Research Letters in Physics

Volume 2008, Article ID 293517, 5 pages

http://dx.doi.org/10.1155/2008/293517

## Structural, Elastic, and Electronic Properties of ReB_{2}:
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.

#### Abstract

The structural, elastic, and electronic properties of the hard material ReB_{2} have been investigated by means of density functional theory. The calculated equilibrium structural parameters of ReB_{2}
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
ReB_{2} 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 OsB_{2} has
been synthesized successfully in experiment [3].
However, the Os lattice expands by approximately 10% upon the incorporation of
B atoms to form OsB_{2} 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 ReB_{2} under ambient
conditions [4], for which the incorporation of
B into the interstitial sites of Re to form ReB_{2} brings only 5%
expansion of the Re lattice and the bulk modulus of ReB_{2} is 360 GPa rivaling
that of the diamond
(442 GPa) [5]. Thus, ReB_{2} is
considered to be one of the most promising candidates for improving the mechanical
properties of OsB_{2}.

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 ReB_{2} 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 ReB_{2} properties, a few results have been reported [7]. Therefore, it is interesting to understand the microcosmic mechanism of
the hardness-related
properties of ReB_{2} from the basic point of view of physics and
applications.

In this work, we performed density
functional calculations on ReB_{2} 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 ReB_{2} were also
calculated for investigating the electronic properties of ReB_{2}.

#### 2. Methods and Models

The density functional calculations
on the structural, elastic, and electronic properties of ReB_{2} were
performed using the CASTEP code [8]. Both lattice and internal
coordinates were optimized to get the equilibrium structure for ReB_{2}. 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 ReB_{2} 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 ReB_{2}, the experimental structure was used as the
initial structure for the calculations [14].

#### 3. Results and Discussions

ReB_{2} 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
2*a*
Wyckoff
sites
and four B atoms occupy the 4*f*
sites, as shown in
Figure 1. The calculated lattice constants of ReB_{2} 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/cm^{3} with an error about 3.1% and
2.2% compared with the experimental value 12.68 g/cm^{3}, respectively [14]. The calculated elastic constants of ReB_{2} 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 ReB_{2}.

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
ReB_{2}, the anisotropy of shear
anisotropy ratio was
calculated by *A* = *C*_{44}/*C*_{66} [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 ReB_{2} 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 ReB_{2},
*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 ReB_{2} 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 ReB_{2} 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 ReB_{2}. To shed more light on the bonding characteristic of ReB_{2},
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 ReB_{2}, 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 ReB_{2} are of
the unusual mixtures of covalent, ionic, and metallic bonding, as observed in
OsB_{2} [22].

The band structure of ReB_{2} 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 ReB_{2}. Since the metallicity is uncommon in
superhard materials, that is, most of the superhard materials are insulators or
semiconductors [23], this special characteristic of ReB_{2} will put
it in a favorable position
in the future application
for electron
conductivity. Figure 4 is the total DOS (TDOS) of ReB_{2} 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 ReB_{2}.
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 ReB_{2}, which is
responsible for the high bulk modulus and hardness.

#### 4. Conclusions

The results and analysis of the
structural, elastic, and electronic properties for ReB_{2} demonstrate
that the hexagonal ReB_{2} with the experimental structural parameters is found to be
metallic, which is uncommon in hard materials and suggests that ReB_{2} 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 ReB_{2} 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.

#### References

- 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 - J. He, L. Guo, X. Guo et al., “Predicting hardness of dense ${\text{C}}_{3}{\text{N}}_{4}$ polymorphs,”
*Applied Physics Letters*, vol. 88, no. 10, Article ID 101906, 3 pages, 2006. View at Publisher · View at Google Scholar - 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 - 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 - J. A. Mcgurty, “Austenitic iron alloys,” April 1978, US Patent no. 4086085. View at Google Scholar
- 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 - X. Hao, Y. Xu, Z. Wu et al., “Low-compressibility and hard materials ${\text{ReB}}_{2}$ and ${\text{WB}}_{2}$: 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - M. Born and K. Huang,
*Dynamical Theory of Crystal Lattices*, Clarendon Press, Oxford, UK, 1965. - J.-Y. Wang and Y.-C. Zhou, “Polymorphism of ${\text{Ti}}_{3}{\text{SiC}}_{2}$ ceramic: first-principles investigations,”
*Physical Review B*, vol. 69, no. 14, Article ID 144108, 13 pages, 2004. View at Publisher · View at Google Scholar - J. Wang, Y. Zhou, T. Liao, and Z. Lin, “First-principles prediction of low shear-strain resistance of ${\text{Al}}_{3}{\text{BC}}_{3}$: a metal borocarbide containing short linear ${\text{BC}}_{2}$ units,”
*Applied Physics Letters*, vol. 89, no. 2, Article ID 021917, 3 pages, 2006. View at Publisher · View at Google Scholar - 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 ${\text{TiSi}}_{2}$,”
*Journal of Applied Physics*, vol. 84, no. 9, pp. 4891–4904, 1998. View at Publisher · View at Google Scholar - 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 - 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 - 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 - 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