  We investigate nonequilibrium excitations and charge transport in
charge-neutral graphene driven with DC electric field by using the
nonequilibrium Green's function technique. Due to the vanishing Fermi surface,
electrons are subject to non-trivial nonequilibrium excitations such as highly
anisotropic momentum distribution of electron-hole pairs, an analog of the
Schwinger effect. We show that the electron-hole excitations, initiated by the
Landau-Zener tunneling with a superlinear IV relation $I \propto E^{3/2}$,
reaches a steady-state dominated by the dissipation due to optical phonons,
resulting in a marginally sublinear IV with $I \propto E$, in agreement with
recent experiments. The linear IV starts to show the sign of current saturation
as the graphene is doped away from the Dirac point, and recovers the
semi-classical relation for the saturated velocity. We give a detailed
discussion on the nonequilibrium charge creation and the relation between the
electron-phonon scattering rate and the electric field in the steady-state
limit. We explain how the apparent Ohmic IV is recovered near the Dirac point.
We propose a mechanism where the peculiar nonequilibrium electron-hole creation
can be utilized in a novel infra-red device.
