Effects of electron–acoustic phonon interaction in the presence of spin–orbit couplings in graphene

We investigate the effects of electron-zone center acoustic phonon interaction on spin–orbit couplings in pristine graphene. We examine the polaron formation and analyse the effects of spin–orbit couplings on polaron formation and electron–phonon couplings. Fröhlich type Hamiltonian within the continuum limit is used to describe the electron–phonon system. With theoretical analysis, we show that, in the presence of electron–phonon interaction, the intrinsic and Rashba spin–orbit couplings of graphene become k dependent and Rashba spin–orbit coupling exhibits different characters with respect to spin orientation. Even there is a symmetry between electron–hole states if only one spin–orbit mechanism is present, electron–phonon coupling induces symmetry breaking between electron–hole states. Moreover, polaronic effect is decreased for both spin states, but it is increased beyond the some critical values of k for spin-up states in the presence of spin–orbit couplings.