# Hexatetra-Carbon: A Novel Two-Dimensional Semiconductor Allotrope of Carbon

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{60}fullerene [8], nanotube [9], carbon nano-cone [10], and nanochain [11], 2D carbon mono-elemental monolayer materials beyond graphene have attracted significant attention from both theoretical and experimental fields of study. However, the electronic structure of graphene limits its application in designing electronic nano-devices due to its semi-metallic gapless nature. Therefore, finding new 2D mono-elemental monolayer materials with a semiconductor behavior is technologically important.

_{8}H

_{8}) is a synthetic hydrocarbon molecule formed by eight carbon atoms positioned at the corner of a cube. It is attached to its three neighboring carbon atoms and a hydrogen atom with tetragonal top and side views [20], 2D graphene, and 3D graphite, which are formed by carbon hexagons. In addition, penta-graphene consists entirely of pentagons of carbon atoms. Moreover, the C

_{60}molecule is formed by 12 pentagons, which are separated by 20 hexagons with a soccer ball shape [8].

## 2. Computational Methods

_{max}= 7, Gmax = 14 Ry

^{1/2}, and l

_{max}= 10. To avoid interlayer interactions, a large vacuum distance of 20 Å along the non-periodic direction was utilized. With regards to dynamic stability, an evaluation of all the calculations was conducted with the Quantum Espresso (QE) package [26]. Furthermore, the Martin–Troullier norm-conserving pseudopotential [27] was used to treat the core electrons, while the valence electronic wave functions were expanded using an energy cut-off of 80 Ry. However, for the investigation of structural properties of bilayer graphene, we considered the van der Waals correction in our calculations.

## 3. Structural Properties and the Stability of Hexatetra-Carbon Monolayer

^{−1}, which is higher than those obtained for silicon [29] (580 cm

^{−1}), MoS2 monolayer [30] (473 cm

^{−1}), and TiC monolayer [31] (810 cm

^{−1}). However, it is lower than the highest phonon frequency of graphene (about 1650 cm

^{−1}) and penta-graphene (about 1600 cm

^{−1}), indicating robust C–C bonds in the predicted monolayer.

_{0})/a

_{0}) and E is the total energy. In addition, V

_{0}is the equilibrium volume of the 2D material evaluated by ${V}_{0}=\frac{3\sqrt{3}{d}^{2}h}{2}$, where d is the adjacent carbon distance in the hexagon ring in the a and b plane, and h is the thickness of the 2D material along the c vector (Figure 1).

_{2}monolayer (265 ± 13 GPa), which are similar to the bilayer [37].

## 4. Electronic Properties

^{2}sigma covalent bonds (1.43 Å) between the C–C atoms in each monolayer of AA-stacked bilayer graphene, as well as a weak p

_{z}-p

_{z}interaction between the monolayers. In comparison, the in-plane bond length of C–C for hexatetra-carbon is 1.56 Å, which is longer than those obtained in AA-stacked bilayer graphene, i.e., the C–C in-plane orbital overlap decreases for the hexatetra-carbon (see Figure 5f). Therefore, to retain its structural stability, the hexatetra-carbon nanostructure compensates this orbital variation by creating interlayer sigma bonds between the neighboring carbons, which are located in the different planes. Moreover, these interlayer bonds would restrict p

_{z}, resulting in the semiconducting nature of 2D hexatetra-carbon.

## 5. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.; Zhang, Y.; Dubonos, S.V.; Grigorieva, I.V.; Firsov, A.A. Electric field effect in atomically thin carbon films. Science
**2004**, 306, 666–669. [Google Scholar] [CrossRef] [PubMed][Green Version] - Butler, S.Z.; Hollen, S.M.; Cao, L.; Cui, Y.; Gupta, J.A.; Gutiérrez, H.R.; Heinz, T.F.; Hong, S.S.; Huang, J.; Ismach, A.F.; et al. Progress, Challenges, and Opportunities in Two-Dimensional Materials Beyond Graphene. ACS Nano
**2013**, 7, 2898–2926. [Google Scholar] [CrossRef] [PubMed] - Molle, A.; Goldberger, J.; Houssa, M.; Xu, Y.; Zhang, S.-C.; Akinwande, D. Buckled two-dimensional Xene sheets. Nat. Mater.
**2017**, 16, 163–169. [Google Scholar] [CrossRef] [PubMed] - Guzmán-Verri, G.G.; Voon, L.C.L.Y. Electronic structure of silicon-based nanostructures. Phys. Rev. B
**2007**, 76, 075131. [Google Scholar] [CrossRef][Green Version] - Xu, M.; Liang, T.; Shi, M.; Chen, H. Graphene-Like Two-Dimensional Materials. Chem. Rev.
**2013**, 113, 3766–3798. [Google Scholar] [CrossRef] - Balendhran, S.; Walia, S.; Nili, H.; Sriram, S.; Bhaskaran, M. Elemental Analogues of Graphene: Silicene, Germanene, Stanene, and Phosphorene. Small
**2015**, 11, 640–652. [Google Scholar] [CrossRef] [PubMed] - Zhang, S.; Xie, M.; Li, F.; Yan, Z.; Li, Y.; Kan, E.; Liu, W.; Chen, Z.; Zeng, H. Semiconducting Group 15 Monolayers: A Broad Range of Band Gaps and High Carrier Mobilities. Angew. Chem. Int. Ed.
**2016**, 55, 1666–1669. [Google Scholar] [CrossRef] - Kroto, H.W.; Heath, J.R.; Brien, S.C.O.; Curl, R.F.; E Smalley, R. C60: Buckminsterfullerene. Nature
**1985**, 318, 162–163. [Google Scholar] [CrossRef] - Iijima, S.; Ichihashi, T. Single-shell carbon nanotubes of 1-nm diameter. Nature
**1993**, 363, 603–605. [Google Scholar] [CrossRef] - Charlier, J.-C.; Rignanese, G.-M. Electronic Structure of Carbon Nanocones. Phys. Rev. Lett.
**2001**, 86, 5970–5973. [Google Scholar] [CrossRef] - Jin, C.; Lan, H.; Peng, L.; Suenaga, K.; Iijima, S. Deriving Carbon Atomic Chains from Graphene. Phys. Rev. Lett.
**2009**, 102, 205501. [Google Scholar] [CrossRef] [PubMed][Green Version] - Li, Y.; Xu, L.; Liu, H.; Li, Y. Graphdiyne and graphyne: From theoretical predictions to practical construction. Chem. Soc. Rev.
**2014**, 43, 2572–2586. [Google Scholar] [CrossRef] [PubMed] - Zhang, S.; Zhou, J.; Wang, Q.; Chen, X.; Kawazoe, Y.; Jena, P. Penta-graphene: A new carbon allotrope. Proc. Natl. Acad. Sci. USA
**2015**, 112, 2372–2377. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wang, Z.; Zhou, X.; Zhang, X.; Zhu, Q.; Dong, H.; Zhao, M.; Oganov, A.R. Phagraphene: A Low-Energy Graphene Allotrope Composed of 5–6–7 Carbon Rings with Distorted Dirac Cones. Nano Lett.
**2015**, 15, 6182–6186. [Google Scholar] [CrossRef][Green Version] - Maruyama, M.; Okada, S. Two-Dimensional sp2Carbon Network of Fused Pentagons: All Carbon Ferromagnetic Sheet. Appl. Phys. Express
**2013**, 6, 095101. [Google Scholar] [CrossRef] - Terrones, H.; Hernández, E.; Grobert, N.; Charlier, J.-C.; Ajayan, P.M. New Metallic Allotropes of Planar and Tubular Carbon. Phys. Rev. Lett.
**2000**, 84, 1716–1719. [Google Scholar] [CrossRef][Green Version] - Jia, T.-T.; Fan, X.; Zheng, M.-M.; Chen, G. Silicene nanomeshes: Bandgap opening by bond symmetry breaking and uniaxial strain. Sci. Rep.
**2016**, 6, 20971. [Google Scholar] [CrossRef][Green Version] - Jia, T.-T.; Zheng, M.-M.; Fan, X.; Su, Y.; Li, S.-J.; Liu, H.-Y.; Chen, G.; Kawazoe, Y. Dirac cone move and bandgap on/off switching of graphene superlattice. Sci. Rep.
**2016**, 6, 18869. [Google Scholar] [CrossRef][Green Version] - Yang, D.-C.; Jia, R.; Wang, Y.; Kong, C.-P.; Wang, J.; Ma, Y.; Eglitis, R.I.; Zhang, H.-X. Novel Carbon Nanotubes Rolled from 6, 6, 12-Graphyne: Double Dirac Points in 1D Material. J. Phys. Chem. C
**2017**, 121, 14835–14844. [Google Scholar] [CrossRef] - Eaton, P.E.; Cole, T.W. Cubane. J. Am. Chem. Soc.
**1964**, 86, 3157–3158. [Google Scholar] [CrossRef] - Abt, R.; Draxl, C.; Knoll, P. Optical response of high temperature superconductors by full potential LAPW band structure calculations. Phys. B Condens. Matter
**1994**, 194–196, 1451–1452. [Google Scholar] [CrossRef] - Blaha, P.; Schwarz, K.; Tran, F.; Laskowski, R.; Madsen, G.K.H.; Marks, L.D. An Augmented Plane Wave+ Local Orbitals Program for Calculating Crystal Properties. J. Chem. Phys
**2020**, 152, 074101. [Google Scholar] [CrossRef] [PubMed] - Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett.
**1996**, 77, 3865–3868. [Google Scholar] [CrossRef] [PubMed][Green Version] - Tran, F.; Blaha, P. Implementation of screened hybrid functionals based on the Yukava potential within the LAPW basis set. Phys. Rev. B
**2011**, 83, 235118. [Google Scholar] [CrossRef][Green Version] - Monkhorst, H.J.; Pack, J.D. Special points for Brillouin-zone integrations. Phys. Rev. B
**1976**, 13, 5188–5192. [Google Scholar] [CrossRef] - Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G.L.; Cococcioni, M.; Dabo, I.; et al. Quantum espresso: A modular and open-source software project for quantum simulations of materials. J. Phys. Condens. Matter
**2009**, 21, 395502. [Google Scholar] [CrossRef] - Troullier, N.; Martins, J.L. Efficient pseudopotentials for plane-wave calculations. Phys. Rev. B
**1991**, 43, 1993–2006. [Google Scholar] [CrossRef] - Alborznia, H.; Naseri, M.; Fatahi, N. Pressure effects on the optical and electronic aspects of T-Carbon: A first principles calculation. Optik
**2019**, 180, 125–133. [Google Scholar] [CrossRef] - Jahangirov, S.; Topsakal, M.; Akturk, E.; Sahin, H.; Ciraci, S. Two- and One-Dimensional Honeycomb Structures of Silicon and Germanium. Phys. Rev. Lett.
**2009**, 102, 236804. [Google Scholar] [CrossRef][Green Version] - Ganatra, R.; Zhang, Q. Few-Layer MoS2: A Promising Layered Semiconductor. ACS Nano
**2014**, 8, 4074–4099. [Google Scholar] [CrossRef] - Li, H.; Zhang, L.; Zeng, Q.; Guan, K.; Li, K.; Ren, H.; Liu, S.; Cheng, L. Structural, elastic and electronic properties of transition metal carbides TMC (TM=Ti, Zr, Hf and Ta) from first-principles calculations. Solid State Commun.
**2011**, 151, 602–606. [Google Scholar] [CrossRef] - Lourenço, M.P.; De Oliveira, C.; Oliveira, A.; Guimaraes, L.; Duarte, H. Structural, Electronic, and Mechanical Properties of Single-Walled Chrysotile Nanotube Models. J. Phys. Chem. C
**2012**, 116, 9405–9411. [Google Scholar] [CrossRef] - Lee, J.-U.; Yoon, D.; Cheong, H. Estimation of Young’s Modulus of Graphene by Raman Spectroscopy. Nano Lett.
**2012**, 12, 4444–4448. [Google Scholar] [CrossRef] [PubMed][Green Version] - Li, Y.; Fleischer, C.M.; Ross, A.E. High Young’s modulus carbon fibers are fouling resistant with fast-scan cyclic voltammetry. Chem. Commun.
**2020**, 56, 8023–8026. [Google Scholar] [CrossRef] [PubMed] - Lourenço, M.P.; Guimarães, L.; da Silva, M.C.; de Oliveira, C.; Heine, T.; Duarte, H.A. Nanotubes With Well-Defined Structure: Single- and Double-Walled Imogolites. J. Phys. Chem. C
**2014**, 118, 5945–5953. [Google Scholar] [CrossRef] - Köhler, T.; Frauenheim, T.; Hajnal, Z.; Seifert, G. Tubular structures of GaS. Phys. Rev. B
**2004**, 69, 193403. [Google Scholar] [CrossRef][Green Version] - Li, Y.; Yu, C.; Gan, Y.; Jiang, P.; Yu, J.; Ou, Y.; Zou, D.-F.; Huang, C.; Wang, J.; Jia, T.; et al. Mapping the elastic properties of two-dimensional MoS2 via bimodal atomic force microscopy and finite element simulation. NPJ Comput. Mater.
**2018**, 4, 49. [Google Scholar] [CrossRef]

**Figure 4.**Band structures of (

**a**) graphene, (

**b**) bilayer graphene, (

**c**) penta-graphene, and (

**d**) hexatetra-carbon.

**Figure 5.**Crystal structure of (

**a**) AA-stacked bilayer graphene, (

**b**) cubane molecule, and (

**c**) hexatetra-carbon (

**d**,

**e**). Top and side views of valence charge density distribution for AA-stacked graphene and (

**f**,

**g**) for hexatetra-carbon obtained by WIEN2K code [21].

**Figure 6.**Cohesive energy of AA-stacked bilayer graphene and hexatetra-carbon versus interlayer distance calculated by Quantum Espresso (due to the interruption in accessing WIEN2k while following up on a reviewer’s comment, we have used Quantum Espresso for Figure 6).

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**MDPI and ACS Style**

Naseri, M.; Jalilian, J.; Salahub, D.R.; Lourenço, M.P.; Rezaei, G. Hexatetra-Carbon: A Novel Two-Dimensional Semiconductor Allotrope of Carbon. *Computation* **2022**, *10*, 19.
https://doi.org/10.3390/computation10020019

**AMA Style**

Naseri M, Jalilian J, Salahub DR, Lourenço MP, Rezaei G. Hexatetra-Carbon: A Novel Two-Dimensional Semiconductor Allotrope of Carbon. *Computation*. 2022; 10(2):19.
https://doi.org/10.3390/computation10020019

**Chicago/Turabian Style**

Naseri, Mosayeb, Jaafar Jalilian, Dennis R. Salahub, Maicon Pierre Lourenço, and Ghasem Rezaei. 2022. "Hexatetra-Carbon: A Novel Two-Dimensional Semiconductor Allotrope of Carbon" *Computation* 10, no. 2: 19.
https://doi.org/10.3390/computation10020019