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The process involving q′ (a phonon very close to the K-point) does not occur because the corresponding electron-phonon coupling matrix element is approximately 0 (see footnote 24 of ref 23), and the process involving q″ has a low weight because, due to the trigonal warping effect, the corresponding phase space in the two-dimensional Brillouin zone is small compared to the phase space for the process involving q.
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The process involving q′ (a phonon very close to the K-point) does not occur because the corresponding electron-phonon coupling matrix element is approximately 0 (see footnote 24 of ref 23), and the process involving q″ has a low weight because, due to the "trigonal warping" effect, the corresponding phase space volume in the two-dimensional Brillouin zone is small compared to the phase space volume for the process involving q.
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16
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33847741697
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We use the code ABINIT ref 17, Wave functions are expanded in plane waves with an energy cutoff at 30 Hartrees. Core electrons are replaced by Trouiller-Martins pseudopotentials in the Kleinman-Bylander form. We use the experimental lattice constant a, 2.459 Å.18
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19
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33847697663
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We use the computer code SELF written by A. Marini
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We use the computer code SELF written by A. Marini (http://www.fisica. uniroma2.it/~self/).
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33847727375
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Private communication
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Lazzeri, M. Private communication.
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Lazzeri, M.1
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21
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33847739266
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Details of the calculations are given in ref 22. We note that the slope of the TO mode in the region between K and M where phonons are excited according to the double-resonant Raman model agrees with other recent ab initio calculations.23,24
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23,24
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23
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19744365772
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Piscanec, S.; Lazzeri, M.; Mauri, F.; Ferrari, A. C.; Robertson, J. Phys. Rev. Lett. 2004, 93, 185503.
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24
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1642585850
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We note that Ferrari et al.7 overcome this difficulty by using the semiempirical parameter dω/dq caption of Table I of ref 7, which is fitted to the experimental dispersion of the D′ peak. We have avoided the use of this semiempirical parameter in order to assess the quantitative validity of the double-resonant Raman model based on first-principles calculations. The good agreement between experimental and theoretical D peak dispersion in ref 6 is based on a corresponding fit of the hopping parameter in the tight-binding band structure and on the parametrization of the phonon dispersion relation
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7 overcome this difficulty by using the semiempirical parameter dω/dq (caption of Table I of ref 7), which is fitted to the experimental dispersion of the D′ peak. We have avoided the use of this semiempirical parameter in order to assess the quantitative validity of the double-resonant Raman model based on first-principles calculations. The good agreement between experimental and theoretical D peak dispersion in ref 6 is based on a corresponding fit of the hopping parameter in the tight-binding band structure and on the parametrization of the phonon dispersion relation.
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33847728574
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For an exact quantitative evaluation of the peak heights, an integration in the two-dimensional Brillouin zone for all transitions of energy ΔE = 2.33 eV would have to be performed.
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For an exact quantitative evaluation of the peak heights, an integration in the two-dimensional Brillouin zone for all transitions of energy ΔE = 2.33 eV would have to be performed.
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Spataru, C. D.; Ismail-Beigi, S.; Benedict, L. X.; Louie, S. G. Phys. Rev. Lett. 2004, 92, 077402.
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Chang, E.; Bussi, G.; Ruini, A.; Molinari, E. Phys. Rev. Lett. 2004, 92, 196401.
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Chang, E.1
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We note that the picture of independent electron-hole pair excitations has been quantitatively very successful for the assignment of nanotube spectra to the chiral indices m,n, However, this success was based on the use of tight-binding band structures with correspondingly fitted parameters. A parameter-free, first-principles calculation of the optical spectra of carbon nanotubes must take into account electron-electron and electron-hole interaction, and we suggest that this may be true for graphite as well
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We note that the picture of independent electron-hole pair excitations has been quantitatively very successful for the assignment of nanotube spectra to the chiral indices (m,n). However, this success was based on the use of tight-binding band structures with correspondingly fitted parameters. A parameter-free, first-principles calculation of the optical spectra of carbon nanotubes must take into account electron-electron and electron-hole interaction, and we suggest that this may be true for graphite as well.
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