  We study the Coulomb drag between two single graphene sheets in intrinsic and
extrinsic graphene systems with no interlayer tunneling. The general expression
for the nonlinear susceptibility appropriate for single-layer graphene systems
is derived using the diagrammatic perturbation theory, and the corresponding
exact zero-temperature expression is obtained analytically. We find that,
despite the existence of a non-zero conductivity in an intrinsic graphene
layer, the Coulomb drag between intrinsic graphene layers vanishes at all
temperatures. In extrinsic systems, we obtain numerical results and an
approximate analytical result for the drag resistivity $\rho_{\textrm{D}}$, and
find that $\rho_{\textrm{D}}$ goes as $T^2$ at low temperature $T$, as $1/d^4$
for large bilayer separation $d$ and $1/n^3$ for high carrier density $n$. We
also discuss qualitatively the effect of plasmon-induced enhancement on the
Coulomb drag, which should occur at a temperature of the order of or higher
than the Fermi temperature.
