Non-adiabatic Kohn-anomaly in a doped graphene monolayer

5 pages, 3 figures, Accepted by Phys. Rev. LettInternational audienceWe compute, from first-principles, the frequency of the E2g, 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