Theory of coherent phonons in carbon nanotubes and graphene nanoribbons

Abstract

We survey our recent theoretical studies on the generation and detection of coherent radial breathing mode (RBM) phonons in single-walled carbon nanotubes and coherent radial breathing like mode (RBLM) phonons in graphene nanoribbons. We present a microscopic theory for the electronic states, phonon modes, optical matrix elements and electron-phonon interaction matrix elements that allows us to calculate the coherent phonon spectrum. An extended tight-binding (ETB) model has been used for the electronic structure and a valence force field (VFF) model has been used for the phonon modes. The coherent phonon amplitudes satisfy a driven oscillator equation with the driving term depending on the photoexcited carrier density. We discuss the dependence of the coherent phonon spectrum on the nanotube chirality and type, and also on the graphene nanoribbon mod number and class (armchair versus zigzag). We compare these results with a simpler effective mass theory where reasonable agreement with the main features of the coherent phonon spectrum is found. In particular, the effective mass theory helps us to understand the initial phase of the coherent phonon oscillations for a given nanotube chirality and type. We compare these results to two different experiments for nanotubes: (i) micelle suspended tubes and (ii) aligned nanotube films. In the case of graphene nanoribbons, there are no experimental observations to date. We also discuss, based on the evaluation of the electron-phonon interaction matrix elements, the initial phase of the coherent phonon amplitude and its dependence on the chirality and type. Finally, we discuss previously unpublished results for coherent phonon amplitudes in zigzag nanoribbons obtained using an effective mass theory.