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 $\sqrt{3}\times \sqrt{3}$ 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 $\sqrt{3}\times \sqrt{3}$ 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 $1\times 1$ 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.

Figure 1

Geometry optimized structures of monolayer (ML) silicene on the $7\times 7$ Ag(111) surface (a) and on the $4\times 4$ Ag(111) surface (b). In (a) and (b) the left and the right panels show the top and the side views, respectively. On the $7\times 7$ Ag(111) and the $4\times 4$ Ag(111), the ML silicene shows the superperiodicity of $3\sqrt{3}\times 3\sqrt{3}$ and $\sqrt{3}\times \sqrt{3}$, respectively. The side views show the corrugation of the silicene layer, where the vertical unit cells are indicated by the solid (black) lines. Top views of the rhombic $\sqrt{3}\times \sqrt{3}$ structures obtained by using the insufficient $\Gamma$-point BZ sampling are shown in (c) and (d), which are mirror symmetric to each other. The Si atoms in the top and bottom vertical positions are depicted by the large red and small blue balls, respectively. The large gray balls depict the positions of the substrate Ag atoms. The simulated lateral unit cells are indicated by the dashed (pink) lines, and the triangles denote the orientations of the protruded Si patterns with respect to Ag(111).

Figure 2

Top and side views of the calculated stable structures (left panels) and corresponding STM images (right panels) of the three rhombic $\sqrt{3}\times \sqrt{3}$ bilayer (BL) silicene on Ag(111). The (a), (b), and (c) are the str1, str2, and str3 (see text), respectively. The large gray balls and the small yellow balls depict the positions of Ag atoms of substrate and the Si atoms of the lower-layer silicene, respectively. The large red balls and small blue balls depict the positions of protruded and unprotruded Si atoms of the upper-layer silicene. The simulated lateral unit cells in the top views are indicated by the dashed (pink) lines, and the vertical unit cells in the side views are indicated by the solid (black) lines. The obtained rhombic $\sqrt{3}\times \sqrt{3}$ periodicity is indicated by the solid (orange) lines.

Figure 3

(a) Calculated reaction energy profile among the three rhombic $\sqrt{3}\times \sqrt{3}$ structures in BL silicene on Ag(111), shown in Figs. 2, 2, and 2. The energy in the vertical axis is defined as $(E-E_{\text{str1}})/n$, where $E$ and $E_{\mathrm{str}1}$ are the total energy of the structure at reaction coordinate and the total energy of str1, respectively. $n$ is the number of atoms in the upper-layer silicene. The transition barrier between str1 and str2 (black dots), str2 and str3 (blue dots), are smaller than that between str1 and str3 (red dots) by about 2 meV/Si. (b) STM images obtained by superimposing the images of two (left panel) and three (right panel) stable structures.

Figure 4

Calculated energy bands (a) and contour plots of squared KS orbitals (b) of the $3\times 3$ BL silicene/$4\times 4$ Ag(111), where the silicene exhibits the $\sqrt{3}\times \sqrt{3}$ structure as shown in Fig. 2. The calculated energy bands (c) and contour plots of squared KS orbitals (d) of the freestanding $3\times 3$ BL silicene peeled from the Ag(111) are also shown. The origin of the energy is set to be $E_{\mathrm{F}}$. The three states that have characters of the mixed $\pi$ (${\pi }^{*}$) are indicated by the blue circles in (a) [(c)], and the corresponding KS orbitals are shown in (b) [(d)]. The insets of (a) and (c) show the enlarged figures for the energy bands around the blue circles for the silicene/Ag(111) and the peeled silicene, respectively. The linear band mainly consisting of the $s$ and $p$ orbitals of Ag is indicated by the black-dotted line. The gray and blue balls depict Ag and Si atoms, respectively.