Group theoretical and topological analysis of the quantum spin Hall effect in silicene

TL;DR

This paper uses group theory and topological analysis to study the quantum spin Hall effect in silicene, revealing how electric fields and Rashba interactions can induce topological phase transitions and non-trivial phases.

Contribution

It introduces a symmetry-based Hamiltonian derivation and a gauge-invariant topological invariant calculation for silicene, highlighting the role of Rashba interactions in topological phases.

Findings

Identification of a new symmetry-allowed term in the Hamiltonian.

Electric fields can induce topological phase transitions in silicene.

Interplay of Rashba terms can generate a non-trivial quantum spin Hall phase.

Abstract

Silicene consists of a monolayer of silicon atoms in a buckled honeycomb structure. It was recently discovered that the symmetry of such a system allows for interesting Rashba spin-orbit effects. A perpendicular electric field is able to couple to the sublattice pseudospin, making it possible to electrically tune and close the band gap. Therefore, external electric fields may generate a topological phase transition from a topological insulator to a normal insulator (or semimetal) and vice versa. The contribution of the present article to the study of silicene is twofold: First, we perform a group theoretical analysis to systematically construct the Hamiltonian in the vicinity of the $K$ points of the Brillouin zone and discover a new, but symmetry allowed term. Subsequently, we identify a tight binding model that corresponds to the group theoretically derived Hamiltonian near the $K$ points. Second, we start from this tight binding model to analyze the topological phase diagram of silicene by an explicit calculation of the $Z_2$ topological invariant of the band structure. To this end, we calculate the $Z_2$ topological invariant of the honeycomb lattice in a manifestly gauge invariant way which allows us to include $S_z$ symmetry breaking terms -- like Rashba spin orbit interaction -- into the topological analysis. Interestingly, we find that the interplay of two Rashba terms can generate a non-trivial quantum spin Hall phase in silicene. This is in sharp contrast to the more extensively studied honeycomb system graphene where Rashba spin orbit interaction is known to compete with the quantum spin Hall effect in a potentially detrimental way.