Saddles, twists, and curls: shape transitions in freestanding nanoribbons.

Efforts to modulate the electronic properties of atomically thin crystalline nanoribbons requires precise control over their morphology. Here, we perform atomistic simulations on freestanding graphene nanoribbons (GNRs) to first identify the minimal shapes as a function of ribbon width, and then develop a core-edge framework based on classical plate theory to explore the effect of size and ribbon elasticity in more general systems. The elastic edge-edge interactions are central to stabilization of the flat phase in ultra-narrow ribbons, and their bifurcation to twisted and bent shapes at critical widths that vary inversely with edge stress. In the case of compressive edge stress, we uncover hitherto ignored saddle shapes that are energetically indistinguishable with twisted shapes in the vicinity of the bifurcation yet dominate the morphological space with increasing width. At much larger widths with negligible edge-edge interactions, rippling instabilities set in, i.e. edge ripples and midline dimples for compressive and tensile edge stresses, respectively. Simulations of tapering GNRs reveal the dynamics of these shape transitions. Our results capture the interplay between geometry and mechanics that sets the morphology of crystalline nanoribbons and also highlight the utility of the core-edge framework in developing a unified understanding of the interplay.