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 ~ 2 nm in diameter shows that dozens of RTMs exhibit following the RBM (~ 150 cm-1 ) up to G-mode (~ 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 (~ 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.