Resonant coherent phonon spectroscopy of single-walled carbon nanotubes

Using femtosecond pump-probe spectroscopy with pulse-shaping techniques, one can generate and detect coherent phonons in chirality-specific semiconducting single-walled carbon nanotubes. The signals are resonantly enhanced when the pump photon energy coincides with an interband exciton resonance, and the analysis of such data provides a wealth of information on the chirality dependence of light absorption, phonon generation, and phonon-induced band-structure modulations. To explain our experimental results, we have developed a microscopic theory for the generation and detection of coherent phonons in single-walled carbon nanotubes using a tight-binding model for the electronic states and a valence force field model for the phonons. We find that the coherent phonon amplitudes satisfy a driven oscillator equation with the driving term depending on photoexcited carrier density. We compared our theoretical results with experimental results on $\text{mod}2$ nanotubes and found that our model provides satisfactory overall trends in the relative strengths of the coherent phonon signal both within and between different $\text{mod}2$ families. We also find that the coherent phonon intensities are considerably weaker in $\text{mod}1$ nanotubes in comparison with $\text{mod}2$ nanotubes, which is also in excellent agreement with experiment.

Figure 1

(Color online) Experimental setup for chirality-selective excitation of coherent lattice vibrations in single-walled carbon nanotubes through ultrafast pulse shaping. The inset shows an example of multiple pulse trains with a repetition rate of 6.15 THz and a central wavelength of 800 nm (1.55 eV).

Figure 2

(Color online) Generation and detection of coherent phonons of the radial breathing mode in single-walled carbon nanotubes. (a) Time-domain transmission modulations due to coherent RBM vibrations in ensemble single-walled carbon nanotube solution that were generated by standard pump-probe spectroscopy without pulse shaping. (b) Fourier transformation of time-domain oscillations with chirality assigned peaks.

Figure 3

(Color online) (Left) Time-domain coherent RBM oscillations selectively excited by multiple pulse trains via pulse shaping with corresponding repetition rates from 6.15 to 7.07 THz. (Right) Fourier transformations of corresponding oscillations, with their dominant nanotube chirality $\left(n,m\right)$ indicated.

Figure 4

(Color online) Differential transmission as a function of time delay at probe wavelengths of 780, 795, and 810 nm for the selective RBM excitation of the (11,3) nanotubes. There is a $\pi$ phase shift between the 780 and 810 nm data. These three wavelengths (from the top to the bottom of the figure) correspond to photon energies above, at, and below the energy of the second exciton resonance, respectively, of (11,3) nanotubes.

Figure 5

(Color online) (a) Differential transmission as a function of time delay together with the pump pulse train (dark blue or dark gray) as well as fit (light blue or light gray up to 4 ps) to exponentially decaying sinusoidal oscillations. [(b),(c)] Data near time zero for two wavelengths corresponding to energies above and below the second exciton resonance, respectively, of (11,3) nanotubes.

Figure 6

(Color online) Extended tight-binding energy dispersion relations for the bonding $\pi$ and antibonding ${\pi }^{\ast }$ bands in graphene along high-symmetry directions in the Brillouin zone are shown as solid black lines. Energy dispersion relations in the nearest-neighbor tight-binding model described in Ref. 1 are shown as dashed-dotted red lines.

Figure 7

Graphene phonon-dispersion relation. The solid circles are the valence force field model of Jishi et al. (Ref. 35) and the solid lines are the best fit results for our seven-parameter valence force field model.

Figure 8

(Color online) The upper panel show the four lowest-lying electronic $\pi$ bands in the one-dimensional Brillouin zone for an (11,0) zigzag nanotube in the ETB model. All the bands are doubly degenerate. The allowed optical transition $E_{ii}$ for light polarized along the tube axis is indicated. The lower panel shows the square of the optical matrix elements defined in Eq. (14) for each of the allowed optical transition.

Figure 9

Computed absorption coefficient as a function of photon energy for all nanotubes $\left(n,m\right)$ in the family for which $2n+m=22$. The photons are assumed to be linearly polarized with the electric polarization vector parallel to the tube axis. Exciton effects are neglected.

Figure 10

(Color online) Computed photoexcited carrier density per unit length for nanotubes in family 22 after photoexcitation by a 50 fs Gaussian laser pulse with a fluence of ${10}^{-5}\text{J}/{\text{cm}}^2$ pumping at the peak of the $E_{22}$ features in Fig. 9.

Figure 11

Phonon-dispersion relations for $\mu =0$ phonons in (11,0) nanotubes. Only the four $q=0$ modes with nonzero frequency can be excited in coherent phonon spectroscopy.

Figure 12

(Color online) Driving function kernel for RBM coherent phonons in the (11,0) nanotube for several optical transitions $E_{ii}$ in the impulsive excitation approximation.

Figure 13

(Color online) Coherent phonon generation in (11,0) nanotubes by photoexcitation at the $E_{11}$ and $E_{22}$ transition energies. The upper panel shows the density of photoexcited electron-hole pairs per unit length, the middle panel shows the coherent phonon driving function, and the bottom panel shows the RBM coherent phonon amplitude.

Figure 14

(Color online) Upper panel: coherent phonon driving functions for RBM oscillations in family 22 nanotubes photoexcited by 50 fs pumps at the broadened $E_{22}$ absorption peaks in Fig. 9. Lower panel: corresponding coherent phonon amplitudes. Curves are offset for clarity.

Figure 15

(Color online) Upper panel: pump laser intensity vs time for a single 50 fs Gaussian pulse and two trains of six 50 fs Gaussian pulses with repetition periods in phase and $180°$ out of phase with the (11,0) RBM phonon period. In all cases, we pump at the (11,0) $E_{22}$ transition energy (2.05 eV) with the same fluence. Note that the intensities of the two pulse trains are magnified by a factor of 6 relative to the intensity of a single Gaussian pulse. Lower panel: corresponding coherent phonon amplitudes for the (11,0) RBM coherent phonon oscillations.

Figure 16

(Color online) The lower panel shows the absorption coefficient in (11,0) nanotubes as a function of photon energy for light polarized along the tube axis. The upper panel is a contour map showing the coherent phonon power spectrum as a function of pump-probe energy and phonon frequency.

Figure 17

(Color online) Coherent phonon spectra for coherent RBM and LO phonons in (11,0) nanotubes photoexcited by an ultrafast $z$-polarized 5 fs pump. The CP spectra at the RBM and LO energies (37.1 and 198 meV) are plotted as a function of pump-probe energy. Two features are seen near $E_{11}$ and $E_{22}$ transitions. The LO curve is multiplied by 1000 and offset for clarity.

Figure 18

(Color online) Coherent phonon intensity at the RBM frequency as a function of pump-probe energy for several $\text{mod}2$ semiconducting nanotubes at the $E_{22}$ transition. The experimental CP spectra are in the right panel and the simulated CP spectra are in the left panel. The upper four curves in each panel are for nanotubes in family 22 and the lower four curves are for tubes in family 25. Each curve is labeled with the chirality $\left(n,m\right)$ and the RBM phonon energy in meV and offset for clarity.

Figure 19

(Color online) Coherent phonon generation in (11,0) $\text{mod}2$ and (13,0) $\text{mod}1$ semiconducting zigzag nanotubes by photoexcitation at the $E_{22}$ transition energy. The upper panel shows the density of photoexcited electron-hole pairs per unit length, the middle panel shows the coherent phonon driving function, and the bottom panel shows the RBM coherent phonon amplitude.

Figure 20

(Color online) Driving function kernel for RBM coherent phonons in the (13,0) nanotube for several optical transitions $E_{ii}$ in the impulsive excitation approximation.

Figure 21

(Color online) Coherent phonon power spectra at the RBM energies in the (13,0) $\text{mod}1$ and (11,0) $\text{mod}2$ semiconducting nanotubes as a function of pump-probe energy. The $\text{mod}1$ CP power spectrum is multiplied by a factor of 5 and offset for clarity.

Figure 22

(Color online) Coherent phonon intensity at the RBM frequency as a function of pump-probe energy for several $\text{mod}1$ semiconducting nanotubes. The upper five curves are for nanotubes in family 26 and the lower five curves are for tubes in family 29. Each curve is labeled with the chirality $\left(n,m\right)$ and the RBM phonon energy in meV. The curves are offset for clarity.