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

TLDR

The effective Hamiltonian is analogous to the graphene quantum spin Hall effect Hamiltonian. The authors derive a general low‑energy effective Hamiltonian with spin‑orbit coupling for silicene and its low‑buckled group‑IVA counterparts (Ge, Sn) using symmetry, tight‑binding, and first‑principles methods. The Hamiltonian predicts a sizable SOC‑induced gap at the Dirac points that grows with buckling, making QSHE observable in low‑buckled Si, Ge, and Sn at experimentally accessible temperatures, and shows that the intrinsic Rashba SOC vanishes at the Dirac point, rendering the QSHE robust.

Abstract

Starting from the symmetry aspects and tight-binding method in combination with first-principles calculation, we systematically derive the low-energy effective Hamiltonian involving spin-orbit coupling (SOC) for silicene, which is very general because this Hamiltonian applies to not only the silicene itself but also the low-buckled counterparts of graphene for other group IVA elements Ge and Sn, as well as graphene when the structure returns to the planar geometry. The effective Hamitonian is the analogue to the first graphene quantum spin Hall effect (QSHE) Hamiltonian. Similar to graphene model, the effective SOC in low-buckled geometry opens a gap at Dirac points and establishes QSHE. The effective SOC actually contains first order in the atomic intrinsic SOC strength $\xi_{0}$, while such leading order contribution of SOC vanishes in planar structure. Therefore, silicene as well as low-buckled counterparts of graphene for other group IVA elements Ge and Sn has much larger gap opened by effective SOC at Dirac points than graphene due to low-buckled geometry and larger atomic intrinsic SOC strength. Further, the more buckled is the structure, the greater is the gap. Therefore, 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 Dirac point $K$, while has nonzero value with $\vec{k}$ deviation from the $K$ point. Therefore, the QSHE in silicene is robust against to the intrinsic Rashba SOC.