Journal
NANO LETTERS
Volume 13, Issue 11, Pages 5692-5697Publisher
AMER CHEMICAL SOC
DOI: 10.1021/nl4033928
Keywords
Graphene; phase field modeling; epitaxial growth; anisotropic diffusion
Categories
Funding
- Balsells Foundation
- NSF Division of Mathematical Sciences [DMS-1217303]
- Royal Society University Research Fellowship by the UK Royal Society
- ARO [W911NF-11-1-0171]
- NSF [CMMI 1308396, DMS 1306179]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1216801] Funding Source: National Science Foundation
- Directorate For Engineering
- Div Of Civil, Mechanical, & Manufact Inn [1308396] Funding Source: National Science Foundation
- Engineering and Physical Sciences Research Council [EP/L003481/1] Funding Source: researchfish
- EPSRC [EP/L003481/1] Funding Source: UKRI
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The epitaxial growth of graphene on copper foils is a complex process, influenced by thermodynamic, kinetic, and growth parameters, often leading to diverse island shapes including dendrites, squares, stars, hexagons, butterflies, and lobes. Here, we introduce a phase-field model that provides a unified description of these diverse growth morphologies and compare the model results with new experiments. Our model explicitly accounts for the anisotropies in the energies of growing graphene edges, kinetics of attachment of carbon at the edges, and the crystallinity of the underlying copper substrate (through anisotropy in surface diffusion). We show that anisotropic diffusion has a very important, counterintuitive role in the determination of the shape of islands, and we present a phase diagram of growth shapes as a function of growth rate for different copper facets. Our results are shown to be in excellent agreement with growth shapes observed for high symmetry facets such as (111) and (001) as well as for high-index surfaces such as (221) and (310).
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