Study of collective radial breathing-like modes in double-walled carbon nanotubes: combination of continuous two-dimensional membrane theory and Raman spectroscopy

Dmitry I Levshov,Alexander V Osadchy,Sergei B Rochal,Matthieu Paillet,Yuri I Yuzyuk,Dmitry V Rybkovskiy,Marina V Avramenko,Thierry Michel,Raul Arenal,Xuan-Tinh Than,Jean-Louis Sauvajol

doi:10.1117/1.jnp.10.012502

Abstract

Radial breathing modes (RBMs) are widely used for the atomic structure characterization and index assignment of single-walled carbon nanotubes (SWNTs) from resonant Raman spectroscopy. However, for double-walled carbon nanotubes (DWNTs), the use of conventional ωRBM(d) formulas is complicated due to the van der Waals interaction between the layers, which strongly affects the frequencies of radial modes and leads to new collective vibrations. This paper presents an alternative way to theoretically study the collective radial breathing-like modes (RBLMs) of DWNTs and to account for interlayer interaction, namely the continuous two-dimensional membrane theory. We obtain an analytical ωRBLM(do,di) relation, being the equivalent of the conventional ωRBM(d) expressions, established for SWNTs. We compare our theoretical predictions with Raman data, measured on individual index-identified suspended DWNTs, and find a good agreement between experiment and theory. Moreover, we show that the interlayer coupling in individual DWNTs strongly depends on the interlayer distance, which is manifested in the frequency shifts of the RBLMs with respect to the RBMs of the individual inner and outer tubes. In terms of characterization, this means that the combination of Raman spectroscopy data and predictions of continuous membrane theory may give additional criteria for the index identification of DWNTs, namely the interlayer distance.

Highlights

A double-walled carbon nanotube (DWNT) has two concentric carbon layers
In our recent work, combining high-resolution transmission electron microscopy (HRTEM), ED, and resonant Raman spectroscopy on individual free-standing DWNTs,[11] we have shown that the strength of this “inner” tube–“outer” tube interaction, which is dependent on the interlayer distance in DWNTs, strongly affects the out-of-phase radial breathing-like mode (RBLM) frequencies
Our individual DWNTs are ultralong, clean from amorphous carbon, and have no signature of the D-band in Raman spectra.[13,14]. Five of these 14 individual DWNTs had both RBLMs observed in the Raman spectra

Summary

Introduction

A double-walled carbon nanotube (DWNT) has two concentric carbon layers. This structure makes it an ideal system for studying the effects of interwall coupling on the physical properties of carbon nanotubes (CNTs). In our recent work, combining high-resolution transmission electron microscopy (HRTEM), ED, and resonant Raman spectroscopy on individual free-standing DWNTs,[11] we have shown that the strength of this “inner” tube–“outer” tube interaction, which is dependent on the interlayer distance in DWNTs, strongly affects the out-of-phase radial breathing-like mode (RBLM) frequencies. We proposed that the positions of the radial breathing-like modes (RBLMs) and the difference between experimental and theoretical calculations within atomistic valence force field model[12] can be used as criteria to evaluate diameters of the inner and outer tubes in nonindex-identified DWNTs. This paper presents an alternative theoretical way to study low-frequency collective modes of DWNTs based on the two-dimensional (2-D) continuous membrane theory. We show that this approach provides a new and straightforward way to quantitatively analyze RBLMs of coupled systems

Experimental Details

Results

Conclusions