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We find that the electronic dispersion in graphite gives rise to double resonant Raman scattering for excitation energies up to 5 eV. As we show, the curious excitation-energy dependence of the graphite D mode is due to this double resonant process resolving a long-standing problem in the literature and invalidating recent attempts to explain this phenomenon. Our calculation for the D-mode frequency shift ( 60 cm(-1)/eV) agrees well with the experimental value.

We investigate the tight-binding approximation for the dispersion of the and * electronic bands in graphene and carbon nanotubes. The nearest-neighbor tight-binding approximation with a fixed ␥ 0 applies only to a very limited range of wave vectors. We derive an analytic expression for the tight-binding dispersion including up to third-nearest neighbors. Interaction with more distant neighbors qualitatively improves the tight-binding picture, as we show for graphene and three selected carbon nanotubes.The band structure of carbon nanotubes is widely modeled by a zone-folding approximation of the graphene and à electronic states as obtained from a tight-binding Hamiltonian. [1][2][3][4][5] The large benefit of this method is the very simple formula for the nanotube electronic bands. While the tight-binding picture provides qualitative insight into the one-dimensional nanotube band structure, it is more and more being used for quantitative comparisons as well. For instance, attempts to assign diameters and chiralities of carbon nanotubes based on optical absorption and Raman data rely heavily on the assumed transition energies. 2,6 Differences between the zone-folding, tight-binding -orbital description and experiment, as observed, e.g., in scanning tunneling measurements, are usually ascribed to ''curvature effects.'' 1 However, the common -orbital tight-binding approach for the nanotube band structure involves two approximations: ͑i͒ zone folding, which neglects the curvature of the wall; and ͑ii͒ the tight-binding approximation to the graphene bands including only first-neighbor interaction. Whereas the first point received some attention in the literature, 7-9 the second approximation is usually assumed to be sufficient.In this paper we compare the tight-binding approximation of the graphene orbitals to first-principles calculations. We show that the nearest-neighbor tight-binding Hamiltonian does not accurately reproduce the and * graphene bands over a sufficiently large range of the Brillouin zone. We derive an improved tight-binding electronic dispersion by including up to third-nearest-neighbor interaction and overlap. The formula for the electronic states we present may readily be used, e.g., in combination with zone folding to obtain the band structure of nanotubes.The first tight-binding description of graphene was given by Wallace in 1947. 10 He considered nearest-and nextnearest-neighbor interaction for the graphene p z orbitals, but neglected the overlap between wave functions centered at different atoms. The other-nowadays better known-tightbinding approximation was nicely described by Saito et al. 4 It considers the nonfinite overlap between the basis functions, but includes only interactions between nearest neighbors within the graphene sheet. To study the different levels of approximation we start from the most general form of the secular equation, the tight-binding Hamiltonian H, and the overlap matrix S, 4where E(k) are the electronic eigenvalues. We used the equivalence of the A and B carbon ...

We present a review of the Raman spectra of graphite from an experimental and theoretical point of view. The disorder-induced Raman bands in this material have been a puzzling Raman problem for almost 30 years. Double-resonant Raman scattering explains their origin as well as the excitation-energy dependence, the overtone spectrum and the difference between Stokes and anti-Stokes scattering. We develop the symmetry-imposed selection rules for double-resonant Raman scattering in graphite and point out misassignments in previously published works. An excellent agreement is found between the graphite phonon dispersion from double-resonant Raman scattering and other experimental methods.

We present a comprehensive study of the chiral-index assignment of carbon nanotubes in aqueous suspensions by resonant Raman scattering of the radial breathing mode. We determine the energies of the first optical transition in metallic tubes and of the second optical transition in semiconducting tubes for more than 50 chiral indices. The assignment is unique and does not depend on empirical parameters. The systematics of the socalled branches in the Kataura plot are discussed; many properties of the tubes are similar for members of the same branch. We show how the radial breathing modes observed in a single Raman spectrum can be easily assigned based on these systematics. In addition, empirical fits provide the energies and radial breathing modes for all metallic and semiconducting nanotubes with diameters between 0.6 and 1.5 nm. We discuss the relation between the frequency of the radial breathing mode and tube diameter. Finally, from the Raman intensities we obtain information on the electron-phonon coupling.

We measured the dispersion of the graphite optical phonons in the in-plane Brillouin zone by inelastic x-ray scattering. The longitudinal and transverse optical branches cross along the Gamma-K as well as the Gamma-M direction. The dispersion of the optical phonons was, in general, stronger than expected from the literature. At the K point the transverse optical mode has a minimum and is only approximately 70 cm(-1) higher in frequency than the longitudinal mode. We show that first-principles calculations describe very well the vibrational properties of graphene once the long-range character of the dynamical matrix is taken into account.

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We present the full in-plane phonon dispersion of graphite obtained from inelastic x-ray scattering, including the optical and acoustic branches, as well as the mid-frequency range between the K and M points in the Brillouin zone, where experimental data have been unavailable so far. The existence of a Kohn anomaly at the K point is further supported. We fit a fifth-nearest neighbour force-constants model to the experimental data, making improved force-constants calculations of the phonon dispersion in both graphite and carbon nanotubes available.