Flat bands in Weaire-Thorpe model and silicene

In order to analytically capture and identify peculiarities in the electronic structure of silicene, Weaire-Thorpe(WT) model, a standard model for treating three-dimensional (3D) silicon, is applied to silicene with the buckled 2D structure. In the original WT model for four hybridized $sp^3$ orbitals on each atom along with inter-atom hopping, the band structure can be systematically examined in 3D, where flat (dispersionless) bands exist as well. For examining silicene, here we re-formulate the WT model in terms of the overlapping molecular-orbital (MO) method which enables us to describe flat bands away from the electron-holesymmetric point. The overlapping MO formalism indeed enables us to reveal an important difference: while in 3D the dipersive bands with cones are sandwiched by doubly-degenerate flat bands, in 2D the dipersive bands with cones are sandwiched by triply-degenerate and non-degenerate (nearly) flat bands, which is consistent with the original band calculation by Takeda and Shiraishi. Thus emerges a picture for why the whole band structure of silicene comprises a pair of dispersive bands with Dirac cones with each of the band touching a nearly flat (narrow) band at $\Gamma$. We can also recognize that, for band engineering, the bonds perpendicular to the atomic plane are crucial, and that a ferromagnetism or structural instabilities are expected if we can shift the chemical potential close to the flat bands.


Introduction
After the physics of graphene has been kicked off, originally by a theoretically prediction for a massless Dirac fermion by Wallace [2] back in the 1950's and by a recent experimental realization by Novoselov and Geim [3], interests are extended to wider class of systems. Then many of theoretical dreams for fancy behaviours of the massless/massive Dirac fermions are experimentally confirmed [4]. History seems to repeat itself, for silicene: the material was theoretically predicted in the early 1990's by one of the present authors [5], and it was synthesised just recently [6,7,8], after a long latent period. Silicene synthesis on Ag and ZrB substrates has been succeeded by Japanese and European groups [6,7,8]. Due to these successes, silicene attracts great attention in both physics and mete rial science fields. In graphene, the carriers, being in two dimensions, are π-electrons arising from the sp 2 hybridization of carbon orbitals. In silicene, by contrast, the honeycomb structure is buckled, so that a sp 3 character of Si is crucially important, which was already pointed out in [5]. Thus silicene is not just a Si analogue of graphene, but multi-orbital character should appear in the electronic states.
Hence silicene has a larger degree of freedom than graphene as a target material for applications and also theoretical considerations. In the present paper, we focus on this multi-orbital feature of silicene. In doing so, we opt for a simplified model, which we propose to introduce as an extension of the Weaire-Thorpe(WT) model, which was originally conceived for a 3D silicon with sp 3 the multi-orbits in a tight-binding type model on a diamond lattice [9]. The WT model gives rise to two characteristically singular dispersions. One is a massless Dirac fermions, and the other is a dispersionless (flat) bands. Both of them can be regarded as topological in origin, as we shall describe here. Dirac fermions in three dimensions are topologically protected [10] but the flat bands are not. However, at a special point the flat bands also exist in 2D as we shall show albebraically, and away from the point we still have nealy flat bands with large density of states (DOS). This kind of nearly flat bands is noticeable in the original calculation by Takeda and Shiraishi [5]. Large enough DOS will induce instabilities of the Fermi surface into symmetry-broken states such as ferromagnetism or structural instabilities.
In the present paper, we first start with a generic overlapping molecular orbital theory due to Hatsugai and Maruyama [1]. Applying this to the WT model enables us to generically treat the flat bands away from the electron-hole symmetric point in multi-orbital models for the first time, while the usual flat-band theories [11,12,13,14] focus on those with the E = 0 electron-hole symmetric point. This enables us to newly interpret why the whole band structure [5] of silicene comprises dispersive bands with Dirac cones and nearly flat bands. The algebraic formulation enables us to pin point an important difference from 3D: while the dipersive bands with cones are sandwitched by doubly-degenerate flat ones in 3D, the dipersive bands with cones are sandwitched by triply-degenerate and nondegenerate (nearly) flat bands in 2D. For the band engineering the bonds perpendicular to the atomic plane are found to be crucial.

Overlapping molecular orbitals and flat bands
Let us start with describing a class of lattice model Hamiltonians that have zero-energy states generically. This was proposed by Hatsugai and Maruyama in a compact form before [1], but is explained here again. Consider fermions on an N-site lattice with the creation operator, c † i for the sites i = 1, · · · , N with {c † i , c j } = δ ij . After shifting the energy of origin by µ, let us consider the case in which the Hamiltonian is expressed as whereN is a number operator of the total fermions, E m ∈ R is the energy level of a molecular orbital m, and C † m is its creation operator and expanded as Note here that if there are flat bands are at the energy zero for the present model, they are at the energy µ for the Hamiltonian H. Then suitably choosing µ, we can describe non-zero energy flat bands algebraically. It is trivial but useful as we demonstrate in Secs.3 and 4.
In this representation, we can regard N as the number of dispersive (i.e., kdependent) bands when we write down the Hamiltonian in the Bloch space. The number of the molecular orbitals, M (the number of terms in this representation) varies from a model to another, as we shall see. In the real-space basis, we do not necessarily require a transnational invariance. Then we have a simple theorem that H − µN has (N − M)fold degenerate zero-energy states when N − M > 0. (To be precise, there may be other zero-energy states by accident.) For its explicit proof, see the footnote 4 of the Ref. [1]. When H is a momentum dependent Hamiltonian in the Bloch basis with transitional invariance, there are (N − M) flat bands at the energy µ when N − M > 0. We can take the orbitals as normalised, ψ † ψ = 1, without loss of generality, but the molecular orbitals have in general overlaps with each other. Then they are not orthogonal with . We can still decompose the Hamiltonian as where P m = ψ m ψ † m is a projection operator with P 2 m = P m . Then they are not orthogonal in general with P m P m ′ = 0 (m = m ′ ). Since P m 's span a one-dimensional space, the dimension of the Hamiltonian, h, is at most M. Still the Hamiltonian is written as an N × N hermitian matrix, which should be redundant. This implies that the space of the dimension N − M has to be null, that is, h has N − M zero eigenvalues.
Following the idea described here, we state the theorem in a slightly extended form, which we shall use later in the present paper. As for the projection P m = P 2 m = P † m , let us define its dimension by TrP m . Since the projection operator has eigenvalues 0 or 1, the dimension coincides with the number of nonzero eigenvalues of P m . Then the number of zero modes, Z, should satisfy a condition We note here that flat bands at the zero energy [11,12,13,14] is discussed in the present argument by considering a square of the Hamiltonian. Also we note here that the overlaping molecular orbitals in real space is discussed in relation to the rigorous treatment of the ferromagnetism on the Hubbard model [12,15]. Then its relation to the present analysis can be an interesting problem and should be discussed in the future.

Weaire-Thorpe model
Weaire and Thorpe considered a simple but multi-orbital tight-binding model for the sp 3 electrons on the diamond lattice, [9] where the original motivation was to treat amorphous silicone. Let us reproduce the model here for later references. We start with sp 3 -hybridized orbitals on a single tetrahedron. The local Hamiltonian for the tetrahedron reads is an annihilation operator of the bond orbitals with energy levels ǫ s for the s orbital and ǫ p for the p orbitals, and Here E 4 is the 4 × 4 unit matrix, and V 1 = 1 4 (ǫ s − ǫ p ). With this bond basis, the WT model for the silicon atoms on the diamond lattice considers only the hopping, denoted by V 2 , between the bond-sharing orbitals. For the diamond lattice (See Fig.1(a)), the Hamiltonian in the Bloch picture of the WT model (H k in the appendix B of [9], but note that here we take the origin of energy at ǫ p , so that the energy in [9] is shifted by −V 1 from ours) is written as and p is a projection onto the space spanned by ψ k . Then applying the discussion in sec.2, it implies 4 × 4 matrix H V (k) has at most one nonzero energy that corresponds to localised molecular orbital at each tetrahedron and there are 3 zero-energy flat bands (in the present choice of the origin of energy).
Based on these observation, we have two representations for the 8 × 8 Hamiltonian as H W T (k) ± V 2 E 8 , where E 8 is an 8 × 8 unit matrix. Although one might think the choice of the origin of the energy to be irrelevant, the whole point here is: we want to deal with flat bands that have nonzero energies. To do that, we can shift the origin of the energy to apply an algebraic argument. Now, if we take the plus sign, we have where P i 's are projections P 2 i = P i and dimP i = Tr Ψ i Ψ † i = Tr Ψ † i Ψ i = 1, 1, 4 for i = 1, 2, 3+. If we count the dimensions, H W T (k) + V 2 E 8 has at most dimP 1 + dimP 2 + dimP 3+ = 1 + 1 + 4 nonzero energy bands, that is, there are 8 − 6 = 2 zero-energy flat bands.
We end up with that the WT Hamiltonian H W T has 4 flat bands at E = ±V 2 . To compare with Ref. [9], we need to shift the energy. Then the flat bands are at −V 1 ± V 2 , since our choice of the H W T is shifted by −V 1 E 8 from the hamiltonian in Ref. [9]. An essential point is that we have succeeded in describing the flat bands at nonzero energies in the WT model algebraically in terms of the overlapping molecular orbitals.

Silicene in a Weaire-Thorpe type model
Now we come to the original aim at describing silicene. We start with an observation that 2D silicene can also be captured in a manner similar to the WT model in 3D. As shown in the Fig.1(b), we have three primitive vectors, e 1 , e 2 , e 3 , from which we have three reciprocal vectors. Corresponding 2D momentum components are given by (k 1 , k 2 ) with k 3 = 0. The Hamiltonian can be obtained from that in 3D by cutting the bonds at the blue bonds in Fig.1. Then the Hamiltonian as a simple extension of the WT model for silicene can be taken as Then we have to note that the bonds normal to the two-dimensional plane, which are originally dangling bonds after the dissection but can be treated with hydrogen termination for instance, should be different from the other Si sites. A simple way to implement this is to modify the Hamiltonian into where the energy ǫ H can be controlled by how the bond is chemically terminated on the silicene surface. Now, an interesting observation, in direct relevance to previous sections, is that, one can precisely apply the consideration in Sec.2, and we can show that the model has exactly flat bands. It follows from a simple observation that where P 3± and P 5 are also projections of dimensions dimP 3± = 3 and dimP 5 = 2 respectively. Since the Hamiltonian is expressed as a linear combination of projection operators again. Similar to the three-dimensional WT model, the wave functions associated with P 1 and P 2 are localised within each tetrahedron, the ones with P 5 are localised within the bonds perpendicular to the plane and the ones with P C 3± extends over the two dimensional plane. Then counting the dimensions, it implies H ǫ H Silicene ±V 2 E 8 has at most dimP 1 + dimP 2 + dimP 3± + dimP 5 = 1 + 1 + 3 + 2 = 7 nonzero-energy bands, that is, there is 8-7=1 flat band at ±V 2 generically.
Further interesting special situation can be ǫ H = ±V 2 . In this case, the coefficient of the projection P 5 for the expansion of H ǫ H =±V 2 Silicene ± V 2 E 8 vanishes. Then additional two dimensional space of the H ±V 2 Silicene ± V 2 E 8 further becomes null and the flat bands at the energies ∓V 2 become three fold degenerate.
As for the signs of the parameters. One has V 1 < 0 since ǫ s < ǫ p and the hopping gains energy, then we expect V 2 < 0. Further it is natural to assume the bonds normal to the plane is close to be dangling (non-bonding) that is ǫ H < 0 for single layer free standing silicene. Then we may consider the case ǫ H = V 2 for a typical situation for free standing silicene. Thus we can see algebraically that the difference in the structure of the Hamiltonian produces the following: while in 3D the dipersive bands with cones are sandwiched by doubly-degenerate flat ones, the band structure in 2D silicene has dipersive bands with cones sand-witched by triply-degenerate and non-degenerate flat bands. Now let us demonstrate the above analytic formulation by numerical results for the band structure in Fig.2. Panel (a) depicts the solvable case ǫ H = V 2 , where we can indeed see the triply-degenerate flat band at E = −V 2 = 1 and the non-degenerate flat band at E = V 2 = −1. Panel (b) depicts a more general case, where we can see that the flat bands become somewhat dispersive with the degeneracy lifted. They confirm the analytic discussion above. Interestingly, the result roughly agrees with the dispersion of silicene [5] reproduced in Fig.3, in terms of both the band width and band multiplicity.
Within the present simple model, one may expect the electrons at the perpendicular bonds can be stabilized by the hydrogen termination. It can be modeled as ǫ H = −V 2 > 0. Then the flat bands originally at −V 2 > 0 move down to V 2 , that is, it is below the energy of the Dirac cones. Then the fermi energy of the silicene is not at the Dirac cones anymore.

Conclusion
After introducing a generic argument for the existence of flat bands, Weair-Thorpe model, originally conceived for 3D silicon, is here extended to 2D silicene. A surprise revealed here is that the flat bands that arise in the WT model for the hybridized sp 3 orbitals also appear, in ideal situations, in silicene, but with different degeneracies in the flat bands due to a hitherto unsuspected algebraic reason. In this picture, the band structure, including the flat ones, are crucially controlled by the out-of-atomic-plane orbits. The flat bands in the idealised model, which should become dispersive in realistic situations, will still have large density of states. If we can shift the chemical potential close to the flat bands, e.g., by chemical doping, interesting phenomena are expected. Among these are (i) structural instabilities, such as those observed experimentally, and (ii) flat-band ferromagnetism [16]. In a broader context, the flat bands in our treatment can be theoretically interesting and important as an importance multi-orbital effect. [13] Moreover, sp 3 characteristic of silicene leads to the variety of hydrogen termination with/without edges, which leads to the appearance of edge states. We believe that these knowledge helps the synthesis of silicene on insulator substrates in the near future.