Linear Plasmon Dispersion in Single-Wall Carbon Nanotubes and the Collective Excitation Spectrum of Graphene

We have measured a strictly linear $\pi$ plasmon dispersion along the axis of individualized single-wall carbon nanotubes, which is completely different from plasmon dispersions of graphite or bundled single-wall carbon nanotubes. Comparative ab initio studies on graphene-based systems allow us to reproduce the different dispersions. This suggests that individualized nanotubes provide viable experimental access to collective electronic excitations of graphene, and it validates the use of graphene to understand electronic excitations of carbon nanotubes. In particular, the calculations reveal that local field effects cause a mixing of electronic transitions, including the “Dirac cone,” resulting in the observed linear dispersion.

Figure 1

(a) Measured loss function of freestanding VA-SWNT at equidistant $q$ from $0.1{\AA }^{-1}$ (top) to $1.0{\AA }^{-1}$ (bottom). TEM micrographs of the cross section (b) and side view (c) of the thin bundled VA-SWNT.

Figure 2

(a) Loss function of the $\pi$ plasmon region at equidistant $q$ ranging from $0.1{\AA }^{-1}$ (top) to $1.0{\AA }^{-1}$ (bottom). Right stack: observed (filled symbols) vs calculated (open symbols) $\pi$ plasmon dispersion for (b) graphite, (c) bundled SWNT [8] vs double-layer graphene, and for (d) VA-SWNT vs graphene. The calculations are averaged over the two in-plane directions $\Gamma K$ and $\Gamma M$.

Figure 3

(a) Loss function of graphene at $q=0.41{\AA }^{-1}$ along $\Gamma M$ calculated from the JDOS (thin dotted line), within the bare RPA (thin solid line) and in RPA including LFE (thick solid line). The latter changes significantly when transitions next to the $K$ point (shaded area) are excluded (thick, dot-dashed line). (b) Dispersion of the peaks in the loss function for different momenta along $\Gamma M$ (solid lines and squares) and $\Gamma K$ (dotted lines and circles).