Electronic, optical, and mechanical properties of superhard cold ... - CMS

APPLIED PHYSICS LETTERS 99, 031901 (2011)

Electronic, optical, and mechanical properties of superhard cold-compressed phases of carbon Haiyang Niu,1 Pengyue Wei,1 Yan Sun,1 Xing-Qiu Chen,1,a) Cesare Franchini,1,2 Dianzhong Li,1 and Yiyi Li1

1 Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang, China 2 Faculty of Physics and Center for Computational Materials Science, University of Vienna, Vienna, Austria

(Received 12 April 2011; accepted 26 June 2011; published online 18 July 2011) By means of standard and hybrid density functional theory, we analyzed the electronic, optical, and mechanical properties of the two discovered superhard orthorhombic (W) and monoclinic (M) phases of carbon, synthesized by cold compression. We demonstrated that both phases exhibit a transparent insulating behaviour with indirect band gaps of about 5.4 eV (W) and 4.5 eV (M), and highly isotropic optical spectra, substantially different to those of the related body centered tetragonal C4 phase. The analysis of the elastic constants and Vickers hardness confirmed that these C 2011 American Institute of Physics. phases are as hard as the second hardest material c-BC2N. V [doi:10.1063/1.3610996] It is well known that at high temperatures (>1300 C) and high pressures (>15 GPa), graphite can be suitably transformed into hexagonal or cubic diamond.1–6 Also, it has been reported that by cold-compression at room temperature, graphite can be transformed, at approximately 17 GPa, into a transparent and superhard phase.7 Cold-compression experiments also detected several peculiar properties accompanying this structural transition:2,7–11 (1) the formation of partial sp3 bonding, (2) an increased electrical resistivity, (3) a sharply dropped optical reflectivity above 15 GPa, (4) an increased optical transmittance above 18 GPa, and (5) a broadening of the higher frequency of the E2g Raman line. Furthermore, a quenchable superhard phase of carbon has been also found by cold-compression of carbon nanotubes.12 Although many features have been revealed clearly, the structure of this phase of carbon, which is reported to be different to those of hexagonal and cubic diamond, remains experimentally unresolved. In order to shed some light on the experimental structural uncertainties, a series of possible candidates have been proposed by first-principles calculations.13–21 The hybrid graphite-diamond structure was first proposed to interpret the high-pressure, low-temperature transparent superhard phases.13,14 However, significant differences between the simulated and experimental x-ray diffraction (XRD) patterns prevent an unambiguous assignment. Conversely, three very recently proposed crystal structures16–18 show a good match with the experimental XRD patterns. They are (1) a monoclinic C2/m phase (M-carbon),16 (2) a body-centered tetragonal C4 phase (C4-carbon),17 and (3) an orthorhombic phase (W-carbon).18 It should be mentioned that much before these predictions, the C4-carbon phase was proposed by Baughman et al.19,20 and subsequently investigated by tight-binding molecular dynamics.21 In addition, the dynamical transformation pathways from graphite to these three phases have been theoretically clarified.18,22 a)

Author to whom correspondence should be addressed. Electronic mail: [email protected].

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In a recent study,23 we found that C4 exhibits strong electronic and optical anisotropy in both the dielectric response and the mechanical properties. In this letter, we aim to understand the relevant differences on the electronic, optical, and mechanical properties of the three different proposed phases (C4, M-carbon, and W-carbon) by means of standard and hybrid density functional theory (DFT) within the Heyd-Scuseria-Ernzerhof (HSE) method24 as implemented in the Vienna ab initio Simulation package (VASP).25,26 We employed the same computational setup discussed in our previous work on C4.23 All structural parameters for W- and M-carbon have been optimized using standard DFT. Due to its demanding computational cost, the HSE method has been only used to calculate electronic and optical properties adopting the DFT derived structures. The elastic and mechanical properties have been investigated within the standard DFT framework. The structural details of W- and M-carbon are schematically displayed in Fig. 1. We noted that these two phases appear very similar as evidenced by the fact that W-carbon is structurally related to M-carbon through a shear deformation along the [001] direction. Detailed analysis demonstrated that both phases primarily consist of four inequivalent quasi-sp3 electronic hybrids. Although the stacking of the four inequivalent carbon atoms in both Wand M-carbon phases is intrinsically different, their local bond lengths and angles for the specific four tetrahedral hybrids are very similar. In comparison with diamond, the overall bonding picture of W- and M-carbon phases appears different: Diamond has an ideal sp3 hybrid arrangement with ˚ and a standard sp3 a carbon-carbon bond length of 1.53 A bond angle of 109.5 , whereas both the M and W phases show a distorted tetrahedral sp3-like hybrid in which the bond lengths and angles are slightly different. For instance, at zero pressure for M-carbon, the bond lengths connecting to C1 are, with increasing length, 1.551, 1.557, 1.573, and ˚ . Similarly, the other bond lengths are, in A ˚ , C2 1.573 A (1.551, 1.573, 1.573, 1.585), C3 (1.505, 1.548, 1.548, 1.584), and C4 (1.505, 1.517, 1.517, 1.557). In addition, we also

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FIG. 1. (Color online) Crystal structures of both W- and M-carbon phases at 0 GPa. W-carbon crystallizes in the orthorhombic space group of Pnma (No. ˚ , b ¼ 2.5255 A ˚ , and 62) with the optimized lattice constants a ¼ 9.0866 A ˚ and with carbon (C) at four inequivalent 4c Wyckoff sites C1: c ¼ 4.1561 A (0.3051, 1/4, 0.5806), C2: (0.1895, 1/4, 0.3061), C3: (0.5207, 0.25, 0.0978), and C4: (0.4635, 1/4, 0.4377). The structure of M-carbon has the monoclinic space group of C2/m (No.12) with the optimized lattice constants a ¼ 9.1899 ˚ , b ¼ 2.5241 A ˚ , and c ¼ 4.1426 A ˚ and b ¼ 97.03 and with C also at four A inequivalent Wyckoff 4i sites C1: (0.0573, 0, 0.8796), C2: (0.2143, 0, 0.0593), C3: (0.4418, 0, 3463) and C4: (0.2713, 0, 0.4147).

noted that both the W and M phases appear to be significantly less anisotropic than the C4 phase.23 The structural analogies between the W and M phases result in a very similar electronic structure, as shown in Fig. 2. Apart from minor differences probably arising from the reduced symmetry, the DOSs of W- and M-carbon resemble closely that of diamond. The band structure shows that both W- and M-carbon phases are wide gap insulators with indirect band gaps of 5.41 eV and 4.55 eV at 15 GPa, respectively. For the W-carbon, the indirect gap is opened between the top of the valence band at the C point and the conduction band minimum at U ¼ (1/2,0,1/2). Similarly, the M-carbon has the highest valence band at C, whereas the conduction band bottom lays at M ¼ (1/2,1/2,1/2). Interest-

FIG. 2. (Color online) HSE calculated band structure and electronic densities of states for W- [panel (a)] and M-carbon [panel (b)] at 15 GPa. The zero energy refers to the top of valence bands. Their densities of states are compared with that of diamond at 15 GPa.

Appl. Phys. Lett. 99, 031901 (2011)

ingly, the calculated band gap (5.29 eV at 15 GPa) of diamond is found to be in between the predicted gaps for Wand M-carbons. In Fig. 3(a), we collect the values of the band gap at 0 and 15 GPa for the four phases under scrutiny. We note that among these four carbon phases, C4 displays the smallest band gap (3.45 eV at 15 GPa) as a result of the substantial anisotropy between the Cpz and Cpx þ py components as discussed in Ref. 23, and that the value of the band gaps does not vary much going from 0 to 15 GPa (about 0.10.2 eV). Figure 3(a) also emphasizes the need to use beyond DFT methods to correctly predict band gaps of carbon allotropes, as manifested by the HSE correction of the band gap of diamond (5.17 eV at 0 GPa) which is significantly underestimated at DFT level (4.12 eV at 0 GPa). We have also computed the optical spectrum of W- and M-carbon along the three a, b, and c axes by calculating the frequency dependent imaginary part of the dielectric matrix.27 The resulting spectra, shown in Figs. 3(b) and 3(c), are almost identical. The only significant difference lies in the high frequency region from x ¼ 18 to 21 eV, where the M-carbon phase exhibits a small shoulder along the Y-channel. Our results show that, unlike the C4 phase,23 both Mand W-carbon phases have highly isotropic absorption spectra, analogous to diamond. Finally, by means of standard DFT, we have investigated the mechanical properties of the four considered phases in terms of the elastic constants and the derived bulk and shear moduli within the Voigt-Reuss-Hill approximation at 0 and 15 GPa (cf. Table I). First, we note that several elastic constants C25, C35, and C46 of monoclinic M-carbon are negative. However, based on the arguments given in Ref. 28, these negative value does not mean an instability because the positive elastic coefficients can be yielded by the actual applied distortions. C4 displays the largest C33 value (1190 GPa) by about 100 GPa larger than that of diamond. The difference can be attributed to the shortest C–C bond length and the associated strongest covalency along the c-axis for the C4 phase. However, the compressibilities along the c-axis for

FIG. 3. (Color online) (a) Comparison between HSE and standard DFT calculated band gaps for the four carbon phases considered (C4, W-carbon, M-carbon, and diamond) at 0 and 15 GPa. (b) and (c) Absorption spectra along the main cartesian axis a, b, and c of both M- and W-carbon phases at 15 GPa.


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TABLE I. DFT calculated elastic constants Cij (GPa), bulk B (GPa), shear G (GPa) and Vicker’s hardness Hv (GPa) for diamond (Dia), C4, W- and M-carbon phases at 0 and 15 GPa. The bulk B and shear G moduli are obtained within Voigt-Reuss-Hill average, respectively. The elastic constants of the M-carbon were calculated based on Ref. 28 and the elastic constants of C4 and diamond are taken from Refs. 23 and 31, respectively. C11 Dia (0 GPa) Dia (expt) C4 (0 GPa) C4 (15 GPa) W (0 GPa) W (15 GPa) M (0 GPa) M (15 GPa)

1067 1079 933 1013 890 977 929 1016

C22

1087 1187 1090 1182

C33

C44

1190 1285 1045 1137 1042 1128

571 578 447 475 522 571 522 543

C55

457 505 452 471

C66

C12

325 330 391 410 389 386

132 124 172 188 53 60 66 58

both M- and W-carbons are just comparable to that of diamond. On the other hand, we also noted that for these phases, the shear moduli are all larger than the corresponding bulk moduli (Table I), thereby resulting in a low Poisson’s ratio, consistent with the strong and directional covalent bonding picture. Therefore, it is expected that all these carbon compounds should display a large hardness29. We have investigated this aspect (see gray symbols in Fig. 4) further by applying our recently introduced empirical scheme30 which correlates the Vickers hardness and the Pugh’s modulus ratio (k ¼ G/B) through the formula Hv ¼ 2ðk2 GÞ0:585 3:

(1)

According to Eq. (1), the Vickers hardness of polycrystalline C4, M- carbon, and W-carbon phases derived from the elastic moduli of Table I are estimated to be 68.9 GPa, 77.6, and 78.5 GPa at 0 GPa, respectively. As pressure increases to 15 GPa, their hardness values are slightly increased. Therefore, it can be safely concluded that these three carbon compounds belong to the superhard class. From Table I, it has been noted that both W- and M-carbon phases have very similar bulk and shear moduli, resulting in the nearly identical Vickers hardness. In particular, from Fig. 4, we see that the Vickers hardness of C4 is comparable to that of the superhard c-BN and both M- and W-carbon phases are as hard as the

FIG. 4. (Color online) Estimation of Vickers hardness for C4, W-carbon, Mcarbon, and diamond, along with a collection of grey data for other representative hard materials. The details on the formalism associated with the evaluation of the hardness can be found in Ref. 30. The solid curve refers to Eq. (1).

C13

59 70 165 203 171 159

C23

91 110 05 98

C15

66 31

C25

174 101

C35

118 2

C46

G

B

Hv

8 2

527 538 421 447 451 488 446 475

444 443 404 440 403 448 400 436

92.6 96.4 68.9 69.4 78.5 79.6 77.6 78.4

second hardest material, BC2N (76 GPa).29 Our computed Vickers hardness of M-carbon (77.6 GPa at 0 GPa) is in good agreement with the value obtained using the Sˇimu˚nek’s model32 (83.1 GPa).16 We have also computed the hardness of W-carbon using the Sˇimu˚nek’s model. The so obtained value, 83.6 GPa, is very close to that of M-carbon and in agreement with the data obtained using Eq. (1). In conclusion, we have performed a computational study of the electronic, optical, and mechanical properties of Mand W-carbon and compared them with those of C4 and diamond. Along with justifying a unified and rational picture of this class of carbon allotropes, we believe that our results will provide a useful guidance for clarifying the still puzzling experimental observations. We are grateful for supports from the “Hundred Talents Project” of CAS, Fellowship for Young International Scientists, and the NSFC (Grant Nos. 51074151 and 51050110444) as well as the computational resources from Beijing Supercomputing Center of CAS. 1

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