Flat bands in silicene, germanene, and stanene with periodic triangular defects.

We study the electronic band structure of low-buckled silicene, germanene, and stanene with triangular defects in the superhoneycomb arrangement by performing first-principles calculations within the framework of density functional theory. Our calculation results show that low buckling structures with triangular defects can generate flat bands. However, these flat bands have partial energy overlap with the adjacent bands. To address this issue, we introduced spin-orbit coupling (SOC), aiming to decouple the flat bands from the adjacent bands. Subsequent calculations confirmed that this method successfully produced isolated flat bands. It is particularly noteworthy that by introducing SOC, we have obtained the highest valence band with a bandwidth of only0.005eVin stanene with a specific configuration, which fully demonstrates the great potential of this system in achieving ultra-flat bands. This study successfully demonstrates that introducing periodic triangular defects and incorporating SOC in low-buckled silicene, germanene, and stanene is an effective strategy for generating isolated flat bands and achieving precise modulation of electronic states, providing a systematic theoretical pathway for the design of two-dimensional flat band materials.