Polytypism of GaAs, InP, InAs, and InSb: An ab initio study

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PANSE, KRIEGNER, AND BECHSTEDT PHYSICAL REVIEW B 84, 075217 (2011) TABLE V. Stacking fault formation energies E (meV/atom) and γ (mJ/m 2 )forthe3C structure. The experimental values are taken from Refs. 67 and 68. ESF ISF E γ E γ Calc. Calc. Calc. Expt. Calc. Expt. GaAs 39.7 46.7 43.9 47 ± 5 51.6 55 ± 5 InP 16.5 18.0 19.6 17 ± 3 21.3 18 ± 3 InAs 31.2 31.7 35.3 30 ± 3 35.9 30 ± 3 InSb 36.6 32.5 39.9 43 ± 4 35.4 38 ± 4 shear stress applied to the 3C crystal. One speaks about an extrinsic stacking fault (ESF) after adding a double layer to the stacking sequence, for example, due to condensation of eigeninterstitials. Adding a C bilayer the resulting stacking sequence is ABCA/C/BCABC.... The occurrence of such stacking faults can also be discussed in terms of a twist by 180 ◦ of the three equivalent bonds between two bilayers in a bonding tetrahedron nonparallel to the [111] axis. Then, besides staggered (cubic) layers also eclipsed (hexagonal) bilayers appear. 64 Within the ANNNI model (4) the formation energies of such a stacking fault E(ISF/ESF) with respect to the infinite stacking of the 3C polytype can easily be derived. Per atom in a two-dimensional cell perpendicular to the [111] axis one finds E ISF = 4J 1 + 4J 2 + 4J 3 , E ESF = 4J 1 + 8J 2 + 8J 3 . With the sign of the interaction parameters in Table IV it becomes obvious that the extrinsic stacking faults are energetically favorable versus the intrinsic stacking faults in agreement with experimental values, as known for silicon crystals. 66 The stacking-fault formation energies γ (ISF/ESF) per unit area follow from E(ISF/ESF) by division with the area √ 3a0 2 /4 of one atom in a (111) plane. The calculated energy results are listed in Table V and compared with measured values. 67,68 The interesting point is that the stacking fault formation energies are not directly related to the bonding energy or the cohesive energy of the compound. Within the In compounds there is a clear correlation with the bond length or the lattice constant. However, the direct proportionality is in contrast to the scaling with the inverse lattice constant found for the stacking fault formation energies of covalent group-IV semiconductors. 64,69 (7) Experimentally it is rather difficult to distinguish between extrinsic and intrinsic stacking faults. In the literature 67,68 only formation energies of stacking faults are given. However, the measured values given in Table V are in excellent agreement within the error bars with the formation energies of intrinsic stacking faults computed within the ANNNI model (see interaction parameters in Table IV). The absolute values of the formation energies for especially extrinsic stacking faults E(ESF) of the order of the thermal energy k B T at room temperature indicate that even in bulk zinc-blende crystals of III-V compounds stacking faults are likely. In nanorods grown in [111] direction stacking fluctuations and therefore the local formation of hexagonal polytypes pH are probable. V. SUMMARY AND CONCLUSIONS In summary, we have performed first-principles calculations for the structural, energetic, and elastic properties of hexagonal polytypes 2H , 4H , and 6H of common III-V semiconductors which crystallize in zinc-blende 3C geometry under ambient conditions. We have observed clear tendencies for the enlargement of the average cation-anion bilayer thickness c/p in [0001] direction but a decrease of the lateral (in-plane) lattice constant a with respect to their ideal values derived from the 3C crystal. On the one hand, the volume per cation-anion pair is almost conserved. On the other hand, the effective ratio 2c/(pa) increases with the hexagonality of the polytype. The agreement with measured lattice-constant variations c/c and a/a for 2H -InAs, 4H -InAs, 2H -InSb, and 4H -InSb is reasonably good. It is improved after inclusion of the relaxation of the atomic positions within the hexagonal unit cells. The energy findings can easily be explained within an ANNNI model taking the interaction up to third-nearestneighbor bilayers into account. The phase diagram constructed within the ANNNI model shows that InP and InAs are closest to the phase boundary between 3C and 6H . A comparison of the charge asymmetry coefficients of the compounds relates the chemical trend for the energetics to the bond ionicity. The ANNNI model also allows the computation of stacking fault formation energies. Excellent agreement is found with measured values. ACKNOWLEDGMENTS The work was financially supported by the Fond zur Förderung der Wissenschaftlichen Forschung (Austria) in the framework of SFB 25, Infrared Optical Nanostructures, and the EU e-I3 ETSF project (GA No. 211956). Grants of computer time from the HLRZ Stuttgart are gratefully acknowledged. * christian.panse@uni-jena.de 1 X. Duan, Y. Huang, Y. Cui, J. Wang, and C. M. Lieber, Nature (London) 409, 66 (2001). 2 J. Wang, M. S. Gudiksen, X. Duan, Y. Cui, and C. M. Lieber, Science 293, 1455 (2001). 3 D. Appell, Nature (London) 419, 553 (2002). 4 A. Mikkelsen, N. Sköld, L. Ouattara, M. Borgström, J. N. Andersen, L. Samuelson, W. Seifert, and E. Lundgren, Nat. Mater. 3, 519 (2004). 5 J. Johansson, L. S. Karlsson, C. Patrik T. Svensson, T. Martensson, B. A. Wacaser, K. Deppert, L. Samuelson, and W. Seifert, Nat. Mater. 5, 574 (2006). 075217-8
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Polytypism of GaAs, InP, InAs, and InSb: An ab initio study

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