First-principles study on competing phases of silicene: Effect of substrate and strain

The stability and electronic structure of competing silicene phases under in-plane compressive stress, either free-standing or on the ZrB$_2$(0001) surface, has been studied by first-principles calculations. A particular ($\sqrt{3}\times\sqrt{3}$)-reconstructed structural modification was found to be stable on the ZrB$_2$(0001) surface under epitaxial conditions. In contrast to the planar and buckled forms of free-standing silicene, in this"planar-like"phase, all but one of the Si atoms per hexagon reside in a single plane. While without substrate, for a wide range of strain, this phase is energetically less favorable than the buckled one, it is calculated to represent the ground state on the ZrB$_2$(0001) surface. The atomic positions are found to be determined by the interactions with the nearest neighbor Zr atoms competing with Si-Si bonding interactions provided by the constraint of the honeycomb lattice.


I. INTRODUCTION
Owing to its ability to be either sp 3 -or sp 2 -hybridized, under ambient conditions, carbon crystallizes in different forms, such as diamond, graphite, graphene, fullerenes and carbon nanotubes. Such diversity does, of course, inspire the search for similar graphitic forms of the element silicon which is just below carbon in the periodic table. Whereas silicon nanotubes have already been reported, 1,2 until recently, the two-and three-dimensional graphitic forms of silicon remained elusive but have been anticipated theoretically. First-principles studies on the stability and the properties of graphitic Si phases can be traced back decades ago 3 and have mostly focused on an atom-thick honeycomb lattice of Si atoms, called silicene, as the Si-counterpart of graphene. In contrast to graphene, however, it is well accepted that silicene is unstable as long as the structure is planar. [3][4][5] Instead, free-standing silicene is predicted to be stable in a so-called "buckled" structure, where the two sub-lattices of the bipartite lattice are at different heights. 5 Like in graphene, charge carriers in free-standing silicene are found to be massless Dirac fermions even if it is buckled. 4,5 The Fermi velocity is calculated to be in the order of 10 6 m/s. 5 Additionally, due to a spin-orbit coupling much stronger than that in graphene, the quantum Hall effect is predicted to occur 5,6 and, owing to its buckled structure, to be switchable by the application of a perpendicular electric field. 7 Since the many fascinating properties and countless potential applications proposed for graphene [8][9][10][11][12][13][14][15][16][17][18] may occur in silicene as well, it is of great importance to uncover all stable structural modifications of silicene.
So far, Si honeycomb structures have been reported to grow on a limited number of substrates, such as Er, Ag, ZrB 2 , and Ir. [19][20][21][22][23][24][25][26][27][28][29] DFT calculations in conjunction with experimental observations point out that all of those two-dimensional forms of silicon are deviating from the expected buckled form of free-standing silicene. [21][22][23][24][25][26][27][28][29] Among them, those on Ag(111) and on ZrB 2 (0001) surfaces are the most exciting ones since the experimental observation of π bands links them to the theoretical concept of silicene. Imposed by epitaxial conditions, the type and degree of buckling can vary by a large amount which in turn influences the electronic properties as well. The intimate relationship between the structure and the electronic properties makes the atomic-scale buckling a highly relevant parameter in the description of silicene. On the other hand, the structural flexibility, related to a mixed sp 2 /sp 3 hybridization, may allow the engineering of desired properties such as the opening of a gap, something that is difficult to achieve in purely sp 2 -hybridized, robust graphene. 28 For silicene on the Ag(111) surface, the growth behavior is strongly dependent on the temperature, the coverage and the deposition rate. [21][22][23][24][25][26] Among the reported superstructures, post-annealing cannot switch one observed phase to another 23 which advocates the presence of non-negligible energy barriers between these phases. It has been proposed that all the observed silicene superstructures exhibit different orientations with respect to the silver substrate which is manifested in particular surface reconstructions. 27 On the contrary, epitaxial silicene on zirconium diboride (0001), formed by the segregation of atoms from the substrate on the surface of ZrB 2 (0001) thin films grown on Si(111) wafers, 28 has the advantage to exhibit a single orientation, to cover the whole sample surface homogeneously, and to be very well reproducible. The latter is a direct consequence of the spontaneous and self-terminating growth mode. Epitaxial silicene on ZrB 2 (0001) is ( unit cell of ZrB 2 (0001). 28 The spontaneous formation of silicene on ZrB 2 (0001) suggests that this particular reconstruction is important for the stability of silicene. The nature of this 3) reconstruction has recently been discussed based on density functional theory (DFT) calculations. 28 Note, however, that the considered phase, whose calculated structural and electronic properties are in partial agreement with the experimental findings, 28,30 has been calculated to be a metastable structure. It deviates from that of the predicted freestanding silicene by the position of one of the atoms forming the bottom sub-lattice lifted up to almost the height of the top sub-lattice. 28 On the other hand, the properties of the ground state that are discussed in the present paper for the first time do not fit with the experimental results. In the ground state proposed by DFT, the structure is "planar-like", closer to that of planar graphene than to the metastable phase. In this structure, five out of the six Si atoms per hexagon reside in a plane characterized by a residual buckling of just  for the studies of free-standing silicene and ZrB 2 (0001) surface, respectively. For the case of silicene on the ZrB 2 (0001) surface, both Brillouin zones have been chosen. In the study of silicene-derived bands, the two symmetric silicene layers on each surface of the slab are considered as two separate layers. Since the supercell is larger than the primitive unit cell chosen for the representation, an additional procedure 37 to unfold the bands into larger zones is needed and has been generalized to the non-orthogonal pseudoatomic basis functions. 38 In order to show the spectral weight related to each eigenstate of the system at a k point, we used the so-called "fat band representation", where the weight was represented by the diameter of a circle. The respective contributions from different orbitals to the spectral weight are given by the diameters of color-coded circles, as specified in the figure captions.
The diameter of grey-colored circle is obtained by summing over all the orbital contributions of an eigenstate. For a better visualization, eigenstates with the same k and separated by an energy differences less than 0.02 eV are considered as being degenerated and represented by a single circle.
In order to assure structural convergence of the ground state of silicene on the ZrB 2 (0001) surface calculated by DFT, additionally, we have performed molecular dynamics calculations starting from different initial structures, the temperature set to 400 K. The same groundstate structure has been found in all of the molecular dynamics simulations. A consistent ground-state structure of silicene on the (2×2) unit cell of the ZrB 2 (0001) surface was also found by using the VASP package with ultrasoft pseudopotentials. 39,40 To analyze the energy barrier between the ground state and a proposed metastable phase, 16 transition images are adopted using the nudged elastic band (NEB) method where the minimum energy path between two configurations can be found. 41 Here, the maximum force is less than 5 × 10 −3 Hartree/Bohr.

III. FREE-STANDING SILICENE
Determined by the sp 2 hybridization similar to graphene, free-standing planar silicene is expected to display comparable electronic properties. 5 As shown in Fig. 1 (a), several hopping parameters can contribute to the electronic band structure of planar silicene. This can well be explained within a tight-binding approach considered here initially. A finite set of hopping integrals related to the σ-bond hopping between two of the sp 2 orbitals (t 1 ), the p z − p z hopping (t 2 ), and the dominant p z − sp 2 hopping (t 3 ) shall be used. In reciprocal space, the Hamiltonian, H( k = k a a * + k b b * ), is given by Eq. (1). Here, a * and b * denote the reciprocal lattice vectors of the unit cell containing two Si atoms. As a consequence of the given symmetry, this Hamiltonian describes several interesting properties of the planar structure of silicene which are discussed below.
Because of the planarity, t 3 is zero. Therefore, H( k) in the p z sub-space is decoupled from that of the other orbitals. As a consequence, this simply leads to the presence of a bonding (π) band and an anti-bonding (π * ) band. The off-diagonal term ( forming a Dirac point, and a linear dispersion can be derived as E( k) = pz ± √ 3|t 2 |aq/2 in the vicinity of the K point, where q = k − K. This is illustrated in Fig. 1   anti-bonding bands defined in the p z sub-space are illustrated in Fig. 1 (b). The buckling gives rise to an additional modification in these electronic states manifested in an avoided crossing of the bonding-type (or anti-bonding-type) bands as shown in Fig. 1 (c).
In the framework of the DFT calculations, in addition to the planar and buckled structures of silicene, we considered the ( √ 3× √ 3)-reconstructed structure as well since it has been predicted to be relevant for epitaxial systems. 27,28 For comparison, we also evaluated the planar and buckled forms within a supercell in the size of the primitive unit cell of this "planar-like" structure. Hereafter, the buckled phase will be referred to as the "regularly-buckled" one in order to distinguish the two non-planar phases. In Fig. 2, the geometrical structures of the planar, the regularly-buckled, and the planar-like phases are plotted together with their total energies as a function of the in-plane lattice constant a of the ( With one Si atom per unit cell protruding out of plane, the planar-like phase is able to sustain a longer in-plane bond length as compared to the planar phase such that it becomes closer to that of the regularly-buckled one. As shown in Fig. 2, in regularly-buckled silicene, the buckling height of 0.69Å at a = 6.35Å decreases to 0.50Å at a = 6.70Å which is a result of a tendency to maintain the bond length for a large range of in-plane strain. For a = 6.20Å to a = 6.60Å, which is close to the value provided by ZrB 2 as a substrate, the total energy of planar-like silicene is in between the planar and regularly-buckled forms. The relationship between the total energy and the lattice constant suggests that under larger in-plane compressive stress, the planar-like phase might take over the regularly buckled phase. Since the total energies of regularly-buckled and planar-like phases are generally lower than that of the planar one, it is important to verify again if the energy gain might be of electronic origin or not. For this purpose, we consider the DFT results for a = 6. While for planar silicene, the p z orbitals do not couple to the sp 2 orbitals, the buckling leads to non-zero hopping integrals between the p z and the neighboring sp 2 orbitals which confer a mixed π−σ character to the originally pure π and π * bands. Unlike one of the σ band marked in green in Figs. 3 (a) and (b) that carries negligible p z weight, the other occupied σ band mixes with the bonding π band such that band crossing is avoided. Although the resulting gap is large, the corresponding bonding and anti-bonding bands are both occupied such that no band energy is gained. Similarly, the energy of the σ band at the Γ point is increased by more than 1 eV. Therefore, the mechanism which stabilizes the regularlybuckled silicene over the planar one is hard to visualize in the electronic band structure itself. It may instead relate to an instability in the phonon part that involves the lattice repulsive potential and the response of electrons to the lattice vibration. 4,5 The electronic structure of planar-like silicene is presented in Fig. 3 (c). A strong breaking of the symmetry causes back-folding and the lifting of the degeneracy of bands. Despite the back-folding, some resemblance to planar silicene is found. In particular, the energies of the σ band at the Γ point are similar. If the origin of the stabilization of the planarlike phase with respect to the planar one in terms of the total energy is not found in the band structure, other energy terms shall be considered. And indeed, it is found that energy is gained mainly from a reduction of the core-core Coulomb repulsion of the nuclei. symmetries. This is also expressed in the splitting of on-site energies and hopping integrals provided in Table I. Note that for the planar-like phase, not only differences between the two sub-lattices but also between individual atoms can be identified. 35Å. On-site energy is denoted by and the t 1 , t 2 , and t 3 are defined in Fig. 1 (a). For the planar-like structure, the protruding Si atom is denoted by Si C. Si B is defined as one of the atoms belonging to the sub-lattice without the protruding Si atom. Si A is defined as one of the two other Si atoms belonging to the sub-lattice of the protruding Si atom. The number in the parenthesis is the one associated with the protruding Si atom. The unit is eV.  Finally, we address the stability of planar-like silicene related to the out-of-plane dis-placement of some of the atoms. In Fig. 4 is shown the total energy as a function of the displacement of the two other Si atoms belonging to the sub-lattice of the protruding Si atom against the remaining four Si atoms of the reconstructed unit cell. The displacement with an equal amount for both atoms is associated with the Γ-point vibration that can lead to two structural modifications with a different type of buckling. As shown in Fig. 4 Table II. From the information thus obtained, favorable in-plane positions for the Si atoms forming silicene on the ZrB 2 (0001) surface can be predicted. The best candidate would be a structure in which all of the Si atoms reside at hollow sites. This is, however, impossible to realize for six Si atoms per (2×2) unit cell since the Si honeycomb lattice prefers a larger lattice constant than the boron sub-lattice of ZrB 2 . On-top sites could also be avoided by locating all six Si atoms in near-bridge sites, that is in positions in between on-top and bridge sites.
However, the deviation from the perfect bridge position would increase the energy too much such that it is preferable to place two Si atoms at hollow sites, three Si atoms at near-bridge sites and one Si atom at an on-top site. This simple analysis provides the in-plane structure model of silicene that has independently been derived from experimental data. 28 If we assume that Si atoms would adopt the height calculated for individual Si atoms Si atom of these two phases on the diboride surface are 1.14 eV and 1.42 eV, respectively. In agreement with the predictions for preferential sites for the adsorption, planar-like, ( √ 3 × √ 3)-reconstructed silicene is calculated to become the most stable epitaxial structure on ZrB 2 (0001). When placing the planar-like silicene phase onto the surface, the original phonon instability mentioned for free-standing, planar-like silicene in section III, is balanced by the gain of energy of those atoms such that the planar-like structure becomes stable against out-of-plane vibrations of the Si atoms above hollow sites. The second, metastable phase, is identified as a deformed version of regularly-buckled silicene where atoms of the top sublattice are in near-bridge sites and those of the bottom sub-lattice in hollow and on-top sites 28 . The on-top Si atom resides at a position higher than those on hollow sites. In the following, we will refer to this structure as the "regularly-buckled-like" structure. p-derived electronic states appear to be almost unaffected by the adsorption of silicene.
While for the discussion of substrate-derived electronic states, the use of the ZrB 2 Brillouin zone is appropriate, silicene-derived states are best discussed using the silicene (1×1) Brillouin zone. The relation between the two is illustrated in Fig. 8 (a). This would then facilitate comparison with the results for free-standing silicene. As shown in Figs. 8 (b) and (c), for both phases, at the silicene (1×1) K Si point and in the vicinity of E F , X-shaped dis-persing bands are now missing. This is very different from the case of free-standing silicene in which cone-like dispersions with p z character can easily be identified. However, for the regularly-buckled-like structure, several bands with partial p z character are still found below E F (black color in Fig. 8 (b)). For the one closest to the Fermi energy, in particular, the In order to understand if these two epitaxial silicene phases on the ZrB 2 (0001) surface can be transformed into each other, molecular dynamics calculations have been performed as well. Regardless of whether the initial structure is close to that of the regularly-buckled-like form or not, the calculations always converge to the planar-like phase corresponding to the ground state. This indicates that the energy barrier from the regularly-buckled-like phase to the planar-like phase is low and has been confirmed by the NEB calculation to be about 15 meV per Si atom in contrast to the deep well at the planar-like configuration. Selected images of transition states and the total energy along the minimum energy path are plotted in Fig. 9. Here, the total energy of the planar-like phase has been set to zero.

VII. DISCUSSION AND SUMMARY
Given that a purely sp 2 -hybridized, planar free-standing silicene is unstable, it is of great interest to understand the mechanisms that stabilize Si honeycomb lattice configurations when placed epitaxially on surfaces. Along this line, we have investigated planar-like and regularly-buckled-like ( √ 3 × √ 3)-reconstructed silicene phases whose occurance has been predicted for epitaxial silicene systems. Additionally, these phases have been compared to their respective free-standing, non-reconstructed forms. For a large range of the in-plane lattice constant, free-standing, ( On the other hand, epitaxial silicene on the ZrB 2 (0001) surface is characterized by the spontaneous formation of stress domains indicating the presence of long range interactions within the two-dimensional layer of the Si ad-atoms. 28 Having the discussion of the theoretical results presented in this paper in mind, the available experimental data suggest that epitaxial strain weakens the interactions with the substrate such that the stability of possible silicene phases on substrates can be reversed. In order to comprehend the impact of epitaxial strain, it may then be necessary to perform large-scale calculations covering an extended parameter set which correctly include the effect of interactions between the experimentally observed domains. Nevertheless, it is interesting to note that the ground state of silicene on ZrB 2 revealed in our DFT calculations, the ( √ 3 × √ 3)-reconstructed planar-like phase, has also been calculated to form on the Ag(111) surface 27 . Compared to that of free-standing silicene, the phase diagram of epitaxial silicene may be thus much richer. Upon increase of a to more than 6.60Å, the planar-like structure relaxes to the planar structure.