Anomaly of optical phonons in bilayer graphene

The frequency shift and broadening of long-wavelength optical phonons are calculated for varying Fermi level in a bilayer graphene. Symmetric modes with displacements of two layers in phase are affected strongly. In the absence of a magnetic field, the broadening disappears and the frequency shift exhibits a logarithmic singularity when the Fermi energy is a half of the phonon energy, and the shift increases in proportion to the Fermi energy for sufficiently high electron density. In magnetic fields, the broadening is resonantly enhanced and the frequency shift exhibits a rapid change when the phonon energy becomes the energy separation of Landau levels between which optical transitions are allowed.