Phonon dispersions of hydrogenated and dehydrogenated carbon nanoribbons

Phonon dispersion relations are presented for carbon nanoribbons along multirow structures of hexagonal rings with zigzag and armchair edges. The dispersions are compared with the $\Gamma K$ and $\Gamma M$ dispersions of graphite as a function of ribbon width $m$. The force constants are obtained from zone folding for those of polycyclic aromatic hydrocarbons (PAHs) based on a transferable force-field model (MO/8 model). The vibrational mode patterns at the $\Gamma$ point are calculated to be longitudinal or transverse in zigzag nanoribbons. All of the normal modes are categorized into the following four groups. (1) The acoustic branches which converge to the origin at $\Gamma$ point allow estimation of the group velocity from their slopes. (2) The acoustic harmonics display systematic vibrational patterns with nonzero phase relations in the width direction. The number of nodes in amplitudes, $k=1-m$, represents dispersions in the width direction. (3) The optical branches which are commensurate with the multirow structures are optical fundamentals, whereas those incommensurate are optical harmonics. Both display systematic frequency changes with respect to the number of nodes $\left(k\right)$ and ribbon widths $\left(m\right)$. The optical harmonics of zigzag ribbons at $\Gamma$ point shift in proportion to the inverse of ribbon width $1∕m$, whereas the acoustic harmonics depend on $1∕\sqrt{m}$. (4) The CH vibrations are less dispersive than these CC vibrations. The CH out-of-plane modes are characterized by strong infrared intensities, and their frequencies are calculated to be higher in zigzag ribbons than in armchair ribbons. This order is in line with the general tendencies for PAHs with solo and duo CH bonds. Hydrogenated and dehydrogenated nanoribbons are calculated to show similar phonon dispersions for their carbon networks. Phonon densities of states (DOSs) are compared between the nanoribbons and PAHs with particular shapes. Prominent DOS peaks are uniquely obtained for zigzag nanoribbons and linear PAHs at $\sim 400$ and $1400{\mathrm{cm}}^{-1}$. Complete selection rules are given for the vibrational transitions in zigzag and armchair nanoribbons according to the irreducible representations for the primitive unit cells.