Radial-tangential mode of single-wall carbon nanotubes manifested by Landau regulation: reinterpretation of low- and intermediate-frequency Raman signals.

The low-frequency Raman signals of single-wall carbon nanotubes (SWNTs), appearing in the range of 100-300 cm, have been interpreted as radial-breathing mode (RBM) comprising pure radial Eigenvectors. Here, we report that most of the low-frequency and intermediate-frequency signals of SWNTs are radial-tangential modes (RTMs) coexisting radial and tangential Eigenvectors, while only the first peak at the low-frequency side is the RBM. Density functional theory simulation for SWNTs of ~ 2 nm in diameter shows that dozens of RTMs exhibit following the RBM (~ 150 cm) up to G-mode (~ 1592 cm) in order with Landau regulation. We specify the RBM and the RTM on Raman spectra obtained from SWNTs, where both appear as prominent peaks between 149 and 170 cm and ripple-like peaks between 166 and 1440 cm, respectively. We report that the RTMs have been regarded as RBM (~ 300 cm) and ambiguously named as intermediate-frequency mode (300-1300 cm) without assignment. The RTMs gradually interlink the RBM and the G-mode resulting in the symmetric Raman spectra in intensity. We reveal high-resolution transmission microscope evidence for a helical structure of SWNTs, informing the typical diameter of commercial SWNTs to be 1.4-2 nm.