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Engineering Mechanics (EM)
Edge-stress-induced warping of graphene sheets and nanoribbons
Graphene, an atomic thin layer of carbon atoms arranged in a honeycomb lattice, is actively being pursued as a material for next-generation electronic devices because of their superior electronic and mechanical properties. Since the electronic structure of graphene is strongly altered by both strain and curvature, the results should have important implications for graphene based devices. In current work, it is shown that the intrinsic edge stress can have a significant influence on the morphology on the graphene sheets (Physical Review Letters, 101, 245501, 2008).
The electronic properties of the finite sheets can vary with the size, shape and orientation of their edges due to their localized edge states. Depending on the edge termination from the periodic honeycomb lattice and edge reconstruction, the edge will be under tensile or compressive stress. In this letter, it is shown that the edge stress have significant influence on the morphology of graphene sheets. It has been found that the edge stress can lead to warping and rippling of graphene sheets, which allows for the reduction of edge energy at the cost of bending of sheets without stretching.
Here, atomistic simulations, analytical and finite element models are utilized to study the phenomena of ripples forming on the edges and how their amplitudes and wave lengths relate to the elastic properties of the graphene sheets. It is found that the finite element model sheets warp at the edges adopt shapes which closely resemble the shapes observed in atomic simulations, conforming the key role played by the edge stresses (Figure 1). A further analytical model is given to explain how the ripples are related to edge stresses and elastic stiffness of the sheets. Based on elastic plate theory, the scaling laws for the amplitude and penetration depth of edge ripples can be identified as a function of wavelength.
The influence of edge stresses is found to be more dramatic at smaller widths, leading to large-scale distortions in the shapes of ribbons. It is demonstrated that the large arching, twisting, sinusoidal modulations are in and out of phase on the two sides of the ribbons (Figures 2 and 3). The present work suggests means to control morphology and hence the electrical and magnetic properties of graphene sheets and nanoribbons by engineering the edge stresses, for example by doping or functionalizing the edges.
Fig.1. Shapes of warped graphene sheets. (a), (b) shapes are from atomistic simulations and (c), (d) are corresponding shapes obtained from finite element simulations which is explicitly account for edge stress of graphene sheets.
Fig.2. Relaxed shapes of zigzag and armchair edges of graphene sheets by atomic simulations (a-f) and corresponding finite element simulations of edge-stress-driven warping of nanoribbons (g-l).
Fig. 3. Relaxed shapes of long strips with narrow widths.
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This page is last updated at: 26-NOV-2010