Fermi energy dependence of the $G$-band resonance Raman spectra of single-wall carbon nanotubes

The Fermi energy dependence of the $G$-band resonance Raman spectra of single-wall carbon nanotubes (SWNTs) is calculated, including the Kohn anomaly effect for metallic tubes. The gate voltage dependence of the $G$-band Raman spectra for SWNTs shows chirality-dependent $G^{+}/G^{-}$ spectra, reflecting their dependence on the eigenvector direction of the optical (LO and TO) phonon modes and the nanotube chirality.

Figure 1

(Color online) (a) The $G$-band spectra for S-SWNTs with the same family number $p=2n+m=28$. (b) el-ph matrix elements ${\mathcal{M}}_{\text{ep}}^{\text{R}}$ vs chiral angle $\theta$ for the LO and TO phonons and for two different family numbers (22 and 28). (c) $\gamma$ vs $\theta$ for members of family $p=28$. (d) Plot of ${\Theta }_{\text{R}}$ vs $\theta$ for three $2n+m$ families of M-SWNTs. (e) The angle ${\Theta }_{\text{R}}$ to the cutting line for the polar coordinate of a $\mathbf{k}$ vector at the van Hove singular point. (f) The angle $\varphi$ between the tube axis and the phonon eigenvector direction for a (12,6) SWNT. The calculated angles $\varphi$ vs $\theta$ for (g) TO and (h) LO phonons fitted by the function given in the text.

Figure 2

(a) The $G$-band spectra of M-SWNTs with the same family number $p=30$ and $E_{\text{F}}=0$. (b) el-ph matrix elements vs $\theta$ for the LO and TO phonons and for two different family numbers. Open circles indicate the ${\mathcal{M}}_{\text{ep}}$ values for the family number $p=30$. (c) $\gamma$ vs $\theta$ for members of $p=30$.

Figure 3

The $G$-band spectra for three M-SWNTs with different chiral angles taken by changing the Fermi energy from $E_{\text{F}}=-0.2$ to 0.2 eV. (a) (10,10). (b) (11,8). (c) (15,0).

Figure 4

(a), (c), and (e) Experimental $G$-band Raman spectra, which are given by the electrochemical doping effect. (a) $V_{\text{g}}=1.5$ to $-1.5\text{V}$. (c) $V_{\text{g}}=1.9$ to $-1.3\text{V}$. (e) $V_{\text{g}}=1.3$ to $-1.3\text{V}$. (b), (d), and (f) Calculated $G$-band Raman spectra taken by changing the Fermi energy $E_{\text{F}}$ (b) 0.45 to $-0.45\text{eV}$, (d) 0.60 to $-0.42\text{eV}$, and (f) 0.39 to $-0.39\text{eV}$. (a) and (b) (11,11), (c) and (d) (22,4), and (e) and (f) (12,0).