Decay and Frequency Shift of Inter and Intravalley Phonons in Graphene -Dirac Cone Migration-

By considering analytical expressions for the self-energies of intervalley and intravalley phonons in graphene, we describe the behavior of D, 2D, and D$'$ Raman bands with changes in doping ($\mu$) and light excitation energy ($E_L$). Comparing the self-energy with the observed $\mu$ dependence of the 2D bandwidth, we estimate the wavevector $q$ of the constituent intervalley phonon at $\hbar vq\simeq E_L/1.6$ ($v$ is electron's Fermi velocity) and conclude that the self-energy makes a major contribution (60%) to the dispersive behavior of the D and 2D bands. The estimation of $q$ is based on an image of shifted Dirac cones in which the resonance decay of a phonon satisfying $q > \omega/v$ ($\omega$ is the phonon frequency) into an electron-hole pair is suppressed when $\mu < (vq-\omega)/2$. We highlight the fact that the decay of an intervalley (and intravalley longitudinal optical) phonon with $q=\omega/v$ is strongly suppressed by electron-phonon coupling at an arbitrary $\mu$. This feature is in contrast to the divergent behavior of an intravalley transverse optical phonon, which bears a close similarity to the polarization function relevant to plasmons.