The radial breathing mode frequency in double-walled carbon nanotubes: an analytical approximation

We study the breathing-like phonon modes of double-walled carbon nanotubes in a simple analytical model by considering the tube walls as coupled oscillators. The force constant of the oscillator coupling is found to be proportional to the inner tube diameter. Thus, only for small-diameter tubes, the shift of the breathing-like phonon mode frequencies (relative to the radial breathing modes of the isolated layers) scales with the tube diameter D. For tubes with larger diameter the in-phase breathing-like phonon mode frequency is inversely proportional to the diameter (like the radial breathing modes of the single wall carbon nanotubes), while the out-of-phase mode approaches asymptotically, as 1/D-2, the graphite B-2g phonon mode frequency.