Intrinsic radial breathing oscillation in suspended single-walled carbon nanotubes

Radial breathing mode (RBM) oscillation is the most characteristic vibration mode in carbon nanotubes. Here we investigate the intrinsic behavior of RBM oscillations of structurally defined single-walled carbon nanotubes (SWNTs) by combining Raman-scattering and electron-diffraction techniques on the same suspended nanotubes. The independent determination of RBM frequencies and nanotube structures allows us to establish conclusively the perfect linear relation between RBM frequencies and inverse nanotube diameters, which has been long speculated to hold in pristine SWNTs. Understanding the intrinsic diameter dependence of SWNT RBM oscillation not only is crucial for reliable Raman characterization of carbon nanotubes, but also enables quantitative probing of SWNT-environment interactions through the RBM oscillation frequency shift.

Figure 1

(a) Schematic illustration for combined TEM electron-diffraction and Raman-scattering techniques on the same suspended nanotubes. (b) Scanning electron micrograph of a SWNT suspended across an open slit on the Si${}_3$N${}_4$/Si substrate. (c) High-resolution transmission electron micrograph of a SWNT. It is seen to be a single-walled character with clean surface. Scale bars in (b) and (c) are 20 μm and 20 nm, respectively.

Figure 2

Electron-diffraction patterns and RBM Raman spectra of three representative SWNTs. (a)–(c) Electron-diffraction patterns of the nanotube allow us to determine their chiral indices as (16,11), (25,12), and (33,23). The corresponding nanotube diameters are 1.84, 2.56, and 3.82 nm, respectively. (d)–(f) RBM Raman spectra of (16,11), (25,12), and (33,23) SWNTs. The RBM frequencies are located at 124, 90, and 61 cm${}^{-1}$, respectively.

Figure 3

Raman spectra covering the $D$-band and $G$-band region for (a) (16,11), (b) (25,12), and (c) (33,23) SWNTs. The $D$-mode Raman peak is not observed, indicating that our SWNTs are high quality with very few defects.

Figure 4

RBM frequencies ${\omega }_{\mathrm{RBM}}$ versus inverse nanotube diameters 1$/d$ (squares). It is described perfectly by a proportional relation of ${\omega }_{\mathrm{RBM}}=228/d$ (cm${}^{-1}$ nm) (dark solid line).