First-principle calculations to investigate the elastic and thermodynamic properties of RBRh 3 (R = Sc, Y and La) perovskite compounds

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1 This article was downloaded by: [Universite de Lorraine] On: 23 February 2012, At: 12:23 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Molecular Physics: An International Journal at the Interface Between Chemistry and Physics Publication details, including instructions for authors and subscription information: First-principle calculations to investigate the elastic and thermodynamic properties of RBRh 3 (R = Sc, Y and La) perovskite compounds F. Litimein a, R. Khenata b c, A. Bouhemadou c d, Y. Al-Douri e & S. Bin Omran c a Laboratoire des Matériaux Magnétiques, Département de Physique, Université de Sidi Bel Abbés, Sidi Bel Abbés 22000, Algérie b Laboratoire de Physique Quantique et de Modélisation Mathématique (LPQ3M), Département de Technologie, Université de Mascara, Mascara, Algérie c Department of Physics and Astronomy, King Saud University, P.O Box 2455, Riyadh 11451, Saudi Arabia d Laboratory for Developing New Materials and their Characterization, Department of Physics, Faculty of Science, University of Setif, Setif, Algeria e Institute of Nono Electronic Engineering, University Malaysia Perlis, Kangar, Perlis, Malaysia Available online: 15 Nov 2011 To cite this article: F. Litimein, R. Khenata, A. Bouhemadou, Y. Al-Douri & S. Bin Omran (2012): First-principle calculations to investigate the elastic and thermodynamic properties of RBRh 3 (R = Sc, Y and La) perovskite compounds, Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 110:2, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Molecular Physics Vol. 110, No. 2, 20 January 2012, RESEARCH ARTICLE First-principle calculations to investigate the elastic and thermodynamic properties of RBRh 3 (R ^ Sc, Y and La) perovskite compounds F. Litimein a, R. Khenata bc, A. Bouhemadou cd, Y. Al-Douri e * and S. Bin Omran c a Laboratoire des Mate riaux Magne tiques, De partement de Physique, Universite de Sidi Bel Abbe s, Sidi Bel Abbe s 22000, Alge rie; b Laboratoire de Physique Quantique et de Mode lisation Mathe matique (LPQ3M), De partement de Technologie, Universite de Mascara, Mascara, Alge rie; c Department of Physics and Astronomy, King Saud University, P.O Box 2455, Riyadh 11451, Saudi Arabia; d Laboratory for Developing New Materials and their Characterization, Department of Physics, Faculty of Science, University of Setif, Setif, Algeria; e Institute of Nono Electronic Engineering, University Malaysia Perlis, Kangar, Perlis, Malaysia (Received 23 September 2011; final version received 20 October 2011) We have performed first-principle calculations using the full-potential linear augmented plane wave (FP-LAPW) method within density functional theory (DFT) to investigate the structural, elastic and thermodynamic properties of the cubic perovskite RBRh 3 (R ¼ Sc, Y and La) compounds. The exchange-correlation potential is treated within the generalized gradient approximation of Perdew Burke Ernzerhof (GGA-PBE). Single-crystal elastic constants are calculated using the total energy variation versus strain technique, then the shear modulus, Young s modulus, Poisson s ratio and anisotropic factor are derived for polycrystalline RBRh 3 using the Voigt Reuss Hill approximations. Analysis of the calculated elastic constants C ij and B/G ratios shows that these compounds are mechanically stable and ductile in nature. Using the quasi-harmonic Debye model, the effect of pressure P and temperature T on the lattice parameter a 0, bulk modulus B 0, thermal expansion coefficient, Debye temperature D and the heat capacity C v for these compounds are investigated for the first time. The computed structural and elastic constants are in good agreement with the available experimental and theoretical data. Keywords: perovskite borides; first-principle calculations; elastic constants; thermodynamic properties; Debye temperature 1. Introduction Theoretical methods have become one of the most active research fields in the physics of matter. It involves using ab initio methods to explain and predict the properties of solids that were previously inaccessible to experiments. Oxygen-based perovskites have been studied intensively because of their interesting physical properties, including superconducting transition, insulating metallic transition, ion conduction characteristics, dielectric properties, ferroelasticity and magnetism [1 4]. Recently, non-oxide perovskitetype compounds such as the ternary rare-earth (R) rhodium borides RBRh 3 and carbides RCRh 3 have also attracted considerable attention due to their high stability and hardness, which make them promising candidates for high-temperature environments, cutting tools and hard coating applications [5,6]. Numerous physical properties of the ternary rareearth (R) rhodium borides and carbides have been investigated experimentally and theoretically [5,7 22]. Experimentally, Shishido et al. [8,13,14] studied the dependency of the lattice parameters and hardness of R rhodium borides and carbides by changing the R atom as well as the boron stochiometry. They found that both the mechanical strength and chemical stability of the RRh 3 B x C 1 x phase essentially depend on the boron content. On the theoretical side, many studies have been carried out using various theoretical methods to understand the variation of the structural, electronic and elastic properties of these compounds as a function of the boron and carbon concentration. Sahara et al. [5,20,21] performed first-principles projected augmented wave (PAW) calculations on the structural and electronic properties of perovskite-type RRh 3 B x and Rh 3 B x C 1 x (R ¼ Sc and Y) by varying x from the total energy calculations. They found that boron doping leads to a continuous decrease in the bulk modulus. Similarly, Kojima et al. [22] studied the electronic structures and the elastic properties of perovskite-type RRh 3 B x. Their study focused on the *Corresponding author. yaldouri@yahoo.com ISSN print/issn online ß 2012 Taylor & Francis

3 122 F. Litimein et al. effect of replacing the Ce atom by Sc, Y and La atoms on the structural and elastic properties using ab initio calculations. Bouhemadou [23] reported the elastic properties of perovskite-type RRh 3 C (R ¼ Sc, Y, La and Lu) under pressure effects using the pseudopotential plane wave (PP-PW) method. From the above it is clear that there has been a considerable amount of experimental and theoretical work performed on rare-earth (R) rhodium boride RBRh 3 compounds. We note that there are limited studies on the elastic properties of these compounds. Moreover, there are no experimental and theoretical data for the thermodynamic properties and the pressure dependence of elastic constants for the investigated compounds. None of the theoretical calculations devoted to these compounds are full potential calculations. Therefore, we were motivated to perform new calculations in order to provide additional reference data for experimentalists and to complete the existing theoretical work on this fascinating class of materials, using the full-potential (linear) augmented plane-wave plus local orbital (FP-(L)APW þ lo) method within the density functional theory (DFT) within the generalized gradient approximation (GGA). The organization of the paper is as follows. The computational method adopted for the calculations is described in Section 2. The most relevant results obtained for the structural, elastic and thermodynamic properties of ScRh 3 B, LaRh 3 B and YRh 3 B are presented and discussed in Section 3. Finally, Section 4 summarizes the main conclusions of our work. 2. Computational method The calculations reported here were carried out using the full-potential APW þ lo method [24,25] with the mixed basis as implemented in the WIEN2K computer package [26]. The exchange-correlation effect was described within the generalized gradient approximation (GGA) of Perdew Burke Ernzerhof (PBE) [27]. In this method, the space is divided into an interstitial region (IR) and non-overlapping muffin tin (MT) spheres centered at the atomic sites. In the IR regions, the basis set consists of plane waves. Inside the MT spheres, the basis set is described by radial solutions of the one-particle Schro dinger equation (at fixed energy) and their energy derivatives multiplied by spherical harmonics. In order to achieve energy eigenvalue convergence, the wave functions in the interstitial region were expanded in plane waves with a cut-off R MT K max ¼ 8, where R MT denotes the smallest muffintin radius and K max gives the magnitude of the largest K vector in the plane wave expansion. The spherical harmonics were expanded up to angular momentum L max ¼ 8 inside the muffin-tin and L max ¼ 4 for the crystal harmonics, to avoid any shape approximations. The self-consistent calculations are considered to be converged when the total energy of the system is stable within 10 5 Ry. The integrals over the Brillouin zone (IBZ) are performed with grids, using the Monkhorst Pack special k-points approach [28]. The thermodynamic properties were investigated using the quasi-harmonic Debye model implemented in the Gibbs program [29]. For solids described by an energy volume (E V) relationship in the static approximation, the Gibbs program allows us to evaluate the Debye temperature to obtain the Gibbs free energy G(V, P, T) and to minimize G to derive the thermal equation of state (EOS) V(P, T). Other macroscopic properties related to P and T can be obtained using certain standard thermodynamic relations. Further details of this procedure can be found elsewhere [29]. 3. Results and discussion 3.1. Structural and elastic properties of RBRh 3 compounds The ternary rare-earth rhodium boride RRh 3 B(R¼Sc, Y and La) compounds crystallize in the cubic perovskite structure with space group Pm3m (#221). The R, B and Rh atoms are positioned at the 1a (0, 0, 0), 1b (0.5, 0.5, 0.5) and 3c (0, 0.5, 0.5) sites of the Wyckoff positions, respectively. The basic procedure in this work is to calculate the total energy as a function of unit-cell volume around the equilibrium cell volume V 0. The calculated total energies versus volume are fitted with Murnaghan s equation of state (EOS) [30] to determine the ground-state properties such as the equilibrium lattice constant a 0, the bulk modulus B 0 and the pressure derivative of the bulk modulus B 0. The results are presented in Table 1, which also contains earlier experimental data and theoretical results for comparison. Our calculated equilibrium lattice constants a 0 are in good agreement with the experimental data. More precisely, the computed lattice constants a 0 overestimate the measured values within 1.0%, which ensures the reliability of the present first-principles computations. This slight overestimation is attributed to the generalized gradient approximation (GGA), which is well known to slightly overestimate the lattice constants compared with the measured values. The bulk modulus B 0 is a factor indicating the resistance to a volume change due to external pressure. The computed bulk moduli values are in accordance with available theoretical data [16,17,19,21]. The calculated bulk modulus decreases in going from ScRh 3 B to

4 Molecular Physics 123 Table 1. Calculated equilibrium lattice constant (a 0 ), unit cell volume (V 0 ), bulk modulus (B) and the pressure derivative of the bulk modulus (B 0 ) for cubic ScRh 3 B, YRh 3 B and LaRh 3 B compared with the available theoretical and experimental data. a 0 (A ) V 0 (Å 3 ) B (kbar) B 0 ScRh 3 B Our work VASP-GGA a VASP-GGA b Exp a (3) Exp b YRh 3 B Our work VASP-GGA a VASP-GGA b VASP-GGA c Exp a (5) Exp b LaRh 3 B Our work VASP-GGA b Exp b Notes: a Refs. [16,17]. b Ref. [21]. c Ref. [19]. LaRh 3 B, suggesting a higher compressibility of LaRh 3 B compared with that of ScRh 3 B. The elastic properties define how a material that undergoes stress deforms and then recovers and returns to its original shape after stress ceases. These properties play an important role in providing valuable information about the binding characteristics between adjacent atomic planes, the anisotropic character of binding and structural stability. For a cubic system there are only three independent elastic constants: C 11, C 12 and C 44. The bulk modulus B 0 of this system is expressed as a linear combination of C 11 and C 12. The elastic constants can be obtained by calculating the total energy as a function of volume-conserving strains that break the cubic symmetry. Details of the calculation can be found in Refs. [31,32]. Table 2 gives our results for the elastic constants C ij at zero pressure compared with those obtained from projected augmented wave (PAW) calculations [19,21]. Good agreement is found between both calculations. It can be observed that the unidirectional elastic constant C 11, which is related to the unidirectional compression along the principal crystallographic directions, is about 78% larger than C 44, indicating that these compounds present a weaker resistance to pure shear deformation than to unidirectional compression. The traditional mechanical stability conditions of the elastic constants of a cubic structure are known to be C 11 C , C , C 11 þ 2C and C 12 5 B 5 C 11 [33]. The calculated elastic constants listed in Table 2 satisfy the above stability conditions and indicate that these compounds are mechanically stable. The elastic anisotropy factor for cubic crystals, which is defined as A ¼ 2C 44 =ðc 11 C 12 Þ, has an important implication in engineering science since it is highly correlated with the possibility of introducing microcracks in a material [34]. For completely isotropic systems, the anisotropy factor A takes the value unity and the deviation from unity measures the degree of elastic anisotropy. The calculated values of the anisotropic factor A are found to be 0.87 for ScBRh 3, 0.78 for YBRh 3 and 0.86 for LaBRh 3, indicating that these compounds are not characterized by a profound anisotropy. Once the single-crystal elastic constants C ij had been calculated, related properties for polycrystals, namely the shear modulus G, the Young s modulus E and the Poisson s ratio, were estimated using the Voigt Reus Hill approximations [35 37]: where G ¼ G V þ G R 2 E ¼ 9BG 3B þ G, G R ¼ 5ðC 11 C 12 ÞC 44 4C 44 þ 3ðC 11 C 12 Þ : ð1þ ¼ 3B þ E 6B, ð2þ, G V ¼ C 11 C 12 þ 3C 44, 5 The calculated values of the elastic moduli for polycrystalline RBRh 3 aggregates are also listed in Table 2. Currently, there are no experimental measurements of the elastic moduli to compare our results with. However, our results are consistent with those of Shishido et al. [16,17], Sahara et al. [21] and Music et al. [18,19]. It is found that ScRh 3 B has a larger shear and Young s modulus than YRh 3 B and LaRh 3 B. As is known, an increasing atomic packing density due to a decrease in the unit cell volume leads to an enhancement of the bulk modulus, Young s modulus and the shear modulus. The shear modulus G represents the resistance to plastic deformation, and the bulk modulus B 0 represents the resistance to fracture. Based on these factors, Pugh [38] introduced the quotient of the shear modulus to bulk modulus (G/B) of polycrystalline phases in order to distinguish between brittle and ductile materials. A low G/B value means better ductility, while a high G/B value implies more brittleness. The critical value, which separates ductile and brittle behavior, ð3þ

5 124 F. Litimein et al. Table 2. Calculated elastic constants C ij (in GPa) and their pressure ij =@P, shear modulus G (in GPa), Young s modulus E (in GPa), anisotropy factor A and Poisson s ratio for cubic ScRh 3 B, YRh 3 B and LaRh 3 B compared with the available theoretical data. C 11 C 12 C =@P G E A ScRh 3 B Our work VASP-GGA a YRh 3 B Our work VASP-GGA a VASP-GGA b LaRh 3 B Our work VASP-GGA a Notes: a Ref. [21]. b Ref. [19]. is about In this work, the calculated G/B values for ScRh 3 B, YRh 3 B and LaRh 3 B are 0.40, 0.42 and 0.38 respectively. Hence, it can be inferred that these compounds are ductile in nature. Furthermore, Frantsevich et al. [39] related the ductility/brittleness behavior to the Poisson s ratio (). For brittle materials 5 1/3, otherwise the material behaves in a ductile manner. Here the calculated values are much smaller than 1/3, categorizing these compounds as ductile materials. The Cauchy pressure, defined as the difference between the two particular elastic constants C 12 C 44, can also serve as an indication of ductility: If the Cauchy pressure is positive (negative), the material is expected to be ductile (brittle) [40 42]. The calculated values for the Cauchy pressure are 64.9 for ScRh 3 B, 54.8 for YRh 3 B and 60.4 for LaRh 3 B, which also highlights the ductile nature of these compounds with a more metallic bonding character. In the following, we study the pressure dependence of the elastic properties. Figure 1 shows the variation of the elastic moduli (C ij, G, E and B) of ScRh 3 B, LaRh 3 B and YRh 3 B as a function of pressure. From Figure 1, we note a linear dependence for all curves in the considered pressure range. Moreover, it is clear that the elastic moduli and bulk modulus increase with increasing pressure. The pressure coefficients of the elastic constants C ij are determined by linear fits. The calculated linear pressure coefficients are listed in Table 2. It is clear that C 11 has a stronger pressure dependence than C 12 and C 44. Moreover, the shear modulus (G) and Young s modulus (E) increase with increasing pressure (see Figure 1), suggesting that the toughness of these compounds can be improved at high pressure. To the best of our knowledge, there are no experimental or theoretical data for the effect of pressure on the elastic properties of ScRh 3 B, LaRh 3 B and YRh 3 B in the literature. Our results can therefore serve as a prediction for future investigations Thermodynamic properties The quasi-harmonic Debye approximation was used to investigate the thermodynamic properties of ScRh 3 B, LaRh 3 B and YRh 3 B. As a first step, a set of total energy calculations versus the primitive cell volume (E V), in the static approximation, was carried out and fitted with the numerical EOS in order to determine the structural parameters at zero temperature and pressure; the macroscopic properties were then derived as a function of pressure and temperature from standard thermodynamic relations. The thermal properties are determined in the temperature range from 0 to 1800 K where the quasi-harmonic model remains fully valid. The pressure effect is studied in the range from 0 to 30 GPa. Figure 2 displays the variation of the lattice parameter and bulk modulus as a function of temperature for RRh 3 B(R¼Sc, Y, La) at zero pressure. The lattice parameter of each compound increases with increasing temperature and the rate of increase is very moderate. One can see that the bulk modulus decreases linearly with temperature for T K and smoothly with increasing temperature for all compounds. The compressibility increases with increasing temperature. The evolution of the lattice parameter and bulk modulus as a function of pressure at temperatures T ¼ 300 K and 1800 K is presented in Figure 3. It is found that the relationship between the bulk modulus of RRh 3 B compounds and pressure is nearly linear at various temperatures. The lattice parameter decreases, whereas the bulk modulus increases, with increasing pressure at a given temperature. This result is due to the fact that the effect of increasing pressure on the material is the same as that of decreasing temperature. Figure 4 presents the effect of the temperature and pressure on the thermal expansion. It can be seen that the thermal expansion coefficient () for each

6 Molecular Physics 125 Figure 1. Pressure dependence of the elastic constants (C 11, C 12 and C 44 ), bulk modulus (B 0 ), shear modulus (G) and Young s modulus (E) for ScRh 3 B, YRh 3 B and LaRh 3 B. compound increases (decreases) with increasing temperature (pressure). For a given temperature, the thermal coefficient decreases strongly with increasing pressure. For a given pressure (Figure 4(a)), the thermal coefficient increases sharply with increasing Figure 2. Variation of the lattice parameter (a) and bulk modulus (b) with temperature for ScRh 3 B, YRh 3 B and LaRh 3 B. temperature up to 400 K. Above this temperature, gradually approaches a linear increase with increasing temperature. At zero pressure and 300 K, the thermal expansions for ScRh 3 B, LaRh 3 B and YRh 3 B are K 1, K 1 and K 1, respectively. The variation of the heat capacity C V versus temperature at zero pressure for ScRh 3 B, LaRh 3 B and YRh 3 B is shown in Figure 5. The heat capacity temperature diagram at 0, 10, 20 and 30 GPa for ScRh 3 B is also presented in the inset of Figure 5. From this figure it can be seen that, with increasing temperature, C V values rapidly increase at low temperature, then slowly increase at high temperature, and tend to the Petit and Dulong [43] limit, which is a constant for all solids at high temperature. From Figure 5, it is clear that when T K, the heat capacity C V depends on both temperature and pressure (C V is proportional to T 3 [44]). The variation of the heat capacity for the studied compounds is similar over a wide range of

7 126 F. Litimein et al. Figure 3. Variation of the lattice parameter (a) and bulk modulus (b) with pressure at different temperatures (300 K solid line and 1800 K dotted line) for ScRh 3 B, YRh 3 B and LaRh 3 B. temperature and pressure. At high temperature, C V approaches approximately 125 J mol 1 K 1. At zero pressure and 300 K the calculated values of C V are found to be , and J mol 1 K 1 for ScRh 3 B, LaRh 3 B and YRh 3 B, respectively. The Debye temperature ( D ) is an important parameter that is characteristic of the thermal properties of solids. The thermal vibrations become more important than the quantum effects. The variation of the Debye temperature as a function of pressure and temperature for the studied compounds is plotted in Figure 6. It can be seen that D is nearly constant from 0 to 200 K, but then decreases almost linearly with increasing temperature. It can also be seen that, when the temperature is fixed, the Debye temperature increases almost linearly with increasing pressure. The calculated values of the Debye temperature at Figure 4. Variation of the thermal expansion coefficient with temperature (a) and pressure (b) for ScRh 3 B, YRh 3 B, and LaRh 3 B. Figure 5. Variation of the heat capacity C V with temperature for ScRh 3 B, YRh 3 B and LaRh 3 B.

8 Molecular Physics 127 Figure 6. Variation of the Debye temperature with temperature at different pressures for ScRh 3 B, YRh 3 B and LaRh 3 B. zero pressure and 300 K are found to be K for ScRh 3 B, K for YRh 3 B and K for LaRh 3 B. It is known that the Debye temperature is proportional to the bulk modulus and the hardest material possesses the highest Debye temperature. This is to say that ScRh 3 B has a higher Debye temperature than YRh 3 B and LaRh 3 B. To our knowledge, there are no experimental or theoretical data available in the literature for the thermodynamic properties of the investigated compounds. Hence our work is a first attempt in this direction. 4. Conclusions In this paper, we have studied the structural, elastic and thermodynamic properties of ScRh 3 B, YRh 3 B and LaRh 3 B using the full-potential (linear) augmented plane wave plus local orbital method within the generalized gradient approximation. The calculated structural properties, including the lattice parameter, bulk modulus and its pressure derivative, are in good agreement with the available theoretical and experimental data. The evaluated elastic parameters allow us to conclude that the investigated compounds are mechanically stable. The pressure dependence of the elastic constant and the derived properties result in an increase in the stiffness and thermal conductivity of these compounds under pressure. Our data also reveal that these compounds behave as ductile materials. Using the quasi-harmonic Debye model, the dependence of the lattice parameter, bulk modulus, thermal expansion coefficient, specific heat capacity and Debye temperature on pressure and temperature has been obtained. To our knowledge, this is the first quantitative theoretical prediction of the thermodynamic properties and pressure dependence of elastic constants for the investigated compounds and awaits experimental confirmation. Acknowledgements This work was supported by FRGS grants Nos and R.K., A.B. and S.O. extend their appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the research group project No. RGP-VPP-088. Y.A. would like to acknowledge TWAS, Italy, for full support of his visit to JUST, Jordan, under a TWAS-UNESCO Associateship. References [1] F. Galasso, Perovskites and High-Tc Superconductors (Gordon and Breach, London, 1990). [2] H. Iwahara, M. Balkanski, T. Takahashi and H.L. Tuller, Solid State Ionics (Elsevier, Amsterdam, 1992). [3] N. Kamegashira, J. Meng, T. Murase, K. Fujita, H. Satoh, T. Shishido and T. Fukuda, in 10th International Ceramics Congress Part A, edited by P. Vincenzini (Techna Srl, Italy, 2003).

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