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

Abstract

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.