Quantization of spin Hall conductivity in two-dimensional topological insulators versus symmetry and spin-orbit interaction

Abstract

The third-rank tensor of the static spin Hall conductivity is investigated for two-dimensional (2D) topological insulators by electronic structure calculations. For highly symmetric hexagonal systems its numerical values are close to the conductance quantum ${e}^{2}/h$, independent of the gap size. 2D crystals with a square Bravais lattice present similar effects, while rectangular translational symmetry yields conductivity values much below ${e}^{2}/h$, showing that a quantum spin Hall phase is not generally characterized by a quantized spin Hall conductivity. Vertical electric fields applied to hexagonal 2D crystals strongly reduce the conductivity, despite the conservation of the quantum spin Hall state up to a critical field strength. Weak symmetry-conserving biaxial but also symmetry-lowering uniaxial strain has a minor influence as long as inverted gaps dictate the topological character. The results are discussed in terms of the atomic geometry and the Rashba contribution to the spin-orbit interaction (SOI) using a tight-binding approximation. Translational and point-group symmetry as well as SOI rule the deviation from the quantization of the spin Hall conductance.