Editor’s Choice
An efficient ab-initio quasiharmonic approach for the thermodynamics of solids

https://doi.org/10.1016/j.commatsci.2016.04.012Get rights and content

Abstract

A first-principles approach called the self-consistent quasiharmonic approximation (SC-QHA) method is formulated to calculate the thermal expansion, thermomechanics, and thermodynamic functions of solids at finite temperatures with both high efficiency and accuracy. The SC-QHA method requires fewer phonon calculations than the conventional QHA method, and also facilitates the convenient analysis of the microscopic origins of macroscopic thermal phenomena. The superior performance of the SC-QHA method is systematically examined by comparing it with the conventional QHA method and experimental measurements on silicon, diamond, and alumina. It is then used to study the effects of pressure on the anharmonic lattice properties of diamond and alumina. The thermal expansion and thermomechanics of Ca3Ti2O7, which is a recently discovered important ferroelectric ceramic with a complex crystal structure that is computationally challenging for the conventional QHA method, are also calculated using the formulated SC-QHA method. The SC-QHA method can significantly reduce the computational expense for various quasiharmonic thermal properties especially when there are a large number of structures to consider or when the solid is structurally complex. It is anticipated that the algorithm will be useful for a variety of fields, including oxidation, corrosion, high-pressure physics, ferroelectrics, and high-throughput structure screening when temperature effects are required to accurately describe realistic properties.

Introduction

Accurately simulating various anharmonic properties, i.e., thermal expansion and thermomechanics, of solids is important for obtaining a deep understanding of their plentiful thermal behaviors and for their realistic applications. The anharmonic properties can be derived from the volume and temperature dependences of the phonon spectra calculated using density-functional theory (DFT) [1]. The most popular approach is the quasiharmonic approximation method (QHA) [2], [3], [4], where only the volume dependence is considered for the phonon anharmonicity, and temperature is assumed to indirectly affect phonon vibrational frequencies through thermal expansion. Here, the phonon spectra of about ten or more volumes are usually required for a typical QHA simulation, and the thermal expansion and thermomechanics are derived by fitting the free energy-volume relationship. In some cases, e.g., at high temperatures, high-order anharmonicity caused by multi-phonon coupling cannot be omitted as in the QHA method, and some more complicated and time-consuming methods, e.g., molecular dynamics [5], [6], [7], [8], [9], [10], [11], [12], self-consistent ab initio lattice dynamics [13], [14], perturbative/nonperturbative renormalized harmonic approximations [15], [16], [17], [18], and vibrational self-consistent field calculations [19], can be used to obtain the temperature-dependent phonons. Nonetheless, approximately ten or more volumes of such phonon spectra are also required to accurately calculate the thermal expansion and thermomechanics with the high-order anharmonicities.

Phonon calculations based on DFT forces are always time consuming, and prior to the actual calculation, various computational parameters [1], [20] also need to be carefully tested to ensure convergence of the vibrational frequencies and anharmonicity, including the pseudopotentials, cutoff energy, k-mesh density, energy and force convergence thresholds, and supercell size in the small-displacement method [21], [22] or the q-mesh density in the density-functional perturbation theory approach [23], [24]. The general rule-of-thumb requiring ten or more volumes will make the anharmonic simulation, even when utilizing the simplest QHA method, rather computationally expensive, especially in some condensed matter fields where a large number of structures must be considered or the compound under study has a large unit cell, low symmetry, and numerous inequivalent atoms:

  • (i)

    In the fields of solid oxidation and corrosion, there are always many compounds (elements, oxides, hydroxides, oxyhydroxides, etc.) to consider [25], [26], [27], [28], [29], [30] and each composition may have many polymorphs [31], [32], [33].

  • (ii)

    In the high-pressure physics field, not only a wide range of volumes but also a large number of complex phases should be examined [34], [35], [36].

  • (iii)

    For the metallic alloys field, the thermodynamics and mechanics of many phases at variable composition and temperature are always of concern [37], [38], [39].

  • (iv)

    In the perovskite oxides [40], [41], [42], ternary ceramics exhibit complex structures and large unit cells. The phonon calculations for an individual structure is already quite time consuming, not to mention the calculation of anharmonic properties in low-symmetry polymorphs.

  • (v)

    In high-throughput screening and materials design [43], [44], [45], [46] when including temperature effects, a huge number of compositions and structures should be calculated with a high efficiency-to-accuracy ratio.

To this end, these diverse fields require an efficient method to accelerate the investigation of the anharmonic properties of related solids at finite temperatures.

In this work, we formulate an ab initio method, called the self-consistent quasiharmonic approximation (SC-QHA) method, for achieving fast anharmonic calculations with high accuracy within the quasiharmonic approximation. Only the phonon spectra of two or three volumes are required in a SC-QHA calculation, which usually is much faster than the conventional QHA method. We carefully test the SC-QHA method using prototypical silicon, diamond, and alumina, and then also study the pressure effect on the anharmonic properties of diamond and alumina. Finally, we apply the SC-QHA method to accurately calculate the thermal expansion and thermomechanics of the structurally complex hybrid-improper ferroelectric Ca3Ti2O7. Apart from the high efficiency, we show that the SC-QHA method is also very convenient for deciphering the microscopic physical origins of lattice dynamical and thermodynamic phenomena. Moreover, it can be readily transferred beyond the quasiharmonic realm to speed up the accurate first-principles simulation of thermal effects for the benefit of multiple fields in condensed-matter physics.

Section snippets

Theoretical basis

The total Gibbs free energy (Gtot=Ftot+PV) of a crystal unit cell is expressed asGtot(P,T)=Fe(V,T)+Fph(V,T)+PV,where P,V=V(P,T), and T are the external pressure, unit-cell volume, and temperature, respectively; Fe and Fph are the electronic and phononic Helmholtz free energies, respectively. To conveniently present the basic algorithm and efficiency of the SC-QHA method, only nonmagnetic insulators are considered here, where the electronic excitation and magnetic excitations are neglected. The

Algorithm benchmarks

The calculated volume thermal-expansion coefficient (α), isobaric heat capacity (Cp=Cv+TVTBTα2) [59], and isothermal bulk modulus (BT) obtained from the 1st-SC-QHA, 2nd-SC-QHA, and QHA methods for silicon (Si), diamond (C), and alumina (Al2O3) under zero pressure and at temperatures from 0 K to the melting point, Tm, are shown in Fig. 3. The low-temperature variations in α(T) and detailed analysis of BT are given in Fig. 4. The available experimental results for Si, C, and Al2O3 [89], [90], [91]

Conclusions

A fast and accurate ab initio method called the self-consistent quasiharmonic approximation (SC-QHA) method has been formulated to calculate various anharmonic properties of solids at finite temperatures. The SC-QHA method not only is about five times per dimension faster than the conventional QHA method, but also aids in the physical analysis of underlying anharmonic mechanisms. Although we showed the superior performance of the SC-QHA method compared to the conventional QHA using nonmagnetic

Contributions

The study was planned, methods formulated, calculations carried out, and the manuscript prepared by L.-F.H. and J.M.R. X.-Z.L. and E.T. performed analyses on Ca3Ti2O7 and Al2O3, respectively. All authors discussed the results, wrote, and commented on the manuscript.

Acknowledgments

L.-F.H. and J.M.R. wish to thank Dr. M. Senn (University of Oxford), and Prof. S.-W. Cheong (Rutgers University) for providing additional experimental data for Ca3Ti2O7, as well as for the helpful e-mail exchanges. L.-F.H. and E.T. were supported by the Office of Naval Research MURI “Understanding Atomic Scale Structure in Four Dimensions to Design and Control Corrosion Resistant Alloys” under Grant No. N00014-14-1-0675. X.-Z.L. and J.M.R. were supported by the National Science Foundation (NSF)

References (124)

  • A. Togo et al.

    Scr. Mater.

    (2015)
  • P. Souvatzis et al.

    Comput. Mater. Sci.

    (2009)
  • B. Beverskog et al.

    Corros. Sci.

    (1996)
  • B. Beverskog et al.

    Corros. Sci.

    (1997)
  • B. Beverskog et al.

    Corros. Sci.

    (1997)
  • B. Beverskog et al.

    Corros. Sci.

    (1997)
  • J. Chivot et al.

    Corros. Sci.

    (2008)
  • X.-q. Li et al.

    Front. Phys.

    (2012)
  • S. Curtarolo et al.

    Comput. Mater. Sci.

    (2012)
  • L.F. Huang et al.

    Solid State Commun.

    (2014)
  • S.L. Shang et al.

    Comput. Mater. Sci.

    (2010)
  • Y. Wang et al.

    Acta Mater.

    (2004)
  • Z.K. Liu et al.

    Scr. Mater.

    (2011)
  • G. Kresse et al.

    Comput. Mater. Sci.

    (1996)
  • R.M. Martin

    Electronic Structure: Basic Theory and Practical Methods

    (2004)
  • S. Baroni et al.

    Rev. Mineral. Geochem.

    (2010)
  • A. Togo et al.

    Phys. Rev. B

    (2010)
  • D. Alfè et al.

    Phys. Rev. B

    (2001)
  • G.J. Ackland

    J. Phys.: Condens. Matter

    (2002)
  • L. Vočadlo et al.

    Phys. Rev. B

    (2002)
  • B. Grabowski et al.

    Phys. Rev. B

    (2009)
  • B. Grabowski et al.

    Phys. Status Solidi B

    (2011)
  • O. Hellman et al.

    Phys. Rev. B

    (2011)
  • L.T. Kong et al.

    Europhys. Lett.

    (2012)
  • O. Hellman et al.

    Phys. Rev. B

    (2013)
  • P. Souvatzis et al.

    Phys. Rev. Lett.

    (2008)
  • B. Rousseau et al.

    Phys. Rev. B

    (2010)
  • I. Errea et al.

    Phys. Rev. Lett.

    (2011)
  • I. Errea et al.

    Phys. Rev. Lett.

    (2013)
  • K.H. Michel et al.

    Phys. Rev. B

    (2015)
  • B. Monserrat et al.

    Phys. Rev. B

    (2013)
  • J. Hafner

    J. Comput. Chem.

    (2008)
  • G. Kresse et al.

    Europhys. Lett.

    (1995)
  • K. Parlinski et al.

    Phys. Rev. Lett.

    (1997)
  • S. Baroni et al.

    Rev. Mod. Phys.

    (2001)
  • P. Giannozzi et al.

    J. Phys.: Condens. Matter

    (2009)
  • B. Beverskog et al.

    J. Electrochem. Soc.

    (1997)
  • A. Belsky et al.

    Acta Crystallogr. B

    (2002)
  • A. Jain et al.

    APL Mater.

    (2013)
  • J. Saal et al.

    JOM

    (2013)
  • A.R. Oganov et al.

    J. Chem. Phys.

    (2006)
  • Y. Wang et al.

    Phys. Rev. B

    (2010)
  • X. Wang et al.

    Nat. Geosci.

    (2015)
  • S.B. Maisel et al.

    Nature

    (2012)
  • X.-L. Yuan et al.

    Front. Phys.

    (2014)
  • J.M. Rondinelli et al.

    MRS Bull.

    (2012)
  • N.A. Benedek et al.

    Dalton Trans.

    (2015)
  • J. Young et al.

    J. Phys.: Condens. Matter

    (2015)
  • S. Curtarolo et al.

    Nat. Mater.

    (2013)
  • J.M. Rondinelli et al.

    APL Mater.

    (2015)
  • Cited by (78)

    • Anisotropic thermal expansion and themomechanic properties of α-phase group-VA monolayers

      2024, Physics Letters, Section A: General, Atomic and Solid State Physics
    View all citing articles on Scopus
    View full text