Silicene on metal and metallized surfaces: ab initio studies

The deposition of silicene on several metals is investigated. For fcc crystals the (111) surfaces while for hexagonal ones the (0001) surfaces are used. The Ca(111)1 × 1 substrate is found to be the most promising candidate. The silicene adsorption on Ca-functionalized Si(111)1 × 1 and 2 × 1 surfaces is also studied. The 1 × 1 substrates lead to overlayer silicene with hexagonal symmetry and Dirac cones. However, the Dirac points are below the Fermi level, and small energy gaps are opened. In the case of 2 × 1 surfaces, strong lattice relaxation occurs. Only rudiments of conical linear bands remain visible.


Introduction
Silicon (Si) is the most important semiconductor because of its wide usage in electronic, optoelectronic and photovoltaic devices. Silicene, the two-dimensional (2D) silicon allotrope with honeycomb symmetry, has recently attracted wide attention due to its graphene-like electronic structure including the presence of Dirac cones near the Fermi level [1][2][3][4]. Theory predicts its stability in a low-buckled phase [3]. Moreover, because of its chemical nature it is compatible with Si-based electronics.
In contrast to graphene, which also arranges in a three-dimensional system, the sheet crystal graphite, for silicene no equivalent system is known. Therefore, silicene has to be prepared by a certain epitaxial technique. Indeed, recent experiments show that epitaxial silicene, or more precisely silicene-like adsorbate layers, can be successfully grown on some metal or halfmetal substrates, such as Ag [5][6][7][8], Ir [9] and ZrB 2 [10]. However, because of the strong adsorbate-substrate interaction the occurrence of Si-derived Dirac cones is controversially discussed in the literature [11][12][13][14][15][16][17]. One alternative seems to be the epitaxial growth on non-metallic substrates with a reduced adsorbate-substrate interaction [18][19][20].
In this paper we simulate silicene deposition on metal substrates or on a metallic layer atop silicon. After the description of the methods in section 2, we summarize the results of a screening of the chosen metal surfaces with a hexagonal geometry in section 3. Since calcium (Ca) appears to be a promising candidate for a metal substrate, we also study thin Ca overlayers on Si(111) substrates in section 4.

Theoretical and numerical methods
The search for suitable substrates is driven by the calculation of atomic geometries, total energies, and electronic structures. All calculations are based on the density functional theory (DFT) as implemented in the Vienna ab initio Simulation Package (VASP) [21]. The 3s and 3p electrons of Si and the 4s electrons of Ca, the favored metal, are included in the simulation as valence electrons. The one-particle wave functions are expanded into plane waves up to a cutoff energy of 350 eV. The projector-augmented-wave (PAW) method [22] is applied to generate the pseudopotentials and describe the electronic states in the PAW spheres. Exchange and correlation are simulated in the framework of a density functional including van der Waals (vdW) interaction according to Dion et al [23] and implemented by Klimeš et al [24]. Dense kpoint meshes [25] are applied to converge the total energies down to deviations smaller than 1 meV/atom and forces below 1 meV/Å.  [26].
The various surfaces are simulated by using symmetric slabs. For investigation of the silicon substrate system, slabs of 18 Si layers are used. For all metallic systems slabs with nine atomic layers are chosen. Because of periodic boundary conditions, we chose a vacuum region between the top layer of a slab and the bottom layer of the slab above, to separate surfaces. This vacuum region is fixed at 15 Å. The Brillouin zone (BZ) of a given repeated slab system is sampled by a 32 × 32 × 1 (16 × 32 × 1) mesh for a 1 × 1 (2 × 1) reconstructed surface. For larger lateral unit cells we use an 8 × 8 × 1 mesh [25]. For freestanding silicene the resulting lateral 2D lattice constant is a = 3.86 Å, while the vertical buckling of such a silicene sheet amounts to Δ = 0.48 Å. These values agree with other vdW-DFT calculations [18]. The Fermi velocity for this silicene is 0.48×10 − m s 6 1 . Quasiparticle corrections may increase this value by about 50% [27,28] but should not considered here.

Screening of metal surfaces
We search for a metallic surface that may be suitable for the silicene epitaxy. We chose to investigate several elemental metals with nuclear number < Z 90 from the periodic table (see figure 1). We exclude the rare earths and actinides. We focus the search on fcc metals with (111) surfaces and hcp metals with [0001] orientation because of their hexagonal symmetry.
Mostly all metals that form silicides (see e.g. [29]), are excluded, because a strong metalsilicene interaction is expected. The corresponding relation of the surface lattice constant to the silicene value (see figure 1) helps to find reasonable coincidence lattices [30]. By allowing rotations around the surface normal and enlargements of the unit cells of silicene and the metal surface, the φ × ( ) n m R 0 silicene layer is related coincidentally to the φ ′ × ′ ′ ( ) n m R 0 Bravais lattice of the metal surface. Here, we use the Wood notation [30]. The lateral lattice constants of the metal surfaces are derived from the experimental lattice constants of the bulk systems [26], while the DFT-vdW value a = 3.86 Å is used for silicene. For further investigations of the silicene/metal combinations, a single coincidence lattice with a small lateral lattice mismatch as well as a small overall cell size is selected from all possible coincidence lattices. We know that freestanding silicene can be biaxially strained up to ±4 % without opening gaps between the Dirac cones [31]. This strain is not exceeded for any studied silicene/metal combination. Silver is excluded from our search, since the silicene deposition on it is thoroughly discussed [11][12][13][14][15][16]32]. However, we followed the tendency for small coincidence lattices, found in the silicene/silver case, as a criterion for favorable combinations of translational symmetries [32].
As a result of the described screening procedure we found promising combinations of silicene monolayers (MLs) on a metal for 1 × 1 silicene on Ca (111) (0001) surfaces have hexagonal symmetry. The crystal structure is indicated (in red) [26], while the relative lateral lattice constant compared to that of silicene is given as a (blue) number in per cent. The elements in red squares form silicides [29].
constants of Ti and Au are very close to that of Ag, coincidence lattices have been studied, which have also been observed experimentally for silicene MLs on Ag(111) (see [32] and references therein). Despite the fact that Ti and Zr may form a silicide (see figure 1) we have studied the seven suggested adsorbate systems by means of minimization of the total energy with respect to the atomic positions for different starting configurations of silicene relative to the metal surface. For Si atoms of the silicene overlayer we found a strong tendency to fill high-symmetry sites on the metallic surface. As a consequence, even for metals where a very weak interaction was anticipated, namely Zn and Cd, the silicene sheet adapts to the surface structure. Because of the lattice match, only for the 1 × 1 surface of Ca is the silicene 1 × 1 starting geometry not destroyed by atomic relaxation.
Consequently the resulting band structures only show silicene-derived Dirac cones in the case of deposition on Ca(111). Despite the less symmetric silicene adsorbate layer, we observe indications for Dirac cones, or more precisely, conical linear bands, also for Au, Cd and Zn substrates. However, similarly to the findings for 3 × 3 silicene on 4 × 4 Ag(111), the corresponding band states are derived from hybridized metal states [13][14][15][16]32]. The most promising metal substrate seems to be the Ca(111) surface that will be investigated in detail in the next section.

Possible Ca-derived surfaces
In the previous section we found the possible stabilization of a silicene 1 × 1 overlayer with a lattice constant a = 3.86 Å on a flat Ca(111)1 × 1 surface with a lattice constant a = 3.92 Å and a tensile biaxial strain of 1.5% in the silicene adsorbate layer. We know that a Si(111)1 × 1 surface is almost lattice-matched with the silicene. However, the clean Si(111) surface cannot be used to grow silicene. Rather, passivation is needed [18]. Indeed, experimental studies showed that the adsorption of Ca atoms leads to a stable and passivated surface, but results in a 2 × 1 reconstruction, i.e., to Ca/Si(111)2 × 1 with a rectangular or oblique 2D Bravais lattice and 0.5 ML of Ca [33,34]. The top and side views of this adsorbate system are illustrated in figure 2.
The result of an total-energy optimization confirms that this 2 × 1 Ca-covered surface phase is formed by π-bonded Seiwatz Si chains [30]. As divalent metal atoms, the Ca atoms are bonded to two Si atoms. The Ca atoms fill the trenches between the π-bonded zig-zag chains of Si atoms, positioned on hollow sites of the silicon layer beneath. This surface is, however, not stable against higher temperatures and some annealing procedures [35]. Nevertheless, according to the total-energy calculation it forms a metastable surface, at least at low temperatures. In order to account for fluctuating Ca coverages and surface translational symmetries in growing an overlayer, we investigate several Ca/Si adsorbates. We use three 2 × 1 Si(111) reconstructions with 0.5, 1 and 1.5 ML of Ca and the 1 × 1 Si(111) surface with 1 ML of Ca. Together with silicene on Ca(111)1 × 1, silicene on five Ca-derived substrates are studied.  In the chosen intervals for Δμ X four of the five studied silicene adsorbate geometries are stable. The silicene on Ca(1.0 ML)/Si(111)1 × 1 structure is energetically favored compared with the Ca(1.0 ML)/Si(111)2 × 1 reconstruction, which therefore does not appear in the phase diagram. The electronic structures are only discussed for the four metastable silicene geometries.

STM images
The filledand empty-state STM images calculated for the four energetically favored adsorbate systems are displayed in figure 5. The perfect hexagonal honeycomb symmetry of the silicene overlayer on a clean Ca(111)1 × 1 surface ( figure 5(a)) and a functionalized Ca(1.0 ML)/Si(111) 1 × 1 surface (figure 5(c)) is clearly demonstrated by the spot distributions. Similarly to silicene on Ag(111) the STM images are determined by the outermost silicene atoms [6-8, 11, 12, 32]. Also the filledand empty-state images are rather similar; only the corrugation heights are different by a factor 2-3. These two images of silicene-covered Ca(111)1 × 1 and Ca(1.0 ML)/ Si(111)1 × 1 surfaces are close to those of freestanding silicene, and raise the hope to find the searched properties of silicene.
The situation is completely different for the other two images presented in figure 5. In figure 5(b), i.e., silicene on Ca(0.5 ML)/Si(111)2 × 1 substrate, the one outstanding silicene atom per unit cell is clearly visible in STM and forms a straight chain together with the outermost atoms of neighboring unit cells. Here a strong contrast between filledand emptystate images exists. In figure 5(d), i.e., silicene on Ca(1.5 ML)/Si(111)2 × 1 substrate, the spots in the STM images are arranged in zig-zag chains due to the different heights of the silicene atoms, which are, nevertheless, arranged in hexagonal rings. This indicates completely different bonding behavior and a decay of the silicene layer into chain structures. The trenches, which are clearly visible between the zig-zag chains, are due to the fact that each second zig-zag chain is closer to the substrate and hence less visible in the STM image. The results presented in

Band structures
The band structures versus the BZs in figure 6 of the four energetically favored silicene overlayers are presented in figure 7. The silicene p z and p xy projections are also given. At first glance, the four band structures look different. However, all investigated adsorbate systems are metallic. The Fermi level crosses several bands. Nevertheless, for the Ca(111)1 × 1 (a) and Ca   . These values are only slightly smaller than the Fermi velocity of freestanding silicene.
In the case of the 2 × 1 translational symmetry in figure 7(b)/(d) for silicene on Ca(0.5 ML/ 1.5 ML) on Si(111)2 × 1, Dirac cones are difficult to identify. The K and K′ points of the 1 × 1 BZ of silicene folded onto the 2 × 1 BZ of the adsorbate system, as illustrated in figure 6, the socalled 'K' point, lies on the high-symmetry line ΓX′ in the smaller BZ. For Ca(0.5 ML)/Si(111) 2 × 1 very flat p z -derived bands are equally distributed in the BZ. For Ca(1.5 ML)/Si(111)2 × 1, on the other hand, there is an indication for two linear bands 0.8 eV below the Fermi level. However, their silicene p z -character, especially of the upper one, is small compared to the two bands closer to the Fermi level around a high-symmetry point ′ X . In order to make the shape of the p z -derived bands clearer, the two uppermost bands near the Fermi level are plotted in figure 8 for k vectors in the reciprocal space. A linear band dispersion can be seen along the high-symmetry line ΓX′. Along the BZ boundary X′M, however, the band dispersion vanishes. The band shape could be described as a Dirac 'valley' instead of a Dirac cone.

Summary and conclusions
We have performed a screening of the (111) surfaces of fcc metals and (0001) surfaces of hexagonal metals in order to find appropriate substrates for silicene. Coincidence lattices were used to minimize strain in the epitaxial silicene. Apart from Ag(111) we found seven promising metal surfaces. However, after lattice relaxation only a silicene 1 × 1 overlayer on the Ca(111) 1 × 1 substrate still shows hexagonal symmetry and Dirac-cone-like features in the band structure.
Because of the promising results for clean Ca surfaces we have also investigated Si(111) surfaces functionalized by Ca atoms. Such substrates are lattice-matched to silicene, too. Because of the bivalent Ca atoms the most stable overlayer is a half Ca ML on a triple-bond Si (111)2 × 1 reconstruction with π-bonded Seiwatz chains. In addition, we have investigated coverages of 1 ML and 1.5 ML Ca atoms on Si(111) for possible deposition of silicene. The Ca (111)1 × 1 and Ca(1.0 ML)/Si(111)1 × 1 surfaces lead to silicene overlayers that exhibit conical linear bands near K and ′ K points but more than 0.5 eV below the Fermi level. The accompanying STM images clearly indicate honeycomb symmetry of the silicene overlayers. The Ca-functionalized Si(111)2 × 1 surfaces lead to the formation of chain motifs in the adsorbed silicene. Dirac-cone-like features are difficult to identify in the corresponding band structure for silicene on Ca(1.5 ML)/Si(111)2 × 1.