The crystal structure of IrB₂: a first-principle calculation

Abstract

First-principle calculations were performed to investigate the structural, elastic, and electronic properties of iridium diboride (IrB₂). It was demonstrated that the new phase of IrB₂ belongs to the monoclinic C2/m space group, and we have named it m-IrB₂. Its structure is energetically much superior to the recently proposed Pmmn-type IrB₂. Further calculations of phonon and elastic constants confirm that m-IrB₂ is dynamically and mechanically stable. The calculated high shear modulus reveals that it is a potentially a material of low compressibility. An analysis of the density of its states and chemical bonding show that the strongly directional covalent B–B and B–Ir bonds in m-IrB₂ make a considerable contribution to its stability.

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1. Introduction

Any material with Vickers hardness H_V above 40 GPa may be defined as a superhard material.¹ Superhard materials are of considerable interest, not only for their hardness, but also because they are ultra-incompressible, and they have high melting temperature, chemical inertness, and high thermal conductivity, and thus, they are important for industrial applications.² The most widely used superhard materials are diamond³ and cubic boron nitride (c-BN),⁴ although the application of these is limited by their natural defects. Finding novel, improved superhard materials is therefore of high priority.

The main principle in the design of hard materials is to combine small strongly covalent bonding light elements, such as boron, carbon, or nitrogen, with large electron-rich transition metals. These compounds usually exhibit high valence electron density and highly directional covalent bonding, making them a rich source of multifunctional superhard materials.^5–9 It has been reported that ReB₂,^10,13 OsB₂ (ref. 11 and 13) and RuB₂ (ref. 12 and 13) have been identified as superhard materials, and theoretical studies support these results. ReB₂ is particularly interesting. OsB₂ and ReB₂ form different lattice structures: OsB₂ has an orthorhombic RuB₂-type structure (space group Pmmn), and ReB₂ crystallizes in a hexagonal ReB₂-type structure (space group P6₃/mmc).^15–17 The hardness of ReB₂ is 48 GPa at a load of 0.49 N.¹⁴ Could the other 5d transition metal borides also present such an interesting nature? Ir and Re are neighboring 5d transition metals, and a pure-phase IrB_1.35 has been produced by X-ray diffraction. The Vickers hardness of IrB_1.35 is 18.2–49.8 GPa, depending on the load, ranging from 0.49 to 9.81 N.¹⁸ In addition, a superhard IrB_1.1 film, with a high Vickers hardness of 43 GPa has also been found.^19,20 Recently, Wang et al. have reported that IrB₂ in the orthorhombic phase, with a Pmmn space group, was most stable at ambient pressures. According to this study, the bulk modulus and shear modulus of IrB₂ were 277 GPa and 108 GPa, respectively.²¹ IrB₂ is thus a good candidate as a superhard material, and it was hoped that a study of its high-pressure structures and phase transitions might amplify the understanding of transition metal compounds and their behavior.

In the present study the structural stability, elastic properties, and electronic structure of iridium diboride (IrB₂) using first-principle calculations based on density functional theory have been explored. A monoclinic C2/m structure IrB₂ has been revealed which is much more energetically preferable to the Pmmn-type structures recently proposed, and it has been named it m-IrB₂. The new phase is dynamically and mechanically stable. The phase stability, electronic structure, and elastic properties of different structures have been compared and analyzed in detail. The quadrilateral networks of the boron layers provide strong covalent bonding, a prerequisite for a hard material of low compressibility.

2. Computational details

In the present study evolutionary variable-cell simulations for IrB₂ were performed at ambient pressure, involving systems containing one to four formula units (FU), as implemented in the USPEX code.^22–24 Most calculations were performed using first-principle plane-wave pseudo-potential density functional theory according to the Vienna ab initio simulation package (VASP).^25–27 The exchange and correlation effects are described by the local-density approximation.²⁸ The k-point sampling in the Brillouin zone (BZ) was performed using the Monkhorst–Pack scheme. For hexagonal structures Γ-centered grids were used. An energy cut-off at 500 eV with 4 × 9 × 5, 6 × 8 × 7, 5 × 8 × 3, 4 × 4 × 9 and 8 × 5 × 6 Monkhorst–Pack grids for the electronic BZ integrations was conducted for the m-IrB₂, C2/m-IrB₂, Cm-IrB₂, Pnnm-IrB₂, and Pmmn-IrB₂ phases, respectively. Formation enthalpy, ΔH, was calculated using eqn (1):

ΔH = E(IrB₂) − E(solid Ir) − 2E(solid B), (1)

where the symbol E represents one FU of total energy for each solid phase, the solid phase of boron being the α-phase. All these were chosen to ensure that enthalpy calculations were fully convergent, with energy differences below 1 meV per atom. For each selected structure the bond lengths, cell parameters, and atomic positions, were fully optimized at different pressures. Elastic constants were obtained by evaluation of the stress tensor generated by small strains, using the density-functional plane wave technique used in the CASTEP code.²⁹ The bulk modulus, shear modulus, Young's modulus, and Poisson's ratio were estimated using the Voigt–Reuss–Hill approximation.³⁰ The phonon calculations were carried out using a super-cell approach, as implemented in the Phonon code.³¹ The theoretical Vickers hardness was estimated using the Chen,³² Gao,³³ and Šimůnek models.³⁴ Details of the convergence tests are described elsewhere.^35–38

3. Results and discussion

3.1 Structure and features

At ambient pressure variable cell structure prediction simulations were performed for IrB₂ with a system containing one to four FUs in the simulation cell. The variable cell simulation showed the monoclinic C2/m structure to be the most stable phase, as illustrated in Fig. 1. There are 12 atoms in the unit cell. The m-IrB₂ unit cell is a monoclinic structure (space group, C2/m) with a = 7.346 Å, b = 2.810 Å and c = 5.837 Å, α = γ = 90°, β = 67.22° and one non-equivalent Ir atom occupies the crystallographic 4i (0.409, 0.000, 0.836) sites and two non-equivalent B atoms occupy the crystallographic 4i (0.669, 0.000, 0.488), and (0.768, 0.5, 0.264) sites at zero pressure. In m-IrB₂, the boron layers form a type of quadrilateral conjugated network, and the Ir atoms are arranged in corrugated hexagonal sheets perpendicular to the c axis. The shortest inter-atomic distance of B–Ir is 2.18 Å, which is less than the sum (2.25 Å) of the covalent radii of the B atom (r = 0.84 Å) and the Ir atom (r = 1.41 Å), suggesting strong covalent bonding.

Fig. 1 Crystal structure of m-IrB₂ in different directions.

3.2 Thermodynamic and dynamic stability

The application of a compound requires an accurate knowledge of its thermodynamic stability, and in particular its phase stability. For further experimental synthesis it was therefore necessary to investigate the relative stability of IrB₂. The formation enthalpies were evaluated using eqn (2):

H = E(IrB₂) − E(Ir) − 2E(B), (2)

where E(B) was calculated from the structure of α-rhombohedral boron.³⁹ Fig. 2 illustrates the calculated formation enthalpy of IrB₂ with different structures under pressures up to 100 GPa. According to these enthalpy calculations, only C2/m-IrB₂, Cm-IrB₂, Pnnm-IrB₂, and m-IrB₂ were competitive structures, and Wang's Pmmn-IrB₂ structure is also included in Fig. 2.

Fig. 2 Formation enthalpy versus pressure for different IrB₂ structures.

As can be seen in Fig. 2, all the structures had a negative formation enthalpy with thermodynamic stability below 100 GPa. With increase in pressure the stability of Pnnm-IrB₂, Cm-IrB₂, and m-IrB₂ gradually decreased, whereas the stability of the other two phases increased. At ambient pressure, the formation enthalpy predicted for m-IrB₂ was much lower than that of the reactants, Ir + 2B, by a factor of 0.37 eV, indicating that it should be directly synthesizable from elemental Ir and B. In Wang's paper, Pmmn-IrB₂ is stated to be the most stable structure, but the monoclinic m-IrB₂ structure is in fact more stable than this. With increasing pressure the enthalpy of m-IrB₂ becomes close to that of Pmmn-IrB₂. At least up to 100 GPa, the enthalpy of m-IrB₂ is in all cases the lowest of the candidate structures.

Dynamic stability is important for structural stability. Phonon structures provide a criterion for judging crystal stability. The phonon–dispersion curves in the whole BZ of m-IrB₂ at ambient conditions are shown in Fig. 3, which shows that there were no imaginary phonon frequencies in the whole BZ, confirming the dynamic stability of the newly proposed m-IrB₂. It is well known that shorter bond lengths contribute to higher phonon frequencies. The phonon frequency of m-IrB₂ (∼30 THz) shown in Fig. 3 was larger than that of Wang's Pmmn-IrB₂ (∼24 THz), and this high phonon frequency indicates that there are shorter bond lengths in m-IrB₂.

Fig. 3 Phonon–dispersion curves of m-IrB₂ at 0 GPa.

3.3 Mechanical stability

The elastic constants were calculated in order to evaluate the mechanical properties and stability of m-IrB₂. The mechanical stability of a crystal requires the strain energy to be positive, which implies that the whole set of elastic constant, C_ij, values satisfies the Born–Huang criterion.⁴⁰ For a monoclinic crystal, the independent elastic stiffness tensor comprises thirteen components: C₁₁, C₂₂, C₃₃, C₄₄, C₅₅, C₆₆, C₁₂, C₁₃, C₂₃, C₁₅, C₂₅, C₃₅, and C₄₆. The criteria for mechanical stability are given by eqn (3) and (4):

C₁₁ > 0, C₂₂ > 0, C₃₃ > 0, C₄₄ > 0, C₅₅ > 0, C₆₆ > 0, (C₃₃C₅₅ − C₃₅²) > 0, (C₄₄C₆₆ − C₄₆²) > 0, (C₂₂ + C₃₃ − 2C₂₃) > 0, [C₂₂(C₃₃C₅₅ − C₃₅²) + 2C₂₃C₂₅C₃₅ − C₂₃²C₅₅ − C₂₅² − C₃₃] > 0, (3)

g = C₁₁C₂₂C₃₃ − C₁₁C₂₃² − C₂₂C₁₃² − C₃₃C₁₂² + 2C₁₂C₁₃C₂₃, {2[C₁₅C₂₅(C₃₃C₁₂ − C₁₃C₂₃) + C₁₅C₃₅(C₂₂C₁₃ − C₁₂C₃₃) + C₂₅C₃₅(C₁₁C₂₃ − C₁₂C₁₃)] − [C₁₅²(C₂₂C₃₃ − C₂₃²) + C₂₅²(C₁₁C₃₃ − C₁₃²) + C₃₅²(C₁₁C₂₂ − C₁₂²)] + C₅₅^g} > 0 (4)

From Table 1 we can confirm that the C_ij of m-IrB₂ fulfills the stability criteria listed previously, suggesting that m-IrB₂ is mechanically stable at ambient pressure. It is therefore apparent that m-IrB₂ is both mechanically and dynamically stable at ambient pressure. Accurate elastic constants are helpful in understanding mechanical properties, and they also provide vital information in predicting the properties of a material. To explore the mechanical properties of m-IrB₂ the elastic constants of m-IrB₂, Pmmn-IrB₂, C2/m-IrB₂, Cm-IrB₂, and Pnnm-IrB₂ were calculated and these are summarized in Table 1. In addition, the Young's modulus, Y, and Poisson's ratio, υ, were obtained from eqn (5) and (6):

Y = (9GB)/(3B + G) (5)

υ = (3B − 2G)/(6B + 2G) (6)

Table 1 Calculated elastic constants, C_ij (GPa), bulk modulus, B (GPa), shear modulus, G (GPa), Young's modulus, Y (GPa), Poisson's ratio, υ, Vicker's hardness, H_v and the B/G ratio, for m-IrB₂, Pmmn-IrB₂, C2/m-IrB₂, Cm-IrB₂, and Pnnm-IrB₂

	C11	C22	C33	C44	C55	C66	C12	C13	C23	B	G	Y	υ	B/G	Hv
m-IrB2	586	556	816	136	201	78	178	311	129	355	172	444	0.29	2.06	13.82
Pmmn-IrB2	369	429	670	45	70	155	246	151	174	290	113	300	0.32	2.56	7.20
C2/m-IrB2	650	480	674	179	137	91	226	285	252	370	151	398	0.32	2.45	9.70
Cm-IrB2	549	453	864	119	179	190	255	222	189	355	177	455	0.28	2.0	14.68
Pnnm-IrB2	553	761	507	127	107	224	346	142	150	344	170	437	0.28	2.02	14.02

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The calculated bulk modulus, shear modulus, Young's modulus, and Poisson's ratio of m-IrB₂ and the reference materials mentioned previously are summarized in Table 1.

As shown in Table 1, the calculated C₃₃ for m-IrB₂ was larger than that for C₁₁ or C₂₂, implying that the resistance to deformation in the c-direction is stronger than in the a- or the b-direction. The calculated bulk modulus of the m-IrB₂ phase was 355 GPa, which is close to the experimental data for OsB₂ (348 GPa),¹¹ indicating its ultra-incompressible structure. It is well known that shear modulus provides a better indicator of potential hardness than bulk modulus. The calculated shear moduli of m-IrB₂, Pmmn-IrB₂, C2/m-IrB₂, Cm-IrB₂, and Pnnm-IrB₂ are 172, 113, 151, 177 and 170 GPa, respectively. It is expected that they will resist shear strain to a large extent. The high Young's moduli of these compounds indicate their ability to resist tension and pressure in the range of elastic deformation.⁴¹

The B/G ratio describes the ductility or brittleness of a material, with 1.75 as the critical value. If the value of B/G lies below 1.75 the material is brittle, otherwise it is ductile.⁴² The calculated B/G ratio of m-IrB₂ was 2.06, implying a ductile nature. The value of Poisson's ratio (υ) indicates the directionality of the covalent bonding. A typical υ value for covalent materials is 0.1 and for metallic materials it is 0.33.⁴³ A small Poisson's ratio contributes to the hardness of a material and indicates a high degree of covalent bonding. Chen's model was used for calculating the hardness:

H_v = 2(K²G)^0.585 − 3, (7)

where K = G/B is Pugh's modulus ratio. Using this model the hardness of m-IrB₂ was calculated to be 14.4 GPa. The calculated hardness was 12.05 GPa using Gao's model and 15.03 GPa using the Šimůnek model. These results, show the value of Vickers hardness by different empirical hardness models lies in the range 12.05–15.03 GPa, thus confirming that m-IrB2 is indeed a hard material, with a mean hardness value of ∼13.82 GPa.

3.4 Electronic structure analysis

The electronic density of states (DOS) and the atom-resolved partial density of states (PDOS) of m-IrB₂ are shown in Fig. 4. An intriguing type of bonding is seen in the PDOS of m-IrB₂, in which the iridium and boron atoms form strong covalent bonds, which is confirmed by the appreciable overlap of the iridium d-electron and the boron p-electron curves. From the PDOS of m-IrB₂ it is seen that the iridium d-electron states contribute most to the DOS, and the boron s-electron states are mainly located at the bottom of the valence bands. The finite electronic DOS at the Fermi level indicates the metallic nature of IrB₂ under ambient pressure.

Fig. 4 Total and partial DOS of m-IrB₂. The Fermi level is indicated by a dashed line.

To gain a more detailed insight into the bonding character of m-IrB₂ the electronic localization function (ELF) was calculated.⁴⁴ The ELF provides a reliable measure of electron pairing and localization. According to its original definition, ELF values lie between 0 and 1, where ELF = 1 corresponds to the perfect localization characteristic of covalent bonds. It should be noted that ELF is not a measure of electron density but is a function of the Pauli principle, and is valuable for distinguishing between metallic, covalent, and ionic bonding. To determine the chemical bonding of m-IrB₂ we plotted its ELF in Fig. 5, with the iso-surface at ELF = 0.74. In addition, strong covalent bonding interaction between the B atoms is seen in m-IrB₂. This is consistent with analysis of the DOS, and as a result this strong covalent bonding will naturally increase the structural stability and high bulk modulus of m-IrB₂.

Fig. 5 Contours of the electronic localization function (ELF) of m-IrB₂.

4. Conclusions

A monoclinic C2/m structure has been shown to be the ground-state structure of IrB₂, using first-principle calculations based on density functional theory. It was proposed that it be named m-IrB₂ and this structure is energetically preferable to the proposed Pmmn structure suggested earlier. The structural stability of IrB₂ was reviewed, and up to pressures of at least 100 GPa, the enthalpy of m-IrB₂ is always the lowest of the candidate structures. According to the calculated phonon dispersions and elastic constants of m-IrB₂, it is concluded that this phase is both dynamically and elastically stable under ambient conditions. Its high bulk and shear moduli imply that m-IrB₂ is potentially a non-compressible and superhard material. In addition, the calculated PDOS results demonstrate that m-IrB₂ is metallic in character. The calculated electronic localization function demonstrates that strongly covalent B–B and Ir–B bonding plays a major role in establishing the incompressibility and hardness of m-IrB₂.

Acknowledgements

We are grateful for financial support from the National Basic Research Program of China (no. 2011CB808200), the Program for Changjiang Scholars and Innovative Research Team in University (no. IRT1132), the National Natural Science Foundation of China (no. 51032001, 11074090, 11404134, 10979001, 51025206, 11104102, 11204100), the National Fund for Fostering Talent in Basic Science (no. J1103202), and the China Postdoctoral Science Foundation (2014M561279, 2013T60314, 2012M511326). Specialized Research Fund for the Doctoral Program of Higher Education (20110061120007, 20120061120008). Some of the calculations were carried out at the High Performance Computing Center (HPCC) of Jilin University.

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