Understanding double-resonant Raman scattering in chiral carbon nanotubes: Diameter and energy dependence of the $D$ mode

摘要

We present a theoretical model to describe the double-resonant scatteringprocess in arbitrary carbon nanotubes. We use this approach to investigate thedefect-induced $D$ mode in CNTs and unravel the dependence of the $D$-modefrequency on the CNT diameter and on the energy of the resonant opticaltransition. Our approach is based on the symmetry of the hexagonal lattice andgeometric considerations, hence the method is independent of the exact modelthat is chosen to describe the electronic band structure or the phonondispersion. We finally clarify the diameter dependence of this Raman mode thatwas controversely discussed in the past and demonstrate that, depending on theexperimental conditions, in general two different dependencies can be measured.We also prove that carbon nanotubes with arbitrary chiral index can exhibit a$D$ mode in their Raman spectrum, in contrast to previous symmetry-basedarguments. Furthermore, we give a direct quantification of thecurvature-induced phonon frequency corrections of the $D$-mode in carbonnanotubes with respect to graphite.