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

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.

Figure 1

Static spin Hall conductivity versus Fermi level for (a) germanene, (b) fluorostanene, (c) stanene, and (d) GeI. The blue dotted line illustrates the quantum $e^2/h$. The hatched gray region indicates the band gap. Zero Fermi level is chosen in midgap position.

Figure 2

Relevant part of the band structures of (a) germanene, (b) iodine-decorated germanene, (c) stanene, and (d) fluorostanene. Red, green, and blue circles depict relative contributions from $s$, $p_x+p_y$, and $p_z$ orbitals, respectively, to the band character. The insets display the gap regions. The valence-band maximum defines the energy zero.

Figure 3

Near gap regions of the band structures of (a) $1S-{\mathrm{MoS}}_2$, (b) $1S-{\mathrm{WS}}_2$, (c) $1T^{\prime }\text{−}{\mathrm{MoS}}_2$, and (d) $1T^{\prime }\text{−}{\mathrm{WS}}_2$. Red, green, blue, and magenta circles depict relative contributions from $s$, $p_x+p_y$, $p_z$, and $d$ orbitals, respectively, to the band character. The valence-band maximum defines the energy zero.

Figure 4

Static spin Hall conductivity versus Fermi level for (a) $1S-{\mathrm{MoS}}_2$, (b) $1S-{\mathrm{WS}}_2$, (c) $1T^{\prime }\text{−}{\mathrm{MoS}}_2$, and (d) $1T^{\prime }\text{−}{\mathrm{WS}}_2$. The blue dotted line illustrates the quantum $e^2/h$. The hatched gray region indicates the band gap. Zero Fermi level is chosen in midgap position. In the $1T^{\prime }$ case the negative ${\sigma }_{xy}^z$ is plotted.

Figure 5

Static spin Hall conductivity, $Z_2$ invariant, and band gap as a function of the applied electric field for germanene and as a function of the compressive biaxial strain for fluorostanene. The blue dotted horizontal line illustrates the quantization value $e^2/h$.

Figure 6

Band structures for (a) germanene, (b) germanene under the influence of external electric field $F_z=2\times {10}^6$ V/cm, (c) under $F_z=3\times {10}^6$ V/cm, and (d) under $F_z=5\times {10}^6$ V/cm. Red, green, and blue circles depict relative contributions from $s$, $p_x+p_y$, and $p_z$ orbitals, respectively, to the band character. The insets display the gap regions. The valence-band maximum defines the energy zero.

Figure 7

Variation of static spin Hall conductivity (a) and band gap (b) as a function of applied uniaxial strain for germanene (black), stanene (green), GeI (red), and SnF (blue) hexagonal crystals. The cyan dotted horizontal line in (a) illustrates the quantization value $e^2/h$.