Collective resonances in carbon nanotubes

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Abstract
A theory is presented to study the plasma oscillations of the full σ+π electron system of carbon nanotubes. Cylindrical charge density was supposed to represent the electron distribution within the shells of the tubule and an integral equation is derived for the perturbation caused by an external electric field. From the solution of this equation the dynamical dipole polarizability is calculated in the energy range above 10 eV. For single-shell nanotubes a double-peaked spectrum resulted resembling the slow tangential and the fast radial dipole active mode of a classical spherical shell of finite width. The peak positions are around 17 and 22 eV for the smallest tubule radius and with increasing radius the low-energy peak shifts to a lower energy, the high-energy peak to a higher energy. For multishell nanotubes a strong shell-shell interaction is found, as a result of which the double-peaked structure of the innermost shell will be the dominant part in the imaginary part of the polarizability. With increasing shell number the low-energy peak shifts to a higher energy, the high-energy peak to lower energy; this is just the opposite tendency found in the radius dependence. For a large enough shell number the two peaks coincide at about 19 eV, and this value is independent of the radius of the innermost shell. Although the numbers mentioned may vary within 3–4 eV depending on the parameters of our models, the tendencies are in overall agreement with the existing electron energy loss spectroscopy measurements and with other theoretical predictions.