Edge states of hydrogen terminated monolayer materials: silicene, germanene and stanene ribbons

We investigate the energy dispersion of the edge states in zigzag silicene, germanene and stanene nanoribbons with and without hydrogen termination based on a multi-orbital tight-binding model. Since the low buckled structures are crucial for these materials, both the $\pi$ and $\sigma$ orbitals have a strong influence on the edge states, different from the case for graphene nanoribbons. The obtained dispersion of helical edge states is nonlinear, similar to that obtained by first-principles calculations. On the other hand, the dispersion derived from the single-orbital tight-binding model is always linear. Therefore, we find that the non-linearity comes from the multi-orbital effects, and accurate results cannot be obtained by the single-orbital model but can be obtained by the multi-orbital tight-binding model. We show that the multi-orbital model is essential for correctly understanding the dispersion of the edge states in tetragen nanoribbons with a low buckled geometry.