Structural tristability and deep Dirac states in bilayer silicene on Ag(111) surfaces

We report on total-energy electronic-structure calculations in the density-functional theory performed for both monolayer and bilayer silicene on Ag(111) surfaces. The root 3 x root 3 structure observed experimentally and argued to be the monolayer silicene in the past [Chen et al., Phys. Rev. Lett. 110, 085504 (2013)] is identified as the bilayer silicene on the Ag(111) surface. The identification is based on our accurate density-functional calculations in which three approximations, the local density approximation, the generalized-gradient approximation, and the van der Waals density-functional approximation, to the exchange-correlation energy have been carefully examined. We find that the structural tristability exists for the root 3 x root 3 bilayer silicene. The calculated energy barriers among the three stable structures are in the range of 7-9 meV per Si atom, indicating possible flip-flop motions among the three. We have found that the flip-flop motion between two of the three structures produces the honeycomb structure in the STM images, whereas the motion among the three does the 1 x 1 structure. We have found that the electron states which effectively follow the Dirac equation in the freestanding silicene couple with the substrate Ag orbitals due to the bond formation, and shift downwards deep in the valence bands. This feature is common to all the stable or metastable silicene layer on the Ag(111) substrate.