Phonon switching and combined Fano-Rice effect in optical spectra of bilayer graphene

Recent infrared measurements of phonon peaks in gated bilayer graphene reveal two striking signatures of electron-phonon interaction: an asymmetric Fano lineshape and a giant variation of the peak intensity as a function of the applied gate voltage. In this Letter we provide a unified theoretical framework which accounts for both these effects and unveils the occurrence of a switching mechanism between the symmetric ($E_g$) and anti-symmetric ($E_u$) phonon mode as dominant channel in the optical response. A complete phase diagram of the optical phonon response is also presented, as a function of both the charge density and the bandgap.