Rashba spin splitting based on trilayer graphene systems

We establish a general Rashba Hamiltonian for trilayer graphene (TLG) by introducing an extrinsic layer-dependent Rashba spin-orbit coupling (SOC) arising from the off-plane inversion symmetry breaking. Our results indicate that the band spin splitting depends strongly on the layer distribution and sign of Rashba SOC as well as the ABA or ABC stacking order of TLG. We find that spin splitting is significantly enhanced as the number of layers of the Rashba SOC with the same sign and magnitude increases. For the spatially separated two Rashba SOCs of the same magnitude but the opposite sign, no spin splitting arises in ABC-TLG due to the preservation of inversion symmetry that ensures the complete cancellation of contributions from the opposite layers, whereas nonzero spin splitting is observed for ABA-TLG due to its own lack of inversion symmetry. We further illustrate that gate voltage is effective to modulate the spin-polarized states near the band edges. Moreover, we use density-functional-theory calculations to verify the Rashba splitting effect in the example of TLG interfaced by Au layer(s), which induce simultaneously the effective terms of Rashba SOC and gate voltage. Our results demonstrate the significance of layer and symmetry in manipulating spin and can be extended to multilayer graphene or other van der Waals interface systems.