Low-energy effective Hamiltonian involving spin-orbit coupling in silicene and two-dimensional germanium and tin

Starting from symmetry considerations and the tight-binding method in combination with first-principles calculation, we systematically derive the low-energy effective Hamiltonian involving spin-orbit coupling (SOC) for silicene. This Hamiltonian is very general because it applies not only to silicene itself but also to the low-buckled counterparts of graphene for the other group-IVA elements Ge and Sn, as well as to graphene when the structure returns to the planar geometry. The effective Hamitonian is the analog to the graphene quantum spin Hall effect (QSHE) Hamiltonian. As in the graphene model, the effective SOC in low-buckled geometry opens a gap at the Dirac points and establishes the QSHE. The effective SOC actually contains the first order in the atomic intrinsic SOC strength ξ₀, while this leading-order contribution of SOC vanishes in the planar structure. Therefore, silicene, as well as the low-buckled counterparts of graphene for the other group-IVA elements Ge and Sn, has a much larger gap opened by the effective SOC at the Dirac points than graphene, due to the low-buckled geometry and larger atomic intrinsic SOC strength. Further, the more buckled is the structure, the greater is the gap. Therefore, the QSHE can be observed in low-buckled Si, Ge, and Sn systems in an experimentally accessible temperature regime. In addition, the Rashba SOC in silicene is intrinsic due to its own low-buckled geometry, which vanishes at the Dirac point K, while it has a nonzero value with deviation of k⃗ from the K point. Therefore, the QSHE in silicene is robust against the intrinsic Rashba SOC.