FIRST-PRINCIPLES STUDY ON MECHANICAL PROPERTIES OF THE PEROVSKITE RBRh3

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3^ème Conférence Internationale sur

le Soudage, le CND et l’Industrie des Matériaux et Alliages (IC-WNDT-MI’12) Oran du 26 au 28 Novembre 2012,

http://www.csc.dz/ic-wndt-mi12/index.php 58

FIRST-PRINCIPLES STUDY ON MECHANICAL PROPERTIES OF THE PEROVSKITE RBRh

3

(R = Sc, Y AND La) ALLOYS

F. Litimein¹, R. Khenata², H. Belkhalfa ³, K. Bougherara⁴, N. Benkhattou¹ and D. Rached¹

1: Laboratoire des Matériaux Magnétiques, Département de physique, Université Djilali

Liabes de Sisi Bel Abbes, 22000 Algérie, [email protected]

2: Laboratoire de Physique Quantique et de Modélisation Mathématique (LPQ3M), Université de Mascara, 29000 Mascara, Algérie, [email protected]

3:Centre de Recherche en soudage et contrôle, Chéraga Alger, Algerie,

[email protected]

4:Département physique, faculté des sciences, Université Djilali Liabes de Sidi Bel Abbes, 22000, Algerie,

Abstract :

The structural, elastic and thermodynamic properties of the cubic perovskite RBRh3 (R=Sc, Y and La) compounds have been calculated using the full-potential linearized-augmented plane wave with the mixed basis FP/APW+lo method. The exchange-correlation potential is treated with the generalized gradient approximation of Perdew-Burke-Ernzerhof (GGA-PBE). The calculated structural properties are in excellent agreement with the available experimental and theoretical data. Single-crystal elastic

constants are calculated using the total energy variation with strain technique, then the shear modulus, Young’s modulus, Poisson’s ratio and anisotropic factor are derived for polycrystalline RBRh3. Ductility behaviour of these compounds is discussed via the elastic constantsC_ij.

Keywords:

Perovskite borides; ab initio calculations; Elastic constants.

Introduction

Ab initio methods, which use only the atomic constants as input parameters to solve the Schrödinger equation, have now become the most powerful probes for investigating important number of physical and chemical properties for atoms, molecules and solids. Needless to mention, that they are also a tool of choice for the prediction of new materials, and they could sometimes replace experiments which are very costly or even impossible in the laboratory.

Oxygen-based perovskites have been largely studied because of their interesting physical properties, including superconducting transition, insulating-metallic transition, ion conduction characteristics, dielectric properties, ferroelasticity and magnetism [1-4]. Recently the non-oxide perovskite-type compounds such as the ternary rare-earth (R) rhodium borides RBRh3 and carbides RCRh3 have also received considerable attention because their high stability and hardness, which make them a promising candidates for high-temperature environments, cutting tools, and hard coating applications [5, 6].

Numerous physical properties of the ternary rare-earth (R) rhodium borides and carbides have been investigated experimentally and theoretically [5, 7-22]. Experimentally, Shishido et al. [8, 13-14]

have studied the dependency of lattice parameters and hardness of R rhodium borides and carbides by changing the R atoms as well as boron stochiometry. They have found that both mechanical strength and chemical stability of the RRh3BxC_1−x phase essentially depend on its boron content. On the theoretical side, many efforts have been made by various theoretical methods to understand the variation of the structural, electronic and elastic properties of these compounds as a function of the boron and the carbon

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http://www.csc.dz/ic-wndt-mi12/index.php 59 concentration. Sahara et al. [5, 20-21] have performed first-principles projected augmented wave (PAW) calculations on the structural and electronic properties of perovskite-type RRh3Bx and Rh3BxC1−x (R=Sc and Y) by varying x from the total energy calculations. They found that the boron doping leads to a continuous decrease in the bulk modulus. In the same way, Kojima et al. [22] studied the electronic structures as well as the elastic properties of perovskite-type RRh3Bx. In their study they focused on the effect of replacing Ce atom by Sc, Y and La atoms on the structural and the elastic properties by using ab initio calculations. Very recently Bouhemadou [23] reported the elastic properties of perovskite-type RRh3C (R=Sc, Y, La and Lu) under pressure effects by using the pseudo-potential plane wave (PP-PW) method.

From the above it is clear that there is considerable experimental and theoretical work on the rare- earth (R) rhodium borides RBRh3 compounds. We note that there exist limited studies on the elastic properties of these compounds. Moreover, there are no experimental and theoretical data for thermodynamic properties and the pressure dependence of elastic constants for the investigated compounds. All the theoretical calculations devoted to these compounds are not full potential calculations. The reasons mentioned above motivate us to perform these calculations in order to provide another reference data for the experimentalist and to complete the existing theoretical works on this fascinating class of materials, using the full-potential augmented plane-wave plus local orbital (FP- APW+lo) method within the density functional theory (DFT).

The organization of this paper is as follows: The computational method we have adopted for the calculations is described in section 2. The most relevant results obtained for the structural, elastic and thermodynamic properties of ScRh3B, LaRh3B and YRh3B compounds are presented and discussed in Section 3. Finally, in section 4 we summarize the main conclusions of our work.

The calculations were performed in the framework of density functional theory (DFT). We have employed the full-potential linearized-augmented plane wave with the mixed basis FP/APW+Lo method [24, 25] as implemented in 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 sets is described by radial solutions of the one particle Schrödinger equation (at fixed energy) and their energy derivatives multiplied by spherical harmonics. In order to achieve energy eigenvalues convergence, the wave functions in the interstitial region were expanded in plane waves with a cut-off R_MTK_max= 8, where R_MT denotes the smallest muffin-tin 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 = 8 inside the muffin-tin and l

max = 4 for the interstitial region, 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 12×12×12 grids, using the Monkhorst-Pack special k-points approach [28].

2. Results and discussions

The ternary rare-earth rhodium borides RRh3B (R = Sc, Y and La) compounds crystallize in the cubic perovskite structure with space group Pm3m (#221). The R, B and Rh atoms are positioned at: 1a (0, 0, 0), 1b (0.5, 0.5, 0.5) and 3c (0, 0.5, 0.5) sites of Wyckoff coordinates, respectively. A series of total energy calculations as a function of primitive cell volume is fitted to Murnaghan's equation of state (EOS)

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http://www.csc.dz/ic-wndt-mi12/index.php 60 [29] to determine the ground-state properties such as the equilibrium lattice constant a0, the bulk modulus B0 and the pressure derivative of the bulk modulus B′. The calculated equilibrium parameters (a0, B0 and B′) for the studied compounds are quoted in Table 1. Results from earlier experimental and theoretical works are quoted for comparison. Our calculated equilibrium lattice constants a0 are in good agreement with the experimental data. The computed lattice constants a0 value overestimates the measured one within 1.0 % which ensures the reliability of the present first-principles computations. This slight overestimation is attributed to our use of the generalized gradient approximation (GGA) which is well- known to slightly overestimate the lattice constants compared to the measured ones. The bulk modulus B0

is a factor indicating the resistance to the volume change due to external pressure. The computed bulk moduli values are in excellent agreement with available theoretical data [16, 17, 19, 21]. The calculated bulk modulus decreases in going from ScRh3B to LaRh3B, suggesting the more compressibility of LaRh3B compared to that of ScRh3B.

Table 1: Calculated equilibrium lattice constant (a0), unit cell volume (V0), bulk modulus (B) and the pressure derivative of bulk modulus (B’) for cubic ScRh3B, YRh3B, and LaRh3B compared to the available theoretical and experimental data

a₀

(Å) V

₀

(Å

³

)

B B’

ScRh₃B

Our work 4.129 70.27 199.63 4.47

VASP-GGA

¹

4.13 70.44 200.1

VASP-GGA

²

4.13 70.44 201

Exp

¹

4.0799(3) 67.91

Exp

²

4.080 67.92

YRh3B

Our work 4.209 74.57 184.23 4.71

VASP-GGA

¹

4.22 75.15 177.4

VASP-GGA

²

4.22 75.15 183

VASP-GGA

⁴

4.216 74.94 181

Exp

¹

4.1675(5) 72.38

Exp

²

4.168 72.41

LaRh3B

Our work 4.289 78.91 167.55 4.66

VASP-GGA

²

4.29 78.95 166

Exp

²

4.251 76.82

1

Ref. [16, 17].

²

Ref. [21].

⁴

Ref. [19].

The elastic constants determine the response of the crystal to the external forces and play an important part in determining the strength of the materials. Values of these constants provide valuable information on the stability and stiffness of materials. There are 21 independent elastic constantsC_ij, but the symmetry of cubic crystal reduces this number to only three independent elastic constants: C11, C12

and C44. The bulk modulus B for cubic systems is expressed as a linear combination of C11 and C12.The elastic constants were obtained using the so-called “total energy-strain” method, as outlined in Refs. [30, 31]. Table 2 gives our results for the elastic constants C_ij at zero pressure compared with those obtained

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http://www.csc.dz/ic-wndt-mi12/index.php 61 from projected augmented wave (PAW) calculations [19, 21]. Good agreement is found between both calculations. It can be observed that the unidirectional elastic constant C11, which is related to the unidirectional compression along the principal crystallographic directions, is about 78% higher than C44, indicating that these compounds present a weaker resistance to the pure shear deformation compared to the resistance to the unidirectional compression. The traditional mechanical stability conditions of the elastic constants of cubic structure are known as to be: C11-C12 >0, C44 >0, C11+2C12 >0 and C12< B < C11

[32]. The calculated elastic constants listed in Table 2 satisfied the above stability condition, indicating that these compounds are mechanically stable.

The elastic anisotropy factor in cubic crystals, which is defined asA

2

C₄₄

/(

C₁₁C₁₂

)

, has an important implication in engineering science since it is highly correlated with the possibility to introduce microcrack in the material [34]. For completely isotropic systems, the anisotropy factor A takes the value of the unity and the deviation from unity measures the degree of elastic anisotropy. The calculated values of the anisotropic factor A are found to be equal to 0.87 for ScBRh3, 0.78 for YBRh3 and 0.86 for LaBRh3, meaning that they are not characterized by a profound anisotropy.

Table2: Calculated elastic constants C_ij (in GPa) and their pressure derivatives P C_ij



 , shear modulus G

(in GPa), Young’s modulus E (in GPa), Poisson’s ratios ν and anisotropy factor A, for cubic ScRh3B, YRh3B, and LaRh3B compared to the available theoretical data.

C11 C12 C44 G E A Ν

P C



 ₁₁ P C



 ₁₂

P C



 ₄₄

ScRh₃ B

Our work

315.8 5

141.4 7

76.53 80.63 213.1 8

0.87 0.67 6.21 3.6 1.12 VASP-

GGA

¹

394 102 77 105

YRh₃ B

Our work

321.7 8

119.7 64.9 77.54 204 0.78 0.68 6.35 3.88 1.06 VASP-

GGA

¹

339 108 67 86

VASP- GGA

²

68 86.9 225.3

LaRh₃B

Our work 261.0 8

120.7 7

60.37 64.27 170.9 5

0.86 0.67 6.20 3.84 1.19 VASP-

GGA

¹

283 106 51 66

1

Ref. [21] .

²

Ref. [19].

Once the single-crystals elastic constants C_ij are calculated, related properties for polycrystals, namely, the shear modulus G, the Young’s modulus E and the Poisson’s ratio ν have been estimated using the Voigt-Reus-Hill approximations [35- 37] by the following relations:

G B E BG

  3

9 (1)

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http://www.csc.dz/ic-wndt-mi12/index.php 62 B

E B

6 3 

  (2) where ,

2

R

V G

G G  ,

5

3 ₄₄

12

11 C C

G_V C  

 and

 



₁₁ ₁₂



44

44 12 11

3 4

5

C C C

C C G_R C





  (3)

The calculated values of the mentioned elastic moduli for polycrystalline RBRh3 aggregates are also listed in Table 2. There are no currently experimental measurements of the elastic moduli to be compared with our results. However, our results are well consistent with those of Shishido et al. [16, 17], Sahara et al.

[21] and Music et al. [18,19]. It is seen that ScRh3B has the highest shear and Young's modulus compared to YRh3B and LaRh3B. As it is known, the increase of the atomic packing densities due to the decrease of unit cell volume leads to the enhancement in bulk modulus, Young's modulus and shear modulus.

Having calculated the shear modulus G and the bulk modulus B, we can now try to elaborate more on the ductile or brittle nature of these compounds. The ductility and brittleness behaviors of materials can be explained from some proposed relationship. Pugh [37] has proposed a simple

relationship that links empirically the plastic properties of materials with their elastic moduli by the ratio B/G. The critical value which separates ductile and brittle materials is around 1.75; if B/G >1 .75, the material behaves in a ductile manner; otherwise the material behaves in a brittle manner. The values of Pugh’s criterion B/G are 2.47, 2.37 and 2.60 for ScRh3B, YRh3B and LaRh3B, respectively. These values are higher than the critical value (1.75), which clearly highlights the ductile nature of these compounds.

We may also refer to Frantsevich et al. [38] who distinguish the ductility and brittleness of materials in terms of Poisson’s ratio (ν). According to Frantsevich rule the critical value of Poisson’s ratio in materials is 0.33. For brittle materials, the Poisson ratio is less than 0.33; otherwise the material behaves in a ductile manner. Here the calculated (ν) values are larger than 1/3 for all compounds, categorizing all compounds as ductile materials. The Cauchy's pressure, defined as the difference between the two particular elastic constants C12-C44 can also serve as an indication of ductility: If the Cauchy's pressure is positive (negative), the material is expected to be ductile (brittle) [39- 41]. The calculated values for Cauchy pressure are 64.9 for ScRh3B, 54.8 for YRh3B and 60.4 for LaRh3B which also reflects the ductile nature of these compounds with a more metallic character of bonding.

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http://www.csc.dz/ic-wndt-mi12/index.php 63 Fig.1: Pressure dependence of the elastic constants (C11

, C

12

and

C44

), bulk modulus (B), shear modulus (G) and Young's modulus (E) for ScRh

3

B, YRh

3

B and LaRh

3

B.

In the following paragraph we shed more light on the pressure dependence of the elastic properties. Figure 1 shows the variation of the elastic moduli (C_ij, G, E and B) of ScRh3B, LaRh3B and YRh3B compounds as a function of pressure. We clearly predict that the elastic moduli increase when the pressure is enhanced. A linear pressure dependence of C_ij, G, E and B curves is found. 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 C11 has the strongest pressure dependence compared to C12 and C44. Moreover, the shear modulus (G) and Young’s modulus (E) increase with increasing pressure (see Fig. 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

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http://www.csc.dz/ic-wndt-mi12/index.php 64 the elastic properties of ScRh3B, LaRh3B and YRh3B compounds given in the literature. Then, our results can serve as a prediction for future investigations.

4. Conclusions

In this work we have performed a detailed investigation on the structural, elastic properties of ScRh3B, YRh3B and LaRh3B compounds using first-principles APW +lo method within GGA

approximation. Firstly, we have determined the ground state properties, including lattice parameter, bulks modulus and its pressure derivatives. An excellent agreement is found between our calculated results and the available theoretical and experimental data. After that we have determined the elastic constants and their pressure dependence using the volume conserving strain technique. The pressure dependence of the elastic constant and their derived properties brings the improvement of stiffness and thermal conductivity of these compounds under pressure. Polycrystalline elastic moduli, namely, shear modulus, Young's modulus, Poisson's ratio were derived from the obtained single-crystal elastic constants. The present results bring the brittle nature for these compounds. To our knowledge this is the first quantitative theoretical prediction for the elastic properties and the pressure dependence of elastic constants for the investigated compounds and still awaits experimental confirmations.

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