Electric Field Effects on the Optical Vibrations in AB-Stacked Bilayer Graphene

Using first-principles methods, we show that an applied perpendicular electric field $E$ breaks the inversion symmetry of AB-stacked bilayer graphene (BLG), thereby slightly mixing the two in-plane high-energy optical vibrations ($E_g$ and $E_u$ modes). The mixed amplitudes increase parabolically with respect to the field strength when $E$$<$2.0 V/nm, and then exhibit linear dependence when $E$$>$2.0 V/nm. In contrast, the mixing effect on the out-of-plane vibrations ($A_{1g}$ and $A_{2u}$ modes) is found to be much stronger, with the mixed amplitudes nearly an order of magnitude larger than those for the in-plane modes. For the two in-plane modes, we then calculate their phonon linewidths and frequency shifts as a function of the electric field as well as the Fermi level. Our results reveal delicate interplay between electrons and phonons in BLG, tunable by the applied fields and charge carrier densities.