Continuum model for long-wavelength phonons in two-dimensional graphite and carbon nanotubes

A continuum model yielding dispersion relations for long-wavelength acoustic and optical phonons in two-dimensional graphite and achiral single-walled carbon nanotubes is developed. Close relations with the theory of thin elastic plates and cylindrical shells are established. The effective thickness of a nanotube determining the limits of validity of the continuum model is estimated. A qualitative analysis of vibrations of chiral nanotubes is presented.

Figure 1

(Color online) Dispersion relations of acoustic phonons for a (10,10) armchair carbon nanotube. The choice of frequency units facilitates observation of the long-wavelength and short-wavelength limiting values given in the text.

Figure 2

(Color online) Dispersion relations of acoustic phonons for a (10,10) armchair carbon nanotube around zero frequency. This figure reveals the role of the corrective terms introduced in the text.

Figure 3

Approximation of the optical phonon dispersion relationship for 2D graphite in the $\mathrm{\Gamma }\text{−}M$ direction.

Figure 4

Calculated dispersion relations of acoustic and optical phonons for a (10,10) armchair carbon nanotube. Here $a=\sqrt{3}{a}_{\mathrm{C}\text{−}\mathrm{C}}=2.46\mathrm{\AA }$.