Small-Angle X-Ray Scattering of Amorphous Germanium: Numerical Modeling


The present work deals with a detailed analysis of the small-angle X-ray scattering of nanoporous atomistic models for amorphous germanium. Structures with spherical nanovoids, others with arbitrarily oriented ellipsoidal ones, with monodisperse and polydisperse size distributions, were first generated. After relaxing the as-generated structure, we compute its radial distribution function, and then we deduce by the Fourier transform technique its X-ray scattering pattern. Using a smoothing procedure, the computed small-angle X-ray scattering patterns are corrected for the termination errors due to the finite size of the model, allowing so, for the first time at our best knowledge, a rigorous quantitative analysis of this scattering. The Guinier’s law is found to be valid irrespective of size and shape of the nanovoids over a scattering vector-range extending beyond the expected limit. A weighted combination of the Guinier’s forms accounts for well the nanovoid size distribution in the amorphous structure. The invariance of the Q-factor and its relationship to the void volume fraction are also confirmed. Our findings support then the quantitative analyses of available small-angle X-ray scattering data for amorphous germanium.

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R. Brahim and A. Chehaidar, "Small-Angle X-Ray Scattering of Amorphous Germanium: Numerical Modeling," Advances in Materials Physics and Chemistry, Vol. 3 No. 1A, 2013, pp. 19-30. doi: 10.4236/ampc.2013.31A003.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] S. R. Elliot, “Physics of Amorphous Materials,” 2nd Edition, Longman Group UK Limited, 1990.
[2] N. J. Shevchik and W. Paul, “Voids in Amorphous Semi- conductors,” Journal of Non-Crystalline Solids, Vol. 16, No. 1, 1974, pp. 55-71. doi:10.1016/0022-3093(74)90068-4
[3] P. D’Antonio and J. H. Konnert, “Small-Angle Scattering Evidence of Voids in Hydrogenated Amorphous Silicon,” Physical Review Letters, Vol. 43, No. 16, 1979, pp. 1161- 1163. doi:10.1103/PhysRevLett.43.1161
[4] T. A. Postol, C. M. Falco, R. T. Kampwirth, I. K. Schuller and W. B. Yellon, “Structure of Amorphous Silicon and Silicon Hydrides,” Physical Review Letters, Vol. 45, No. 8, 1980, pp. 648-652. doi:10.1103/PhysRevLett.45.648
[5] A. J. Leadbetter, A. A. M. Rashid, R. M. Richardson, A. F. Wright and J. C. Knights, “Small-Angle X-Ray Scattering and Neutron Scattering Studies of Plasma-Deposited Amorphous Silicon-Hydrogen Films,” Solid State Communications, Vol. 33, No. 9, 1980, pp. 973-977. doi:10.1016/0038-1098(80)90293-8
[6] R. Bellisent, A. Chenevas-Paule and M. Roth, “Medium Range Order Investigation in a-Si:H by Small Angle Neutron Scattering,” Journal of Non-Crystalline Solids, Vol. 59-60, 1983, pp. 229-232. doi:10.1016/0022-3093(83)90563-X
[7] A. H. Mahan, D. L. Williamson, B. P. Nelson and R. S. Crandall, “Characterization of Microvoids in Device-Quality Hydrogenated Amorphous Silicon by Small-Angle X-Ray Scattering and Infrared Measurements,” Physical Review B, Vol. 40, No. 17, 1989, pp. 12024-12027. doi:10.1103/PhysRevB.40.12024
[8] D. L. Williamson, S. Roorda, M. Chicoine, R. Tabti, P. A. Stolk, S. Acco and F. W. Saris, “On the Nanostructure of Pure Amorphous Silicon,” Applied Physics Letters, Vol. 67, No. 2, 1995, pp. 226-228. doi:10.1063/1.114675
[9] C. A. Guy, A. C. Wright, R. N. Sinclair, R. J. Stewart and F. Jansen, “A Small Angle Neutron Scattering (SANS) Study of Hydrogen in Hydrogenated Amorphous Silicon,” Journal of Non-Crystalline Solids, Vol. 196, 1996, pp. 260-266. doi:10.1016/0022-3093(96)80011-1
[10] D. L. Williamson, D. W. M. Marr, E. Iwaniczko and B. P. Nelson, “Small-Angle Neutron Scattering Studies of Hot- Wire CVD a-Si:H,” Thin Solid Film, Vol. 430, No. 1-2, 2003, pp. 192-196. doi:10.1016/S0040-6090(03)00109-3
[11] A. C. Wright, A. C. Hannon, R. N. Sinclair, T. M. Brunier, C. A. Guy, R. J. Stewart, M. B. Strobel and F. Jansen, “Neutron Scattering Studies of Hydrogenated, Deuterated and Fluorinated Amorphous Silicon,” Journal of Physics: Condensed Matter, Vol. 19, 2007, Article ID: 415109. doi:10.1088/0953-8984/19/41/415109
[12] D. L. Young, P. Stradins, Y. Xu, L. M. Gedvilas, E. Iwaniczko, Y. Yan, H. M. Branz and Q. Wang, “Nano-structure Evolution in Hydrogenated Amorphous Silicon during Hydrogen Effusion and Crystallization,” Applied Physics Letters, Vol. 90, 2007, Article ID: 081923. doi:10.1063/1.2435959
[13] A. Chenevas-Paule, R. Bellisent, M. Roth and J. I. Pankove, “Correlation between Stabler Wronski Effect and Medium Range Order in a-Si:H by SANS,” Journal of Non-Crystalline Solids, Vol. 77-78, 1985, pp. 373-376.
[14] Y. P. Bork, P. A. Fedders, R. E. Norbeg, D. J. Leopold, K. D. Mackenzie and H. W. Paul, “Deuteron Magnetic Resonance in a-Ge:D, H and a-Si:D,” Journal of Non-Crystalline Solids, Vol. 77-78, 1985, pp. 715-718.
[15] Y. P. Chou and S. C. Lee, “Evidence for the Void Size Related IR Absorption Frequency Shifts in Hydrogenated Amorphous Germanium Films,” Solid State Communications, Vol. 113, No. 2, 2000, pp. 73-75. doi:10.1016/S0038-1098(99)00443-3
[16] M. Vanecek, J. Holoubek and A. Shah, “Optical Study of Microvoids, Voids, and Local Inhomogeneities in Amorphous Silicon,” Applied Physics Letters, Vol. 59, No. 18, 1991, pp. 2237-2239. doi:10.1063/1.106081
[17] S. Chattopadhyay, S. N. Sharma, R. Banerjee, D. M. Bhusari, S. T. Kshirsagar, Y. Chen and D. L. Williamson, “Short-Range Order, Microstructure and Their Correlation with Light-Induced Degradation in Hydrogenated Amorphous Silicon Deposited at High Growth Rates by Cathode Heating Technique,” Journal of Applied Physics, Vol. 76, No. 9, 1994, pp. 5208-5212. doi:10.1063/1.357239
[18] D. Han, G. Yue, K. Wang, J. Baugh, Y. Wu, Y. Xu and Q. Wang, “Nanovoid-Related Large Redshift of Photoluminescence Peak Energy in Hydrogenated Amorphous Silicon,” Applied Physics Letters, Vol. 80, No. 1, 2002, pp. 40-42. doi:10.1063/1.1431396
[19] J. S. Kuster, M. O. Thompson, D. C. Jacobson, J. M. Poate, S. Roorda, W. C. Sinke and F. S. Paepen, “Density of Amorphous Silicon,” Applied Physics Letters, Vol. 64, No. 4, 1994, pp. 437-439. doi:10.1063/1.111121
[20] M. A. Wank, R. A. C. M. M. Van Swaaij, P. Kudlacek, M. C. M. Van de Sanden and M. Zeman, “Hydrogenated Amorphous Silicon Deposited under Accurately Controlled Ion Bombardment Using Pulse-Shaped Substrate Biasing,” Journal of Applied Physics, Vol. 108, No. 10, 2010, Article ID: 103304. doi:10.1063/1.3505794
[21] O. Glatter and O. Karatky, “Small Angle X-Ray Scattering,” Academic Press, London, 1982.
[22] A. C. Wright and M. F. Thorpe, “Eighty Years of Random Networks,” Physica Status Solidi B, 2013, pp. 1-6. doi:10.1002/pssb.201248500
[23] G. T. Barkema and N. Mousseau, “High-Quality Continuous Random Networks,” Physical Review B, Vol. 62, No. 8, 2000, pp. 4985-4990. doi:10.1103/PhysRevB.62.4985
[24] R. L. C. Vink, G. T. Barkema, M. A. Stijnman and R. H. Bisseling, “Device-Size Atomistic Models of Amorphous Silicon,” Physical Review B, Vol. 64, No. 24, 2001, pp. 245214-24519. doi:10.1103/PhysRevB.64.245214
[25] R. Biswas, I. Kwon, A. M. Bouchard, C. M, Soukoulis and G. S. Grest, “Intense Small Wave-Vector Scattering from Voids in Amorphous Silicon: A Theoretical Simulation,” Physical Review B, Vol. 39, No. 8, 1989, pp. 5101- 5106. doi:10.1103/PhysRevB.39.5101
[26] A. Chehaidar and T. Chermiti, “Microstructural Analysis of Nanoporous Paracrystalline Atomistic Models of Amorphous Silicon,” Journal of Non-Crystalline Solids, Vol. 353, No. 18-21, 2007, pp. 1766-1772. doi:10.1016/j.jnoncrysol.2007.02.009
[27] P. Biswas, R. Atta-Fynn, S. Chakraborty and D. A. Drabold, “Real Space Information from Fluctuation Electron Microscopy: Application to Amorphous Silicon,” Journal of Physics: Condensed Matter, Vol. 19, No. 45, 2007, Article ID: 455202. doi:10.1088/0953-8984/19/45/455202
[28] G. Opletal, T. C. Peterson, I. K. Snook and D. G. McCulloch, “Modeling of Structure and Porosity in Amorphous Silicon Systems Using Monte Carlo Methods,” Journal of Chemical Physics, Vol. 126, No. 21, 2007, Article ID: 214705. doi:10.1063/1.2743029
[29] S. Chakraborty, D. C. Bobela, P. C. Taylor and D. A. Drabold, “Voids in Hydrogenated Amorphous Silicon: A Comparison of ab Initio Simulations and Proton NMR Studies,” Materials Research Society Symposium Proceedings, Vol. 1066, 2008. doi:10.1557/PROC-1066-A11-02
[30] C. R. Miranda, K. V. Tretiakov and S. Scandolo, “A Computational Study of Elastic Properties of Disordered Systems with Voids,” Journal of Non-Crystalline Solids, Vol. 352, No. 40-41, 2006, pp. 4283-4286. doi:10.1016/j.jnoncrysol.2006.07.036
[31] M. M. J. Treacy, J. M. Gibson and P. J. Keblinski, “Paracrystallites Found in Evaporated Amorphous Tetrahedral Semiconductors,” Journal of Non-Crystalline Solids, Vol. 231, No. 1-2, 1998, pp. 99-110. doi:10.1016/S0022-3093(98)00371-8
[32] Adam M. R. de Graff and M. F. Thorpe, “The Long- Wavelength Limit of the Structure Factor of Amorphous Silicon and Vitreous Silica,” Acta Crystallographica Section A, Vol. 66, 2010, pp. 22-31. doi:10.1107/S0108767309045206
[33] H. Rücker and M. Methfessel, “Anharmonic Keating Model for Group-IV Semiconductors with Application to the Lattice Dynamics in Alloys of Si, Ge, and C,” Physical Review B, Vol. 52, No. 15, 1995, pp. 11059-11072. doi:10.1103/PhysRevB.52.11059
[34] R. B. Brahim and A. Chehaidar, “Microstructural Analysis of Atomistic Models of Si-Rich Amorphous Silicon- Germanium Alloys,” Journal of Non-Crystalline Solids, Vol. 357, No. 6, 2011, pp. 1531-1537. doi:10.1016/j.jnoncrysol.2010.12.008
[35] E. Lorch, “Neutron Diffraction by Germania, Silica and Radiation-Damaged Silica Glasses,” Journal of Physics C: Solid State Physics, Vol. 2, No. 2, 1969, pp. 229-237. doi:10.1088/0022-3719/2/2/305
[36] A. Guinier and G. Fournet, “Small-Angle Scattering of X- Rays,” John Wiley & Sons, New York, 1955.

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