Probing the phonon dispersion relations of graphite from the double-resonance process of Stokes and anti-Stokes Raman scatterings in multiwalled carbon nanotubes

The Stokes and anti-Stokes Raman spectra of a multiwalled carbon nanotube (MWNT) sample are studied here by four excitation energies and the observed Raman modes are assigned based on the double resonance Raman effect and the previous results in graphite whiskers. There exists frequency discrepancy between Stokes and anti-Stokes lines (FDSA) of many Raman modes in MWNT’s and the discrepancy values are strongly dependent on the excitation energy, in which the FDSA value of the $D^{\prime }$ mode even changes from a positive value $(9\mathrm{}{\mathrm{cm}}^{-1},$ 1.58 eV) to a negative value (-11 ${\mathrm{cm}}^{-1},$ 2.54 eV). The laser-energy dependence of the FDSA values of some modes in MWNT’s is attributed to the nonlinear frequency dependence of Stokes and anti-Stokes Raman lines of these modes on the excitation energy. Raman results and the theoretical analysis of the intravalley and intervalley double resonance processes of Stokes and anti-Stokes Raman scatterings both show that the frequency of an anti-Stokes peak excited by ${\varepsilon }_L$ is equal to that of the corresponding Stokes peak excited by a laser excitation of ${\varepsilon }_L+ħ{\omega }_S$ where $ħ{\omega }_S$ is the phonon energy of the Raman mode. Stokes and anti-Stokes double-resonance Raman scatterings have been used to probe the phonon dispersion relations of graphite. The Raman data of the well-known disorder-induced D mode are in good agreement with the theoretical results.