  We analyze the origin of spin-orbit coupling (SOC) in fluorinated graphene
using Density Functional Theory (DFT) and a tight-binding model for the
relevant orbitals. As it turns out, the dominant source of SOC is the atomic
spin-orbit of fluorine adatoms and not the impurity induced SOC based on the
distortion of the graphene plane as in hydrogenated graphene. More
interestingly, our DFT calculations show that SOC is strongly affected by both
the type and concentrations of the graphene's carriers, being enhanced by
electron doping and reduced by hole doping. This effect is due to the charge
transfer to the fluorine adatom and the consequent change in the
fluorine-carbon bonding. Our simple tight-binding model, that includes the SOC
of the $2p$ orbitals of F and effective parameters based on maximally localized
Wannier functions, is able to account for the effect. The strong enhancement of
the SOC induced by graphene doping opens the possibility to tune the spin
relaxation in this material.
