Plasma wave instabilities in nonequilibrium graphene

We study two-stream instabilities in a nonequilibrium system in which a stream of electrons is injected into doped graphene. As with equivalent nonequilibrium parabolic band systems, we find that the graphene systems can support unstable charge-density waves whose amplitudes grow with time. We determine the range of wave vectors $\mathbit{q}$ that are unstable, and their growth rates. We find no instability for waves with wave vectors parallel or perpendicular to the direction of the injected carriers. We find that, within the small-wave-vector approximation, the angle between $\mathbit{q}$ and the direction of the injected electrons that maximizes the growth rate increases with increasing $|\mathbit{q}|$. We compare the range and strength of the instability in graphene to that of two- and three-dimensional parabolic band systems.