Anharmonicity and thermal properties of solids

January 10, 2018 to January 12, 2018


  • Francois Bottin (Commissariat à l'Énergie Atomique, France)
  • Matthieu Verstraete (University of Liege, Belgium)
  • Olle Hellman (California Institute of Technology (Caltech), Sweden)
  • Johann Bouchet (CEA-DIF, France)





Institut Henri-Poincaré


The thermal properties of solids are one of the most important features studied in condensed matter. In numerous fields of research, the behavior of materials as a function of temperature is a key quantity for both fundamental and practical or applied purposes. Specifically, in the context of first principles electronic structure theory, this is reflected in the building of an equation of state, the stabilization of high temperature phases due to entropy effects, the calculation of the electron-phonon coupling and the interplay between the lattice vibrations and the electronic structure, the phonon-phonon scattering and its effect on thermal conductivity, and in the evolution of elastic properties as a function of temperature.

For a long time, the thermal evolution of material properties has been evaluated in the framework of classical simulations. A large number of methods has been proposed and implemented successfully, in order to take temperature/entropy effects into account explicitly. In particular, the calculation of the free energy via thermodynamic integration, and the direct computation of the Interatomic Force Constants (IFC). However, these methods often fail to reproduce effects coming from the modification of the electronic structure. In numerous cases, electronic and atomic structures need to be evaluated on equal (quantum) footing.

Due to their historically prohibitive computational cost, the works evaluating the temperature dependency of electronic and lattice dynamics from an ab initio point of view have been much scarcer. Apart from some pioneering works, these calculations mainly extrapolated ground state results to non-zero temperature by means of the so called quasi-harmonic approximation (QHA) [K. Albe 1997] employing either the state of the art “density functional perturbation theory” [X. Gonze et al. 1997, S. Baroni et al. 2001] or finite difference calculations, as implemented for example in the popular PHONOPY package [A. Togo and I. Tanaka 2015]. Over the same period, several theoretical works showed that the variation of the electronic structure (gap, absorption spectrum, etc…) with temperature plays a crucial role in numerous materials. Many features appearing at non-zero temperature can not be deduced from ground state 0K calculations:
- the stability of high-temperature phases, which are not metastable at 0 K, such as the bcc phase of Zr and Li [P. Souvatzis et al. 2008; O. Hellman et al. 2011] or Pu [X. Dai et al. 2003],
- the importance of the anharmonic contributions and the failure of the QHA to describe some phase transitions, such as α-U at low temperature [A. Dewaele et al. 2013],
- the high temperature behavior of optical phonon modes and the thermal conductivity in PbTe [O. Delaire et al. 2012; A. H. Romero et al. 2015]

Consequently, the capture of the temperature effects by means of ab initio calculations has become an active field of research and many powerful new methods have emerged.

Over the past two decades, the advent of massively parallel supercomputers and electronic codes with a high scaling efficiency [P. Giannozzi et al. 2004, F. Bottin et al. 2008, A. Maniopoulou et al. 2012…], have opened the way to simulate materials at non-zero temperature. The number of atoms included in supercells, as well as the number of configurations or the length of simulations (needed to achieve ergodicity), became sufficient to account for thermal properties in realistic systems.

However, the thermal quantities extracted from such simulations were often scarce and reduced to straightforward thermodynamic ones -- temperature, pressure… -- or indirectly deduced from the molecular dynamic trajectory through a tedious procedure and, generally, limited accuracy: vDOS (via the velocity autocorrelation function), elastic constants (via finite differences) [B. Martorell et al. 2013], free energy (via thermodynamic integration) [D. Alfè et al. 2001]… This process was not satisfactory because it did not offer a global framework to compute all the quantities from a unique starting point.

In the past ten years, strong efforts have been made to take into account explicit temperature effects and major advances have been obtained. New methods capturing the thermal properties of solids at non-zero temperature are now available and can be applied in ab initio calculations. These approaches combine ideas including finite large displacements, molecular dynamics sampling, self consistent harmonic theories, and different force fitting schemes. The most widely used methods include Self-Consistent Ab-Initio Lattice Dynamics (SCAILD) [P. Souvatzis et al. 2008, P. Souvatzis et al. 2009, W. Luo et al. 2010], Stochastic Self-Consistent Harmonic Approximation (SSCHA) [I. Errea et al. 2014, I. Errea et al. 2014, L. Paulatto et al. 2015, M. Borinaga et al. 2016], Temperature Dependent Effective Potential (TDEP) [O. Hellman et al. 2011, O. Hellman 2013, P. Steneteg et al. 2013, J. Bouchet et al. 2015], Anharmonic LAttice MODEl (ALAMODE) [Tadano et al. 2014, Tadano et al. 2015], Compressive Sensing Lattice Dynamics [L. J. Nelson et al. 2013, F. Zhou et al. 2014]. Other methods obtain anharmonic contributions via a derivation of the Gibbs energy [A. Glensk et al. 2015], or a series expansion of the interatomic forces constants [K. Esfarjani et al. 2008, K. Esfarjani et al. 2011, J. Shiomi et al. 2011]. A large number of new, intrinsically temperature dependent, phenomena can now be captured: the modification of the phonons density of states (vDOS) and free energy, the (T,P) phase transition boundaries, the evolution of elastic constants or Grueneisen coefficients, the phonon lifetimes, the thermal conductivity….

In particular, non-trivial evolutions of the phonons modes ω(V,T) (and also of the second, third and fourth order IFCs) as a function of the temperature can now be reproduced. This key quantity can show three kinds of behavior: (i) harmonic ω(V,T) ≡ ω(T=0K), if no evolution is considered, (ii) quasi-harmonic ω(V,T) ≡ ω(V(T),T=0K), if the evolution can be implicitly taken into account through the evolution of the volume V(T) (i.e. through the thermal expansion), and (iii) anharmonic ω(V,T) ≡ ω(V(T),T) when the temperature dependency is explicit. Modern theories try to capture these latter effects and account for them in the phonon spectrum. With respect to previous approaches, the theories are now more user friendly (not too cumbersome) and rely on a smaller number of configurations/time steps (not too time consuming). Given the diversity and complexity of the theoretical landscape, CECAM level meetings of developers are essential to keep everyone abreast of the state of the art, push forward novel ideas, and project a strategy for the community.

In the future, several major advances can be expected in this field or research, including:
- The construction of a classical potential able to reproduce all the properties and allotropic phases in a (T,P) phase diagram.
- The use of phonon spectra and IFCs to accelerate the ab initio simulations (learn on the fly) or to increase the size of a simulation supercell (QM/MM)…
- The proper incorporation of the influence of defects (vacancy, dislocation, surface…)
- Improved vibrational properties of alloys, amorphous and liquid materials

The focus of this workshop is a hot topic of research, central to the interests of many fundamental and applied fields of research (nano and meso electronics, thermal barrier coatings, sound and phonon engineering, geophysics, etc…). The DFT community has invested strongly in the field, and has been very dynamic in the development of new methods, applications to high throughput studies, and to important complex thermal evolutions. This proposal is timely and important for the community as a whole.


1. Karsten Albe, Theoretical study of boron nitride modifications at hydrostatic pressures, Phys. Rev. B 55, 6203 (1997)
2. Xavier Gonze and Changyol Lee, Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory, Phys. Rev. B 55, 10355 (1997)
3. D. Alfè, G. D. Price, and M. J. Gillan, Thermodynamics of hexagonal-close-packed iron under Earth’s core conditions, Phys. Rev. B 64, 045123 (2001)
4. Stefano Baroni, Stefano de Gironcoli, Andrea Dal Corso, and Paolo Giannozzi, Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys. 73, 515 (2001)
5. X. Dai, S.Y. Savrasov, G. Kotliar, A. Migliori, H. Ledbetter and E. Abrahams, Calculated phonon spectra of plutonium at high temperatures, Science 300, 953 (2003)
6. P. Giannozzi, F. de Angelis and R. Car, First-principle molecular dynamics with ultrasoft pseudopotentials: Parallel implementation and application to extended bioinorganic systems, J. Chem. Phys. 120 (2004) 5903.
7. 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)
8. P. Souvatzis and S. P. Rudin, Dynamical stabilization of cubic ZrO 2   by phonon-phonon interactions: Ab initio calculations, Phys. Rev. B 78, 184304 (2008)
9. François Bottin, Stéphane Leroux, Andrew Knyazev, Gilles Zérah, Large-scale ab initio calculations based on three levels of parallelization, Comp. Mater. Sci. 42, 329 (2008)
10. Keivan Esfarjani and Harold T. Stokes, Method to extract anharmonic force constants from first principles calculations, Phys. Rev. B 77, 144112 (2008)
11. P. Souvatzis, O. Eriksson, M.I. Katsnelson, and S.P. Rudin, The self-consistent ab initio lattice dynamical method, Comp. Mater. Sci. 44, 888 (2009)
12. W. Luo, B. Johansson, O. Eriksson, S. Arpan, P. Souvatzis, M. I. Katsnelson and R. Ahuja, Dynamical stability of body center cubic iron at the Earths core conditions, PNAS 107, 9962-9964 (2010).
13. O. Hellman, I. A. Abrikosov, and S. I. Simak, Lattice dynamics of anharmonic solids from first principles, Phys. Rev. B 84, 180301(R) (2011)
14. Keivan Esfarjani, Gang Chen, and Harold T. Stokes, Heat transport in silicon from first-principles calculations, Phys. Rev. B 84, 085204 (2011)
15. Junichiro Shiomi, Keivan Esfarjani, and Gang Chen, Thermal conductivity of half-Heusler compounds from first-principles calculations, Phys. Rev. B 84, 104302 (2011)
16. Asimina Maniopoulou, Erlend R.M. Davidson, Ricardo Grau-Crespo, Aron Walsh, Ian J. Busha, C. Richard A. Catlow, Scott M. Woodley, Introducing k-point parallelism into VASP, Comp. Phys. Comm. 183, 1696 (2012)
17. Olle Hellman, Peter 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)
18. Ion Errea, Matteo Calandra, and Francesco Mauri, First-Principles Theory of Anharmonicity and the Inverse Isotope Effect in Superconducting Palladium-Hydride Compounds, Phys. Rev. Lett. 111, 177002 (2013)
19. Peter Steneteg, Olle Hellman, Olga Yu. Vekilova, Nina Shulumba, Ferenc Tasnádi, and Igor A. Abrikosov, Temperature dependence of TiN elastic constants from ab initio molecular dynamics simulations, Phys. Rev. B 87, 094114 (2013)
20. Olle Hellman and I. A. Abrikosov, Temperature-dependent effective third-order interatomic force constants from first principles, Phys. Rev. B 88, 144301 (2013)
21. A. Dewaele, J. Bouchet, F. Occelli, M. Hanfland, and G. Garbarino, Refinement of the equation of state of α  -uranium, Phys. Rev. B 88, 134202 (2013)
22. Benjamí Martorell, Lidunka Vočadlo, John Brodholt, and Ian G. Wood, Strong premelting effect in the elastic properties of hcp-Fe under inner-core conditions, Science 342, 466 (2013)
23. Ion Errea, Matteo Calandra, and Francesco Mauri, Anharmonic free energies and phonon dispersions from the stochastic self-consistent harmonic approximation: Application to platinum and palladium hydrides, Phys. Rev. B 89, 064302 (2014)
24. Lance J. Nelson, Gus L. W. Hart, Fei Zhou (周非), and Vidvuds Ozoliņš, Compressive sensing as a paradigm for building physics models, Phys. Rev. B 87, 035125 (2013)
25. A. Glensk, B. Grabowski, T. Hickel, and J. Neugebauer, Breakdown of the Arrhenius Law in Describing Vacancy Formation Energies: The Importance of Local Anharmonicity Revealed by Ab initio Thermodynamics, Phys. Rev. X 4, 011018 (2014)
26. Terumasa Tadano, Yoshihiro Gohda, and Shinji Tsuneyuki, Anharmonic force constants extracted from first-principles molecular dynamics: applications to heat transfer simulations, J. Phys. Condens. Matter 26, 2254012 (2014)
27. Fei Zhou (周非), Weston Nielson, Yi Xia, and Vidvuds Ozoliņš, Lattice Anharmonicity and Thermal Conductivity from Compressive Sensing of First-Principles Calculations, Phys. Rev. Lett. 113, 185501 (2014)
28. Atsushi Togo and Isao Tanaka, First principles phonon calculations in materials science, Scr. Mater. 108, 1-5 (2015)
29. Terumasa Tadano and Shinji Tsuneyuki, Self-consistent phonon calculations of lattice dynamical properties in cubic SrTiO3 with first-principles anharmonic force constants, Phys. Rev. B 92, 054301 (2015)
30. Terumasa Tadano, Yoshihiro Gohda, and Shinji Tsuneyuki, Impact of Rattlers on Thermal Conductivity of a Thermoelectric Clathrate: A First-Principles Study, Phys. Rev. Lett. 114, 095501 (2015)
31. J. Bouchet and F. Bottin, Thermal evolution of vibrational properties of α-U, Phys. Rev. B 92, 174108 (2015)
32. Agnès Dewaele, Vincent Stutzmann, Johann Bouchet, François Bottin, Florent Occelli, and Mohamed Mezouar, High pressure-temperature phase diagram and equation of state of titanium, Phys. Rev. B 91, 134108 (2015)
33. A. Glensk, B. Grabowski, T. Hickel and J. Neugebauer, Understanding Anharmonicity in fcc Materials: From its Origin to ab initio Strategies beyond the Quasiharmonic Approximation, PRL 114, 195901 (2015)
34. Lorenzo Paulatto, Ion Errea, Matteo Calandra, and Francesco Mauri, First-principles calculations of phonon frequencies, lifetimes, and spectral functions from weak to strong anharmonicity: The example of palladium hydrides, Phys. Rev. B 91, 054304 (2015)
35. Andrew Ian Duff, Theresa Davey, Dominique Korbmacher, Albert Glensk, Blazej Grabowski, Jörg Neugebauer, and Michael W. Finnis, Improved method of calculating ab initio high-temperature thermodynamic properties with application to ZrC, Phys. Rev. B 91, 214311 (2015)
36. A. H. Romero, E. K. U. Gross, M. J. Verstraete, and Olle Hellman, Thermal conductivity in PbTe from first principles, Phys. Rev. B 91, 214310 (2015)
37. Miguel Borinaga, Ion Errea, Matteo Calandra, Francesco Mauri, and Aitor Bergara, Anharmonic effects in atomic hydrogen: Superconductivity and lattice dynamical stability, Phys. Rev. B 93, 174308 (2016)