Quantum Spin Hall and Quantum Valley Hall Effect in Trilayer Graphene and their topological structure.

The present study pertains to the tri-layer graphene in the presence of spin-orbit coupling (SOC) to probe as quantum spin /valley Hall effect. The spin Chern-number C_s for energy-bands of tri-layer graphene having the essence of intrinsic SOC is analytically calculated. We find that for each valley and spin, C_s is three times larger in tri-layer graphene as compared to single layer graphene. We also study the tri-layer graphene in the presence of both electric-field and intrinsic SOC and investigate that tri-layer graphene goes through a phase transition from quantum spin Hall to a quantum valley Hall state when the strength of the electric field exceeds the intrinsic SOC strength. The robustness of associated topological bulk-state of tri-layer graphene are evaluated by adding various perturbations such as Rashba SO interaction α_R, and exchange-magnetization M. In addition, we consider a theoretical model, where only one of the outer layers in tri-layer graphene has the essence of intrinsic SOC, while the other two layers have zero intrinsic SOC.