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First principles study of elastic and mechanical properties of TlBr and TlCl compounds
Journal of Molecular Structure 1200 (2020) 127150
Contents lists available at ScienceDirect
Journal of Molecular Structure journal homepage: http://www.elsevier.com/locate/molstruc
First principles study of elastic and mechanical properties of TlBr and TlCl compounds ur b, G. Ug ur b, E. Güler a, * M. Güler a, S¸. Ug a b
Ankara Hacı Bayram Veli University, Department of Physics, Ankara, Turkey Gazi University, Department of Physics, Ankara, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 July 2019 Received in revised form 25 September 2019 Accepted 27 September 2019 Available online 30 September 2019
We performed first principles calculations to determine the typical cubic elastic constants (C11, C12 and C44), bulk modulus (B), shear modulus (G) and Young’s modulus (E) of TlBr and TlCl compounds. We employed two different GGA functionals, the Perdew-Wang 1991 (PW91) and the Perdew-BurkeErnzerhof (PBE) for the exchange and correlation energy to present calculations. Our results for the elastic constants, bulk modulus, shear modulus and Young’s modulus of TlBr and TlCl compounds were compared both with former experiments and theoretical data. Present results are found to be satisfactory and better than some earlier literature data. © 2019 Elsevier B.V. All rights reserved.
Keywords: TlBr TlCl DFT GGA Elastic Mechanical
1. Introduction In the last few decades, the computational modeling of materials has been a powerful and fast tool to address the unclear issues of physical interest. On the other hand, elastic, mechanical, phonon and thermodynamical properties of materials are considerable facts which are important for recent industry and technological applications. These properties can be predicted through computational surveys. The results of these computations help to predict new materials and can replace the very expensive experiments that are even impossible in the laboratory [1e6]. Among the technologically important materials, thallium halides such as thallium chloride (TlCl) and thallium bromide (TlBr) have received great attention for their fundamental physics and optoelectronic applications. Their high dielectric constants make them very useful. For applications as radiation detectors and as new optical fibre crystals. These compounds (TlCl and TlBr) both crystallize in the cubic CsCl (B2) structure. Recently, extensive computational literature has been published on the structural, elastic, optical, electronic and phonon properties of TlCl and TlBr [7e12] to capture best
* Corresponding author. E-mail address: [email protected] (E. Güler). https://doi.org/10.1016/j.molstruc.2019.127150 0022-2860/© 2019 Elsevier B.V. All rights reserved.
experimental results. We have performed a study of the structural, elastic, electronic and phonon properties of TlCl and TlBr, to provide a better basis for further experimental and theoretical investigations. Du and Singh performed DFT calculations to study the electronic-structure and lattice-dynamics of thallium halides with two generalized gradient approximations GGA. They employed PBE exchange-correlation functionals for total-energy and latticedynamics calculations, while they Engel-Vosko GGA was used for electronic-structure calculation of thallium halides. They concluded that Born effective charges in TlBr and TlCl compounds are more than twice larger than the nominal ionic charges [7]. Besides, Tiwari and his coworkers studied the harmonic dynamical behavior of thallium halides (TlBr and TlCl) by using the new van der Waals three-body force shell model (VTSM) and concluded that the inclusion of van der Waals interaction is very essential for the complete description of the phonon dynamical behavior of TlBr and TlCl ur et al. employed local density compounds [8]. In 2013, Ug approximation (LDA) of DFT to TlBr and TlCl compounds and found somewhat overestimating cubic elastic constants values [9]. Tiwari et al. also investigated the lattice dynamics of TlBr with van der Waals three-body force shell model (VTSM) by modifying the three-body force shell model (TSM) for this type of crystals. They hold good agreement between the theoretical and experimental results [10]. Cowley and Okazaki measured the frequencies of a
2
M. Güler et al. / Journal of Molecular Structure 1200 (2020) 127150
representative set of normal modes of vibration of a single crystal of TlBr at room temperature and at 100 K using inelastic neutron scattering techniques and found that the most of the frequencies are almost independent of temperature, but some modes seem to display marked broadening of the neutron groups at the higher temperatures [11]. Amran et al. reported ab initio calculations of the optical properties of TlBr and TlCl binary semiconductor compounds using the self-consistent scalar relativistic full potential linear augmented plane wave band method (FP-LAPW) within the local density approximation (LDA) including the generalized gradient approximation (GGA). They predicted optical constants of these compounds and obtained well agreement within the available experimental data [12]. As very clear from the above the literature review, many experimental and theoretical attempts have been performed to understand the lattice dynamical, optical and phonon properties of TlBr and TlCl compounds. However, there still is not enough study especially on the elastic and mechanical properties of TlBr and TlCl compounds. Therefore, in this work we focused on the elastic and mechanical properties of TlBr and TlCl compounds to contribute the existing but scant literature of these compounds. The following part of the paper gives the computational details in “Section II”. We also compare our results with the experimental data and previous theoretical findings through the results and discussion part of the paper in “Section III”. Finally, “Section IV” summarizes the main findings of this work in the conclusions.
Fig. 1. C11 elastic constant of TlBr compound.
2. Computational details We employed the first-principles computational approach to present calculations based on the density functional theory (DFT). All calculations are performed with the Cambridge Serial Total Energy Package (CASTEP) code [13]. The details of the parameter settings during our calculations were summarized as in the followings. Exchange correlation functional is selected to be as generalized gradient approximation of Perdew-Burke-Ernzerhof (GGA-PBE) [14] and Perdew-Wang 1991 (PW-91) [15]. As well, pseudo-potential was chosen as the ultra soft pseudo-potential with 500 eV cut-off energy and k mesh was set to be as 7 7 7 using Monkhorst-Pack scheme through the Broyden-FletcherGoldfarb-Shanno (BFGS) geometry optimization. The selfconsistent convergence total energy was chosen to be as 5 106 eV/atom where maximum force, maximum displacement and maximum stress were set to be as 0.01 eV/Å, 5 104 Å and 0.02 GPa, respectively.
Fig. 2. C12 elastic constant of TlBr compound.
3. Results and discussion As is well known, for a cubic material, due to crystal symmetry, there exists only distinct elastic constants namely C11, C12 and C44 which can be used to elucidate the structural (dynamical) stability of a given crystal structure [16e20]. Physically, C11 defines the longitudinal elastic behavior, whereas C12 and C44 expose the offdiagonal and shear elastic feature of cubic crystals related with shearing, respectively [16e20]. Fig. 1, Fig. 2 and Fig. 3 shows the typical cubic elastic constants C11, C12 and C44 of TlBr. As seen in the comparative bar graphs, presently employed GGA the PerdewWang 1991 (PW91) functional provides satisfactory results for the C11, C12 and C44 elastic constants which are about experiments [21] and better than previously overestimating local density approximation (LDA) results. Furthermore, presently employed other GGA Perdew-Burke-Ernzerhof (PBE) functional slightly underestimates the experimental [21] results of the cubic elastic constants of the TlBr compound. Similarly, Fig. 4, Fig. 5 and Fig. 6 denote the comparative bar graphs for cubic elastic constants of TlCl
Fig. 3. C44 elastic constant of TlBr compound.
compound. For C11 elastic constant of TlCl compound, presently employed both functionals (PBE and PW91) provide reasonable results with experiments where they quietly underestimate the experimental values of C12 and C44 elastic constants of TlCl compound. As another significant result, according to Born structural
M. Güler et al. / Journal of Molecular Structure 1200 (2020) 127150
Fig. 4. C11 elastic constant of TlCl compound.
3
From mechanical view point of the materials, bulk modulus (B), is an essential elastic constant connected to the bonding strength and is used as a primary parameter for the calculation of a material’s hardness. In addition, shear modulus (G), is the measure of the resistivity of a material after applying a shearing force. As well, Young’s modulus (E), defines the amount of a material’s resistance to uniaxial tensions [16e20]. All these three distinct moduli (B, G and E) are valuable parameters for classifying the mechanical properties of materials. Table 1 and Table 2 Summarize our present findings for some mechanical properties of TlBr and TlCl compounds, respectively. As is clear, from Table 1, computed values with PW91 functional for B, G and E of TlBr compound are highly reasonable when compared with the former experimental [21] results. Again, presently obtained PBE functional somewhat underestimate the experimental data of B, G and E of TlBr compound. For TlCl compound, both functionals underestimate the experimental findings of B, G, E where PW91 functional provides much closer results to experiments than PBE. Ductility and brittleness play a major role during materials production. So, we also evaluated the ductile (brittle) characteristics of TlBr and TlCl compounds. The adjectives brittle and ductile imply the two separate mechanical properties of solids when they subjected to external stress. In general, brittle materials are not deformable or less deformable before fracture. Contrarily, ductile materials are deformable a lot before fracture. For a separation, Pugh ratio is a determinative limit for ductile (brittle) behavior of materials and has a common use in literature. If the B/G ratio of a material is about 1.75 and higher, the material is ductile; otherwise, the material becomes brittle [16e20]. B/G ratio analysis of TlBr gives 2.87 and TlCl gives 2.40 after PW91 calculations. These values (2.87 for TlBr and 2.40 for TlCl) strongly suggest that both compounds (TlBr and TlCl) have a ductile character.
4. Conclusions
Fig. 5. C12 elastic constant of TlCl compound.
Following findings can be concluded from our present work given as in below: The employed PW91 exchange correlation functional used in this work, well capture the considered elastic and mechanical properties of both TlBr and TlCl compounds (Figs. 1e6 and Tables 1 and 2). From a quantitative evaluation, the obtained results of this work approximate the former experimental valuesdin particular for elastic constants and other mechanical properties of the studied compounds. Both compounds satisfy the mechanical and cubic stability conditions with their ductile character. Presently obtained better results of this work may especially be helpful to future studies regarding the high-pressure elastic, mechanical, phononic and thermodynamic and other related properties of TlBr and TlCl
Table 1 A comparison for some mechanical properties of TlBr compound. Parameter
Exp [21]
This study GGA PW 91
This study GGA PBE
B (GPa) G (GPa) E (GPa)
22.40 8.90 23.58
22.34 7.78 30.41
13.72 4.90 12.96
Fig. 6. C44 elastic constant of TlCl compound.
stability, cubic elastic constants must satisfy C11eC12 > 0, C11 > 0, C44 > 0, C11 þ 2 C12 > 0 and cubic stability exemplary, C12
Table 2 A comparison for B, G and E values of TlCl compound. Parameter
Exp [21]
This study GGA PW 91
This study GGA PBE
B (GPa) G (GPa) E (GPa)
23.77 9.32 24.73
20.34 8.47 33.02
16.32 6.31 16.64
4
M. Güler et al. / Journal of Molecular Structure 1200 (2020) 127150
compounds. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
M. Güler, E. Güler, Crystals 7 (2017) 161, and references therein. M. Güler, E. Güler, Crystals 7 (2017) 164, and references therein. E. Güler, M. Güler, Chin. J. Phys. 53 (2015), 040807. E. Güler, M. Güler, Braz. J. Phys. 45 (2015) 296e301. E. Güler, M. Güler, Ann. Mater. Sci. Eng. (2013), 525673. E. Güler, M. Güler, Int. J. Mod. Phys. B 29 (2015) 15550222. M.H. Du, D.J. Singh, Phys. Rev. B 81 (2010) 144114. S.K. Tiwari, L.J. Shukla, K.S. Upadhyaya, Pramana - J. Phys. 74 (2010) 793. € Ko €ker, Turkish physical society, in: 30th International G. Ugur, S. Ugur, O. Conference on Physics, 2013. [10] S.K. Tiwari, L.K. Pandey, L.J. Shukla, K.S. Upadhyaya, Phys. Scr. 80 (2009), 065603. [11] E.R. Cowley, A. Okazaki, Proc. R. Soc.Lond. Ser A Math. Phys. Sci. 300 (1460)
(1967) 45. [12] N. Amrane, M. Benkraouda, F. Hamed, World J. Condens. Matter Phys. 1 (2011) 63. [13] S.J. Clark, M.D. Segall, C.J. Pickard, P.J. Hasnip, M.J. Probert, K. Refson, M.C. Payne, Zeitschrift fuer Kristallographie 220 (5e6) (2005) 567e570. [14] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1997) 1396, 3865 e Published 28 October 1996; Erratum Phys. Rev. Lett. 78. [15] Y. Wang, J.P. Perdew, Phys. Rev. B 44 (24) (1991) 13298. [16] M. Güler, E. Güler, Mater. Res. Express 3 (2016), 075901. [17] M. Güler, E. Güler, Braz. J. Phys. 46 (2016) 192e197. [18] E. Güler, M. Güler, Appl.Phys. A 119 (2) (2015) 721e726. [19] E. Güler, M. Güler, Math. Res. 17 (5) (2014) 1268e1272. [20] M. Güler, E. Güler, J. Optoelectron. Adv. Mater. 16 (2014) 1222e1227. €rnstein Numerical Data and Functional [21] A.G. Every, A.K. McCurdy, Landolt-Bo Relationships in Science and Technology, New Series, Group III: Crystal and Solid State Physics. Low Frequency Properties of Dielectric Crystals, Subvolume a: Second and Higher Order Elastic Constants, vol. 29, Springer-Verlag, 1992.
Contents lists available at ScienceDirect
Journal of Molecular Structure journal homepage: http://www.elsevier.com/locate/molstruc
First principles study of elastic and mechanical properties of TlBr and TlCl compounds ur b, G. Ug ur b, E. Güler a, * M. Güler a, S¸. Ug a b
Ankara Hacı Bayram Veli University, Department of Physics, Ankara, Turkey Gazi University, Department of Physics, Ankara, Turkey
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 July 2019 Received in revised form 25 September 2019 Accepted 27 September 2019 Available online 30 September 2019
We performed first principles calculations to determine the typical cubic elastic constants (C11, C12 and C44), bulk modulus (B), shear modulus (G) and Young’s modulus (E) of TlBr and TlCl compounds. We employed two different GGA functionals, the Perdew-Wang 1991 (PW91) and the Perdew-BurkeErnzerhof (PBE) for the exchange and correlation energy to present calculations. Our results for the elastic constants, bulk modulus, shear modulus and Young’s modulus of TlBr and TlCl compounds were compared both with former experiments and theoretical data. Present results are found to be satisfactory and better than some earlier literature data. © 2019 Elsevier B.V. All rights reserved.
Keywords: TlBr TlCl DFT GGA Elastic Mechanical
1. Introduction In the last few decades, the computational modeling of materials has been a powerful and fast tool to address the unclear issues of physical interest. On the other hand, elastic, mechanical, phonon and thermodynamical properties of materials are considerable facts which are important for recent industry and technological applications. These properties can be predicted through computational surveys. The results of these computations help to predict new materials and can replace the very expensive experiments that are even impossible in the laboratory [1e6]. Among the technologically important materials, thallium halides such as thallium chloride (TlCl) and thallium bromide (TlBr) have received great attention for their fundamental physics and optoelectronic applications. Their high dielectric constants make them very useful. For applications as radiation detectors and as new optical fibre crystals. These compounds (TlCl and TlBr) both crystallize in the cubic CsCl (B2) structure. Recently, extensive computational literature has been published on the structural, elastic, optical, electronic and phonon properties of TlCl and TlBr [7e12] to capture best
* Corresponding author. E-mail address: [email protected] (E. Güler). https://doi.org/10.1016/j.molstruc.2019.127150 0022-2860/© 2019 Elsevier B.V. All rights reserved.
experimental results. We have performed a study of the structural, elastic, electronic and phonon properties of TlCl and TlBr, to provide a better basis for further experimental and theoretical investigations. Du and Singh performed DFT calculations to study the electronic-structure and lattice-dynamics of thallium halides with two generalized gradient approximations GGA. They employed PBE exchange-correlation functionals for total-energy and latticedynamics calculations, while they Engel-Vosko GGA was used for electronic-structure calculation of thallium halides. They concluded that Born effective charges in TlBr and TlCl compounds are more than twice larger than the nominal ionic charges [7]. Besides, Tiwari and his coworkers studied the harmonic dynamical behavior of thallium halides (TlBr and TlCl) by using the new van der Waals three-body force shell model (VTSM) and concluded that the inclusion of van der Waals interaction is very essential for the complete description of the phonon dynamical behavior of TlBr and TlCl ur et al. employed local density compounds [8]. In 2013, Ug approximation (LDA) of DFT to TlBr and TlCl compounds and found somewhat overestimating cubic elastic constants values [9]. Tiwari et al. also investigated the lattice dynamics of TlBr with van der Waals three-body force shell model (VTSM) by modifying the three-body force shell model (TSM) for this type of crystals. They hold good agreement between the theoretical and experimental results [10]. Cowley and Okazaki measured the frequencies of a
2
M. Güler et al. / Journal of Molecular Structure 1200 (2020) 127150
representative set of normal modes of vibration of a single crystal of TlBr at room temperature and at 100 K using inelastic neutron scattering techniques and found that the most of the frequencies are almost independent of temperature, but some modes seem to display marked broadening of the neutron groups at the higher temperatures [11]. Amran et al. reported ab initio calculations of the optical properties of TlBr and TlCl binary semiconductor compounds using the self-consistent scalar relativistic full potential linear augmented plane wave band method (FP-LAPW) within the local density approximation (LDA) including the generalized gradient approximation (GGA). They predicted optical constants of these compounds and obtained well agreement within the available experimental data [12]. As very clear from the above the literature review, many experimental and theoretical attempts have been performed to understand the lattice dynamical, optical and phonon properties of TlBr and TlCl compounds. However, there still is not enough study especially on the elastic and mechanical properties of TlBr and TlCl compounds. Therefore, in this work we focused on the elastic and mechanical properties of TlBr and TlCl compounds to contribute the existing but scant literature of these compounds. The following part of the paper gives the computational details in “Section II”. We also compare our results with the experimental data and previous theoretical findings through the results and discussion part of the paper in “Section III”. Finally, “Section IV” summarizes the main findings of this work in the conclusions.
Fig. 1. C11 elastic constant of TlBr compound.
2. Computational details We employed the first-principles computational approach to present calculations based on the density functional theory (DFT). All calculations are performed with the Cambridge Serial Total Energy Package (CASTEP) code [13]. The details of the parameter settings during our calculations were summarized as in the followings. Exchange correlation functional is selected to be as generalized gradient approximation of Perdew-Burke-Ernzerhof (GGA-PBE) [14] and Perdew-Wang 1991 (PW-91) [15]. As well, pseudo-potential was chosen as the ultra soft pseudo-potential with 500 eV cut-off energy and k mesh was set to be as 7 7 7 using Monkhorst-Pack scheme through the Broyden-FletcherGoldfarb-Shanno (BFGS) geometry optimization. The selfconsistent convergence total energy was chosen to be as 5 106 eV/atom where maximum force, maximum displacement and maximum stress were set to be as 0.01 eV/Å, 5 104 Å and 0.02 GPa, respectively.
Fig. 2. C12 elastic constant of TlBr compound.
3. Results and discussion As is well known, for a cubic material, due to crystal symmetry, there exists only distinct elastic constants namely C11, C12 and C44 which can be used to elucidate the structural (dynamical) stability of a given crystal structure [16e20]. Physically, C11 defines the longitudinal elastic behavior, whereas C12 and C44 expose the offdiagonal and shear elastic feature of cubic crystals related with shearing, respectively [16e20]. Fig. 1, Fig. 2 and Fig. 3 shows the typical cubic elastic constants C11, C12 and C44 of TlBr. As seen in the comparative bar graphs, presently employed GGA the PerdewWang 1991 (PW91) functional provides satisfactory results for the C11, C12 and C44 elastic constants which are about experiments [21] and better than previously overestimating local density approximation (LDA) results. Furthermore, presently employed other GGA Perdew-Burke-Ernzerhof (PBE) functional slightly underestimates the experimental [21] results of the cubic elastic constants of the TlBr compound. Similarly, Fig. 4, Fig. 5 and Fig. 6 denote the comparative bar graphs for cubic elastic constants of TlCl
Fig. 3. C44 elastic constant of TlBr compound.
compound. For C11 elastic constant of TlCl compound, presently employed both functionals (PBE and PW91) provide reasonable results with experiments where they quietly underestimate the experimental values of C12 and C44 elastic constants of TlCl compound. As another significant result, according to Born structural
M. Güler et al. / Journal of Molecular Structure 1200 (2020) 127150
Fig. 4. C11 elastic constant of TlCl compound.
3
From mechanical view point of the materials, bulk modulus (B), is an essential elastic constant connected to the bonding strength and is used as a primary parameter for the calculation of a material’s hardness. In addition, shear modulus (G), is the measure of the resistivity of a material after applying a shearing force. As well, Young’s modulus (E), defines the amount of a material’s resistance to uniaxial tensions [16e20]. All these three distinct moduli (B, G and E) are valuable parameters for classifying the mechanical properties of materials. Table 1 and Table 2 Summarize our present findings for some mechanical properties of TlBr and TlCl compounds, respectively. As is clear, from Table 1, computed values with PW91 functional for B, G and E of TlBr compound are highly reasonable when compared with the former experimental [21] results. Again, presently obtained PBE functional somewhat underestimate the experimental data of B, G and E of TlBr compound. For TlCl compound, both functionals underestimate the experimental findings of B, G, E where PW91 functional provides much closer results to experiments than PBE. Ductility and brittleness play a major role during materials production. So, we also evaluated the ductile (brittle) characteristics of TlBr and TlCl compounds. The adjectives brittle and ductile imply the two separate mechanical properties of solids when they subjected to external stress. In general, brittle materials are not deformable or less deformable before fracture. Contrarily, ductile materials are deformable a lot before fracture. For a separation, Pugh ratio is a determinative limit for ductile (brittle) behavior of materials and has a common use in literature. If the B/G ratio of a material is about 1.75 and higher, the material is ductile; otherwise, the material becomes brittle [16e20]. B/G ratio analysis of TlBr gives 2.87 and TlCl gives 2.40 after PW91 calculations. These values (2.87 for TlBr and 2.40 for TlCl) strongly suggest that both compounds (TlBr and TlCl) have a ductile character.
4. Conclusions
Fig. 5. C12 elastic constant of TlCl compound.
Following findings can be concluded from our present work given as in below: The employed PW91 exchange correlation functional used in this work, well capture the considered elastic and mechanical properties of both TlBr and TlCl compounds (Figs. 1e6 and Tables 1 and 2). From a quantitative evaluation, the obtained results of this work approximate the former experimental valuesdin particular for elastic constants and other mechanical properties of the studied compounds. Both compounds satisfy the mechanical and cubic stability conditions with their ductile character. Presently obtained better results of this work may especially be helpful to future studies regarding the high-pressure elastic, mechanical, phononic and thermodynamic and other related properties of TlBr and TlCl
Table 1 A comparison for some mechanical properties of TlBr compound. Parameter
Exp [21]
This study GGA PW 91
This study GGA PBE
B (GPa) G (GPa) E (GPa)
22.40 8.90 23.58
22.34 7.78 30.41
13.72 4.90 12.96
Fig. 6. C44 elastic constant of TlCl compound.
stability, cubic elastic constants must satisfy C11eC12 > 0, C11 > 0, C44 > 0, C11 þ 2 C12 > 0 and cubic stability exemplary, C12
Table 2 A comparison for B, G and E values of TlCl compound. Parameter
Exp [21]
This study GGA PW 91
This study GGA PBE
B (GPa) G (GPa) E (GPa)
23.77 9.32 24.73
20.34 8.47 33.02
16.32 6.31 16.64
4
M. Güler et al. / Journal of Molecular Structure 1200 (2020) 127150
compounds. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
M. Güler, E. Güler, Crystals 7 (2017) 161, and references therein. M. Güler, E. Güler, Crystals 7 (2017) 164, and references therein. E. Güler, M. Güler, Chin. J. Phys. 53 (2015), 040807. E. Güler, M. Güler, Braz. J. Phys. 45 (2015) 296e301. E. Güler, M. Güler, Ann. Mater. Sci. Eng. (2013), 525673. E. Güler, M. Güler, Int. J. Mod. Phys. B 29 (2015) 15550222. M.H. Du, D.J. Singh, Phys. Rev. B 81 (2010) 144114. S.K. Tiwari, L.J. Shukla, K.S. Upadhyaya, Pramana - J. Phys. 74 (2010) 793. € Ko €ker, Turkish physical society, in: 30th International G. Ugur, S. Ugur, O. Conference on Physics, 2013. [10] S.K. Tiwari, L.K. Pandey, L.J. Shukla, K.S. Upadhyaya, Phys. Scr. 80 (2009), 065603. [11] E.R. Cowley, A. Okazaki, Proc. R. Soc.Lond. Ser A Math. Phys. Sci. 300 (1460)
(1967) 45. [12] N. Amrane, M. Benkraouda, F. Hamed, World J. Condens. Matter Phys. 1 (2011) 63. [13] S.J. Clark, M.D. Segall, C.J. Pickard, P.J. Hasnip, M.J. Probert, K. Refson, M.C. Payne, Zeitschrift fuer Kristallographie 220 (5e6) (2005) 567e570. [14] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1997) 1396, 3865 e Published 28 October 1996; Erratum Phys. Rev. Lett. 78. [15] Y. Wang, J.P. Perdew, Phys. Rev. B 44 (24) (1991) 13298. [16] M. Güler, E. Güler, Mater. Res. Express 3 (2016), 075901. [17] M. Güler, E. Güler, Braz. J. Phys. 46 (2016) 192e197. [18] E. Güler, M. Güler, Appl.Phys. A 119 (2) (2015) 721e726. [19] E. Güler, M. Güler, Math. Res. 17 (5) (2014) 1268e1272. [20] M. Güler, E. Güler, J. Optoelectron. Adv. Mater. 16 (2014) 1222e1227. €rnstein Numerical Data and Functional [21] A.G. Every, A.K. McCurdy, Landolt-Bo Relationships in Science and Technology, New Series, Group III: Crystal and Solid State Physics. Low Frequency Properties of Dielectric Crystals, Subvolume a: Second and Higher Order Elastic Constants, vol. 29, Springer-Verlag, 1992.