Helical inter −π−−π− band collective excitations in single-walled carbon nanotubes with chiral symmetry

We present a detailed theoretical study of the collective π-electronicπ-electronic excitations associated with electron-density fluctuations (π-plasmonsπ-plasmons) in individual single-walled carbon nanotubes (SWCNTs) possessing chiral symmetry. Unlike previous theoretical treatments of this problem, based on a cylindrical-symmetry model, we use a different approach, initiated by White et al. [Phys. Rev. B 47 (1993) 5485], which properly takes account of genuine helical and rotational symmetries of the chiral SWCNTs. Taking into account the Coulomb interaction between π-electronsπ-electrons within the random-phase-approximation, we derive an explicit expression for the dynamic-dielectric-response function of the SWCNTs in helical representation and apply it to the numerical calculation of the dispersion and damping of the π-plasmonsπ-plasmons in a number of representative SWCNTs with chiral symmetry. Multiple (up to eight) well-defined inter-π-band-plasmoninter-π-band-plasmon modes, with wave vectors lying in the direction of the helical line of such nanotubes, are shown to exist for each value of the quantum angular momentum L   independent of whether the nanotubes are metallic or semiconducting. These π-plasmonπ-plasmon modes, which we refer to as helical π-plasmonsπ-plasmons, are found to have several unexpected dispersive features, making them distinctly different from those obtained previously within the above-mentioned cylindrical-symmetry model. In particular, none of the helical π-plasmonπ-plasmon branches starts from a zero value of the helical wave-number qHqH, and in all the calculated spectra of the helical π-plasmonsπ-plasmons, there exists the highest-lying branch of the dispersive π-plasmonπ-plasmon modes, which either extends up to the energy of about 15 eV or makes its start with this energy. This highest-lying branch in the spectrum of the π-plasmonsπ-plasmons is found to have the largest spectral weight in the electron-energy losses, leading to the appearance of the dominant 15-eV-resonant peak in the low-loss spectra of the nanotubes under consideration. This possibly suggests an explanation, on the microscopic level, to the origin of the intensive resonant feature at ∼15eV, observed in recent electron-energy-loss-spectroscopy experiments on isolated SWCNTs.