Calculated dependence of vibrational band frequencies of single-walled and double-walled carbon nanotubes on diameter

Abstract

We have used density functional theory (DFT) at the B3LYP/6-31G level to calculate Raman and IR spectra of zigzag ( n,0) single-walled carbon nanotubes (SWNTs) and ( n,0) and (2 n,0) double-walled carbon nanotubes (DWNTs), for n ranging from 6 to 19 and 6 to 8, respectively. In the low frequency RBM region, calculated Raman spectra of SWNTs indicate that there are three vibrational modes, with symmetries A 1g, E 1g and E 2g, whose frequencies depend strongly on nanotube diameter. The E 2g mode is not only diameter dependent, but also depends on whether the number of hexagons formed in the circumference direction of the CNT is even or odd. Two IR spectral modes (of A 2u and E 1u symmetries) are found in calculated IR spectra that show strong diameter dependence. Also, three Raman bands with E 1g, A 1g and E 2g symmetries found are to exist in the G-band region. For this latter case, computed spectra indicate that while Raman bands with A 1g symmetry essentially remain constant for even number of hexagons formed in the circumference direction, (e.g., (0,2 n)-type CNTs with band position 1526 ± 0.5 cm −1), bands corresponding to odd number of hexagons, i.e., (0,2 n + 1)-type CNTs, are diameter dependent. The frequencies of the E 1g and E 2g modes (in the G-band region) are not only strongly diameter dependent, but also converge towards one another with increasing tube diameter. This latter type of behavior can lead to erroneous classification of nanotubes as metallic or semiconducting, since partially overlapping bands in the G-band region might result in bands that appear to have diffuse shoulders, a characteristic of metallic SWNTs. The RBMs for DWNTs are also strongly diameter dependent and are blue-shifted relative to their corresponding RBMs in the spectra of SWNTs. The relative distance between RBMs vibrational modes in the spectrum of a selected DWNT is larger than that for the corresponding SWNTs. The electron density for small-sized DWNT, e.g., (6,0)&(12,0), indicates an intratube (inner-outer tube) σ-bonding in the excited state.