Table of Contents
Research Letters in Physics
Volume 2008, Article ID 293517, 5 pages
Research Letter

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

School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China

Received 23 September 2007; Accepted 16 December 2007

Academic Editor: Wai-Yim Ching

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.


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 kc/ka, and anisotropic factors and AB compared with the experimental results.
Figure 1: The crystal structure for ReB2. Black and grey balls represent Re and B atoms, respectively.

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 kc/ka 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 kc/ka 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, AB = 1.91% and AG = 1.36% with LDA calculation and AB = 1.94% and AG = 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].

Figure 2: (a) The total charge density contour (unit in e/angstrom3) for ReB2. (b) The difference charge density contour (unit in e/angstrom3) for ReB2. The charge density shown here is the result from GGA calculations. The bonding between Re and B exhibits the strong directionality throughout the plane. Black and grey balls represent Re and B atoms, respectively.

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.

Figure 3: The band structure of ReB2. The dashed line represents the Fermi level.
Figure 4: Total DOS of ReB2 and partial DOS of B-2s, B-2p, Re-5p, and Re-5d from the GGA calculations. The dashed line represents the Fermi level.

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.


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.


  1. R. B. Kaner, J. J. Gilman, and S. H. Tolbert, “Designing superhard materials,” Science, vol. 308, no. 5726, pp. 1268–1269, 2005. View at Publisher · View at Google Scholar
  2. J. He, L. Guo, X. Guo et al., “Predicting hardness of dense C3N4 polymorphs,” Applied Physics Letters, vol. 88, no. 10, Article ID 101906, 3 pages, 2006. View at Publisher · View at Google Scholar
  3. R. W. Cumberland, M. B. Weinberger, J. J. Gilman, S. M. Clark, S. H. Tolbert, and R. B. Kaner, “Osmium diboride, an ultra-incompressible, hard material,” Journal of the American Chemical Society, vol. 127, no. 20, pp. 7264–7265, 2005. View at Publisher · View at Google Scholar
  4. H.-Y. Chung, M. B. Weinberger, J. B. Levine et al., “Synthesis of ultra-incompressible superhard rhenium diboride at ambient pressure,” Science, vol. 316, no. 5823, pp. 436–439, 2007. View at Publisher · View at Google Scholar
  5. J. A. Mcgurty, “Austenitic iron alloys,” April 1978, US Patent no. 4086085. View at Google Scholar
  6. M. H. Manghnani, K. Katahara, and E. S. Fisher, “Ultrasonic equation of state of rhenium,” Physical Review B, vol. 9, no. 4, pp. 1421–1431, 1974. View at Publisher · View at Google Scholar
  7. X. Hao, Y. Xu, Z. Wu et al., “Low-compressibility and hard materials ReB2 and WB2: prediction from first-principles study,” Physical Review B, vol. 74, no. 2, Article ID 224112, 5 pages, 2006. View at Publisher · View at Google Scholar
  8. M. D. Segall, P. J. D. Lindan, M. J. Probert et al., “First-principles simulation: ideas, illustrations and the CASTEP code,” Journal of Physics: Condensed Matter, vol. 14, no. 11, pp. 2717–2744, 2002. View at Publisher · View at Google Scholar
  9. J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Physical Review Letters, vol. 77, no. 18, pp. 3865–3868, 1996. View at Publisher · View at Google Scholar
  10. D. M. Ceperley and B. J. Alder, “Ground state of the electron gas by a stochastic method,” Physical Review Letters, vol. 45, no. 7, pp. 566–569, 1980. View at Publisher · View at Google Scholar
  11. J. P. Perdew and A. Zunger, “Self-interaction correction to density-functional approximations for many-electron systems,” Physical Review B, vol. 23, no. 10, pp. 5048–5079, 1981. View at Publisher · View at Google Scholar
  12. D. Vanderbilt, “Soft self-consistent pseudopotentials in a generalized eigenvalue formalism,” Physical Review B, vol. 41, no. 11, pp. 7892–7895, 1990. View at Publisher · View at Google Scholar
  13. H. J. Monkhorst and J. D. Pack, “Special points for Brillouin-zone integrations,” Physical Review B, vol. 13, no. 12, pp. 5188–5192, 1976. View at Publisher · View at Google Scholar
  14. S. J. La Placa and B. Post, “The crystal structure of rhenium diboride,” Acta Crystallographica, vol. 15, part 2, pp. 97–99, 1962. View at Publisher · View at Google Scholar
  15. M. Born, “On the stability of crystal lattices. I,” Proceedings of the Cambridge Philosophical Society, vol. 36, pp. 160–172, 1940. View at Google Scholar
  16. M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, Oxford, UK, 1965.
  17. J.-Y. Wang and Y.-C. Zhou, “Polymorphism of Ti3SiC2 ceramic: first-principles investigations,” Physical Review B, vol. 69, no. 14, Article ID 144108, 13 pages, 2004. View at Publisher · View at Google Scholar
  18. J. Wang, Y. Zhou, T. Liao, and Z. Lin, “First-principles prediction of low shear-strain resistance of Al3BC3: a metal borocarbide containing short linear BC2 units,” Applied Physics Letters, vol. 89, no. 2, Article ID 021917, 3 pages, 2006. View at Publisher · View at Google Scholar
  19. P. Ravindran, L. Fast, P. A. Korzhavyi, B. Johansson, J. Wills, and O. Eriksson, “Density functional theory for calculation of elastic properties of orthorhombic crystals: application to TiSi2,” Journal of Applied Physics, vol. 84, no. 9, pp. 4891–4904, 1998. View at Publisher · View at Google Scholar
  20. D. Sanchez-Portal, E. Artacho, and J. M. Soler, “Projection of plane-wave calculations into atomic orbitals,” Solid State Communications, vol. 95, no. 10, pp. 685–690, 1995. View at Publisher · View at Google Scholar
  21. R. S. Mulliken, “Electronic population analysis on LCAO-MO molecular wave functions. I,” Journal of Chemical Physics, vol. 23, no. 10, pp. 1833–1840, 1955. View at Publisher · View at Google Scholar
  22. H. Gou, L. Hou, J. Zhang, H. Li, G. Sun, and F. Gao, “First-principles study of low compressibility osmium borides,” Applied Physics Letters, vol. 88, no. 22, Article ID 221904, 3 pages, 2006. View at Publisher · View at Google Scholar
  23. F. D. Murnaghan, “The compressibility of media under extreme pressures,” Proceedings of the National Academy of Sciences of the United States of America, vol. 30, no. 9, pp. 244–247, 1944. View at Publisher · View at Google Scholar