Long-wavelength optical phonons in single-walled boron nitride nanotubes

A unified macroscopic continuum model is proposed to calculate the long-wavelength optical phonons of single-walled boron nitride nanotubes. The effect of the polarity in the boron nitride nanotubes on optical vibrations is considered by this model. We obtain the dispersion relation of the optical phonons. The avoided crossing between longitudinal and transverse optical phonon branches arises from the indirectly coupling between different phonon branches. The diameter dependence of the optical phonon frequency is derived. Our results are in agreement well with the experiment ones [R. Arenal, A.C. Ferrari, S. Reich, L. Wirtz, J.-Y. Mevellec, S. Lefrant, A. Rubio, A. Loiseau, Nano Lett. 6 (2006) 1812].