Resonant Raman intensity of the RBM and A1 G modes of single-walled carbon nanotubes

Fig. 1. Resonance Raman profile of the radial-breathing mode (RBM) and the A1 G modes of chiral single-walled carbon nanotubes (SWNTs) belonging to the Mod0, Mod1 and Mod2 groups of tubes[5]. The sequence of the optical transitions is that of the π-band tight-binding model. Notice the complicated shape of the features of the RRP of the different optical transitions. Although this circumstance poses a serious task before the approximate description of the RRP, such a description can effectively be realized[5].

The Raman scattering of light in single-walled carbon nanotubes is usually observed under resonant conditions, i.e., when the laser photon energy is close to an optical transition of a nanotube[1,2]. The quantum-mechanical description of the Raman scattering process can be done considering the system of electrons, photons and phonons, and their interactions[3,4]. The Raman intensity of the most resonant Stokes process is given by . Here, , ; is the energy of the initial state, is the incident photon energy (laser excitation); , , are the energies of the intermediate (a, b) and final (f) states of the system; and are momentum matrix elements for incident and scattered photons, resp.; is the electron-phonon matrix element [5]; γ is the sum of the halfwidths of conduction and valence states [6]. The dependence of the Raman intensity on the laser excitation energy for a given tube is the resonance Raman profile (RRP). It consists of features, corresponding to the different optical transitions (Fig. 1). The shape of the RRP for a given optical transition Eii consists of an in-coming and an out-going resonances at energies and, where Eph is the phonon energy. In the case of the radial-breathing mode (RBM), and the two resonances overlap, giving rise to a single, bell-like peak. For the A1 G-band modes, normally and the two resonances may not overlap. For metallic tubes, the closeness of the two components of the warping effect splitting may lead in addition to a quite complicated shape of the RRP[5]. References: 1. V. N. Popov, New J. Phys. 6 (2004) 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. 5. V. N. Popov and Ph. Lambin, Phys. Rev. B 73 (2006) 165425. 6. V. N. Popov and Ph. Lambin, Phys. Rev. B 74 (2006) 075415.

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Valentin Popov

January 28, 2006