Nonadiabatic Kohn Anomaly in a Doped Graphene Monolayer

We compute, from first principles, the frequency of the $E_{2g}$, $\mathbf{\Gamma }$ phonon (Raman $G$ band) of graphene, as a function of the charge doping. Calculations are done using (i) the adiabatic Born-Oppenheimer approximation and (ii) time-dependent perturbation theory to explore dynamic effects beyond this approximation. The two approaches provide very different results. While the adiabatic phonon frequency weakly depends on the doping, the dynamic one rapidly varies because of a Kohn anomaly. The adiabatic approximation is considered valid in most materials. Here, we show that doped graphene is a spectacular example where this approximation miserably fails.