Resonant Raman intensity of the radial-breathing mode of single-walled carbon nanotubes - approximations

Fig. 1. Maximum Raman intensity.

For the calculation of the resonant Raman intensity of any nanotube, one can start from a model of the electronic band structure of the nanotube[1,2]. Then, the resonance Raman profile can be calculated from the quantum-mechanical expression derived in third-order perturbation theory[3,4]. Two approximation schemes of different complexity are usually applied for modelling the features of the resonance Raman profiles (RRPs) of the radial-breathing mode (RBM) due to different optical transitions. I. First of all, one assumes that the features due to the different optical transitions are enough far away so that no interference between them takes place. 1. Additionally, one can assume the matrix element of the momentum and electron-phonon coupling as wavevector-independent. The remaining summation over the wavevector can be accomplished analytically. Then, the maximum intensity of each feature of a given RRP can be presented as[3,4] , where is a weakly tube-dependent quantity. 2. Further simplification of the expression of the RRP can be achieved assuming that the p's and D's are not only wavevector-independent but are also tube-independent. In this case the maximum intensity is given by[3,4] The results of the calculation of (or ) and for all 300 SWNTs in the radius range from 2 Ǻ to 12 Ǻ for transition energies up to 3.5 eV are shown in Figs. 1 and 2 (Refs. [3,4]). References: 1. V. N. Popov, New J. Phys. 6 (2004) 1-17. 2. V. N. Popov and L. Henrard, Phys. Rev. B 70 (2004) 115407. 3. V. N. Popov, L. Henrard, and Ph. Lambin, Nano Letters 4 (2004) 1795-1799. 4. V. N. Popov, L. Henrard, and Ph. Lambin, Phys. Rev. B 72 (2005) 035436.

Fig. 2. Maximum Raman intensity (or which has qualitatively the same behavior).

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Valentin Popov

July 22, 2005