The effect of ${}^{13}$C isotope impurities on the phonon properties of graphene is discussed theoretically. We calculated the values of the phonon lifetimes due to isotope impurity scattering for all values of densities, isotopic masses, and for all wave vectors using second-order perturbation theory. We found that for natural concentrations of ${}^{13}$C, the contribution of isotopic scattering to the phonon lifetime of the optical modes is negligible when compared to the electron-phonon interaction. Nevertheless, for atomic concentrations of ${}^{13}$C as high as $\rho =0.5$ both contributions become comparable. Our results are compared with recent experimental results and we find good agreement both in the ${}^{13}$C atomic density dependence of the lifetime as well as in the calculated spectral width of the G-band. Due to phonon scattering by ${}^{13}$C isotopes, some graphene phonon wave functions become localized in real space. Numerical calculations show that phonon localized states exist in the high-energy optical phonon modes and in regions of flat phonon dispersion. In particular, for the case of in-plane optical phonon modes, a typical localization length is on the order of 3 nm for ${}^{13}$C atomic concentrations of $\rho \approx 0.5$. Optical excitation of phonon modes may provide a way to experimentally observe localization effects for phonons in graphene.

• Received 12 April 2012

DOI:https://doi.org/10.1103/PhysRevB.85.245406