Chirality dependence of coherent phonon amplitudes in single-wall carbon nanotubes

We simulate the ultrafast dynamics of laser-induced coherent phonons in single-wall carbon nanotubes (SWNTs). In particular, we examine the coherent phonon amplitude of the radial breathing mode (RBM) as a function of excitation energy and chirality. We find that the RBM coherent phonon amplitudes are very sensitive to changes in excitation energy and are strongly chirality dependent. We discuss how the SWNT diameter changes in response to femtosecond laser excitation and under what conditions the diameter of a given SWNT will initially increase or decrease. An effective-mass theory for the electron-phonon interaction gives a physical explanation for these phenomena.

Figure 1

The coherent RBM phonon amplitude $Q_{\mathrm{m}}$ for an $(11,0)$ zigzag tube as a function of laser excitation energy $E_{\mathrm{laser}}$. For clarity, $Q_{\mathrm{m}}$ is plotted in units of $0.0259$Å. A positive (negative) sign of the vibration amplitude denotes a vibration whose initial phase corresponds to an expanding (shrinking) diameter. The absorption coefficient versus $E_{\mathrm{laser}}$ is shown for comparison with the $Q_{\mathrm{m}}$ behavior.

Figure 2

(Color online) RBM electron-phonon matrix elements of (a) $(11,0)$ and (b) $(13,0)$ zigzag nanotubes within the ETB approximation.

Figure 3

(Color online) Electron and hole components of the ETB $M_{\text{el-ph}}$ shown by solid and dashed lines, respectively, for (a) $(11,0)$ and (b) $(13,0)$ zigzag nanotubes, as a function of $k$. The matrix elements for $E_{11}$ and $E_{22}$ are shown in black and red, respectively.

Figure 4

(Color online) Cutting lines for (a) (11,0) and (b) (13,0) zigzag nanotubes near the graphene K point. Black and red solid lines denote the $E_{11}$ and $E_{22}$ cutting lines, respectively, while the dotted lines correspond to higher cutting lines. The angle $\Theta \left(\mathit{k}\right)$ is measured counterclockwise from a line perpendicular to the cutting lines, with the positive direction of the line to the right of the K point. Here $\Theta \left(\mathit{k}\right)$ is shown for a $k$ point on the $E_{22}$ cutting line for both SWNTs. The difference between the type-I and type-II families can be understood from the position of the $E_{11}$ or $E_{22}$ cutting lines relative to the K point (Ref.22).

Figure 5

(Color online) RBM electron-phonon matrix elements of (a) $(11,0)$ and (b) $(13,0)$ nanotubes calculated within the effective-mass theory using $g_{\mathrm{off}}=6.4\mathrm{eV}$. In (a) and (b), the matrix elements near $k=0$ are comparable with the results in Fig. 2. (c) shows the matrix elements of an $(11,0)$ nanotube calculated within the ETB model for interactions up to the fourth-nearest neighbors. The results including fourth-nearest neighbors exactly reproduce the results in Fig. 2a.

Figure 6

(Color online) The lattice response of SWNTs with diameters in the range 0.7–1.1$\mathrm{nm}$ is mapped onto the unrolled graphene lattice specifiying the tube chiralities $(n,m)$. In this map $Q_m$ is expressed in terms of $\sqrt{\hslash /2M{\omega }_{\mathrm{RBM}}}$. Red and blue colored hexagons denote the SWNTs whose vibrations start by increasing or decreasing their diameters, respectively. For clarity, the shrinking tubes (blue colored hexagons) are also specified by underlining their (n,m). The laser excitation energies are selected within the range 1.5–3.0$\mathrm{eV}$. For each $(n,m)$ tube, the corresponding $E_{ii}$ (in eV) found within this energy region is listed on each hexagon with the label $E_{ii}$. The calculated results for the $(7,4)$ and $(6,6)$ nanotubes are not shown in this figure because their $E_{11}^{\mathrm{L}}>3.0\mathrm{eV}$ and the $(6,6)$ tube gives a negligibly small $Q_{\mathrm{m}}$.