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(-1), 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 similar to 2 nm in diameter shows that dozens of RTMs exhibit following the RBM (similar to 150 cm(-1)) up to G-mode (similar to 1592 cm(-1)) 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(-1) and ripple-like peaks between 166 and 1440 cm(-1), respectively. We report that the RTMs have been regarded as RBM (similar to 300 cm(-1)) and ambiguously named as intermediate-frequency mode (300-1300 cm(-1)) 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.