-like structure of Sc atoms reduced the magnitude of spontaneous polarization [K. Shimada et al.: Semicond. Sci. Technol. 27 (2012) 105014]. The magnitude of the spontaneous polarization of ScxAlyGa 1-x-yN is larger than that of Yx AlyGa1-x-yN in a wide mole fraction range of Ga. We found the nonlinearity and dependence of the atomic arrangement of Sc in the alloys. The band-gap energies at have the same characteristics as the spontaneous polarization. The band-gap energies of Scx AlyGa1-x-yN are also larger than those of Yx Al yGa1-x-yN in the wide mole fraction range of Ga. The band structures of Scx AlyGa1-x-yN have a direct gap at and form a flat band around the valence band top originating from the hybridization of the Sc 3d and N 2p electrons.
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