Figure 1

Electronic Raman scattering (ERS) may occur in terms of first-order processes or second-order processes, and intravalley interactions (AV) or intervalley interactions (EV). For simplicity, we only draw the direct Coulomb interactions in this diagram, while the exchange Coulomb interaction processes are shown in Figs. 2b and 2c. In EV, the interaction between the photoexcited carrier in the parabolic band and the electron(s) in the linear band takes place in different valleys ($K$ and $K^{\prime }$ points). It should be noted that we do not consider a direct Coulomb interaction with a large momentum transfer $\mathbf{q}$ connecting the $K$ and $K^{\prime }$ points, because its matrix element is two orders of magnitude smaller than that of $\mathbf{q}$ within the same valley [compare the direct interaction described in the legend of Fig. 2d with the exchange interaction described in the legend of Fig. 2e for the EV case].Reuse & Permissions

Figure 2

(a) The direct (AV), (b) the exchange (AV), and (c) the direct (red) and exchange (green) (EV) Coulomb interaction process. (d) and (e) contribution from the direct Coulomb interaction [Eq. (2)] and the exchange Coulomb interaction [Eq. (3)], respectively, in the exciton-exciton matrix element ${\mathcal{M}}_{\mathrm{ex}-\mathrm{ex}}^{\pm }$ [Eq. (4)] of a (23,14) tube ($d_t=2.53$ nm). The label AV (EV) inside each panel shows the states, in which two electrons lie in the same (different) valley. The one-dimensional (1D) wave vector $\mathbf{q}$ is projected on the 1D SWNT cutting lines and is expressed in terms of the translational vector length $T$. We have $T=0.46$ nm for the (23, 14) tube.Reuse & Permissions

Figure 3

(a) Calculated result (this work) and (b) experimental results (adapted from Ref. 23) for the Raman intensity versus scattered photon energy $\hslash {\omega }_s$ for a (23,14) tube where we have the calculated $M_{22}^L=2.10$ eV, and the experimental $M_{22}^L=2.08$ eV and $M_{22}^H=2.20$ eV. Here $M_{22}^L$ and $M_{22}^H$ are the splitting of the transition energy in mSWNTs due to the trigonal warping effect.[5] The laser excitation energies $E_L$ are taken as 2.00, 2.07, 2.10, 2.14, and 2.20 eV.Reuse & Permissions

Figure 4

(a) Calculated Raman spectra for a (23,14) SWNT with $E_L=2.14$ eV. The total intensity shown is represented by the solid line. The dashed lines show contributions from the RBM and $G$ modes, while the dotted blue line is the contribution from the ERS. Each line shape for the RBM, the $G$ modes, and the ERS is Lorentzian. The inset shows the $G$-band spectra after subtracting the ERS spectrum. Filled squares are calculated results and the solid line shows the BWF fitting [Eq. (8)]. (b) The asymmetric factor ($1/q_{\mathrm{BWF}}$), and (c) spectral width ($\Gamma$) and peak position (${\omega }_0$) of the $G$ band as a function of resonance condition ($E_L-M_{22}^L$) for the (23,14) tube. The solid and dashed arrows are given as a guide for the corresponding axes.Reuse & Permissions

Figure 5

(a) The phase factor $\Theta \left(\mathbf{k}\right)=\arctan (k_y/k_x)$ in Eq. (A9) where $k_x$ ($k_y$) axis is defined by circumferential (axis) direction of a zigzag SWNT.[29] (b) Schematic figure of one-body electron wave functions at $|\mathbf{k}c\rangle$ and $|\mathbf{k}v\rangle$ as defined by Eq. (A7). For simplicity, we plot the atomic orbitals only along a one-dimensional line connecting the $A$ and $B$ atoms ($x$ axis). (c) Schematic figure of the Coulomb integral Eq. (A11) $K_{\mathbf{k}c,{\mathbf{k}}^{\prime }c,\mathbf{k}c,{\mathbf{k}}^{\prime }v}$ at $\mathbf{q}=0$. The two-body wavefunction $|\mathbf{k}a\left(x\right);{\mathbf{k}}^{\prime }b\left(y\right)\rangle ,(a,b=c,v)$ is substituted from (b). Generally, $|v_{ss}\left(0\right)|>|v_{s{s}^{\prime }}\left(0\right)|$ with $[s,s^{\prime }\in \left(A\mathrm{or}B\right),s^{\prime }\ne s]$ as defined in Eq. (A12). The red (white) color shows a positive (negative) value. From the approximation in Eq. (A11), the dominant matrix elements constituting $K_{\mathbf{k}c,{\mathbf{k}}^{\prime }c,\mathbf{k}c,{\mathbf{k}}^{\prime }v}$ come from the diagonal part.Reuse & Permissions

Figure 6

The LO exciton-phonon matrix element from the $n$th state to the lowest exciton state ${\mathcal{M}}_{\mathrm{ex}-\mathrm{ph}}^{0,n}$ and the exciton-photon matrix element ${\mathcal{M}}_{\mathrm{ex}-\mathrm{op}}^{n,i}$ from the initial state to the $n$th exciton state.[35]Reuse & Permissions