  Elemental monolayers of the group 14 with a buckled honeycomb structure,
namely silicene, germanene, stanene, and plumbene, are known to demonstrate a
spin splitting as a result of an electric field parallel to their high symmetry
axis which is capable of tuning their topological phase between a quantum spin
Hall insulator and an ordinary band insulator. We perform first-principles
calculations based on the density functional theory to quantify the
spin-dependent band gaps and the spin splitting as a function of the applied
electric field and extract the main coefficients of the invariant Hamiltonian.
Using the linear response theory and the Wannier interpolation method, we
calculate the spin Hall conductivity in the monolayers and study its
sensitivity to an external electric field. Our results show that the spin Hall
conductivity is not quantized and in the case of silicene, germanene, and
stanene degrades significantly as the electric field inverts the band gap and
brings the monolayer into the trivial phase. The electric field induced band
gap does not close in the case of plumbene which shows a spin Hall conductivity
that is robust to the external electric field.
