Symmetry properties of vibrational modes in graphene nanoribbons

We present symmetry properties of the lattice vibrations of graphene nanoribbons with pure armchair (AGNR) and zigzag edges (ZGNR). In non-symmorphic nanoribbons the phonon modes at the edge of the Brillouin zone are twofold degenerate, whereas the phonon modes in symmorphic nanoribbons are non-degenerate. We identified the Raman-active and infrared-active modes. We predict 3N and 3(N+1) Raman-active modes for N-ZGNRs and N-AGNRs, respectively (N is the number of dimers per unit cell). These modes can be used for the experimental characterization of graphene nanoribbons. Calculations based on density functional theory suggest that the frequency splitting of the LO and TO in AGNRs (corresponding to the E2g mode in graphene) exhibits characteristic width and family dependence. Further, all graphene nanoribbons have a Raman-active breathing-like mode, the frequency of which is inversely proportional to the nanoribbon width and thus might be used for experimental determination of the width of graphene nanoribbons.Comment: 10 pages, 5 figure

Chirality distribution and transition energies of carbon nanotubes

From resonant Raman scattering on isolated nanotubes we obtained the optical transition energies, the radial breathing mode frequency and Raman intensity of both metallic and semiconducting tubes. We unambiguously assigned the chiral index (n_1,n_2) of approximately 50 nanotubes based solely on a third-neighbor tight-binding Kataura plot and find omega_RBM=214.4cm^-1nm/d+18.7cm^-1. In contrast to luminescence experiments we observe all chiralities including zig-zag tubes. The Raman intensities have a systematic chiral-angle dependence confirming recent ab-initio calculations.Comment: 4 pages, to be published in Phys. Rev. Let

The strength of the radial-breathing mode in single-walled carbon nanotubes

We show by ab initio calculations that the electron-phonon coupling matrix element M of the radial breathing mode in single-walled carbon nanotubes depends strongly on tube chirality. For nanotubes of the same diameter the coupling strength |M|^2 is up to one order of magnitude stronger for zig-zag than for armchair tubes. For (n,m) tubes M depends on the value of (n-m) mod 3, which allows to discriminate semiconducting nano tubes with similar diameter by their Raman scattering intensity. We show measured resonance Raman profiles of the radial breathing mode which support our theoretical predictions

Graphene on Si(111)7x7

We demonstrate that it is possible to mechanically exfoliate graphene under ultra high vacuum conditions on the atomically well defined surface of single crystalline silicon. The flakes are several hundred nanometers in lateral size and their optical contrast is very faint in agreement with calculated data. Single layer graphene is investigated by Raman mapping. The G and 2D peaks are shifted and narrowed compared to undoped graphene. With spatially resolved Kelvin probe measurements we show that this is due to p-type doping with hole densities of n_h \simeq 6x10^{12} cm^{-2}. The in vacuo preparation technique presented here should open up new possibilities to influence the properties of graphene by introducing adsorbates in a controlled way.Comment: 8 pages, 7 figure

Phonons in graphene with point defects

The phonon density of states (DOS) of graphene with different types of point defects (carbon isotopes, substitution atoms, vacancies) is considered. Using a solvable model which is based on the harmonic approximation and the assumption that the elastic forces act only between nearest neighboring ions we calculate corrections to graphene DOS dependent on type and concentration of defects. In particular the correction due to isotopic dimers is determined. It is shown that a relatively small concentration of defects may lead to significant and specific changes in the DOS, especially at low frequencies, near the Van Hove points and in the vicinity of the K-points of the Brillouin zone. In some cases defects generate one or several narrow gaps near the critical points of the phonon DOS as well as resonance states in the Brillouin zone regular points. All types of defects are characterized by the appearance of one or more additional Van Hove peaks near the (Dirac) K points and their singular contribution may be comparable with the effect of electron-phonon interaction. Besides, for low frequencies and near the critical points the relative change in density of states may be many times higher than the concentration of defects.Comment: 19 pages, 7 figure

Exciton binding energies in carbon nanotubes from two-photon photoluminescence

One- and two-photon luminescence excitation spectroscopy showed a series of distinct excitonic states in single-walled carbon nanotubes. The energy splitting between one- and two-photon-active exciton states of different wavefunction symmetry is the fingerprint of excitonic interactions in carbon nanotubes. We determine exciton binding energies of 0.3-0.4 eV for different nanotubes with diameters between 0.7 and 0.9 nm. Our results, which are supported by ab-initio calculations of the linear and non-linear optical spectra, prove that the elementary optical excitations of carbon nanotubes are strongly Coulomb-correlated, quasi-one dimensionally confined electron-hole pairs, stable even at room temperature. This alters our microscopic understanding of both the electronic structure and the Coulomb interactions in carbon nanotubes, and has direct impact on the optical and transport properties of novel nanotube devices.Comment: 5 pages, 4 figure

The phonon dispersion of graphite by inelastic x-ray scattering

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.Comment: 7 pages; submitted to Phys. Rev.

Dependence of exciton transition energy of single-walled carbon nanotubes on surrounding dielectric materials

We theoretically investigate the dependence of exciton transition energies on dielectric constant of surrounding materials. We make a simple model for the relation between dielectric constant of environment and a static dielectric constant describing the effects of electrons in core states, $\sigma$ bonds and surrounding materials. Although the model is very simple, calculated results well reproduce experimental transition energy dependence on dielectric constant of various surrounding materials.Comment: 5pages, 4 figure

Intersubband decay of 1-D exciton resonances in carbon nanotubes

We have studied intersubband decay of E22 excitons in semiconducting carbon nanotubes experimentally and theoretically. Photoluminescence excitation line widths of semiconducting nanotubes with chiral indicess (n, m) can be mapped onto a connectivity grid with curves of constant (n-m) and (2n+m). Moreover, the global behavior of E22 linewidths is best characterized by a strong increase with energy irrespective of their (n-m) mod(3)= \pm 1 family affiliation. Solution of the Bethe-Salpeter equations shows that the E22 linewidths are dominated by phonon assisted coupling to higher momentum states of the E11 and E12 exciton bands. The calculations also suggest that the branching ratio for decay into exciton bands vs free carrier bands, respectively is about 10:1.Comment: 4 pages, 4 figure