Optically active cross-band transition in double-walled carbon nanotube and its impact on Raman resonances

Double-walled carbon nanotubes (DWNTs), often regarded as the simplest one-dimensional (1D) van der Waals moiré superlattices, are the ideal structures to explore the interlayer coupling, including mechanical and electronical coupling, at the nanoscale in 1D systems. By combining spatial modulation spectroscopy (SMS) and resonant Raman scattering (RRS) experiments on well-identified individual free-standing DWNTs with the calculations of their optical absorption by using the recursion method with non-orthogonal tight-binding (NTB) basis functions, we are able to unambiguously assign the nature, namely mirror transition or cross transition, of each optical transitions in DWNTs. Our study identifies and allows to determine the conditions of the mixing of the electronic states leading to the appearance of cross transitions in the optical absorption of DWNTs, beyond of the so-called Koshino's criteria previously derived. On the other hand, the striking behavior of the resonant excitation profile (REP) of the Raman active modes around the cross transition reveals the strong sensitivity of the modes intensity on the electronic interlayer coupling in DWNTs, emphasizing the relevance of Raman spectroscopy for studying the electronic and mechanical coupling at the nanoscale.