Absence and presence of Dirac electrons in silicene on substrates

We report on the total-energy electronic-structure calculations on the basis of the density-functional theory that clarify atomic and electronic structures of the silicene on the Ag(111), the hexagonal BN, and the hydrogen-processed Si(111) surfaces. On the Ag(111) surfaces which are most commonly used as substrates for the silicene in current experiments, we find several stable and metastable structures with the $4\times 4$, $\sqrt{13}\times \sqrt{13}$, and $2\sqrt{3}\times 2\sqrt{3}$ periodicities with respect to the $1\times 1$ Ag(111) lateral cell within the total-energy difference of 70 meV per Si atom. Those stable structures show the excellent agreement with the scanning tunneling microscopy measurement in their structural characteristics. The metastable structures with comparable total energies await experimental observations. In all the stable and metastable structures, the silicene is buckled substantially so that the $\pi$ state rehybridizes with the $\sigma$ state, leading to the $\pi +$ state, and then the linear energy dispersion peculiar to the Dirac electrons disappears in several cases associated with the opening of the energy gap. Moreover, we find that the substantial mixing of the $\pi +$ state, generated in such a way, with the states of the Ag atoms in the substrate converts the $\pi +$ state to the mixed $\pi +$ state and thus makes the state shift downwards or upwards, eventually annihilating Dirac electrons near the Fermi level. The absence of Dirac electrons caused in this way is found to be common to all the stable and metastable structures of the silicene on the Ag(111) substrates. We also find that the interaction between the $\pi +$ and the substrate orbitals should be weak enough to preserve Dirac electrons and at the same time be sizable to keep the system stable. We then propose two specific substrates as good candidates for the silicene with Dirac electrons, i.e., hexagonal BN and the hydrogen-processed Si(111) surface. We clarify that the silicene on those substrates are stable enough with the binding energy comparable to or twice that of the graphite and preserve Dirac electrons near the Fermi level.

Figure 1

Calculated energy bands and Kohn-Sham (KS) orbitals of the stable low-buckled freestanding silicene. The energy bands are obtained (a) by the LDA calculations and (b) by our $s{p}^3$ TB calculations. $E_{\mathrm{F}}$ is shown by short-dashed lines. The KS orbitals of the $\pi$ and ${\pi }^{*}$ states which are rehybridized with $\sigma$ and degenerate at $E_{\mathrm{F}}$, indicated by the blue square in (a), are shown in (c). The contour values are normalized to the maximum.

Figure 2

Calculated STM images (right parts) and top views of atomic structures (left parts) of the silicene on Ag(111) surfaces. The stable [(a), (c), and (e)] and the metastable [(b), (d), and (f)] structures with the $4\times 4$ periodicity [(a) and (b)], the $\sqrt{13}\times \sqrt{13}$ periodicity [(c) and (d)], and the $2\sqrt{3}\times 2\sqrt{3}$ periodicity [(e) and (f)] are shown. Each circle of the left parts depicts the position of the Si atom in the silicene layer: The positions of the topmost Si and the bottommost Si are depicted by the largest (red) and the smallest (blue) circles; the intermediate-size circles depict the Si atoms between the top and the bottom. The large gray balls depict the positions of the substrate Ag atoms. The lateral unit cells are indicated by the orange lines.

Figure 3

Calculated energy bands of the stable structures for the 4 $\times$ 4 (a) and the $\sqrt{13}\times \sqrt{13}$ (b) silicene/Ag(111) surfaces obtained by LDA. States marked by the square at either $\Gamma$ or $K$ in (a) and (b) have characters of mixed $\pi +$ and $\pi {+}^{*}$ as shown in Fig. 6 (see text). Energy bands of the stable structures for the $4\times 4$ silicene/Ag(111) calculated by GGA and vdW-DF are also shown in (c) and (d), respectively. As marked by the black dots, it is clearly shown that one of the mixed $\pi +$ ($\pi {+}^{*}$) states for the $4\times 4$ structure possesses the linear energy dispersion.

Figure 4

Calculated energy bands of the 3 $\times$ 3 silicene layer peeled from the stable 4 $\times$ 4 Ag surface (a) and the $\sqrt{7}\times \sqrt{7}$ silicene layer peeled from the stable $\sqrt{13}\times \sqrt{13}$ Ag surface (c). The contour plots of the KS orbitals of the 4 and 2 states near $E_{\mathrm{F}}$ marked by the squares in (a) and (c) are shown in (b) and (d), respectively.

Figure 5

Energy bands of the $3\times 3$ silicene layer peeled from the $4\times 4$ Ag surface (a) and the $\sqrt{7}\times \sqrt{7}$ silicene layer peeled from the $\sqrt{13}\times \sqrt{13}$ Ag surface (b), obtained by our TB model. States marked by squares at the $\Gamma$ and $K$ point have characters of $\pi +$ and $\pi {+}^{*}$.

Figure 6

Contour plots of the KS orbitals of the mixed $\pi +$ ($\pi {+}^{*}$) states (see text) in the silicene/Ag(111) on the plane perpendicular to the silicene sheet. (a) The KS orbitals of the four mixed $\pi +$ ($\pi {+}^{*}$) states marked in Fig. 3a in the $4\times 4$ silicene/Ag(111) at $\Gamma$ point. (b) The KS orbitals of the two mixed $\pi +$ ($\pi {+}^{*}$) states marked in Fig. 3b in the $\sqrt{13}\times \sqrt{13}$ silicene/Ag(111) at $K$ point. The blue and gray balls depict Si and Ag atoms, respectively.

Figure 7

Calculated energy bands of the peeled $3\times 3$ silicene with H atoms attached below the Si atoms (a) and of the same silicene but flattened and with the H atoms removed (b). The KS orbitals for the four mixed $\pi +$ ($\pi {+}^{*}$) states marked by the square at $\Gamma$ in (a) are shown on the plane perpendicular to the silicene sheet in (c). These mixed states shift upward to $E_{\mathrm{F}}$ in (b) depicted as a square. The blue and pink balls depict Si and H atoms, respectively.

Figure 8

Calculated energy bands (left) and contour plots of the KS orbitals of the mixed $\pi$ (${\pi }^{*}$) states (right) of the AA stacked silicene bilayer with interlayer distance of 3.4 Å. The KS orbitals for the mixed $\pi$ (${\pi }^{*}$) states marked by the square are shown on the plane perpendicular to the silicene layer. As indicated by the black-dotted lines, the energy bands of the mixed $\pi$ (${\pi }^{*}$) states possess the linear energy dispersion near the $K$ point.

Figure 9

Calculated energy bands (left) and the contour plots of the KS orbitals of mixed $\pi$ (${\pi }^{*}$) states (right) for the ideally hexagonal planar $3\times 3$ silicene on the $4\times 4$ Ag surface. As marked by the blue-dotted lines, the energy bands of the mixed $\pi$ (${\pi }^{*}$) states possess the linear energy dispersion.

Figure 10

Calculated energy bands for the stable structure of the $4\times 4$ silicene/Ag(111) (a) and for the clean Ag(111) surface with the $4\times 4$ periodicity (b). The green-dashed line corresponds to the BZ position where the energy of the linear band (indicated by the blue-dashed line) in the ARPES experiment (Ref. 15) reaches $-3.0$ eV. The sequence of the states marked by the black dots shows the quasilinear energy dispersion.

Figure 11

KS orbitals of the quasilinear energy bands for the stable structure of the $4\times 4$ silicene/Ag(111) (a) and of the clean Ag(111) surface with the $4\times 4$ periodicity (b). The KS orbitals corresponding to the states marked by the blue squares in Fig. 10 are shown. Left: Isovalue surface viewed along the [01$\overline{1}$] direction at its value of 33$\%$ of the maximum value. Right: The contour plot of the KS orbital on (1$\overline{1}$1) plane. The blue and gray balls depict Si and Ag atoms, respectively.

Figure 12

Calculated energy bands and contour plots of the mixed $\pi +$ ($\pi {+}^{*}$) states at $\Gamma$ point of the silicene on h-BN with the bridge configuration. Enlarged energy bands near $E_{\mathrm{F}}=0$ are shown in the inset. The KS orbitals of the mixed $\pi +$ ($\pi {+}^{*}$) states marked by the square are shown on the plane perpendicular to the silicene layer in the right panel. The top view of the optimized structure is shown in the inset of the left panel, where the lateral unit cell is indicated by the blue lines and the blue, green, and gray balls depict the Si atoms, B atoms, and N atoms, respectively.

Figure 13

Calculated energy bands (a) of the silicene on the hydrogen-covered Si(111) surface [inset of (a)] and contour plots of the mixed $\pi +$ ($\pi {+}^{*}$) states (b). Enlarged energy bands near $E_{\mathrm{F}}=0$ are shown in another inset and the distributions of the mixed $\pi +$ ($\pi {+}^{*}$) states at $K$ point marked by the square are shown on the plane perpendicular to the surface. The blue and pink balls in the inset depict Si and H atoms, respectively.