Article pubs.acs.org/IC
Crystal Structures, Stabilities, Electronic Properties, and Hardness of MoB2: First-Principles Calculations Li-Ping Ding,† Peng Shao,*,† Fang-Hui Zhang,† Cheng Lu,*,‡,§ Lei Ding,† Shu Ya Ning,† and Xiao Fen Huang∥ †
Department of Optoelectronic Science & Technology, College of Electrical & Information Engineering, Shaanxi University of Science & Technology, Xi’an 710021, China ‡ Department of Physics, Nanyang Normal University, Nanyang 473061, China § Beijing Computational Science Research Center, Beijing 100084, China ∥ Physics Department, Sichuan Normal University, Chengdu 610068, China S Supporting Information *
ABSTRACT: On the basis of the first-principles techniques, we perform the structure prediction for MoB2. Accordingly, a new ground-state crystal structure WB2 (P63/mmc, 2 fu/cell) is uncovered. The experimental synthesized rhombohedral R3̅m and hexagonal AlB2, as well as theoretical predicted RuB2 structures, are no longer the most favorite structures. By analyzing the elastic constants, formation enthalpies, and phonon dispersion, we find that the WB 2 phase is thermodynamically and mechanically stable. The high bulk modulus B, shear modulus G, low Poisson’s ratio ν, and small B/G ratio are benefit to its low compressibility. When the pressure is 10 GPa, a phase transition is observed between the WB2-MoB2 and the rhombohedral R3̅m MoB2 phases. By analyzing the density of states and electron density, we find that the strong covalent is formed in MoB2 compounds, which contributes a great deal to its low compressibility. Furthermore, the low compressibility is also correlated with the local buckled structure. recently.17 Compared with the traditional superhard materials,18−21 boron-rich tungsten possesses more ultrahardness, higher melting points, and stronger resistance to oxidation. In the periodic table, the element sitting one column above tungsten (W) is molybdenum (Mo). It has been tested to have low compressibility. Thus, it would be very interesting to investigate the mechanical properties for Mo borides. MoB2 is first proposed to have a hexagonal structure (AlB2type and space group P6/mmm)22,23 which is formed by planar metal and boron sheets, but this crystal structure is the subject of continuing debate. The later researchers prefer to employ the rhombohedral structure (space group R3̅m, No. 166),24,25 in which the Wyckoff positions of Mo, B1, and B2 are 6c (0, 0, 0.5758), 6c (0, 0, 0.6817), and 6c (0, 0, 0.1677), respectively. Even more recently, Chen et al.26 found that the orthorhombic RuB2-type structure has lower energy than the hexagonal AlB2type structure. Apart from these investigations on the crystal structures of MoB2 compounds, some model works on Mo−B compounds also have given insights into their dynamical, thermodynamic, and mechanical properties.27−30 However, the
1. INTRODUCTION As is known to all, superhard materials can be formed two ways. One is the combination of light elements, namely, boron, carbon, nitrogen, and oxygen. Good examples of this type of covalent compounds are diamond,1 c-BN,2 BC2N,3 B6O,4 and BC5.5 The other widely accepted way is the exploration of covalent compounds consisting of light elements combined with transition metals (TMs). Numerous researchers have investigated structures and performance of this class of compounds, such as the synthesized TM carbides, TM oxides, TM nitrides, and TM borides. The results showed that this class of materials has high hardness6−9 indeed. However, most of them are unstable under ambient temperature and pressure, with the exception of TM borides. This may be due to boron itself having very high hardness. Therefore, TM borides, which can achieve superhard conditions, have attracted considerable attention.10,11 The first synthesized transition metal boride is OsB212 obtained by the arc dissolution method. Two years later, ReB2 is predicted to be the superhard material.13 However, their respective application and prime cost are questioned.14−16 One hence pursues relatively inexpensive TM borides to overcome the above shortcomings. One of the typical examples is the synthesis of boron-rich tungsten (WB2) © XXXX American Chemical Society
Received: April 13, 2016
A
DOI: 10.1021/acs.inorgchem.6b00899 Inorg. Chem. XXXX, XXX, XXX−XXX
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Inorganic Chemistry
Table 1. Optimized Equilibrium Lattice Parameter a, b, and c (Å); Density ρ (g/cm3); Unit Cell Volume V (Å3) per Formula Unit; and Difference in Total Energy for Various MoB2, with the Available Experimental and Theoretical Values structure
space group R3̅m
166 expta exptb theora theorc AlB2 exptb exptd expte theorf ReB2 theorf WB2 theorg OsB2 RuB2 theorf MoSi2 CaF2
P6/mmm
P63/mmc P63/mmc Pmmn Pmmn I4/mmm Fm3̅m
a
b
c
ρ
V
ΔE
3.013 3.012 3.011 3.012 3.019 3.026 3.046 3.050 3.004 3.000 2.922 2.906 3.013 3.041 4.637 4.637 4.616 2.716 5.110
3.013 3.012 3.011 3.012 3.019 3.026 3.046 3.050 3.004 3.000 2.922 2.906 3.013
20.928 20.958 20.943 20.958 20.961 3.313 3.071 3.066 3.172 3.321 7.725 7.700 13.955 14.097 4.226 4.226 4.210 7.552 5.110
7.117
27.427
0.028
7.430
26.273
0.452
6.837
28.551
0.309
7.119
27.422
0.000
6.837 6.839
28.545 28.545
0.289 0.289
7.008 5.851
27.858 33.362
2.645 4.216
2.914 2.913 2.897 2.716 5.110
a
Reference 46, CASTEP: GGA. bReference 39 cReference 38, VASP: GGA. dReference 23 eReference 47 fReference 26, CASTEP: GGA-PBE. Reference 38, CPMD: GGA.
g
Table 2. Calculated Elastic Constants (GPa), Bulk Modulus B (GPa), Shear Modulus G (GPa), Young Modulus Y (GPa), B/G, Poisson’s Ratio ν, and Elastic Anisotropy Index AU for MoB2a structure
C11
C12
C13
R3m ̅ theorb theorc theord exptd theore AlB2 theorc theorf theore ReB2 theorf WB2 OsB2 RuB2 theorf MoSi2 CaF2
565 609 602 640
120 123 131
550 585 612 627 613 518 531 578 458 459 472 274 98
126 144 136 120 120 172 178 133 209 207 211 270 255
C22
C23
C33
C44
162 152 156
666 718 739 758
227 239 232
180 217 236 231 220 87 88 152 141 140 143 172
624 422 416 398 391 855 876 681 793 792 818 677
230 120 150 174 168 247 247 221 171 171 162 −822 −175
513
60
496 495 513
66 63 60
C55
289 290 303
C66
213 214 242 −119
B 297 310 312 320(2) 314(11) 299 304 315 313 304 284 294 300 284 282 292 264 202
G 225 238 239
216 153 169 192 463 228 240 226 212 213 234 −164 −136
ν
AU
539 569 572
0.20 0.19 0.20
0.01
1.99
522 393 429
0.28 0.27
0.49
1.25
540
0.18
0.36
1.33 1.34 1.32
542 509 510
0.20 0.20 0.20
0.01 0.59 0.59
−1.61 −1.49
−620 −526
0.89 0.93
1.10
B/G 1.32 1.30
Y
a
In addition, the previous experiment and theoretical values are also available. bReference 37, VASP: GGA. cReference 49, VASP: GGA-PAW. d Reference 46, CASTEP: GGA. eReference 39, CASTEP: GGA-PBE. fReference 26, CASTEP: GGA-PBE.
we have established the pressure-induced phase transition diagram of MoB2 at zero temperature. Furthermore, their mechanical and electronic properties were also investigated to further understand the hardness, especially the contribution of the covalent bond on hardness.
most favorite structures and mechanical behaviors of MoB2 compounds were not well-understood until now. As is known to all, the crystal structures of materials are very important to their mechanical properties. Hence, there is a pressing need to explore structural stabilities, mechanical properties, and phase transition for MoB2. In the present work, we systematically investigate the crystal structures of MoB2 by using first-principles calculations. Accordingly, a new ground-state WB2-type structure was found, which is evidently lower in energy than hexagonal AlB2-type, rhombohedral R3m ̅ , and orthorhombic RuB2-type structures by 452, 28, and 289 meV, respectively. In addition,
2. COMPUTATIONAL DETAILS In the framework of density functional theory (DFT), we performed all the calculations by using the Cambridge Serial Total Energy Package (CASTEP) code.31 The Vanderbilt US-PP32 is employed with the 650 eV cutoff energies for rhombohedral R3m ̅ (OsB2, MoSi2, and RuB2) structures, 750 eV for hexagonal AlB2-type (ReB2 and WB2) structures, as well as 850 eV for fluorite CaF2 structure. The K-points B
DOI: 10.1021/acs.inorgchem.6b00899 Inorg. Chem. XXXX, XXX, XXX−XXX
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Inorganic Chemistry of 13 × 13 × 2 for rhombohedral and WB2 structures, 13 × 13 × 12 for hexagonal AlB2-type structure, 13 × 13 × 4 for ReB2 structure, 7 × 12 × 8 for OsB2 and RuB2 structures, and 10 × 10 × 4 for MoSi2 structure are generated using the Monkhorst−Pack scheme.33 The exchange and correlation functionals are carried out by the Perdew− Burke−Ernzerhof within generalized gradient approximation (GGAPBE).34 The tolerances for geometry optimization are set as the difference in total energy being within 5 × 10−6 eV/atom, the maximum stress within 0.02 GPa, and the maximum ionic displacement within 5 × 10−4 Å, respectively. The formation enthalpies are defined by ΔH = E(MoB2) − E(Mo) − 2E(B), in which E(MoB2) is the energy of various stable MoB2 crystal, and E(Mo) and E(B) are the respective total energies of pure bcc-Mo and α-B.
Figure 1. Optimized crystal structures: (a) rhombohedral R3̅m MoB2, (b) WB2-MoB2, (c) AlB2-MoB2, and (d) RuB2-MoB2.
3. RESULTS AND DISCUSSION 3.1. Determination of Crystal Structure. The rhombohedral R3̅m and hexagonal AlB2-type structures24,35,36 for MoB2 have been synthesized experimentally, and many other structures were proposed by previous theoretical studies.26,37−39 Therefore, we first selected the two experimental structures to ensure the reliability of the present calculations. In addition, some other chemically related structures (ReB2,40 WB2,41 OsB2,13 RuB2,42 and MoSi243) were also considered. The reason is that the structures of chemically related compounds are always similar. Following this idea, Wang et al.44 have predicted that IrN2 possesses the IrP2 structure successfully. Besides, we also adopted the fluorite CaF2 structure45 because it is usually used when the form of the studied crystal is similar to AlB2. Table 1 lists the calculated lattice lengths (a, b, and c), densities (ρ), cell volumes per formula unit (V), and relative energies of MoB2 with different structures. As can be seen from Table 1, hexagonal WB2-type structure has the lowest total energy, indicating that the experimentally synthesized rhombohedral R3m ̅ and hexagonal AlB2-type structures are no longer favorite structures. This is different from the result of Hayami et al.38 WB2-MoB2 is lower in energy (28 meV) than the rhombohedral R3̅m structure in our calculations. In addition, the WB2-MoB2 structure is also more energetically favorable than the theoretical obtained RuB2-MoB2. We hope that our results will be further verified in the future. The calculated lattice parameters of rhombohedral, hexagonal AlB2-type, ReB2type, WB2-type, and RuB2-type structures are in accord with the previous experimental and theoretical values,23,26,37−39,46,47 verifying the reliability of our calculations. Furthermore, the elastic constants of these structures are calculated and listed in Table 2. According to Table 2, we found that MoSi2-type and CaF2-type structures are mechanically unstable due to their elastic constants not satisfying the stability criteria.48 Thus, they are ruled out first. To deeply understand the structural properties of MoB2, the structures of experimentally synthesized rhombohedral R3̅m and hexagonal AlB2-type, and theoretically predicted RuB2MoB2 and WB2-MoB2 structures, are optimized and shown in Figure 1. As is shown in Figure 1a, the structure of rhombohedral R3̅m can be described as the stacking sequence of AbA′bAcA′cAaA′a... along the c axis, where a, b, and c stand for the close-packed Mo layers, and A and A′ are two kinds of different hexagonal boron networks. The A′ layer, which is sandwiched between two of the same Mo layers (e.g., aa or bb or cc), is a planar graphite-like sheet, while the A layer consists of a puckered hexagonal three-dimensional (3D) boron network and sits between two different Mo layers. In the
puckered hexagonal boron rings, the boron atoms are alternately placed upward and downward, and each boron atom is located at the center of the Mo trigonal pyramids. For our predicted hexagonal WB2-MoB2 structure, it is energetically much more stable than the experimentally synthesized rhombohedral R3̅m and hexagonal P6/mmm structures. As shown in Figure 1b, the stacking order of WB2-MoB2 structure is AcA′cAbA′b... along the crystallographic c axis. The distance between the two adjacent same Mo layers (b and b, c and c) is 3.18 Å, while that between two different layers (b and c) is 4.42 Å. As for hexagonal AlB2-type MoB2 (Figure 1c), layers of Mo and B atoms alternate, and the B layer is a planar hexagonal network, similar to the A′-type boron of above rhombohedral structure. The positions of Mo atoms of the close-packed Mo layers of AlB2-MoB2 are directly above or below the centers of boron six-membered rings. The predicted RuB2-MoB2, which has the identical structure with OsB2-MoB2, is 289 meV lower in energy than the most stable structure WB2-MoB2. In RuB2MoB2 structure, the boron sheets are formed by six boat-like rings, and the folding Mo layers alternately stack along the c direction. It contains two MoB2 formula units (fu) in a unit cell and can be regarded as the orthorhombic Pmmn space group. The shortest B−B bond is 1.737 Å, which is shorter than those of superhard materials RuB2 (1.902 Å) and OsB2 (1.878 Å).17 Thus, the high hardness for RuB2-MoB2 can be predicted. 3.2. Mechanical Stability. As mentioned above, if the elastic constants of the crystal structure meet the mechanical stability criteria,48 it is mechanically stable and vice versa. Therefore, the accurate elastic constants could be used to estimate the hardness of materials and help us to understand well the maromechanical properties. On the basis of the strain− stress method, the elastic constants for various structural MoB2 are obtained and listed in Table 2. Obviously, all the considered structures are mechanically stable except for MoSi2-MoB2 and CaF2-MoB2. The calculated elastic constants of rhombohedral R3m ̅ , AlB2-MoB2, ReB2-MoB2, and RuB2-MoB2 are in accord with the experimental and theoretical values.26,37,39,46,49 The extremely large C33 values are found for ReB2-MoB2 (855 GPa), OsB2-MoB2 (793 GPa), and RuB2-MoB2 (792 GPa) phases. These values are larger than C11 of c-BN (773 GPa),50 suggesting that they are difficult to compress along the c axis. In addition, we found that the C11 and C33 values of rhombohedral R3̅m and WB2-MoB2 are almost equal, indicating their isotropic linear incompressibility. The same conclusion is also found between OsB2-MoB2 and RuB2-MoB2. On the basis of the elastic constants, the bulk modulus B and shear modulus G are calculated by the Voigt−Reuss−Hill approximations.51 The Young’s modulus Y and Poisson’s ratio ν C
DOI: 10.1021/acs.inorgchem.6b00899 Inorg. Chem. XXXX, XXX, XXX−XXX
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Inorganic Chemistry are calculated from the formulas Y = 9GB/(3B + G) and v = (3B − 2G)/[2(3B + G)],48 respectively. The results are listed in Table 2. From Table 2, we can see that the B, Y, and ν of rhombohedral, AlB2-type, ReB2-type, and RuB2-type MoB2 agree well with the available experimental and theoretical values. All the mechanically stable compounds possess large bulk moduli, which are higher than those of commonly used hard materials SiC52 and Al2O3.53 The materials have high bulk moduli indicating their high incompressibility. Among these structures of MoB2, the lowest-energy WB2-structured MoB2 has the largest Young modulus, 542 GPa, revealing its strong ability to resist tension and pressure. The shear modulus (G) is a better indicator of potential hardness compared to the bulk modulus. It can quantify the resistance to the shear deformation. From Table 2, the shear modulus values of rhombohedral R3̅m, AlB2-MoB2, ReB2-MoB2, OsB2-MoB2, and RuB2-MoB2 are 225, 228, 226, 212, 213 GPa, respectively. Therefore, we expect that they are able to withstand the shear strain to a great extent. AlB2-MoB2 has the relatively small shear moduli (153 GPa) indicating its low resistibility of deformation at a constant volume. Furthermore, the Poisson’s ratio (ν) is an essential parameter to express the degree of directionality for the covalent bond. The directionality of the bonding can affect the hardness of material.54 The small Poisson’s ratio of these mechanically stable MoB2 implies their strong covalent bonds. In light of Pugh’s criterion,55 the ratio B/G of a material could be used in determing whether it is brittle (B/G < 1.75) or ductile (B/G > 1.75). Apparently, only the ratio of hexagonal AlB2-MoB2 is larger than 1.75, indicating its ductile nature, whereas the other structures of MoB2 are all brittle. Overall, the large elastic modulus, small value of B/G, and Poisson’s ratio indicate that MoB2 with rhombohedral R3̅m, WB2, ReB2, OsB2, and RuB2 structures are potential superhard materials. The elastic anisotropy of varying degree for a material is valuable for layered diborides. The elastic anisotropy of crystals can be represented many different ways, for instance, by utilizing the calculated elastic constants A = 2C44/(C11 − C12).56−58 Here, a universal elastic anisotropy index AU = 5GV/ GR + BV/BR − 614 is used to estimate the elastic anisotropy of MoB2. In this expression, B and G, with subscripts V and R, denote the bulk modulus and shear modulus, with Voigt and Reuss approximations, respectively. The calculated anisotropic indexes of MoB2 are tabulated in Table 2. It is found that rhombohedral R3̅m and WB2-type MoB2 have superior elastic isotropic natures due to their AU values slightly deviating from zero, whereas the other structures are anisotropic. 3.3. Thermodynamic Stability. The knowledge of thermodynamic stability for a compound, in particular the phase stability,59−61 is required for its application. Thermodynamic stability could be analyzed by the formation enthalpies of the ground state. In principle, the more thermodynamically stable structure has the lower formation enthalpy. The calculated formation enthalpies of various MoB2 structures with the pressure ranging from 0 to 100 GPa are plotted in Figure 2. From Figure 2, we can see that all the mechanically stable structures MoB2 are also thermodynamically stable toward dissociation into the corresponding constituent elements. Their formation enthalpies decrease gradually with the increase of pressure, indicating that high pressure is beneficial to their thermodynamic stabilities. At ambient pressure, our predicted WB2-MoB2 is the most stable phase. However, it transforms to the experimentally synthesized
Figure 2. Calculated enthalpy as a function of pressure.
rhombohedral R3̅m phase when the pressure is above 10 GPa. This may help to synthesize the high-pressure phase of rhombohedral R3̅m. Besides, it is surprisingly found that the formation enthalpies of these two phases are very close to each other above 10 GPa. Furthermore, the enthalpy−pressure curves of OsB2-MoB2 and RuB2-MoB2 merge together in the whole range of pressure, which further uncovers that these two phases belong to the same structure. The phase transition between ReB2-MoB2 and AlB2-MoB2 occurs at 80 GPa, and that between RuB2-MoB2 (OsB2-MoB2) and AlB2-MoB2 occurs at 95 GPa, indicating that AlB2-MoB2 is more energetically stable under a high pressure. As is well-known, the phonon can be used to estimate structural stability. In this case, we have carefully calculated the phonon dispersion for the mechanically and thermodynamically stable phases (rhombohedral R3m ̅ , AlB2-MoB2, ReB2-MoB2, WB2-MoB2, OsB2-MoB2, and RuB2-MoB2) at ambient pressure. All the results are shown in Figure 3. It is found that the AlB2MoB2 phase is structurally unstable due to the imaginary phonon frequency that exists in phonon dispersion spectra (Figure 3c), whereas the other phases are stable structures. 3.4. Electronic Structure Analysis. To investigate the electronic properties of MoB2 with various different structures, the densities of states (DOS) have been studied and plotted in Figure 4 in which the dotted line indicates Fermi level EF. From Figure 4, it is clearly found that the finite N(EF) occus at the Fermi level in all the compounds. That is to say, these structured MoB2 materials exhibit metallic behavior in their crystalline phases. This metallic behavior may be caused by the metallic bonding between Mo atoms. In addition, the typical feature of these compounds is the presence of a pseudogap (a sharp valley around the Fermi energy) in the total DOS, which separates the bonding and antibonding states.62 The structural stabilities of these compounds would be enhanced with the presence of a pseudogap, indicating that the strong interaction is formed between Mo and B atoms. In the case of a partial DOS, the DOS at the Fermi level of these compounds mainly comes from the Mo-4d orbitals. From the bottom of the valence band to the Fermi level, it is clearly found that the Mo4d and B-2p orbitals are energetically degenerate, implying that the orbital hybridization between molybdenum and boron atoms is strong. Thus, the strong covalent bonding is formed in MoB2 compounds, which may magnify their bulk modulus and shear modulus. However, the spatial separation among molybdenum and boron atoms lowers the hybridization effect. As a consequence, there are two types of bonds formed in these MoB2 compounds. The interactions of B−B and Mo−B are covalent bonds, while the other one is ionic bond. D
DOI: 10.1021/acs.inorgchem.6b00899 Inorg. Chem. XXXX, XXX, XXX−XXX
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Inorganic Chemistry
Figure 3. Phonon dispersion curves: (a) rhombohedral R3m ̅ MoB2, (b) AlB2-MoB2, (c) ReB2-MoB2, (d) WB2-MoB2, (e) OsB2-MoB2, and (f) RuB2MoB2.
found that are two different bonding characters (covalent and ionic) emerge in these MoB2 structures. As is well-known, the hardness differs from the bulk and shear modulus. In addition, compared to the ionic component, it is assumed that the small metallic component has a stronger negative impact on hardness.64 Therefore, the evaluation of the Vicker hardness is very necessary on the basis of the semiempirical model, which has been successfully used to estimate the Vicker hardness of various transition metal borides.65 In this model, we considered the role of the metallic bond on hardness. The expression is written as follows
To further understand the bonding mechanism of various MoB2 species, the Mulliken populations and bond lengths in unit lattice for those stable MoB2 structures have been analyzed and summarized in Table 3. In addition, the corresponding difference charge density around Mo−B and B−B bonds and electron densities of those structures are presented in Figures 5 and 6, respectively. In Figure 5, the red and yellow area corresponds to an increase of the electron density (Δρ > 0), whereas the electron density decreases in the blue and green area (Δρ < 0). In Figure 6, the blue area shows a high electron density region, and electron density becomes overlapped when the covalent bond is formed. We all know that the negative values of Mulliken population present antibonding character and the positive values correspond to the bonding character. Moreover, if the Mulliken population value is close to zero, the ionic interaction between two bonding atoms may increase. According to this principle, we can see from Table 3 that the large electron densities locate between two nearest boron atoms, as well as between the Mo atom and its neighboring boron atoms because of their large value of the positive Mulliken population. Indeed, the large electron densities between them can be easily found from the plotted difference charge density and electron density maps (see Figures 5 and 6). Meanwhile, the large positive Mulliken population also suggests a strong covalent bond,63 which may contribute to their high hardness. 3.5. Hardness. The large bulk modulus and shear modulus, small B/G ratios, low Poisson’s ratios, and strong covalent bonds may contribute to the high hardness of MoB2. However, by analyzing the electron density states and electron density, we
Hν(Gpa) = 699Pνb−5/3 exp( −3005fm1.553 )
(1)
where νb is the volume of bond (vμb = (dμ)3/∑[(dν)3Nνb]), P is the value of Mulliken population, and f m is a factor of 0.026N (E F) ). In νb and f m, dμ is the bond length ne bond, Nbν is the number of bond. N(EF) is the
metallicity ( fm =
of the μ-type electron density of states at the Fermi level, and ne stands for the valence electrons. For the complex compound which contains more than two types of chemical bonds in their unit cell, the hardness could be defined as the average hardness of all binary systems. μ
μ
Hν(Gpa) = [∏ (Hνμ)n ]1/ ∑ n
μ
(2)
μ
Here, n denotes the number of μ-type bonds in the chemical compound. Using eq 2, we estimate the hardness of various MoB2. The hardness and bond parameter are calculated and tabulated in Table 3. Meanwhile, the calculated Vicker hardness E
DOI: 10.1021/acs.inorgchem.6b00899 Inorg. Chem. XXXX, XXX, XXX−XXX
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Inorganic Chemistry
Figure 4. Total and partial density of states (DOS): (a) rhombohedral R3̅m MoB2, (b) AlB2-MoB2, (c) ReB2-MoB2, (d) WB2-MoB2, (e) OsB2MoB2, and (f) RuB2-MoB2. The Fermi level is at zero.
values based on Gao’s66 and Chen’s67 hardness models are also shown in Table 3 as comparison. As we can see, the experimentally synthesized rhombohedral structure (space group R3m ̅ , No. 166) MoB2 contains three types of bonding bonds (two types B−B and one type B−Mo) with bond lengths 1.740, 1.847, and 2.359 Å, respectively. In our calculation of hardness, the two types of antibonding bonds for B−Mo are not considered. The Mulliken overlap population for the two types of B−B bonds are 2.41e and 2.09e, respectively. The two values are significantly larger than that of B−Mo. Thus, the strong covalent B−B bond dominates the hardness of this phase. A similar conclusion is also obtained from the other phases. The obtained Vicker hardness value of rhombohedral MoB2 is 22.8 GPa (24.1 GPa) which agrees well with the previous theoretical and experimental values,37,39,68 demonstrating that it is a hard material. According to the obtained Vicker hardness, the other phases, such as ReB2-, WB2-, OsB2-, and RuB2-MoB2, are also hard materials due to their hardness values being comparable with that of rhombohedral MoB2. Furthermore, it is worth mentioning that the hardness of our predicted lowest-energy WB2-MoB2 is the largest (29.0 GPa) within the considered structures, which is comparable to those of α-SiO2 (30.6 GPa) and β-Si3N4 (30.3 GPa).66 With the comparison between WB2-MoB2 and the rhombohedral phase, one can found that the high hardness of WB2-MoB2 originates from the strong covalency B−Mo and the local structure of conjoint covalent bonds. Additionally, a similar situation is found for their bond lengths of bonding B−B and B−Mo (see
Table 3), but the Mulliken population of covalent bond B−Mo in rhombohedral phase is smaller than that of WB2-MoB2.
4. CONCLUSIONS To explore new hard materials, the structures, stabilities, compressibilities, electronic structures, and hardnesses of molybdenum diborides have been investigated. All the results are summarized as follows. (1) Among these considered structures, a new hexagonal structure WB2 (space group P63/ mmc, No. 194, Z = 4) is uncovered to be the lowest-energy structure for MoB2 compound. The analyses of mechanical stability and thermodynamical stability further confirmed that it is indeed the most stable structure at ambient conditions. (2) The calculated elastic constants C11 and C33 of the lowestenergy WB2-MoB2 are very close to those of the experimentally synthesized rhombohedral R3̅m phase MoB2, indicating their isotropic linear incompressibility. In addition, their low compressibilities are further confirmed by their large bulk modulus (B), shear modulus (G), low Poisson’s ratio (ν), and small B/G ratio. (3) Formation enthalpies reveal that high pressure is beneficial to the stability of the crystal. When the pressure is up to 10 GPa, the MoB2 phase transforms from WB2 to rhombohedral R3̅m. Namely, rhombohedral R3̅m is a highpressure phase. (4) On the basis of our semiempirical method including the metallic contributions, the lowest-energy WB2MoB2 is found to possess the largest Vicker hardness (29.0 GPa) within our considered structures. The analyses of density of states and electron density indicate that the strong covalent F
DOI: 10.1021/acs.inorgchem.6b00899 Inorg. Chem. XXXX, XXX, XXX−XXX
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Inorganic Chemistry Table 3. Calculated Bond Parameters and Vickers Hardnessa bond
d
B−B B−B B−Mo B−Mo B−Mo
1.740 1.847 2.359 2.371 2.212
B−B B−Mo B−Mo
1.800 2.246 2.336
B−B B−B B−Mo B−Mo B−Mo
1.739 1.851 2.215 2.351 2.357
B−B B−B B−Mo B−Mo B−B
1.781 1.912 2.209 2.233 2.957
B−B B−B B−Mo B−Mo B−B
1.737 1.919 2.201 2.250 2.953
νb Structure 4.561 5.456 11.367
P 166 2.41 2.09 0.64 −0.01 −0.23
Structure ReB2 5.315 2.16 −0.26 11.618 0.56 Structure WB2 4.522 2.43 5.454 2.07 −0.23 11.174 1.66 −0.03 Structure OsB2 2.856 0.64 3.533 1.28 5.449 0.04 5.629 0.37 −0.41 Structure RuB2 2.652 0.69 3.576 1.26 5.395 0.02 5.763 0.37 −0.41
f m (×10−3)
Hν
0 0 3.468
22.8 24.1 31.5 24.4b 24.2c 22.0d
0
22.6 24.3 33.9
1.864 0 0 5.705
0 0 2.024 2.024
0 0 1.891 1.891
Figure 6. Electron density plots for various MoB2.
dral R3̅m shows that high hardness is also correlated with the local buckled structure.
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29.0 29.9 31.0
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.6b00899. Crystallographic information for R3̅m-MoB2 (CIF) Crystallographic information for RuB2-MoB2 (CIF) Crystallographic information for WB2-MoB2 (CIF)
22.4 24.7 29.6
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AUTHOR INFORMATION
Corresponding Authors
24.8 26.7 30.3
*E-mail:
[email protected] (P.S.). Phone/fax: +86 29 86168320. *E-mail:
[email protected] (C.L.).
The first three values of hardness are obtained by our model and Gao’s and Chen’s hardness models, respectively. bReference 37. c Reference 39. dReference 68.
Notes
a
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (nos. 11547147 and 11304167), Natural Science Foundations of Shaanxi Province (nos. 2016JQ1003 and 2016JQ1028), Scientific Research Plan Projects of Shaanxi Education Department (no. 16JK1098), the Shaanxi University of Science & Technology Key Research Grant (nos. BJ15-07 and 2016BJ-01), Postdoctoral Science Foundation of China (no. 20111317), and Open Project of State Key Laboratory of Superhard Materials (no. 201405).
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DOI: 10.1021/acs.inorgchem.6b00899 Inorg. Chem. XXXX, XXX, XXX−XXX
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DOI: 10.1021/acs.inorgchem.6b00899 Inorg. Chem. XXXX, XXX, XXX−XXX
