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Efficient first-principles approach to Gibbs free energy with thermal expansion

Kota Hashimoto*, Tomonori Tanaka, and Yoshihiro Gohda†

  • *Contact author: hashimoto.k.d594@m.isct.ac.jp
  • †Contact author: gohda@mct.isct.ac.jp
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Phys. Rev. B 111, 224309 – Published 27 June, 2025

DOI: https://doi.org/10.1103/6qsr-xzgb

Abstract

We propose a method to evaluate the Gibbs free energy from constant-volume first-principles phonon calculations. The volume integral of the pressure is performed by determining the volume and the bulk modulus in equilibrium at finite temperatures, where the pressure and its volume derivative are evaluated utilizing first-principles calculations of the Grüneisen parameter without varying the volume. We validate our method for fcc Al by comparing it with the conventional quasiharmonic approximation. Furthermore, we integrate our method with self-consistent phonon theory and apply it to calculations for bcc Ti, hcp Ti, and tetragonal ZrO2. We demonstrate the accuracy and computational efficiency of our method by comparing results with those obtained from directly volume-varied self-consistent phonon calculations. In all cases, our method accurately evaluates the free energy change caused by thermal expansion using only constant-volume phonon calculations.

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References (76)

  1. L.-Q. Chen, Phase-field models for microstructure evolution, Annu. Rev. Mater. Res. 32, 113 (2002).
  2. I. Steinbach, Phase-field models in materials science, Model. Simul. Mat. Sci. Eng. 17, 073001 (2009).
  3. R. M. Martin, Electronic Structure (Cambridge University Press, Cambridge, 2004).
  4. Z.-K. Liu, Thermodynamics and its prediction and CALPHAD modeling: Review, state of the art, and perspectives, Calphad 82, 102580 (2023).
  5. Y. Harashima, K. Tamai, S. Doi, M. Matsumoto, H. Akai, N. Kawashima, M. Ito, N. Sakuma, A. Kato, T. Shoji, and T. Miyake, Data assimilation method for experimental and first-principles data: Finite-temperature magnetization of (Nd,Pr,La,Ce)2(Fe,Co,Ni)14B, Phys. Rev. Mater. 5, 013806 (2021).
  6. A. Yamamura, S. Sakane, M. Ohno, H. Yasuda, and T. Takaki, Data assimilation with phase-field lattice Boltzmann method for dendrite growth with liquid flow and solid motion, Comput. Mater. Sci. 215, 111776 (2022).
  7. R. Kikuchi, A theory of cooperative phenomena, Phys. Rev. 81, 988 (1951).
  8. C. M. Van Baal, Order-disorder transformations in a generalized Ising alloy, Physica 64, 571 (1973).
  9. J. M. Sanchez and D. de Fontaine, The FCC Ising model in the cluster variation approximation, Phys. Rev. B 17, 2926 (1978).
  10. J. M. Sanchez, F. Ducastelle, and D. Gratias, Generalized cluster description of multicomponent systems, Physica A 128, 334 (1984).
  11. T. Mohri, J. M. Sanchez, and D. De Fontaine, Short range order diffuse intensity calculations in the cluster variation method, Acta Metall. 33, 1463 (1985).
  12. R. Kikuchi, CVM entropy algebra, Prog. Theor. Phys. Suppl. 115, 1 (1994).
  13. S. Enomoto, S. Kou, T. Abe, and Y. Gohda, Subphase exploration for SmFe12-based permanent magnets by Gibbs energies obtained with first-principles cluster-expansion method, J. Alloys Compd. 950, 169849 (2023).
  14. M. T. Dove, Introduction to Lattice Dynamics (Cambridge University Press, Cambridge, 1993).
  15. S. Nishino and Y. Gohda, Structures of Sm–Cu intermetallics with Fe as subphase candidates in SmFe12-based permanent magnets studied by first-principles thermodynamics, Jpn. J. Appl. Phys. 62, 030902 (2023).
  16. L. Mauger, M. S. Lucas, J. A. Muñoz, S. J. Tracy, M. Kresch, Y. Xiao, P. Chow, and B. Fultz, Nonharmonic phonons in α-iron at high temperatures, Phys. Rev. B 90, 064303 (2014).
  17. T. Tanaka and Y. Gohda, First-principles study of magnetism-dependent phonons governed by exchange ligand field, J. Phys. Soc. Jpn. 89, 093705 (2020).
  18. T. Tanaka and Y. Gohda, Prediction of the Curie temperature considering the dependence of the phonon free energy on magnetic states, npj Comput. Mater. 6, 184 (2020).
  19. P. Pavone, K. Karch, O. Schütt, W. Windl, D. Strauch, P. Giannozzi, and S. Baroni, Abinitio lattice dynamics of diamond, Phys. Rev. B 48, 3156 (1993).
  20. B. B. Karki, R. M. Wentzcovitch, S. de Gironcoli, and S. Baroni, High-pressure lattice dynamics and thermoelasticity of MgO, Phys. Rev. B 61, 8793 (2000).
  21. N. Mounet and N. Marzari, First-principles determination of the structural, vibrational and thermodynamic properties of diamond, graphite, and derivatives, Phys. Rev. B 71, 205214 (2005).
  22. E. T. Ritz and N. A. Benedek, Interplay between phonons and anisotropic elasticity drives negative thermal expansion in PbTiO3, Phys. Rev. Lett. 121, 255901 (2018).
  23. A. Togo, L. Chaput, I. Tanaka, and G. Hug, First-principles phonon calculations of thermal expansion in Ti3SiC2, Ti3AlC2, and Ti3GeC2, Phys. Rev. B 81, 174301 (2010).
  24. L.-F. Huang, X.-Z. Lu, E. Tennessen, and J. M. Rondinelli, An efficient ab-initio quasiharmonic approach for the thermodynamics of solids, Comput. Mater. Sci. 120, 84 (2016).
  25. N. S. Abraham and M. R. Shirts, Thermal gradient approach for the quasi-harmonic approximation and its application to improved treatment of anisotropic expansion, J. Chem. Theory Comput. 14, 5904 (2018).
  26. P. B. Allen, Anharmonic phonon quasiparticle theory of zero-point and thermal shifts in insulators: Heat capacity, bulk modulus, and thermal expansion, Phys. Rev. B 92, 064106 (2015).
  27. P. B. Allen, Theory of thermal expansion: Quasi-harmonic approximation and corrections from quasi-particle renormalization, Mod. Phys. Lett. B. 34, 2050025 (2020).
  28. A. Bakare and A. Bongiorno, Enhancing efficiency and scope of first-principles quasiharmonic approximation methods through the calculation of third-order elastic constants, Phys. Rev. Mater. 6, 043803 (2022).
  29. R. Masuki, T. Nomoto, R. Arita, and T. Tadano, Anharmonic Grüneisen theory based on self-consistent phonon theory: Impact of phonon-phonon interactions neglected in the quasiharmonic theory, Phys. Rev. B 105, 064112 (2022).
  30. R. Masuki, T. Nomoto, R. Arita, and T. Tadano, Full optimization of quasiharmonic free energy with an anharmonic lattice model: Application to thermal expansion and pyroelectricity of wurtzite GaN and ZnO, Phys. Rev. B 107, 134119 (2023).
  31. S. Rostami and X. Gonze, Approximations in first-principles volumetric thermal expansion determination, Phys. Rev. B 110, 014103 (2024).
  32. O. Hellman, P. Steneteg, I. A. Abrikosov, and S. I. Simak, Temperature dependent effective potential method for accurate free energy calculations of solids, Phys. Rev. B 87, 104111 (2013).
  33. O. Hellman and I. A. Abrikosov, Temperature-dependent effective third-order interatomic force constants from first principles, Phys. Rev. B 88, 144301 (2013).
  34. A. H. Romero, E. K. U. Gross, M. J. Verstraete, and O. Hellman, Thermal conductivity in PbTe from first principles, Phys. Rev. B 91, 214310 (2015).
  35. S. Kadkhodaei, Q.-J. Hong, and A. van de Walle, Free energy calculation of mechanically unstable but dynamically stabilized bcc titanium, Phys. Rev. B 95, 064101 (2017).
  36. S. Kadkhodaei and A. van de Walle, Software tools for thermodynamic calculation of mechanically unstable phases from first-principles data, Comput. Phys. Commun. 246, 106712 (2020).
  37. P. Souvatzis, O. Eriksson, M. I. Katsnelson, and S. P. Rudin, Entropy driven stabilization of energetically unstable crystal structures explained from first principles theory, Phys. Rev. Lett. 100, 095901 (2008).
  38. T. Tadano and S. Tsuneyuki, Self-consistent phonon calculations of lattice dynamical properties in cubic SrTiO3 with first-principles anharmonic force constants, Phys. Rev. B 92, 054301 (2015).
  39. T. Tadano and S. Tsuneyuki, First-principles lattice dynamics method for strongly anharmonic crystals, J. Phys. Soc. Jpn. 87, 041015 (2018).
  40. A. A. Quong and A. Y. Liu, First-principles calculations of the thermal expansion of metals, Phys. Rev. B 56, 7767 (1997).
  41. A. van de Walle and G. Ceder, The effect of lattice vibrations on substitutional alloy thermodynamics, Rev. Mod. Phys. 74, 11 (2002).
  42. B. Zhang, X. Li, and D. Li, Assessment of thermal expansion coefficient for pure metals, Calphad 43, 7 (2013).
  43. Y. Wang, J. J. Wang, H. Zhang, V. R. Manga, S. L. Shang, L.-Q. Chen, and Z.-K. Liu, A first-principles approach to finite temperature elastic constants, J. Phys.: Condens. Matter 22, 225404 (2010).
  44. A. Togo and I. Tanaka, First principles phonon calculations in materials science, Scr. Mater. 108, 1 (2015).
  45. E. T. Ritz, S. J. Li, and N. A. Benedek, Thermal expansion in insulating solids from first principles, J. Appl. Phys. 126, 171102 (2019).
  46. R. Masuki, T. Nomoto, R. Arita, and T. Tadano, Ab initio structural optimization at finite temperatures based on anharmonic phonon theory: Application to the structural phase transitions of BaTiO3, Phys. Rev. B 106, 224104 (2022).
  47. E. Grüneisen, Theorie des festen zustandes einatomiger elemente, Ann. Phys. 344, 257 (1912).
  48. F. Birch, The effect of pressure upon the elastic parameters of isotropic solids, according to Murnaghan's theory of finite strain, J. Appl. Phys. 9, 279 (1938).
  49. F. Birch, Finite elastic strain of cubic crystals, Phys. Rev. 71, 809 (1947).
  50. T. Katsura and Y. Tange, A simple derivation of the Birch–Murnaghan equations of state (EOSs) and comparison with EOSs derived from other definitions of finite strain, Minerals 9, 745 (2019).
  51. G. C. Fletcher and M. Yahaya, The electronic Gruneisen parameter for the transition metals, J. Phys. F: Met. Phys. 9, 1529 (1979).
  52. B. Grabowski, T. Hickel, and J. Neugebauer, Ab initio study of the thermodynamic properties of nonmagnetic elementary fcc metals: Exchange-correlation-related error bars and chemical trends, Phys. Rev. B 76, 024309 (2007).
  53. F. Körmann, A. Dick, B. Grabowski, B. Hallstedt, T. Hickel, and J. Neugebauer, Free energy of bcc iron: Integrated abinitio derivation of vibrational, electronic, and magnetic contributions, Phys. Rev. B 78, 033102 (2008).
  54. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Cengage Learning, 2011).
  55. See Supplemental Material at http://link.aps.org/supplemental/10.1103/6qsr-xzgb for details of the results using the VIP method, which includes Refs. [75, 76].
  56. P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  57. G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  58. G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
  59. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  60. J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Restoring the density-gradient expansion for exchange in solids and surfaces, Phys. Rev. Lett. 100, 136406 (2008).
  61. K. Tolborg and A. Walsh, Exploring the high-temperature stabilization of cubic zirconia from anharmonic lattice dynamics, Crystal Growth Design 23, 3314 (2023).
  62. C. Verdi, F. Karsai, P. Liu, R. Jinnouchi, and G. Kresse, Thermal transport and phase transitions of zirconia by on-the-fly machine-learned interatomic potentials, Npj Comput. Mater. 7, 156 (2021).
  63. M. Delarmelina, M. G. Quesne, and C. R. A. Catlow, Modelling the bulk properties of ambient pressure polymorphs of zirconia, Phys. Chem. Chem. Phys. 22, 6660 (2020).
  64. T. Tadano, Y. Gohda, and S. Tsuneyuki, Anharmonic force constants extracted from first-principles molecular dynamics: applications to heat transfer simulations, J. Phys.: Condens. Matter 26, 225402 (2014).
  65. Y. S. Touloukian, R. K. Kirby, R. E. Taylor, and P. D. Desai, Thermal Expansion: Metallic Elements and Alloys (ntrs.nasa.gov, 1975).
  66. J. H. Jung, A. Forslund, P. Srinivasan, and B. Grabowski, Dynamically stabilized phases with full ab initio accuracy: Thermodynamics of Ti, Zr, Hf with a focus on the hcp-bcc transition, Phys. Rev. B 108, 184107 (2023).
  67. Y. Zhang, H.-X. Chen, L. Duan, J.-B. Fan, L. Ni, and V. Ji, A comparison study of the born effective charges and dielectric properties of the cubic, tetragonal, monoclinic, ortho-I, ortho-II and ortho-III phases of zirconia, Solid State Sci. 81, 58 (2018).
  68. Y. Oba, T. Tadano, R. Akashi, and S. Tsuneyuki, First-principles study of phonon anharmonicity and negative thermal expansion in ScF3, Phys. Rev. Mater. 3, 033601 (2019).
  69. X. Zhang, B. Grabowski, F. Körmann, C. Freysoldt, and J. Neugebauer, Accurate electronic free energies of the 3d, 4d, and 5d transition metals at high temperatures, Phys. Rev. B 95, 165126 (2017).
  70. A. Togo, L. Chaput, T. Tadano, and I. Tanaka, Implementation strategies in phonopy and phono3py, J. Phys.: Condens. Matter 35, 353001 (2023).
  71. R. P. Feynman, Statistical Mechanics: A Set of Lectures (CRC Press, Boca Raton, FL, 2018).
  72. A. T. Dinsdale, SGTE data for pure elements, Calphad 15, 317 (1991).
  73. K. Momma and F. Izumi, VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data, J. Appl. Crystallogr. 44, 1272 (2011).
  74. G. P. Srivastava, The Physics of Phonons, 2nd ed. (CRC Press, Boca Raton, FL, 2023).
  75. G. A. Slack and S. F. Bartram, Thermal expansion of some diamondlike crystals, J. Appl. Phys. 46, 89 (1975).
  76. N. Igawa and Y. Ishii, Crystal structure of metastable tetragonal zirconia up to 1473 K, J. Am. Ceram. Soc. 84, 1169 (2001).

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Phys. Rev. B 111, 224309– Published 27 June, 2025
  • Received 27 September 2024
  • Revised 27 May 2025
  • Accepted 12 June 2025
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