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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.