ARTICLE pubs.acs.org/JPCC
Novel High-Pressure Phase of RhB: First-Principles Calculations Qianqian Wang, Zhisheng Zhao, Lifang Xu, Li-Min Wang, Dongli Yu, Yongjun Tian, and Julong He* State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, Hebei Province, China ABSTRACT: A new high-pressure phase of RhB has been predicted using a newly developed particle swarm optimization algorithm based on first-principles calculations. The new phase belongs to the orthorhombic Pnma space group (FeB type); it transforms from hexagonal RhB (anti-NiAs type) when the pressure exceeds 22 GPa. The high-pressure phase is both mechanically and dynamically stable, as verified by the calculations of its elastic stiffness constants and phonon dispersion. Further calculations predicate this high-pressure phase to be semimetallic, with excellent ductility (B/G = 3.56) and high Vickers hardness (25 GPa).
1. INTRODUCTION Hard materials have attracted much attention because of their excellent performance in technological applications, such as abrasives, cutting tools, and wear-resistant coatings. To search for novel superhard materials, much effort has been devoted to solids formed by light elements such as BC2N, C3N4, and BxCy.18 Great achievements have been made in the past decades. Diamond-like BC2N was synthesized with a Vickers hardness of 76 GPa, ranking second only to diamond.1 The Vickers hardness of synthesized cubic BC5 was reported to be up to 71 GPa.4 What makes these materials superhard are the strong 3D covalent bonding networks formed by the light elements.9,10 As another way to design hard materials, compounds consisting of transition metals and light elements (i.e., B, C, N, and O), such as PtN2, IrN2, ReB2, WB4, PtC, and RuC, and so on, have aroused the interest of scientists.1117 It is because the compounds formed by transition metal and light elements usually possess high valence electron density and directional covalent bonds.9 Moreover, d valence electrons are considered to contribute to the hardness of transition-metal compounds.18 Recently, bulk RhB1.1 was synthesized using an electron beam apparatus at vacuum condition; the measured Vickers hardness was 722.6 GPa, depending on the loads ranging from 0.49 to 9.81 N.19 Later, RhB1.1 film (film thickness of 1.0 μm) was deposited on SiO2 substrates, and the film was considered superhard because of its 44 GPa Vickers hardness.20 As a matter of fact, RhB1.1 is just a notation, and the ideal structure formula is RhB.21 In the following text, we only consider RhB with the ideal stoichiometric ratio, and RhB1.1 will be denoted with anti-NiAs-RhB. The previous syntheses of RhB were carried out under either ambient pressure or zero pressure.1921 In the current study, we report a new phase with higher density and hardness, which could be obtained under high pressure. At ambient pressure, RhB crystallized into the anti-NiAs structure, which is exceptive because many transition-metal compounds have the NiAs structure.2123 To our knowledge, r 2011 American Chemical Society
compounds with an anti-NiAs structure are rare and limited to RhB, PtB, and the high-pressure phase of FeO.21,22 In the NiAs structure, the Ni atoms occupy the corners of a primitive hexagonal cell, and As atoms are located in the center of the hexagonal prisms formed by Ni atoms. Because the positions of the two kinds of atoms are inequivalent, the anti-NiAs structure is then formed by exchanging the positions of cations and anions in the NiAs structure. With similar structures, the NiAs structure has attracted much attention because of its widespread phase transition into the MnP structure under some conditions such as temperature and pressure,2426 whereas little or no investigation has been made on the anti-NiAs structured compounds. This led us to the idea that a similar phase transition may occur in the anti-NiAs structure, which may bring about novel properties. In the current Article, we investigate the effect of pressure on the structure of anti-NiAs RhB crystals. The RhB structure was investigated using a crystal structure prediction technique through the particle swarm optimization (PSO) algorithm,27 which requires only the chemical compositions for a given compound at given pressure. On the basis of the simulations, above 22 GPa RhB crystals adopt the orthorhombic FeB type structure. Further calculations are performed to study the properties of this highpressure phase.
2. CALCULATION METHODS To search for potential crystal structures, we employed the PSO technique implemented in the CALYPSO (Crystal structure AnaLYsis by PSO) code. On the basis of an efficient global minimization of free-energy surfaces merging totalenergy calculations via PSO technique, the approach requires only chemical compositions for a given compound to predict stable or metastable structures under given external conditions Received: May 31, 2011 Revised: August 29, 2011 Published: September 02, 2011 19910
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Figure 1. Structure of RhB in the ground state (a) and in the high-pressure phase (b). The relaxations were preformed at zero pressure. The lattice parameters obtained at zero pressure for anti-NiAs-RhB are a = 3.391 Å and c = 4.151 Å, with Rh and B atoms occupying the 2c (1/3, 2/3, 1/4) and 2a (0, 0, 0) positions, respectively. In the FeB-RhB, the lattice parameters are a = 5.675 Å, b = 3.345 Å, and c = 4.172 Å, with Rh and B atoms taking the 4c (0.325, 0.25, 0.126) and 4c (0.474, 0.25, 0.619) positions, respectively. The pink spheres represent the B atoms, and the green ones represent the Rh atoms.
3. RESULTS AND DISCUSSION 3.1. Structure Identification. 3.1.1. Phase Transition under Pressure. The simulations were carried out within pressure range
Figure 2. Enthalpy differences between the reacting substance (Rh + αB) and the FeB-RhB relative to anti-NiAs-RhB as a function of pressure. The inset demonstrates the volume decrease as the pressure goes up.
(e.g., \pressure).27 Detailed descriptions and employment of this algorithm have been described in previous studies.2731 The underlying structural relaxations and electronic band structure calculations were performed in the framework of density functional theory, as carried out in the VASP code.32 Generalized gradient approximation (GGA) within the Perdew BurkeEruzerhof (PBE) functional form was adopted for the exchange and correlation function.33 The PAW pseudopotentials are expanded with the energy cutoff of 450 eV for the plane wave basis set.34 For Rh and B, the 4d85s1 and 2s2p1 electrons were considered as the valence electrons, respectively. The k-point samplings in the Brillouin zone are 10 10 7 for the anti-NiAsRhB and 5 8 6 for the FeB-RhB based on the MonkhorstPack method.35 In calculating the phonon frequencies, we adopted the direct supercell method, which uses the forces obtained through the HellmannFeynaman theorem.36 The calculation of the elastic constant and bond population was carried out using the CASTEP code.37
of 0300 GPa. At 0 GPa, we reproduced the anti-NiAs (P63/mmc) structure possessing the lowest energy for RhB, as shown in Figure 1a. This structure is in accordance with the experimental result and confirms the accuracy of our calculation.21 At 50 GPa, we found a new RhB phase, which has an orthorhombic (Pnma) FeB-type structure (denoted with FeB-RhB), as shown in Figure 1b. Above 50 GPa, no further phase transition was observed in our calculations. Precise relaxations were performed for the two types of RhB structures and the reacting substance (Rh + αB) in the pressure range of 0100 GPa. Figure 2 presents the enthalpy curves of the FeB-RhB structure and the reacting substance (Rh + αB) relative to the anti-NiAs-RhB within the given pressure range. At ambient pressure, the enthalpy of anti-NiAs-RhB is much lower than that of the reacting substance (Rh + αB) by ∼0.75 eV per formula, indicating that it can be directly synthesized by elemental Rh and B, which is in accordance with the previous experimental result.21 Owing to the higher enthalpy of FeB-RhB compared with antiNiAs-RhB within the pressure range of 022 GPa, FeB-RhB will not appear until the pressure reaches 22 GPa. When the pressure is higher than 22 GPa, the phase transition from anti-NiAs type to FeB type occurs because in such case FeB-RhB is more energetically stable. The volume decrease of 1.2%, as shown in Figure 2 inset, suggests that the P63/mmc f Pnma phase transition is a first-order transition. The density of the FeB-RhB is 9.537 g/cm3, higher than that of the anti-NiAs-RhB (9.140 g/cm3). 3.1.2. Mechanical and Dynamical Stability. To be mechanically stable, the elastic constant of the FeB-RhB should satisfy the generalized elastic stability criteria for orthorhombic crystals. For orthorhombic crystals, there are nine independent elastic constants. The criteria for the mechanical stability of orthorhombic structures are given as follows: Cii > 0, (i = 1, 2..., 6), [C11 + C22 + C33 + 2(C12 + C13 + C23)] > 0, (C11 + C22 2C12) > 0, (C11 + C33 2C13) > 0, and (C22 + C33 2C23) > 0.38 19911
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The calculated elastic constants of the FeB-RhB are summarized in Table 1. Clearly, the FeB-RhB satisfies the elastic stability criteria. Dynamic stability is also important for the lattice, as the appearance of soft phonon modes will lead to the distortion of the crystal. Phonon dispersions in the whole Brillouin zone of the FeB-RhB are plotted in Figure 3. The inexistence of imaginary frequencies indicates the dynamic stability of the new phase. Therefore, being mechanically and dynamically stable at zero pressure, the FeB-RhB is recoverable at ambient pressure, which is important for the technological applications of the new phase. 3.2. Electronic Structures. The electronic properties of both phases were investigated by analyzing their electronic band structure and density of states (Figure 4). The anti-NiAs-RhB exhibits obvious metallic character, with one valence band and one conduction band crossing the Fermi level, as shown in Figure 4a. The band structure of FeB-RhB (Figure 4b) shows semimetallic property, with a slight indirect overlap between the first conduction band and the last valence band. Both of the structures present a deep valley (pseudo gap), which separates the RhB bonding and antibonding states in the partial density Table 1. Elastic Constants, Bulk Modulus (GPa), Shear Modulus (GPa), and B/G Value of RhB crystal FeB-RhB
C11 C22 C33 C44 C55 C66 C12 C13 C23
B
366 386 458 184 98 109 249 289 299 310
anti-NiAs-RhB 438
342 172
223 256
G B/G
of states (PDOS). The location of the Fermi level at the pseudo-gap-center indicates the structure’s stability. Below and above the Fermi level, the Rh 4d orbital and B 2p orbital show some extent of hybridization. However, near the Fermi level, the Rh 4d orbital dominates the conduction properties, whereas the B atoms contribute little. 3.3. Incompressibility and Ductility. The mechanical properties of the two structures were investigated because they are important for applications. The calculated bulk modulus of antiNiAs-RhB and FeB-RhB is 296 and 310 GPa (Table 1), respectively, larger than that of B6 O (B = 270 GPa).39 Both phases can be grouped into incompressible materials. The compression behavior of the FeB-RhB is presented in Figure 5. Compressible anisotropy is obvious because the incompressibility along the a and b axes is comparable to, but much lower than, that along the c axis. Interestingly, the incompressibility along the c axis is even larger than that of diamond, but the bulk incompressibility of the FeB-RhB is much lower than that of diamond. The ratio value of B/G is commonly used to describe the ductility or brittleness of materials, with 1.75 as the critical value.38 A B/G value higher (or lower) than the criteria is considered to be ductile (or brittle). The B/G value of anti-NiAs-RhB is 2.90, exceeding the critical value and implying its ductile nature. The ductility of the high-pressure phase FeB-RhB is better than that of anti-NiAs-RhB, with a B/G value of 3.56, which is comparable to that of RuC (B/G = 3.95).15
87 3.56
296 102 2.90
Figure 3. Phonon dispersion of FeB-RhB at zero pressure.
Figure 5. Normalized lattice parameters and unit cell volumes of RhB as a function of pressure compared with those of diamond.
Figure 4. Band structure and partial densities of states for anti-NiAs-RhB (a) and FeB-RhB (b) at ambient pressure. 19912
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Table 2. Calculated Volume, Bond Parameter, and Hardness of RhB, Where Nμ Represents the Bond Number in a Unit Cell of μ Type Bond
a
bond-type
dμ (Å)
Nμ
Nμe
fμm ( 103)
41.32
RhB
2.216
12
0.344
1.287
79.20
BB RhB (I)
2.075 2.223
2 4
0.301 0.385
RhB (II)
2.239
4
RhB (III)
2.280
RhB (IV)
2.284
RhB (V) BB
crystal
V (Å3/unit cell)
anti-NiAs-RhB FeB-RhB
Hμv (GPa)
Hv cal. (GPa)
Hv expt. (GPa)
16.480
16.05
16.2 (0.98 N)a
0.754 0.263
13.677 28.563
25.05
0.377
0.263
27.664
4
0.357
0.263
25.499
8
0.355
0.263
25.271
2.386
8
0.312
0.263
20.789
1.967
4
0.410
0.141
27.873
Ref 15.
Figure 6. Phase transition route. (a,c) Anti-NiAs-RhB structure as seen from different directions and (b,d) FeB-RhB in the corresponding direction. Atoms surrounded by the dashed red parallelograms in panels b and d formed similar units before and after the phase transition. Arrows in each unit cell represent the direction of the atom translation relative to the anti-NiAs-RhB.
3.4. Hardness. The Vickers hardness of both structures was calculated using our hardness formula, which takes the ionic and
metallic components and d valence electrons into consideration. The formula of the Vickers hardness of covalent solids18 is as 19913
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0:55
Hv ðGPaÞ ¼ ANe2=3 d2:5 exp 1:191fi 32:2fm
where A is 350 for the BB covalent bond and 1051 for the RhB bond; Ne is the electron density; fi is the Phillips ionicity of the RhB bonds, which can be calculated by fi = (fh)0.735 = [1 exp(|Pc P|/P)]0.735; d is the bond length; and fm is a factor of metallicity. A detailed description of the calculation method has been presented in previous studies.15,18 In the present work, there are still two points to emphasize. First, in the anti-NiAs-RhB, all RhB bonds are located in the BRh6 octahedra, which are structurally similar to the PtN6 octahedron of p-PtN2.18 Therefore, we choose 0.57 for the Pc value of the RhB bond in the anti-NiAs-RhB structure. Using the calculated RhB bond population of 0.37, we obtain a Phillips ionicity of 0.53 for the RhB bond. With respect to fi of the RhB bond in the FeB-RhB, we adopt the same value as that of the anti-NiAs-RhB. Second, note that in the electron density calculation, we consider four valence electrons in the Rh atom that take part in the bonding of RhB, as half-filled d shells are a comparably stable state according to Hund’s rules.15 The calculated volume, bond parameter, and hardness of the two RhB crystals are listed in Table 2. The calculated hardness of 16.05 GPa for anti-NiAs-RhB is consistent with the experimental value.19 The calculated hardness of FeB-RhB is 25 GPa, which is comparable to the experimental hardness of ReB2 (26.6 GPa).12 The higher level of hardness of the FeB-RhB is attributed to its faint metallicity, wherein the PDOS of FeB-RhB at the Fermi level is much smaller than that of anti-NiAs-RhB. 3.5. Phase Transition Route. The FeB-RhB can be considered to be the distortion of the anti-NiAs-RhB, which can have three steps of translation in different directions and some subtle modifications, as shown in Figure 6. First, the Rh atom should be translated in about half of the BB bond length along the c axis. Therefore, in the c-axis direction, the Rh atoms and related B atoms lie in the same plane compared with the alternate stacking of Rh and B planes perpendicular to the c axis (Figure 6a,b). Second, Rh atoms marked with arrows move along the b axis in less than one unit (Figure 6c,d). Finally, the B atoms in one line move up every other B atom along the b axis, but the movement of these B atoms in adjacent boron planes parallel to the bc plane deviates from the bc plane in different directions, as plotted in Figure 6c,d. As a result, the B atoms in the FeB-RhB form zigzag chains in contrast with the linear chains in the anti-NiAs-RhB. The atom translations are accompanied by the break of RhB bonds in the anti-NiAs-RhB structure and the formation of more new RhB bonds in the new phase. In the end, the FeB-RhB form five types of RhB bonds with different bond lengths in contrast with only one type of RhB bond in the anti-NiAs-RhB. The Rh atoms are six-fold-coordinated in the anti-NiAs-RhB, and their coordination number increases to seven in the FeB-RhB after the phase transition. The same trend occurs in the coordination number of B atoms, which increases from eight to nine. The increase in the coordinate numbers for Rh and B atoms leads to a density growth of 4.34%.
4. CONCLUSIONS In summary, using the PSO algorithm, a new orthorhombic phase of RhB at high pressure is found. At 22 GPa, the anti-NiAsRhB transforms into the FeB-RhB. Phonon dispersions, elastic
constant, and PDOS analysis suggest the stability of the new structure. The calculated band structures and PDOS results demonstrate that the anti-NiAs-RhB is metallic, whereas the high-pressure phase FeB-RhB is semimetallic. After phase transition, the increase in the coordinate numbers of Rh and B atoms results in the enhancement of density and hardness. The density increases from 9.140 (anti-NiAs-RhB) to 9.537 g/cm3 (FeB-RhB). The calculated Vickers hardness of FeB-RhB is 25 GPa, much higher than that of the anti-NiAs-RhB (16 GPa). The superior performance and recoverability of FeB-RhB under ambient conditions make this new material suitable for technological applications.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT This work was supported by NSFC (grant nos. 51072176, 51072174, 50821001, and 91022029), FANEDD (grant no. 2007B36), and by NBRPC (grant no. 2011CB808205). ’ REFERENCES (1) Solozhenko, V. L.; Dub, S. N.; Novikov, N. V. Diamond Relat. Mater. 2001, 10, 2228–2231. (2) Li, Q; Wang, M; Oganov, A. R.; Cui, T; Ma, Y. M; Zou, G. T. J. Appl. Phys. 2009, 105, 053514. (3) Malkow, T. Mater. Sci. Eng., A. 2001, 302, 311–324. (4) Solozhenko, V. L.; Kurakevych, O. O.; Andrault, D.; Godec, Y. L.; Mezouar, M. Phys. Rev. Lett. 2009, 102, 015506. (5) Xu, L. F.; Zhao, Z. S.; Wang, L. M.; Xu, B.; He, J. L.; Liu, Z. Y.; Tian, Y. J. J. Phys. Chem. C. 2010, 114, 22688–22690. (6) Li, Q; Wang, H; Tian, Y. J; Xia, Y; Cui, T; He, J. L; Ma, Y. M; Zou, G. T. J. Appl. Phys. 2010, 108, 023507. (7) Brazhkin, V. V.; Lyapin, A. G. Philos. Mag. A. 2002, 82, 231–253. (8) Mcmillan, P. F. Nat. Mater. 2002, 1, 19–25. (9) Kaner, R. B.; Gilman, J. J.; Tolbert, S. H. Science 2005, 308, 1268. (10) Gao, F. M.; He, J. L.; Wu, E. D.; Liu, S. M.; Yu, D. L.; Li, D. C.; Zhang, S. Y.; Tian, Y. J. Phys. Rev. Lett. 2003, 91, 015502. (11) Crowhurst, J. C.; Goncharov, A. F.; Sadigh, B.; Evans, C. L.; Morrall, P. G.; Ferreira, J. L.; Nelson, A. J. Science 2006, 311, 1275–1278. (12) Gu, Q. F.; Krauss, G.; Steurer, W. Adv. Mater. 2008, 20, 3620–3626. (13) Wang, M.; Li, Y. W.; Cui, T.; Ma, Y. M.; Zou, G. T. Appl. Phys. Lett. 2008, 93, 101905. (14) Ono, S.; Kikegawa, T.; Ohishi, Y. Solid State Commun. 2005, 133, 55–59. (15) Zhao, Z. S.; Wang, M.; Cui, L.; He, J. L.; Yu, D. L.; Tian, Y. J. J. Phys. Chem. C 2010, 114, 9961–9964. (16) Li, Y. W; Wang, H; Li, Q; Ma, Y. M; Cui, T; Zou, G. T. Inorg. Chem. 2009, 48, 9904. (17) Zhang, M; Wang, M; Cui, T; Ma, Y. M; Niu, Y. L; Zou, G. T. J. Phys. Chem. Solids 2008, 69, 2096. (18) Guo, X. J.; Liu, Z. Y.; Yu, D. L.; He, J. L.; Liu, R. P.; Xu, B. J. Appl. Phys. 2008, 104, 023503. (19) Rau, J. V.; Latini, A. Chem. Mater. 2009, 21, 1407–1409. (20) Latini, A.; Rau, J. V.; Teghil, R.; Generosi, A.; Albertini, V. R. ACS Appl. Mater. Interfaces 2010, 2, 581–587. (21) Aronsson, B.; Stenberg, E.; Aselius, J. Acta Chem. Scand. 1960, 14, 733–741. (22) Fang, Z.; Terakura, K.; Sawada, H.; Miyazaki, T.; Solovyev, I. Phys. Rev. Lett. 1998, 81, 1027–1030. 19914
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