The ground-state structure and physical properties of ReB₃ and IrB₃ predicted from first principles

Abstract

ReB₃ has been synthesized and was reported to have symmetry of P6₃/mmc [Acta Chem. Scand. 1960, 14, 733]. However, we find that this structure is not stable due to its positive formation energy. In 2009, IrB_1.35 and IrB_1.1 were synthesized and were considered to be superhard [Chem. Mater. 2007, 21, 1407; ACS Appl. Mater. Interfaces 2010, 2, 581]. Inspired by these results, we explored the possible crystal structures of ReB₃ and IrB₃ by using the developed particle swarm optimization algorithm. We predict that P6̄m2-ReB₃ and Amm2-IrB₃ are the ground-state phases of ReB₃ and IrB₃, respectively. The stability, elastic properties, and electronic structures of the predicted structures were studied by first-principles calculations. The negative calculated formation enthalpies for P6̄m2-ReB₃ and P6₃/mmc-ReB₃ indicate that they are stable and can be synthesized under ambient pressure. Their dynamical stability is confirmed by calculated phonon dispersion curves. The predicted P6₃/mmc-ReB₃ has the highest hardness among these predicted structures. The calculated density of state shows that these predicted structures are metallic. The chemical bonding features of the predicted ReB₃ and IrB₃ were investigated by analyzing their electronic localization function.

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I. Introduction

The term hardness was created by the French scientist R. A. F de Réaumur in 1722, and it has been a fundamental mechanical property of materials since then.¹ Nowadays, ultra-incompressible hard materials are extremely useful in industry due to their excellent performance, such as for cutting tools, drilling tools, hard coatings, and abrasives. Great efforts have been devoted to search for new superhard materials. One approach is to synthesize light element compounds which contain boron, carbon, nitrogen, or oxygen, for instance, diamond, cubic boron nitride (c-BN), and carbon nitrides, etc.^2–4 It is well known that diamond is the hardest material with a measured hardness of 60–120 GPa.² However, at high temperature, diamond is not only unstable in the presence of oxygen, but also reacts easily with iron-containing materials. These light element compounds are usually synthesized under high temperature and high pressure conditions, which leads to high production costs. Another approach in the search for new superhard materials is performed by inserting light elements into the lattice of a transition metal, for example ReB₂.^5–7 The introduction of light elements improves the hardness of the transition metal by forming a strong covalent interaction between the TM and the light elements, and between the light elements. These compounds usually possess high electron density and strong covalent bonds. Notably, the transition metal borides can be synthesized under ambient pressure, which reduces the cost of synthesis. The synthesis of ReB₂ and its high hardness has inspired the passion for exploring new transition-metal compounds with high hardness.^5–7 Some novel transition-metal borides have been successfully synthesized under ambient pressure, such as OsB₂, TaB₂, CrB₄, and WB₄.^8–11 They all have high bulk modulus which is not far from that of diamond. Therefore, transition metal borides could be promising surperhard materials.

5d transition metal triborides have also attracted great attention. For example, the structure of WB₃ with high hardness has been explored and is discussed in many articles.^12–15 In 1960, Aronsson et al. reported that they had synthesized ReB₃.¹⁶ However, this structure was suggested to be unstable.^17,18 Up to now, the structure of ReB₃ is still an open question. In 2009, IrB_1.35 was synthesized with iridium powder and boron powder taken in molar ratio of 1 : 1.5. Its Vickers hardness decreases from 49.8 to 18.2 GPa as the load is increased from 0.49 to 9.81.¹⁹ After that, IrB_1.1 was deposited on SiO₂ substrates, and the film was considered to be superhard due to its high Vickers hardness of 43 GPa.²⁰ Iridium borides with integral elements ratios, IrB and IrB₂, were investigated by Zhao et al. and Wang et al.^21,22 Pnma-IrB and the orthorhombic OsB₂-type structure of IrB₂ were considered to be the most energetically stable.

The hardness of TM borides can be improved by increasing the concentration of boron in them. Therefore, in this work, we explore the possible structures of ReB₃ and IrB₃ through the method of the particle-swarm optimization (PSO).^23,24 The bonding features, the electronic structure, and the mechanical properties of the predicted structures of ReB₃ and IrB₃ were investigated by first-principles calculations.

II. Computational detail

In this work, we used CALYPSO to search the potential crystal structures of ReB₃ and IrB₃ with 1–4 formula units (f.u.) in each simulation cell at 0 GPa.^23,24 The advantage of this approach is that it needs only the chemical compositions of the given compounds at specified external conditions and does not need any structural information. The underlying structure relaxations and properties calculations were performed with the projector-augmented-wave method (PAW) based on density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP).^25,26 The Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA) was used to treat the exchange–correlation function. Geometry optimization of the structures was achieved through the conjugate gradient algorithm method with a plane wave cutoff energy of 400 eV, and the forces on each ion were well-converged to be less than 5 meV per atom. Both the lattice constants and the atomic positions of the predicted structures were relaxed. For the k-point samplings in the Brillouin zone for the hexagonal structures, Γ centered grids were used, and for other structures, the Monkhorst–Pack scheme was used. Formation enthalpy was calculated from the equation ΔH = H(MB₃) − 3H(solid B) − H(M). The solid boron is α-B₁₂ at 0 GPa. The elastic constants were obtained by the strain–stress method. With the obtained elastic constants C_ij, the polycrystalline bulk modulus B and shear modulus G were estimated using the Voigt–Reuss–Hill approximation.²⁷ Young’s modulus Y and Poisson’s ratio ν were obtained by using the equations Y = (9GB)/(3B + G) and ν = (3B − 2G)/(6B + 2G), respectively. The Debye temperature was calculated from the bulk modulus, shear modulus and so on.

To check the reliability of the method used in this work, we explored the structures of WB₃ and OsB₃. The previous reported ground-state structure of OsB₃ with P6̄m2 symmetry has been successfully predicted by us through the PSO method.²⁸ Two new structures with symmetries of P3̄m1 and P6/mmm were predicted at the same time. The positive formation energies indicate that they are metastable at 0 GPa. Moreover, the negative and lowest formation energy of P6̄m2-OsB₃ show that it is the ground-state phase of OsB₃. For WB₃, at 0 GPa, the earliest discovered P6₃/mmc-WB₃ has been predicted again by us, substantiating the reliability of our method.^13,29,30 Moreover, we found some other previous reported structures of WB₃ with symmetries of R3̄m, P6̄m2, and P3̄m1 by this method.^12,15 Among these structures, the R3̄m phase possesses the lowest formation energy, which indicates that the R3̄m phase is the ground-state structure of WB₃.

III. Results and discussions

A. Structural properties and stability

We searched the potential crystal structures of ReB₃ and IrB₃ using the PSO methodology. The relaxed lattice parameters, density, and atom volume are tabulated in Table 1, and the structures are shown in Fig. 1 and 2, respectively.

Table 1 Calculated formation energy per unit (ΔH), optimized equilibrium lattice parameters a and c (Å), density (ρ in g cm⁻³), and atom volume (V in ų per atom) of ReB₃ and IrB₃

	Space group	ΔH	a	c	ρ	V
a Ref. 18, VASP.b Ref. 16, experiment.
ReB3	P6̄m2 (187)	−0.287	2.9235	4.5953	10.736	8.50
ReB3	P63/mmc (194)	−0.012	2.9044	9.3134	10.925	8.36
	P63/mmc (194)^a	0.7300	3.0960	7.5730
	P63/mmc (194)^b		2.9000	7.4750
ReB3	P3̄m1 (164)	0.0222	2.8743	4.6734	10.929	8.35
IrB3	Amm2 (38)	−0.671	5.3106	2.8456	10.739	8.73
IrB3	P63/mmc (194)	−0.018	2.9324	9.2820	10.855	8.64
IrB3	P6̄m2 (187)	−0.004	2.9282	4.6450	10.878	8.62
IrB3	P3̄m1 (164)	0.1585	2.8727	4.6665	11.249	8.34

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Fig. 1 Crystal structures of ReB₃. The blue and gray spheres represent Re and B atoms, respectively. (a) ReB₃ (P6̄m2): no. 187, the Re atom is at 1b (1, 0, 0.5000), and the B atom has two sites: 2h (0.3333, 0.6667, 0.1805), 1a (1, 0, 0); (b) ReB₃ (P6₃/mmc): no. 194, the Re atom is at 2a (0, 0, 0), and the B atom has two sites: 4f (0.3333, 0.6667, 0.6626), 2b (0, 1, 0.25); (c) ReB₃ (P3̄m1): no. 164, the Re atom is at 1a (0, 0, 0), and the B atom has two sites: 2d (0.6667, 0.3333, 0.3320), 1b (0, 0, 0.5000); (d) experimental ReB₃ (P6₃/mmc): no. 194.

Fig. 2 Crystal structures of IrB₃. The green and gray spheres represent Ir and B atoms, respectively. (a) IrB₃ (Amm2): no. 38, the Ir atom is at: 4d (0, 0.8228, −0.8126), and the B atom has four sites: 4d (0, 0.0933, −0.1879), 2b (0.5, 0, −0.3081), 2b (0.5, 0, −0.6512), 4e (0.5, 0.1602, −0.4684); (b) IrB₃ (P6₃/mmc): no. 194, the Ir atom is at: 2a (0, 0, 0), and the B atom has two sites 2b (0, 1, 0.7500), 4f (0.3333, 0.6667, 0.1644); (c) IrB₃ (P6̄m2): no. 187, the Ir atom is at: 1d (0.3333, 0.6667, 0.5000) and the B atom has two sites: 1c (0.3333, 0.6667, 0), 2i (0.6667, 0.3333, 0.8288); (d) IrB₃ (P3̄m1): no. 164, the Ir atom is at: 1b (0, 0, 0.5000) and the B atom has two sites 2d (0.3333, 0.6667, 0.1443), 1a (0, 0, 0).

ReB₃ was first synthesized by Aronsson et al. and was suggested to be hexagonal with symmetry of P6₃/mmc.¹⁶ However, several theoretical studies on the experimental ReB₃ suggested that the experimental ReB₃ easily decomposes during the course of synthesis.^17,18 Gao et al. supposed that the antibonding feature of all Re–B bonds leads to the instability of ReB₃. By using the PSO method, we found a new structure of P6₃/mmc-ReB₃, which is lower in energy than the experimental one. At zero pressure, the negative formation energy of the predicted P6₃/mmc-ReB₃ indicates that it is a stable structure. The experimental structure is composed of a puckered hexagonal planar boron layer. The shortest Re–B and B–B bonds in it are 2.34 and 1.86 Å, respectively. The predicted P6₃/mmc-ReB₃ is composed of puckered hexagon-mesh layers with half of the boron atoms in the hexagon-mesh layers being replaced by a pair of vertical boron dimers, and the nearest boron layers tend to offset each other. The boron dimers in the predicted P6₃/mmc-ReB₃ may have increased strength along the c-axis compared with the experimental one. We also calculated the bond lengths of the predicted P6₃/mmc-ReB₃. In this structure, each rhenium atom is coordinated by four boron atoms, with bond lengths of 2.26–2.32 Å. The boron atoms in it can be divided into two types. The first type is coordinated by one boron atom and one rhenium atom, with bond lengths of 1.85 (B–B) and 2.26 (Re–B) Å, respectively. The second type is coordinated by two boron atoms and two rhenium atoms, with bond lengths of 1.85 (B–B) and 2.32 (Re–B) Å, respectively. The shortest Re–B and B–B bond lengths are 2.26 and 1.85 Å, respectively. They are all shorter than the experimental ones. Such shorter Re–B and B–B bonds in the predicted P6₃/mmc-ReB₃ mean that there is a stronger covalent interaction between the adjacent atoms, which is helpful to increase its stability.

To further explore the origin of the lower energy of the predicted P6₃/mmc-ReB₃ than the experimental structure, we calculated their density of states (DOS) and electronic localization function (ELF), and display them in Fig. 3 and 4, respectively. As seen in Fig. 3, there is a DOS peak at the Fermi level of the experimental structure, indicating that the experimental structure is not so stable. However, a pseudogap appears near the Fermi level of the DOS of the predicted P6₃/mmc-ReB₃, indicating the high stability of the predicted P6₃/mmc-ReB₃. Moreover, the broad DOS peak from −8 eV to 0 eV indicates the strong hybridization between Re-d and B-p states in the predicted P6₃/mmc-ReB₃, which will increase its stability. The pseudogap of the experimental ReB₃ appears at −3.3 eV, which indicates that excessive electrons exist in the experimental structure. ELF can directly show the bonding features of the materials. As seen in Fig. 4(a) and (c), the largest value of ELF between the adjacent B atoms in experimental ReB₃ is much smaller than that of the predicted P6₃/mmc-ReB₃. Thus, the covalency of the B–B bond in the predicted P6₃/mmc-ReB₃ is stronger than that in the experimental structure, which is also helpful for increasing the stability of the predicted P6₃/mmc-ReB₃. Therefore, the stronger covalency of the Re–B and B–B bonding in the predicted P6₃/mmc-ReB₃ will lead to the higher stability of the predicted P6₃/mmc-ReB₃ than the experimentally synthesized ReB₃.

Fig. 3 Calculated total and partial DOS of ReB₃. The Fermi level is at zero.

Fig. 4 Contours of the electronic localization function (ELF) of (a) experimental ReB₃ (P6₃/mmc), (b) ReB₃ (P6̄m2), (c) ReB₃ (P6₃/mmc), (d) ReB₃ (P3̄m1), (e) IrB₃ (Amm2), (f) IrB₃ (P6̄m2), (g) IrB₃ (P6₃/mmc), (h) IrB₃ (P3̄m1).

Some new structures have been found at the same time, and the symmetries are P6̄m2 and P3̄m1, respectively. P6̄m2-ReB₃ has a puckered graphite like B layer with half of the boron atoms being replaced by a pair of vertical B₂ dimers. The rhenium atom is coordinated by three boron atoms, with bond lengths of 2.24–2.30 Å. There are two types of boron atoms in this compound. One is coordinated by one rhenium atom and two boron atoms, with bond lengths of 2.24 (Re–B), 1.66 (B–B), and 1.88 (B–B) Å, respectively. The other is coordinated by one rhenium atom and two boron atoms, with bond lengths of 2.30 (Re–B), 1.88 (B–B), and 1.88 (B–B) Å, respectively. The values of the shortest Re–B and B–B bond lengths are much smaller than those in the P6₃/mmc one. Therefore, the P6̄m2 phase may be more stable than the P6₃/mmc one, which is in accordance with the results of the calculated formation energies in Table 1. Therefore, the P6̄m2 phase is the ground-state phase of ReB₃. The P3̄m1 phase contains a puckered hexagonal planar boron layer formed by B–B bonds. In this phase, each rhenium atom is coordinated by three boron atoms. Moreover, these are two types boron atoms in the P3̄m1 phase. One is coordinated by one boron atom and one rhenium atom, and the other is coordinated by two boron atoms and one rhenium atom. The positive formation energy of this phase implies that it will be prone to decompose in the course of synthesis.

For IrB₃, four structures have been predicted. Their symmetries are Amm2, P6₃/mmc, P6̄m2, and P3̄m1, respectively. The Amm2 phase is the most stable structure because of the negative and lowest formation energy at 0 GPa. Therefore, it is the ground-state phase of IrB₃. The boron atoms in Amm2-IrB₃ form a stable triangle. The shortest Ir–B and B–B bond lengths are 2.14 and 1.71 Å, respectively. The P6₃/mmc phase has a similar structure to P6₃/mmc-ReB₃. Each iridium atom is coordinated by four boron atoms. There are two types of boron atoms in P6₃/mmc-IrB₃. One is coordinated by one iridium atom and two boron atoms, with bond lengths of 2.28 (Re–B), 1.59 (B–B), and 1.87 (B–B) Å, respectively. The other is coordinated by two iridium atoms and two boron atoms, with bond lengths of 2.32 (Re–B) and 1.87 (B–B) Å, respectively. The shortest Ir–B and B–B bond lengths are 2.28 and 1.59 Å, respectively. The negative formation energy and the short B–B bond indicate that it is stable. The P6̄m2 phase has a consistent architecture with P6̄m2-ReB₃. Each iridium atom is coordinated by three boron atoms. There are also two types boron atoms in the P6̄m2 phase. One is coordinated by one iridium atom and two boron atoms, with bond lengths of 2.32 (Re–B) and 1.78 (B–B) Å, respectively. The other is coordinated by one iridium atom and two boron atoms with bond lengths of 2.28 (Re–B), 1.59 (B–B), and 1.87 (B–B) Å, respectively. The shortest Ir–B and B–B bond lengths are 2.28 and 1.59 Å, respectively. P3̄m1-IrB₃ is similar to P3̄m1-ReB₃ in structure. The shortest B–B bond length is 1.79 Å. The positive formation energy indicates that it is not stable at 0 GPa.

We calculated the formation enthalpies of these predicted compounds under a pressure range from 0 GPa to 100 GPa shown in Fig. 5. All the enthalpy–pressure curves show that with increasing of pressure, the stabilities of these structures gradually increase, suggesting that high pressure is helpful to their stabilities. For ReB₃, the P6̄m2 phase and the P6₃/mmc phase are thermodynamically stable at 0 GPa. Notably, the P6̄m2 phase is always the most stable phase under the whole studied pressure range. The P3̄m1 phase becomes a thermodynamically stable phase when the pressure is above 5 GPa. For IrB₃, the Amm2 phase is the most stable phase under the whole studied pressure range. The P3̄m1 phase becomes stable when the pressure is above 40 GPa.

Fig. 5 Relative formation enthalpy–pressure of (a) ReB₃ and (b) IrB₃.

To compare the stability of the predicted ReB₃ with ReB₂ and IrB₃ with IrB₂, we calculated the enthalpy of each predicted structure relative to that of P6₃/mmc-ReB₂ (ref. 31) and Pmmn-IrB₂,²² respectively, as shown in Fig. 6. The enthalpies of Re, Ir, B, ReB₂, and IrB₂ in this figure are our calculation results at the corresponding pressures. From Fig. 6(a), at low pressure, ReB₂ is more stable than all the structures of ReB₃, which indicates that these ReB₃ structures may decompose into ReB₂ and B at low pressure. P6̄m2-ReB₃ becomes more stable than ReB₂ when the pressure is increased to 37 GPa. From Fig. 6(b), under low pressure, IrB₂ is more stable than all the structures of IrB₃. However, IrB₃ with phases of Amm2, P6₃/mmc, P6̄m2, and P3̄m1 become more stable than IrB₂ as the pressure is increased to 7 GPa, 28 GPa, 33 GPa, and 73 GPa, respectively. Thus, P6̄m2-ReB₃, and IrB₃ with symmetry of Amm2, P6₃/mmc, P6̄m2, and P3̄m1 can be synthesized at high pressure.

Fig. 6 Calculated enthalpy versus pressure of ReB₃ and IrB₃ relative to ReB₂ and IrB₂, respectively: (a) ReB₃ relative to the ReB₂ (b) IrB₃ relative to the IrB₂.

B. Dynamical stability and elastic properties

Dynamic stability is very important for a predicted new structure to exist, because the appearance of soft phonon modes can lead to its distortion. For example, Cheng et al. calculated the phonon dispersion curves of WB₃ with symmetry of P6₃/mmc, R3̄m, P6̄m2, and P3̄m1, and found no imaginary phonon frequency appearing in the whole Brillouin zone of them, illustrating that they are dynamically stable.¹⁵ In view of this, we calculated the phonon dispersion curves at 0 GPa to check the dynamical stabilities of the currently predicted structures of ReB₃ and IrB₃. The phonon dispersion curves of ReB₃ and IrB₃ are shown in Fig. 7 and 8, respectively. We can see that no imaginary phonon frequency appears in the whole Brillouin zone of these predicted structures. This confirms that all of these predicted compounds are dynamically stable at 0 GPa. Comparing these phonon dispersion curves, for long wave-length limitation, the transverse acoustic wave velocity of P6₃/mmc-ReB₃ is largest. In other words, P6₃/mmc-ReB₃ has the highest shear wave velocity among these compounds. That is to say, P6₃/mmc-ReB₃ may have the strongest ability against applied shear deformation. The highest shear modulus of P6₃/mmc-ReB₃ among these structures in Table 2 confirms this conclusion.

Fig. 7 Phonon dispersion curves of ReB₃ at 0 GPa.

Fig. 8 Phonon dispersion curves of IrB₃ at 0 GPa.

Table 2 Calculated elastic constants (in GPa), bulk modulus (B, B₀ in GPa), shear modulus (G in GPa), Young’s modulus (Y in GPa), Poisson’s ratio ν, Debye temperature (Θ_D in K), and hardness (H_v in GPa) of ReB₃ and IrB₃

	C11	C12	C13	C33	C44	C66	B(B0)	G	B/G	Y	ν	ΘD	Hv
ReB3 (P6̄m2)	567	130	175	905	229	218	333(320)	239	1.39	579	0.21	764	30
ReB3 (P63/mmc)	665	128	123	809	217	269	321(320)	258	1.24	610	0.18	789	37
ReB3 (P3̄m1)	613	177	122	795	194	218	318(317)	228	1.39	552	0.21	744	29
IrB3 (Amm2)	475	277	120	486	184	164	285(286)	157	1.82	398	0.27	617	16
IrB3 (P63/mmc)	398	172	223	643	164	113	297(283)	143	2.08	370	0.29	590	13
IrB3 (P6̄m2)	414	182	206	666	125	116	298(286)	133	2.24	347	0.31	570	11
IrB3 (P3̄m1)	556	176	147	666	131	190	302(302)	178	1.70	446	0.25	651	19
Diamond	1047	128			560		434	564	0.77	1181	0.05

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Mechanical stability is one necessary condition for the existence of a crystal. Accurate elastic constants can not only help us to understand the mechanical properties but also provide very useful information to estimate the hardness of a compound. To be mechanically stable, the elastic stiffness constants of a given crystal should satisfy the generalized elastic stability criteria.³² The calculated elastic constants using the strain–stress method are listed in Table 2. Obviously, the elastic constants of all the studied compounds can satisfy the mechanical stability criteria, indicating that they are all mechanically stable. The positive eigenvalues of the elastic constants matrix for these compounds further prove that they are elastically stable. The elastic constants of these predicted compounds are drawn in Fig. 9. As seen in this figure, C₄₄, C₅₅, and C₆₆ for these compounds are close. The values of C₃₃ for these predicted compounds are much larger than those of C₁₁ and C₁₂, except for Amm2-IrB₃. C₁₁, C₂₂, and C₃₃ for these compounds are larger than 400 GPa, indicating strong incompressibility along the a-axis, b-axis, and c-axis, respectively. Among these compounds, P6̄m2-ReB₃ has an extremely large C₃₃ which is much larger than that of C₁₁, indicating that the bond strength along the [001] direction is much stronger than that along the [100] direction. C₄₄ is an important indicator for the hardness of materials. All the studied structures have large C₄₄ values. Notably, P6̄m2-ReB₃ possesses the largest C₄₄ value, indicating its relatively strong strength against shear deformation.

Fig. 9 Calculated elastic constants of ReB₃ and IrB₃.

Bulk and shear moduli are important indicators of the hardness of a material. The calculated elastic constants were used to estimate the bulk and shear modulus of ReB₃ and IrB₃ using the Voigt–Reuss–Hill (VRH) approximation. Moreover, the bulk modulus (B₀) was also calculated using the third order Birch–Murnaghan equation, and the obtained values are consistent with the bulk modulus (B), implying the reliability of the present theoretical method. As is known, a high bulk modulus of a material illustrates its strong ability to resist volume deformation caused by an applied load. Apparently, all the predicted compounds in Table 2 have a large bulk modulus (above 260 GPa), indicating that they are difficult to compress. Compared with the bulk modulus, the shear modulus (G) is a much better parameter to indicate the hardness of a material. From Table 2, all structures of ReB₃ possess high shear modulus. Among them, P6₃/mmc-ReB₃ has the largest shear modulus (258 GPa), indicating that it can withstand the largest extent shear strain. In contrast, the shear moduli of all the structures of IrB₃ are below 200 GPa, indicating the relative low resistance to shape change at a constant volume.

We also list the ratio value of B/G, Young’s modulus Y, and Poisson’s ratio ν. The ratio value of B/G is a criterion to describe the ductility or brittleness of materials with 1.75 as the critical value.³² A B/G value higher (or lower) than the criteria is considered to be ductile (or brittle). From Table 2, except Amm2-IrB₃, P6₃/mmc-IrB₃, and P6̄m2-IrB₃, the B/G values of the other predicted phases are under the critical value, implying their brittle nature. The Young’s modulus, Y, indicates the capability of resisting tension and pressure in the range of elastic deformation. A large value of Y manifests a stiff material. From Table 2, we can see that P6₃/mmc-ReB₃ will be much stiffer than the other studied compounds due to its largest Y (610 GPa).

Poisson’s ratio ν is the ratio between transverse and longitudinal strain in the elastic loading direction. Thus, it can be used to describe the degree of directionality for the covalent bond. Materials with a small Poisson’s ratio are more easily compressed than sheared (small B/G), whereas those with a high Poisson’s ratio resist compression in favor of shear (large B/G).³³ A small Poisson’s ratio implies a strong degree of directionality of covalent bonding in a material. As shown in Table 2, all the studied compounds have small Poisson’s ratios ν. In particular, P6₃/mmc-ReB₃ (0.18) has a smaller ν than those of ReB₂ (0.21 (ref. 34) and 0.19 (ref. 18)) and WB₄ (0.34 (ref. 35)), indicating its strong degree of directionality of covalent bonding in P6₃/mmc-ReB₃.

As a fundamental parameter, the Debye temperature correlates with many physical properties of a solid, such as specific heat, elastic constant, and melting temperature. At low temperature, the calculated Debye temperature from elastic constants is the same as the one which is determined from specific heat measurements. In this work, we calculated the Debye temperature of the studied compounds and list them in Table 2. Debye temperature can represent the characterization and the microhardness of a solid.^36,37 P6₃/mmc-ReB₃ has the highest Debye temperature of 789 K, which is comparable to that of ReB₂ (755.5 K,³⁸ 858.3 K³⁹). Therefore, P6₃/mmc-ReB₃ may have the largest hardness among these predicted compounds.

The hardness of a material is the intrinsic resistance to deformation when a force is applied, which depends on the loading force and the quality of the sample (i.e., the presence of defects such as vacancies and dislocations).^40,41 In this work, we adopted the first-principle model of Chen et al.^41,42 to figure out the theoretical hardness of the predicted ReB₃ and IrB₃. The hardness is defined by:

H_v (GPa) = 2(k²G)^0.585 − 3, (1)

where H_v, k, and G are the Vickers hardness (GPa), Pugh modulus ratio k = G/B,⁴³ and shear modulus (GPa), respectively. Table 2 lists the calculated hardness of the predicted structures. P6₃/mmc-ReB₃ possesses a large hardness of 37 GPa, which is close to the superhard limit of 40 GPa, indicating that it is a promising superhard material. P6̄m2-ReB₃ also possesses a large hardness of 30 GPa next to P6₃/mmc-ReB₃. We hope the predicted P6₃/mmc-ReB₃ can be the excellent candidate of superhard materials.

C. Electronic structure and chemical bonding

To further understand the electronic structures and chemical bonding of these triborides, we calculated the total and partial density of states at zero pressure, and display them in Fig. 3 and 10. Clearly, the finite electronic DOS at the Fermi level indicates that all the predicted structures are metallic. It is easy to see that the peaks below −9 eV are mainly attributed to B-s states with a slight contribution from B-p, M (Re and Ir)-p, and M-d states. The states above −9 eV mainly originate from M-d and B-p orbitals with small contributions from M-p and B-s orbitals. The DOSs of M-d and B-p have a similar shape, indicating a strong hybridization between M-5d and B-2p orbitals. This hybridization could herald strong M–B bonding in these compounds, especially in P6̄m2-ReB₃. Another typical feature of DOS is a pseudogap, which is the borderline between bonding and antibonding states.⁴⁴ For P6̄m2-ReB₃, and Amm2-IrB₃, the presence of a pseudogap at the Fermi level suggests that their high stability. The pseudogap also implies the relatively stronger bonding between M and B atoms in these structures. For P6₃/mmc-IrB₃, the pseudogap appears below the Fermi energy, indicating full and partial occupancy of the bonding and antibonding states, respectively.

Fig. 10 Calculated total and partial DOS of IrB₃. The Fermi level is at zero.

To gain a deeper understanding of the bonding features, we calculated the electronic localization function (ELF) of all the structures we predicted, which can characterize electron pairing and localization. ELF is based on the Hartree–Fock pair probability of parallel spin electrons and is widely used to describe and visualize chemical bonding in molecules and solids.⁴⁵ ELF values are scaled between 0 and 1, the upper limit ELF = 1 corresponds to the perfect localization characteristic of covalent bonds or lone pairs, while ELF = 0.5 represents an electron-gas-like pair probability with values of this order indicating regions with bonds of a metallic character, and ELF = 0 is typical for a vacuum (no electron density) or areas between atomic orbitals. Therefore, the ELF is useful in distinguishing metallic, covalent, and ionic bonds, and should not be equated with electron density. The color contour maps of the calculated ELF are shown in Fig. 4. In this figure, high electron localization can be seen in the region between adjacent B atoms, indicating the existence of strong covalent B–B bonding. This strong covalent B–B bonding may result in their large hardness. In P6̄m2-ReB₃, the largest value of ELF between B atoms is 0.805, indicating a very strong covalent B–B bonding in it. The local maximum values of ELF between Re and B atoms reach 0.75, indicating the covalent bonding feature between Re and B atoms. A similar situation appears in P6₃/mmc-ReB₃. These strong covalent B–B bonding and Re–B bonding can be the source of their high hardness and stability. However, in these compounds, the local maximum values of ELF between M and B atoms are very close to the B sites, indicating partial covalent and ionic interactions between M and B atoms. Moreover, the values of ELF near the center of the M and B atoms are all close to 0.5, signifying a partially metallic feature in the M–B bond.

IV. Conclusion

Using the developed particle swarm optimization algorithm, we predicted the possible crystal structures of ReB₃ and IrB₃, and investigated their structures, phase stability, dynamical stability, elastic properties, and electronic properties by using first principles calculations based on density functional theory. We predicted two structures of ReB₃ with symmetry of P6₃/mmc and P6̄m2. They are more stable than the experimental one. Their negative formation energies suggest that they can exist at atmospheric pressure. The ground-state phases of ReB₃ and IrB₃ are P6̄m2-ReB₃ and Amm2-IrB₃, respectively. These compounds are dynamically stable at zero pressure. The calculated elastic constants indicate they are all mechanically stable. The high bulk modulus of these compounds indicate their low compressibility. The high shear moduli of P6̄m2-ReB₃ and P6₃/mmc-ReB₃ suggest their strong ability against shape change at a constant volume. The calculated DOSs show that they are all metallic. The results of the calculated electronic localization function show the strong covalent B–B bond in these structures, which makes a large contribution to their high hardness. The analysis of DOSs and chemical bonding shows that the strong covalent bonds in these compounds make great contributions to their stabilities. We hope these calculations can stimulate extensive experimental work on these predicted triborides.

Acknowledgements

This research was sponsored by the National Natural Science Foundation of China (no. 21071045 and 51371076) and the Program for Innovative Research Team (in Science and Technology) in University of Henan Province (no. 13IRTSTHN017).

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