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

Abstract

We have measured a strictly linear π 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 (LFE) cause a mixing of electronic transitions, including the 'Dirac cone', resulting in the observed linear dispersion. PACS numbers: 73.20.Mf,78.20.Bh Single-wall carbon nanotubes (SWNT) and its parent compound graphene are archetypes of low dimensional systems with strongly anisotropic and unique electronic properties which make them interesting for both fundamental research and as building blocks in nanoelectronic applications [1]. Their electronic bandstructure is frequently studied. In graphene, the linear band dispersion at the Fermi level, the 'Dirac cone', leads to unique characteristics in nanoelectronic devices [2]. One can expect a strong analogy between graphene and isolated SWNT for excitations along the sheet and along the tube axis, respectively. Within the zone-folding model, i.e. neglecting curvature effects, the graphene bandstructure is sliced along parallel lines when the sheet is rolled up into a cylinder. The result are characteristic van Hove singularities (VHS) in the density of states (DOS) [3]. Bulk (i.e. bundled) SWNT show an optical absorption peak at ∼ 4.5 eV due to transitions of the π electrons [4]. In vertically aligned SWNT (VA-SWNT) one finds the same peak position for onaxis polarization and an additional peak for perpendicular polarization at ∼ 5.2 eV [5]. Further information can be obtained from collective electronic excitations (plasmons) beyond the optical limit [6] (i.e. momentum transfer q > 0). Angle resolved electron energy loss spectroscopy (EELS) assesses the detailed plasmon dispersion [7,8], but it is so far missing for freestanding isolated sp 2 carbon systems. Models based on the homogeneous electron gas [9], or the tight-binding scheme [10,11] have been used to describe these excitations. The former are however bound to metallic systems. The latter have provided valuable insight and predictions for the properties of isolated sheets, tubes, and assemblies of these objects; in particular, they have predicted an almost linear plasmon dispersion for isolated systems. However, the tight binding results neglect screening beyond the π bands, and they depend on parameters that hide the underlying complexity. No realistic parameter-free calculations have been performed to predict the plasmon dispersion in these systems, nor has its origin been analyzed. Instead, ab initio spectroscopy calculations have dealt with absorption spectra (q → 0) for SWNT [12,13,14],...

TDDFT study of time-dependent and static screening in graphene

“…For example, the π plasmon dispersion in graphene and carbon nanotubes splits if Q is in the → M direction but does not split if it is in the → K direction. [37][38][39] This means we should average Q over the high symmetry directions.…”

Empirical evaluation of attractive van der Waals potentials for type-purified single-walled carbon nanotubes

“…Transitions associated with the σ/π electronic states emerge in the 12-to-20 eV range. [34][35][36][37][38] Since we are interested in a generic chiral mixture of specific electronic types, our approach is to model the Drude regime with type-specific dielectric spectroscopy data and the π regime with type-specific UV-Vis-NIR spectroscopy data. While the anisotropy and type dependence of the π plasmon is experimentally accessible, less is known about the σ + π plasmon.…”

“…For the nanotubes, the σ + π features are thus deduced from EELS of sparse mixed-type films extrapolated to zero momentum transfer. 35,36 We use the common π plasmon to bridge our measured optical spectra to the EELS spectra, with the f -sum rule and the measured ratio of mean diameters as a constraint. For the PDMS substrate, we use spectra based on low-loss EELS measured in the context of cryo-TEM and referenced to the well-known dielectric response function of amorphous ice.…”

Many‐body effects in optical response of graphene‐based structures

“…For graphene, the LF effects cause a mixing of electronic transitions thus substantially changing the loss function at large momentum transfer. [39] …”