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Novel ultra-incompressible phases of Ru2C
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 175505 (http://iopscience.iop.org/0953-8984/27/17/175505) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 192.231.202.205 This content was downloaded on 16/05/2015 at 19:12
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 175505 (6pp)
doi:10.1088/0953-8984/27/17/175505
Novel ultra-incompressible phases of Ru2C Jian Lu1 , Feng Hong1,4 , Wenjun Lin1 , Wei Ren1 , Yinwei Li2,4 and Yanfa Yan3 1
Department of Physics, SHU-Solar E R& D Lab, Shanghai University, Shanghai 200444, People’s Republic of China 2 School of Physics and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, People’s Republic of China 3 Department of Physics and Astronomy, The University of Toledo, Toledo 43606, OH, USA E-mail: [email protected] and yinwei [email protected] Received 17 December 2014, revised 28 February 2015 Accepted for publication 16 March 2015 Published 15 April 2015 Abstract
The crystal structures of Ru2 C were extensively investigated using the structure searching method coupled with first-principles calculations. In contrast to the previously proposed P -3m1 phase, two energetically more favorable structures with space groups P -31m and P 63 /mmc were found, which are stable at pressure ranges of 0–32 GPa and 32–50 GPa, respectively. The dynamical stabilities of both phases at ambient and high pressures are confirmed by the phonon dispersions. Further calculations indicate that the two new predicted phases are ultra-incompressible metals due to a strong covalent Ru–C bond. Keywords: first-principles calculations, structure searching, dynamical stabilities (Some figures may appear in colour only in the online journal)
space group P -3m1 was proposed for Ru2 C. However, a fully optimization of the P -3m1 structure results in quite different lattice parameters (a = 2.8849 Å and c = 4.8769 Å) [16] with experiment. Moreover, subsequent theoretical study [17] revealed the phonon instability of P -3m1−Ru2 C below 30 GPa. As is often the case in the TM–LE compounds, it is difficult to determine the crystal structure of Ru2 C in experiment due to the small scattering cross section of carbon. Considering the controversy between experimental and theoretical results, the question about the crystal structure of synthesized Ru–C compounds still remains open. Here, using a recently developed crystal structure prediction method based on the particle swarm optimization [18], we have extensively investigated the stable structures of Ru2 C at 0, 5 and 14.2 GPa. Two dynamically stable structures with the space groups P 63 /mmc and P -31m were found to be energetically more favorable than the previously proposed P -3m1 structure. Ru2 C is suggested to be an ultra-incompressible material due to the relatively large bulk modulus.
1. Introduction
Superhard materials are of great significance in the range of science and technology due to their superior properties, such as high stiffness, high hardness, high thermal conductivity and high melting point. It is long believed that these materials only belong to strongly covalent bonded compounds formed by light elements (B, C, N, and O), such as, diamond [1], cubic BN [2], BC2 N [3], B2 CO [4] etc. Recently, a new class of materials composed of heavy transition metal (TM) and light element (LE) were considered to be potential superhard/hard materials [5–7]. One of the typical TM–LEs is transition metal carbides (TMCs) since they show very high bulk modulus, for example 242 GPa of TiC [8], 223 GPa of ZrC [9], 439 GPa of WC [10], 303 GPa of PtC [11] and 316 GPa of RuC [12– 15]. Recently, a new Ru–C compound with stoichiometry of Ru2 C was synthesized at 5 GPa and 2000 K [16]. Based on the temperature quenched high pressure x-ray diffraction (HPXRD) pattern at 14.2 GPa, a hexagonal structure with a = 2.534 Å and c = 4.147 Å [16] was considered for Ru2 C at ambient pressure. Combining with the firstprinciple calculations, a trigonal Fe2 N-type structure with 4
2. Computational methods
Structure searches for Ru2 C were performed through the intelligent crystal structure AnaLYsis by particle swarm
Authors to whom any correspondence should be addressed.
0953-8984/15/175505+06$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 175505
J Lu et al
Figure 2. Enthalpies of the predicted P -31m and P 63 /mmc structure relative to the experimental proposed P -3m1 structure as a function of pressure.
calculations were performed using density functional theory (DFT) within the Perdew–Burke–Ernzerhof (PBE) [25] generalized gradient approximation (GGA) as implemented in the VASP package [23, 24]. For the structure predictions, an energy cutoff of 500 eV for the plane-wave basis set and Monkhorst–Pack [26] Brillouin zone sampling grid with the resolution of 2π ×0.07 Å−1 was used. A more accurate optimization was performed with energy cutoff of 600 eV and Monkhorst–Pack k meshes of 12×12×9, 12×12×7, 8×8×8 for the P -3m1, P 63 /mmc and P -31m structure, respectively, to ensure that enthalpy calculations are well converged to better than 1 meV/atom. The phonon dispersions were computed based on the supercell approach using the PHONOPY [27] code with 4×4×2, 4×4×1, 2×2×2 supercells for the P 3m1, P 63 /mmc and P -31m structure, respectively. Single crystal elastic constants were determined from evaluation of stress tensor generated small strain, and the bulk and shear modulus were thus estimated by using the Voigt−Reuss−Hill approximation [28, 29]. The theoretical Vickers hardness was estimated by using the Chen’s model [29].
Figure 1. Crystal structures of Ru2 C with the P -3m1 (a), the P 63 /mmc (b) and the P -31m structures (c). Big blue and small red spheres represent Ru and C atoms, respectively. Table 1. Structural parameters of Ru2 C within P -3m1, P 63 /mmc and P -31m space group at ambient pressure.
Lattice parameters (Å)
Atomic coordinates
3. Results and discussion
a = 2.82 Ru1 1a (1/3, 2/3, 0.265) (2.8849 [16], 2.81 [17]) Ru2 1a (2/3, 1/3, 0.735) c = 5.01 C 1a (0, 0, 0) (4.8769 [16], 4.98 [17]) P 63 /mmc a = 2.77 Ru 4f (1/3, 2/3, 0.113) c = 10.21 C 2b (0, 0, 0.25) P -31m a = 4.93 Ru 6k (0.652, 0, 0.238) C1 1a (0, 0, 0) c = 4.98 C2 2d (1/3, 2/3, 0.5 ) P -3m1
We performed crystal structure predictions with simulation cells containing 1 to 4 Ru2 C formula units (f.u.) at pressures of 0, 5 and 14.2 GPa. In contrast to the P -3m1 structure for the synthesized Ru2 C proposed by previous study [16], we found two completely new energetically competitive structures with space groups P 63 /mmc and P -31m. The crystal structures of three phases mentioned above are shown in figure 1 and the corresponding structural parameters are listed in table 1. It is found that the predicted P 63 /mmc structure and previous proposedP -3m1 both stack with Ru–C–Ru trilayer along the c-axis with each Ru atom coordinated by three C atoms. However, different from AAA. . . stacked layers of the Ru sublattice in the P -3m1 structure, the stacking layers of the Ru sublattice in the P 63 /mmc structure is ABAB. . . In addition, the Ru–C bond length in the P 63 /mmc phase is 2.125 Å at ambient
optimization methodology [18, 19], as implemented in the CALYPSO code [19]. This approach has been benchmarked on a variety of known systems [19] and has made several successful predictions of high pressure structures of, for instance, Xe–Fe/Ni [20], W–B [21], and H2 S systems [22]. The underlying ab initio structural relaxations and electronic 2
J. Phys.: Condens. Matter 27 (2015) 175505
J Lu et al
Figure 3. (a)–(c) show the calculated phonon dispersions of Ru2 C with the P -3m1, the P 63 /mmc and the P -31m structure at ambient pressure, respectively. Table 2. The calculated elastic constants Cij (GPa), bulk modulus B (GPa), shear modulus G (GPa), and B/G ratio at ambient pressure.
C11 C33 C44 C66 C12 C13 B P 63 /mmc P -31m ZB-RuC [12] I 4mm-RuC [12]
592 665 220 500 481 151 330 81 396 700 76 75
G
B/G
152 210 333 191 1.74 198 239 315 139 2.27 198 242 74 3.27 307 199 316 66 4.79
pressure, slightly longer than 2.101 Å in the P -3m1 structure. In the P -31m structure, each C atom is coordinated by six Ru atoms forming edge-shared Ru6 C octahedron with Ru–C distance of 2.069 Å (figure 1(c)). The relative stabilities with pressure were obtained by the calculated enthalpies of three structures of Ru2 C in the pressure range 0–50 GPa, as shown in figure 2. It is clearly seen that both of the two new predicted phases are much more energetically favorable than the previously proposed P -3m1 phase in the whole pressure range we studied. The P -31m structure is the most stable phase at 0 GPa, indicating that the P -31m phase is the ground state of Ru2 C at ambient pressure. The P -31m structure is stable up to 32 GPa, above which the P 63 /mmc structure is more stable. Therefore, according to our enthalpy results, a P -31m → P 63 /mmc phase transition was obtained for Ru2 C. In order to check the structural stabilities of the two new phases of Ru2 C, the phonon dispersions were calculated and figure 3 gives the results at 0 GPa. The phonon dispersion of the previous proposed P -3m1 structure is also presented. In figure 3(a), it is clearly seen that the P -3m1 structure is dynamically unstable at 0 GPa due to the existence of the imaginary phonon modes around G point, in accordance with Sun’s results of instabilities below 30 GPa for P -3m1 structure [17]. In contrast, the P 63 /mmc and P -31m structures are dynamically stable at 0 GPa since no imaginary phonon frequency is found in the whole Brillouin zone. The phonon dispersions of the P 63 /mmc and P -31m structure at high pressures (see figures A1 and A2) also reveal that all the frequencies are positive, indicating that the two new phases are dynamically stable at pressure range of 0–50 GPa.
Figure 4. Band structures and partial density of states of Ru2 C for the P 63 /mmc (a) and the P -31m structure (b). Horizontal lines are the Fermi levels.
After checking the structural stabilities, we then performed further studies on the mechanical properties of Ru2 C, since these properties are very important for the potential industrial and technological applications. Due to the instability of the P -3m1 phase at ambient pressure which limits its industrial applications, we only discuss the mechanical properties of the P 63 /mmc and P -31m phase below, and these are tabulated in table 2. For the hexagonal crystals, the mechanical stability requires the elastic constants satisfying the Born criteria [30] C44 > 0, 3
C11 > |C12 | ,
(C11 + 2C12 ) C33 > 2C13 C13 .
J. Phys.: Condens. Matter 27 (2015) 175505
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Figure 5. The total charge density contours of the P 63 /mmc phase on the (1,−2,0) plane (a) and P -31m phase on the (1,1,−1) plane (b).
Table 2 has shown that the whole set of the elastic constants for both the P 63 /mmc and P -31m phases satisfy the above conditions, implying that the two phases are mechanically stable at ambient pressure. Compared with the corresponding stable phase of RuC [12] at ambient pressure, the two novel phases of Ru2 C are also ultra-incompressible with large bulk modulus of 333 GPa and 315 GPa for the P 63 /mmc and P 31m structure, respectively. In addition, we have calculated the hardness of the two phases using Chen’s model [29]. The estimated hardness of the P 63 /mmc and the P -31m phases are 19.5 GPa and 10.8 GPa, respectively, indicating that Ru2 C is a hard material. It should be noted that B/G (bulk/shear modulus ratio) for Ru2 C in the P -31m phase is above 2.2. Since a high or low B/G value is often associated with ductility or brittleness, and the critical value that separates ductility and brittleness is 1.75 [31], Ru2 C is supposed to be a ductile material. In order to understand the mechanical properties of the P 63 /mmc and P -31m phase, the band structure and partial densities of states (DOS) were calculated, as shown in figure 4. Figure 4 has shown that both of the two phases are metallic. In addition, a distinct feature for the P -31m phase in figure 4(b), is that Fermi level locates in a pseudogap [32] separated by the Ru–C bonding and antibonding states. Note that the Fermi level locating close to the minimum of the pseudogap rather than a DOS peak helps to enhance the stability of the P -31m phase [7]. It is also found that Ru-d orbital has a strong hybridization with C-p orbital, indicating the existence of covalent Ru–C bonds in these two compounds. Moreover, the total charge density (figure 5) clearly reveals strong directional covalent Ru–C bonds in the P -31m and P 63 /mmc phase, which may explain the high bulk modulus of Ru2 C. It should be pointed out that though both P -31m and P 63 /mmc structure are predicted to be more stable in our studies, the lattice constants of both phases are very different from the experimental results [16], thus the most possible structure of synthesized compounds is still unclear. The XRD patterns also show the discrepancy between our calculated results and experimental data. Figure 6 shows
Figure 6. Simulated XRD patterns at 14.2 GPa for fixed-cell and fully optimized P -3m1 structure and the newly predicted P 63 /mmc and P -31m structure along with the experimental data [16].
the XRD patterns simulated at 14.2 GPa, the same pressure of experiment [16]. One can clearly see that the simulated XRD patterns of two new phases fit badly to the experimental data [16], P -3m1 structure gives the same conclusion as well. It should be mentioned that our results are based on fullrelaxed calculations, the optimized values of c−axis of the previously proposed P -3m1 (c = 4.98 Å) structure along with our newly predicted phases (c = 10.21/4.98 Å) differ significantly from the experimental one (c = 4.035 Å [16]). Interestingly, if the experimental lattice constants are used for the P -3m1 structure, the simulated XRD pattern fits well with experimental measurements. While the consistence of XDR pattern could still not let us to get the conclusion that the synthesized compound has the P -3m1 structure, since both our results and previous calculations [17] have shown that this phase is dynamically unstable at least below 30 GPa. In a survey of literature, a similar discrepancy of lattice parameters between experiments and calculations 4
J. Phys.: Condens. Matter 27 (2015) 175505
J Lu et al
Figure A1. (a)–(c) show the calculated phonon dispersions of Ru2 C with P 63 /mmc structure at 10 GPa, 30 GPa and 50 GPa, respectively.
Figure A2. (a)–(c) show the calculated phonon dispersions of Ru2 C with P -31m structure at 10 GPa, 30 GPa and 50 GPa, respectively.
can be found in the case of Pt–N compound [33–35]. The synthesized Pt–N compound was initially supposed to have the stoichiometry of PtN [33]. However, first-principles calculations found that the optimized lattice parameters of the dynamically stable rock salt structure [34] of PtN disagree with the experimental results. Subsequently, both experimental and theoretical studies [35, 36] agreed that the actual stoichiometry of Pt–N compound is PtN2 . The same situation happened in W–B compound, in which WB3 [37] was finally determined instead of experimental initially proposed WB4 [38]. From this point of view, we supposed that the experimental synthesized Ru–C compound has a stoichiometry other than Ru2 C or is even a mixed phase. Further investigations are demanded to validate this proposal.
structure predictions. The P -31m and P 63 /mmc structures are energetically more favorable than the experimental proposed P -3m1 structure at pressure range of 0–50 GPa. The two new structures are mechanically and dynamically stable by checking the elastic constants and phonon dispersions. A phase transformation of P -31m → P 63 /mmc was uncovered at 32 GPa. The electronic DOS and total charge density suggest that Ru2 C with the two structures is partially covalently bonded metal and ultra-incompressible material. The question about the possible structure of synthesized Ru–C compound still remains open and needs further investigations. Acknowledgments
The authors thank H Gao, H Y Liu, X Tang, C L Tang and W J Li for fruitful discussions. This work was supported by National Natural Science Foundation of China (Nos 11104117 and 11204111) and high performance computing platform of Shanghai University.
4. Conclusions
In summary, we have predicted two new phases of Ru2 C with space groups P -31m and P 63 /mmc by using first-principle 5
J. Phys.: Condens. Matter 27 (2015) 175505
J Lu et al
[18] Wang Y C, Lv J, Zhu L and Ma Y M 2012 Comput. Phys. Commun. 183 2063–70 [19] Wang Y C, Lv J, Zhu L and Ma Y M 2010 Phys. Rev. B 82 094116 [20] Zhu L, Liu H Y, Pickard Chris J, Zou G T and Ma Y M 2014 Nature Chem. 6 644–8 [21] Li Q, Zhou D, Zheng W T, Ma Y M and Chen C F 2013 Phys. Rev. Lett. 110 136403 [22] Li Y W, Hao J, Liu H Y and Ma Y M 2014 J. Chem. Phys. 140 174712 [23] Kresse G and Furthm¨uller J 1996 Phys. Rev. B 54 11169–86 [24] Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865–8 [25] Kresse G and Joubert D 1999 Phys. Rev. B 59 1758–75 [26] Monkhorst H J and Pack J D 1976 Phys. Rev. B 13 5188–92 [27] Togo A, Oba F and Tanaka I 2008 Phys. Rev. B 78 134106 [28] Hill R 1952 Proc. Phys. Soc. Lond. A 65 349–54 [29] Chen X Q, Niu H Y, Franchini C, Li D Z and Li Y Y 2011 Phys. Rev. B 84 121405 [30] Wu Z J, Zhao E J, Xiang H P, Hao X F, Liu X J and Meng J 2007 Phys. Rev. B 76 054115 [31] Pugh S. F. 1954 Phil. Mag. 45 823 [32] Vajeeston P, Ravindran P, Ravi C and Asokamani R 2001 Phys. Rev. B 63 045115 [33] Gregoryanz E, Sanloup C, Somayazulu M, Badro J, Fiquet G, Mao H K and Hemley R 2004 Nature Mater. 3 294 [34] Patil S, Khare S, Tuttle B, Boarding J and Kodambakara S 2006 Phys. Rev. B 73 104118 [35] Crowhurst J C, Goncharov A F, Sadigh B, Evans C L, Morrall P G, Ferreira J L and Nelson A J 2006 Science 311 1275–8 [36] Yu R and Zhang X F 2005 Appl. Phys. Lett. 86 121913 [37] Zhang R F, Legut D, Lin Z J, Zhao Y S, Mao H K and Veprek S 2012 Phys. Rev. Lett. 108 255502 [38] Gu Q F, Krauss G and Steurer W 2008 Adv. Mater. 20 3620–6
Appendix
The calculated phonon dispersions of Ru2 C with P 63 /mmc and P -31m structures are shown in figures A1 and A2, respectively. References [1] Occelli F, Farber D L and Toullec R L 2013 Nature Mater. 2 151–4 [2] Zheng J C 2005 Phys. Rev. B 72 052105 [3] He D W, Zhao Y S, Daemen L, Qian J, Shen T D and Zerda T W 2002 Appl. Phys. Lett. 81 643–5 [4] Li Y W, Li Q and Ma Y M 2011 Europhys. Lett. 95 66006 [5] Chung H Y, Weinberger M B, Levine J B, Kavner A, Yang J M, Tolbert S H and Kaner R B 2007 Science 316 (5823) 436–9 [6] Tse J S, Klug D D, Uehara K, Li Z Q, Haines J and Leger J M 2000 Phys. Rev. B 72 10029–34 [7] Zhang M G, Yan H Y, Zhang, G T and Wang H 2012 J. Phys. Chem. C 116 4293–7 [8] Dubrovinskaia N A, Dubrovinsky L S, Saxena S K, Ahuja R and Johansson B 1999 J. Alloys Compounds 289 24 [9] Chang R and Graham L J 2004 J. Appl. Phys. 37 3778 [10] Lee M and Gilmore R S 1982 J. Mater. Sci. 17 2657 [11] Ono S, Kikegawa T and Ohishi Y 2005 Solid State Commun. 133 55 [12] Hao J, Tang X, Li W J and Li Y W 2014 Europhys. Lett. 105 46004 [13] Zhao Z, Wang M, Cui L, He J, Yu D and Tian Y 2010 J. Phys. Chem. C 114 996 [14] Kempter C P and Nadler M R 1960 J. Chem. Phys. 33 1580 [15] Kempter C P 1964 J. Chem. Phys. 41 1515 [16] Sanjay Kumar N R, Chandra Shekar N V, Chandra S, Basu J, Divakar R and Sahu P Ch 2012 J. Phys.: Condens. Matter 24 362202 [17] Sun W W, Chakraborty S and Ahuja R 2013 Appl. Phys. Lett. 103 251901
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Novel ultra-incompressible phases of Ru2C
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 175505 (http://iopscience.iop.org/0953-8984/27/17/175505) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 192.231.202.205 This content was downloaded on 16/05/2015 at 19:12
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 175505 (6pp)
doi:10.1088/0953-8984/27/17/175505
Novel ultra-incompressible phases of Ru2C Jian Lu1 , Feng Hong1,4 , Wenjun Lin1 , Wei Ren1 , Yinwei Li2,4 and Yanfa Yan3 1
Department of Physics, SHU-Solar E R& D Lab, Shanghai University, Shanghai 200444, People’s Republic of China 2 School of Physics and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, People’s Republic of China 3 Department of Physics and Astronomy, The University of Toledo, Toledo 43606, OH, USA E-mail: [email protected] and yinwei [email protected] Received 17 December 2014, revised 28 February 2015 Accepted for publication 16 March 2015 Published 15 April 2015 Abstract
The crystal structures of Ru2 C were extensively investigated using the structure searching method coupled with first-principles calculations. In contrast to the previously proposed P -3m1 phase, two energetically more favorable structures with space groups P -31m and P 63 /mmc were found, which are stable at pressure ranges of 0–32 GPa and 32–50 GPa, respectively. The dynamical stabilities of both phases at ambient and high pressures are confirmed by the phonon dispersions. Further calculations indicate that the two new predicted phases are ultra-incompressible metals due to a strong covalent Ru–C bond. Keywords: first-principles calculations, structure searching, dynamical stabilities (Some figures may appear in colour only in the online journal)
space group P -3m1 was proposed for Ru2 C. However, a fully optimization of the P -3m1 structure results in quite different lattice parameters (a = 2.8849 Å and c = 4.8769 Å) [16] with experiment. Moreover, subsequent theoretical study [17] revealed the phonon instability of P -3m1−Ru2 C below 30 GPa. As is often the case in the TM–LE compounds, it is difficult to determine the crystal structure of Ru2 C in experiment due to the small scattering cross section of carbon. Considering the controversy between experimental and theoretical results, the question about the crystal structure of synthesized Ru–C compounds still remains open. Here, using a recently developed crystal structure prediction method based on the particle swarm optimization [18], we have extensively investigated the stable structures of Ru2 C at 0, 5 and 14.2 GPa. Two dynamically stable structures with the space groups P 63 /mmc and P -31m were found to be energetically more favorable than the previously proposed P -3m1 structure. Ru2 C is suggested to be an ultra-incompressible material due to the relatively large bulk modulus.
1. Introduction
Superhard materials are of great significance in the range of science and technology due to their superior properties, such as high stiffness, high hardness, high thermal conductivity and high melting point. It is long believed that these materials only belong to strongly covalent bonded compounds formed by light elements (B, C, N, and O), such as, diamond [1], cubic BN [2], BC2 N [3], B2 CO [4] etc. Recently, a new class of materials composed of heavy transition metal (TM) and light element (LE) were considered to be potential superhard/hard materials [5–7]. One of the typical TM–LEs is transition metal carbides (TMCs) since they show very high bulk modulus, for example 242 GPa of TiC [8], 223 GPa of ZrC [9], 439 GPa of WC [10], 303 GPa of PtC [11] and 316 GPa of RuC [12– 15]. Recently, a new Ru–C compound with stoichiometry of Ru2 C was synthesized at 5 GPa and 2000 K [16]. Based on the temperature quenched high pressure x-ray diffraction (HPXRD) pattern at 14.2 GPa, a hexagonal structure with a = 2.534 Å and c = 4.147 Å [16] was considered for Ru2 C at ambient pressure. Combining with the firstprinciple calculations, a trigonal Fe2 N-type structure with 4
2. Computational methods
Structure searches for Ru2 C were performed through the intelligent crystal structure AnaLYsis by particle swarm
Authors to whom any correspondence should be addressed.
0953-8984/15/175505+06$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 175505
J Lu et al
Figure 2. Enthalpies of the predicted P -31m and P 63 /mmc structure relative to the experimental proposed P -3m1 structure as a function of pressure.
calculations were performed using density functional theory (DFT) within the Perdew–Burke–Ernzerhof (PBE) [25] generalized gradient approximation (GGA) as implemented in the VASP package [23, 24]. For the structure predictions, an energy cutoff of 500 eV for the plane-wave basis set and Monkhorst–Pack [26] Brillouin zone sampling grid with the resolution of 2π ×0.07 Å−1 was used. A more accurate optimization was performed with energy cutoff of 600 eV and Monkhorst–Pack k meshes of 12×12×9, 12×12×7, 8×8×8 for the P -3m1, P 63 /mmc and P -31m structure, respectively, to ensure that enthalpy calculations are well converged to better than 1 meV/atom. The phonon dispersions were computed based on the supercell approach using the PHONOPY [27] code with 4×4×2, 4×4×1, 2×2×2 supercells for the P 3m1, P 63 /mmc and P -31m structure, respectively. Single crystal elastic constants were determined from evaluation of stress tensor generated small strain, and the bulk and shear modulus were thus estimated by using the Voigt−Reuss−Hill approximation [28, 29]. The theoretical Vickers hardness was estimated by using the Chen’s model [29].
Figure 1. Crystal structures of Ru2 C with the P -3m1 (a), the P 63 /mmc (b) and the P -31m structures (c). Big blue and small red spheres represent Ru and C atoms, respectively. Table 1. Structural parameters of Ru2 C within P -3m1, P 63 /mmc and P -31m space group at ambient pressure.
Lattice parameters (Å)
Atomic coordinates
3. Results and discussion
a = 2.82 Ru1 1a (1/3, 2/3, 0.265) (2.8849 [16], 2.81 [17]) Ru2 1a (2/3, 1/3, 0.735) c = 5.01 C 1a (0, 0, 0) (4.8769 [16], 4.98 [17]) P 63 /mmc a = 2.77 Ru 4f (1/3, 2/3, 0.113) c = 10.21 C 2b (0, 0, 0.25) P -31m a = 4.93 Ru 6k (0.652, 0, 0.238) C1 1a (0, 0, 0) c = 4.98 C2 2d (1/3, 2/3, 0.5 ) P -3m1
We performed crystal structure predictions with simulation cells containing 1 to 4 Ru2 C formula units (f.u.) at pressures of 0, 5 and 14.2 GPa. In contrast to the P -3m1 structure for the synthesized Ru2 C proposed by previous study [16], we found two completely new energetically competitive structures with space groups P 63 /mmc and P -31m. The crystal structures of three phases mentioned above are shown in figure 1 and the corresponding structural parameters are listed in table 1. It is found that the predicted P 63 /mmc structure and previous proposedP -3m1 both stack with Ru–C–Ru trilayer along the c-axis with each Ru atom coordinated by three C atoms. However, different from AAA. . . stacked layers of the Ru sublattice in the P -3m1 structure, the stacking layers of the Ru sublattice in the P 63 /mmc structure is ABAB. . . In addition, the Ru–C bond length in the P 63 /mmc phase is 2.125 Å at ambient
optimization methodology [18, 19], as implemented in the CALYPSO code [19]. This approach has been benchmarked on a variety of known systems [19] and has made several successful predictions of high pressure structures of, for instance, Xe–Fe/Ni [20], W–B [21], and H2 S systems [22]. The underlying ab initio structural relaxations and electronic 2
J. Phys.: Condens. Matter 27 (2015) 175505
J Lu et al
Figure 3. (a)–(c) show the calculated phonon dispersions of Ru2 C with the P -3m1, the P 63 /mmc and the P -31m structure at ambient pressure, respectively. Table 2. The calculated elastic constants Cij (GPa), bulk modulus B (GPa), shear modulus G (GPa), and B/G ratio at ambient pressure.
C11 C33 C44 C66 C12 C13 B P 63 /mmc P -31m ZB-RuC [12] I 4mm-RuC [12]
592 665 220 500 481 151 330 81 396 700 76 75
G
B/G
152 210 333 191 1.74 198 239 315 139 2.27 198 242 74 3.27 307 199 316 66 4.79
pressure, slightly longer than 2.101 Å in the P -3m1 structure. In the P -31m structure, each C atom is coordinated by six Ru atoms forming edge-shared Ru6 C octahedron with Ru–C distance of 2.069 Å (figure 1(c)). The relative stabilities with pressure were obtained by the calculated enthalpies of three structures of Ru2 C in the pressure range 0–50 GPa, as shown in figure 2. It is clearly seen that both of the two new predicted phases are much more energetically favorable than the previously proposed P -3m1 phase in the whole pressure range we studied. The P -31m structure is the most stable phase at 0 GPa, indicating that the P -31m phase is the ground state of Ru2 C at ambient pressure. The P -31m structure is stable up to 32 GPa, above which the P 63 /mmc structure is more stable. Therefore, according to our enthalpy results, a P -31m → P 63 /mmc phase transition was obtained for Ru2 C. In order to check the structural stabilities of the two new phases of Ru2 C, the phonon dispersions were calculated and figure 3 gives the results at 0 GPa. The phonon dispersion of the previous proposed P -3m1 structure is also presented. In figure 3(a), it is clearly seen that the P -3m1 structure is dynamically unstable at 0 GPa due to the existence of the imaginary phonon modes around G point, in accordance with Sun’s results of instabilities below 30 GPa for P -3m1 structure [17]. In contrast, the P 63 /mmc and P -31m structures are dynamically stable at 0 GPa since no imaginary phonon frequency is found in the whole Brillouin zone. The phonon dispersions of the P 63 /mmc and P -31m structure at high pressures (see figures A1 and A2) also reveal that all the frequencies are positive, indicating that the two new phases are dynamically stable at pressure range of 0–50 GPa.
Figure 4. Band structures and partial density of states of Ru2 C for the P 63 /mmc (a) and the P -31m structure (b). Horizontal lines are the Fermi levels.
After checking the structural stabilities, we then performed further studies on the mechanical properties of Ru2 C, since these properties are very important for the potential industrial and technological applications. Due to the instability of the P -3m1 phase at ambient pressure which limits its industrial applications, we only discuss the mechanical properties of the P 63 /mmc and P -31m phase below, and these are tabulated in table 2. For the hexagonal crystals, the mechanical stability requires the elastic constants satisfying the Born criteria [30] C44 > 0, 3
C11 > |C12 | ,
(C11 + 2C12 ) C33 > 2C13 C13 .
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Figure 5. The total charge density contours of the P 63 /mmc phase on the (1,−2,0) plane (a) and P -31m phase on the (1,1,−1) plane (b).
Table 2 has shown that the whole set of the elastic constants for both the P 63 /mmc and P -31m phases satisfy the above conditions, implying that the two phases are mechanically stable at ambient pressure. Compared with the corresponding stable phase of RuC [12] at ambient pressure, the two novel phases of Ru2 C are also ultra-incompressible with large bulk modulus of 333 GPa and 315 GPa for the P 63 /mmc and P 31m structure, respectively. In addition, we have calculated the hardness of the two phases using Chen’s model [29]. The estimated hardness of the P 63 /mmc and the P -31m phases are 19.5 GPa and 10.8 GPa, respectively, indicating that Ru2 C is a hard material. It should be noted that B/G (bulk/shear modulus ratio) for Ru2 C in the P -31m phase is above 2.2. Since a high or low B/G value is often associated with ductility or brittleness, and the critical value that separates ductility and brittleness is 1.75 [31], Ru2 C is supposed to be a ductile material. In order to understand the mechanical properties of the P 63 /mmc and P -31m phase, the band structure and partial densities of states (DOS) were calculated, as shown in figure 4. Figure 4 has shown that both of the two phases are metallic. In addition, a distinct feature for the P -31m phase in figure 4(b), is that Fermi level locates in a pseudogap [32] separated by the Ru–C bonding and antibonding states. Note that the Fermi level locating close to the minimum of the pseudogap rather than a DOS peak helps to enhance the stability of the P -31m phase [7]. It is also found that Ru-d orbital has a strong hybridization with C-p orbital, indicating the existence of covalent Ru–C bonds in these two compounds. Moreover, the total charge density (figure 5) clearly reveals strong directional covalent Ru–C bonds in the P -31m and P 63 /mmc phase, which may explain the high bulk modulus of Ru2 C. It should be pointed out that though both P -31m and P 63 /mmc structure are predicted to be more stable in our studies, the lattice constants of both phases are very different from the experimental results [16], thus the most possible structure of synthesized compounds is still unclear. The XRD patterns also show the discrepancy between our calculated results and experimental data. Figure 6 shows
Figure 6. Simulated XRD patterns at 14.2 GPa for fixed-cell and fully optimized P -3m1 structure and the newly predicted P 63 /mmc and P -31m structure along with the experimental data [16].
the XRD patterns simulated at 14.2 GPa, the same pressure of experiment [16]. One can clearly see that the simulated XRD patterns of two new phases fit badly to the experimental data [16], P -3m1 structure gives the same conclusion as well. It should be mentioned that our results are based on fullrelaxed calculations, the optimized values of c−axis of the previously proposed P -3m1 (c = 4.98 Å) structure along with our newly predicted phases (c = 10.21/4.98 Å) differ significantly from the experimental one (c = 4.035 Å [16]). Interestingly, if the experimental lattice constants are used for the P -3m1 structure, the simulated XRD pattern fits well with experimental measurements. While the consistence of XDR pattern could still not let us to get the conclusion that the synthesized compound has the P -3m1 structure, since both our results and previous calculations [17] have shown that this phase is dynamically unstable at least below 30 GPa. In a survey of literature, a similar discrepancy of lattice parameters between experiments and calculations 4
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Figure A1. (a)–(c) show the calculated phonon dispersions of Ru2 C with P 63 /mmc structure at 10 GPa, 30 GPa and 50 GPa, respectively.
Figure A2. (a)–(c) show the calculated phonon dispersions of Ru2 C with P -31m structure at 10 GPa, 30 GPa and 50 GPa, respectively.
can be found in the case of Pt–N compound [33–35]. The synthesized Pt–N compound was initially supposed to have the stoichiometry of PtN [33]. However, first-principles calculations found that the optimized lattice parameters of the dynamically stable rock salt structure [34] of PtN disagree with the experimental results. Subsequently, both experimental and theoretical studies [35, 36] agreed that the actual stoichiometry of Pt–N compound is PtN2 . The same situation happened in W–B compound, in which WB3 [37] was finally determined instead of experimental initially proposed WB4 [38]. From this point of view, we supposed that the experimental synthesized Ru–C compound has a stoichiometry other than Ru2 C or is even a mixed phase. Further investigations are demanded to validate this proposal.
structure predictions. The P -31m and P 63 /mmc structures are energetically more favorable than the experimental proposed P -3m1 structure at pressure range of 0–50 GPa. The two new structures are mechanically and dynamically stable by checking the elastic constants and phonon dispersions. A phase transformation of P -31m → P 63 /mmc was uncovered at 32 GPa. The electronic DOS and total charge density suggest that Ru2 C with the two structures is partially covalently bonded metal and ultra-incompressible material. The question about the possible structure of synthesized Ru–C compound still remains open and needs further investigations. Acknowledgments
The authors thank H Gao, H Y Liu, X Tang, C L Tang and W J Li for fruitful discussions. This work was supported by National Natural Science Foundation of China (Nos 11104117 and 11204111) and high performance computing platform of Shanghai University.
4. Conclusions
In summary, we have predicted two new phases of Ru2 C with space groups P -31m and P 63 /mmc by using first-principle 5
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Appendix
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