Phonons in Graphitic Tubules

Abstract

The phonon dispersion and density of states of graphitic tubules with various helicities and diameters are calculated with a tight-binding molecular-dynamics method (TBMD). Compared with graphite, phonon modes in tubules are softened by the curvature. Unique features of the graphitic tubule, with special emphasis on low-frequency modes, are discussed. The symmetry of phonon modes is analyzed, and infrared (IR) and Raman active modes are identified. It is found that not only the frequencies but also the number of IR and Raman modes depend on tubule helicity. Sound velocities in graphitic tubules are also calculated as functions of tubule helicity and diameter.