Edge-Stress-Induced Warping of Graphene Sheets and Nanoribbons

We show that edge stresses introduce intrinsic ripples in freestanding graphene sheets even in the absence of any thermal effects. Compressive edge stresses along zigzag and armchair edges of the sheet cause out-of-plane warping to attain several degenerate mode shapes. Based on elastic plate theory, we identify scaling laws for the amplitude and penetration depth of edge ripples as a function of wavelength. We also demonstrate that edge stresses can lead to twisting and scrolling of nanoribbons as seen in experiments. Our results underscore the importance of accounting for edge stresses in thermal theories and electronic structure calculations for freestanding graphene sheets.

Figure 1

(a) Graphene sheet with zigzag (1) and armchair (2) edges. (b) Tensile forces exerted on a planar sheet by edges with compressive edge stresses.

Figure 2

(a),(b) Shapes of warped $16.62\mathrm{nm}\times 16.57\mathrm{nm}$ graphene sheets obtained by relaxing perturbed graphene lattice with the AIREBO potential. The amplitudes of the waves along the zigzag edges are larger than those along the armchair edges. (c),(d) Warped shapes of $16.62\mathrm{nm}\times 16.57\mathrm{nm}$ graphene sheets obtained from finite element simulations that explicitly account for edge stresses.

Figure 3

Out-of-plane perturbation $\zeta =A\sin (k{x}_1)e^{-x_2/l}$ applied to a semi-infinite planar sheet.

Figure 4

(a)–(f) Relaxed shapes of $7.62\mathrm{nm}\times 2.37\mathrm{nm}$ nanoribbons. The longer and shorter sides are terminated by zigzag and armchair edges, respectively. (g)–(m) Finite element simulations of edge-stress-driven warping in $7.62\mathrm{nm}\times 2.37\mathrm{nm}$ nanoribbons.

Figure 5

Relaxed shapes of $11.49\mathrm{nm}\times 1.12\mathrm{nm}$ nanoribbons. The longer and shorter sides are terminated by zigzag and armchair edges, respectively.