(a) Phonon band of graphene: the LA-mode tangents to the LO-mode at the K point. There are only LA and LO modes around 150 meV corresponding to the center of phononic bandgap of CNTs. (b) Blue lines are contours near the K point with equal phonon energy in the phonon band of graphene. Dirac’s cone is isotropic close to the K point and becomes distorted away from the K point. (c) Surface plot of energy difference between LA and LO modes. Center black region indicates no energy difference at the K point. As the color becomes warmer, the energy difference becomes larger.
