  Motivated by recent experiments, we investigate the electron-vibron coupling
in suspended carbon nanotube quantum dots, starting with the electron-phonon
coupling of the underlying graphene layer. We show that the coupling strength
depends sensitively on the type of vibron and is strongly sample dependent. The
coupling strength becomes particularly strong when inhomogeneity-induced
electronic quantum dots are located near regions where the vibronic mode is
associated with large strain. Specifically, we find that the longitudinal
stretching mode and the radial breathing mode are coupled via the strong
deformation potential, while twist modes couple more weakly via a mechanism
involving modulation of the electronic hopping amplitudes between carbon sites.
A special case are bending modes: for symmetry reasons, their coupling is only
quadratic in the vibron coordinate. Our results can explain recent experiments
on suspended carbon nanotube quantum dots which exibit vibrational sidebands
accompanied by the Franck-Condon blockade with strong electron-vibron coupling.
