Application of Anharmonic Correlated Debye Model in Investigating Anharmonic XAFS Thermodynamic Properties of Crystalline Silicon
Main Article Content
Abstract
In this work, the temperature dependence of the anharmonic X-ray absorption fine structure (XAFS) and thermodynamic properties of the crystalline silicon (c-Si) have been investigated. The thermodynamic parameters are derived from the influence of the absorbing and backscattering atoms of all their nearest neighbors in the crystalline lattice with thermal vibrations. The Debye-Waller factor and thermal expansion coefficient in the anharmonic XAFS of c-Si were calculated in explicit forms using the anharmonic correlated Debye (ACD) model. This calculation model is developed from the many-body perturbation approach and correlated Debye model using the anharmonic effective potential. The numerical results of c-Si in temperature ranging from 0 to 1500 K are in good agreement with those obtained by the other theoretical procedures and experiments at several temperatures. The analytical results showed that the ACD model is useful in analyzing the experimental XAFS data on c-Si.
Keywords:
XAFS Debye-Waller factor; thermal expansion coefficient; crystalline silicon; anharmonic correlated Debye model.
References
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[4] J. J. Rehr, F. D. Vila, J. J. Kas, N. Y. Hirshberg, K. Kowalski, B. Peng, Equation of Motion Coupled-cluster Cumulant Approach for Intrinsic Losses in X-ray Spectra, Journal of Chemical Physics, Vol. 152, 2020,
pp. 174113, https://doi.org/10.1063/5.0004865.
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[8] J. J. Rehr, R. C. Albers, Theoretical Approaches to X-ray Absorption Fine Structure, Reviews of Modern Physics, Vol. 72, No. 3, 2000, pp. 621-654, https://doi.org/10.1103/RevModPhys.72.621.
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[10] E. D. Crozier, J. J. Rehr, R. Ingalls, Amorphous and Liquid Systems, in: D. C. Koningsberger, R. Prins (Eds.),
X-ray Absorption: Principles, Applications, Techniques of XAFS, SXAFS and XANES, John Wiley & Sons, New York, 1988. pp. 373-442, https://www.wiley.com/en-gb/exportProduct/pdf/9780471875475 (accessed on: May 1st, 2022).
[11] G. Bunker, Application of the Ratio Method of EXAFS Analysis to Disordered Systems, Instruments and Methods in Physics Research, Vol. 207, No. 3. 1983, pp. 437-444, https://doi.org/10.1016/0167-5087(83)90655-5.
[12] L. Tröger, T. Yokoyama, D. Arvanitis, T. Lederer, M. Tischer, K. Baberschke, Determination of Bond Lengths, Atomic Mean-square Relative Displacements, and Local Thermal Expansion By Means of Soft-x-ray Photoabsorption, Physical Review B, Vol. 49, No. 2, 1994, pp. 888-903, https://doi.org/10.1103/PhysRevB.49.888.
[13] T. Yokoyama, K. Kobayashi, T. Ohta, A. Ugawa, Anharmonic Interatomic Potentials of Diatomic and Linear Triatomic Molecules Studied by Extended X-ray Absorption Fine Structure, Physical Review B, Vol. 53, No. 10, 1996, pp. 6111-6122, https://doi.org/10.1103/PhysRevB.53.6111.
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[18] N. V. Hung, C. S. Thang, N. B. Duc, D. Q. Vuong, T. S. Tien, Temperature Dependence of Theoretical and Experimental Debye-Waller Factors, Thermal Expansion and XAFS of Metallic Zinc, Physica B: Condensed Matter, Vol. 521, 2017, pp. 198-203, http://dx.doi.org/10.1016/j.physb.2017.06.027.
[19] N. V. Hung, C. S. Thang, N. B. Duc, D. Q. Vuong, T. S. Tien, Advances in Theoretical and Experimental XAFS Studies of Thermodynamic Properties, Anharmonic Effects and Structural Determination of Fcc Crystals, European Physical Journal B, Vol. 90, 2017, pp. 256, https://doi.org/10.1140/epjb/e2017-80383-1.
[20] M. G. Voronkov, Silicon era, Russian Journal of Applied Chemistry, Vol. 80, No. 12, 2007, pp. 2190-2196, https://doi.org/10.1134/S1070427207120397.
[21] E. Epstein, Silicon, Annual Review of Plant Physiology and Plant Molecular Biology, Vol. 50, 1999,
pp. 641-664, https://doi.org/10.1146/annurev.arplant.50.1.641.
[22] J. Emsley, Nature’s Building Blocks: An A-Z Guide to the Elements, Second ed., Oxford University Press, New York, 2011, https://archive.org/details/naturesbuildingb0000emsl (accessed on: May 1st, 2022).
[23] W. Heywang, K. H. Zaininger, Silicon: the Semiconductor Material, in: P. Siffert, E. F. Krimmel (Eds.), Silicon: Evolution and Future of a Technology, Springer Verlag, Heidelberg, 2004, pp. 25-42, https://doi.org//10.1007/978-3-662-09897-4_2.
[24] N. V. Hung, N. B. Duc, D. Q. Vuong, N. C. Toan, T. S. Tien, Advances in EXAFS Studies of Thermodynamic Properties and Anharmonic Effects Based on Debye-waller Factors, Applications to Semiconductors, Vacuum, Vol. 169, 2019, pp. 108872, https://doi.org/10.1016/j.vacuum.2019.108872.
[25] T. S. Tien, L. V. Hoang, Temperature Dependence of Anharmonic EXAFS Oscillation of Crystalline Silicon, Journal of Tan Trao University, Vol. 6, No. 19, 2020, pp. 95-102, https://doi.org/10.51453/2354-1431/2020/435.
[26] M. Benfatto, C. R. Natoli, A. Filipponi, Thermal and Structural Damping of the Multiple-Scattering Contributions to the X-ray-absorption Coefficient, Physical Review B, Vol. 40, No. 14, 1991, pp. 9626-9635, https://doi.org/10.1103/PhysRevB.40.9626.
[27] N. V. Hung, N. B. Trung, B. Kirchner, Anharmonic Correlated Debye Model Debye-waller Factors, Physica B: Condensed Matter, Vol. 405, No. 11, 2010, pp. 2519-2525, https://doi.org/10.1016/j.physb.2010.03.013.
[28] N. V. Hung, T. T. Hue, H. D. Khoa, D. Q. Vuong, Anharmonic Correlated Debye Model High-order Expanded Interatomic Effective Potential and Debye-waller Factors of Bcc Crystals, Physica B: Condensed Matter,
Vol. 503, No. 15, 2016, pp. 174-178, http://dx.doi.org/10.1016/j.physb.2016.09.019.
[29] T. S. Tien, Analysis of EXAFS Oscillation of Monocrystalline Diamond-semiconductors Using Anharmonic Correlated Debye Model, European Physical Journal Plus, Vol. 136, 2021, pp. 539, https://doi.org/10.1140/epjp/s13360-021-01378-z.
[30] T. S. Tien, Effect of The Non-Ideal Axial Ratio C/A on Anharmonic EXAFS Oscillation of H.C.P. Crystals, Journal of Synchrotron Radiation, Vol. 28, 2021, pp. 1544-1557, https://doi.org/10.1107/S1600577521007256.
[31] P. M. Morse, Diatomic Molecules According to the Wave Mechanics, II. Vibrational Levels, Physical Review,
Vol. 34, 1929, pp. 57-64, https://doi.org/10.1103/PhysRev.34.57.
[32] L. A. Girifalco, V. G. Weizer, Application of the Morse Potential Function to Cubic Metals, Physical Review,
Vol. 114, No. 3, 1959, pp. 687-690, https://doi.org/10.1103/PhysRev.114.687.
[33] N. V. Hung, L. H. Hung, T. S. Tien, R. R. Frahm, Anharmonic effective potential, effective local force constant and EXAFS of hcp crystals: Theory and comparison to experiment, International Journal of Modern Physics B, Vol. 22, No. 29, 2008, pp. 5155-5166, https://doi.org/10.1142/S0217979208049285.
[34] N. V. Hung, T. S. Tien, N. B. Duc, D. Q. Vuong, High-order Expanded XAFS Debye-Waller Factors of HCP crystals based on classical anharmonic correlated Einstein model, Modern Physics Letters B, Vol. 28, No. 21, 2014, pp. 1450174, https://doi.org/10.1142/S0217984914501747.
[35] N. V. Hung, J. J. Rehr, Anharmonic correlated Einstein-model Debye-Waller factors, Physical Review B,
Vol. 56, No. 1, 1997, pp. 43-46, https://doi.org/10.1103/PhysRevB.56.43.
[36] N. V. Hung, P. Fornasini, Anharmonic Effective Potential, Correlation Effects, and EXAFS Cumulants Calculated from a Morse Interaction Potential for fcc Metals, Journal of the Physical Society of Japan, Vol. 76, No. 8, 2007, pp. 084601, https://doi.org/10.1143/JPSJ.76.084601.
[37] S. H. Simon, The Oxford Solid State Basics, First ed., Oxford University Press, Oxford, 2013, https://www-thphys.physics.ox.ac.uk/people/SteveSimon/book.html (accessed on: May 1st, 2022).
[38] G. Beni, P. M. Platzman, Temperature and Polarization Dependence of Extended X-ray Absorption Fine-structure Spectra, Physical Review B, Vol. 14, No. 4, 1976, pp. 1514-1518, https://doi.org/10.1103/PhysRevB .14.1514.
[39] G. D. Mahan, Many-Particle Physics, Second ed., Plenum, New York, 1990, https://doi.org/10.1007/978-1-4613-1469-1.
[40] T. S. Tien, Advances in studies of the temperature dependence of the EXAFS amplitude and phase of FCC crystals, Journal of Physics D Applied Physics, Vol. 53, No. 11, 2020, pp. 315303, https://doi.org/10.1088/1361-6463/ab8249.
[41] T. S. Tien, Investigation of the Anharmonic EXAFS Oscillation of Distorted HCP Crystals Based on Extending Quantum Anharmonic Correlated Einstein Model, Japanese Journal of Applied Physics, Vol. 60, 2021,
pp. 112001, https://doi.org/10.35848/1347-4065/ac21b3.
[42] P. Chaloner, Organic Chemistry: A Mechanistic Approach, First ed., CRC Press, Boca Raton, 2015, https://doi.org/10.1201/b17689.
[43] R. A. Swalin, Theoretical Calculations of the Enthalpies and Entropies of Diffusion and Vacancy Formation in Semiconductors, Journal of Physics and Chemistry of Solids, Vol. 18, No. 4, 1961, pp. 290-296, https://doi.org/10.1016/0022-3697(61)90120-2.
[44] J. D. Schall, G. Gao, J. A. Harrison, Elastic Constants of Silicon Materials Calculated as A Function of Temperature Using A Parametrization of the Second-generation Reactive Empirical Bond-order Potential, Physical Review B, Vol. 77, 2008, pp. 115209, https://doi.org/10.1103/PhysRevB.77.115209.
[45] Y. Okada, Y. Tokumaru, Precise determination of Lattice Parameter and Thermal Expansion Coefficient of Silicon between 300 and 1500 K, Journal of Applied Physics, Vol. 56, No. 2, 1984, pp. 314-320, http://dx.doi.org/10.1063/1.333965.