# High-Throughput Computation of New Carbon Allotropes with Diverse Hybridization and Ultrahigh Hardness

^{1}

^{2}

^{*}

**—**Nano-Theory)

## Abstract

**:**

^{3}C–C bonding. All structures were globally optimized at the first-principles level. The thermodynamic stability of some selected carbon allotropes was further validated by computing their phonon dispersions. The predicted carbon allotropes possess a broad range of Vickers’ hardness. This wide range of Vickers’ hardness is explained in detail in terms of both atomic descriptors such as density, volume per atom, packing fraction, and local potential energy throughout the unit cell, and global descriptors such as elastic modulus, shear modulus, and bulk modulus, universal anisotropy, Pugh’s ratio, and Poisson’s ratio. For the first time, we found strong correlation between Vickers’ hardness and average local potentials in the unit cell. This work provides deep insight into the identification of novel carbon materials with high Vickers’ hardness for modern applications in which ultrahigh hardness is desired. Moreover, the local potential averaged over the entire unit cell of an atomic structure, an easy-to-evaluate atomic descriptor, could serve as a new atomic descriptor for efficient screening of the mechanical properties of unexplored structures in future high-throughput computing and artificial-intelligence-accelerated materials discovery methods.

## 1. Introduction

## 2. Computational Procedure and Methods

- Step 1: Initially, the RG
^{2}code generated 1598 carbon allotropes in total with different hybridization states. - Step 2: We perform first-principles calculations to fully optimize those structures with low Monkhorst-pack k-mesh. The k-mesh in low-resolution DFT calculations depends on total number of atoms in the cell. Specifically, for numbers of atoms $<10$, $11\u201330$, $30\u201350$, and $>50$, the k-mesh was $8\times 8\times 8$, $4\times 4\times 4$, $2\times 2\times 2$, and $1\times 1\times 1$, respectively. After this step, we had 1576 carbon allotropes.
- Step 3: We continued to perform first-principles calculations to fully optimize the structures that were successfully optimized in the previous step, with high Monkhorst-pack k-mesh. The k-mesh in high-resolution DFT calculations depends on the length of lattice of the cell. Specifically, the product of the k-mesh in each direction and lattice size was approximately 60 Å. This was equivalent to the k-mesh of $16\times 16\times 16$ for diamond with an 8-atom conventional cell, which was high enough for global structure optimization. After this step, we had 1461 carbon allotropes.
- Step 4: After global structure optimization was finished, we cross-checked the 1461 carbon structures and also compared the structures with those downloaded from the SACADA database [29]. We found that some of the finally optimized structures had been already identified or reported in previous studies. After cross-checking and screening, we had 1105 new and unique carbon allotropes.
- Step 5: We finally calculated the elastic constants with conventional unit cells for all 1105 unique structures. Again, the k-mesh in each direction was determined by the same procedure as in Step 3. After this step, we successfully obtained the elastic constants of 1105 carbon allotropes.
- Step 6: Some structures had unreasonable universal anisotropy [32] so we decided to only report the carbon allotropes with universal anisotropy between 0 and 3. Finally, 904 carbon allotropes remained from all the screening processes.

^{3}, and ${\mathrm{sp}}^{2}/{\mathrm{sp}}^{3}$ mixture hybridizations were formed [28,33,34,35]. The input parameters of RG

^{2}mainly included the target space symmetry group(s), elements, number of inequivalent atoms in the unit cell, number of bonded atoms for each element, and bond feature information (e.g., bond angle, bond length, and the tolerance for their derivation). With these input parameters, the RG

^{2}package built the correct labeled quotient graph. In this article, different structures with different numbers of carbon structures with different hybridization states were arbitrarily distributed in a stochastic cell which had an arbitrary symmetry and lattice constant [27]. The number of symmetrically independent atoms usually ranged from 2 to 10, with majority between 3 and 7. A structure was initially generated with an equivalent number of carbon atoms based on symmetry. The code then computed the distance matrix of all the carbon atoms in that specifically generated structure and built the labeled quotient graph (LQG) based on ${\mathrm{sp}}^{2}$ or ${\mathrm{sp}}^{3}$ hybridizations. The generated structures with reasonable LQGs based on the bond lengths and angles could be relaxed by the code. The new structures produced by the code were named according to parameters used to predict the structures in the code, and those parameters (characters or numbers) used in the names were separated by a hyphen. The naming process took place from left to right as the following: space group number, number of nonequivalent atoms in the unit cells, element names (always C in this work), ring or loop structure in carbon local ID, and possibly one more hyphen to distinguish two IDs.

## 3. Results and Discussion

#### 3.1. Ground-State Energy and Thermodynamic Stability

#### 3.2. Pearson Correlation

## 4. Conclusions

^{3}. First-principles calculations were performed to optimize the structures and calculate Vickers’ hardness. Atomic descriptors such as packing fraction, density, and volume per atom along with mechanical properties such as bulk, shear, and elastic moduli; universal anisotropy; Poisson’s ratio; and Pugh’s ratio were utilized to explain the wide range of Vickers’ hardness results for the ductile and brittle carbon allotropes. The relationships between the average of local potential in the carbon allotrope unit cell and the anisotropy of a carbon allotrope with Vickers’ hardness were reported for the first time in this article, to the best of our knowledge. We believe that this work adds more insight and understanding to Vickers’ hardness in terms of finding various descriptors to explain Vickers’ hardness and accelerating the process of discovering unexplored novel superhard materials with captivating properties.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**2 × 2 unit cell structures of (

**a**) 135-3-40-C-r89-np-id355667 with pure ${\mathrm{sp}}^{2}$ hybridization, (

**b**) 136-3-20-C-r689-np-id545421_1 with hybrid ${\mathrm{sp}}^{2}/{\mathrm{sp}}^{3}$ hybridization, (

**c**) 179-3-36-C-r6789-p-id545421 with pure ${\mathrm{sp}}^{3}$ hybridization, and (

**d**) 165-3-28-C-r68-np-id545421 with pure ${\mathrm{sp}}^{3}$ hybridization.

**Figure 2.**Phonon dispersion of (

**a**) 135-3-40-C-r89-np-id355667, (

**b**) 136-3-20-C-r689-np-id545421_1, (

**c**) 179-3-36-C-r6789-p-id545421, (

**d**) 165-3-28-C-r68-np-id545421.

**Figure 3.**Pearson correlation matrix between Vickers’ hardness from Tian’s model ($V{H}_{Tian}$), universal anisotropy (AU), shear modulus ($G$), Pugh’s ratio ($k$), bulk modulus ($B$), elastic modulus ($E$), Poisson’s ratio ($\nu $), density (ρ), volume per atom (VPA), packing fraction (PF), and average local potential (Avg. LOCPOT).

**Figure 4.**2D map of bulk modulus vs. shear modulus, with Pugh’s ratio as color bar to explain hardness and show the relationship between all the mentioned properties.

**Figure 5.**2D map of bulk modulus vs. shear modulus with Poisson’s ratio as color bar to explain hardness and show the relationship between all the mentioned properties.

Materials | Hybridizations | Vickers’ Hardness, (GPa) | Ground-State Energy, (eV) | Average Local Potential, (eV) |
---|---|---|---|---|

(a) 135-3-40-C-r89-np-id355667 | ${\mathrm{sp}}^{2}$ | 30.88994 | −341.2203 | −11.7709 |

(b) 136-3-20-C-r689-np-id545421_1 | Hybrid ${\mathrm{sp}}^{2}$/${\mathrm{sp}}^{3}$ | 69.35087 | −175.5362 | −12.7828 |

(c) 179-3-36-C-r6789-p-id545421 | ${\mathrm{sp}}^{3}$ | 90.11029 | −320.6033 | −12.9213 |

(d) 165-3-28-C-r68-np-id545421 | ${\mathrm{sp}}^{3}$ | 90.10081 | −245.7932 | −13.1631 |

Materials | Vickers’ Hardness, (GPa) | Universal Anisotropy | Bulk Modulus, (GPa) | Elastic Modulus, (GPa) | Poisson’s Ratio | $\mathbf{Density},(\mathbf{kg}/{\mathbf{m}}^{3})$ | Volume Per Atom | Packing Fraction | Average Local Potential, (eV) |
---|---|---|---|---|---|---|---|---|---|

206-1-16-C-r0-np-id355667 | 104.302 | 0.00457 | 385.28 | 1060.73 | 0.04114 | 3.5534 | 5.61272 | 0.25595 | −13.467 |

181-1-6-C-r0-p-id224838_1 | 94.8507 | 0.04505 | 429.82 | 1108.01 | 0.07036 | 3.9125 | 5.71263 | 0.2515 | −13.185 |

154-1-6-C-r0-p-id224838 | 94.36159 | 0.044264 | 431.9735 | 1109.813 | 0.071805 | 3.496109 | 5.704695 | 0.251855 | −13.195 |

180-1-12-C-r0-p-id224838 | 94.16793 | 0.055942 | 428.2763 | 1102.033 | 0.071136 | 3.485896 | 5.72141 | 0.251119 | −13.169 |

182-1-12-C-r6x-p-id224838 | 94.13454 | 0.084798 | 433.4282 | 1111.486 | 0.072599 | 3.487636 | 5.718554 | 0.251244 | −13.169 |

Materials | Vickers’ Hardness, (GPa) | Universal Anisotropy | Bulk Modulus, (GPa) | Elastic Modulus, (GPa) | Poisson’s Ratio | $\mathbf{Density},(\mathbf{kg}/{\mathbf{m}}^{3})$ | Volume Per Atom | Packing Fraction | Average Local Potential (eV) |
---|---|---|---|---|---|---|---|---|---|

224-3-72-C-r69-np-id355667 | 2.99986 | 1.56873 | 99.07801 | 87.23116 | 0.353262 | 1.561686 | 12.77097 | 0.112502 | −8.092 |

131-2-48-C-r68x-np-id224838 | 3.082382 | 2.944065 | 170.7716 | 125.9887 | 0.37704 | 2.074342 | 9.61473 | 0.149433 | −9.773 |

207-3-72-C-r689-np-id224838 | 1.162879 | 1.085925 | 66.67588 | 42.13926 | 0.394666 | 1.276296 | 15.62666 | 0.091943 | −6.915 |

222-3-112-C-r6x-np-id355667 | 1.541133 | 1.974846 | 118.8571 | 70.42662 | 0.401245 | 1.477887 | 13.4951 | 0.106465 | −7.750 |

155-3-54-C-r6x-p-id355667 | 3.318981 | 2.33672 | 69.3944 | 72.49172 | 0.325894 | 1.541569 | 12.93762 | 0.111052 | −8.030 |

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**MDPI and ACS Style**

Al-Fahdi, M.; Rodriguez, A.; Ouyang, T.; Hu, M.
High-Throughput Computation of New Carbon Allotropes with Diverse Hybridization and Ultrahigh Hardness. *Crystals* **2021**, *11*, 783.
https://doi.org/10.3390/cryst11070783

**AMA Style**

Al-Fahdi M, Rodriguez A, Ouyang T, Hu M.
High-Throughput Computation of New Carbon Allotropes with Diverse Hybridization and Ultrahigh Hardness. *Crystals*. 2021; 11(7):783.
https://doi.org/10.3390/cryst11070783

**Chicago/Turabian Style**

Al-Fahdi, Mohammed, Alejandro Rodriguez, Tao Ouyang, and Ming Hu.
2021. "High-Throughput Computation of New Carbon Allotropes with Diverse Hybridization and Ultrahigh Hardness" *Crystals* 11, no. 7: 783.
https://doi.org/10.3390/cryst11070783