  We explore the relaxation of photo-excited graphene by solving a transient
Boltzmann transport equation with electron-phonon (e-ph) and electron-electron
(e-e) scattering. Simulations show that when the excited carriers are relaxed
by e-ph scattering only, a population inversion can be achieved at energies
determined by the photon energy. However, e-e scattering quickly thermalizes
the carrier energy distributions washing out the negative optical conductivity
peaks. The relaxation rates and carrier multiplication effects are presented as
a function of photon energy and dielectric constant.
