The effect of electron-A1g phonon coupling in spin-orbit-coupled graphene

We investigate the effects of electron-A(1g) phonon and Rashba spin-orbit couplings on the electronic spectrum of pristine graphene charge carriers. We introduce a Frohlich-type Hamiltonian describing the electron-phonon interaction in spin-orbit-coupled graphene, and propose a diagonalisation procedure to solve this Hamiltonian based on the Lee-Low-Pines theory which includes two successive unitary transformations. By means of these transformations, our results show that, in spin-orbit-coupled graphene, electron-A1g phonon interaction lifts the degeneracy of two zero-gap branches and enhances the existing gap between other two branches. It is also found that the magnitude of the splitting in ungapped branches depends on the strength of spin-orbit interaction, and thus the emerging gap can be tuned by spin-orbit interaction.