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We present the results of the symmetry classification of the electron energy bands in graphene and silicene using group theory algebra and the tight-binding approximation. The analysis is performed both in the absence and in the presence of the spin-orbit coupling. We also discuss the bands merging in the Brillouin zone symmetry points and the conditions for the latter to become Dirac points.

Since graphene was first isolated experimentally [

The present work has two aspects: a pragmatic and a pedagogical one. The first aspect is connected with the recent synthesis of silicene, the counterpart of graphene for silicon, with buckled honeycomb geometry. This novel two-dimensional material has attracted recently considerable attention, both theoretically [9,10] and experimentally, due to its exotic electronic structure and promising applications in nanoelectronics as well as its compatibility with current silicon-based electronic technology. So we present the symmetrty analysis of the silicene electron bands.

The pedagogical aspect is connected with the fact that different approaches to the symmetry classification, even if giving the same results, are based on different methods of applications of group theory. Thus in our previous paper [

We also generalize the symmetry classification by taking into account the spin-orbit coupling both for graphene and for silicene. This, to the best of our knowledge, wasn’t done before even for graphene. Though in graphene the spin-orbit coupling is very weak, the problem is interesting in principle. One can expect that in silicenr the coupling is stronger, and it will become even more so for graphene related materials from heavier elements, provided they can be synthesized.

To remind to a reader a few basic things, important for the symmetry classification of the bands in any crystal, consider a point sub-group R of the space group characterizing the symmetry of a crystal (we restrict ourselves with the consideration of symmorphic space groups). Any operation of the group R (save the unit transformation) takes a general wavevector k into a distinct one. However, for some special choices of k some of the operations of the group R will take k into itself rather than into a distinct wavevector. These particular operations are called the group of k; it is a subgroup of the group R. Points (lines) in the Brillouin zone for which the group of the wavevector contains elements other than the unit element are called symmetry points (lines). We may use a state (states) corresponding to such a special wavevector to generate a representation for the group of k [11,12]. In this paper we consider crystals with the hexagonal Brillouin zone. In this case the symmetry points are Γ—the center of the Brillouin zone, the points K which are corners of the Brillouin zone and the points M which are the centers of the edges of the Brillouin zone.

We’ll deal with the materials with a basis of two atoms per unit cell, and we’ll search for the solution of Schroedinger equation as a linear combination of the functions

where are atomic orbitals, labels the sublattices, and is the radius vector of an atom in the sublattice.

A point symmetry transformation of the functions is a direct product of two transformations: the transformation of the sub-lattice functions, where

and the transformation of the orbitals. Thus the representations realized by the functions (1) will be the direct product of two representations. Generally, these representations will be reducible. To decompose a reducible representation into the irreducible ones it is convenient to use equation

which shows how many times a given irreducible representation is contained in a reducible one [

where is the dimensionality of the irreducible representation and is the operator corresponding to a transformation. The operator projects a given function to the linear space of the representation. For a one dimensional representation the operator thus gives basis of the representation.

Our tight-binding model space includes four atomic orbitals:. Notice that we assume only symmetry of the basis functions with respect to rotations and reflections; the question how these functions are connected with the atomic functions of the isolated carbon atom is irrelevant.

The Hamiltonian of graphene being symmetric with respect to reflection in the graphene plane, the bands built from the orbitals decouple from those built from the orbitals. The former are odd with respect to reflection, the latter are even. In other words, the former form bands, and the latter form bands.

The group of wave vector at the point is, at the point is, at the lines is [8,15]. The representations of the groups and can be obtained on the basis of identities

the irreducible representations of the group are presented in the

The irreducible representations of the group are presented in the

the representations of the group can be classified as symmetric (g) or antisymmetric (u) with respect to inversion. Thus each representation of the group, say, begets two representations of the group: and.

Notice that the orbitals (or) realize representation both of the group and of the group, hence the representations of the groups realized by the functions will be identical to those realized by the sub-lattice functions.

Let us start from the symmetry analysis at the point. Because the transformations change sublattices, the characters corresponding to these transfor of the group.

Mations are equal to zero. The transformations leave the sub-lattices as they were. Hence from

Taking into account the symmetry of the states relative to reflection in the plane of graphene, we obtain that at the point the functions (here and further on, when this is not supposed to lead to a misunderstanding, we’ll suppres the index in) realize and representations of the group, characterizing band; the functions realize and representations of the group, characterizing band.

Acting by projection operators and on a function, we obtain that the irreducible representation is realized by symmetric combination of the and orbitals, and the irreducible representation by the antisymmetric combination. One can expect that the first case occurs in the hole band, and the second in the electron band.

The orbitals realize representation of the group [

Taking into account the symmetry of the states relative to reflection in the plane of graphene, we obtain that at the point the functions realize and representations of the group, characterizing bands.

To find wavefunctions, realizing each of the irreducible representations, we apply the projection operators and obtain

where if, and vice versa. Thus representation is realized by symmetric (antisymmetric) combinations of orbitals. One can expect that the first representation is realized at the hole band, and the second at the valence band.

Now let us perform the symmetry analysis at the point. The representation of the group realized by the functions is determined by the transformation law of the exponentials under the symmetry operations. Rotation of the radius vector by the angle anticlockwise, is equivalent to rotation of the vector in the opposite direction, that is to substitution of the three equivalent corners of the Brillouin zone: where

, and

. The rotation multiplies each basis vector by the factor. Using Eqation (3), we obtain

Hence, the functions realize irreducible representation of the group.

Taking into account the symmetry of the states relative to reflection in the plane of graphene, we obtain representation of the group, realized by the functions (merging bands), and representation, realized by the functions, characterizing bands.

The orbitals realize representation of the group [

taking into account the symmetry of the states relative to reflection in the plane of graphene, we obtain representations, and of the group, realized by the functions, characterizing bands.

Acting by projection operators, we obtain that the representation is realized by the vector space with the basis vector, and the representation is realized by the vector space with the basis vector. The vector spaces realizing representations and being found, the representation is obviously realized by the vector space spanned by the vectors.

Because the irreducible representation is realized both by and functions, these representations should be considered together. According to Wigner theorem [

The symmetry of the electron bands at the points and being determined, the symmetry at the lines follows unequivocally from the compatibility relations, presented in

The results of this section are presented on

The difference between silicene (or symmetrically equivalent to it buckled graphene) and graphene for our consideration is due solely to the decreased symmetry of the former. The group of the wavevector at the point in silicene is, at the point –(this is also the point group of silicene). The representations of the group we can obtain on the basis of identity

The direct product has twice as many representations as the group, half of them being symmetric (denoted by the suffix), and the other half antisymmetric (suffix) with respect to inversion. The characters of the representations of the group are presented in the

The symmetry analysis in silicene parallels that in graphene, so we’ll be brief.

At the point the functions realize graphene, so we’ll be brief.

Orbitals and realize representation, and the orbitals realize representation of the group. Thus at the point the functions (and) realize representation of the group. Reducible representation realized by the functions can be decomposed into the irreducible ones:

So when the symmetry is reduced by going from graphene to silicene, the representations and turn into and. Representation and two representations turn into three representations. Loosing the reflection in plane symmetry, we can not claim now that one representation is realized exclusively by orbitals. All the representations mix orbitals.

At the point the the functions realizes reducible representation of the group:

Orbitals realize, orbitals –, and orbitals –representations of the group. Thus the functions (1) realize reducible representation of the group which can be decomposed as

So when the symmetry is reduced by going from graphene to silicene, the representations and turn into the representations and respectively.

The band structure of silicene is being different from that of graphene, the merging of the bands is no different. The statement becomes clear when comparing

In the absence of spin-orbit coupling, electron spin can be taken into account in a trivial way: each band we considered was doubly spin degenerate.

When the spin-orbit coupling is taken into account, the symmetry, and the representations realized by the sub-

lattice functions (2) remaim the same. However, instead of atomic orbitals we should consider atomic terms. So for the case of hybridization, enumerates states from doublets and quartet.

Due to the semi-integer value of the angular momentum we have to consider double-valued representations realized by the atomic terms (and by the crystal wave functions). We remind that in this case it is convenient to introduce the concept of a new element of the group (denoted by); this is a rotation through an angle about an arbitrary axis, and is not the unit element, but gives the latter when applied twice:.

The characters of the rotation by angle applied to the term is

With respect to the inversion the character is

where the sign plus corresponds to the s states, and the sign minus to the p states. Finally, the charecters corresponding to reflection in a plane and rotary reflection through an angle are found writing these symmetry transformations as

Both in graphene and in silicene we’ll restrict ourselves by the symmetry analysis at the point.

The sub-lattice functions realize representation of point group. The electron terms realize two-valued representations of the group, which are presented in

Doublet realizes representation of the group; doublet realizes representation, quartet

realizes representation twice (We decided to use chemical notation for the single-valued representation, and BSW notation for double-valued representations [

Thus at the point four bands realize representation of the group each, and four bands realize representation each. In particular, we obtained the (well known) result that the four-fold degeneracy (including spin) of the bands merging at the point is partially removed by the spin-orbit coupling, and only two-fold (Kramers) degeneracy is left.

The two-valued representations of are presented in

(For the same reasons as for ordinary representations, two complex conjugate two-valued representations must be regarded as one physically irreducible representation of twice the dimension).

Thus at the point four bands which realize representation of the group each, and four bands realize representation each.

In this final part of the paper we would like to clarify the relation between the symmetry and the existence of dirac points.

According to the classical approach [18,19], the merging of the bands at a point is connected with the multi (higher than one)-dimensional representation of the space group, realized in this point. Looking for a linear dispersion point in the vicinity of the merging point we may use the degenerate perturbation theory. Let a two-dimensional irreducible representation is realized at a point. Expanding the wavefunction with respect to the basis of the representation

for the expansion coefficients we obtain equation in the form

where (of course, we need the absence of inversion symmetry at the point, for the matrix elements to be different from zero). The dispersion law is given by the equation

where are cartesian indexes. Equation (24) should contain only combinations of wavevector components which are invariant with respect to all elements of the group. In the case when the group does not have any vector invariants, and the only tensor invariant is the quantity, we obtain the dispersion law

which, like it was shown by Dirac himself in 1928, guaranties that Equation (25) is Dirac equation, in the sense the the matrices and satisfy anticommutation relations

where is the unity matrix.

To be more specific, consider the groups of wavevector at the point; in graphene it is, and in silicene it is. In both cases, to find the dispersion law at the point it is enough to study invariants of the group. And we can easily check up that both conditions, necessary for the existence of the Dirac point, are satisfied. This explains, in particular, why the band calculations show the existence of Dirac points in silicene [9,10], which has a lower symmetry than graphene.

In general, the role of the tight binding approximation in symmetry classification of the bands in graphene, like its role in symmetry classification of bands in other crystals, is only auxiliary. The approximation greatly helps in the classification and sheds additional light on the nature of the bands. but one must remember that there are more important things that this or that approximation and this is symmetry.

This paper presents an applications of group theory to very important cases of graphene and silicene.

Discussions with J. L. Manes, V. Falko and Hua Jiang were very illuminating for the author.