  Functionalizing graphene was recently shown to have a dramatic effect on the
electronic properties of this material. Here we investigate spatial ordering of
adatoms driven by the RKKY-type interactions. In the ordered state, which
arises via a Peierls-instability-type mechanism, the adatoms reside mainly on
one of the two graphene sublattices. Bragg scattering of electron waves induced
by sublattice symmetry breaking results in a band gap opening, whereby Dirac
fermions acquire a finite mass. The band gap is found to be immune to the
adatoms' positional disorder, with only an exponentially small number of
localized states residing in the gap. The gapped state is stabilized in a wide
range of electron doping. Our findings show that controlled adsorption of
adatoms or molecules provides a route to engineering a tunable band gap in
graphene.
