  We investigate the doping of AA-stacked graphene bilayers. Applying a mean
field theory at zero temperature we find that, at half-filling, the bilayer is
an antiferromagnetic insulator. Upon doping, the homogeneous phase becomes
unstable with respect to phase separation. The separated phases are an undoped
antiferromagnetic insulator and a metal with a non-zero concentration of charge
carriers. At sufficiently high doping, the insulating areas shrink and
disappear, and the system becomes a homogeneous metal. The conductivity changes
drastically upon doping, so the bilayer may be used as a switch in electronic
devices. The effects of finite temperature are also discussed.
