Comparative studies on total energetics of nonequivalent hexagonal polytypes for group IV semiconductors and group III n
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We report the results of the systematic investigation into correlations between energetics and hexagonal stacking configurations for carbon, silicon, SiC, BN, AlN, GaN, and InN polytypes with sp3-bonded networks. The atomistic geometry, energetics, and electronic structure for these compounds with up to the periodic stacking length of L 5 8 have been carefully calculated based on the density functional theory within the generalized gradient approximation (GGA). Using the Axial Next-Nearest-Neighbor Ising model extracted from the GGA calculations, we have also studied the energetics for more than 6 million kinds of nonequivalent stacking polytypes with up to L 5 30, whose configurations have been deduced by the efficient polytype generation algorithm [E. Estevez-Rams and J. Martinez-Mojicar, Acta Crystallogr., Sect. A: Found. Crystallogr. 64, 529 (2008)], and illustrated some trends of structural and energetic properties for these compounds. I. INTRODUCTION
Some group IV semiconductors (carbon, silicon, and SiC) and group III–V compounds crystallize in a cubic and/or a hexagonal form and exhibit polytypism, in which the nearest neighbor atoms are tetrahedrally coordinated and have sp3 hybrid characters. The polytypes are characterized by a stacking sequence with a given repeat unit along the hexagonal c-axis direction and are theoretically possible to have endless permutations of the sequences.1 Among the polytypes for these elements and compounds, the silicon carbide (SiC) systems have been the most attractive and motivated tremendous amounts of theoretical and experimental investigations of their fundamental and technological properties for many years.2–9 The SiC polytypes differ only in the stacking order of double layers of carbon (C) and silicon (Si), and more than 200 polytypes have experimentally been determined to date.10 From the theoretical view, the problem of mathematically generating all nonequivalent polytypes for a given length (L) shares common problems with the counting problem. Some mathematicians and/or crystallographers have ardently been challenging this issue.1 Because of the exponential explosion with L, smart counting procedures that avoid the necessity of generating all the sequences have been necessary to effectively count the number of nonequivalent polytypes for a given length of L. Estevez-Rams and Martinez-Mojicar1 have recently presented an efficient polytype generation algorithm and have released a computational source code generating the hk notations11 a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2012.206 J. Mater. Res., Vol. 28, No. 1, Jan 14, 2013
for nonequivalent polytypes on the International Union of Crystallography electronic archives, so that we can use their method conveniently. Using the first-principles calculations within the density functional theory (DFT), some papers have reported the correlations between energetics and stacking configurations in the polytypes for group IV semiconductors and group III–V compounds
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