Temperature dependence of radial breathing mode Raman frequency of single-walled carbon nanotubes

Recent high-temperature studies of Raman-active modes in single-walled carbon nanotube (SWNT) bundles report a softening of the radial and tangential band frequencies with increasing sample temperature. A few speculations have been proposed in the past to explain the origin of these frequency downshifts. In the present study, based on experimental data and the results of molecular dynamics simulations, we estimate the contributions from three factors that may be responsible for the observed temperature dependence of the radial breathing mode frequency $[{\ensuremath{\omega}}_{\mathrm{RBM}}(T)].$ These factors include thermal expansion of individual SWNTs in the radial direction, softening of the C-C (intratubular) bonds, and softening of the van der Waals intertubular interactions in SWNT bundles. Based on our analysis, we find that the first factor plays a minor role due to the very small value of the radial thermal expansion coefficient of SWNTs. On the contrary, the temperature-induced softening of the intra- and intertubular bonds contributes significantly to the temperature-dependent shift of ${\ensuremath{\omega}}_{\mathrm{RBM}}(T).$ For nanotubes with diameters $(d)g~1.34\mathrm{nm},$ the contribution due to the radial thermal expansion is \ensuremath{\leqslant}4% over the temperature range used in this study. Interestingly, this contribution increases to \ensuremath{\geqslant}10% in the case of nanotubes having $dl~0.89\mathrm{nm}$ due to the relatively larger curvature of these nanotubes. The contributions from the softening of the intra- and intertubular bonds are approximately equal. These two factors together contribute a total of about \ensuremath{\sim}95% and 90%, respectively, for SWNTs having $dg~1.34\mathrm{nm}$ and \ensuremath{\leqslant}0.89 nm.