Exciton-photon, exciton-phonon matrix elements, and resonant Raman intensity of single-wall carbon nanotubes

Within the framework of the tight-binding model, we have developed exciton-photon and exciton-phonon matrix elements for single-wall carbon nanotubes. The formulas for first-order resonance and double-resonance Raman processes are discussed in detail. The lowest-energy excitonic state possesses an especially large exciton-photon matrix element compared to other excitonic states and continuum band states because of its localized wave function with no node. Unlike the free-particle picture, the photon matrix element in the exciton picture shows an inverse diameter dependence but no tube type or chirality dependences. As a result, the optical absorption intensity shows a strong diameter dependence but no tube type or chirality dependences. Moreover, the continuum band edge can be determined from the wave function or exciton-photon matrix element. For the radial breathing mode (RBM) and $G$-band modes, the phonon matrix elements in the exciton and free-particle pictures are almost the same. As a result, the intensity for the Kataura plots for the RBM or $G$-band modes by the exciton and free-particle pictures show similar family patterns. However, the excitonic effect has greatly increased the diameter dependence and magnitude of the intensities for the RBM and $G$ band by enhancing the diameter dependence and magnitude of the photon matrix element. Therefore, excitons have to be considered in order to explain the strong diameter dependence of the Raman signal observed experimentally.

Figure 1

(a) Electron and (b) hole scattering processes in the ex-ph matrix element for the first and second terms of Eq. (8). The matrix element for (a) and (b) is determined by the electron and hole matrix elements weighted by the wave function coefficients from the initial and final states.

Figure 2

First-order resonance Raman Stokes processes: (a) incident resonance process and (b) scattered resonance process.

Figure 3

Double-resonance Stokes Raman processes: (a) incident resonance process and (b) scattered resonance process. The energy difference between the $E_{11}\left(A_2^0\right)$ and $E_{11}\left(E^0\right)$ states is less than $100\mathrm{meV}$ and the $E_{11}\left(A_2^0\right)$ state lies lower in energy (Ref. 12).

Figure 4

Electron (a) and hole (b) processes for an ex-ph matrix element between a bright $A_2$ state and a dark $E$ state. $c_1$ $\left(c_2\right)$ and $v_1$ $\left(v_2\right)$ are used to label the first (second) conduction and valence bands in energies, respectively.

Figure 5

$M_{\mathrm{ex}\text{−}\mathrm{op}}$ for $E_{22}\left(A_{2u}^n\right)$ states for an (8,0) tube. The continuum band edge $E_c$ is indicated. The dashed lines are a guide to the eyes. The dotted line shows the density of states, and the exciton states are also indicated. The inset is a zoom-in of part of the main figure, but uses different symbols, i.e., solid line instead of white circles.

Figure 6

$M_{\mathrm{el}\text{−}\mathrm{op}}$ and $Z_{\mathbf{k}}$ in $k$ space measured from $k_{22}$ for an (8,0) tube. (a) $M_{\mathrm{el}\text{−}\mathrm{op}}$, (b) the wave function coefficient $Z_{\mathbf{k}}$ for the $E_{22}\left(A_{2u}^0\right)$ (dashed line) and for the continuum band edge $E_c$ state (solid line), and (c) $Z_{\mathbf{k}}$ for a continuum band state with an energy $E_c+0.1\mathrm{eV}$. In (b) $Z_{\mathbf{k}}$ for the $E_{22}\left(A_{2u}^0\right)$ state is multiplied by 5.

Figure 7

(a) $M_{\mathrm{ex}\text{−}\mathrm{op}}$ for $E_{22}\left(A_2^n\right)$ states in a (6,5) tube with $L=200\mathrm{nm}$, and (b) the normalized $M_{\mathrm{ex}\text{−}\mathrm{op}}∕N^{1∕2}$ for $E_{22}\left(A_2^1\right)$ states for all S-SWNTs with $0.6<d_t<1.6\mathrm{nm}$. Filled and open circles are for SI and SII SWNTs, respectively. The arrow indicates the $\theta$ decreasing direction from armchair (A) to zigzag (Z) SWNTs. In (a), even $n$ states have larger $M_{\mathrm{ex}\text{−}\mathrm{op}}$ than odd $n$ states and thus in the energy range $E-E_{22}–E_c$ the circles form two lines.

Figure 8

$M_{\mathrm{ex}\text{−}\mathrm{op}}$ and $Z_{\mathbf{k}}$ for various states in a (7,4) M-SWNT. (a) $M_{\mathrm{ex}\text{−}\mathrm{op}}$ for $E_{11\mathrm{L}}\left(A_2^n\right)$ with $L=200\mathrm{nm}$ and (b) $Z_k$ for $E_{11\mathrm{L}}\left(A_2^0\right)$ (solid line). $Z_k$ for $E_{22}\left(A_2^0\right)$ for a (6,5) SWNT is also shown (dashed line). All $E_{11\mathrm{L}}\left(A_2^n\right)(n>2)$ states belong to continuum band states.

Figure 9

Tube diameter dependence of $M_{\mathrm{ex}\text{−}\mathrm{op}}∕N^{1∕2}$, $M_{\mathrm{el}\text{−}\mathrm{op}}$, and ${\ell }_{\mathbf{k}}$ in SWNTs with $0.6<d_t<1.6\mathrm{nm}$. Filled, open, and crossed circles are for SI-, SII-, and M-SWNTs, and (a) $M_{\mathrm{ex}\text{−}\mathrm{op}}$ for $E_{22}\left(A_2^0\right)$, $E_{11}\left(A_2^0\right)$ of S-SWNTs and $E_{11\mathrm{L}}\left(A_2^0\right)$ of M-SWNTs, (b) $M_{\mathrm{el}\text{−}\mathrm{op}}$ at $k_{22}$, and (c) ${\ell }_{\mathbf{k}}$ for $E_{22}\left(A_2^0\right)$. The arrows in (b) and (c) indicate the directions along which $\theta$ is decreasing (A, armchair side; Z, zigzag side).

Figure 10

RBM $M_{\mathrm{ex}\text{−}\mathrm{ph}}$ for $E_{22}\left(A_2^n\right)$, $M_{\mathrm{el}\text{−}\mathrm{ph}}$ at $k_{22}$, and $\mid Z_{\mathbf{k}}{\mid }^2$ for the $E_{22}\left(A_2^0\right)$ and $E=E_c$ states for an (8,0) tube. (a) $M_{\mathrm{ex}\text{−}\mathrm{ph}}$ as a functions of $E-E_{22}$. (b) $M_{\mathrm{el}\text{−}\mathrm{op}}$, and (c) $Z_{\mathbf{k}}$ as functions and of $k-k_{22}$ in units of $1∕a$. $\mid Z_{\mathbf{k}}{\mid }^2$ for the $E_c$ state is a $\delta$-like function at $k_{22}$. $\mid Z_{\mathbf{k}}{\mid }^2$ for $E_{22}\left(A_2^n\right)$ has been multiplied by 40. For both the $E_{22}\left(A_2^0\right)$ and $E_c$ states, $\mid Z_{\mathbf{k}}{\mid }^2$ is normalized to the units taken for the integration of $k$.

Figure 11

RBM matrix elements $M_{\mathrm{ex}\text{−}\mathrm{ph}}$ (filled) for $E_{22}\left(A_2^0\right)$ and $M_{\mathrm{el}\text{−}\mathrm{ph}}$ (open) around $k_{22}$ for all S-SWNTs with $0.6<d_t<1.6\mathrm{nm}$ for (a) SI and (b) SII tubes. The arrows indicate the direction along which $\theta$ is decreasing (A, armchair side; Z, zigzag side).

Figure 12

LO and TO mode matrix elements $M_{\mathrm{ex}\text{−}\mathrm{ph}}$ for $E_{22}\left(A_2^0\right)$ (filled) and $M_{\mathrm{el}\text{−}\mathrm{ph}}$ at $k_{22}$ (open) for all S-SWNTs with $0.6<d_t<1.6\mathrm{nm}$. LO mode for (a) SI tubes and (b) SII tubes. TO mode for (c) SI tubes and (d) SII tubes. In (c) and (d), $M_{\mathrm{ex}\text{−}\mathrm{ph}}=M_{\mathrm{el}\text{−}\mathrm{ph}}=0$ for zigzag tubes. The arrows indicate the direction along which $\theta$ is decreasing (A, armchair side; Z, zigzag side).

Figure 13

$M_{\mathrm{el}\text{−}\mathrm{ph}}$ (solid lines) for LO and TO modes in 1D $k$ space for a (6,5) tube. $\mid Z_{\mathbf{k}}{\mid }^2$ (dashed line) for the $E_{22}\left(A_2^0\right)$ state, and a line tangential to the curve for the TO mode at the right side of $k_{22}$ (dot) are also shown (long-dashed line).

Figure 14

RBM Raman intensity per length for SWNTs with $0.6<d_t<1.6\mathrm{nm}$. (a) and (b) are for $E_{22}\left(A_2^0\right)$ and free $e\text{−}h$ at $k_{22}$ in S-SWNTs, respectively. Filled and open circles are for SI and SII tubes. (c) and (d) are for $E_{11\mathrm{L}}\left(A_2^0\right)$ and free $e\text{−}h$ at $k_{11\mathrm{L}}$ in M-SWNTs, respectively. The intensity in the free el-ph case has been multiplied by 200. The arrows indicate the $\theta$ decreasing direction (A, armchair side; Z, zigzag side). In (d), within a family the armchair tube has a larger intensity than its neighbor due to a node effect.

Figure 15

$G$-band Raman intensities per length in S-SWNTs with $0.6<d_t<1.6\mathrm{nm}$. (a) and (b) are for the LO mode with (a) for $E_{22}\left(A_2^0\right)$ excitonic states and (b) for free $e\text{−}h$ $k_{22}$ states. Filled and open circles are for SI and SII tubes. (c) and (d) are for the TO mode with (c) for $E_{22}\left(A_2^0\right)$ excitonic states and (d) for free $e\text{−}h$ states at $k_{22}$. The intensity in the free $e\text{−}h$ case has been multiplied by 100 (b) and 1000 (d). The arrows indicate the $\theta$ decreasing direction (A, armchair side; Z, zigzag side).

Figure 16

The Raman intensity ratio between the TO and LO modes, $I_{\mathrm{TO}}∕I_{\mathrm{LO}}$, in S-SWNTs with $0.6<d_t<1.6\mathrm{nm}$. (a) and (b) are for the exciton and free $e\text{−}h$ pictures, respectively. The arrows indicate the $\theta$ decreasing direction (A, armchair side; Z, zigzag side).