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, and then employ a core-edge framework based on classical plate theory to quantify the width dependence in more general systems. The elastic edge-edge interactions force ultra-narrow ribbons to be flat, which then bifurcate to twisted and bent shapes at critical widths that vary inversely with edge stress. Compressive edge stresses results in twisted and saddle shapes that are energetically indistinguishable in the vicinity of the bifurcation. Increasing widths favor the saddle shapes with (longitudinal) ribbon curvatures that vary non-linearly with width and edge stress. Positive edge stresses result in a flat-to-curled transition with similar scalings. At large widths with negligible edge-edge interactions, rippling instabilities set in, i.e. edge ripples and midline dimples for compressive and tensile edge stresses. Our results highlight the utility of the core-edge framework in developing a unified understanding of the interplay between geometry and mechanics that sets the morphology of crystalline nanoribbons.