Resonant Raman intensity of the totally symmetric phonons of single-walled carbon nanotubes

The resonant Raman intensity of the totally symmetric phonons of all single-walled carbon nanotubes in the radius range from $2.5\mathrm{\AA }\text{to}11.5\mathrm{\AA }$ was calculated within a symmetry-adapted nonorthogonal tight-binding model. The obtained intensity of these modes exhibits radius and chirality dependence. The simulated Raman spectra of samples of moderate-diameter tubes with a Gaussian diameter distribution show characteristic peaks at about 1540, 1570, and $1593{\mathrm{cm}}^{-1}$ originating from $LO$ phonons of metallic tubes, $TO$ phonons of metallic and semiconducting tubes, and $LO$ phonons of semiconducting tubes, respectively. This general behavior of the Raman spectra corresponds to that of the normally measured spectra.

Figure 1

The calculated resonance Raman profiles of three nanotubes (16, 7), (15, 8), and (14, 9) with radius $\approx 8\mathrm{\AA }$. It is seen that the $A_1^{LO}$ modes have highest intensity followed by the RBM ones and those of the $A_1^{TO}$ modes. The Raman profile shape depends on the three quantities ${\gamma }_e$, $E_o$, and $\Delta E_{ii}$. The more characteristic cases are: ${\gamma }_e⪡\Delta E_{ii}⪡E_o$ (a), ${\gamma }_e⪡\Delta E_{ii}\sim E_o$ (b), ${\gamma }_e⪡E_o⪡\Delta E_{ii}$ (c), ${\gamma }_e\sim E_o\sim \Delta E_{ii}$ (d), and ${\gamma }_e\sim E_o⪡\Delta E_{ii}$ (e). For discussion, see text.

Figure 2

The calculated $J_{ii}$ of the RBM of metallic tubes for the first six optical transitions. “$A$,” “$Z0$,” and “$C0$” stand for armchair, zigzag, and chiral tubes, the latter two of type Mod0. The panel for the most intesively studied transitions $E_{11}$ and $E_{22}$ is enlarged to show the family arrangement of the points for $2L_1+L_2=\mathrm{const}$. The points of several families are connected by dotted lines for clarity.

Figure 3

The calculated $J_{ii}$ of the RBM of semiconducting tubes for the first six optical transitions. “$Z1$,” “$C1$,” “$Z2$,” and “$C2$ stand for zigzag and chiral tubes of types Mod1 and Mod2. The panel for $E_{22}$ is enlaged to show the family patterns for $2L_1+L_2=\mathrm{const}$. The points of several families are connected by dotted lines.

Figure 4

The calculated $J_{ii}$ of the $A_1^{TO}$ $G$-band modes of metallic tubes. Notice the almost equal values of $J_{ii}$ in each pair of transitions (11, 22), (33, 44), etc.

Figure 5

The calculated $J_{ii}$ of the $A_1^{TO}$ $G$-band modes of semiconducting tubes. Notice the almost equal values of $J_{ii}$ in each pair of transitions (11, 22), (33, 44), etc.

Figure 6

The calculated $J_{ii}$ of the $A_1^{LO}$ $G$-band modes of metallic tubes. The sign of $J_{ii}$ alternates with the transition index $ii$ as ${(-1)}^i$ or ${(-1)}^{i+1}$ depending on the tube type.

Figure 7

The calculated $J_{ii}$ of the $A_1^{LO}$ $G$-band modes of semiconducting tubes. The sign of $J_{ii}$ alternates as for metallic tubes.

Figure 8

The simulated $G$-band region of the Raman spectra of a nanotube sample with an average radius $(6.7\pm 0.9)\mathrm{\AA }$ and a number of commonly used laser excitation energies. The NTB optical transitions were upshifted by $0.3\mathrm{eV}$ in this calculation. Note that the “metallic” $G$-subband shifts to lower frequencies with the increase of the laser excitation energy. The rate of this shift is about that normally measured and estimated, and commonly explained with double-resonance processes.