  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.
