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Substrate-modulated morphology of graphene
We delineate a general theoretical framework to determine the substrate-regulated graphene morphology through energy minimization. We then apply such a framework to study the graphene morphology on a substrate with periodic surface grooves. Depending on the substrate surface roughness and the graphene–substrate interfacial bonding energy, the equilibrium morphology of graphene ranges from (1) closely conforming to the substrate, to (2) remaining flat on the substrate. Interestingly, in certain cases, the graphene morphology snaps between the above two limiting states. Our quantitative results envision a promising strategy to precisely control the graphene morphology over large areas. The rich features of the substrate-regulated graphene morphology (e.g. the snap-through instability) can potentially lead to new design concepts of functional graphene device components.
T. Li, Z. Zhang, Substrate-regulated morphology of graphene, Journal of Physics D: Applied Physics, 43, 075303 (2010).
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Teng, I quickly scanned
Teng, I quickly scanned through your paper and will read it more carefully over the next few days. Meanwhile, I have a question for you. In my quick reading, although you mention it in your leading paragaph, I did not see how you incoprorate thermal fluctuations? Do you expect your conclusions to change if you allow the sheet to fluctuate corresponding to the temperature rather than impose a pre-determined corrugation profile on the sheet?
Effect of thermal fluctuation
Pradeep,
Your question on the effect of thermal flutuation on graphene morphology parallels one of the reviewer's comments. In short, we expect such an effect to be minor. Detailed discussion in our response to that comment is included below. We later ran some MD simulations of a related configuration at room temperature, which yield similar morphology with thermal fluctuations of much smaller amplitude than that of the morphologic features. Hope this is helpful.
Response to the comment:
The snap-through instability is a thermodynamic transition where total free energy of the system minimizes. The robustness of the snap-through instability of graphene on substrates can be further justified by considering the suppression of intrinsic random thermal fluctuations in graphene by the substrate regulation. As estimated in Ref [1], the graphene-SiO2 interaction energy is ~6 meV/Å2, which significantly prevails the stored strain energy density in the graphene due to the random corrugation (~1 meV/Å2). The estimated graphene-substrate interaction energy is thus sufficient to overcome the energy cost of the corrugations needed to for graphene to follow the SiO2 morphology. This estimation is supported by quite a few direct experimental observations of the substrate-regulated graphene at atomic-resolution, as referred in the manuscript (Refs. [19-23] therein). Ref[2] reported the ultraflat graphene morphology on the atomically flat terraces of cleaved mica substrate surfaces. The high-resolution atomic force microscopy images in that paper offer strong evidence of the suppression of any existing intrinsic ripples in graphene. Since the snap-through instability of graphene on substrates originates from the substrate regulation on graphene morphology, the existing theoretical and experimental evidence described above justifies the robustness of such a morphologic instability of graphene on substrates.
Ref [1]: Ishigami M, Chen J H, Cullen W G, Fuhrer M S and Williams E D 2007 Nano letters 7 1643-8.
Ref [2]: Lui C H, Liu L, Mak K F, Flynn G W, Heinz T F 2009 Nature 462 339-41.