Rippling instabilities in suspended nanoribbons

Morphology mediates the interplay between the structure and electronic transport in atomically thin nanoribbons such as graphene as the relaxation of edge stresses occurs preferentially via out-of-plane deflections. In the case of end-supported suspended nanoribbons that we study here, past experiments and computations have identified a range of equilibrium morphologies, in particular, for graphene flakes, yet a unified understanding of their relative stability remains elusive. Here, we employ atomic-scale simulations and a composite framework based on isotropic elastic plate theory to chart out the morphological stability space of suspended nanoribbons with respect to intrinsic (ribbon elasticity) and engineered (ribbon geometry) parameters, and the combination of edge and body actuation. The computations highlight a rich morphological shape space that can be naturally classified into two competing shapes, bendinglike and twistlike, depending on the distribution of ripples across the interacting edges. The linearized elastic framework yields exact solutions for these rippled shapes. For compressive edge stresses, the body strain emerges as a key variable that controls their relative stability and in extreme cases stabilizes coexisting transverse ripples. Tensile edge stresses lead to dimples within the ribbon core that decay into the edges, a feature of obvious significance for stretchable nanoelectronics. The interplay between geometry and mechanics that we report should serve as a key input for quantifying the transport along these ribbons.

Figure 1

(a) Schematic illustration of a suspended nanoribbon, simply supported or clamped depending on the end supports. Experimental observations of (b) rippling in multilayer graphene ribbons[36] and (c) edge rippling in hollow BN nanoribbons.[37] The images are reproduced with the authors’ permissions. (d) and (e) Similar shapes observed in atomic-scale computations on $l\approx 20–25$ nm long, armchair-terminated graphene nanoribbons (AGNRs) with (d) compressive and (e) tensile edge stresses. The critical wave number $k^{*}w$ is the ratio of the ribbon width to ripple wavelength. In all cases, both bendinglike (symmetric, S) and twistlike (antisymmetric, AS) shapes are observed. The atoms are colored based on the scaled magnitude of their out-of-plane displacements. The additional effect of applied compressive and tensile strains for ${\tau }_e<0$ is shown in bottom two plots in (d). The insets show details of the bent morphology at and around the ribbon midline.

Figure 2

(a) Morphological stability diagram showing the scaled edge stress ${\tau }_{e}w/D$ as a function of the wave number $kw$ for relaxed (${\epsilon }_{xx}=0$, light shading) and intrinsically strained [${\epsilon }_{xx}^0=-2{\tau }_e/\left(Sw\right)$, dark shading)] nanoribbons with compressive edge stresses. Here as well as in the following figures, the critical curves for bendinglike and twistlike buckling are indicated by black and red lines, respectively. (Inset) Magnified plot for small wave numbers $kw$ that shows the bifurcation between the two morphological classes for strained ribbons. (b) The ribbon profile obtained from the analytical solution[48] for relaxed ribbons plotted as the scaled deflection $f\left(y\right)/f_{\max }$ vs scaled width $y/w$ for the three different wave numbers. For both classes, the rippling localizes to the edges with increasing wave number. Schematic illustrations of the shapes for some of the profiles are also shown. All plots are based on Poisson's ratio $\nu =0.17$ corresponding to that for graphene.[47]

Figure 3

(a) and (b) Comparison of the analytically predicted stability of of bendinglike and twistlike shapes and the buckled shapes observed in atomic-scale simulations for (a) relaxed and (b) intrinsically strained AGNRs, plotted as $k$ vs $w$. The representative atomic-configurations for symmetrically rippled ribbons with widths $w=0.72$, $w=1.93$, and $5.12$ nm are also shown (insets).

Figure 4

(a) and (b) Iso-(edge)stress critical contours for (a) bendinglike and (b) twistlike rippling of strained ribbons with compressive and tensile edge stresses. The contour plot associated with classical Euler buckling (${\tau }_e=0$) is also plotted. The solid white lines are isostrain plots for intrinsically strained nanoribbons summarized in Figs. 2 and 5a. (c) The critical curves for strained unreconstructed AGNRs with compressive edge stress ${\tau }_e=10.5$ eV/nm. (d) Ribbon profiles $f\left(y\right)/f_{\max }$ for some of the critical points below (${\epsilon }_{xx}=0.15\%$, $k^{*}w=0.5$) as well as in the vicinity of the peak strain (${\epsilon }_{xx}=1\%$, $k^{*}w=7.5,37$).

Figure 5

(a) Stability diagram and (b) ribbon profiles as in Fig. 2, but for intrinsically strained nanoribbons with tensile edge stresses. For clarity, the dimpled morphologies predicted by our analysis for $k^{*}w=5$ and 50 are shown schematically.