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

Spin splitting and spin Hall conductivity in buckled monolayers of the group 14: First-principles calculations