The stability of two-dimensional (2D) layers and membranes is the subject of a long-standing theoretical debate. According to the so-called Mermin–Wagner theorem1, long-wavelength fluctuations destroy the long-range order of 2D crystals. Similarly, 2D membranes embedded in a 3D space have a tendency to be crumpled2. These fluctuations can, however, be suppressed by anharmonic coupling between bending and stretching modes meaning that a 2D membrane can exist but will exhibit strong height fluctuations2,3,4. The discovery of graphene, the first truly 2D crystal5,6, and the recent experimental observation of ripples in suspended graphene7 make these issues especially important. Besides the academic interest, understanding the mechanisms of the stability of graphene is crucial for understanding electronic transport in this material that is attracting so much interest owing to its unusual Dirac spectrum and electronic properties8,9,10,11. We address the nature of these height fluctuations by means of atomistic Monte Carlo simulations based on a very accurate many-body interatomic potential for carbon12. We find that ripples spontaneously appear owing to thermal fluctuations with a size distribution peaked around 80 Å which is compatible with experimental findings7 (50–100 Å). This unexpected result might be due to the multiplicity of chemical bonding in carbon.
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Mermin, N. D. Crystalline order in two dimensions. Phys. Rev. 176, 250–254 (1968).
Nelson, D. R., Piran, T. & Weinberg, S. (eds) Statistical Mechanics of Membranes and Surfaces (World Scientific, Singapore, 2004).
Nelson, D. R. & Peliti, L. Fluctuations in membranes with crystalline and hexatic order. J. Physique 48, 1085–1092 (1987).
Le Doussal, P. & Radzihovsky, L. Self-consistent theory of polymerized membranes. Phys. Rev. Lett. 69, 1209–1212 (1992).
Novoselov, K. S. et al. Two-dimensional atomic crystals. Proc. Natl Acad. Sci. USA 102, 10451–10453 (2005).
Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).
Meyer, J. C. et al. The structure of suspended graphene membrane. Nature 446, 60–63 (2007).
Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).
Zhang, Y., Tan, J. W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005).
Geim, A. K. & Novoselov, K. S. The rise of graphene. Nature Mater. 6, 183–191 (2007).
Katsnelson, M. I. Graphene: Carbon in two dimensions. Mater. Today 10, 20–27 (2007).
Los, J. H., Ghiringhelli, L. M., Meijer, E. J. & Fasolino, A. Improved long-range reactive bond-order potential for carbon. I. Construction. Phys. Rev. B 72, 214102 (2005).
Car, R. & Parrinello, M. Unified approach for molecular dynamics and density-functional theory. Phys. Rev. Lett. 55, 2471–2474 (1985).
Ghiringhelli, L. M., Los, J. H., Meijer, E. J., Fasolino, A. & Frenkel, D. Modeling the phase diagram of carbon. Phys. Rev. Lett. 94, 145701 (2005).
Chandler, D. Introduction to Modern Statistical Mechanics Chaps 3 and 6 (Oxford Univ. Press, New York, 1987).
Carlsson, J. M. & Scheffler, M. Structural, electronic, and chemical properties of nanoporous carbon. Phys. Rev. Lett. 96, 046806 (2006).
Bowick, M. J. in Statistical Mechanics of Membranes and Surfaces (eds Nelson, D. R., Piran, T. & Weinberg, S.) Ch. 11 (World Scientific, Singapore, 2004).
Abramovitz, F. F. & Nelson, D. R. Diffraction from polymerized membranes. Science 249, 393 (1990).
Niclow, R., Wakabayashi, N. & Smith, H. G. Lattice dynamics of pyrolytic graphite. Phys. Rev. B 5, 4951–4962 (1972).
Tarjus, G., Kivelson, S. A., Nussinov, Z. & Viot, P. The frustrated-based approach of supercooled liquids and the glass transition: A review and critical assessment. J. Phys. Condens. Matter 17, R1143–R1182 (2005).
Katsnelson, M. I. & Fasolino, A. Solvent-driven formation of bolaamphiphilic vesicles. J. Phys. Chem. B 110, 30–32 (2006).
Manyuhina, O. V. et al. Anharmonic magnetic deformation of self-assembled molecular nanocapsules. Phys. Rev. Lett. 98, 146101 (2007).
Lubensky, T. C. & MacKintosh, F. C. Theory of “rippled” phases of liquid bilayers. Phys. Rev. Lett. 71, 1565–1568 (1993).
Mounet, N. & Marzari, N. First-principles determination of the structural, vibrational and thermodynamic properties of diamond, graphite, and derivatives. Phys. Rev. B 71, 205214 (2005).
Morozov, S. V. et al. Strong suppression of weak localization in graphene. Phys. Rev. Lett. 97, 016801 (2006).
Castro Neto, A. H. & Kim, E. A. Charge inhomogeneity and the structure of graphene sheets. Preprint at <http://arxiv.org/cond-mat/0702562> (2007).
Geim, A. K. & Katsnelson, M. I. Electronic scattering on microscopic corrugations in graphene. Phil. Trans. R. Soc. A (in the press); preprint at <http://arxiv.org/arXiv:0706.2490> (2007).
We are grateful to D. Nelson, J. C. Maan, A. Geim, K. Novoselov and J. Meyer for helpful discussions. This work was supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM), the Netherlands.
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Fasolino, A., Los, J. & Katsnelson, M. Intrinsic ripples in graphene. Nature Mater 6, 858–861 (2007). https://doi.org/10.1038/nmat2011
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