Raman-active modes in finite and infinite double-walled carbon nanotubes

The calculation of the nonresonant Raman spectrum of double-walled carbon nanotubes (DWCNT's) is performed in the framework of the bond polarization theory, using the spectral moment method. The calculation of the Raman spectrum of DWCNT's as a function of the diameter and chirality of the inner and outer tubes allows us to derive the diameter dependence of the frequencies of the breathing-like modes (BLM's) and tangential-like modes (TLM's) in a large diameter range. In particular, the diameter dependence of the frequencies of the two radial breathing-like modes (RBLM's) resulting from the in-phase and counterphase coupled motions of the totally symmetric radial breathing mode (the so-called RBM) of the inner and outer tubes is discussed. The Raman spectrum of a bundle of identical DWCNT's is also calculated. The dependence of the spectrum with the size of the bundle is analyzed. Additional breathing-like modes are predicted in DWCNT bundles of finite size. The majority of these peaks rapidly vanish when the tube length increases, except a totally symmetric mode that subsists in the DWCNT crystal.