We investigate the tight-binding approximation for the dispersion of the $\pi$ and ${\pi }^{*}$ electronic bands in graphene and carbon nanotubes. The nearest-neighbor tight-binding approximation with a fixed ${\gamma }_0$ 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.

Two-dimensional nanomaterials such as MoS₂ are of great interest both because of their novel physical properties and their applications potential. Liquid exfoliation, an important production method, is limited by our inability to quickly and easily measure nanosheet size, thickness or concentration. Here we demonstrate a method to simultaneously determine mean values of these properties from an optical extinction spectrum measured on a liquid dispersion of MoS₂ nanosheets. The concentration measurement is based on the size-independence of the low-wavelength extinction coefficient, while the size and thickness measurements rely on the effect of edges and quantum confinement on the optical spectra. The resultant controllability of concentration, size and thickness facilitates the preparation of dispersions with pre-determined properties such as high monolayer-content, leading to first measurement of A-exciton MoS₂ luminescence in liquid suspensions. These techniques are general and can be applied to a range of two-dimensional materials including WS₂, MoSe₂ and WSe₂.

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 $\Gamma \mathrm{\text{−}}K$ as well as the $\Gamma \mathrm{\text{−}}M$ 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 $\approx 70{\mathrm{c}\mathrm{m}}^{-1}$ 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.

We present scanning tunneling microscopy (STM) images of single-layer graphene crystals examined under ultrahigh vacuum conditions. The samples, with lateral dimensions on the micrometer scale, were prepared on a silicon dioxide surface by direct exfoliation of crystalline graphite. The single-layer films were identified by using Raman spectroscopy. Topographic images of single-layer samples display the honeycomb structure expected for the full hexagonal symmetry of an isolated graphene monolayer. The absence of observable defects in the STM images is indicative of the high quality of these films. Crystals composed of a few layers of graphene also were examined. They exhibited dramatically different STM topography, displaying the reduced threefold symmetry characteristic of the surface of bulk graphite.

Excitonic effects in the linear and nonlinear optical properties of single-walled carbon nanotubes are manifested by photoluminescence excitation experiments and ab initio calculations. One- and two-photon spectra showed a series of exciton states; their energy splitting is the fingerprint of excitonic interactions in carbon nanotubes. By ab initio calculations we determine the energies, wave functions, and symmetries of the excitonic states. Combining experiment and theory we find binding energies of $0.3–0.4\mathrm{eV}$ for nanotubes with diameters between 6.8 and $9.0\mathrm{\AA }$.