Raman characterization of electronic transition energies of metallic single-wall carbon nanotubes

Using a confocal micro-Raman system, spectra showing the splitting of optical transitions due to trigonal warping effect are presented for metallic single-wall carbon nanotubes (SWNT’s). Our results indicate that the intensity variations between different optical transitions can be attributed primarily to the differences in the magnitude of the electron-phonon coupling matrix elements. Our approach will allow the study of the magnitude of electron-phonon matrix elements as well as quantum interference effects between different transitions in metallic SWNT’s.

Figure 1

(Color online) ETB$+$MB calculations of the optical transition energies versus ${\omega }_{\mathrm{RBM}}$. The blue circles and the red squares are semiconducting and metallic SWNT’s, respectively. The $2n+m=\mathrm{const}$ families are denoted by solid lines and the $2n+m$ family numbers are also indicated. The yellow diamonds are experimental $E_{11\mathrm{H}}^{\mathrm{M}}$ and $E_{11\mathrm{L}}^{\mathrm{M}}$ optical transition energies.

Figure 2

RBM signals at the same spatial position at two different laser excitation energies (a) $2.54\mathrm{eV}$ and (b) $2.16\mathrm{eV}$. The ${\omega }_{\mathrm{RBM}}=240{\mathrm{cm}}^{-1}$ feature appears in both spectra.

Figure 3

(Color online) The resonance profile for (a) $(10,4)$ SWNT, (b) $(18,0)$ SWNT, and (c) $(15,3)$ SWNT. The red circles are the experimental integrated RBM intensities. The SWNT intensities were calibrated against that of the $520\text{−}{\mathrm{cm}}^{-1}$ Si peak. The blue diamonds are calculations using ETB$+$MB theory, and their horizontal positions correspond to the resonance energy. Regarding the intensity of the $(10,4)$ SWNT, if we match the $E_{11\mathrm{L}}^{\mathrm{M}}$ and $E_{11\mathrm{H}}^{\mathrm{M}}$ results, we can then see that the experimental and theoretical results agree well (see text). The black line is the calculated RBM resonance profile using Eq. (1). The blue drop lines indicates the $E_{11\mathrm{L}}^{\mathrm{M}}$ and $E_{11\mathrm{H}}^{\mathrm{M}}$ values used for the calculated resonance profile. Note that the $E_{11\mathrm{L}}^{\mathrm{M}}$ and $E_{11\mathrm{H}}^{\mathrm{M}}$ differ from the peak center by $E_{\mathrm{ph}}∕2$ for the Stokes process. The experimental points are downshifted by $72\mathrm{meV}$ from the theoretical predictions for the $(10,4)$ SWNT. All SWNT’s show $\Gamma =35\mathrm{meV}$.