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The size dependence of the energy gap of CNRs and CNTs is shown. The size of the CNR is simply the ribbon width, whereas for the CNT it is the circumference of the tube. At matched width, CNRs have significantly smaller band gaps.
(Color online) The phonon dispersion relations for a (4,0) CNT (left) and (4,0) CNR (right) are shown in the first Brillouin zone. The lowest energy modes are similar; degeneracy lifting and energy reduction are evident for the higher modes.
(Color online) The lowest two phonon modes of the (4,0) CNR are shown. Arrows represent in-plane displacements, the discs represent out-of-plane displacements. Solid discs are displacements out of the paper and open discs displacements into the paper. The size of the discs and length of the arrows indicate relative magnitude.
(Color online) The third and fourth phonon modes of the (4,0) CNR. Note that the fourth (out-of-plane) mode clearly has nonperiodic boundary conditions, not possible in the CNT.
Comparison of CNR and CNT mobilities at matched band gap. Due to the reduced CNR band gap, CNTs have considerably higher mobilities at matched band gap. Phonon limited electron mobility in silicon under strong inversion is also shown.
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