Double-resonant Raman scattering in graphene, few-layer graphene, and carbon nanotubes

In this work, we present an analysis of double-resonant two-phonon and phonon-defect Raman modes in graphene, few-layer graphene and carbon nanotubes. Graphene is a single-layer of sp2-hybridized carbon atoms that arrange in a hexagonal lattice and has attracted tremendous scientific interest since its first experimental preparation in 2004. Carbon nanotubes can be considered as a rolled-up piece of graphene with periodic boundary conditions along its circumference. The study of both systems is largely based on optical spectroscopy, namely Raman spectroscopy, since this approach enables a thorough characterization of their electronic and vibrational properties. Raman scattering and especially double-resonant Raman scattering are also the experimental methods of choice for the analyses presented here. Although the concept of double-resonant Raman scattering has been introduced more than a decade ago, the precise scattering processes of certain Raman modes are still controversially discussed and thus need clarification. The present work can be separated into two main parts. The first part will, both theoretically and experimentally, investigate the double-resonant 2D mode in Bernal-stacked bilayer graphene. The 2D mode is a two-phonon process, in which two transverse optical (TO) phonons with opposite momentum from the edges of the Brillouin zone (K points) are scattered. In contrast to single-layer graphene, this Raman mode presents a complex lineshape in bilayer graphene and was subject of many theoretical and experimental works. However, a consistent explanation of the different contributions to the characteristic peak shape is still lacking. By calculating the two-dimensional double-resonant scattering cross-section of this Raman mode completely from first principles, we will unravel the dominant contributions and provide a precise analysis of the different scattering processes. In fact, by proving that the 2D mode in bilayer graphene is composed of three dominant contributions, we contradict all previous works on this topic. Moreover, we show that the dominant contribution to the Raman scattering cross-section stems from so-called inner scattering processes with phonons from the K − Γ high-symmetry direction. A very important, but often neglected, key feature of double-resonant Raman scattering is quantum interference between different scattering paths. We will demonstrate that the characteristic lineshape of the 2D mode in bilayer graphene is dictated by interference effects. In fact, the observed 2D-mode lineshape strongly depends on the electronic broadening, i.e., the inverse lifetime of the electronic states, and can be manipulated by external parameters, such as doping, defects, strain, or the laser excitation energy. We furthermore demonstrate that the splitting between both TO phonon branches in bilayer graphene along K − Γ is of great importance for a correct analysis of the 2D-mode lineshape. Using our correct assignment of the different peaks in the 2D-mode spectrum, we present an approach to directly extract the TO phonon splitting from experimentally measured Raman spectra. Furthermore, we show that the splitting of the electronic bands can be also obtained using the presented analysis. Finally, we discuss the po-