Silicene-derived phases on Ag(111) substrate versus coverage: Ab initio studies

Silicene is systematically investigated as an epitaxial overlayer on an Ag(111) substrate based on the ab initio density functional theory. The geometry and stability of five silicene-silver adsorbate systems with four coincidence lattices, $\sqrt{7}\times \sqrt{7}$ on $\sqrt{13}\times \sqrt{13}$, $3\times 3$ on $4\times 4$, $2\times 2$ on $\sqrt{7}\times \sqrt{7}$, and $\sqrt{7}\times \sqrt{7}$ on $2\sqrt{3}\times 2\sqrt{3}$, are related to the Si coverage, biaxial strain, and preparation conditions. Their phase diagram is calculated for varying chemical potential of the Si reservoir. The scanning tunneling microscopy images calculated for the optimized atomic geometries agree with those observed experimentally. The destruction of the original honeycomb symmetry and the strong adsorbate-substrate interaction significantly influence the electronic structure. Four peeled-off silicene sheets show conical linear bands, with small gaps. However, the band edges of the $3\times 3$ on $4\times 4$ geometry cannot be explained in terms of gap opening between Dirac cones for symmetry reasons. We confirm the conclusion that the linear bands observed by ARPES are due to folded Ag bands.

Figure 1

Top (left) and side (right) view of the relaxed silicene and first Ag(111) layer for the five geometries (a) $\sqrt{7}\times \sqrt{7}$ on $\sqrt{13}\times \sqrt{13}$ I, (a) $\sqrt{7}\times \sqrt{7}$ on $\sqrt{13}\times \sqrt{13}$ II, (c) $3\times 3$ on $4\times 4$, (d) $2\times 2$ on $\sqrt{7}\times \sqrt{7}$, and (e) $\sqrt{7}\times \sqrt{7}$ on $2\sqrt{3}\times 2\sqrt{3}$. The uppermost (lower) Si atoms are indicated by red (blue) circles while the first-layer Ag atoms are indicated by gray dots. Si atoms in intermediate positions are indicated by yellow circles. The crystallographic directions are related to the [111]-oriented cubic silver substrate. The unit cell of the adsorbate is displayed by dashed lines. Characteristic lateral and vertical distances are also given.

Figure 2

Phase diagram of silicene on Ag(111) surface. The adsorbate formation energy $\gamma$ (1) is plotted as a function of the Si chemical potential ${\mu }_{\mathrm{Si}}$. Its deviations $\Delta {\mu }_{\mathrm{Si}}$ are related to ${\mu }_{\mathrm{Si}}^{\mathrm{ref}}$ of the free standing silicene (lower axis) or diamond silicon (upper axis). The background colors indicate the range of the most stable adsorbate system. Coverage $\Theta$ and strain $\varepsilon$ are indicated. Additionally, the difference to the surface formation energies $\Delta \gamma$, relative to the that of the $4\times 4$ structure, is shown in the inset.

Figure 3

Total energy of free standing silicene vs its lattice constant. Three structures are studied with three (normal free standing silicene, red crosses), one (yellow dots), and zero (flat silicene, blue squares) buckled atoms.

Figure 4

Filled (left) and empty-state (right) STM images calculated for the (a) $\sqrt{7}\times \sqrt{7}$ on $\sqrt{13}\times \sqrt{13}$ I, (b) $\sqrt{7}\times \sqrt{7}$ on $\sqrt{13}\times \sqrt{13}$ II, (c) $3\times 3$ on $4\times 4$, (d) $2\times 2$ on $\sqrt{7}\times \sqrt{7}$, and (e) $\sqrt{7}\times \sqrt{7}$ on $2\sqrt{3}\times 2\sqrt{3}$ geometries of silicene on Ag(111). The bias voltages are fixed at $\mp 1$ V. Uppermost Si atoms (red dots) and Si atoms in the sheet closer to the substrate (blue dots), as well as intermediate Si atoms (yellow dots) are indicated. The unit cell of the silicene overlayer is displayed by dashed lines.

Figure 5

Arrangement of the small Brillouin zones (blue lines) for the (a)/(b) $\sqrt{13}\times \sqrt{13}$, (c) $4\times 4$, (d) $\sqrt{7}\times \sqrt{7}$, and (e) $2\sqrt{3}\times 2\sqrt{3}$ silver slab structures compared to the $1\times 1$ silicene BZ (red lines) and the Ag(111)$1\times 1$ BZ (green dashed lines). The high-symmetry points in the small (large) silicene BZs are indicated by blue (red) letters.

Figure 6

Band structures of the silicene/Ag(111) adsorbate systems (a) $\sqrt{7}\times \sqrt{7}$ on $\sqrt{13}\times \sqrt{13}$ I, (b) $\sqrt{7}\times \sqrt{7}$ on $\sqrt{13}\times \sqrt{13}$ II, (c) $3\times 3$ on $4\times 4$, (d) $2\times 2$ on $\sqrt{7}\times \sqrt{7}$, and (e) $\sqrt{7}\times \sqrt{7}$ on $2\sqrt{3}\times 2\sqrt{3}$. The left panels show the band structures of complete silicene-covered silver slabs centered around the high-symmetry point onto which a corner point of ideal silicene should be folded (see Fig. 5). An energy range in the proximity of the Fermi level (energy zero) is chosen. The strength of the Si $p_{xy}$ or Si $p_z$ orbital character is indicated by symbols of varying size. Ag-derived bands in the adsorbate systems are represented as gray lines. The right panels display the band structures of the peeled-off silicene sheets. The bands obtained by fitting Eq. ((2)) are indicated by red lines.

Figure 7

Local density of states of five silicene-silver adsorbate systems: (a) $\sqrt{7}\times \sqrt{7}$ on $\sqrt{13}\times \sqrt{13}$ I, (b) $\sqrt{7}\times \sqrt{7}$ on $\sqrt{13}\times \sqrt{13}$ II, (c) $3\times 3$ on $4\times 4$, (d) $2\times 2$ on $\sqrt{7}\times \sqrt{7}$, and (e) $\sqrt{7}\times \sqrt{7}$ on $2\sqrt{3}\times 2\sqrt{3}$. For the purpose of comparison the spectrum is also shown for (f) free standing silicene. Two positions 2 Å above chemically different silicon atoms are studied. The colors are chosen in correspondence to Fig. 1. The blue dashed line is calculated for a silicon atom close to the substrate and the red solid line for an outward buckled Si atom. The Fermi level of the adsorbate system is used as energy zero.