Probing zone-boundary optical phonons in doped graphene

We use first-principles density-functional theory to determine the adiabatic frequency shift of the $A_1^{\prime }\text{−}\mathbf{K}$ and $E^{\prime }\text{−}\mathbf{K}$ phonons in a monolayer graphene as a function of both electron and hole doping. Compared to the results for the $E_{2g}\text{−}\mathbf{\Gamma }$ phonon (Raman $G$ band), the results for the $A_1^{\prime }\text{−}\mathbf{K}$ phonon are dramatically different, while those for the $E^{\prime }\text{−}\mathbf{K}$ phonon are not so different. Furthermore, we calculate the frequency shifts, as a function of the charge doping of the $(\mathbf{K}+\Delta \mathbf{K})$ phonons responsible for the Raman 2D band—a key fingerprint of graphene, where $\mid \Delta \mathbf{K}\mid$ is determined by the double-resonance Raman process.