Low-energy effective Hamiltonian involving spin-orbit coupling in Silicene and Two-Dimensional Germanium and Tin

TL;DR

This paper derives a comprehensive low-energy Hamiltonian for silicene and related materials, revealing how spin-orbit coupling induces quantum spin Hall effects with larger gaps than graphene, and discusses the robustness of these effects.

Contribution

It provides a systematic derivation of the effective Hamiltonian including SOC for silicene and similar materials, highlighting the impact of buckling and atomic SOC strength on the quantum spin Hall effect.

Findings

Effective SOC opens a sizable gap at Dirac points in silicene and related materials.

The quantum spin Hall effect is observable at accessible temperatures due to larger SOC-induced gaps.

Intrinsic Rashba SOC in silicene does not destroy the QSHE, ensuring robustness.

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.