Kinematics of Inverse Compton Scattering from Dirac Carriers in Graphene

We examine the kinematics of Compton and inverse Compton scattering by Dirac carriers in graphene. Unlike photons in vacuum, which propagate at the speed of light c, graphene carriers have a characteristic velocity vF, with vF much less than c. This difference means that the standard Compton scattering formula cannot be applied to graphene simply by replacing c with vF. We first examine Compton-like scattering for a gapless Dirac carrier initially at the Dirac point and for a gapped carrier at the band edge, both corresponding to zero in-plane momentum. We then consider scattering from an energetic gapless Dirac carrier. By combining energy conservation with the graphene dispersion and in-plane momentum conservation, we obtain a quadratic equation for the scattered photon energy within the model considered. This relation is the main kinematic result of the work. We identify inverse Compton-like scattering by the transfer of energy from the carrier to the photon, subject to the simultaneous conservation of energy and in-plane momentum. We also discuss how this process differs kinematically from Doppler-shifted emission, Cerenkov-like radiation, harmonic generation, photon drag, and other nonlinear optical processes. Finally, we propose a time-resolved pump-probe arrangement in which the scattered photon spectrum is the primary observable, while time- and angle-resolved photoemission spectroscopy (TR-ARPES) provides an independent probe of the nonequilibrium carrier distribution. These results provide a kinematic framework for analysing inverse Compton-like scattering in graphene and for identifying the conditions under which energy can be transferred from energetic Dirac carriers to photons.