Monoclinic C2/m-20 carbon: a novel superhard sp³ carbon allotrope

Abstract

Based on density function theory with the ultrasoft pseudopotential scheme in the frame of the generalized gradient approximation, the structural, mechanical, and electronic properties of C2/m-20 carbon have been systematically investigated in this paper. The elastic constants and phonon spectra calculations show that C2/m-20 carbon is mechanically stable and dynamically stable at 0 GPa and 100 GPa. From our calculations, we found that C2/m-20 carbon has a larger bulk modulus of 412 GPa, a larger shear modulus of 463 GPa, a larger Young’s modulus of 1010 GPa, and a hardness of 70.6 GPa, which means that it is a superhard material with potential technological and industrial applications. C2/m-20 carbon exhibits greater anisotropy than C2/m-16 carbon and M-carbon using Young’s modulus and the universal anisotropic index. In addition, C2/m-20 carbon is an indirect and wide semiconductor with a band gap of 5.10 eV. The most extraordinary thing is that the band gap increases with increasing pressure.

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Group 14 elements (C, Si, and Ge) exist as various stable and metastable allotropes, some of which have been widely applied in industry. Searches for low-energy monoclinic allotropes of carbon have been performed, using the evolutionary metadynamics technique^1–4 and Universal Structure Prediction: Evolutionary Xtallography (USPEX),^5–8 or post-graphite superhard phases synthesized via cold-compressing graphite.^9–17 Carbon is a unique element in the natural world as it adopts a wide range of structures, from superhard wide semiconductors, such as, diamond and lonsdaleite, to ultrasoft semimetals, for example, graphite, and even superconductors, like intercalated graphite, fullerenes, and doped diamond.^18–22 An important and useful direction in carbon research is the discovery of carbon allotropes with advanced mechanical and electronic properties. Hardness is an important property that determines a lot of the technological applications of materials. Diamond is important. Why is it important? Just because it has the greatest hardness in the natural world. There are many superhard carbon allotropes that have been predicted by researchers, such as, O-carbon,²³ W-carbon,²⁴ bct-C4,²⁵ oC16-carbon,²⁶ P- and R-carbon,²⁷ F-carbon,²⁸ C-carbon,²⁹ lonsdaleite,³⁰ Z-carbon-1, Z-carbon-2, Z-carbon-3, Z-carbon-7 and Z-carbon-10,³¹ orthorhombic C32,³² H-carbon,^33,34 Hex-C₂₄,³⁵ yne-diamonds,³⁶ S-carbon,³⁷ Imma-carbon,^38–40 P222₁ carbon,⁴¹ C2/m-16 carbon,⁴² and so on. M-carbon was first proposed by Oganov and Glass,⁴³ and then Li et al.⁴⁴ reported this novel phase of carbon; this novel phase of carbon possesses a monoclinic C2/m structure (M-carbon, 8 atoms per cell), with a larger bulk modulus and a larger hardness. Recently, Zhou et al.³¹ attained nine new polymorphic phases of carbon besides the known crystalline structures using an evolutional algorithm, including seven monoclinic symmetry carbon allotropes: Z-carbon-1, Z-carbon-2, Z-carbon-3, Z-carbon-5, Z-carbon-6, Z-carbon-7 and Z-carbon-10. But only Z-carbon-1, Z-carbon-2, Z-carbon-3, Z-carbon-7 and Z-carbon-10 are potential superhard materials, unlike Z-carbon-5 and Z-carbon-6 which have a hardness of 34.1 GPa and 34.2 GPa, respectively. Not long ago, we reported a novel monoclinic phase carbon allotrope, C2/m-16 carbon,⁴² including its structural, mechanical and electronic properties using first-principles calculations. We found that C2/m-16 carbon is a potential superhard material with a hardness of 59.5 GPa and a semiconductor with a wide and indirect band gap of 4.20 eV.

In this paper, we will report a novel monoclinic phase carbon allotrope, namely C2/m-20 carbon. We will have performed detailed first-principles calculations of the structural, mechanical and electronic properties of C2/m-20 carbon based on density functional theory. Prediction of the physical properties for the new carbon allotrope was performed using density functional theory (DFT)^45,46 based on the Cambridge Serial Total Energy Package (CASTEP) code⁴⁷ with the Vanderbilt ultrasoft pseudopotential. The exchange and correlation are approximated by the generalized gradient approximation (GGA) parameterized by Perdew, Burke and Ernzerhof (PBE).⁴⁸ The total energy convergence tests showed that convergence to below 1 meV per atom was achieved with the above calculation parameters. The Broyden–Fletcher–Goldfarb–Shanno (BFGS)⁴⁹ minimization scheme was used for geometry optimization. In the structure calculation, plane-wave basis sets with an energy cut-off at 400 eV are used for C2/m-20 carbon, C2/m-16 carbon and M-carbon. The 2s²2p² electrons of the carbon atoms were regarded as valence electrons. A proper k-point grid of 7 × 16 × 8, 13 × 7 × 8 and 4 × 16 × 10 in the Brillouin zone is obtained with respect to the Monkhorst–Pack method⁵⁰ for the C2/m-20, C2/m-16 and M phases for carbon, respectively. The self-consistent convergence of the total energy is 5 × 10⁻⁶ eV per atom; the maximum force on the atom is 0.01 eV Å⁻¹, the maximum ionic displacement is within 5 × 10⁻⁴ Å and the maximum stress is within 0.02 GPa.

The novel monoclinic symmetry carbon allotrope, namely C2/m-20 carbon, contains sp³-hybridized covalent bonds. The crystal structures of C2/m-20 carbon, C2/m-16 carbon and M-carbon are illustrated in Fig. 1a–f. C2/m-20 carbon, C2/m-16 carbon and M-carbon all belong to the C2/m phase. In their crystal structures, six-membered carbon rings exist in C2/m-16 carbon and C2/m-20 carbon. In addition, C2/m-16 carbon and C2/m-20 carbon also have five-membered carbon rings and seven-membered carbon rings, while M-carbon only has five-membered carbon rings and seven-membered carbon rings. Furthermore, C2/m-20 carbon has six-membered carbon ring layers inserted into the five-membered carbon rings and seven-membered carbon rings, and the six-membered carbon rings in C2/m-16 carbon are zigzag carbon rings. The optimized crystal parameters of C2/m-16 carbon, C2/m-20 carbon and M-carbon are listed in Table 1 and compared to the available calculated and experimental data for diamond. The optimized crystal parameters of C2/m-16 carbon, C2/m-20 carbon, M-carbon and diamond are in excellent agreement with previously calculated and experimental data. Moreover, we also calculated the crystal density of three monoclinic symmetry carbon allotropes and diamond, and the values are also listed in Table 1. The crystal density of the three monoclinic symmetry carbon allotropes is much more closed, and slightly smaller than that of diamond.

Fig. 1 The crystal structure of C2/m-16 carbon (a), C2/m-20 carbon (b), M-carbon (c), C2/m-16 carbon along the [010] direction (d), C2/m-20 carbon along the [010] direction (e) and M-carbon along the [010] direction (f). At zero pressure, C2/m-20 carbon contains five inequivalent crystallographic sites, occupying the 4i (0.07488, 0.0, −0.51860), 4i (−0.50848, 0.0, −0.33322), 4i (0.75150, 0.0, −0.04876), 4i (−0.99004, 0.0, −0.90598) and 4i (−0.36506, 0.0, −0.73151) positions, respectively.

Table 1 The calculated lattice constants (Å), β (°), and density (g cm⁻³) of the monoclinic symmetry carbon allotropes

	a	b	c	β	ρ
a Ref. 53.b Ref. 44.c Ref. 65 – experiment.
C2/m-20	9.011	2.519	8.707	143.458	3.389
	9.019^a	2.524	8.716	143.430
C2/m-16	4.721	5.046	7.154	146.109	3.363
	4.726^a	5.053	7.161	146.079
M	9.083	2.520	4.141	97.068	3.355
	9.089^b	2.496	4.104	96.960
Diamond	3.564				3.525
	3.567^c				3.516

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The bond lengths of the C–C bonds in C2/m-20 carbon are 1.497 Å, 1.516 Å, 1.522 Å, 1.531 Å, 1.539 Å, 1.554 Å, 1.559 Å, 1.621 Å and 1.627 Å, the average bond length is 1.552 Å. There are five different bond lengths (1.509 Å, 1.521 Å, 1.533 Å, 1.589 Å and 1.688 Å) in C2/m-16 carbon, and the average bond length is 1.567 Å, while, there are eight different bond lengths (1.499 Å, 1.520 Å, 1.528 Å, 1.534 Å, 1.542 Å, 1.555 Å, 1.570 Å and 1.627 Å) in M-carbon, and the average bond length is 1.551 Å. It is slightly greater than that of diamond (1.535 Å), and P222₁-carbon (1.412 Å)⁴¹ at ambient pressure. We calculated the hardness using Lyakhov and Oganov’s model,⁵¹ the calculated results are shown in Table 2. The average bond length of the C–C bonds in C2/m-16 carbon is the longest of the three monoclinic symmetry carbon allotropes, so its hardness is the smallest (59.5 GPa). While the mean bond length of the C–C bonds in M-carbon is slightly smaller than that of C2/m-20 carbon, but the hardness of M-carbon (66.6 GPa) is smaller than that of C2/m-20 carbon (70.6 GPa). The reason for this phenomenon may be that there are more C–C bonds in C2/m-20 carbon than in C2/m-16 carbon. Based on the elastic modulus and other related values, we also evaluated the hardness (H_v) using two other different empirical models: the Chen et al. model⁵² and Ma et al. model;⁵³ the calculated results of the Chen et al. model and Ma et al. model are also shown in Table 2. The results of Lyakhov and Oganov’s model are slightly smaller than those of the Chen et al. model and Ma et al. model. As in Lyakhov and Oganov’s model, the smallest hardness out of the monoclinic structures and diamond using the Ma et al. model is for C2/m-16 (84.9 GPa), and the greatest hardness is for diamond (89.5 GPa). For the Chen et al. model, among the C2/m-20, C2/m-16, and M phases and diamond, the greatest hardness is still for diamond (89.5 GPa), but the smallest hardness is for the C2/m-20 phase. The main reason for this situation is that an empirical formula may be used to estimate the value of a material’s hardness too high or too low. Most researchers agree on the definition that “superhard” materials are those with a H_v value exceeding 40 GPa.⁵⁴ Although there are slight differences between the results of the empirical models above, all of them far exceed 40 GPa, indicating that C2/m-20 is a superhard material.

Table 2 The elastic modulus (GPa) and hardness (GPa) of the monoclinic symmetry carbon allotropes

	B	G	E	v	HOganov	HMa	HChen
a Ref. 66 – experiment.b Ref. 51.
C2/m-20	412	463	1010	0.091	70.6	85.9	80.1
C2/m-16	398	458	993	0.084	59.5	84.9	81.9
M	398	456	990	0.086	66.6		81.3
Diamond	439	528	1131	0.071	89.7	89.5	94.2
	442^a				91.0^b

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To study the incompressibility, we calculated the dependence of the lattice parameters on pressure for the C2/m-16 carbon, M-carbon, C2/m-20 carbon and P222₁-carbon, the results of which are shown in Fig. 2. It is obvious that the P222₁-carbon presents excellent incompressibility along the a-direction as the pressure is increased up to 100 GPa, but slightly weaker incompressibility along the c-direction. While for the three novel monoclinic symmetry (C2/m-16 carbon, C2/m-20 carbon and M carbon) carbon allotropes, C2/m-16 carbon presents stronger incompressibility along the a-direction and c-direction, and C2/m-20 carbon presents excellent incompressibility along the b-direction. Fig. 2d illustrates the dependence of the cell volume V on pressure, from 0 to 100 GPa for C2/m-16 carbon, C2/m-20 carbon, M-carbon, P222₁ carbon, c-BN and diamond. Reasonably, diamond exhibits the strongest incompressibility on cell volume. Of the three novel monoclinic carbon allotropes, C2/m-20 carbon exhibits the greatest incompressibility. The thermodynamic stability of C2/m-20 carbon is examined by a direct enthalpy comparison with the known experimental carbon allotrope diamond and theoretical carbon allotropes, such as other monoclinic symmetry carbon allotropes (Fig. 3). The C2/m-20 phase has a high thermodynamic stability and is energetically more favorable than the C2/m-16 phase above ambient pressure (Fig. 3). Apart from the experimental diamond, the C2/m-20 phase is energetically the most favourable compared with the recently proposed monoclinic structures (C2/m-16 and M phases). To confirm the dynamic stability of the C2/m-20 phase, the calculated phonon spectra of the C2/m-20 phase at 0 GPa and 100 GPa are shown in Fig. 4. No imaginary frequencies were observed throughout the whole Brillouin zone, which gives direct proof of their dynamical stability at ambient pressure, and no imaginary frequency was observed throughout the entire Brillouin zone (see Fig. 4b), indicating its dynamic stability at 100 GPa.

Fig. 2 The lattice constants a/a₀ (a), b/b₀ (b), c/c₀ (c) and primitive cell volume V/V₀ (d) for compression as functions of pressure for C2/m-16 carbon, C2/m-20 carbon, P222₁-carbon and M-carbon, c-BN and diamond.

Fig. 3 Enthalpy of C2/m-20 carbon and other carbon allotropes relative to graphite as a function of pressures.

Fig. 4 Phonon spectra for C2/m-20 carbon at 0 GPa (a) and 100 GPa (b).

The elastic properties of the proposed structure for the C2/m-20 phase and the previously proposed structures of the C2/m-16 and M phases were investigated. The calculated results of the elastic moduli are listed in Table 2, and the elastic constants are listed in Table 3, and are compared to the hardest known material diamond. Young’s modulus E and Poisson’s ratio v are taken as: E = 9BG/(3B + G), and v = (3B − 2G)/(6B + 2G).^55,56 Firstly, the elastic constants of C2/m-20 carbon satisfied the mechanical stability criteria^57,58 and indicated that it is mechanically stable. Secondly, the elastic moduli of C2/m-20 carbon are the greatest of the three monoclinic symmetry carbon allotropes. So this also explains why C2/m-20 carbon has the greatest hardness of the three monoclinic symmetry carbon allotropes. For a monoclinic crystal, there are thirteen independent components, i.e. C₁₁, C₂₂, C₃₃, C₄₄, C₅₅, C₆₆, C₁₂, C₁₃, C₂₃, C₁₅, C₂₅, C₃₅ and C₄₆. The elastic constants and elastic moduli versus pressure for C2/m-20 carbon are shown in Fig. 5. All of the elastic constants C_ij increase at different rates under increasing pressure, except for C₃₅. C₁₁ of C2/m-20 carbon increases the fastest of all of the elastic constants, and C₁₅ grows the slowest.

Table 3 The calculated elastic constants (GPa) of the monoclinic symmetry carbon allotropes and diamond

	C2/m-16	C2/m-20	M	Diamond
a Ref. 59 – experiment.
C11	1022	1105	926	1055	1076^a
C22	1022	1249	1085
C33	1015	928	1046
C44	482	452	522	569	577
C55	482	398	453
C66	457	472	394
C12	78	38	43	119	125
C13	116	108	147
C23	72	80	80
C15	26	41	67
C25	54	16	−29
C35	−1	8	22
C46	71	60	−6

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Fig. 5 Elastic constants and elastic moduli of C2/m-20 carbon as a function of pressure.

It is well known that mechanical anisotropy is an important implication in engineering science and crystal physics. So we investigated the anisotropy in Young’s modulus using ELAM codes.^42,59 For all of the probable directions of C2/m-20 carbon, we made plane projections of the directional dependence of the Young’s modulus versus pressure, as shown in Fig. 6a. The black, red and blue lines represent the Young’s modulus at 0 GPa, 50 GPa and 100 GPa, respectively, and the solid line, dash line and dash dot line represent the Young’s modulus at the xy plane, xz plane and yz plane, respectively. The maximal values are 1241 GPa, 1413 GPa and 1655 GPa for the C2/m-modulus at the xy plane, xz plane and yz plane, respectively. The maximal values are 1241 GPa, 1413 GPa and 1655 GPa for C2/m-20 carbon at 0 GPa, 50 GPa and 100 GPa, respectively. The minimal values are 870 GPa, 1045 GPa and 1145 GPa for C2/m-20 carbon at 0 GPa, 50 GPa and 100 GPa, respectively. The maximal and minimal values for M-carbon are 1094 GPa and 849 GPa, and the maximal and minimal values for C2/m-16 carbon are 1102 GPa and 864 GPa, respectively. The E_max/E_min ratios for C2/m-16 carbon, M-carbon and C2/m-20 carbon are 1.275, 1.289 and 1.426 at 0 GPa, respectively. That is to say, C2/m-20 carbon shows the greatest anisotropy in Young’s modulus. As shown in Fig. 6a, as the pressure increases, the anisotropy decreases first, then the trend changes to an increase. In order to understand the trend in the pressure, the universal anisotropic index A^U, and the E_max/E_min ratio versus pressure are described in Fig. 6b. The A^U, and the E_max/E_min ratio changing trends are the same. At P = 10 GPa, they simultaneously arrived at the lowest point, and they exhibit a slow descending tendency between 40 and 50 GPa.

Fig. 6 2D representation of Young’s modulus for C2/m-20 carbon (a) (the black, red and blue lines represent Young’s modulus at 0 GPa, 50 GPa and 100 GPa, respectively and the solid line, dash line and dash dot line represent Young’s modulus at the xy plane, xz plane and yz plane, respectively). The universal anisotropic index A^U and E_max/E_min of C2/m-20 carbon as a function of pressure (b).

It is well known that electronic structure determines the fundamental physical and chemical properties of materials. Fig. 7 shows the band structure of C2/m-20 carbon at 0 GPa and 100 GPa. It can be clearly seen that this structure is a semiconductor with an indirect band gap of approximately 4.04 eV at 0 GPa and 4.52 eV at 100 GPa. It is known that a band gap calculated with DFT is usually underestimated by 30–50%; the true band gap must be larger than the calculated results. In consideration of this problem, we calculated the band gap using the HSE06 hybrid function.^60,61 The electronic band structure calculation utilizing the HSE06 hybrid function shows that the C2/m-20 carbon is an indirect band gap semiconductor with a band gap of 5.10 eV at 0 GPa and 5.72 eV at 100 GPa. The valence band maximum is located at the G point and the conduction band minimum is located at the Z point. The most extraordinary thing is that the band gap increases with the increasing pressure. This trend has been explained by Oliver Tschauner et al.⁶² using the Bader charge analysis and other articles.^42,63,64

Fig. 7 Electronic band structure for C2/m-20 carbon at 0 GPa (a) and 100 GPa (b).

Conclusions

This paper reports a detailed investigation of the structural, mechanical and electronic properties of a novel monoclinic symmetry phase C2/m-20 carbon using DFT within the ultrasoft pseudopotential scheme in the frame of GGA. We find that C2/m-20 carbon is mechanically stable satisfying the criteria set by Born for a monoclinic structure. In addition, phonon calculations reveal that C2/m-carbon is dynamically stable at 0 GPa and 100 GPa. The C2/m-20 carbon is the most favorable among the three monoclinic symmetry carbon allotropes (C2/m-20 carbon, C2/m-16 carbon and M-carbon). C2/m-20 carbon shows the greatest anisotropy in Young’s modulus among the three monoclinic symmetry carbon allotropes. Furthermore, in the three monoclinic symmetry carbon allotropes discussed in this article, C2/m-20 carbon has the greatest elastic modulus and hardness. Finally, the band structure calculation shows that C2/m-20 carbon is a wide and indirect band gap semiconductor with a band gap of 5.10 eV.

Acknowledgements

This work was supported by the Fund for Talents of Yunnan Province, China (Grant No. KKSY201403006) and the Natural Science Foundation of China (No. 61563005).

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