Opening of a Gap in Graphene Due to Supercell Potential: Group Theory Point of View ()
1. Introduction
Graphene is a two-dimensional crystal of carbon atoms, which form a honeycomb lattice with the point symmetry described by the group
. The first Brillouin zone (BZ) has a hexagonal form, and the conduction band touches the valence band in six BZ corners which form two non-equivalent triads of BZ corners, 
and
. One of the routes toward tailoring the electronic properties of graphene is through the adsorption of metals [1] [2] . Recently, several types of adatoms were used to dope graphene in attempts to tailor properties of graphenebased devices [3] -[7] . The gap opening in the high symmetry points for the hexagonal lattices due to interaction with the interface was considered in [8] . In Ref. [9] , it was shown that, using ab initio density functional ory calculations, the adsorption of an alkali-metal submonolayer ongraphene occupying every third hexagon of the honeycomb lattice in a commensurate 
arrangement induces an energy gap in the spectrum of graphene. We decided to analyze this opening of the gap in the framework of the group theory. In our previous publications [10] -[12] we summed up the classification of the energy bands in graphene on the basis of the point group analysis. The only fact from that analysis we need in the present work, is the fact that the little group at the point 
is 
and the bands 
and 
realize 
representation of the group. The same can be said about the point
. This fact by itself means that the bands 
and 
touch each other at the points 
and
, and the electron states in the vicinity of these points are described by massless Dirac equation [11] .
For the purpose of the present paper, we should put the abovementioned fact into the framework of the theory of the space group symmetry [13] [14] , from which we will need only a few basic ideas. According to the theory of the space group symmetry, the bands 
and 
should be considered at the points 
and 
(these two points can be considered as the stars of the wave vector
, and designated 
together, thus realizing a 4-dimensional representation of the 
group. Due to the identity
(1)
any element of the group 
can be presented as an element of the group
, or as a product of 
and such element. Representation of the point group 
realized at the point 
defines representation of the space group realized at
. The matrix representing an element 
is a super-matrix 
(2)
super-indices 1 and 2 referring to the points 
and 
respectively. The matrix representing an element 
is
(3)
We will not need the exact form of the non-diagonal matrix elements in Equation (3); what we need is the fact that the trace of the matrix 
is equal to zero. Naturally, when we consider dispersion in grapheme as it is, space group symmetry point of view adds very little in comparison to point group symmetry point of view, because the Hamiltonian, which has the symmetry
, is block-diagonal.
Now we apply the group theory to analyze what happens at the points 
in grapheme with a perfectly commensurate superlattice potential (which appears either because of the substrate or because of the absorbed atoms), which has the same point symmetry 
as graphene. We consider explicitly a 
superlattice, known as the Kekule distortion of the honeycomb lattice [9] . In this case we may consider the Bril-louine zone (BZ) of the superlattice as the folding of the original [9] [15] . The folding leads to the identification of the corners of the original BZ (
and
) with the center 
of the new BZ. The Hamiltonian is no longer block diagonal and, because the points 
and 
are now identical, has the full symmetry
. We thus observe a paradox situation: due to decrease of the translational symmetry the point symmetry of the Hamiltonian has increased.
Because of the symmetry of the Hamiltonian, we need to decompose representation realized by Matrices (2) and (3) with respect to the irreducible representations of the group
.To obtain the decomposition, it is convenient to use equation
(4)
which shows how many times a given irreducible representation 
is contained in a reducible one [16] . In Equation (4) 
is the number of elements in the group, 
is the character of an operator 
in the irreducible representation 
and 
is the character of the operator 
in the representation being decomposed. Actually, even without using Equation (4), just by inspection of the two lowest line of Table 1 we obtain the decomposition
(5)
We see that due to supercell potential two degenerate Dirac points disappear. At the point 
we have two merging bands realizing representation 
and another two merging bands realizing representation
. We
Table 1. Characters table for irreducible representations of 
point groups.
may expect that representation
, as being more symmetrical, is realized at the top of the valence band, and the representation 
is realized at the bottom of the conduction band.
To get the form of the energy spectrum of the electrons in the vicinity of the point 
let us consider both the 
term [17] and the supercell potential as a perturbation. The effective Hamoltonian is
(6)
where
(7)
and 
reduces to two independent real constants
(8)
The specific form of the operator 
follows from the symmetry of the base functions realizing representations 
and
. By shifting origin of the energy axis these two constants can be chosen as
(9)
Forming and solving the secular equation from these matrix elements, we obtain
(10)
the sign plus corresponding to an upper pair of bands, and the sign minus corresponding to a lower pair of bands.
To resolve between the branches in each pair we should take into account 
corrections to the operator
. The first order in 
corrections is equal to zero because the symmetry group contains the center of inversion. To the second order in 
we have 
(11)
where 
is an Hermitian tensor operator (symmetrical in the suffixes 
and
). These include the corrections from the terms linear in 
in the Hamiltonian in the second-order perturbation theory and the corections from the terms quadratic in 
in the first-order perturbation theory [17] . Notice that 
is small relative to both 
(because we consider the states in the vicinity of the point
), and with respect to
.
The relations exist between the matrix elements of the operator because of the requirements of symmetry. As regards their transformation law under the symmetry operations, the wave functions which form the basis of the representation 
(1, 2 branches) can be taken in the form
(12)
and the wave functions which form the basis of the representation 
can be taken in the form
(13)
From this, we easily conclude that in the first case the matrix elements of the 
reduce to three independent real constants
(14)
The matrix elements of the operator 
are
(15)
In the second case the matrix elements of the 
also reduce to three independent real constants
(16)
The matrix elements of the operator 
are
(17)
Forming and solving the secular equation from these matrix elements, we obtain the 
branches of the spectrum
(18)
The sign of 
is determined by the fact whether we are dealing with the lower or upper bands. The formula for the 
branches of the spectrumcan is obtained similarly.
The folding of the BZ, together with the destruction of previously existing gapless Dirac points, leads to appearance of the new ones. In fact, the new BZ is still a hexagon, and the same symmetry arguments used for graphene can be used to explain appearance of the gapless Dirac points at the corners of the new BZ
. However, these new Dirac points are situated deep below or high above the Fermi level, hence, less of them manifest themselves than Dirac points of unreconstructed graphene.