Temperature effect on the phonon dispersion stability of zirconium by machine learning driven atomistic simulations

It is well known that conventional harmonic lattice dynamics cannot be applied to energetically unstable crystals at 0 K, such as high temperature body centered cubic (BCC) phase of crystalline Zr. Predicting phonon spectra at finite temperature requires the calculation of force constants to the third, fourth and even higher orders, however, it remains challenging to determine to which order the Taylor expansion of the potential energy surface for different materials should be cut off. Molecular dynamics, on the other hand, intrinsically includes arbitrary orders of phonon anharmonicity, however, its accuracy is severely limited by the empirical potential field used. Using machine learning method, we developed an inter-atomic potential for Zirconium crystals for both the hexagonal closed-packed (HCP) phase and the body centered cubic phase. The developed potential field accurately captures energy-volume relationship, elastic constants and phonon dispersions. The instability of BCC structure is found to originate from the double-well shape of the potential energy surface where the local maxima is located in an unstable equilibrium position. The stabilization of the BCC phase at high temperature is due to the dynamical average of the low-symmetry minima of the double well due to atomic vibrations. Molecular dynamics simulations are then performed to stochastically sample the potential energy surface and to calculate the phonon dispersion at elevated temperature. The phonon renormalization in BCC-Zr is successfully captured by the molecular dynamics simulation at 1188 K.

[1]  Herzig,et al.  Phonon dispersion of the bcc phase of group-IV metals. II. bcc zirconium, a model case of dynamical precursors of martensitic transitions. , 1991, Physical review. B, Condensed matter.

[2]  Gábor Csányi,et al.  Accuracy and transferability of Gaussian approximation potential models for tungsten , 2014 .

[3]  R. Kondor,et al.  On representing chemical environments , 2012, 1209.3140.

[4]  Stefano de Gironcoli,et al.  Phonons and related crystal properties from density-functional perturbation theory , 2000, cond-mat/0012092.

[5]  Atsuto Seko,et al.  First-principles interatomic potentials for ten elemental metals via compressed sensing , 2015, 1505.03994.

[6]  M. Kanatzidis,et al.  Ultralow thermal conductivity and high thermoelectric figure of merit in SnSe crystals , 2014, Nature.

[7]  I. Tanaka,et al.  First principles phonon calculations in materials science , 2015, 1506.08498.

[8]  Gabor Csanyi,et al.  Achieving DFT accuracy with a machine-learning interatomic potential: thermomechanics and defects in bcc ferromagnetic iron , 2017, 1706.10229.

[9]  G. Kresse,et al.  From ultrasoft pseudopotentials to the projector augmented-wave method , 1999 .

[10]  Gábor Csányi,et al.  Development of a machine learning potential for graphene , 2017, 1710.04187.

[11]  Tianli Feng,et al.  Four-phonon scattering significantly reduces intrinsic thermal conductivity of solids , 2017 .

[12]  Volker L. Deringer,et al.  Machine learning based interatomic potential for amorphous carbon , 2016, 1611.03277.

[13]  R. Kondor,et al.  Gaussian approximation potentials: the accuracy of quantum mechanics, without the electrons. , 2009, Physical review letters.

[14]  J. Behler Perspective: Machine learning potentials for atomistic simulations. , 2016, The Journal of chemical physics.

[15]  V. Kharchenko,et al.  Ab-initio calculations for the structural properties of Zr-Nb alloys , 2012, 1206.7035.

[16]  Gábor Csányi,et al.  Gaussian approximation potentials: A brief tutorial introduction , 2015, 1502.01366.

[17]  Bo Qiu,et al.  Anharmonicity and necessity of phonon eigenvectors in the phonon normal mode analysis , 2015 .

[18]  J. Zarestky,et al.  Temperature dependence of the normal vibrational modes of hcp Zr , 1978 .

[19]  Anharmonicity in the High-Temperature Cmcm Phase of SnSe: Soft Modes and Three-Phonon Interactions. , 2016, Physical review letters.

[20]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[21]  Wu Li,et al.  ShengBTE: A solver of the Boltzmann transport equation for phonons , 2014, Comput. Phys. Commun..

[22]  Volker L. Deringer,et al.  Gaussian approximation potential modeling of lithium intercalation in carbon nanostructures. , 2017, The Journal of chemical physics.

[23]  Christopher M. Bishop,et al.  Neural networks for pattern recognition , 1995 .

[24]  Terumasa Tadano,et al.  Empirical interatomic potentials optimized for phonon properties , 2017, npj Computational Materials.

[25]  Igor A. Abrikosov,et al.  Temperature-dependent effective third-order interatomic force constants from first principles , 2013, 1308.5436.

[26]  Pressure dependence of the elastic moduli in aluminum-rich Al-Li compounds. , 1992, Physical review. B, Condensed matter.

[27]  M I Katsnelson,et al.  Entropy driven stabilization of energetically unstable crystal structures explained from first principles theory. , 2008, Physical review letters.

[28]  Turab Lookman,et al.  Developing an interatomic potential for martensitic phase transformations in zirconium by machine learning , 2018, npj Computational Materials.

[29]  Noam Bernstein,et al.  Machine Learning a General-Purpose Interatomic Potential for Silicon , 2018, Physical Review X.

[30]  M. Payne,et al.  The Gaussian Approximation Potential , 2010 .

[31]  H. Ohno,et al.  Interatomic potential construction with self-learning and adaptive database , 2017 .

[32]  Michele Parrinello,et al.  Generalized neural-network representation of high-dimensional potential-energy surfaces. , 2007, Physical review letters.

[33]  Journal of Chemical Physics , 1932, Nature.

[34]  Burke,et al.  Generalized Gradient Approximation Made Simple. , 1996, Physical review letters.

[35]  Johansson,et al.  Elastic constants of hexagonal transition metals: Theory. , 1995, Physical review. B, Condensed matter.

[36]  G. Kresse,et al.  Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set , 1996 .

[37]  Georg Kresse,et al.  Ab initio Force Constant Approach to Phonon Dispersion Relations of Diamond and Graphite , 1995 .

[38]  Ronggui Yang,et al.  Thermal conductivity modeling of hybrid organic-inorganic crystals and superlattices , 2017 .

[39]  D. J. Hooton Anharmonische Gitterschwingungen und die lineare Kette , 1955 .

[40]  Germany,et al.  Neural network interatomic potential for the phase change material GeTe , 2012, 1201.2026.

[41]  Cristina H. Amon,et al.  Predicting phonon dispersion relations and lifetimes from the spectral energy density , 2010 .

[42]  M. Born,et al.  The Stability of Crystal Lattices , 1940, Mathematical Proceedings of the Cambridge Philosophical Society.

[43]  P C Howell,et al.  Comparison of molecular dynamics methods and interatomic potentials for calculating the thermal conductivity of silicon. , 2012, The Journal of chemical physics.