Photoexcited electron relaxation processes in single-wall carbon nanotubes

The relaxation processes for photoexcited electrons in single-wall carbon nanotubes (SWNTs) are discussed in connection with recent photoluminescence excitation (PLE) spectroscopy experiments. The electron-phonon $\left(e\text{−}\mathrm{ph}\right)$ interaction of SWNTs is calculated for each phonon mode as a function of photoexcited electron energy. Because of the cylindrical surface of a SWNT, the twisting and radial breathing phonon modes are responsible for the $e\text{−}\mathrm{ph}$ interaction in the lower phonon energy region. Optic phonon modes are also responsible for the relaxation of photoexcited electrons. Compared with graphite, the relaxation of electrons is much faster in SWNTs because of their cylindrical shape. We identify and explain some aspects of both photoluminescent and Raman processes appearing in PLE plots for SWNTs.

Figure 1

Four different kinds of photoexcited electron relaxation processes which connect the two conduction bands ($E_1$ and $E_2$) at the $K$ and $K^{\prime }$ points of the two-dimensional Brillouin zone. ABAV (intraband, intravalley), EBAV (interband, intravalley), ABEV (intraband intervalley), EBEV (interband, intervalley) scattering processes.

Figure 2

Possible ABAV processes. (a) PL, (b) two-phonon emission, (c) hot-electron emission, (d) incident resonance one-phonon Raman process, (e) scattered one-phonon Raman process, (f) hot-electron Raman process.

Figure 3

Schematic view of PLE diagram; (a) to (f) correspond to the processes of Fig. 2. Lines $P$ and $Q$ are Raman and Rayleigh processes, respectively.

Figure 4

(a) Internanotube transition from a smaller diameter SWNT with a larger $E_{11}$ to a SWNT with a smaller $E_{11}^{\prime }$. (b) On the vertical line at $E_{\mathrm{PL}}=E_{11}^{\prime }$ some “dummy spots” D are observed which do not originate from a phonon mode of the smaller $E_{11}^{\prime }$ SWNT.

Figure 5

The atomic deformation potential vectors defined by Eq. (11) as a function of the wave-function position of $\varphi (\mathbf{r}-{\mathbf{R}}_{s^{\prime }})$ for a (10,10) nanotube. The origins of $\mathbf{\nabla }v(\mathbf{r}-r_{\sigma })$ and $\varphi (\mathbf{r}-{\mathbf{R}}_s)$ are fixed for each figure and are denoted by an open square and circle in (a) and (c). (a) and (b) [side view of (a)] are the case having the largest value of the atomic deformation potential vector, while (c) and (d) [side view of (c)] are the case having the second largest value of the atomic deformation potential vector.

Figure 6

The electron relaxation rate dependence on the initial electron state energy for a (6,5) nanotube; (a) and (b) correspond to ABAV and ABEV processes, respectively. In (a), the lines labeled from 1 to 6 are for the RBM, TW, oTO, LO, LA, and TO phonon modes, respectively. In (b), the lines labeled from 1 to 6 are, respectively, for the oTA, iTA, oTO, TO, LA, and LO zone-boundary modes. LA, iTA, TW, RBM, oTA, oTO, LO, and TO represent longitudinal acoustic, in-plane transverse acoustic, twisting, radial breathing, out-of-plane transverse acoustic, out-of-plane transverse optical, longitudinal optical, and transverse optical modes, respectively.

Figure 7

RBM and $G$-band Raman intensity dependence on laser energy for a (6,5) SWNT. The calculated $E_{11}$ value for the SWNT is about $1.1\mathrm{eV}$.

Figure 8

The singularities in the relaxation rate near $E_{11}$ for a (6,5) SWNT. The peaks labeled from 1 to 9 correspond to the acoustic, RBM, oTA (near $K$), oTO (near $K$), oTO, iTA (near $K$), iLA (near $K$), TO, and LO phonons, respectively.

Figure 9

The PL intensity by the one-phonon emission process. The peaks labeled from 1 to 9 correspond to the same phonon modes as are labeled in Fig. 8.

Figure 10

PL due to Raman processes. The thickness of the lines represents the PL intensity. The results for the RBM and $G$-band (LO mode) are labeled.