Fig. 1. Resonance Raman profile of the RBM of chiral (C) SWNTs
belonging to the Mod0, Mod1 and Mod2 groups of tubes. The sequence of the
optical transitions is that of the π-band tight-binding model.
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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. Each feature can then be characterized by its height (or maximum intensity) Im and shape function F(EL) (Fig. 2). 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. |
Fig. 2. The shape of the RRP, associated with an optical transition is
due to an in-coming and out-going resonances, which overlap partially in the
case of the RBM, giving rise to a single, bell-like peak. This peak can be
characterized by its height (or maximum intensity) Im and the shape function F(EL). The
figure shows the peak, due to transitionE33
of nanotube (13,6) with radius of about 6.6 Ǻ, calculated with constant γ = 0.03 eV and Eph = 190 cm-1.
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Valentin Popov