The Schwinger mechanism and graphene

The Schwinger mechanism, the production of charged particle-antiparticle pairs in a macroscopic external electric field, is derived for 2+1 dimensional theories. The rate of pair production per unit area for four species of massless fermions, with charge $q$, in a constant electric field $E$ is given by $ \pi^{-2} \hbar^{-3/2} \tilde{c}^{-1/2} (q E)^{3/2} $ where $\tilde{c}$ is the speed of light for the two-dimensional system. To the extent undoped graphene behaves like the quantum field-theoretic vacuum for massless fermions in 2+1 dimensions, the Schwinger mechanism should be testable experimentally. A possible experimental configuration for this is proposed. Effects due to deviations from this idealized picture of graphene are briefly considered. It is argued that with present day samples of graphene, tests of the Schwinger formula may be possible.