Definition Peculiarities of Energy of Vacancy Formation In 4d-Transition Metals from First Principles

Автори

In this paper a study of the temperature dependence of the vacancy formation energy in bulk fcc 4d-transition metals Ag and Pd using DFT was presented. Peculiarity of this work is the use of experimental values of the lattice parameters for the respective temperatures. This paper discusses the various contributions to the vacancy formation energy and shows that they can play an important role. It was shown that thermal excitation has a significant impact on the vacancy formation energy at high temperatures. The possibility of the existence of the compensation effect, i.e. the simultaneous changes in the contributions of the free energy and the vacancy formation energy in the fcc 4d-transition metals Ag and Pd, which were investigated from first principles. Taking in to account free oscillation energy and electronic thermal excitation depending on the temperature allows obtaining a qualitative picture of the effect of thermal expansion. The vacancy formation energy is in good agreement with previous theoretical and experimental studies. The effect of mutual compensation of different contributions to the vacancy formation energy can explain the constant value of the vacancy formation energy at any temperature and justifies the neglect of the temperature dependence of the simulation properties.

Publication year: 
2014
Issue: 
4
УДК: 
538.9:539.1
С. 127–132., Іл. 2. Табл. 1. Бібліогр.: 39 назв.
References: 

1. K.N. Grew and W.K.S. Chiu, “Review of Modeling and Simulation Techniques Across the Length Scales for the Solid Oxide Fuel Cell,” J. Power Sources, vol. 199, pp. 1—13, 2012.
2. T.R. Mattsson and A.E. Mattsson, “Calculating the vacancy formation energy in metals: Pt, Pd, and Mo”, Phys. Rev. B, vol. 66, p. 214110, 2002.
3. K.F. McCarty et al., “Vacancies in. Solids and the Stability of Surface Morphology”, Nature (London), vol. 412, p. 622, 2001.
4. Gh.A. Nematollahi et al., “Thermodynamics of carbon solubility in ferrite and vacancy formation in cementite in strained pearlite”, Acta Materialia, vol. 61, p. 1773, 2013.
5. L. Ventelon et al., “Ab initio investigation of radiation defects in tungsten: Structure of self-interstitials and specificity of di-vacancies compared to other bcc transition metals”, J. Nuclear Mater., vol. 425, p. 16, 2012.
6. B. Grabowski et al., “Formation energies of point defects at finite temperatures”, Phys. Status Solidi B, vol. 248, p. 1295, 2011.
7. P.A. Korzhavyi et al., “First-principles calculations of the vacancy formation energy in transition and noble metals”, Phys. Rev. B, vol. 59, p. 11693, 1999.
8. A.E. Mattsson et al., “Electronic surface error in the Si interstitial formation energy”, Ibid, vol. 77, p. 155211, 2008.
9. R. Nazarov et al., “Vacancy formation energies in fcc metals: influence of exchange-correlation functionals and correction schemes”, Ibid, vol. 85, p. 144118, 2012.
10. A.J. Hatt et al., “Harmonic and anharmonic properties of Fe and Ni:Thermal expansion, exchange-correclation errors, and magnetism”, Ibid, vol. 82, p. 134418, 2010.
11. M. Mantina et al., “First-principles calculation of selfdiffusion coefficients”, PRL, vol. 100, p. 215901, 2008.
12. T.R. Mattsson et al., “Quantifying the anomalous self-diffusion in molybdenum with first-principles simulations”, Phys. Rev. B, vol. 80, p. 224104, 2009.
13. D. Simonovic and M.H.F. Sluiter, “Impurity diffusion activation energies in Al from first principles”, Ibid, vol. 79, p. 054304 2009.
14. P. Hohenberg and W. Kohn, “Inhomogeneous Electron Gas”, Ibid, vol.136, p. B864, 1964.
15. W. Kohn and L.J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects”, Phys. Rev., vol. 140, p. A1133, 1965.
16. P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials”, J. Phys.: Condens. Matter, vol. 21, p. 395502, 2009.
17. P.E. Blochl, “Projector augmented-wave method”, Phys. Rev. B, vol. 50, p. 17953, 1994.
18. J. Perdew et al., “Restoring the density-gradient expansion for exchange in solids and surfaces”, Phys. Rev. Lett., vol. 100, p. 136406, 2008.
19. H.J. Monkhorst and J.D. Pack, “On Special Points for Brillouin Zone Integrations”, Phys. Rev. B, vol. 13, p. 5188, 1976.
20. M.J. Gillan, “Calculation of the vacancy formation energy in aluminium”, J. Phys.: Condens. Matter, vol. 1, p. 689, 1989.
21. B. Grabowski et al., “Formation energies of point defects at finite temperatures”, Phys. Status Solidi B, vol. 248, p. 1295, 2011.
22. D.E. Turner et al., “Energetics of vacancy and substitutional impurities in aluminum bulk and clusters”, Phys. Rev. B, vol. 55, p. 13842, 1997.
23. A.V. Ruban, V.I. Razumovskiy, “First-principles based thermodynamic model of phase equilibria in bcc Fe-Cr alloys”, Ibid, vol. 86, p. 174111, 2012.
24. O.I. Gorbatov et al., “The role of magnetism in Cu precipitation in α-Fe”, Ibid, vol. 88, p. 174113, 2013.
25. G. Kresse et al., “Ab initio Force Constant Approach to Phonon Dispersion Relations of Diamond and Graphite”, Europhys. Lett., vol. 32, p. 729, 1995.
26. A. Togo et al., “First-principles calculations of the ferroelastic transition between rutile-type and CaCl2-type SiO2 at high pressures”, Phys. Rev. B, vol. 78, p. 134106, 2008.
27. A. Togo. (2009). Phonopy v.1.8.5 [Online]. Avaliable: http://phonopy.sourceforge.net
28. Y. Kraftmakher, “Equilibrium vacancies and thermophysical properties of metals”, Phys. Rep., vol. 299, p. 79, 1998.
29. H.M. Polatoglou et al., “Vacancy-formation energies at the (111) surface and in bulk Al, Cu, Ag, and Rh”, Phys. Rev. B, vol. 48, p. 1877, 1993.
30. T. Korhonen et al., “Vacancy-formation energies for fcc and bcc transition metals”, Phys. Rev. B, vol. 51, p. 9526, 1995.
31. M.J. Mehl and D.A. Papaconstantopoulos, “Applications of a tight-binding total-energy method for transition and noble metals: Elastic constants, vacancies, and surfaces of monatomic metals”, Phys. Rev. B, vol. 54, p. 4519, 1996.
32. J.L. Campbell et al., “Temperature dependence of positron trapping in silver and nickel”, J. Phys. F: Met. Phys., vol. 7, p. 1985, 1977.
33. A.E. Mattsson et al., “The AM05 density functional applied to solids”, J. Chem. Phys., vol. 128, p. 084714, 2008.
34. I-K Suh et al., “High-temperature thermal expansion of six metallic elements measured by dilatation method and X-ray diffraction Mo, Ag”, J. Mater. Sci., vol. 23, p. 757, 1988.
35. J.W. Arblaster, “Crystallographic Properties of Palladium”, Platinum Metals Rev., vol. 56, p. 181, 2012.
36. X. Tang and B. Fultz, “First-principles study of phonon linewidths in noble metals”, Phys. Rev. B, vol. 84, p. 054303, 2011. 3
7. C.V. Pandya et al., “Lattice Mechanical Properties of Pd, Pt and Ni — A Model Potential Approach”, J. Korean Physical Soc., vol.
38, p. 377, 2001.
38. V.A. Korshunov, “Determination of the phonon density of states from the thermodynamic functions of a crystal: Nickel, palladium, and platinum”, Soviet Physics J., vol. 22, is. 8, pp. 903—905, 1979.
39. V.L. Moruzzi et al., “Calculated thermal properties of metals”, Phys. Rev. B, vol. 37, p. 790, 1988.

References [transliteration]: 

1. K.N. Grew and W.K.S. Chiu, “Review of Modeling and Simulation Techniques Across the Length Scales for the Solid Oxide Fuel Cell,” J. Power Sources, vol. 199, pp. 1–13, 2012.
2. T.R. Mattsson and A.E. Mattsson, “Calculating the vacancy formation energy in metals: Pt, Pd, and Mo”, Phys. Rev. B, vol. 66, p. 214110, 2002.
3. K.F. McCarty et al., “Vacancies in. Solids and the Stability of Surface Morphology”, Nature (London), vol. 412, p. 622, 2001.
4. Gh.A. Nematollahi et al., “Thermodynamics of carbon solubility in ferrite and vacancy formation in cementite in strained pearlite”, Acta Materialia, vol. 61, p. 1773, 2013.
5. L. Ventelon et al., “Ab initio investigation of radiation defects in tungsten: Structure of self-interstitials and specificity of di-vacancies compared to other bcc transition metals”, J. Nuclear Mater., vol. 425, p. 16, 2012.
6. B. Grabowski et al., “Formation energies of point defects at finite temperatures”, Phys. Status Solidi B, vol. 248, p. 1295, 2011.
7. P.A. Korzhavyi et al., “First-principles calculations of the vacancy formation energy in transition and noble metals”, Phys. Rev. B, vol. 59, p. 11693, 1999.
8. A.E. Mattsson et al., “Electronic surface error in the Si interstitial formation energy”, Ibid, vol. 77, p. 155211, 2008.
9. R. Nazarov et al., “Vacancy formation energies in fcc metals: influence of exchange‐correlation functionals and correction schemes”, Ibid, vol. 85, p. 144118, 2012.
10. A.J. Hatt et al., “Harmonic and anharmonic properties of Fe and Ni:Thermal expansion, exchange-correclation errors, and magnetism”, Ibid, vol. 82, p. 134418, 2010.
11. M. Mantina et al., “First-principles calculation of self-diffusion coefficients”, PRL, vol. 100, p. 215901, 2008.
12. T.R. Mattsson et al., “Quantifying the anomalous self-diffusion in molybdenum with first-principles simulations”, Phys. Rev. B, vol. 80, p. 224104, 2009.
13. D. Simonovic and M.H.F. Sluiter, “Impurity diffusion activation energies in Al from first principles”, Ibid, vol. 79, pP. 054304 2009.
14. P. Hohenberg and W. Kohn, “Inhomogeneous Electron Gas”, Ibid, vol.136, p. B864, 1964.
15. W. Kohn and L.J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects”, Phys. Rev., vol. 140, p. A1133, 1965.
16. P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials”, J. Phys.: Condens. Matter, vol. 21, p. 395502, 2009.
17. P.E. Blochl, “Projector augmented-wave method”, Phys. Rev. B, vol. 50, p. 17953, 1994.
18. J. Perdew et al., “Restoring the density-gradient expansion for exchange in solids and surfaces”, Phys. Rev. Lett., vol. 100, p. 136406, 2008.
19. H.J. Monkhorst and J.D. Pack, “On Special Points for Brillouin Zone Integrations”, Phys. Rev. B, vol. 13, p. 5188, 1976.
20. M.J. Gillan, “Calculation of the vacancy formation energy in aluminium”, J. Phys.: Condens. Matter, vol. 1, p. 689, 1989.
21. B. Grabowski et al., “Formation energies of point defects at finite temperatures”, Phys. Status Solidi B, vol. 248, p. 1295, 2011.
22. D.E. Turner et al., “Energetics of vacancy and substitutional impurities in aluminum bulk and clusters”, Phys. Rev. B, vol. 55, p. 13842, 1997.
23. A.V. Ruban, V.I. Razumovskiy, “First-principles based thermodynamic model of phase equilibria in bcc Fe-Cr alloys”, Ibid, vol. 86, p. 174111, 2012.
24. O.I. Gorbatov et al., “The role of magnetism in Cu precipitation in α-Fe”, Ibid, vol. 88, p. 174113, 2013.
25. G. Kresse et al., “Ab initio Force Constant Approach to Phonon Dispersion Relations of Diamond and Graphite”, Europhys. Lett., vol. 32, p. 729, 1995.
26. A. Togo et al., “First-principles calculations of the ferroelastic transition between rutile-type and CaCl2-type SiO2 at high pressures”, Phys. Rev. B, vol. 78, p. 134106, 2008.
27. A. Togo. (2009). Phonopy v.1.8.5 [Online]. Avaliable: http://phonopy.sourceforge.net
28. Y. Kraftmakher, “Equilibrium vacancies and thermophysical properties of metals”, Phys. Rep., vol. 299, p. 79, 1998.
29. H.M. Polatoglou et al., “Vacancy-formation energies at the (111) surface and in bulk Al, Cu, Ag, and Rh”, Phys. Rev. B, vol. 48, p. 1877, 1993.
30. T. Korhonen et al., “Vacancy-formation energies for fcc and bcc transition metals”, Phys. Rev. B, vol. 51, p. 9526, 1995.
31. M.J. Mehl and D.A. Papaconstantopoulos, “Applications of a tight-binding total-energy method for transition and noble metals: Elastic constants, vacancies, and surfaces of monatomic metals”, Phys. Rev. B, vol. 54, p. 4519, 1996.
32. J.L. Campbell et al., “Temperature dependence of positron trapping in silver and nickel”, J. Phys. F: Met. Phys. , vol. 7, p. 1985, 1977.
33. A.E. Mattsson et al., “The AM05 density functional applied to solids”, J. Chem. Phys., vol. 128, p. 084714, 2008.
34. I-K Suh et al., “High-temperature thermal expansion of six metallic elements measured by dilatation method and X-ray diffraction Mo, Ag”, J. Mater. Sci., vol. 23, p. 757, 1988.
35. J.W. Arblaster, “Crystallographic Properties of Palladium”, Platinum Metals Rev. , vol. 56, p. 181, 2012.
36. X. Tang and B. Fultz, “First-principles study of phonon linewidths in noble metals”, Phys. Rev. B, vol. 84, p. 054303, 2011.
37. C.V. Pandya et al., “Lattice Mechanical Properties of Pd, Pt and Ni – A Model Potential Approach”, J. Korean Physical Soc., vol. 38, p. 377, 2001.
38. V.A. Korshunov, “Determination of the phonon density of states from the thermodynamic functions of a crystal: Nickel, palladium, and platinum”, Soviet Physics J., vol, 22, is. 8, pp. 903–905, 1979.
39. V.L. Moruzzi et al., “Calculated thermal properties of metals”, Phys. Rev. B, vol. 37, p. 790, 1988.

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