  We report on theoretical investigations of the spin-orbit coupling effects in
fluorinated graphene. First-principles density functional calculations are
performed for the dense and dilute adatom coverage limits. The dense limit is
represented by the single-side semifluorinated graphene, which is a metal with
spin-orbit splittings of about 10 meV. To simulate the effects of a single
adatom, we also calculate the electronic structure of a $10 \times 10$
supercell, with one fluorine atom in the top position. Since this dilute limit
is useful to study spin transport and spin relaxation, we also introduce a
realistic effective hopping Hamiltonian, based on symmetry considerations,
which describes the supercell bands around the Fermi level. We provide the
Hamiltonian parameters which are best fits to the first-principles data. We
demonstrate that, unlike for the case of hydrogen adatoms, fluorine's own
spin-orbit coupling is the principal cause of the giant induced local
spin-orbit coupling in graphene. The $sp^3$ hybridization induced transfer of
spin-orbit coupling from graphene's $\sigma$ bonds, important for hydrogenated
graphene, contributes much less. Furthermore, the magnitude of the induced
spin-orbit coupling due to fluorine adatoms is about $1000$ times more than
that of pristine graphene, and 10 times more than that of hydrogenated
graphene. Also unlike hydrogen, the fluorine adatom is not a narrow resonant
scatterer at the Dirac point. The resonant peak in the density of states of
fluorinated graphene in the dilute limit lies 260 meV below the Dirac point.
The peak is rather broad, about 300 meV, making the fluorine adatom only a
weakly resonant scatterer.
