  Making devices with graphene necessarily involves making contacts with
metals. We use density functional theory to study how graphene is doped by
adsorption on metal substrates and find that weak bonding on Al, Ag, Cu, Au and
Pt, while preserving its unique electronic structure, can still shift the Fermi
level with respect to the conical point by $\sim 0.5$ eV. At equilibrium
separations, the crossover from $p$-type to $n$-type doping occurs for a metal
work function of $\sim 5.4$ eV, a value much larger than the graphene work
function of 4.5 eV. The numerical results for the Fermi level shift in graphene
are described very well by a simple analytical model which characterizes the
metal solely in terms of its work function, greatly extending their
applicability.
