## Abstract

The general properties of long wavelength triangular graphene superlattices are studied. It is shown that Dirac points with and without gaps can arise at a number of high-symmetry points of the Brillouin zone. The existence of gaps can lead to insulating behaviour at commensurate fillings. Strain and magnetic superlattices are also discussed.

## 1. Introduction

Graphene is a two-dimensional metal when carriers are induced by an electric field (Novoselov *et al.* 2004, 2005; Geim & Novoselov 2007; Castro Neto *et al.* 2009). A gap at the Fermi level has been observed by scanning tunnelling microscopy measurements (Zhang *et al.* 2008; Li *et al.* 2009; see also Wehling *et al.* 2008). We analyse the gaps induced by a periodic structure, and the possibility that these gaps are generated spontaneously.

Graphene superlattices have been observed in graphene layers grown on transition metals (Vázquez de Parga *et al.* 2008; Borca *et al.* 2010; see also Oshima & Nagashima 1997; NDiaye *et al.* 2006; Marchini *et al.* 2007; Jiang *et al.* 2008; Martoccia *et al.* 2008; Pan *et al.* 2008; Usachov *et al.* 2008). Superlattices are also found in graphene grown by the decomposition of SiC (Zhou *et al.* 2007). In general, graphene superlattices can have interesting properties, such as highly anisotropic transport properties (Park *et al.* 2008*a*) or Dirac points at finite energies (Park *et al.* 2008*b*; Brey & Fertig 2009; Tiwari & Stroud 2009; Arovas *et al.* 2010; Barbier *et al.* 2010). In general, the study of the properties of graphene superlattices has attracted great interest, owing to the many novel features their electronic spectra can show (Pedersen *et al.* 2008; Abedpour *et al.* 2009; Bliokh *et al.* 2009; Gibertini *et al.* 2009; Park *et al.* 2009; Rosales *et al.* 2009; Shytov *et al.* 2009; Snyman 2009; Balog *et al.* 2010; da Silva-Araújo *et al.* 2010; Guinea *et al.* 2010). In the following, we analyse the general properties of the spectra of graphene superlattices with a two-dimensional triangular periodicity. These superlattices share the symmetries of the graphene lattice, and are commonly found in graphene layers grown on metallic surfaces. As discussed below, these superlattices can show a gap at the Fermi energy for a number of commensurate fillings. It seems probable that they can be formed spontaneously on very uniform substrates, such as BNi, or as a result of intrinsic instabilities of graphene.

In this paper, we study the general properties of triangular graphene superlattices created by a scalar potential, followed by a discussion of strain and magnetic superlattices.

## 2. Brillouin zone of triangular graphene superlattices

We define the lattice vectors of the graphene lattice as
2.1
A triangular superlattice is described by the unit vectors
2.2
where *n*_{1} and *n*_{2} are arbitrary integers different from zero.

There are three types of high-symmetry points in the Brillouin zone of a triangular lattice, **Γ**, M and K. There are two inequivalent *K* points, *K* and *K*′, at the corners of the hexagonal Brillouin zone, and three inequivalent M points, at the centres of the edges. Time reversal exchanges K and K′, while leaving the **Γ** point and the three M points unchanged (Mañes *et al.* 2007). The vectors that define these points are such that
2.3

The low-energy states of graphene lie close to the K and K′ of the original Brillouin zone. The positions of these points in the superlattice Brillouin zone are determined by
2.4
Thus, when 2*n*_{1}+*n*_{2} is a multiple of three, the graphene K and K′ points will be mapped onto the **Γ**_{S} point of the superlattice Brillouin zone. Otherwise, they will be mapped onto the corners of the Brillouin zone, *K*_{S} and *K*′_{S}. Examples of superlattice Brillouin zones are given in figure 1.

## 3. Dispersion near high-symmetry points

### (a) The model

We study superlattices induced by a modulation of the on-site energy of the *π* orbitals. We assume that it is a weak perturbation of the graphene Dirac equation, except in cases where degeneracies occur. We consider the Fourier components of the potential with lowest wavevector, of modulus
3.1
We write the potential as the sum of symmetric part *V*_{G}, and an antisymmetric part, **Δ**_{G}, with respect to the interchange of sublattices. We neglect for the moment the short wavelength components that mix the two inequivalent Dirac points of the unperturbed graphene layer.

We analyse the changes in the Fermi velocity near the Dirac energy induced by the superlattice potential, and the points in the lowest bands of the superlattice where degeneracies persist when *V*_{G}≠0 and **Δ**_{G}=0. As discussed below, this situation gives rise to a new set of Dirac equations at finite energies.

### (b) Dirac energy at the *Γ*_{S} point

We consider the case when the K and K′ points of the graphene Brillouin zone are mapped onto the **Γ**_{S} point of the superlattice Brillouin zone. Using lowest order perturbation theory, we find, near the **Γ**_{S} point, a renormalization of the Fermi velocity (Park *et al.* 2008*a*,*b*)
3.2
where *ϕ*_{k,G} is the angle between vectors ** k** and

**. For the hexagonal superlattice considered here we find the isotropic reduction in the Fermi velocity 3.3 There is a twofold degeneracy at the three M**

*G*_{S}points, if

**Δ**_{G}=0. The energy of these states is

*v*

_{F}

*G*/2. At finite distances from the M

_{S}points, we can write an effective hamiltonian, 3.4 which gives an anisotropic Dirac equation with a gap, 3.5

At the *K*_{S} and *K*′_{S} points, there are three degenerate levels for *V*_{G}=**Δ**_{G}=0, with energy . When *V*_{G}≠0, these three levels are split into a doublet, with energy , and a singlet, at . Expanding around the *K*_{S} point, the effective hamiltonian for the doublet is
3.6
This is the two-dimensional Dirac equation with a mass term. The dispersion relation is
3.7

There are two sets of degenerate bands, derived from the K and K′ points of the Brillouin zone of graphene. This degeneracy will be broken by short wavelength terms in the superlattice potential.

### (c) Dirac energy at the *K*_{S} and *K*′_{S} points

The renormalization of the Fermi velocity near the *K*_{S} and *K*′_{S} points is the same as in equation (3.3).

There are doubly degenerate states, even when *V*_{G}≠0, at the six inequivalent points at positions *L*_{S}=*K*_{S}/2, as shown in figure 2. The energy of these points is *ϵ*_{LS}=*v*_{F}*G*/2. Expanding around these points, we find an effective anisotropic Dirac equation, given by equation (3.4).

There is another set of doubly degenerate states at the M_{S} points. The two states arise from the K and K′ points of the original graphene Brillouin zone. The degeneracy persists when *V*_{G}≠0 and **Δ**_{G}≠0, and is broken only by short wavelength components of the superlattice potential. When these components are finite, an effective anisotropic Dirac equation will arise similar to that in equation (3.4).

For *V*_{G}=**Δ**_{G}=0, there are six degenerate states at the **Γ**_{S} point. The long-range part of the superlattice potential will hybridize states that are derived from the K and K′ points of the original graphene Brillouin zone. We obtain two sets of isotropic Dirac equations, described by equation (3.6), and two degenerate states. The short-range part of the Dirac equation will break these degeneracies.

### (d) Results

We analyse the bands induced by an *N*×*N* superlattice. The hopping matrix between *π* orbitals in neighbouring carbon atoms is *t*=3 eV. The bands for *V*_{G}=0.3 eV and **Δ**_{G}=0 are shown in figure 3. The bands show Dirac points at the M_{S} and *K*_{S} points. When **Δ**_{G} is increased to **Δ**_{G}=0.1 eV, a gap appears between successive bands, as shown in figure 4. The density of states for those two cases is shown in figure 5. Note that the potential breaks the electron–hole symmetry of clean graphene, and the gaps are not of the same magnitude for positive and negative energies.

The results are in reasonable agreement with the analytical description in the previous section. A gap of order 2**Δ**_{G} is induced at the M_{S} point. In order for this gap to be possible, the following inequalities must be satisfied:
3.8
The scaling properties of the Dirac equation imply that, if the dimension of the superlattice is increased, , with *λ*<1, then a rescaling of the superlattice potential, , will lead to the same band structure, with energies scaled as .

## 4. Strain superlattices

A superlattice can also be produced by inducing strains, which modulate the interatomic hoppings. The corresponding perturbation can be seen as a gauge field, ** A**, which shifts locally the momentum (Vozmediano

*et al.*2010). A simple case is when the strains are due to height modulations,

*h*(

**), which can be induced by a substrate. In terms of the Fourier components of the modulation,**

*r**h*

_{G}, and allowing for the relaxation of the in-plane displacements, the effective gauge field can be written as (Guinea

*et al.*2008) 4.1 where ,

*t*≈ 3 eV is the hopping between nearest neighbour

*π*orbitals,

*a*≈ 1.4 Å is the distance between nearest neighbour carbon atoms, and the tensor is the Fourier transform of the functions 4.2 and

*λ*and

*μ*are the elastic Lamé coefficients of graphene. The field in equation (4.1) has opposite signs in the two valleys in the Brillouin zone.

The calculation of the effective magnetic field induced by the gauge field in equation (4.1) is simplified when, as in the previous sections, only one component, , in a superlattice is considered. The tensor in equation (4.2) has non-zero components for all combinations of the type *G*_{k}+*G*_{l}. When the gauge field ** A**(

**) is parallel to**

*G***, the vector potential can be gauged away, and does not induce an effective magnetic field. This implies that out of the 18 possible values of the vector**

*G*

*G*_{k}+

*G*_{l}, only six possible values contribute to the effective magnetic field. These vectors are given by and the vectors equivalent to it by a symmetry transformation. After some algebra, we obtain for the effective magnetic field 4.3 The superlattice defined by the effective magnetic field has a unit vector of length , so that the area of its unit cell is smaller than the area of the unit cell of the original superlattice by a factor 1/3.

Using equation (4.3) and *G*=2*π*/*L*, where *L* is the length of the unit vector of the superlattice, we can write the magnetic length associated with the maximum effective field in the system as
4.4
where is the maximum value of *h*(** G**), assuming

*h*

_{G}>0. For values nm and

*L*≈ 40 nm, we find

*l*

_{B}≈ 14 nm, so that the effective field is such that T.

## 5. Magnetic superlattices

A superlattice can also be induced by a spatially modulated magnetic field. A combination of a modulated magnetic field and a scalar potential opens a gap at the Dirac energy, and the resulting insulator is a quantum Hall system (Snyman 2009), with chiral currents at the boundaries (Haldane 1988). Here, we obtain this effect using second-order perturbation theory, instead of the arguments used in Snyman (2009). As in the previous sections, we assume the simplest periodicity compatible with the superlattice hexagonal symmetry
5.1
The eigenstates of the unperturbed Hamiltonian at the K and K′ points of the Brillouin zone can be written as |0〉_{A}=(1,0) and |0〉_{B}=(0,1), of which each component of the spinor corresponds to one sublattice. These states are hybridized with states |*G*_{±}〉_{K}=(1,±e^{iϕG}) and |*G*_{±}〉_{K′}=(1,∓*e*^{−iϕG}), with energies *ϵ*_{G}=±*v*_{F}|** G**| and e

^{iϕG}=(

*G*

_{x}+i

*G*

_{y})/|

**|.**

*G*The energies of states |0〉_{A} and |0〉_{B} are modified in different ways by virtual hoppings into states |*G*_{±}〉_{K} and |*G*_{±}〉_{K′}, leading to gaps in both valleys. Moreover, the gaps have different signs, showing that the time reversal symmetry in the system is broken, and that a quantum Hall phase has been induced. The gap can be written as
5.2
where *A*_{G} is the vector potential, which we define as
5.3
where *Φ*_{0}=*eh*/*c* is the quantum unit of flux. Using this expression, we finally obtain
5.4
in agreement with Snyman (2009).

## 6. Self-consistent opening of a gap

The previous analysis shows that a gap can open at finite energies in graphene in the presence of a superlattice potential with a staggered component. When the number of carriers is such that only a small number of subbands are completely filled and the rest are completely empty the electronic energy will be lowered in the presence of the gap. A lattice distortion that leads to the appropriate potential will be energetically favourable if the gain in electronic energy exceeds the formation energy of the distortion, as in the Peierls instability in one dimension.

In graphene on top of a metal or other substrate with a large dielectric constant, as in Vázquez de Parga *et al.* (2008), out-of-plane displacements lead to changes in the on-site energies of the *π* orbitals. An electron in a given carbon atom experiences the image potential owing to the screening. A change in position of **Δ***z* leads to a change in the image potential of order *ee****Δ***z*/(4*d*^{2}), where *e**=*e*(*ϵ*_{0}−1)/(*ϵ*_{0}+1) is the image charge, *ϵ*_{0} is the dielectric constant of the substrate and *d* is the distance to the substrate. A vertical displacement of **Δ***z*∼1 Å when the graphene layer is at a distance *d* ≈ 3 Å of the substrate can lead to shifts of the on-site energies of order 0.1 eV. The electronic gain of energy owing to the existence of a gap, per unit cell, is then *E*_{elec} ≈ **Δ**_{G} ≈ *ee****Δ***zd*^{−2}.

The elastic energy per unit cell required to create a staggered distortion of amplitude **Δ***z* is of order *E*_{elas} ≈ *κ***Δ***z*^{2}*a*^{−2}, where *κ* ≈ 1 eV is the bending rigidity of graphene.

A gap will exist above a threshold for the superlattice potential, . The area of the Brillouin zone of the supercell, , should be close to the area within the Fermi surface of the unperturbed graphene, *πk*^{2}_{F}. Hence, a spontaneous staggered distortion is favoured if
6.1
The last equation in (6.1) implies that *G*∝*k*_{F}, and from the first two we obtain that . Hence, a spontaneous distortion is energetically favoured for carrier densities such that
6.2
and the staggered distortion, with inverse wavelength *G*∼*k*_{F} of the order
6.3
For a metal, we have and *e**=*e*. For *a* ≈ 2 Å, *d* ≈ 3 Å and *κ* ≈ 1 eV, we find that a staggered corrugation is energetically favoured for carrier densities cm^{−2}, leading to maximum deformations of 0.2 Å. For SiO_{2}, where *ϵ*_{0}=3.9 and *e** ≈ 0.6*e*, the corrugations take place for carrier densities cm^{−2}, and maximum deformations of 0.15 Å.

The formation of these long wavelength modulations is only possible in systems with a high degree of order, as the superlattice features will be reduced by disorder. For instance, weak scatterers with concentration *n*_{imp}, which change the on-site potential by an amount of order **Δ**_{G}, lead to a mean free path . A large concentration of these defects, *n*_{imp}∼*a*^{−2}, will suppress the formation of a superlattice, while little affecting the conductivity, .

## 7. Conclusion

We have analysed the formation of Dirac points and gaps at high-energy points of triangular graphene superlattices. We have shown that, in some cases, a gap can be formed over the entire Fermi surface, making graphene insulating. We have discussed instabilities that might give rise to the spontaneous formation of gaps of this type. General properties of strain and magnetic superlattices have also been discussed.

## Acknowledgements

We appreciate useful conversations with J. L. Mañes, R. Miranda and A. Vázquez de Praga. Funding from MICINN (Spain) through grants FIS2008-00124 and CONSOLIDER CSD2007-00010 is gratefully acknowledged. T.L. acknowledges funding from INDEX/NSF (US).

## Footnotes

One contribution of 12 to a Theme Issue ‘Electronic and photonic properties of graphene layers and carbon nanoribbons’.

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