Strength of radial breathing mode in single-walled carbon nanotubes

We show by ab initio calculations that the electron-phonon coupling matrix element ${\mathcal{M}}_{e\text{−}\mathit{ph}}$ of the radial breathing mode in single-walled carbon nanotubes depends strongly on tube chirality. For nanotubes of the same diameter the coupling strength ${\mid {\mathcal{M}}_{e\text{−}\mathit{ph}}\mid }^2$ is up to one order of magnitude stronger for zigzag tubes than for armchair tubes. For $(n_1,n_2)$ tubes ${\mathcal{M}}_{e\text{−}\mathit{ph}}$ depends on the value of $(n_1-n_2)\mathrm{mod}3$, which allows us to discriminate semiconducting nanotubes with similar diameter by their Raman scattering intensity. We show measured resonance Raman profiles of the radial breathing mode which support our theoretical predictions.

Figure 1

Calculated band-energy changes (absolute value) per unit change in radius for zigzag (circles) and armchair tubes (triangles). Lines are linear fits of the data corresponding to the indicated $k$ value [as defined in the horizontal axis of Fig. 2c]; they include the origin as a data point. Inset: Band-energy change $\mid \partial E_b\left(\mathbf{k}\right)∕\partial \mathbf{u}\mid$ for the first optical transition of nanotubes with diameter $\approx 8\mathrm{\AA }$ as a function of the chiral angle.

Figure 2

(a) Detail of the Brillouin zone (BZ) of graphene around $K$ with the band lines of a (19,0) nanotube and the $\Gamma \text{−}KM$ line. (b) Electronic bands of graphene along $\Gamma \text{−}K\text{−}M$ in equilibrium (solid line) and under a deformation of $0.1\mathrm{\AA }$ corresponding to the RBM of a (19,0) tube (dashed line, difference enhanced $\times 10$). (c) Calculated $\partial E_b\left(\mathbf{k}\right)∕\partial \mathbf{u}$ for a (17,0) (squares) and a (19,0) (circles) tube, and of graphene deformed to simulate the RBM of a (19,0) tube (solid line) and a (17,0) nanotube (dotted line).

Figure 3

(a) Measured resonance Raman profiles (symbols) and fits (lines) of two different nanotubes. (b) Raman spectrum with a laser energy of $1.65\mathrm{eV}$.