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

TL;DR

This paper analytically examines electron diffraction in graphene, linking diffraction intensities to structural properties like thickness and atomic vibrations, and measures the Debye-Waller factor for monolayer graphene.

Contribution

It provides an analytic model for electron diffraction in multi-layer graphene and experimentally measures the Debye-Waller factor, connecting diffraction data with atomic displacement.

Findings

Single-layer graphene scatters only 0.5% of incident electrons.

Debye-Waller factor of monolayer graphene is consistent with Debye model estimates.

Finite size effects stabilize graphene against melting, reducing the need for ripples.

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.