Analytical calculation of electron’s group velocity surfaces in uniform strained graphene
Abstract
Electron group velocity for graphene under uniform strain is obtained analitically by using the TightBinding approximation. Such closed analytical expressions are useful in order to calculate electronic, thermal and optical properties of strained graphene. These results allow to understand the behavior of electrons when graphene is subjected to strong strain and nonlinear corrections, for which the usual Dirac approach is not longer valid. Some particular cases of uniaxial and shear strain were analized. The evolution of the electron group velocity indicates a break up of the trigonal warping symmetry, which is replaced by a warping consistent with the symmetry of the strained reciprocal lattice. The Fermi velocity becomes strongly anisotropic, i.e, for a strong pure shearstrain (20% of the lattice parameter), the two inequivalent Dirac cones merge and the Fermi velocity is zero in one of the principal axis of deformation. We found that nonlinear terms are essential to describe the effects of deformation for electrons near or at the Fermi energy.
I Introduction
Graphene was the first twodimensional (2D) crystal discovered Novoselov et al. (2004). It has been broadly studied due to the observed peculiar physical properties Geim (2007); S. Das Sarma (2007); Geim (2009); Novoselov (2011). The electronic properties are mainly determined by electrons at the Fermi energy Geim (2009). For graphene, such electrons have momentum near or at the high symmetry points of Brillouin Zone (BZ). This behavoir can be modeled by a Dirac Hamiltonian Castro Neto et al. (2009), where electrons behave as massless Dirac fermions with a Fermi velocity m/s, which plays the role of the speed of light. In this model, the Fermi velocity is a constant parameter. However, this is not longer true when graphene has corrugations (curved space) or is stretched, since these deformations give rise to a spacedependent Fermi velocity Pereira et al. (2009), suggesting changes in the electronic conductivity. Furthermore, in the case of stretching, a bandgap opening is observed Pereira et al. (2009); Cocco et al. (2010); Sena et al. (2012). Such results open the possibility for doing “strain engineering” in order to tailor the electronic properties and thus control the electron transport Pereira and Castro Neto (2009); Guinea (2012); Jiang et al. (2013); Gradinar et al. (2013); BarrazaLopez et al. (2013); Hung Nguyen et al. ().
Several theoretical approaches have been proposed to study deformations in graphene Castro Neto et al. (2009); Winkler and Zülicke (2010); Linnik (); Sloan et al. (2013); Kerner et al. (2012); Pacheco Sanjuan et al. (2014a, b). The most common one is a combination of the TightBinding Hamiltonian (TB) and linear elasticity to derive a Dirac effective equation Castro Neto et al. (2009). Under such approach, pseudomagnetic fields appear, although lattice deformations were not included in the original derivation Kitt et al. (2012). In the case of strain, recent works have included these considerations starting from different treatments BarrazaLopez et al. (2013); Sloan et al. (2013); Kitt et al. (2012, 2013). Still, there are some problems with such approach OlivaLeyva and Naumis (2013) because a common confusion is the assumption that the Dirac cone tips in the new deformed lattice coincides with the strained high symmetry points and .
In the present work, we calculate analitically the group velocity surfaces for the important case of uniform strain, which can be solved without the usual perturbative analysis of the Dirac equation. For this goal, the TB approximation has been used. The Fermi velocities are obtained by looking at the appropiate points in the reciprocal space. Thus, our results are more general and include the Dirac theory of strain as a limiting case. Additionally, we found that even for a realistic value of pure shear strain, a mixed DiracSchröedinger behavior can arise, suggesting that the Dirac theory has to be modified. In fact, this behavior has been obtained in other cases Pereira et al. (2009); RomanTaboada and Naumis (2014).
The layout of this paper is the following. In Sec. 2, we describe electron behavior in graphene under uniform strain. Then a dispersion relation is obtained using TB aproximations. For this relation, we display the surfaces and contour plots for the particular case of pure shear strain. In Sec. 3, we derive the group velocity for electrons, and analyze the pure uniaxial and pure shear strain cases. Finally, in Sec. 4, we give our conclusions.
Ii Electrons in Strained Graphene
Graphene is formed by a single atomic layer of carbon atoms arranged in an hexagonal estructure, as shown in Fig. 1. The structure can be described in terms of two triangular sublattices, A and B, with a basis of two atoms per unit cell. The lattice unit vectors are given by and and the three nearestneighbor vectors can be written as , and Castro Neto et al. (2009). Likewise, there are two reciprocallattice vectors given by and , and two inequivalent special points at the corners of the graphene BZ, called high symmetry points and Castro Neto et al. (2009). For unstrained graphene, the tips of the Dirac cones (or the Dirac points ) are located at the and points.
In the case of a uniform strain, if the vector represents the positions of the carbon atoms in the undeformed graphene, its deformed counterpart is given by , where is the identity matrix and is the uniform strain tensor. The lattice unit and nearestneighbor vectors are thus as follows, and , while the reciprocallattice vectors are deformed as, . The new high symmetry points in the corners of the first BZ of the strained reciprocal lattice are obtained by construction of the WignerSeitz primitive cell and can written in general as
(1) 
with
and
where and are the and components of the deformed reciprocal vectors ().
To obtain the electronic properties of graphene under uniform strain, we use the nearestneighbor TB Hamiltonian OlivaLeyva and Naumis (2013)
(2) 
where runs over all sites of the deformed honeycomb lattice and the hopping integral varies due to the modification in the carboncarbon distances as Ribeiro et al. (), with and eV is the hopping energy for unstrained graphene Castro Neto et al. (2009). The operators and correspond to creating and annihilating electrons on the sublattice position and sublattice position , respectively. Now, using the Fourier representation for these operators, the previous Hamiltonian can be written as OlivaLeyva and Naumis (2013)
(3) 
which finally leads to the closed dispersion relation for graphene under uniform strain OlivaLeyva and Naumis (2013)
(4) 
It can be stressed that in our work, the latest equation has been developed more explicitly, which leads to
(5) 
where
and
Let us now explore the strain effects on the energy dispersion relation. As it was previously explained in the introduction, when an uniform strain is applied the reciprocal lattice is also strained. Thus, the first BZ is modified, i.e., its original hexagonal form is varied to a polygonal form, as shown for the particular cases of pure shear strain along the armchair direction: , (see red lines Fig. 2 a)) and (see red lines Fig. 2 b)). In the same figure 2, along with the first BZ, we present the contour plot of the energy obtained from Eq. (5).
Once the first BZ and the energy surfaces are obtained, we need to locate the Dirac points using the condition , which corresponds to electrons at the Fermi level. By applying this condition, the Dirac points are indicated as pink circles in Fig. 2. The most important conclusion from the figure is that such points are no longer located at the high symmetry points and (red circles) of the corners of the first BZ (Eq. 1), since they are shifted to the saddle point. This plot ilustrates an issue that has been disregarded in the literature concerning graphene.
As deformation increases (up to 20), the Dirac points merge into the saddle point and a gap opens, which is consistent with the results in references Pereira et al. (2009); Montambaux et al. (2009). Furthermore, in this critical point, the dispersion relation is linear along one direction ((relativistic Dirac behavior) and quadratic along the other one (nonrelativistic Schrödinger behavior). Therefore, the Dirac theory needs to be modified.
Summarizing the above, the effects caused in graphene under uniform strain are the following:

i) The Dirac points are shifted from the corners of the strained BZ.

ii) The Dirac equation is no longer suitable for long strain (), since for particular cases a DiracSchrödinger behavior is observed and furthermore one might expect significant nonlinear corrections. It follows that the anisotropic Fermi velocity is not longer valid in these regimens. Therefore, we must consider a more general velocity to understand the electron behavior. This is done through the calculation of the group velocity, as we will discuss in the following section.
Iii Group velocity
In the literature, the basic properties of electron transport phenomena in a crystal are described in terms of Bloch waves with wave vectors Mizutani (2001). Using these waves, we can build a dispersive wave packet with a certain group velocity. It can be shown quite generally that the mean electron velocity is given by the group velocity of the wave packet Mizutani (2001)
(6) 
where is the gradient operator in space. From this equation, the realspace motion of the electron can be described. Here we are interested in the behavior of electrons in graphene under uniform strain. Thus, by substituting Eq. (5) into Eq. (6) we obtain
(7) 
The components and of are given by
(8) 
where and . The group velocity norm is given by .
In the Fig. 3 we plot the surfaces and contour of the group velocity norm for pure graphene. It is important to note that at low energies and in the vicinity of the Dirac points, is isotropic, coinciding with the Fermi velocity. However, as we move away from the Dirac point (corresponding to nonlinear corrections to the Dirac cone) a trigonal warping appears, giving rise to an anisotropic behavior (see contour Fig. 3). Furthermore, it can be observed that in the directions where the trigonal warping appear, increases, while in other directions decreases drastically. These results do not appear when the Dirac theory is used. Therefore, if we want a more complete understanding of the behavior of electrons in the energy bands, nonlinear corrections and directions should be considered, since strain effects enhance such features, as we discuss below.
We analyze the particular cases of pure uniaxial and pure shear strain with , and , , as shown in Fig. 4 and Fig. 5, respectively. In these Fig. 4 and Fig. 5, the contours plots of the velocity norm are presented. Overimposed to these contours, we present the first BZ of the strained reciprocal lattice with red lines. Likewise, the high symmetry points (Eq. 1) are indicated with red circles and the pink circles correspond to the position of the Dirac point where the Fermi energy is located. Notice again how the points do not have the same position as the high symmetry points and . The effect for shear strain is much more pronounced.
On the other hand, the effects caused by the deformation in the velocity surfaces are the following:

ii) The surfaces do not display the trigonal symmetry, and instead, they present the symmetry of the corresponding strained reciprocal lattice.

iii) The trigonal warping observed in pure graphene disappears, as in (Fig. 4 a)). It reappears in (Fig. 4 b)) but with deformed angles which follow the symmetry of the strained reciprocal lattice. This new warping is strongly modulated, and it touches the Dirac points (Fig. 4 b)). Since the warping is associated with nonlinear terms, this suggests that nonlinearity is important in order to describe such cases. As a result, a pure Dirac equation is not longer valid. For the shear strain, in Fig. 5 a) the trigonal warping is deformed, until in Fig. 5 b) becomes two lobules reflecting the symmetry of the strained reciprocal space.
Iv Conclusions
We have obtained the electron’s group velocity for graphene under uniform strain using the tightbinding approximation. Our results indicate that the velocity is strongly anisotropic and that the trigonal warping is deformed to follow the symmetry of the strained reciprocal lattice. As strain increases, this warping touches the Dirac points. Thus, we found that nonlinearity is very important in order to describe electrons in a proper way near the Fermi energy for strain, since the trigonal warping observed in graphene touches the Dirac point and gets modulated by the symmetry of the strained reciprocal lattice. As a result, a Dirac equation kind of approximation is not longer valid for such cases.
Finally, our closed analytical expressions for the electron velocities are useful in order to calculate electronic, thermal and optical properties of strained graphene.
Acknowledgments
G. Naumis thanks the program DGAPAPASPA for a sabatical shoolarship. W. GómezArias thanks CONACyT for a master schoolarship. This work was funded by DGAPAPAPIIT proyect 102513. Calaculations were performed at DGTICUNAM supercomputer center.
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