Hexagonally Warped Dirac Cones and Topological Phase Transition in Silicene Superstructure

Hexagonally Warped Dirac Cones and Topological Phase Transition in Silicene Superstructure


Silicene is a monolayer of silicon atoms forming a two-dimensional honeycomb lattice. We investigate the topological properties of a silicene superstructure generated by an external periodic potential. The superstructure is a quantum spin-Hall (QSH) insulator if it is topologically connected to silicene. It is remarkable that two inequivalent K and K’ points in the silicene Brillouin zone are identified in certain superstructures. In such a case two Dirac cones coexist at the same Dirac point in the momentum space and they are hexagonally warped by the Coulomb interaction. We carry out a numerical analysis by taking an instance of the () superstructure on the () structure of the Ag substrate. We show that it is a QSH insulator, that there exists no topological phase transition by external electric field, and that the hexagonally warping occurs in the band structure.

I Introduction

Topological insulator is a new state of matter which is characterized by an insulating bulk and surface edge modes(1); (2). It is robust against perturbations such as disorders and impurities as far as the gap does not close. It is an interesting and important question whether the topological insulator is robust against a formation of a superstructure in a periodic potential. This is nontrivial since it changes a global structure of the material such as the Brilloin zone. It is intriguing that superstructures have been realized in silicene on top of the Ag substrate(3); (4); (5). There may well be other ways to create periodic potentials. Silicene is an interesting play ground where many types of superstructures are naturally materialized.

Silicene is a sheet of silicon atoms replacing carbons in graphene. It could follow the trend of graphene and attract much attention(3); (4); (5); (6); (7); (8); (9); (10); (11); (12); (13); (14). It would open new perspectives for applications, especially due to its compatibility with Si-based electronics. Almost every striking property of graphene could be transferred to this innovative material. Indeed, it has Dirac cones akin to graphene. It has additionally a salient feature, that is a relatively large spin-orbit (SO) gap, which provides a mass to Dirac electrons and realizes a detectable quantum spin Hall (QSH) effect(8); (10). The QSH insulator is a two-dimensional topological insulator with helical gapless edge modes(1); (2). Furthermore a topological phase transition occurs from the QSH insulator to the trivial band insulator in the electric field(10). In this paper we address the problem if a superstructure is a QSH insulator as well and if there exists a similar topological phase transition.

In silicene the states near the Fermi energy are orbitals residing near the inequivalent K and K’ points at opposite corners of the hexagonal Brillouin zone. It is folded into a reduced Brillouin zone in a superstructure. We can generally argue that a superstructure is a QSH insulator provided it is topologically connected to silicene, namely, the gap is open and does not close as the periodic potential is continuously switched off. We also study the problem whether a silicene superstructure undergoes a topological phase transition by applying external electric field as in free-standing silicene(10). Furthermore, we point out an intriguing possibility that the K and K’ points are identified by the folding. When it happens, there exists only one Dirac point where two Dirac cones coexist, and this is experimentally detectable. In addition to general arguments, to demonstrate these novel phenomena we carry out a numerical analysis by taking an instance of the () superstructure on the () structure of the Ag substrate(15).

This paper is composed as follows. In Section II, we present how a superstructure is constructed from a honeycomb lattice. We derive the condition that the K and K’ points coincide. The (33) superstructure is the simplest example where this coincidence occurs. In Section III, we postulate the tight-binding Hamiltonian to describe a superstructure. Making a numerical analysis of the band structure of the (33) superstructure, we explicitly see that two degenerate Dirac cones are merged into two undegenerate hexagonally warped cones in the presence of infinitesimal Coulomb interactions. The mechanism of the hexagonal warping is explained based on the effective low-energy Dirac theory. In Section IV we investigate the topological properties of superstructures in three different ways, i.e., the topological connectedness, the bulk-edge correspondence and the Dirac theory. We present a general criterion when the topological phase transition may occur in superstructures. In the instance of the (33) superstructure, it is shown to be topologically connected to silicene, and hence it is a QSH insulator. We also show that there exists no topological phase transition in external electric field. Section V is devoted to discussions.

Ii Superstructure

Figure 1: (Color online) Illustration of (a) () superstucture and (b) () superstructure. The shaded area spanned by vectors and represents one unit cell of a superstructure.

We start with a study of a superstructure made on the honeycomb lattice (Fig.1). The honeycomb lattice is specified by the two basis vectors and . It has two inequivalent sites (A and B sites) per unit cell. There are three B sites adjacent to one A site. The A and B sites are generated by


where is a vector connecting adjacent A and B sites, and are integers. They span the the A-sublattice and the B-sublattice. A superstructure is specified by two translational vectors and  defined by


with and integers subject to . The choice of the angles generates the same superstructure due to the hexagonal symmetry. There are two types: The type-I superstructure is generated by choosing , while the type-II superstructure is generated by the space inversion of the type-I. When , type-I and type-II are identical. We show two examples by choosing and in Fig.1.

By a geometrical analysis of the honeycomb lattice we have


The structure thus generated is customarily referred to as the () superstructure. The number of silicon atoms per unit cell is given by


The angle between the vector and the -axis is given by


For instance, the choices and generate the () superstructure with and and the () superstructure with and , respectively, as illustrated in Fig.1.

Figure 2: (Color online) Brillouin zone of silicene and the ) superstructure. Its shape is honeycom (blue lines) or equivalently rhombus (large orange lines) for silicene, and rhombus (red lines) for the superstructure. It is constructed by folding the Brillouin zone of silicene. The K and K’ points are identified, which we denote by K̄ and take at the center of the Brillouin zone (the shaded area). The band structure along the horizontal thin lines (h1,h2) and vertical thin lines (v1,v2) are shown in Fig.3.

The basis vectors of the reciprocal lattice are given by solving the relations in the honeycomb lattice (Fig.2). The Brillouin zone is made by and in the reciprocal lattice with opposite sides being identified. A similar construction is carried out for a superstructure. The reciprocal vectors and  of a superstructure is given by


where is the unit vector perpendicular to the silicene plane. The Brillouin zone of a superstructure is constructed from that of silicene by identifying two momentum vectors and provided they are different only by the principal reciprocal vectors and , as illustrated in Fig.2.

There may occur an intriguing phenomenon. In general, when there are two integers such that


the K and K’ points are identified. We call it the K̄ point, and choose . Two Dirac cones coexist at the K̄ point. This happens indeed in the superstructure, where the Brillouin zone is obtained by folding that of silicene three times along the -axis and three times along the -axis. The area of the Brillouin zone is of that of silicene, and its shape is rhombus (Fig.2). This superstructure is simplest and most interesting, which we investigate as an explicit example.

Iii Tight-Binding Hamiltonian

The superstructure system is described by the second-nearest-neighbor tight-binding model constructed as follows. The Hamiltonian consists of two parts. The basic part is the Hamiltonian of silicene,(16)


where creates an electron with spin polarization at site , and run over all the nearest/next-nearest neighbor hopping sites. The first term represents the usual nearest-neighbor hopping with the transfer energy eV. The second term represents the intrinsic SO coupling with meV, where is the Pauli matrix of spin, with and the two bonds connecting the next-nearest neighbors. Additionally there exist two types of Rashba SO couplings ( and ) to describe silicene(11). They are quite small, and play no important roles in the present analysis, as far as we have numerically checked. To avoid unnecessary complications, we neglect them.

Figure 3: (Color online) (a1,a2) Band structure of silicene along the horizontal lines (h1,h2) in Fig.2. (b1,b2) Band structure of silicene along the vertical lines (v1,v2) in Fig.2. (a3,b3) 3-folded band structure obtained from (a1,a2) together with shifts along the axis, and the one obtained from (b1,b2) together with shifts along the -axis. They give the band structure of the () superstructure shown in Fig.5(a1) and (b1) with . The horizontal arrow in (a3,b3) indicates the first Brillouin zone.

The second part is the periodic potential term that introduces a superstructure to the honeycomb lattice. The periodicity is such that the band structure satisfies


where are the principal reciprocal vectors (6). It is to be emphasized that the band structure is simply constructed by folding that of silicene when , as illustrated in Fig.3 for an instance of the superstructure.

iii.1 An Explicit Example and Numerical Results

Figure 4: (Color online) Unit cell of the (33) superstructure. There are silicon atoms per unit cell, in which atoms (red) are in higher position and atoms (blue) are in lower position. There are A and  B sites in higher position. Their vertical sepation is Å. The basis vectors are represented by .

Superstructures have been materialized on the Ag substrate. The structure of the substrate acts as a periodic potential to silicene and generate a superstructure on silicene. Such an effect can be formulated into the periodic potential term .

Silicene available now is mostly grown on the Ag(111) substrate. The most common one is the () superstructure synthesized on the () Ag-structure(3); (4), though many other superstructures are possible due to the commensurability with the Ag substrate(17); (18). There are silicon atoms on top of silver atoms in this structure. The unit cell contains silicon atoms. The superstructure contain two sublattices: There are () atoms in the higher (lower) sublattice which is Å (Å) above the Ag surface, as illustrated in Fig.4. This structure has been observed in scanning tunneling microscopy (STM) images, where only the atoms in the higher sublattice are visible(3); (4). We may summarize the effect from the Ag substrate as a chemical potential difference between silicon atoms in higher and lower sublattices. Additionally we may apply the electric field perpendicular to the plane. It generates a staggered sublattice potential with Å between them.

The basic nature of the above superstructure is summarized into the periodic potential term,




where the summation is taken over higher sites: is a constant of the order of eV according to the first-principle calculation(18).

Figure 5: (Color online) Band structure of the () superstructure (a1,a2) along the -axis and (b1,b2) along the -axis, where (a1,b1) for and (a2,b2) for . Band structure (a1,b1) agrees with (a3,b3) in Fig.3. The vertical axis is the energy in unit of . The horizontal axis is momentum or . We have set for illustration.

The total Hamiltonian is , where the spin is a good quantum number. Hence, the Hamiltonian is decomposed into the spin sectors indexed by . Since the number (4) of silicon atoms per unit cell is in the (33) superstructure, we analyze the 18-band tight-binding model. We show numerically calculated band structures of the superstructure for and in Fig.5. They have qualitatively very similar behaviors though there are some quantitative difference.

Figure 6: (Color online) Contour plot of two trigonally warped Dirac cones at the K̄ point (a) for and (b) for . They are originally present at the K and K’ points in silicene.

We also show the contour plot of the band structures (Fig.6). We find two trigonally warped Dirac cones at the center K̄ of the Brillouin zone. The origin of these two cones is clear: There are Dirac cones at the K and K’ points in silicene, but they are placed on the same point in the superstructure since the K and K’ points are identified. There appears a new feature because these two Dirac cones now cross each other. The level crossing will be transformed into the level anticrossing when Dirac electrons interact among themselves. We show the contour plots of the Dirac cones in Fig.7, by introducing an infinitesimal on-site Coulomb interaction. The two Dirac cones merges and transformed into two hexagonally warped cones as illustrated in Fig.7. They are no longer degenerate, and must be experimentally detectable by angle resolved photoemission spectroscopy (ARPES).

iii.2 Dirac Theory

To understand this phenomenon of the hexagonal warping analytically, we analyze the low energy Dirac theory of the Hamiltonian (8),


where is the Pauli matrix of pseudospin associated with the A and B sites. Two Dirac cones are labelled by , which are originally present at the K point () and the K’ point ( in silicene. The way of trigonal warping is represented by the factor with the phase being opposite () between the two types of electrons(19). We assume in the analytical treatment for simplicity. Nevertheless, we are able to see clearly how two trigonally warped Dirac cones are transformed into two undegenerate hexagonally warped Dirac cones.

Figure 7: (Color online) Contour plots of two hexagonally warped Dirac cones. Trigonally warped Dirac cones [Fig.6] emerge when two crossing levels are transformed into two anticrossing levels by an infinitesimal on-site Coulomb interaction. (a) We have taken for simplicity. (b) The Dirac cones are modified slightly when .

When we introduce the on-site Coulomb interaction, the Hamiltonian and are mixed into


where we have chosen the basis , and


with representing the Coulomb energy. Up to the first order in , the two bands are explicitly given by


where is the azimuthal angle in the momentum space, . The degeneracy has been resolved. We can check that these two band structures reproduce the hexagonally warped Dirac cones in Fig.7 quite well.

Iv Topological Numbers

The topological quantum numbers(1); (2) are the Chern number and the index. If the spin is a good quantum number, the index is identical to the spin-Chern number modulo . They are defined when the state is gapped and given by


where is the summation of the Berry curvature in the momentum space over all occupied states of electrons with . The characteristic feature is that they are unchanged even if some parameters are continuously switched off in the Hamiltonian provided that the gap does not close(20); (21).

Figure 8: (Color online) Band gap as a function of . We show for the () superstructure in the solid curve and for free-standing silicene in the dashed lines. The band gap closes at with for the free-standing silicene (), but not closes for the () superstructure (eV). The vertical axis is the energy in unit of . We have set for illustration.

The spin-Chern number is defined by the integration of the Berry curvature over the conduction band indexed in the Brilloin zone for each spin sector,


Here, the Brilloin zone is that of the superstructure. The number of conduction bands is when the system is half filled. By making use of the band structure (9) we may equivalently integrate the Berry curvature over one conduction band in the extended Brilloin zone , which is the Brilloin zone of silicene,


Recall that conduction bands in the Brilloin zone are merged into one conduction band in the extended Brilloin zone , as illustrated in Fig.3.

It is well known that silicene is a SQH insulator characterized by the topological numbers,


It is an important question whether a superstructure is also a topological insulator. We investigate this problem in the following three ways. All of them produce the same result on this problem.

iv.1 Adiabatic Analysis

We take the example of the () superstructure by investigating our Hamiltonian system . We have explicitly calculated the band gap of the () superstructure by taking the potential term (10), whose result is given in Fig.8. It is surprising that the band gap never closes as a function of . Let us first choose the external electric field to cancel the internal chemical potential difference , namely, . Then, the Hamiltonian of the system is reduced to that of silicene, and hence the topological numbers are given by (21). We then continuously change till . As we have seen, the gap is kept open during this process. Hence, the topological quantum numbers are kept unchanged during this continuous process. Namely the () superstructure is topologically connected to silicene. We conclude that the superstructure is a QSH insulator.

Figure 9: (Color online) Band structure of a zigzag nanorbibbon made of the () superstructure (a) for and (b) for . The width contains 8 unit cells of the superstructure. The red lines indicate the doubly degenerate edge modes with up and down spins. The vertical axis is the energy in unit of . The horizontal axis is the momentum . The points and are identified, and give the K̄ point, while is the point. We have set for illustration. Each red line connects the K̄ point in the conduction and valence bands after winding once a circle of the whole Brillouin zone. (c) Illustration of a nanoribon with the width containing 3 unit cells of the superstructure. The red and blue arrows indicate the degenerate gapless edge modes with up and down spins.

When we apply external electric field , silicene undergoes a topological phase transition(10). However, the () superstructure does not since the gap does not close. We shall see the reason why this is the case in subsection IV.3.

iv.2 Edge States Analysis

The bulk-edge correspondence(1); (2) is well known to characterize the topological insulators. Namely, a topological insulator is characterized by the appearance of gapless modes on edges. The reason why gapless modes appear in the edge of a topological insulator is understood as follows. The topological insulator has a nontrivial topological number, the index(16), which is defined only for a gapped state. When a topological insulator has an edge beyond which the region has the trivial index, the band must close and yield gapless modes in the interface. Otherwise the index cannot change its value across the interface.

We have numerically calculated the band structure of a zigzag nanoribbon geometry in Fig.9(a,b), where the width contains 8 unit cells of the superstructure. Two zero-energy edge modes (indicated by red lines) are clearly seen: Each line represents the two-fold degenerate helical modes propagating one edge as indicated in Fig.9(c). The band structure of a nanoribbon with is just the one obtained by folding that of a silicene nanoribbon as in Fig.9(a). It is modified for , but the modification is only slight even when we take a quite large value for , as shown in Fig.9(b). The essential feature is that the edge state, connecting the conduction and valence bands at the same K̄ point, is topologically protected since it winds a circle of the whole Brillouin zone as in Fig.9. Namely, as far as the band is open, though the band structure is modified by changing , the gapless edge modes connecting the conduction and valence bands continue to exist. We remark that each zero mode originally connects the K and K’ points in silicene and is topologically protected. This property is not lost even when they are identified as the K̄ point. Consequently, the superstructure is a QSH insulator.

iv.3 Dirac Theory Analysis

We have found that there exists no topological phase transition in the () superstructure by the change of the external electric field . We present an analytic argument by calculating the topological numbers based on the effective Dirac theory. This analysis is valid for a generic superstructure. We assume that the buckled structure consists of two sublattices. Indeed, there is a report(18) that this is the case for all structures grown on Ag(1,1,1). Let the numbers of A sites and B sites be and , respectively, in the higher sublattice per unit cell which contains silicon atoms. It yields a contribution to the Dirac mass within the mean-field approximation,


It has been shown(12); (1); (2) that the topological charges are determined by the sign of the Dirac mass .

In the () superstructure, since we have and , the superstructure is a QSH insulator. Furthermore, since it does not depend on , there occurs no phase transitions by the change of . This gives an analytic reasoning why the gap does not close in Fig.8.

The condition that the phase transition occurs (12) is that , which determines the critical field,


In the case of free-standing silicene, we have , , , , and hence


as agrees with the previous result(10). In general it is necessary that for a topological phase transition to occur in a superstructure described by the potential term (10).

V Discussions

Figure 10: (Color online) Cotour plot of Dirac cones with structural deformation by transfer-energy difference for (a) and (b) . We set and . By comparing these with Fig.6, the modification of the band structure is found to be quite small.

In conclusion, we have investigated the topological properties of a superstructure made from the silicene honeycomb lattice by a periodic potential term . The superstructure is a QSH insulator if it is topologically connected to silicene. The concept of the topological connectedness may be formulated as follows. We consider the Hamiltonian,


Provided the gap is open in the system for any values of () as is continuously changed from to , the two systems and are topologically connected. For the instance of the potential (10) we have used the electric field instead of as such a continuous parameter to show the topological connectedness.

We have investigated the problem if the topological phase transition occurs as in silicene(10). We have also derived the condition (7) that the K and K’ points become identical in superstructures. In this case, there occurs an intriguing phenomenon that two Dirac cones coexist at a single point (K̄ point). They exhibit hexagonal warping in the presence of the Coulomb interaction, which must be observed by ARPES.

We have carried out a numerical analysis and confirm these observations by taking an instance of the () buckled superstructure together with the potential term given by (10). In so doing we have assumed that all transfer energys are the same. In principle, there may be two different transfer energyies, and , where is the transfer energy between two lower sites (indicated as blue balls) and is the transfer energy between lower and higher sites (indicated as blue and red balls) in Fig.4, although there is no report that there is a significant difference between them. Note that there is no term as seen in Fig.4. We have calculated the band structure with different transfer energy in Fig.10. It is clearly seen that there is no qualitative difference even if we assume very different transfer energies such as . A possible transfer-energy difference does not cause any qualitative change of the band structure. In particular, the gap does not close as a function of . We conclude that such a transfer-energy difference does not destroy the topological properties of the superstructure.

Silicene superstructures are grown on the Ag substrate. The most common one is the () superstructure on the () Ag-structure. There are several possible superstructures obtained in this way. Here we give some correspondences between them(17); (18):

Si-superstructure Ag-structure p q ratio Refs.
(3); (4)

It is interesting that the ratios of the Ag-structure and Si-superstructures are around . There would be other ways to construct superstructures by introducing periodic potentials to silicene. Our analysis of topological properties in superstructure is applicable to any of them.

I am very much grateful to G.L. Lay, S. Hasegawa and N. Nagaosa for many fruitful discussions on the subject. This work was supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture No. 22740196.


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