Intensity of the resonance Raman excitation spectra of single-wall carbon nanotubes

The electron-phonon matrix elements are calculated for the radial breathing mode (RBM) and the $G$-band $A$ symmetry mode of single-wall carbon nanotubes. The RBM intensity decreases with increasing nanotube diameter and chiral angle. The RBM intensity at van Hove singular $k$ points is larger outside the two-dimensional Brillouin zone around the $K$ point than inside the Brillouin zone. For the $G$ band $A$ symmetry mode, the matrix element shows that all semiconducting nanotubes have nonzero LO mode intensity, and the LO mode generally has a larger intensity than the TO mode, while the ratio of the intensity of the LO mode to that of the TO mode decreases with increasing chiral angle. In particular, zigzag nanotubes have zero intensity for the TO mode, and armchair nanotubes have zero intensity for the LO mode. Using the matrix elements thus obtained, the resonance Raman excitation profiles are calculated for nanotube samples under different broadening factor $\gamma$ regimes. For semiconducting nanotubes, the excitation profiles for the RBM are consistent with experiments. For metallic nanotubes, a quantum interference effect in the Raman intensity is found for both the RBM and LO modes. For the RBM and LO modes, different kinds of excitation profiles are discussed for nanotube samples in the large and small $\gamma$ regimes by considering the electron-phonon matrix element and the trigonal warping effect. For nanotube samples in the large $\gamma$ regime, a shift in the energy of the peak in the RBM intensity relative to the corresponding peak in the joint density of states is found.

Figure 1

The chiral angle dependence of the e-ph matrix element $M$ of the RBM for (a) the $E_{22}^S$ transition for semiconducting nanotubes, and (b) the $E_{11}^M$ transition for metallic nanotubes. The matrix element $M$ is in units of $\sqrt{\hslash ∕\left(N_u{m}_C\right)}$ with $N_u$ and $m_C$ the number of graphene unit cells in the SWNT and the mass of a carbon atom. (a) Upper and lower curves are for SI and SII nanotubes, and solid, dashed, and long-dashed curves are for transition energies 1.5, 2.0, and 2.5 eV, respectively. The inset shows the vHS positions in the 2D BZ. The SI vHSs are outside the 2D BZ of graphite and the SII vHSs are inside the 2D BZ, defined by the symmetry points $K$ and $M$. (b) Upper and lower curves are for vHSs with lower $E_{11L}^M$ and higher $E_{11H}^M$ energies, and solid and dashed curves are for transition energies 2.0 and 2.5 eV, respectively. The dots in (a) and (b) indicate the node positions for the corresponding transition energies.

Figure 2

The chiral angle dependence of the e-ph matrix element of the $G$ band mode with $A$ symmetry: (a) LO mode and (b) TO mode for the $E_{22}^S$ transition. Solid and dashed curves are for the transition energies 1.5 and 2.0 eV, respectively. The matrix element is given in units of $\sqrt{\hslash ∕\left(N_u{m}_C\right)}$ with $N_u$ the number of graphite unit cells in the SWNT and $m_C$ the mass of a carbon atom. (a) Upper and lower sets of curves are for SI and SII nanotubes, respectively. (b) Thick and thin curves are for SI and SII tubes, respectively.

Figure 3

Calculated Raman intensity in the Stokes process for the RBM vs excitation energy $E_L$ for (a) an SI (13,5) tube and (b) an SII (13,6) tube and using $\gamma =0.06\mathrm{eV}$.

Figure 4

The RBM Raman intensity vs excitation energy $E_L$ for an SI (13,5) tube. Solid and dashed lines are for Stokes and anti-Stokes processes, respectively, and the plots are for $\gamma =0.06\mathrm{eV}$ (see text for normalization of the anti-Stokes curve).

Figure 5

Raman intensity in the Stokes process for the RBM vs excitation energy $E_L$ for (a) an SI (13,5) tube and (b) an SII (13,6) tube and using $\gamma =0.006\mathrm{eV}$.

Figure 6

Illustration of the four kinds of RRPs in the Stokes process for the RBM in metallic nanotubes (a) (8,5), (b) (13,4), (c) (12,6), and (d) (11,8). The profiles calculated by eliminating the interference effect between the contributions from $E_{11L}^M$ and $E_{11H}^M$ and the profiles calculated by taking $\mid M\mid$ as the electron-phonon matrix element are also shown by dashed and long-dashed curves. The joint density of states are given to show the vHS peak splitting energy. The broadening factor is $\gamma =0.06\mathrm{eV}$.

Figure 7

Illustration of the two kinds of RRPs in the Stokes process for the RBM in metallic nanotubes (a) (11,8) and (b) (13,4) calculated with $\gamma =0.006\mathrm{eV}$. The joint density of states are given to show the vHS peak positions.

Figure 8

Raman intensity for the Stokes process for the $G$ band $A$ symmetry mode vs excitation energy $E_L$ for (a) an SI (13,5) and (b) an SII (13,6) semiconducting tube where $\gamma =0.06\mathrm{eV}$ is used in the calculations. The dashed line in (a) is the calculated profile by eliminating the interference effect between the $E_{11}^S$ and $E_{22}^S$ bands.

Figure 9

Raman intensity for the Stokes process for the $G$ band $A$ symmetry mode vs excitation energy $E_L$ for (a) an SI (13,5) tube and (b) an SII (13,6) tube where $\gamma =0.006\mathrm{eV}$ is used in the calculations.

Figure 10

Illustration of the four kinds of RRPs for the Stokes process for the LO mode in metallic nanotubes (a) (9,0), (b) (9,3), (c) (12,3), and (d) (11,8). The profiles calculated by eliminating the interference effect and the profiles calculated by taking $\mid M\mid$ as the electron-phonon matrix element are also shown by dashed and long-dashed curves. The joint density of states are given to show the vHS peak splitting energy. The broadening factor $\gamma =0.06\mathrm{eV}$ is used in the calculations.

Figure 11

Illustration of the two kinds of RRPs calculated for the Stokes process for the LO mode in metallic nanotubes for the (a) (12,6) and (b) (13,4) SWNTs and using $\gamma =0.006\mathrm{eV}$. The joint density of states curves are plotted to show the vHS peak position.

Figure 12

Classification of the SII and metallic nanotubes according to their characteristics in the RBM Raman spectra. Curves 1 and 2 delineate three regions (see text).