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…

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  National Science Foundation