Collective excitations in metallic graphene tubules

The collective modes of coaxial-graphene-tubule systems are calculated using both classical and quantum-mechanical approaches. While the derived results are valid for any number of coaxial tubules, the numerical results for the dispersion relations of the plasmons are presented for a system consisting of four graphene tubules for various values of the azimuthal quantum number m. The lowest modes (m=0) are found to be quasiacoustic, like the one-dimensional plasmons, whereas the higher modes are of optical nature. For each m, the number of observable modes is equal to the number of tubules having a metallic character. The effect of the intertubule interactions is a broadening of the frequency range of the m modes. This broadening remains important so long as the plasmon momentum is significantly smaller than the Fermi momentum.