title: Dark-field transmission electron microscopy and the Debye-Waller factor of graphene.
abstract: 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 selected 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. The diffraction peak intensities also depend on the mean-square displacement of atoms from their ideal lattice locations, which is parameterized by a Debye-Waller factor. We measure the Debye-Waller factor of a suspended monolayer of exfoliated graphene and find a result consistent with an estimate based on the Debye model. For laboratory-scale graphene samples, finite size effects are sufficient to stabilize the graphene lattice against melting, indicating that ripples in the third dimension are not necessary.
authors.affiliation: Department of Physics and Astronomy, University of California, Los Angeles, California, 90095, USA.|California NanoSystems Institute, University of California, Los Angeles, California, 90095, USA.
authors.forename: Brian
authors.initials: B
authors.lastname: Shevitski
authors.affiliation: Department of Physics and Astronomy, University of California, Los Angeles, California, 90095, USA.|California NanoSystems Institute, University of California, Los Angeles, California, 90095, USA.|Microelectronics Technology Department, The Aerospace Corporation, Los Angeles, California 90009, USA.
authors.forename: Matthew
authors.initials: M
authors.lastname: Mecklenburg
authors.affiliation: Department of Physics and Astronomy, University of California, Los Angeles, California, 90095, USA.|California NanoSystems Institute, University of California, Los Angeles, California, 90095, USA.
authors.forename: William A
authors.initials: WA
authors.lastname: Hubbard
authors.affiliation: Department of Physics and Astronomy, University of California, Los Angeles, California, 90095, USA.|California NanoSystems Institute, University of California, Los Angeles, California, 90095, USA.
authors.forename: E R
authors.initials: ER
authors.lastname: White
authors.affiliation: Department of Physics and Nanoscience Technology Center, University of Central Florida, Orlando, FL, 32816, USA.
authors.forename: Ben
authors.initials: B
authors.lastname: Dawson
authors.affiliation: Department of Physics and Nanoscience Technology Center, University of Central Florida, Orlando, FL, 32816, USA.
authors.forename: M S
authors.initials: MS
authors.lastname: Lodge
authors.affiliation: Department of Physics and Nanoscience Technology Center, University of Central Florida, Orlando, FL, 32816, USA.
authors.forename: Masa
authors.initials: M
authors.lastname: Ishigami
authors.affiliation: Department of Physics and Astronomy, University of California, Los Angeles, California, 90095, USA.|California NanoSystems Institute, University of California, Los Angeles, California, 90095, USA.
authors.forename: B C
authors.initials: BC
authors.lastname: Regan