# Analytical Model of CVD Growth of Graphene on Cu(111) Surface

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## Abstract

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## 1. Introduction

- The surface-catalyzed pyrolysis of hydrocarbons that lead to numerous carbon species on the surface;
- The aggregation and nucleation of these species on the metal surface, which produces graphene nuclei;
- The growth of the nuclei due to the attachment of feeding species;
- The coalescence of the flakes when high coverage is reached.

- Simulations are performed in the range of up to several hundred nanometers, whereas the typical sizes of the nuclei observed in experiments are in the micrometer scale [30];
- kMC calculations are usually limited to the steady growth of a single graphene nucleus, so there is no information on the nucleation step included. Consequently, no size distribution or nuclei density can be obtained, which are known to be crucial characteristics of CVD graphene;
- Simulations on the hexagonal kMC lattice are complicated by the formation of the pentagon edge (the pairwise closing of the Klein edge of graphene on the Cu(111) surface) and its opening by the next attaching particle.

## 2. Theory and Computational Details

- $\mathrm{C}$ and ${\mathrm{C}}_{2}$ adparticles which have high mobility on the surface and play the role of feeding species for nucleation and growth;
- Small ${\mathrm{C}}_{\mathrm{n}}$ clusters of various geometries which cannot yet be associated with the solid crystalline graphene phase;
- Well-defined graphene nuclei which are immobile on the surface and grow due to the attachments of $\mathrm{C}$ and ${\mathrm{C}}_{2}$ particles.

#### 2.1. Growth Rate

#### 2.2. Kinetic Model of Growth

#### 2.3. Nucleation Rate

#### 2.4. Details of DFT Calculations

## 3. Results and Discussion

#### 3.1. DFT Calculations

#### 3.2. Analytical Kinetic Model

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CVD | chemical vapor deposition |

BEEF–vdW | Bayesian error estimation functional with van der Waals correlation |

DFT | density functional theory |

D2 / D3 | D2 / D3 dispersion correction of Grimme |

kMC | kinetic Monte Carlo |

ML | monolayer (unit of the surface concentration) |

PBE | Perdew–Burke–Ernzerhof functional |

## Appendix A. Expressions for the Rate Constants

## References

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**Figure 1.**Schematic illustration of the model discussed in the paper. Top view on the surface of Cu(111) with graphene edge. ${\mathrm{CH}}_{4\left(\mathrm{g}\right)}$ corresponds to gas-phase methane, while all species labeled with * are adsorbed on the surface. Methane dissociation and all following processes are taking place on the surface and are surface-catalyzed.

**Figure 2.**(

**a**) Depiction of graphene ribbons in the simulation cell used in this work. (

**b**) Depiction of the smaller model used in [19].

**Figure 3.**Attachment barrier of ${\mathrm{C}}_{2}$ to the graphene zigzag edges on the Cu(111) surface calculated with respect to two individualized species. All structures involved are shown. The barrier is calculated in the following manner: ${E}_{\mathrm{Barrier}}={E}_{\mathrm{TS}}-{E}_{{\mathrm{C}}_{2}}-{E}_{\mathrm{ribbon}}+{E}_{\mathrm{surface}}$.

**Figure 4.**(

**a**) Effective oversaturation, which depends simultaneously on oversaturation/ concentration of $\mathrm{C}$ and ${\mathrm{C}}_{2}$ (see Equation (7)), as a function of time. It reaches zero at 0.615 seconds, which corresponds to the duration of the stage when nucleation is not possible. Starting from 0.615 seconds, the solid phase becomes stable against the lattice gas of adparticles. (

**b**) Nucleation rate as a function of time. Note different time scales—the major amount of nuclei are formed in a short period of time—from 0.95 seconds to 1.1 seconds, when the effective oversaturation reaches its maximum values. (

**c**) ${\mathrm{C}}_{2}$ concentration profile. One can see that the effective oversaturation reaches zero, when ${\mathrm{C}}_{2}$ concentration is orders of magnitude higher than the equilibrium one ${c}_{eq}\left({\mathrm{C}}_{2}\right)$ (which is around $5\times {10}^{-5}$ ML (monolayer), Table S2). (

**d**) Size distribution of graphene flakes built at 1.5 seconds. Note that the shape and width of this curve will not change after the end of active nucleation; just the absolute size of the particles will increase.

Reaction | BEEF-vdW | PBE-D3 | Ref. [19] |
---|---|---|---|

${\mathrm{CH}}_{4\left(\mathrm{g}\right)}\to {\mathrm{C}}^{*}+2{{\mathrm{H}}_{2\left(\mathrm{g}\right)}}^{a}$ | 1.94 | 1.48 | 1.63 |

${\mathrm{C}}^{*}+{\mathrm{C}}^{*}\to {\mathrm{C}}_{2}^{*}$ | 0.25 | 0.55 | 0.25 |

${\mathrm{C}}_{2}^{*}\to {\mathrm{C}}^{*}+{\mathrm{C}}^{*}$ | 3.97 | 3.62 | 2.75 |

${\mathrm{C}}^{*}+\mathrm{edge}\to \mathrm{edge}-\mathrm{C}$ | 0.57 | 0.68 | 1.27 |

$\mathrm{edge}-\mathrm{C}\to {\mathrm{C}}^{*}+\mathrm{edge}$ | 1.76 | 1.36 | 1.57 |

${\mathrm{C}}_{2}^{*}+\mathrm{edge}\to \mathrm{edge}-{\mathrm{C}}_{2}$ | 1.21 | 1.32 | 0.58 |

$\mathrm{edge}-{\mathrm{C}}_{2}\to {\mathrm{C}}_{2}^{*}+\mathrm{edge}$ | 2.31 | 2.30 | 2.19 |

^{a}Barriers are given for the reaction ${\mathrm{CH}}_{4\left(\mathrm{g}\right)}\to {\mathrm{CH}}_{3}^{*}+{\mathrm{H}}^{*}$ because it determines the ${\mathrm{C}}^{*}$ production rate J. All energies in the table are given as enthalpies. Graphical representation of Gibbs free energies on the path ${\mathrm{CH}}_{4}^{*}\to {\mathrm{C}}^{*}+2{\mathrm{H}}_{2\left(\mathrm{g}\right)}$ can be found in Figure S2.

**Table 2.**${\mathrm{C}}_{2}$ attachment/detachment barriers in eV for different ribbon widths calculated using the BEEF-vdW functional.

Ribbon Width | Attachment Barrier (eV) | Detachment Barrier (eV) |
---|---|---|

two-ring | 1.21 | 2.40 |

three-ring | 1.32 | 2.38 |

four-ring | 1.25 | 2.28 |

five-ring | 1.21 | 2.31 |

**Table 3.**Values of kinetic parameters calculated with BEEF-vdW barriers for $T=1300\phantom{\rule{0.222222em}{0ex}}\mathrm{K}$ and $p\left(\right)open="("\; close=")">{\mathrm{CH}}_{4}$. Parameters A and U are fitted from experimental data on size distribution Ref. [30].

Kinetic Parameter | Units | Value | Physical Meaning |
---|---|---|---|

J | $\left(\right)$ | $7.32\times {10}^{17}$ | Rate of $\mathrm{C}$ production |

${k}_{d}$ | $\left(\right)$ | $2.71\times {10}^{-8}$ | Rate constant of $\mathrm{C}$ dimerization |

${k}_{at}\left(\mathrm{C}\right)$ | $\left(\right)$ | $6.94\times {10}^{-10}$ | Rate constant of $\mathrm{C}$ attachment to a flake |

${k}_{det}\left(\mathrm{C}\right)$ | $\left(\right)$ | $7.26\times {10}^{5}$ | Rate constant of $\mathrm{C}$ detachment from a flake |

${k}_{at}\left(\right)open="("\; close=")">{\mathrm{C}}_{2}$ | $\left(\right)$ | $1.62\times {10}^{-12}$ | Rate constant of ${\mathrm{C}}_{2}$ attachment to a flake |

${k}_{det}\left(\right)open="("\; close=")">{\mathrm{C}}_{2}$ | $\left(\right)$ | $3.09\times {10}^{3}$ | Rate constant of ${\mathrm{C}}_{2}$ detachment from a flake |

A | $\left(\right)$ | $2.0\times {10}^{22}$ | Rate of nucleation assuming zero nucleation barrier |

U | - | 15 | Value of $U/ln(\xi +1)$ - defines nucleation barrier for the given effective oversaturation $\xi $ |

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**MDPI and ACS Style**

Popov, I.; Bügel, P.; Kozlowska, M.; Fink, K.; Studt, F.; Sharapa, D.I.
Analytical Model of CVD Growth of Graphene on Cu(111) Surface. *Nanomaterials* **2022**, *12*, 2963.
https://doi.org/10.3390/nano12172963

**AMA Style**

Popov I, Bügel P, Kozlowska M, Fink K, Studt F, Sharapa DI.
Analytical Model of CVD Growth of Graphene on Cu(111) Surface. *Nanomaterials*. 2022; 12(17):2963.
https://doi.org/10.3390/nano12172963

**Chicago/Turabian Style**

Popov, Ilya, Patrick Bügel, Mariana Kozlowska, Karin Fink, Felix Studt, and Dmitry I. Sharapa.
2022. "Analytical Model of CVD Growth of Graphene on Cu(111) Surface" *Nanomaterials* 12, no. 17: 2963.
https://doi.org/10.3390/nano12172963