Phonon localization in single wall carbon nanotube: Combined effect of ¹³C isotope and vacancies

The combined effect of ¹³C isotope doping and vacancies on the phonon properties of a single-wall carbon nanotube is theoretically investigated using the forced oscillation method. The phonon density of states (PDOS) is calculated for all (0%–100%) ¹³C isotope contents and wide (0%–30%) vacancy concentrations. We found a redshift of the Raman active E_2g peak in the PDOS with increasing isotope contents, while the disappearance of the E_2g peak and the appearance of a new sharp peak in the low-energy region with increasing combined defects. Both ¹³C isotope and combined defects cause the localization of the high-energy optical phonons. We calculated the typical mode patterns for the in-plane longitudinal optical phonon to visualize the localization phenomena elaborately at the presence of isotope and vacancies. The calculated average localization length shows an asymmetric behavior with increasing ¹³C isotope concentrations which is in good agreement with the ¹³C isotope dependence localization length of single-layer graphene. We noticed that a typical localization length is on the order of ∼1 nm at 70% isotope concentrations. The combined effect of ¹³C isotope and vacancies shows an abruptly decreasing localization length with increasing defect densities. These results are important to understand the heat conduction as well as nanoscopic vibrational studies such as tip-enhanced Raman spectra in carbon nanotubes where the local phonon energies may be mapped.