Intrinsic ripples in graphene

Abstract

The stability of two-dimensional (2D) layers and membranes is subject of a long standing theoretical debate. According to the so called Mermin-Wagner theorem, long wavelength fluctuations destroy the long-range order for 2D crystals. Similarly, 2D membranes embedded in a 3D space have a tendency to be crumpled. These dangerous fluctuations can, however, be suppressed by anharmonic coupling between bending and stretching modes making that a two-dimensional membrane can exist but should present strong height fluctuations. The discovery of graphene, the first truly 2D crystal and the recent experimental observation of ripples in freely hanging graphene makes these issues especially important. Beside the academic interest, understanding the mechanisms of stability of graphene is crucial for understanding electronic transport in this material that is attracting so much interest for its unusual Dirac spectrum and electronic properties. Here we address the nature of these height fluctuations by means of straightforward atomistic Monte Carlo simulations based on a very accurate many-body interatomic potential for carbon. We find that ripples spontaneously appear due to thermal fluctuations with a size distribution peaked around 70 \AA which is compatible with experimental findings (50-100 \AA) but not with the current understanding of stability of flexible membranes. This unexpected result seems to be due to the multiplicity of chemical bonding in carbon.Comment: 14 pages, 6 figure

A low-temperature method to produce highly reduced graphene oxide

“…The RGO films show much narrower Raman peaks than the GO film. Although a prominent D peak is typically an indication of disorder in the Raman of peeled graphene, our observations are consistent with previous reports of RGO materials 28 . A greater ratio of the Raman D/G intensity of the RGO Na-NH 3 films compared with the GO film indicates an increase in the number of smaller sp 2 domains 33,34 .…”

“…2a,b). A line scan of the height profile shows an average topographic height of B0.91 nm, consistent with the typical height reported for singlelayer graphene on mica substrate (B0.5-1.0 nm) 28 . Transmission electron microscopy (TEM) study was further used to investigate the morphology and crystalline structure of RGO Na-NH3 sheet.…”

Structure of graphene and its disorders: a review

“…a substrate with a matching lattice). Indeed, the atomically thin film can be significantly stabilized by partially decoupled bending and stretching modes [188]. By using TEM, a monolayer graphene was found to be freely suspended on a microfabricated metallic scaffold in both vacuum and air at room temperature, with random elastic deformations in the third dimension [9].…”

“…As observed under STM [189], these spontaneous ripples on the suspended graphene are dynamic. Plus, the density of the ripples is higher near the edges or defects due to the amplified asymmetry of the bond lengths at these regions [188,190]. The amplitude of ripples is limited by the strain energy induced by its perpendicular displacement, and increases with the size of graphene sheet [191].…”

“…The amplitude of ripples is limited by the strain energy induced by its perpendicular displacement, and increases with the size of graphene sheet [191]. The spatial distributions of ripples can be derived from the equation [188]: …”