  Spin-orbit splitting in graphene on Ni, Au, or Ag (111) substrates was
examined on the basis of density-functional theory. Graphene grown on the three
metals was found to have Rashba splitting of a few or several tens of meV. The
strong splitting obtained on Au or Ag substrates was mainly ascribed to
effective hybridization of graphene $p_{z}$ state with Au or Ag $d_{z^{2}}$
states, rather than charge transfer as previously proposed. Our work provides
theoretical understandings of the metal-induced Rashba effect in graphene.
