Dark-field transmission electron microscopy and the Debye-Waller factor of graphene

Brian Shevitski, Matthew Mecklenburg, William A. Hubbard, E. R. White, Ben Dawson, M. S. Lodge, Masa Ishigami, and B. C. Regan

Phys. Rev. B 87, 045417 – Published 15 January 2013

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 multilayer 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 multilayer 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.

Received 30 June 2012

DOI:https://doi.org/10.1103/PhysRevB.87.045417

Authors & Affiliations

Brian Shevitski^1,2,*, Matthew Mecklenburg^1,2,3,†, William A. Hubbard^1,2, E. R. White^1,2, Ben Dawson⁴, M. S. Lodge⁴, Masa Ishigami⁴, and B. C. Regan^1,2,‡

¹Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA
²California NanoSystems Institute, University of California, Los Angeles, California 90095, USA
³Microelectronics Technology Department, The Aerospace Corporation, Los Angeles, California 90009, USA
⁴Department of Physics and Nanoscience Technology Center, University of Central Florida, Orlando, Florida 32816, USA

^*Authors contributed equally to this work.
^†Authors contributed equally to this work.
^‡regan@physics.ucla.edu

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 87, Iss. 4 — 15 January 2013

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Unit cells and primitive vectors are shown for monolayer, AB-stacked bilayer, and ABC-stacked trilayer graphene. On the far right is the graphene reciprocal lattice with large and small spheres representing strong and weak peaks, respectively.Reuse & Permissions
Figure 2
The selected area diffraction pattern (a) from CVD graphene suspended over a nominally 3- $μ$ m hole reveals the presence of two distinct grains. No grain structure is visible in the bright-field image (b), where most of the contrast is generated by residual PMMA. Dark-field images (c)–(f) are generated using an objective aperture to separately select both the 1 $^{st}$ and 2 $^{nd}$ order diffraction peaks, as indicated by dashed and solid circles on the diffraction pattern (a), respectively. All four dark-field images (c)–(f) were acquired with the same exposure time (5 s) and are displayed using identical contrast scales to permit fair comparison. In the images (c)–(d) acquired with the 1 $^{st}$ order peaks, the grain boundary is barely visible. The images (e)–(f) acquired with the 2 $^{nd}$ order peaks show much better contrast with obvious moiré substructure. Furthermore, image (f) clearly reveals a lattice-aligned bilayer island (arrow) that is not visible in image (d). The scale bars represent 500 nm.Reuse & Permissions
Figure 3
A composite dark-field image (a) of a single graphene crystal on a holey carbon grid is formed using electrons from the 2 $^{nd}$ order peak, which is indicated by the solid circle on the diffraction pattern (b). Graphene over a hole, such as the one indicated by the yellow triangle in (a), is fully suspended. Raman spectra (c), which were acquired from the regions designated by the colored circles in (a) before transfer to the grid, confirm the thickness of each region. Higher-magnification dark-field images (d) and (e) of the region within the black circle in (a) are acquired using the 1 $^{st}$ and 2 $^{nd}$ order peaks, respectively. The same area is shown again as a false color legend in the plot (f), which depicts the average normalized intensity of each disconnected region of (d) and (e) as a function of the square of the number of layers.Reuse & Permissions
Figure 4
The diffraction pattern (a) was acquired from the monolayer graphene suspended over the hole indicated by the yellow triangle in Fig. 3a. The integrated intensity of each diffraction peak is fit to Eq. (9), where the strong peaks have been normalized by dividing by a factor of 4 (see Table ). The peak intensities diminish more quickly with increasing $G$ than can be explained by the atomic form factor (static lattice) and zero-point motion ( $T = 0$ expectation) alone. The number of measured peaks of each order is indicated at each radius.Reuse & Permissions

Physical Review B

covering condensed matter and materials physics