IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER J. Phys.: Condens. Matter 20 (2008) 015204 (5pp) doi:10.1088/0953-8984/20/01/015204 Finite-size effect on the Raman-active modes of double-walled carbon nanotubes K Sbai 1 , A Rahmani 1,3 , H Chadli 1 and J-L Sauvajol 2 1 Equipe de Physique Informatique et Mod´ elisation des Syst` emes, Universit´ e MY Ismail, Facult´ e des Sciences, BP 11201, Zitoune, 50000 Mekn` es, Morocco 2 Laboratoire des Colloides, Verres et Nanomat´ eriaux (UMR CNRS 5587), Universit´ e Montpellier II, 34095 Montpellier Cedex 5, France Received 27 July 2007, in final form 4 October 2007 Published 29 November 2007 Online at stacks.iop.org/JPhysCM/20/015204 Abstract The dependence of the breathing-like phonon modes (BLM) and tangential-like phonon modes (TLM) of individual, finite and infinite bundles of double-walled carbon nanotubes (DWCNTs) as a function of the relative lengths of the inner ( L i ) and outer ( L o ) tubes is calculated by using the spectral moments method in the framework of the bond-polarization theory. Depending on the relative lengths of the inner ( L i ) and outer ( L o ) tubes, additional modes are evidenced in the BLM region. These modes must be considered in the analysis of the experimental data. 1. Introduction Carbon nanotubes [1] have become a standard material in nanotechnology and a variety of applications using nanotubes have been proposed for investigation. Single-walled carbon nanotubes (SWCNTs) have been intensively studied during the past decade. One of the most important subjects for nanotube technology is to characterize nanotubes in a simple and quick way. Among the techniques extensively used to characterize these materials, Raman spectroscopy provides important information regarding both the electronic and the phonon spectrum of carbon nanotubes [2]. To derive this information, the Raman data have been correlated with theoretical predictions [3–5]. In this context, in the framework of the bond-polarization theory and using the spectral moments method [6, 7], we previously calculated the nonresonant polarized Raman spectra of both chiral and achiral SWCNTs as a function of the tube diameter and length [8]. Double-walled carbon nanotubes (DWCNTs), which consist of two concentric cylindrical graphene layers, are synthesized by a catalytic chemical vapor deposition method (CCVD) [9–11] and by thermal conversion of fullerene C 60 encapsulated in SWCNTs [12]. Phonons in DWCNTs have been extensively investigated by Raman scattering [13–17]. In this context, the Raman responses of infinite DWCNTs have been calculated in the framework of the bond-polarization theory, using the spectral moments method [18]. Relations have been derived which describe the dependence of the radial breathing-like mode (RBLM) frequencies with the diameter of 3 Author to whom any correspondence should be addressed. the inner and outer tubes. It was found that the frequencies in the breathing-like mode (BLM) and tangential-like mode (TLM) regions of DWCNTs significantly differ from those calculated for single-walled carbon nanotubes. For diameters of inner tubes ( D i ) and outer tubes ( D o ) in the range 0.6– 2.2 nm and 1.2–3.2 nm, respectively, we found, unlike the expressions in [18], that the diameter dependence frequencies of the in-phase (low-frequency component: ω LF ) and out-of- phase (high-frequency component: ω HF ) RBLM was described by the following relations: ω LF (cm −1 ) =−734.3/ D i + 118.1/ D 2 i + 946.1/ D o + 564.8/ D 2 o (1) and ω HF (cm −1 ) = 128.0 + 188.6/ D i − 300.0/ D o + 295.6/ D 2 o . (2) These expressions well describe the diameter dependence of both RBLMs in the range investigated. As expected for infinite diameters, the in-phase mode, which correspond to a transverse acoustic mode, has a zero frequency and the out- of-phase mode, which correspond to the breathing mode of a bilayer of graphene, has a frequency of 128 cm −1 , close to that of the B 2g graphite phonon mode [19]. In this paper we extend our previous study by considering DWCNTs with inner and outer tubes of different lengths. Indeed, some experiments show short portions of inner tubes inside outer tubes. Contrary to that, after different treatments, inner tubes can go beyond outer tubes. Because these different structures exist in a real sample, we calculate the nonresonant Raman spectra of individual and bundled DWCNTs consisting 0953-8984/08/015204+05$30.00 © 2008 IOP Publishing Ltd Printed in the UK 1