Resonant coherent phonon spectroscopy of single-walled carbon nanotubes

Abstract

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 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 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 mod 2 families. We also find that the coherent phonon intensities are considerably weaker in mod 1 nanotubes in comparison with mod 2 nanotubes, which is also in excellent agreement with experiment.

Theory of coherent phonons in carbon nanotubes and graphene nanoribbons

“…The lowest frequency coherent phonons that can be photoexcited in SWNTs using ultrafast laser pulses are coherent RBM phonons with phonon wavevector q = 0 corresponding to a mode in which the diameter of the nanotube periodically expands and contracts. These coherent RBM phonons have been studied experimentally by several groups [51][52][53][54][55][56]. In addition to the lower frequency coherent RBM modes, higher frequency q = 0 coherent G mode phonons have also been observed [57,58].…”

“…Recently, microscopic theories have been developed for the generation and detection of coherent phonons in SWNTs and GNRs using an extended tight-binding model for electronic states, valence force field models for the phonons, and deformation potential coupling between electrons and phonons [47,54]. As in bulk non-polar semiconductors, the CP amplitudes in SWNTs and GNRs are found to satisfy a driven oscillator equation with the driving term depending on the photoexcited carrier densities.…”

“…Further details concerning the evaluation of the electron-phonon matrix elements in SWNTs and GNRs can be found in the appendices of [54] and [47].…”

“…For zigzag and armchair GNRs, the Fermi golden rule imaginary dielectric function is given in [47]. For SWNTs we have [54] …”

“…In our MVFF model, we include bond stretching, in-plane bond bending, out-of-plane bond bending, and bond twisting potentials. Our MVFF model for SWNTs has seven force constants [54], four due to bond-stretching interactions out to fourth nearest neighbor shells and one each from the remaining three interactions. We obtained force constants for the MVFF model by fitting our MVFF results for graphene to the VFF results shown in figure 10.…”