Analytical RPA response of Carbon and BN single-walled nanotubes: Application to EELS and wave loss spectra

Abstract

We calculate the local RPA (Random Phase Approximation) polarizability of single-walled carbon and BN nanotubes and apply our results to the loss spectra of electrons (EELS: Electron Energy Loss Spectrum) and photons in the case of excitations. We take into account crystal local field effects (CLFE) by developing a two-order matricial treatment. For spatially periodic tubules, the whole calculation (one-electron and RPA steps) is analytical and leads to simple formulas. We compare our results to other RPA-type ones where CLFE are neglected (fig.1). We also examine opened tubule where the analytical derivation is only possible for the one-electron polarizability, the RPA step being made numerically. We operate a characterization of the excitation modes by determining the local phase-shifts and the displaced charges on each site. The general conclusion is that EELS and electromagnetic wave spectra exhibit common excitation modes with differences in intensities depending on the conditions of the interaction. An interesting result is given by EELS with an incident beam direction parallel to the tubule axis when the impact parameter is varied: for a central interaction or zero impact parameter (respectively, for a large impact parameter) the spectrum is closely similar to the wave spectrum with parallel polarization (respectively, with perpendicular polarization). The evolution of the various modes with tubule size is also studied.