Mechanical properties and stability of metallic materials at extreme conditions, particularly high and ultra high pressures, are of increasing interest due to its significance for the practical applications in various fields of industry including energy industry, machinery as well as in geology and resource-extraction and many more. The mechanical stability and properties of metallic materials under high pressures are also of great interest for the fundamental science, e.g. due to the rapid development of the experimental methods in this field. Indeed the highest static pressure obtained in the laboratory at present is 600 GPa [1]. These achievements motivate the use the computer modeling to study the high-pressure behavior of materials. Moreover opportunities provided by ab initio calculations help to obtain the fundamental knowledge and discover new effects in materials, which occur at extreme conditions.
The present study is dedicated to the investigation of the mechanical and dynamical stability, as well as elastic, dynamical and electronic properties of the transition metals at high pressure. We calculate their elastic constants in the framework of the Density Functional Theory (DFT), using state-of-the-art methods, as well as methods specially developed for this study. Within the former, the elastic constants are obtained as the second derivatives of the free energy with respect to the components of the infinitesimal strain tensor [2]. However this method does not allow to take into account the contribution of the nonlinear elastic effects, which are described by the higher order elastic constants. In the method developed in [3] the elastic constants of second and third order are obtained from the energy-stress relation as second and third derivatives of the total energy, however with respect to the Lagrangian finite strain tensor. The energy values, required for the analysis are obtained ab inito, using DFT.
Knowing the higher order elastic constants, one can apply the Landau theory of phase transitions and group theory to predict the symmetry of the high pressure phase. These predictions are limited to the special type of phase transitions – elastic phase transitions which are associated with the loss of stability of the crystal lattice to the uniform shear strain. The high pressure phase in this case has a lower symmetry than the parent phase. The phase transitions of this type are diffusionless and are analogues to the well-known martensitic phase transitions.
Elastic phase transitions under pressure are common in the alkali metals [4]. The (bcc rhombohedral) phase transition in V at ~70 GPa [5] shows that they are possible in transition metals as well. The mechanism of this transition was studied using the developed method. The obtained values of second, third and the estimation of fourth order elastic constants were used in the Landau theory analysis and provided the value of the critical pressure as well as of the type and the high-pressure structure of the phase transition that agree with experiment quite well.
The softening of the elastic constants does not always lead to the elastic phase transition, however it may be a precursor for another transformation with a different mechanism. This case is described in [6]. Mo shows significant softening of the elastic constant C’, which does not result into the elastic phase transition, instead the softening indicates the low energy barrier for the bccdhcp transition. Combined with the phonon dispersions investigation the analysis of the pressure dependence of allowed us to understand the mechanism of this transformation.
Important in itself, the character of the elastic behavior may help to discover and understand the peculiarities of the electronic structure. For example Ru [7] demonstrates the outstanding mechanical stability with increasing pressure. This feature makes Ru an interesting material for the application in the high-pressure experiments. The fundamental reason for such behavior was found in the specific form of the pseudogap of the electronic density of states. Ta [8] and Nb both demonstrate the pronounced softening of the elastic constants with pressure. In case of Nb the softening is present in , in case of Ta in the both and . The literature search and preliminary results show that also these peculiarities originate from the specific features of the electronic structure of these elements.
For all the mentioned elements the full sets of the elastic constants of second and third order, as well as the phonon dispersions for V, Mo and Ru in the gigapascal pressure range are calculated [3, 6, 7, 8, 9].
This work was supported by Grant of Russian Federation Ministry for Science and Education (grant No. 14.Y26.31.0005).
References.
[1] Dubrovinsky, L., Dubrovinskaia, N., Prakapenka, V. B., & Abakumov, A. M. (2012). Nat. Commun., 3, 1163.
[2] G. Grimvall, B. Magyari-Kцpe, V. Ozolinš, K.A. Persson, Rev. Mod. Phys. 84 (2012) 945–986.
[3] Krasilnikov, O. M., Vekilov, Y. K., Mosyagin, I. Y., Isaev, E. I., & Bondarenko, N. G. (2012). J. Phys.: Condens. Matter, 24(19), 195402.
[4] Maksimov E. G., Magnitskaya M. V., Fortov V. E. (2005). Phys. Usp. 48 761–780.
[5] Ding, Y., Ahuja, R., Shu, J., Chow, P., Luo, W., & Mao, H. (2007). Phys. Rev. Lett., 98(8), 085502.
[6] Krasilnikov, O. M., Belov, M. P., Lugovskoy, a. V., Mosyagin, I. Y., & Vekilov, Y. K. (2014). COMMAT, 81, 313–318.
[7] A. V. Lugovskoy, M. P. Belov, Y. K. Vekilov, and O. M. Krasilnikov, (2014), J. Phys.: Conf. Ser. 490, 012059.
[8] Krasil’nikov, O. M., Vekilov, Y. K., & Mosyagin, I. Y. (2012). JETP, 115(2), 237–241.
[9] Krasilnikov, O. M., Vekilov, Y. K., Lugovskoy, A. V, Mosyagin, I. Y., Belov, M. P., & Bondarenko, N. G. (2014). Journal of Alloys and Compounds, 586, S242–S245.