Ab initio structure determination of n-diamond

A systematic computational study on the crystal structure of n-diamond has been performed using first-principle methods. A novel carbon allotrope with hexagonal symmetry R32 space group has been predicted. We name it as HR-carbon. HR-carbon composed of lonsdaleite layers and unique C3 isosceles triangle rings, is stable over graphite phase above 14.2 GPa. The simulated x-ray diffraction pattern, Raman, and energy-loss near-edge spectrum can match the experimental results very well, indicating that HR-carbon is a likely candidate structure for n-diamond. HR-carbon has an incompressible atomic arrangement because of unique C3 isosceles triangle rings. The hardness and bulk modulus of HR-carbon are calculated to be 80 GPa and 427 GPa, respectively, which are comparable to those of diamond. C3 isosceles triangle rings are very important for the stability and hardness of HR-carbon.

Scientific RepoRts | 5:13447 | DOi: 10.1038/srep13447 favorable than the other candidates. The simulated XRD patterns, Raman and ELNES spectra can match the experimental data. We call this novel hexagonal carbon allotrope as HR-carbon which has an incompressible atomic arrangement due to unique C 3 isosceles triangle rings. It is stable in the range of 0 GPa up to 50 GPa at least. C 3 isosceles triangle rings of HR-carbon are critical for the stability and hardness of HR-carbon.

Results and Discussion
A novel candidate structure for n-diamond, named it as HR-carbon, has been predicted by using USPEX method [50][51][52] with 15 carbon atoms in the simulation cell. HR-carbon has hexagonal lattice R32 symmetry as depicted in Fig. 1(a). It is made of exclusively three-dimensional sp 3  There are 45 atoms in the unit cell. In rhombohedral representation, it has a 15 atoms rhombohedral unit cell. The crystal structure of HR-carbon is distinct from that of diamond. The HR-carbon can be regarded as a modulated graphite phase composed of lonsdaleite layers and C 3 isosceles triangle rings layers with stacking sequence of ABCABCABC…along the crystallographic c axis of hexagonal lattice. The close-packed A and B layers are lonsdaleite layers. The C layers composed of isolated C 3 isosceles triangle rings are sandwiched between two AB layers with unique twisted sp 3 covalent bonds with bond angle 60°, 107.15°, and 140.04° which are much different from those of standard sp 3 covalent bonds of diamond (bond angle 109.47°). This finding suggests a novel combination form of carbon atoms to construct post graphite phases. The structural type of HR-carbon is also consistent with previous theoretical suggestion that n-diamond should have cubic or rhombohedral space group 46 . Enthalpy calculations suggest that HR-carbon is much stable than the previously proposed candidates of n-diamond Fig. 1(b) 47 . And HR-carbon becomes stable relative to graphite at above 14.2 GPa which is almost equal to the phase transition pressure of cold-compressed graphite phase (14 GPa) 3 . Its bulk modulus and shear modulus are 428 and 471 GPa, respectively. Furthermore, no imaginary frequencies are observed throughout the whole Brillouin zone in phonon dispersion Fig. 2(b), confirming dynamically structural stability of HR-carbon. The highest phonon frequency of HR-carbon (41.5 THz) is very close to that of diamond (40 THz) 53 , which reflects the diamond-like structural bonding character of HR-carbon. The calculated electronic band structure and density of state (DOS) at ambient pressure reveal that the HR-carbon is semiconductor with direct band gap 4.6 eV Fig. 2(a). Strong hybridization between s and p orbitals of DOS indicates the presence of strong covalent bonds in HR-carbon. The HR-carbon can be expected to have good mechanical properties among compressed graphite phases.
To confirm the consistency of HR-carbon and n-diamond, we simulate the XRD pattern of HR-carbon and compare with the experimental results Fig. 3 27,49 . Our simulated XRD patterns for HR-carbon, reactants, and products are consistent with experimental XRD patterns. Three individual peaks of diamond can be easily indexed in the experimental XRD pattern Fig. 3(a). And the peaks of HR-carbon at ~46° and 51° can explain the experimentally observed forbidden diamond peaks. The relatively weaker peaks of HR-carbon in the range of 54°-95° have merged into the background. While the peaks at ~42°, 76° and 93° contribute to the peaks broadening of diamond at ~45°, 75° and 90°. The peaks of HR-carbon in the range of 20°-40° have been covered by the peaks of amorphous carbon (a-C). Furthermore, we also compare our simulated XRD pattern with that of n-diamond synthesized by Fe-catalysed carbon black methods 50 . The peaks of diamond, graphite, NaCl, and Fe have been indexed in the experimental XRD pattern Fig. 3(b). Four characteristic peaks of experimental XRD at 50°, 54°, 77°, and 84° can be explained very well by the peaks of HR-carbon in the same range. The peaks of HR-carbon at ~24° and 26°, ~33°, and ~45° contribute the peak broadening of graphite (~26°), NaCl (~32°) and Fe (~45°), respectively. The peaks of HR-carbon at ~42° can explain the experimentally observed shoulder peak at ~43°. The other peaks of HR-carbon have merged to the background. It is noted that the  other candidate structures can also simulate the experimental XRD patterns. However, they are all thermodynamically unstable with respect to HR-carbon. We also compare the XRD pattern of HR-carbon with those of previously proposed post-graphite phases as shown in Fig. 4. Nineteen inequivalent structures (M-carbon 7 , W-carbon 11 , Bct-C 4 8 , X-carbon 18 , O-carbon 21 (H-carbon 20 , R-carbon 19 ), C-carbon 16 (S-carbon 20 ), F-carbon 12 (J-carbon 22 ), T12-carbon 13 , Z-carbon 15 (Cco-C8 9 , oC16-II 10 ), P-carbon 19 (Z4-A3B1 14 ), oC32-carbon 23 , M585-carbon 25 , R3-carbon 24 , mP16-carbon 6 , mS32-carbon 6 , oP20-carbon 6 , oP24-I-carbon 6 , oP24-II-carbon 6 and oP28-carbon 6 ) have been considered. It can be found that only the XRD of HR-carbon at ~45°, ~50°, ~54°, ~76° and ~84° can explain the experimentally observed characteristic peaks of n-diamond (45°, 51°, 55°, 78° and 84°) 27,49 . So HR-carbon is the most possible candidate structure of n-diamond.
To confirm reliability of our structure, the ELNES and Raman of HR-carbon was calculated and compared with the experimental results 32,38 . The ELNES spectra are very useful to distinguish the various hybridized covalent bonds in carbon materials. Strikingly, the XRD results indicate a coexistence of n-diamond, amorphous carbon, diamond and some other carbon allotropes, consistent with Konyashin's conclusion that the experimental sample is a mixture of n-diamond and other carbon allotropes 44 . Obvious π * feature of experimental ELNES indicates the presence of sp 2 -hybridized covalent bonds. However, no π * features are found in the ELNES of diamond and HR-carbon, indicating the sample should contain graphite or a-C because those are composed of sp 2 -hybridized covalent bonds. Moreover, the a-C was already detected by XRD because of the presence of broad peak in range of 20-30° Fig. 3(a). In order to compare to experimental data, we calculate the fitting average values (FAV) of HR-carbon and possible carbon allotropes (diamond, graphite and amorphous carbon) (Fig. 5). It can be found that the FAV of HR-carbon and diamond cannot match the experimental ELNES. The FAV of HR-carbon and graphite/a-C are very similar to each other. The two main differences between those two spectra are found in range of 280-290 eV and 315-330 eV. Obvious π * feature can be found in the range of 280-290 eV in those two situation. However, the intensity of π * peak of FAV of HR-carbon and a-C is stronger than that of π * peak of FAV of HR-carbon and graphite and can match the experimental data very well. But only the peaks of FAV of HR-carbon and graphite in the range of 315-330 eV can match the experimental third broad peak. It is noteworthy that the presence of graphite was not reported in Li's experimental XRD pattern 27 because the main peaks of graphite and amorphous carbon merge together in the XRD pattern and some weak peaks have been covered by the background. Moreover, only the valley of ELNES of HR-carbon in range of 305-310 eV can explain the experimental valley between the second and third board peaks at 310 eV very well. This valley is an important key to confirm the existence of HR-carbon. So the ELNES of HR-carbon together with those of amorphous carbon, diamond, and graphite can explain the experimental ELNES very well. The comparison of theoretical and experimental Raman spectra is shown in Fig. 6. Previous experiments confirmed that the experimental sample is a mixture of n-diamond and other carbon allotropes. So it is difficult to distinguish the Raman bands  of n-diamond from the experimental Raman spectrum. Because the diamond-like carbon allotropes have characteristic peaks in the range of ~1300-1400 cm −1 which are located in the range of Raman bands of diamond 54 . However, the HR-carbon has three independent characteristic peaks at ~1300 cm −1 which can match the experimentally observed three peaks at about 1350 cm −1 . These also confirm that HR-carbon is the most possible candidate structure of n-diamond.
While the graphite transforms to sp 3 -hybridized carbon allotropes such as diamond and coldcompressed graphite phase, the hardness of materials enhances much more. So it is worth expecting that HR-carbon has high hardness among compressed graphite phases. The calculated bulk modulus (B 0 ) of HR-carbonis 427 GPa which is very close to that of diamond (466 GPa). The Vickers hardness H v of HR-carbon (80 GPa), estimated by the Chen's hardness model 55 , is much larger than that of c-BN (~62 GPa). However, it is smaller than that of diamond (~95 GPa). HR-carbon has very similar atomic arrangement to that of diamond except the C 3 isosceles triangle rings. So the hardness gap of HR-carbon and diamond could be attributed to the unique C 3 isosceles triangle rings. To confirm the effect of C 3 isosceles triangle rings for structural stability, we remove the C 3 isosceles triangle rings from the crystal structure of HR-carbon. This new structure was named as modulated HR-carbon (see supplementary Figure S1 and Table S1). We check the mechanical and dynamical properties of modulated HR-carbon. The bulk modulus of modulated HR-carbon is only 344 GPa which is much smaller than that of HR-carbon. So due to unique C 3 isosceles triangle rings in the C layer of HR-carbon, the HR-carbon has stronger incompressibility. We also calculated the phonon dispersion of modulated HR-carbon to compare with that of HR-carbon Fig. 2(b). The red dashed line of the phonon spectrum is the imaginary phonon mode of modulated HR-carbon. The imaginary frequencies at H and Γ points indicate that the C 3 isosceles triangle rings are very important for the dynamic stability of HR-carbon (see Supplementary Figure S1).
In an effort to assess the effect of C 3 isosceles triangle rings for the hardness of HR-carbon, we calculated the Mulliken overlap population (MOP) and bond length which can give a quantitative description for the covalent bonds of HR-carbon as summarized in Table I. The average bond length and MOP of HR-carbon is 1.53 Å and 0.81, respectively, which is comparable with those of diamond (1.531 Å, 0.75). It is noteworthy that the bond length of C 3 isosceles triangle rings is 1.53 Å which is equal to the average bond length. However, the MOP of C 3 isosceles triangle rings is only 0.61 which is much smaller than the average MOP of HR-carbon. So the bond strength of C 3 isosceles triangle rings is weaker than that of the other bonds in HR-carbon. The bond strength of C 3 isosceles triangle rings is not the mainly reasons for the high hardness of HR-carbon. However, it can be found that the bond3 has the smallest bond length (1.508 Å) and largest MOP (0.88), indicating the bond strength of bond3 is the strongest one in HR-carbon. It makes the C 3 isosceles triangle ring stable between Layer A and B and limits the mobility of Layer A and B along a and b directions, resulting in high hardness of HR-carbon.

Conclusion
In conclusion, we have found a novel diamond-like carbon allotrope, HR-carbon, which is a likely candidate structure for n-diamond. The simulated XRD, Raman and ELNES of HR-carbon can reproduce the experimental results. HR-carbon is a semiconductor with an incompressible atomic arrangement. It is stable over graphite above 14.2 GPa. This allotrope possesses high hardness (80 GPa) and high bulk modulus (427 GPa), which are comparable to those of diamond. Unique C 3 isosceles triangle rings are very important for the stability and hardness of HR-carbon.

Methods
Using the USPEX method, the variable-cell structure prediction of n-diamond was performed at 10, 20, 30, 50 and 100 GPa, respectively [50][51][52] . The simulation cell containing 2 ~ 20 carbon atoms were selected. The structural relaxation were performed within the density functional theory, carried out within the Vienna ab initio simulation package (VASP) 56,57 . The projector augmented wave method was used 58 . The 2s 2 2p 2 electrons are treated as valence electrons. The local density approximation (LDA) was employed 59 . The tested plane-wave cutoff energy was taken as 1100 eV. A Gamma-point-centered k-mesh of 7 × 7 × 2 k-point sampling was used for the calculations. The geometries were optimized when the remanent Hellmann-Feynman forces on the ions are less than 0.01 eV/Å. The ELNES, Raman and Mulliken population calculations were performed by CASTEP code 60

Table 1. The bond length (d) and Mulliken overlap population (MOP) of bonds in HR-carbon.
the supercell-core-excited approach. A supercell containing 64 atoms was used to avoid the interaction of adjacent core holes. The phonon spectra were calculated using the direct supercell method, performed by PHONON software.