New Crystal Structures of IrB and IrB2: First-Principles Calculations

Article pubs.acs.org/JPCC

New Crystal Structures of IrB and IrB2: First-Principles Calculations De Yu Wang, Bing Wang, and Yuan Xu Wang* Institute for Computational Materials Science, School of Physics and Electronics, Henan University, Kaifeng 475004, People’s Republic of China ABSTRACT: Superhard IrB1.35 [Chem. Mater. 2009, 21, 1407] and IrB1.1 film [ACS Appl. Mater. Interfaces 2010, 2, 581] have been synthesized in experiment, but the structural formulas of iridium borides with integral ratio between Ir and B atoms are still undefined up to now. Here, we use a combination of particle-swarm optimization technique and first-principles calculations to explore the crystal structures of IrB and IrB2. We demonstrate that the new phase P1−IrB belongs to the orthorhombic Pnma space group, while P5−IrB2 (space group Pmmn) has a same structure type with OsB2. At the pressure of about 5 GPa, a phase transition occurs between the Pnma and anti-NiAs phases for P1−IrB. Further phonon and elastic constants calculations imply that both P1−IrB and P5−IrB2 are dynamically and mechanically stable and are potential low compressible materials because of their high bulk moduli. The analysis of density of states and chemical bonding indicates that the formation of strong covalent bonding in these compounds contributes greatly to their stabilities.

I. INTRODUCTION Synthesizing and designing hard materials are of great scientific interests due to their importance in fundamental science and technological applications.1 Although diamond is the hardest known material with a measured hardness at 60−120 GPa,2 it is unstable in the presence of oxygen at moderate temperatures and reacts easily with iron-containing materials. Cubic boron nitride (c-BN) is second to the hardness of diamond but has to be synthesized under high pressure and high temperature conditions, which causes great cost of synthesis. Therefore, great efforts have been devoted to exploring new types of hard and ultraincompressible materials. One common way to develop hard materials is by combining transition metals with light elements, namely, boron, carbon, nitrogen, or oxygen. The compounds formed by transition metals and light elements usually possess high valence electron density and directional covalent bonds,3 and these covalent bonds are strong enough to inhibit creation and movement of dislocations, greatly improving the mechanical properties. On the other hand, d valence electrons are considered to contribute to the hardness of transition-metal compounds.4 Recently, several transitionmetal borides (TMBs), such as ReB2, OsB2, TaB2, CrB4, and WB4,5−11 have been successfully synthesized under ambient pressure. The obtained results revealed that they possess high bulk and shear moduli. Moreover, TMBs can be synthesized under ambient pressure, which leads to the low-cost synthesis condition and is beneficial to their applications. Therefore, TMBs are good candidates as hard materials. In a survey of the literature, we found that experimentalists have synthesized a phase-pure IrB1.35 (space group C2/m) using an electron beam apparatus at vacuum condition, with iridium powder and boron powder taken in molar ratio of 1:1.5.12 The © 2012 American Chemical Society

measured Vickers hardness of IrB1.35 was 18.2−49.8 GPa, depending on the loads ranging from 0.49 to 9.81 N, which is comparable to the hardness of ReB2 (20.8−49.9 GPa).13 Later, IrB1.1 film (0.4 μm) was deposited on SiO2 substrates, and the film was considered to be superhard because of its high Vickers hardness 43 GPa.14 However, there is no experimental report on the synthesis of iridium borides with integral ratio between Ir and B atoms. Moreover, detailed theoretical investigation of IrB and IrB2 is also seldom. Zhao et al.15 studied the hexagonal P63/mmc structure for IrB and two different hexagonal phases (P63/mmc and P6/mmm) and an orthorhombic Pmmn (OsB2type) structure for IrB2 using the first-principles calculations, and they found that the hexagonal P63/mmc IrB is elastically stable and that the orthorhombic OsB2-type structure for IrB2 is most energetically stable. In the present work, we report two new phases for IrB and IrB2, obtained by using the newly developed particle swarm optimization (PSO) approach for crystal structure prediction.16 This method has successfully predicted several structures, which were subsequently confirmed by independent experiments.16−19 Our results show that the predicted phase for IrB belongs to the orthorhombic Pnma space group, while the structure type of the new phase for IrB2 (space group Pmmn) is the same as that of synthesized OsB2. Both of the two phases are dynamically and mechanically stable. In addition, other structures for IrB and IrB2 are also studied for comparison. Received: July 5, 2012 Revised: August 23, 2012 Published: September 25, 2012 21961

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Table I. Calculated Formation Enthalpy (ΔH in eV) per Unit, Optimized Equilibrium Lattice Parameters a, b, and c (Å), and Unit Cell Volume (V in Å3) per fu of IrB and IrB2 IrB

IrB2

a

structure

space group

ΔH

a

b

c

V

P1−IrB anti-NiAs−IrB anti-NiAs−IrBa WC−IrB FeB−IrB CsCl−IrB P5−IrB2 OsB2−IrB2 OsB2−IrB2a OsB2−IrB2b ReB2−IrB2 ReB2−IrB2a WB2−IrB2 AlB2−IrB2 AlB2−IrB2a

Pnma P63/mmc

−0.425 −0.393

2.870

−0.383 −0.269 0.305 −0.279 −0.279

7.021 3.995 3.986 2.718 3.520

22.31 21.25

P-6m2 Pnma Pm-3m Pmmn Pmmn

P63/mmc

−0.111 −0.149

P63/mmc P6/mmm

0.417 1.503

4.428 3.505 3.508 3.052 6.914 2.741 3.152 3.149 3.153 3.136 3.071 3.072 2.979 3.107 3.107

3.542 4.548 4.550 4.547 4.486

4.042 4.044 4.042 3.971 7.078 7.074 15.045 3.262 3.259

21.93 21.55 20.59 28.97 28.97 28.97 27.93 28.90 28.91 28.91 27.26 27.25

Reference 15, VASP, PAW-GGA. bReference 32, CASTEP, USPP-GGA.

II. COMPUTATIONAL DETAILS The newly developed crystal structure prediction method through the PSO approach was employed to search for potential crystal structure, as implemented in the CALYPSO code.20,21 This approach was designed to predict stable or metastable crystal structures requiring only chemical compositions of a given compound at specified external conditions, unbiased by any known structural information. The underlying structural relaxations and electronic structure calculations were performed by using the density functional theory implemented in the Vienna ab initio simulation package (VASP).22−24 Exchange-correlation terms were treated by Perdew−Burke− Ernzerhof in generalized gradient approximation (GGA).25 The projector-augment wave (PAW) method was employed with a plane-wave cutoff energy of 500 eV for all phases. The k-point samplings in the Brillouin zone were performed using the Monkhorst−Pack scheme.26 For hexagonal structures, the Γcentered grids were used. The 5d76s2 and 2s22p1 were considered as the valence electrons for the Ir and B atoms, respectively. In the geometrical optimization, all forces on the atoms were converged to less than 5 meV/Å. Formation enthalpy was calculated from ΔH = E(IrBn) − E(solid Ir) − nE(solid B), where the symbol E represents one formula unit (fu) total energy of each solid phase. The solid phase of boron is from its α phase.27 Elastic stiffness constants were calculated by strain−stress method. From the calculated elastic constants Cij, the polycrystalline bulk modulus B and shear modulus G were further estimated using the Voigt−Reuss−Hill (VRH) approximation.28

Table I lists the calculated formation enthalpies, lattice parameters, and unit cell volumes of IrB and IrB2 with considered structures. As is well known, the negative values of the formation enthalpies indicate that they are thermodynamically stable. From Table I, we can see that all the considered structures for IrB have negative formation enthalpies except the CsCl-type structure; the lowest formation enthalpy of P1−IrB indicates that it is most thermodynamically stable at ambient conditions. P1−IrB contains four IrB fu in a unit cell, in which the Ir and B atoms occupy the Wyckoff 4c (0.3455, 0.25, 0.63361) and 4c (0.37308, 0.25, 0.95037) sites, respectively. Figure 1a plots the structure of P1−IrB. It is seen that P1−IrB

III. RESULTS AND DISCUSSION A. Structure and Feature. In the present work, the structures of CsCl (space group Pm-3m), WC (space group P6m2), FeB (space group Pnma), anti-NiAs (space group P63/ mmc), and a predicted new phase, P1−IrB (space group Pnma), have been considered for IrB. For the crystal structure of IrB2, we possess five possible structures, ReB2 (space group P63/ mmc), AlB2 (space group P6/mmm), OsB2 (space group Pmmn), WB2 (space group P63/mmc), and a predicted phase, P5−IrB2 (space group Pmmn), obtained by the PSO method.

Figure 1. Optimized crystal structures: (a) P1−IrB (1 × 2 × 1 supercell), (b) anti-NiAs−IrB, (c) P5−IrB2 (2 × 2 × 2 supercell). The large green and small blue balls represent Ir and B atoms, respectively.

exhibits a layered structure along the b axis, and the adjacent layers are connected by a parallelogram consisted of Ir and B atoms. In the parallelogram, the shortest interatomic distance of B−Ir, 2.16 Å, is smaller than the sum (2.25 Å) of the covalent radii of B atom (r = 0.84 Å) and Ir atom (r = 1.41 Å), suggesting a strong covalent bonding feature. The calculated lattice parameters of anti-NiAs−IrB are in excellent agreement 21962

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Figure 2. Calculated enthalpy as a function of pressure for (a) IrB and (b) IrB2.

with the previous ones,15 and the calculated formation enthalpy of anti-NiAs−IrB is second to that of P1−IrB, illustrating its good structural stability. In the NiAs structure, the Ni atoms occupy the corners of a primitive hexagonal cell, and the As atoms are located in the center of the hexagonal prisms formed by the Ni atoms, each As atom has six neighboring Ni atoms. Then, the anti-NiAs structure is formed by exchanging the positions of cations and anions, which is adopted for IrB (Figure 1b). For IrB2, the predicted P5−IrB2 is the most thermodynamically stable phase among the considered structures, which has the same structure type with OsB2. P5−IrB2 belongs to the orthorhombic Pmmn space group containing two IrB2 fu in a unit cell (a = 3.152 Å, b = 4.548 Å, and c = 4.042 Å), in which Ir and B atoms occupy the Wyckoff 2a (0, 0, 0.66392) and 4e (0.5, 0.30067, 0.85033) sites, respectively. Figure 1c shows a supercell structure of P5−IrB2. It is seen that both the layers of Ir atoms and those of B atoms are undulated. The B-atom sheets consist of boatlike six rings, again conjugated. The Ir atom arrangement is that of corrugated hexagonal sheets, with a stacking sequence VVV··· perpendicular to the c axis. Each Ir atom has two plus four Ir atoms neighbors within the layer. Within the corrugated boron atom layer, the shortest B−B bond is 1.81 Å, which is comparable to the ones in RuB2 and OsB2.29−31 To further confirm that P5−IrB2 has the same structure with OsB2, the OsB2 structure was also considered for IrB2. As seen in Table I, the calculated lattice parameters for ReB2−IrB2, OsB2−IrB2, and AlB2−IrB2 phases are in good agreement with the previous results.15,32 The calculated lattice constants and formation enthalpy of P5−IrB2 are very close to those of OsB2−IrB2, and the relative large difference of formation enthalpy for OsB2−IrB2 between the present work and ref32 may result from their different pseudopotential. The negative values of the formation enthalpies for ReB2−IrB2 and OsB2−IrB2 indicate that they are thermodynamically stable. Although the hexagonal AlB2 structure, which has good structural stability, has been adopted by many transitionmetal diborides, such as CrB2, TiB2, MoB2, MnB2, TaB2, and HfB2,33−36 the positive formation enthalpy indicates that it is not available for IrB2 at ambient condition. B. Formation Enthalpy and Dynamical Stability Considerations. It is known that the application of a

compound requires accurate knowledge of the thermodynamic stability of all relevant phases, in particular the phase stability.37−40 Therefore, it is necessary to investigate the relative stability of iridium borides for further experimental synthesis. Figure 2 plots the calculated formation enthalpy of IrB and IrB2 with different structures under the pressure up to 100 GPa. From Figure 2a, we can see that a stable phase under high pressure is not necessarily stable under atmospheric pressure. With the increase of pressure, the stabilities of P1−IrB and WC−IrB are gradually decreased, while the stabilities of the other three phases are increased. P1−IrB is the most stable phase at ambient pressure, but it transforms to the anti-NiAs− IrB phase when the pressure is above 5 GPa, indicating that anti-NiAs−IrB is a high pressure phase. This is different from RhB,41 which belongs to the anti-NiAs phase at ambient pressure and transforms to the FeB-type structure when the pressure above 22 GPa. The phase transition from WC−IrB to FeB−IrB occurs when the pressure is higher than 35 GPa, because in such a case FeB−IrB is more energetically stable. The phase CsCl−IrB becomes thermodynamically stable when the pressure is above 52 GPa, indicating that it is a highpressure phase. As seen from Figure 2b, the stabilities of WB2− IrB2, AlB2−IrB2, P5−IrB2, OsB2−IrB2, and ReB2−IrB2 are gradually enhanced with the increase of pressure, suggesting that pressure is helpful to their stabilities. However, the enthalpies of WB2−IrB2 and AlB2−IrB2 are still positive under considered pressure range, which indicates that much higher pressure or appropriate temperature is needed for their stabilities. The enthalpy−pressure curves of OsB2−IrB2 and P5−IrB2 merge together among considered pressure, which further indicates that they belong to the same structure. To check the dynamical stabilities of the currently predicted P1−IrB and P5−IrB2, we have calculated their phonon dispersion curves. As seen in Figure 3, the absence of any imaginary phonon frequency in the whole Brillouin zone for P1−IrB and P5−IrB2 gives direct proof of their dynamical stabilities at ambient condition. Moreover, it is worth noticing that the highest phonon frequency of P1−IrB displayed in Figure 3a is very large (∼92 THz). The crystal structures of BC542 and YH343 have been studied, and these reports noted that the shorter bond lengths contribute to higher phonon 21963

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they have the identical structure. It is worth noticing that the incompressibility along the c axis of ReB2−IrB2 and P5−IrB2 is very large, which is similar to that of RuB2 and OsB2,46−49 suggesting that a similar feature in the electronic structure contributes to high C33 value. By use of the calculated elastic constants, the polycrystalline bulk modulus and shear modulus are determined by the VRH approximation. The Young’s modulus is obtained by the equation Y = (9GB)/(3B + G). As seen in Table II, the calculated bulk modulus (B) agrees well with that directly obtained from the fitting results (B0) of the third-order Birch− Murnaghan equation of state,50 which further verifies the good accuracy of our elastic calculations. The calculated bulk modulus values of P1−IrB (274 GPa), anti-NiAs−IrB (316 GPa), and P5−IrB2 (277 GPa) are comparable to those of B2CO51 and OsB4,52 indicating their strong ability to resist volume deformation. The comparable bulk moduli of P5−IrB2 and ReB2−IrB2 result from their similar effective density of valence electrons, 0.519 and 0.518 e/Å3, respectively. The large Young’s moduli of these compounds indicate their strong capability of resisting tension and pressure in the range of elastic deformation. It is well known that the shear modulus of a material quantifies its resistance to the shear deformation and is a more accurate predictor of hardness than bulk modulus. The calculated shear moduli of P1−IrB, anti-NiAs−IrB, and P5−IrB2 are 124, 144, and 108 GPa, respectively; it is expected that they can withstand the shear strain to a large extent. The relative small value of G for P5−IrB2 may result from its double Ir layers, alternating with covalent B layers. The weak Ir−Ir metallic bonds within the layers likely reduce the resistance of IrB2 to large shear deformations in the easy-slip direction. Moreover, the calculated elastic moduli (B and G) of antiNiAs−IrB is much larger than that of RhB (B = 296 GPa, G = 102 GPa),41 which probably suggests that anti-NiAs−IrB is a more promising hard material candidate than RhB. The B/G ratio reflects the ductility or brittleness of a material, with 1.75 as the critical value.53 B/G less than 1.75 means the material is brittle, otherwise ductile. The calculated B/G ratio of P1−IrB, anti-NiAs−IrB, and P5−IrB2 are 2.21, 2.19, and 2.56, respectively, implying their ductile nature. D. Electronic Structure Analysis. To further understand the properties of P1−IrB, anti-NiAs−IrB, and P5−IrB2, the total and partial density of states (DOS) were calculated, as displayed in Figure 4. It is seen that the Ir 5d states of these three compounds contribute most to their total DOS and dominate the DOS at the Fermi level, and the B 2s electrons

Figure 3. Phonon dispersion curves of (a) P1−IrB and (b) P5−IrB2 at zero pressure.

frequencies. Therefore, we deduce that the high phonon frequency of P1−IrB may result from the short bond lengths in this compound. This high phonon frequency means higher coupling-weighted phonon momentum and may lead to a strong electron−phonon interaction in P1−IrB. C. Elastic Properties. The mechanical stability is a necessary condition for a crystal to exist. Accurate elastic constants are helpful to understand the mechanical properties and also provide very useful information to estimate the hardness of a material. The strain−stress method was used to obtain their elastic constants. A small finite strain was applied on the optimized structure, and the atomic position was optimized. Then, the elastic constants were obtained by fitting the stress of the strained structure and the strains. Since CsCl− IrB, WB2−IrB2, and AlB2−IrB2 are thermodynamically unstable at ambient condition and WC−IrB is mechanically unstable, we will not discuss them any more. As seen in Table II, the studied compounds P1−IrB, anti-NiAs−IrB, FeB−IrB, ReB2−IrB2, OsB2−IrB2, and P5−IrB2 satisfy the mechanical stability criteria for crystal,44,45 indicating that they are elastically stable. Moreover, the calculated eigenvalues of the obtained elastic constant matrix of these materials are all positive, which also suggests that they are elastically stable. The large values of C11, C22, and C33 of these compounds indicate that they are extremely difficult to be compressed along the a-axis, b-axis, and c-axis, respectively. The calculated elastic constants for antiNiAs−IrB, OsB2−IrB2, and ReB2−IrB2 are consistent with the available theoretical values.15,32 The calculated lattice constants of OsB2−IrB2 are very close to that of P5−IrB2, indicating that

Table II. Calculated Elastic Constants (GPa), Bulk Modulus B0 and B (GPa), Shear Modulus G (GPa), and Young Modulus Y (GPa) of IrB and IrB2 P1−IrB FeB-IrB anti-NiAs−IrB anti-NiAs−IrBa P5−IrB2 OsB2−IrB2 OsB2−IrB2a OsB2−IrB2b ReB2−IrB2 ReB2−IrB2a a

C11

C12

C13

C22

C23

C33

C44

C55

C66

B0

B

G

B/G

Y

514 368 521 525 353 351 345 397 326 315

108 253 234 235 239 242 241 259 198 179

333 265 267 264 138 138 139 186 248 254

368 337

183 205

106 176

269 276 309

68 68 67 24

140 142 137 151

282 282

274 260 316 317 277 277 274 307 283 280

124 121 144 146 108 108 104 92 113 109

2.21 2.15 2.19

171 172 170 170

55 150 163 164 69 70 62 73 124 126

227 162

416 418 414 453

438 336 335 338 676 670 669 747 705 695

323 314 375 377 288 288 277 251 299

254

2.56 2.56

2.50 2.57

Reference 15, VASP, PAW-GGA. bReference 32, CASTEP, USPP-GGA. 21964

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The electronic localization function (ELF)54 offers a reliable measure of electron pairing and localization. According to its original definition, the ELF values are scaled between 0 and 1, where ELF = 1 corresponds to the perfect localization characteristic of covalent bonds or lone pairs. To analyze the chemical bonding of P1−IrB, anti-NiAs−IrB, and P5−IrB2, we plot their ELF in Figure 5, with the isosurface at ELF = 0.65. A partially covalent bonding interaction between the Ir and B atoms in these three compounds is indicated with the ELF maxima biased toward the B atoms (with short bond distances about 2.16, 2.26, and 2.16 Å for P1−IrB, anti-NiAs−IrB, and P5−IrB2, respectively). This is consistent with the analysis of their DOSs. Moreover, a strong covalent bonding interaction between B atoms is also seen in P1−IrB and P5−IrB2 displayed in parts a and c of Figure 5. As a result, these strong covalent bonding will surely increase the structural stabilities and high bulk moduli of P1−IrB, anti-NiAs−IrB, and P5−IrB2.

IV. CONCLUSION To conclude, by use of the PSO technique for crystal structure prediction based on first-principles calculations, we obtained two new orthorhombic phases for IrB (P1−IrB, space group Pnma) and IrB2 (P5−IrB2, space group Pmmn). The structure type of predicted P5−IrB2 is same as that of synthesized OsB2. According to the calculated phonon dispersions and elastic constants of P1−IrB and P5−IrB2, we find that both of the two phases are dynamically and elastically stable. For P1−IrB, there is a phase transition between the Pnma and anti-NiAs phases at the pressure of about 5 GPa. The high bulk moduli reveal that P1−IrB, anti-NiAs−IrB, and P5−IrB2 are potential low compressible materials. The analysis of DOS and chemical bonding indicate that the formation of strong covalent bonding in these compounds makes great contributions to their stabilities. The current theoretical prediction would inevitably stimulate future experimental synthesis of IrB and IrB2.

Figure 4. Total and partial DOS. The Fermi level is at zero.

mainly locate at the bottom of their valence bands. The states at the Fermi levels indicate the metallic behavior of these borides here. Therefore, they may be used as hard conductors. As seen from parts a and b of Figure 4, the typical feature of these two borides is that there is a deep valley, namely, the pseudogap at the Fermi level, which is the borderline between the bonding and antibonding states.33 The presence of pseudogap will surely increase structural stabilities of P1−IrB and anti-NiAs−IrB, which also implies that strong B−Ir covalent bonding exist in these two borides. From Figure 4c, we can see that there is a strong orbital hybridization between Ir 5d and B 2p orbitals for P5−IrB2 in comparison with that of P1−IrB and anti-NiAs−IrB, indicating a strong covalent interaction between the B and Ir atoms in P5−IrB2. Compared with IrB, the neighbor B atoms around the Ir atom in P5−IrB2 obviously increase. There are five B atoms and six B atoms around each Ir atom in P1−IrB and in anti-NiAs−IrB, respectively. However, there are eight B atoms around each Ir atom in P5−IrB2. Moreover, the B−B bond is strongly strengthened in P5−IrB2 (1.95 Å in P1−IrB, 2.0 Å in anti-NiAs−IrB, and 1.81 Å in P5−IrB2). The higher boron composition in P5−IrB2 induces stronger covalent bondings. Therefore, we deduced that the enhanced orbital hybridization of P5−IrB2 may originate from the increased content of B atoms from IrB to IrB2.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was sponsored by the National Natural Science Foundation of China (Grant No. 21071045), the Program for New Century Excellent Talents in University (Grant No. NCET-10-0132), and the fund of Henan University (Grant No. SBGJ090508).

Figure 5. ELF of three-dimensional framework for (a) P1−IrB (2 × 1 × 1 supercell), (b) anti-NiAs−IrB, (c) P5−IrB2 (2 × 2 × 2 supercell) with the value of isosurface at ELF = 0.65. The large green and small blue spheres represent the Ir and B atoms, respectively. 21965

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