Quantum spin Hall and quantum valley Hall effects in trilayer graphene and their topological structures

Majeed Ur Rehman^{1,2}, A A Abid^{1,2}

1. ICQD, Hefei National Laboratory for Physical Sciences at Microscale, Department of Physics, University of Science and Technology of China, Hefei 230026, China;
2. Key Laboratory of Geospace Environment(Chinese Academy of Sciences), Department of Geophysics and Planetary Science, University of Science and Technology of China, Hefei 230026, China

The present study pertains to the trilayer graphene in the presence of spin orbit coupling to probe the quantum spin/valley Hall effect. The spin Chern-number C_{s} for energy-bands of trilayer graphene having the essence of intrinsic spin-orbit coupling is analytically calculated. We find that for each valley and spin, C_{s} is three times larger in trilayer graphene as compared to single layer graphene. Since the spin Chern-number corresponds to the number of edge states, consequently the trilayer graphene has edge states, three times more in comparison to single layer graphene. We also study the trilayer graphene in the presence of both electric-field and intrinsic spin-orbit coupling and investigate that the trilayer graphene goes through a phase transition from a quantum spin Hall state to a quantum valley Hall state when the strength of the electric field exceeds the intrinsic spin coupling strength. The robustness of the associated topological bulk-state of the trilayer graphene is evaluated by adding various perturbations such as Rashba spin-orbit (RSO) interaction α_{R}, and exchange-magnetization M. In addition, we consider a theoretical model, where only one of the outer layers in trilayer graphene has the essence of intrinsic spin-orbit coupling, while the other two layers have zero intrinsic spin-orbit coupling. Although the first Chern number is non-zero for individual valleys of trilayer graphene in this model, however, we find that the system cannot be regarded as a topological insulator because the system as a whole is not gaped.

Corresponding Authors:
Majeed Ur Rehman
E-mail: majeedqau@live.com

Cite this article:

Majeed Ur Rehman, A A Abid Quantum spin Hall and quantum valley Hall effects in trilayer graphene and their topological structures 2017 Chin. Phys. B 26 127304

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