Intrinsic spin Hall conductivity in one-, two-, and three-dimensional trivial and topological systems

Ab initio relativistic band structure calculations are performed for the frequency-dependent spin Hall conductivity of three- (3D), two- (2D) and one-dimensional (1D) materials such as bulk semiconductors, atomically thin crystals, and their nanoribbons. Besides the influence of the dimensionality we also study differences between trivial and topological insulators (TIs). The frequency dependence of the conductivity is governed by the band-structure details, while its static value scales with the spin-orbit interaction in 3D but is quantized in units of ${e}^{2}/h$ for 2D TIs. 1D topological edge states influence the conductivity mainly for vanishing frequencies.