Brian Shevitski, Matthew Mecklenburg, William A. Hubbard, E. R. White, Ben Dawson, M. S. Lodge, Masa Ishigami, B. C. Regan
Graphene's structure bears on both the material's electronic properties and fundamental questions about long range order in two-dimensional crystals. We present an analytic calculation of select area electron diffraction from multi-layer graphene and compare it with data from samples prepared by chemical vapor deposition and mechanical exfoliation. A single layer scatters only 0.5% of the incident electrons, so this kinematical calculation can be considered reliable for five or fewer layers. Dark-field transmission electron micrographs of multi-layer graphene illustrate how knowledge of the diffraction peak intensities can be applied for rapid mapping of thickness, stacking, and grain boundaries. We measure the Debye-Waller factor of a suspended monolayer of exfoliated graphene and find that the in-plane mean-square atomic displacement is $54\pm 3$ pm$^2$. This result is consistent with an estimate based on the Debye model that shows that finite size effects are sufficient to stabilize the graphene lattice against melting.
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http://arxiv.org/abs/1207.0040
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