The radial breathing mode frequency in double‐walled carbon nanotubes: an analytical approximation

Abstract

We study the breathing-like phonon modes of double-walled carbon nanotubes in a simple analytical model by considering the tube walls as coupled oscillators. The force constant of the oscillator coupling is found to be proportional to the inner tube diameter. Thus, only for small-diameter tubes, the shift of the breathing-like phonon mode frequencies (relative to the radial breathing modes of the isolated layers) scales with the tube diameter D. For tubes with larger diameter the in-phase breathing-like phonon mode frequency is inversely proportional to the diameter (like the radial breathing modes of the single wall carbon nanotubes), while the out-of-phase mode approaches asymptotically, as 1/D 2 , the graphite B 2g phonon mode frequency.Due to the remarkable physical properties and potential applications carbon nanotubes (CNTs) have been investigated intensively since their discovery by Iijima [1]. In particular, the vibrational properties of single-wall carbon nanotubes (SWCNTs) [2] have been studied extensively by many research groups [3][4][5]. More recently, experimental studies of double-walled CNTs have appeared [6,7] focussing on the role of the wall-to-wall interaction for the so-called radial breathing mode of the tubes. For isolated single-walled tubes this frequency is generally taken to be a good indicator of the tube diameter D. For double-walled tubes the simple 1/D dependence of the frequency is modified by the presence of the additional wall-to-wall interaction. Popov and Henrard [8] studied the breathing-like phonon modes (BLMs) of double-and multi-walled CNTs within a valence-force field model and a continuum model, showing clearly the effect of this additional interaction. Here we present a very simple, analytically solvable model of two coupled oscillators by which we are able to explain the main features of the BLMs of double-walled carbon nanotubes (DWCNTs) as predicted in Ref. [8]. In particular, we derive simple expressions in the small and large-diameter limit from which one may estimate the diameter of inner and outer tubes from experimental Raman frequencies.A DWCNT consists of two co-axial SWCNTs, the walls of which are assumed to be at a distance d/2 = 3.44 Å and the diameters of which are D 1 = D and D 2 = D + d. The frequency of the radial breathing mode (RBM) of isolated SWCNT scales inversely with the tube diameter (ω = α/D) as was originally

Quantum-coupled radial-breathing oscillations in double-walled carbon nanotubes

“…The collective RBM oscillations of a unit-length DWNT can be modelled by a coupled mechanical oscillator with five parameters: m i , k i , m o , k o and k c (Fig. 3a) 25 . The unit-length mass of the inner-and outerwalls (m i and m o ) can be obtained directly from the tube chiral indices, and the unit-length intrinsic force constant of the two walls (k i and k o ) can be obtained from RBM frequencies of isolated SWNTs (Supplementary Note S1).…”

Vibrational Modes of Nano-Template Viruses

“…It has been shown that continuum approach works remarkably well even for nanometer thick semiconductor structures and single wall carbon nanotubes. [16][17] Due to the fact that the length of the viruses is much larger than their diameter, we model TMV and M13 as infinite cylinders (see Fig. 1).…”

Symmetry Based Fundamentals of Carbon Nanotubes

“…Accordingly, the elements of submatrices D αβ are much less than those of D αβ (unless α = β), and may be considered as a perturbation to the dynamical matrix of the noninteracting walls. This invokes fruitful interpretation of CDWNT modes in terms of the modes of the isolated layers [54,65]. …”

“…2.19), being mutually well separated by frequency. The frequencies of the resulting [65] in-phase and out-ofphase breathing-like (BL) modes are found by substituting frequencies (2.37) in (2.42). The numerical results [66], well fitted by κ BL = −246404 + 38799D, are in accordance with earlier predictions [54] and match the experimen- tal results [69][70][71].…”