Anomaly of optical phonon in monolayer graphene

The frequency shift and broadening of long-wavelength optical phonons due to interactions with electrons are calculated in a monolayer graphene as a function of the electron density. The broadening disappears and the frequency shift exhibits a logarithmic singularity when the Fermi energy is half of the energy of the optical phonon. The shift increases in proportion to the Fermi energy for sufficiently high electron density.