Influence of the phonon-mediated spin-orbit coupling in graphene decorated with adatoms*

Graphene covered with certain heavy adatoms has been predicted( Conan Weeks et al., Phys. Rev. X 1, 021001 (2011)) to become a topological insulator by virtue of a proximity-effect induced spin-orbit coupling. In addition, the adatoms also induce a coupling between the electron spin and the phonons. Using group theory and tight-binding models, we systematically investigate the coupling between the electron spin and in-plane lattice phonons. We discuss the consequences of this coupling for the dynamics of electrons on the graphene $\pi$ band.