Elsevier

Computer Physics Communications

Volume 220, November 2017, Pages 20-30
Computer Physics Communications

Ab initio calculations of pressure-dependence of high-order elastic constants using finite deformations approach

https://doi.org/10.1016/j.cpc.2017.06.008Get rights and content

Abstract

We present a description of a technique for ab initio calculations of the pressure dependence of second- and third-order elastic constants. The technique is based on an evaluation of the corresponding Lagrangian stress tensor derivative of the total energy assuming finite size of the deformations. Important details and parameters of the calculations are highlighted. Considering body-centered cubic Mo as a model system, we demonstrate that the technique is highly customizable and can be used to investigate non-linear elastic properties under high-pressure conditions.

Introduction

There is a notable increasing interest in higher order elastic constants (HOEC) of solids [1], [2], [3], [4], [5], [6]. The study of nonlinear elasticity helps to reveal the details of complex behavior of materials. For example, investigating the trends of HOECs one can understand mechanisms of structural instabilities [3] or incorporate proper description of nonlinear elasticity in case where there is a discrepancy between more general theory and experiment [4].

Usually the elasticity theory is considered in an approximation of infinitesimal deformations [7]. Such approach is the most logical choice, when the applied deformation is small compared to inter-atomic distances of undeformed material. But inter-atomic potentials in real materials are anharmonic. The explicit accounting of anharmonicity in study of solids becomes more and more prominent. This is quite evident when theoretical investigation of materials is performed for realistic conditions including the extreme ones, which are, for instance, of interest for cutting edge studies [8], [9]. Standard theoretical methods of defining components of elastic constants tensor typically operate in a linear limit of elasticity theory.

Although there are comprehensive works present in the literature, which are dedicated to finite deformation and calculation of higher order elastic constants [10], [11] the techniques, applied in these works do not take into account the external conditions, e.g. pressure and temperature. We also would like to note, that there is a lack of systemic study of numerical aspects of higher order elastic constants calculation. In these work we expand and clarify most important numeric aspects of methodology of non-linear elastic properties investigation for solids under pressure, using finite strain theory to calculate second- and third-order elastic constants under pressure. Doing so we concentrate on the algorithm and the strategy for finding optimal calculation parameters, both regular convergence tests in regard to simulation parameters, as well as internal algorithm parameters, such as deformation range. The importance of careful adjustment of the parameters naturally comes from the fact, that even the second order elastic constants require highly converged mesh of the k-points and plane wave cut-off energy [12]. Obviously the importance of this aspect increases, if the HOEC are calculated. Although discussion of convergence of third and even fourth order elastic constants calculation parameters is present in [10], [11], we additionally focus on the impact of external loading and its effect on the calculation procedure.

To demonstrate the application of this technique we choose molybdenum, a 6th group transitional metal with a body-centered cubic structure. We investigate the second and third order elastic constants in the pressure range of up to 300 GPa. The investigation of elastic properties at pressures this high is motivated by the recent breakthrough in experimental techniques of achieving static pressures [13], [14], [15]. The measurement of elastic constants at extreme static compression is a challenging task that contains some uncertainties [16]. This makes theoretically obtained elastic constants trends under pressure a valuable addition to the knowledge about studied materials. We would also like to mention, that transition metals, including molybdenum, are often used in high pressure experiments [13], [14], [15], [17], [18].

Section snippets

Method

We start with an outline of basic ideas of elasticity theory behind the proposed method. In this work we focus on calculation-specific technical issues and then give a detailed outline of the computational algorithm. A short description of a conventional infinitesimal strains method can also be found in this section for completeness. A detailed theoretical description, including the full set of deformation schemes, can be found in [19], [20], [21], [22].

Results and discussion

Using the described method, second- and third-order elastic constants were calculated in the range of volumes corresponding to pressures from 0 GPa up to 300 GPa.

Second-order constants of bcc molybdenum, C˜ and C˜44 are shown in Fig. 6. These constants were obtained using distortions from Table 1, namely S3: (η,η,0,0,0,0) and S5: (0,0,0,η,0,0). Exact expressions for distortion vectors can be found in Appendix B. In the same range of volumes, second-order elastic constants c and c44 were also

Conclusions

Higher order elastic constants can be used to describe lattice dynamics, electronic structure peculiarities, acoustic properties of material, or anharmonic effects in the vicinity of structural transformations. However, their calculations under pressure is not a trivial task. This paper presents a detailed description and careful verification of an algorithm for calculation of high-order elastic constants under pressure based on finite deformation technique, applied to molybdenum. We calculate

Acknowledgments

We are grateful for the support provided by the Swedish Foundation for Strategic Research (SSF) program SRL Grant No. 10-0026, the Swedish Research Council Grant Nos. 2015-04391 and 2014-4750, the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Linköping University (Faculty Grant SFO-Mat-LiU No 2009 00971), as well as by the Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST “MISIS”

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