EELS for Graphite, Graphene and SWCNTs: Plasmons and LFE.

We have investigated the momentum and frequency resolved electron energy-loss function of carbon systems with different dimensionality; namely Graphite, Graphene, and Carbon Nanotubes. To this end, first principles calculations [1] have been performed in the framework of (time-dependent) density functional theory using the Random Phase Approximation and taking into account crystal local field effects (LFE). First, we studied the in-plane dispersion of the pi-plasmon in isolated graphene. LFE were found to strongly mix electronic transitions over a large range of energies, which results in a nearly linear plasmon dispersion. A comparison with electron energy-loss (EELS) measurements on individualised single wall carbon nanotubes (SWNT) reveals an identical, linear behaviour for the on-axis plasmon. This validates the use of graphene to understand electronic excitations of carbon nanotubes and vice versa. Second, the energy-loss function of graphite was studied for momentum transfers q beyond the first Brillouin zone. Our calculations show that near Bragg reflections the spectra can change drastically for very small variations in q. This discontinuity in the dynamic structure factor is caused by LFE: the dielectric function of the crystal cannot be reduced to a simple scalar (known as macroscopic dielectric function). Instead, we have to consider an effective 2x2 matrix that describes the coupling between excitations at small and large momentum transfers (known as the two plasmon-band model [2]). The effect is confirmed by inelastic x-ray scattering (IXS) measurements on graphite. [1] AbInit: www.abinit.org, DP-code: www.dp-code.org [2] K. Sturm, L. E. Oliveira, Phys. Rev. B, 22 (1980), 6268