ARTICLE pubs.acs.org/JPCC
Structural, Mechanical Stability, and Physical Properties of Iridium Carbides with Various Stoichiometries: First-Principles Investigations Xiang Li, Xiang Po Du, and Yuan Xu Wang* Institute for Computational Materials Science, School of Physics and Electronics, Henan University, Kaifeng 475004, People’s Republic of China ABSTRACT:
Iridium carbides with various stoichiometries were investigated by using the first-principles method. It is found that iridium carbides with the iridium germanide structures usually exhibit a better elastic property. Among these studied iridium carbides, tetragonal IrC2 with the RhSn2 structure has the largest bulk modulus and trigonal IrC4 with the IrGe4 structure is found to exhibit the largest shear modulus and smallest Poisson’s ratio. Moreover, Ir4C5 with the Ir4Ge5 structure is a semiconductor and has high bulk and shear moduli. We also found that the shear modulus of iridium carbides with the iridium germanide structures basically increases with the increased carbon content, except Ir4C5; this is due to the fact that only Ir4C5 with the Ir4Ge5 structure exhibits a semiconductor property, whereas other structures all present metallic. The eight structures for IrC4 are all ductile, except for the IrGe4 structure of IrC4 showing brittleness. On the basis of the comparison of the crystal structure, elastic constants, elastic moduli, and electronic structure for iridium carbides, we expect that Ir4C5 and IrC4 with the iridium germanide structures are potential ultrahard materials.
I. INTRODUCTION Attempts to synthesize or theoretically predict new superhard and ultra-incompressible materials that are widely used in various industrial applications are the subject of intensive current research activities. One common way to develop superhard materials is by combining transition metals (TMs) with small covalent main group elements, namely, boron, carbon, nitrogen, or oxygen. Therefore, great efforts have been devoted to synthesizing various compounds constituted by transition metals and these light elements, such as transition metal nitrides,13 carbides,49 and borides,1012 both theoretically and experimentally. It is known to all that some transition-metal compounds with rich in small covalent main elements are superhard materials. For example, WB4 is the light element (LE)-rich material, and the measured average hardness is very high, exceeding 46.2 GPa,10 and MoB3 exhibits the large Vickers hardness of 31.8 GPa.13 IrN21 with a high bulk modulus and OsN414 with a high shear modulus also have been well investigated by the firstr 2011 American Chemical Society
principles method. However, there are few theoretical and experimental reports about transition-metal carbides with high carbon content. In fact, transition-metal carbides are of great fundamental interest and technological importance due to their excellent properties, such as high hardness, high melting point, chemical stability, good wear, and oxidation resistance. Therefore, studying transition-metal carbides with various stoichiometries especially with high carbon content is of great significance. We need to investigate these kinds of carbides with various stoichiometries on the theory aspect for providing a guidance to better look for and synthesize potential hard materials or ultraincompressible materials. In this paper, we investigate the structural, mechanical stability, and physical properties of the iridium carbides with various Received: December 28, 2010 Revised: February 28, 2011 Published: March 22, 2011 6948
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Table 1. Calculated Equilibrium Lattice Parameters, a, b, and c (Å); Unit Cell Volume (V in Å3); and Relative Total Energy of per Chemical Formula Unit (E in eV) of Iridium Carbides with Various Stoichiometries structure space group
a
b
c
V
E
Ir5C3 Rh5Ge3
Pbam
4.9014 12.0405
Ir4C5 Ir4Si5
P121/m1
5.5266
2.7754 12.5097 91.78
0
Ir4Ge5
P-4c2
4.9732
4.9732 14.8623 91.90
0.28
OsSi2
Cmca
8.8137
5.4300
9.5623 28.60
0
OsGe2
C12/m1
6.6749
2.8990
6.9009 28.65
0.96
RhSn2
I4/mcm
4.4988
4.4988
5.1739 26.18
1.05
I4/mmm Im-3m
3.3224 4.8258
3.3224 4.8258
5.3600 29.57 5.8219 88.91
1.46
Ir3C7
LaC2 Ir3Ge7
IrC3
TcP3
Pnma
13.4651
2.5750
4.2869 37.16
IrC4
RuP4
P-1
3.9355
5.4280
6.1489 40.73
0
IrSn4
Aba2
5.8111
3.9985
4.5637 47.67
0.686
CdP4
P121/c1
4.0448
3.6939
5.5674 40.96
0.838
ReP4
Pbca
6.8355
8.9198
7.0192 53.50
1.897
IrGe4
P3121
4.8258
4.8258
5.8219 39.14
2.232
PtSn4 PtPb4
Ccca P4/nbm
3.9209 4.6314
9.2037 4.6314
4.7626 42.97 3.8696 41.50
2.265 6.341
WB4
P63/mmc
5.5761
5.5761
6.2669 42.19 11.347
IrC2
3.1198 92.06
stoichiometries based on the known of transition-metal crystal structures. Nine IrC structures have been investigated using ab initio calculations based on the structures of known transitionmetal compounds,15 so here we do not consider it in detail any more. We hope our result can stimulate experimental research to synthesize potentially superhard and ultra-incompressible compounds.
II. COMPUTATIONAL DETAILS The calculations presented in this study were performed within the density functional theory (DFT), using the projector-augmented wave (PAW) method16,17 as implemented in the Vienna Ab initio Simulation Package (VASP).1820 The local density approximation (LDA)21 was used with the PAW potential. The structure was optimized with the conjugate gradient algorithm method, and a plane-wave cutoff energy of 400 eV was employed. Structural optimization was performed until the energy change of each atom was less than 0.005 eV; the forces on atoms were less than 0.02 eV/Å. The k-point samplings in the Brillouin zone were performed using the MonkhorstPack scheme22 for iridium carbides. The strainstress method was used to obtain the elastic constants. From the calculated elastic constants, cij, the polycrystalline bulk modulus, B, and shear modulus, G, were further estimated using the VoigtReussHill (VRH) approximation.23 III. RESULTS AND DISCUSSION A. Determination of Crystal Structure. Iridium carbides have not been synthesized in experiment, and there are few theoretical studies about them. As is well-known, the chemically related compounds may have a similar structure. For example, Wang et al. have successfully predicted IrN2 to be the IrP2 structure following this idea.24 Therefore, we first selected structures for iridium carbides with the synthesized iridium silicides (Ir4Si5,25 IrSi326) and iridium germanides (Ir4Ge5,27
Figure 1. Optimized structures of iridium carbides: (a) the Ir4Ge5 structure of Ir4C5, (b) the RhSn2 structure of IrC2, (c) the Ir3Ge7 structure of Ir3C7, and (d) the IrGe4 structure of IrC4. The red and black spheres represent iridium and carbon atoms, respectively.
Ir3Ge7,28 IrGe429), respectively. Moreover, some other structures of iridium carbides were also calculated as follows: the Rh5Ge330 structure for Ir5C3, the Ru2Ge331 structure for Ir2C3, the Ir5Sn732 structure for Ir5C7, and four potential crystal structures for both IrC2 (OsSi2,33 OsGe2,34 LaC2,35 RhSn236) and IrC3 (IrP3,37 IrAl3,38 TcP3,39 RhSn340) were determined, and the considered structures for IrC4 are based on the tetragonal PtPb441 structure (PtPb4IrC4), hexagonal WB442 structure (WB4IrC4), orthorhombic structures (PtSn4,32 ReP4,43 IrSn444), monoclinic CdP445 (CdP4IrC4), and triclinic RuP446 (RuP4IrC4). In these potential structures, Ru2Ge3Ir2C3, Ir5Sn7Ir5C7, and IrC3 with IrSi3, RhSn3, IrP3, and IrAl3 structures are all elastically unstable through our calculations and were ruled out. Table 1 lists the computed equilibrium lattice parameters, unit cell volume, and relative total energy of Ir5C3, Ir4C5, IrC2, Ir3C7, IrC3, and IrC4 with these potential structures. From this table, it can be seen that Ir4Si5, OsSi2, and RuP4 structures have the lowest total energy for Ir4C5, IrC2, and IrC4, respectively. The compound with the smallest volume exhibits the highest density and a good elastic property. As seen from Table 1, iridium carbides with the synthesized iridium germanide structures basically exhibit a smaller volume comparing with other structures, except Ir4C5. Ir4Si5 and RhSn2 structures have the smallest cell volume for the Ir4C5 phase and IrC2 phase, respectively. To well understand the elastic properties, the value of the cell volume is not enough and the elastic constant should be investigated. B. Structure and Elastic Properties. The optimized crystal structures of Ir4C5, Ir3C7, and IrC4 in the iridium germanide-type structures and IrC2 with the RhSn2 structure are shown in Figure 1. We found that only IrGe4IrC4 does not have covalent CC dimers, and its value for the average Mulliken overlap population (MOP) of CC bonds close to zero indicates that there is no significant interaction between the electronic populations of the C atoms. Among these compounds, Ir4Ge5Ir4C5 has the shortest IrC bonds (2.0275 Å) and the largest value of 6949
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Table 2. Calculated Elastic Constants (in GPa), Bulk Modulus (B in GPa), Shear Modulus (G in GPa), and Poisson’s Ratio (v) of Iridium Carbides with Various Stoichiometries structure
c11
c12
c13
c22
c33
Ir5C3
Rh5Ge3
415
219
285
366
466
Ir4C5
Ir4Ge5 Ir4Si5
685 455
221 211
246 211
685 469
542 466
c44
G
G/B
ν
307.7
84.1
0.27
0.375
369.3 297.4
197.8 137.6
0.54 0.46
0.273 0.300
c55
c66
B
72
91
55
195 123
195 121
86 123
RhSn2
599
316
187
599
827
67
67
185
377.7
136.9
0.36
0.338
OsSi2
864
331
198
530
314
79
14
310
295.2
101.2
0.34
0.346
Ir3C7
Ir3Ge7
510
279
279
510
510
182
182
182
356.0
169.9
0.48
0.294
IrC4
IrGe4
648
215
195
648
591
164
164
217
343.6
211.3
0.61
0.250
RuP4
305
241
242
606
441
148
183
144
269.9
140.5
0.52
0.278
CdP4
387
208
343
320
510
122
240
165
285.2
134.8
0.47
0.296
448
282
250
448
496
179
179
83
328
129
0.39
0.33
IrC2
WB442
MOP (0.63). The high MOP indicates a high degree of covalency in the IrC bonds. The appearance of CC dimers along a, b, and c axes may contribute to its incompressibility in these directions for Ir3Ge7Ir3C7. In IrGe4IrC4, the C atoms are dimerized with the shortest CC distance of 1.4854 Å and the next shortest, 1.4946 Å. The bond length of CC in IrGe4IrC4 is shorter than the shortest BB bond (1.68 Å) in WB4 and the CC bond (1.53 Å) in diamond. The strength of the bond can be characterized by using the average overlap population per unit volume of the bond.47 The strong covalent bonding between C atoms is also confirmed by the calculated MOP of the CC bonds in IrGe4IrC4 (0.96 and 0.91). Previous studies have shown that the three-dimensional BB network in WB4 is the key for its high hardness,42 so we predict that the covalent CC bonding in IrGe4IrC4 should be helpful to its high hardness. Moreover, the length of each bond between Ir and C in IrC4 is shorter than the WB bonds in WB4. The strong IrC covalent bonds should contribute to its elastic properties and strength. The mechanical stability is a necessary condition for a crystal to exist. The accurate elastic constants can help us to well understand the macromechanical properties and also provide very useful information to estimate the hardness of the material. To check the mechanical stability of iridium carbides, their elastic constants were calculated by the strainstress method and are listed in Table 2. Obviously, the obtained elastic constants of iridium carbides with these structures listed in Table 2 all satisfy the mechanical stability criteria for the crystal, which are given in ref 48, suggesting mechanical stability of the ten compounds. Moreover, we calculated the eigenvalues of the elastic constant matrix of these materials and found that all eigenvalues are positive, which also indicates that they are elastically stable. As is well-known, the elastic constant c44 is an important parameter indirectly governing the indentation hardness. As shown in Table 2, Ir4C5, Ir3C7, and IrC4 with the iridium germanide structures all have large c44 values. Besides, the elastic constant c44 of Ir4Ge5Ir4C5 with the space group P-4c2 has the largest value, indicating its relatively strong shear strength. The bulk modulus depends mainly on the average valence electron density and is less related to structural detail. Thus, it is necessary to calculate the average valence electron density. The calculated average valence electron density for trigonal IrGe4IrC4 is 0.639 e/Å3, which is larger than that of WB4 (0.465 e/Å3).42 Thus, IrC4 with the IrGe4 structure is expected to show a smaller compressibility than WB4. Generally speaking, superhard materials possess high bulk modulus to support the volume decrease caused by an applied load and high shear modulus to restrict deformation in a
direction different from the applied load. Table 2 lists the calculated elastic properties of iridium carbides with various stoichiometries. For comparison with IrC4, the bulk modulus, shear modulus, and Poisson’s ratio of WB442 are also tabulated in the same table. Apparently, it can be seen that the listed compounds all have a large bulk modulus. The high bulk modulus shows that these materials are difficult to compress. Among these compounds, IrC2 with the RhSn2 structure exhibits the largest bulk modulus, 377.7 GPa, which indicates that this structure is the most difficult to compress. For OsC2, previously theoretical study showed that Cmca in the OsSi2-type structure is the most energetically stable phase, but its bulk modulus (273 GPa) and shear modulus (82 GPa) are smaller than those of the OsB2 structure with the Pmmn space group. Therefore, OsSi2 is not the hardest among the considered structures.9 Meanwhile, the bulk modulus and shear modulus of OsC2 with the OsSi2 structure9 are smaller than those of IrC2 with the OsSi2 structure. Our calculations also indicate that IrC2 with the most energetically stable structure OsSi2 has a smaller bulk modulus (295.2 GPa) and shear modulus (101.2 GPa) than those of the RhSn2 structure. It is interesting to note that the Ir4Ge5Ir4C5 exhibits a better elastic property especially having a higher bulk modulus than that of Ir4Si5Ir4C5, although Ir4Si5Ir4C5 has a smaller cell volume. The calculated bulk modulus of IrGe4IrC4 by the VRH approximation is 344 GPa and is larger than that of other type structures, which is even larger than that of WB4, 328 GPa.42 The calculated bulk modulus agrees well with that (346 GPa) determined by fitting pressures and cell volumes with the thirdorder BirchMurnaghan equation of state,49 suggesting the reliability of our calculation method. As we know, shear modulus provides a much better correlation with hardness than bulk modulus. As the carbon content is increased, the shear modulus of iridium carbides with iridium germanide structures increases, except Ir4C5. Among them, Ir4Ge5Ir4C5 also exhibits a large shear modulus (197.8 GPa) and is only 13.5 GPa lower than that of the IrGe4IrC4 (211.3 GPa). In particular, the shear modulus for IrC4 with the IrGe4, RuP4, and CdP4 structures are all larger than that of WB4 (129 GPa),42 especially the IrGe4 structure of IrC4. It suggests that IrGe4IrC4 can withstand the shear strain to a large extent. Gu et al. synthesized WB4 and measured the average Vickers hardness; the value is very hard, exceeding 46 GPa.10 Thus, it is to be expected that the hardness of IrGe4IrC4 is maybe higher than that of WB4. The value of Poisson’s ratio is indicative of the degree of directionality of the covalent bonding, and the quantity is calculated using the following relationship: 6950
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Figure 2. Total and partial DOS: (a) the trigonal Ir4Ge5 structure of Ir4C5, (b) the Ir3Ge7 structure of Ir3C7, (c) the RhSn2 structure of IrC2, and (d) the OsSi2 structure of IrC2. The Fermi level is at zero.
Figure 3. Total and partial DOS: (a) the trigonal IrGe4 structure of IrC4 and (b) the RuP4 structure of IrC4. The Fermi level is at zero.
ν¼
3B 2G 2 ð3B þ GÞ
ð1Þ
The relative directionality of the bonding in the material also has an important effect on its hardness and can be determined by the value of G/B.50 The value of Poisson’s ratio and G/B of iridium carbides with various stoichiometries were calculated and are listed in Table 2. Obviously, it can be seen that the order for G/B of iridium carbides is IrGe4IrC4 > Ir4Ge5Ir4C5 > Ir3Ge7Ir3C7 > RhSn2IrC2 > Rh5Ge3Ir5C3 and the order for ν of iridium carbides is IrGe4IrC4 < Ir4Ge5Ir4C5 < Ir3Ge7Ir3C7 < RhSn2IrC2 < Rh5Ge3Ir5C3. It is found that increasing the carbon content in iridium carbides with the iridium germanide
structures can enhance the degree of directionality of the covalent bonding and hardness, but Ir4C5 with the Ir4Ge5 structure is a special compound. It exhibits larger G/B and smaller ν values than that of RhSn2IrC2 and Ir3Ge7Ir3C7, comparable to that of IrGe4IrC4. The small Poisson’s ratio ν (0.25) for trigonal IrGe4IrC4 indicates a strong degree of covalent bonding, which contributes to its high hardness. Moreover, the calculated ratio G/B for IrGe4IrC4 in the P3121 phase (0.61) is much larger than that of WB4 (0.39),42 indicating the more pronounced directional bonding between the ions in trigonal IrGe4IrC4. A high (low) B/G value51 is associated with ductility (brittleness), and the critical value is about 1.75. Only the ratio B/G of trigonal IrGe4IrC4 is smaller than 1.75, indicating its brittleness, whereas the other structures of IrC4 are all ductile. We found that increasing the carbon content may cause the weak ductile property, because the ratio B/G gradually decreases with increased carbon content. The listed compounds are all ductile except TcP3IrC3 and IrGe4IrC4. In a word, the large elastic modulus, large value of the G/B, and small Poisson’s ratio show that Ir4Ge5Ir4C5 and IrGe4IrC4 are potential ultrahard materials. C. Electronic Structure Analysis. To further understand the property of iridium carbides with various stoichiometries, the total and partial density of states (DOS) were calculated and are shown in Figures 2 and 3. According to Figures 2 and 3, we found that only Ir4C5 with the Ir4Ge5 structure is a semiconductor and is different from the metallic behavior of other iridium carbides. Figure 4 plots the calculated band structure of Ir4C5 with the Ir4Ge5 structure, which displays that this compound has an indirect band gap of 0.5 eV, indicating a rare property of semiconductivity among the transition-metal light-element compounds. The semiconductor behavior of Ir4Ge5Ir4C5 may come from the strong covalent IrC bonding, which can be seen from the short bond length and the large bond populations (0.63) of its IrC bond. Ir4Ge5Ir4C5 has a large bulk modulus and shear modulus, which may be related to this semiconductor character. Through our calculations, we found that increasing the carbon content in iridium carbides with the iridium germanide structures can enhance the degree of directionality of the covalent and hardness, except Ir4Ge5Ir4C5. We conclude that this special semiconductor property may be the reason that Ir4Ge5Ir4C5 does not follow this rule. Thus, the semiconductor behavior of Ir4Ge5Ir4C5 may be helpful to its stability and high hardness. Figure 2 depicts that IrC2 with RhSn2 and OsSi2 structures are metallic due to the finite DOS at the Fermi level. The material 6951
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increase with the increasing carbon content, except for Ir4C5. Ir4Ge5Ir4C5 is a semiconductor with an indirect band gap of 0.5 eV, which is different from other studied iridium carbides. The small ratio of B/G for TcP3IrC3 and IrGe4IrC4 shows that they are brittle materials. The calculation results show that trigonal IrGe4IrC4 is not the most energetically stable phase but is a high mechanical stability conductor with a higher bulk modulus, the highest shear modulus, and the smallest Poisson’s ratio. With these physical properties, tetragonal Ir4Ge5Ir4C5 and trigonal IrGe4IrC4 are attractive for advanced superhard and ultra-incompressible materials.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
Figure 4. Band structure of Ir4C5 with the Ir4Ge5 structure. The high symmetry k points Z, A, M, Γ, R, and X in the figure represent the points (0, 0, 0.5), (0.5, 0.5, 0.5), (0.5, 0.5, 0), (0, 0, 0), (0, 0.5, 0.5), and (0, 0.5, 0), respectively.
stability is related to both the position and the value of N(Ef) of the DOS at the Fermi level.52 Compared with the OsSi2 structure, the value of N(Ff) of the DOS at the Fermi level for IrC2 with the RhSn2 structure is smaller, indicating the stronger stability of the RhSn2 structure. For the DOS of IrC4 with the trigonal IrGe4 structure and the RuP4 structure, there is a deep valley called a pseudogap around the Fermi level, which results from the strong hybridization (Ir d and C p) and leads to a separation between the bonding and antibonding states. The presence of a pseudogap will surely increase the stability of compounds. Comparing with the RuP4 structure of IrC4, it is clearly seen that the DOS of the Ir 5d orbital and C 2p orbital have a similar shape for the trigonal IrGe4IrC4 phase, which indicates that there is a strong hybridization between the Ir d orbital and C p orbital. Therefore, a strong covalent IrC bonding exists in the IrGe4IrC4 phase, which will contribute to its high incompressibility and high shear strength. As seen from Figure 3, a lower N(Ef) (2.15 electrons/eV) value for trigonal IrGe4IrC4 than that of triclinic RuP4IrC4 and WB442 indicates the strong stability of IrGe4IrC4. The appearance of a pseudogap and low states around the Fermi level of the DOS of IrGe4IrC4 would be beneficial to its stability and high shear modulus, consistent with the calculated large shear modulus of IrGe4IrC4.
IV. CONCLUSION In summary, iridium carbides with various stoichiometries based on the structures of synthesized transition-metal compounds are studied by the first-principles calculations. Among these compounds, iridium carbides with iridium germanium structures have the large bulk modulus, B, shear modulus, G, and the elastic constant. All positive eigenvalues of the elastic constant matrix confirm that they are elastically stable. RhSn2IrC2 has the largest bulk modulus among these carbides, indicating its incompressibility. For iridium carbides with iridium germanium structures, the value of the shear modulus and G/B
’ ACKNOWLEDGMENT This research was sponsored by the National Natural Science Fundation of China (No. 21071045) and the fund of Henan University (No. SBGJ090508). ’ REFERENCES (1) Crowhurst, J. C.; Goncharov, A. F.; Sadigh, B.; Evans, C. L.; Morrall, P. G.; Ferreira, J. L.; Nelson, A. J. Science 2006, 311, 1275. (2) Young, A. F.; Sanloup, C.; Gregoryanz, E.; Scandolo, S.; Hemley, R. J.; Mao, H. K. Phys. Rev. Lett. 2006, 96, 155501. (3) Jiang, C.; Lin, Z.; Zhao, Y. Phys. Rev. Lett. 2009, 103, 185501. (4) Zhao, E. J.; Wang, J. P.; Meng, J.; Wu, Z. J. J. Comput. Chem. 2010, 31, 2883. (5) Ono, S.; Kikegawa, T.; Ohishi, Y. Solid State Commun. 2005, 133, 55. (6) Chen, Z. W.; Gu, M. X.; Sun, C. Q.; Zhang, X. Y.; Liu, R. P. Appl. Phys. Lett. 2007, 91, 061905. (7) Zhao, Z. S.; Wang, M.; Cui, L.; He, J. L.; Yu, D. L.; Tian, Y. J. J. Phys. Chem. C 2010, 114, 9961. (8) Du, X. P.; Wang, Y. X. J. Appl. Phys. 2010, 107, 053506. (9) Cai, J.; Zhao, E. J.; Wu, Z. J. Comput. Mater. Sci. 2009, 46, 1098. (10) Gu, Q. F.; Krauss, G.; Steurer, W. Adv. Mater. 2008, 20, 3620. (11) Qin, J. Q.; He, D. W.; Wang, J. H.; Fang, L. M.; Lei, L.; Li, Y. J.; Hu, J.; Kou, Z. L.; Bi, Y. Adv. Mater. 2008, 20, 4780. (12) Chung, H. Y.; Yang, J. M.; Tolbert, S. H.; Kaner, R. B. J. Mater. Res. 2008, 23, 1797. (13) Zhang, M. G.; Wang, H.; Wang, H. B.; Cui, T.; Ma, Y. M. J. Phys. Chem. C 2010, 114, 6722. (14) Zhao, W. J.; Xu, H. B.; Wang, Y. X. Phys. Status Solidi RRL 2009, 3, 272. (15) Zhao, Z. S.; Xu, L. F.; Wang, M.; Cui, L.; Wang, L. M.; He, J. L.; Tian, Y. J. Phys. Status Solidi RRL 2010, 4, 230. (16) Kresse, G.; Joubert, D. Phys. Rev. B 1999, 59, 1758. (17) Bl€ochl, P. E. Phys. Rev. B 1994, 50, 17953. (18) Kresse, G.; Hafner, J. Phys. Rev. B 1993, 47, 558. (19) Kresse, G.; Hafner, J. J. Phys.: Condens. Matter 1994, 6, 8245. (20) Kresse, G.; Furthm€uller, J. Phys. Rev. B 1996, 54, 11169. (21) Ceperley, D. M.; Alder, B. J. Phys. Rev. Lett. 1980, 45, 566. (22) Monkhorst, H. J.; Pack, J. Phys. Rev. B 1976, 13, 5188. (23) Hill, R. Proc. Phys. Soc. A 1952, 65, 349. (24) Wang, Y. X.; Arai, M.; Sasaki, T.; Fan, C. Z. Phys. Rev. B 2007, 75, 104110. (25) Engstr€om, I.; Zackrisson, F. Acta Chem. Scand. 1970, 24, 2109. (26) Finnie, L. N. J. LessCommon Met. 1962, 4, 24. (27) Flieher, G.; V€ollenkle, H.; Nowotny, H. Monatsh. Chem. 1968, 99, 877. (28) Schubert, K.; Pfisterer, H. Z. Metallkd. 1950, 41, 433. 6952
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dx.doi.org/10.1021/jp112308t |J. Phys. Chem. C 2011, 115, 6948–6953
