Vibrational Analysis of Carbon Nanotube Based Nanomechanical Resonators

A vibrational analysis of three types of carbon nanotube-based nanomechanical resonator is presented. Harmonic vibrational frequencies and the associated normal modes are evaluated through diagonalization of the full mass-weighted Hessian matrix, where a very large mass is assigned to the relevant carbon atoms to represent the constraints arising as a consequence of the different resonator configurations. The vibrational frequencies are determined for carbon nanotubes of different dimensions, and the response of the resonators to an applied mass is studied. For the flexural modes that are relevant for mass-sensing resonator devices, the calculations show the resonant frequency to increase as the tube diameter increases. For the longest nanotubes studied, the frequencies for cantilever and bridged resonators are very similar, and double-walled nanotubes have resonant frequencies that lie between the frequencies of the component single-walled nanotubes. The vibrational modes for a shuttle resonator have also been determined, and the lowest frequency mode was found to correspond to the relative rotation of the nanotubes with frequencies in the range of 70-120 GHz. The calculations predict a sensitivity of up to 1030 Hz/g although the response of the flexural modes of suspended nanotubes is dependent on the location of the adsorbed mass, while the response based upon the relative rotational motion in double-walled nanotubes is independent of the position of the adsorbed mass