Symmetry of the High-Energy Modes in Carbon Nanotubes

Abstract

Introduction. Group theory predicts 15 or 16 Raman active modes for single-walled nanotubes, irrespective of their diameter [1]. In the carbon-bond stretching region ( 1600 cmy1) several peaks are observed in the Raman spectra. Based on lattice dynamical calculations these peaks have been assigned to A1g, E1g and E2g modes [1,2]; ab initio calculations also find modes of these symmetries at these energies [3]. Calculations of the Raman polarizability predict the E1g modes to be the strongest [4,5]. No polarized Raman spectra from crystals of nanotubes have been published to confirm this assignment. Recently, Sun et al. produced samples with aligned small-diameter tubes (d 5 A) embedded in a host with cylindrical holes [6]. From their spectra they identify an A1g, an E1g, and an E2g mode. Single crystals not being available, a way to determine the symmetries of modes in Raman scattering is through depolarization ratios (D) [7]. While in general it is not possible to uniquely derive the sign or magnitude of the tensor elements, one can distinguish traceless tensors (E1g and E2g) from fully symmetric tensors (A1g).