FIG. 3: (a) Armchair edge ripple amplitude versus ribbon width for λ = 50 ˚ A. Inset: Schematics of ripple formation along the armchair and zigzag edge for λ = 50 ˚ A. (b) Armchair (red) and zigzag (blue) edge ripple amplitude as a function of λ . Light blue band shows the typical range of thermal fluctuation. edge warping with displacement µ$_{e}$ = a sin(2 πx/λ ) of amplitude a and wavelength λ , which decays exponentially into the sheet as e − y/l (See inset of Fig. 3), where l is the decay length, Shenoy et al. have shown that minimization of strain energy leads to characteristic length scales of such warping instability as l ≈ 0 . 23 λ and a ≈ √ ( − λτ$_{e}$ ) / (1 . 37 E$_{b}$ + 14 . 8 E$_{e}$/λ ). Using their empirical-potential values of τ$_{e}$ , E$_{e}$ and E$_{s}$ , they estimated that the warping magnitude of armchair edge is smaller than that of zigzag edge and both are larger than typical thermal fluctuations[10]. Our first-principle predictions, however, are different in several ways. First, absent from empirical prediction, the quantum oscillation of τ$_{e}$ of armchair edge gives rise to an oscillating armchair edge warping amplitude for given wavelength as a function of nanoribbon width, as shown in Fig. 3a. Second, opposite to empirical prediction, the warping amplitude of 5