Structural, Elastic, and Electronic Properties of ReB<sub>2</sub>: A First-Principles Calculation

Long, Run; Dai, Ying; Jin, Hao; Huang, Baibiao

doi:https://doi.org/10.1155/2008/293517

Physics Research International

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Research Letter | Open Access

Volume 2008 | Article ID 293517 | https://doi.org/10.1155/2008/293517

Structural, Elastic, and Electronic Properties of ReB₂: A First-Principles Calculation

Run Long,¹Ying Dai,¹Hao Jin,¹and Baibiao Huang¹

Academic Editor: Wai-Yim Ching

Received23 Sept 2007

Accepted16 Dec 2007

Published28 Feb 2008

Abstract

The structural, elastic, and electronic properties of the hard material ReB₂ have been investigated by means of density functional theory. The calculated equilibrium structural parameters of ReB₂ are in agreement with the experimental results. Our result of bulk modulus shows that it is a low compressible material. Furthermore, the elastic anisotropy is discussed by investigating the elastic stiffness constants. The charge density and the electronic properties indicate that the covalent bonding of Re-B and B-B plays an important role in formation of a hard material. The good metallicity and hardness of ReB₂ might serve as hard conductors.

1. Introduction

Superhard materials are useful in a variety of industrial applications for fast machining and drilling. Intense experimental and theoretical efforts have been focused on the possibility of synthesizing and designing novel superhard material [1, 2]. Recently, the ultra-incompressible hard material OsB₂ has been synthesized successfully in experiment [3]. However, the Os lattice expands by approximately 10% upon the incorporation of B atoms to form OsB₂ and undergoes a distortion to an orthorhombic phase. In searching for much more hard materials, Chung et al. attempted to use the element Re, which lays directly next to the left of Os in the periodic table of the elements. They successfully synthesized the hard material ReB₂ under ambient conditions [4], for which the incorporation of B into the interstitial sites of Re to form ReB₂ brings only 5% expansion of the Re lattice and the bulk modulus of ReB₂ is 360 GPa rivaling that of the diamond (442 GPa) [5]. Thus, ReB₂ is considered to be one of the most promising candidates for improving the mechanical properties of OsB₂.

The pure Re has a large bulk modulus of 360 GPa while its hardness is only 1.3–3.2 GPa [6]. The high bulk modulus of Re is mainly due to the high valence electron density, while the low hardness is related to the metallic bonds because they are the same in all directions and therefore poor at resisting either plastic or elastic shape deformations. However, the ReB₂ is an ultra-incompressible (360 GPa) as well as a hard (30.1–55.5 GPa) material [4]. Though first-principles quantum mechanical calculations can provide robust explanations of ReB₂ properties, a few results have been reported [7]. Therefore, it is interesting to understand the microcosmic mechanism of the hardness-related properties of ReB₂ from the basic point of view of physics and applications.

In this work, we performed density functional calculations on ReB₂ to study its structural, elastic, and electronic properties related to the electron configuration of this material. The equilibrium structural parameters, bulk modulus, and elastic stiffness constants were calculated and compared with experimental results, and its elastic anisotropy and compressibility were discussed. The charge density, band structure, and density of states (DOS) of ReB₂ were also calculated for investigating the electronic properties of ReB₂.

2. Methods and Models

The density functional calculations on the structural, elastic, and electronic properties of ReB₂ were performed using the CASTEP code [8]. Both lattice and internal coordinates were optimized to get the equilibrium structure for ReB₂. The exchange-correlation function was treated by both the generalized gradient approximation (GGA-PBE) [9] and the local density approximation (LDA-CAPZ) [10, 11]. The Vanderbilt ultrasoft pseudopotential [12] was used with the cutoff energy of 350 eV, and the k-point meshes of for hexagonal ReB₂ were generated using the Monkhorst-Pack scheme [13]. The energy convergence was checked with increasing cutoff energy to 400 eV, which brought only 10 meV error. In the geometry optimization, all forces on atoms were converged to be less than 0.03 eV/Å; the maximum ionic displacement was within 0.001 Å, and the total stress tensor was reduced to the order of 0.05 GPa. For ReB₂, the experimental structure was used as the initial structure for the calculations [14].

3. Results and Discussions

ReB₂ has a hexagonal lattice (space group P63/mmc, No.194) with the experimental lattice parameters a = b = 2.900 and [14]. In the hexagonal structure, two Re atoms occupy the 2a Wyckoff sites and four B atoms occupy the 4f sites, as shown in Figure 1. The calculated lattice constants of ReB₂ within both LDA and GGA are shown in Table 1, which shows that the optimized lattice constants are larger using GGA method than that of using LDA method. In addition, the calculated lattice parameters of a and c with GGA are in good agreement with the experimental data [14], differing only by 0.66% and 0.92%, respectively. By contrast, the results using LDA are larger than the experimental values by 1.0% and 1.1. The calculated densities within both LDA and GGA are 13.08 and 12.96 g/cm³ with an error about 3.1% and 2.2% compared with the experimental value 12.68 g/cm³, respectively [14]. The calculated elastic constants of ReB₂ at zero pressure are also shown in Table 1. All of them satisfy the criteria for stability of hexagonal structure based on the Born-Huang criterion [15, 16], which indicates that this structure of the material is stable. The calculated values for bulk modulus using LDA and GGA are 360 and 353 GPa, respectively, both of which are in good agreement with the experimental results (360 GPa) [4]. This value rivals that of diamond (442 GPa) [5], indicating the incompressibility characteristic of hexagonal ReB₂.

Table 1

Calculated structural parameters of ReB₂ using LDA and GGA: a(Å), b(Å), c(Å), density (g/cm³), elastic constants (GPa), bulk modulus B (GPa), and shear anisotropy A = C₄₄/C₆₆, the ratio for linear compressibility coefficients k_c/k_a, and anisotropic factors and A_B compared with the experimental results.

The elastic anisotropy (A) of crystals can exert great effects on the properties of physical mechanisms, such as anisotropic plastic deformation, crack behavior, and elastic instability. Thereby, it is important to calculate elastic anisotropy in order to obtain a deep understanding for the properties of such material. For hexagonal ReB₂, the anisotropy of shear anisotropy ratio was calculated by A = C₄₄/C₆₆ [17], and the results using the LDA and GGA approached 1.02. Additionally, the knowledge of elastic coefficients also provides a method to evaluate the linear compressibility. The ratio for linear compressibility coefficients k_c/k_a of hexagonal crystal is defined as [18]. A value of 1.0 implies the isotropic compressibility, and the deviation from 1.0 is the value that measures the anisotropy for the linear compressibility along the c and a directions. The k_c/k_a of ReB₂ is 0.58 for LDA and 0.57 for GGA, respectively, which means that the compressibility along the c axis is much smaller than that along the a axis, coinciding with the calculated elastic constants very well along different axes (shown in Table 1). In addition, the percentages of anisotropy in compressibility and shear were also calculated by bulk modulus factors and shear modulus factors [19], respectively, in which B and G represented the bulk and shear modulus, and the subscripts and denoted the Voigt and Reuss approximations. For hexagonal ReB₂, A_B = 1.91% and A_G = 1.36% with LDA calculation and A_B = 1.94% and A_G = 1.55% with GGA calculation, respectively. These results indicate that ReB₂ is slightly anisotropic in compressibility, which is consistent with the experimental phenomenon [4].

It is known that the hardness of a material is mainly determined by the bonding type in the system. To investigate the chemical bonding characteristic of the system, the electronic structure of ReB₂ such as the electron density, band structure, and density of sates (DOS) was calculated with GGA and LDA at zero pressure. We find that the band structure, density of states, and charge density calculated by both the LDA and GGA methods at the corresponding theoretical equilibrium lattice constants show the similar patterns, and hence we just give the GGA results in this paper. The total charge density plot (the plot in the plane including B1, B2, B3, B4, Re1, and Re2) is shown Figure 1 as well as Figures 2(a) and 2(b). There are some electrons between Re atoms and their neighbor B atoms, and a strong directional bonding exists between them, indicating covalent bonding in ReB₂. To shed more light on the bonding characteristic of ReB₂, we plot the difference charge density (the difference between total density and superposition of atomic densities) in Figure 2(b). Clearly, two neighbor B atoms form a very strong covalent bond, which indicates that the formation of these directional covalent bonds leads to the high hardness. Population analysis in CASTEP is performed using a projection of the plane wave states onto a localized basis using a technique described by Sanchez-Portal et al. [20], and population analysis of the resulting projected states is then performed using the Mulliken formalism [21]. This technique is widely used in the analysis of electronic structure calculations performed with linear combination of atomic orbitals (LCAO) basis sets. From the Mulliken population analysis for ReB₂, the net charges of B and Re are and 0.65, respectively. It indicates a charge transfer from Re to B atoms, which revealed a partial ionic contribution to the Re-B bonding. On the other hand, along the axis, metal atoms form linear chains and the metal-metal distance is very large and nearly equals the half of the cell parameter of the c axis, that is, 3.075 Å. This might be an implication of certain weak metallic bonding between metal atoms. Therefore, our result suggests that the bonds of ReB₂ are of the unusual mixtures of covalent, ionic, and metallic bonding, as observed in OsB₂ [22].

(a)

(b)

The band structure of ReB₂ is shown in Figure 3 and the dashed line represents the Fermi level . It can be seen that the occupied and unoccupied levels form one band and the levels near are crossing and extended, indicating a metallic characteristic in ReB₂. Since the metallicity is uncommon in superhard materials, that is, most of the superhard materials are insulators or semiconductors [23], this special characteristic of ReB₂ will put it in a favorable position in the future application for electron conductivity. Figure 4 is the total DOS (TDOS) of ReB₂ and partial DOS (PDOS) of B-2s, B-2p, Re-5p, and Re-5d with GGA calculation. Figure 4(e) shows that locates in the extended states with substantially density of states, which implies some good metallicity of ReB₂. Additionally, in the DOS (see Figure 4(e)), the deep valley near the , named as pseudogap, can be ascribed to the strong hybridization that leads to a separation of the bonding states. The partial DOS in Figure 4 shows that the total DOS near is mainly contributed by the strong hybridization between the Re-5d and B-2p states, and the contributions of Re-5p and B-2s states are quite small. This strong hybridization of Re-5d and B-2p states also indicates the strong covalent bonding of Re-B. It is obvious that both of the analyses of charge density and DOS demonstrate the covalent bonding characteristic of ReB₂, which is responsible for the high bulk modulus and hardness.

4. Conclusions

The results and analysis of the structural, elastic, and electronic properties for ReB₂ demonstrate that the hexagonal ReB₂ with the experimental structural parameters is found to be metallic, which is uncommon in hard materials and suggests that ReB₂ is a potential hard conductor. Its bulk modulus calculated by LDA and GGA is large enough to compare with that of diamond. The bonds of ReB₂ are of the unusual mixtures of covalent, ionic, and metallic bonding. The electron structure indicates that the covalent bonding of Re-B and B-B is mainly responsible for the high hardness.

Acknowledgments

This work is supported by the National Science Foundation of China under Grant 10774091, National Basic Research Program of China (973 program, 2007CB613302), and the Fund for Doctoral Program of National Education 20060422023.

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Copyright © 2008 Run Long et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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