Abstract— In this study, using tight binding model a simple analytical approach has been proposed to investigate the energy dispersion of graphene under the conditions of different planner strain distribution. Here the change in the angle between the primitive unit vectors due to application of external strain has been taken into consideration to propose the approach. From our proposed model it is found that graphene under relaxed or symmetrical strain distribution is a zero bandgap semiconductor. However a band gap is opened as the asymmetrical strain is applied to it. It is seen that upto a certain level of strain (i.e. 12.2 % parallel to carbon-carbon bond and 7.3% perpendicular to carbon-carbon bond) the band gap of graphene increases and then begin to fall . So, four different assumptions have been made for angular change of primitive unit vectors for four different regions of applied strain (i.e. before and after the strain of 12.2 % parallel to carbon-carbon bond & before and after the strain of 7.3% perpendicular to carbon-carbon bond). The result obtained in the present study are compared and found an excellent agreement, with more or less 96% accuracy with that of determined from first principle technique.
Keywords—Graphene, planner strain, tight binding model, energy dispersion, band-gap. I. INTRODUCTION
Graphene, a strictly two-dimensional material having unusual and interesting properties  is a rapidly rising star on the horizon of material science and condensed matter physics. It is a material of interest in semiconductor industry because of its exceptionally high crystal and electronic quality, excellent transport properties (i.e. high electron mobility  and high thermal conductivity), and as it is planner, it is capable of extreme device scaling comparing with silicon technology. However these excellent properties are associated with a major drawback; graphene is a zero bandgap semiconductor or semimetal -. For large scale manufacturing, the absence of bandgap is the most difficult engineering issue to solve. The zero bandgap revels that it is impossible to switch graphene based device from the conductive to the nonconductive state. So it can not be used in the logic circuit.
As the zero bandgap property of graphene limits its application in practical fields, scientists are working to find out the methods to open the bandgap in graphene. To solve this problem several methods have been proposed, such as graphene nanoribbin using quantum confinement effect in its transverse direction -, bilayer graphene introducing symmetry breaking between two carbon layers via an external electric field , , by the process of doping - and by the process of external strain ,.
To investigate the bandgap opening by the above methods, several techniques have been applied for calculating the band structure of graphene such as first principal calculation, tight binding modeling, k.p method etc. All of them are performed earlier using the software simulation or numerical techniques, which require a huge computational complexity and time consuming and need high capacity super computer. In our study we have proposed a simple analytical approach to investigate the energy dispersion of graphene under different planner strain condition. Using the proposed method the bandgap opening is calculated under the application of asymmetrical strain parallel and perpendicular to the carbon-carbon bond in graphene. The results obtained from the proposed method is compared with the result published by the first principle method and found to be in good agreement with more or less 96 % accuracy.
Graphene is a honeycomb lattice of regular hexagonal structure. But it loses its regular hexagonal structural symmetry under uniaxial/shear strain. When planar stress is applied to graphene, the position of carbon atoms shift relative to each other. As a result the vector position of lattice point changes. To explain this, the angle between a1 and a2 is considered here as θ instead of assuming 60o which is true for ideal or relaxed graphene structure. The eﬀect in the tight-binding Hamiltonian is that the parameters of tight-binding scales changes accordingly. The strained lattice structure of graphene is shown in Fig.1. We have used the simple nearest Neighbor tight binding model. Here each Carbon atom is
σ bonded with three of its nearest neighbor Carbon atoms.
Fig.1 : The direct lattice structure of graphene under strained condition The primitive unit vectors can be represented by
The separation of the carbon atoms (A and B) can be represented by three vectors R1, R2, R3
From Tight-binding energy dispersion model the formula of energy dispersion is given by 
Here is a fitting parameter which is often called the nearest neighbor overlap energy or hopping integral. The value of varies from 2.7eV to 3.3eV.
(2) This is the generalized equation for the energy dispersion of graphene. Here is the angle between the primitive unit vectors. For the unstrained or relaxed condition, the value of the angle = 60o. In this case the π bands overlap at direct point or K point of the two dimensional brillouin zone.
Fig.2(a) energy dispersion of relaxed graphene and (b) the corresponding brillouin zone.
We have investigated the electronic structure of graphene under different planar strain distributions by the tight-binding (TB) approach. The graphene has been strained in three different ways . These are : (i) symmetrical strain distribution (keeping the hexagonal symmetry unchanged) as shown in fig. 3.1(a) , (ii) asymmetrical strain distribution parallel to
C-C bonds as shown in Fig. 3.1(b) , (iii) asymmetrical distribution perpendicular to C-C bonds as shown in Fig.3.1(c).
Fig 3(a) Graphene system with symmetrical strain distribution, (b) asymmetrical strain distribution perpendicular to C-C bonds, and (c) asymmetrical strain distribution parallel to C-C bonds. Corresponding primitive cells in black, reciprocal lattices in green dashed and Brillouin zones in green grey are illustrated below the deformed lattices. Γ, K, M, R and S are the high symmetrical points. Lx and Ly are the half of the diagonal lengths of the primitive cells in parallel and perpendicular direction of the carbon-carbon bond.
As the strain is applied to the graphene, it causes the deformation of the regular hexagonal structure of it . It also causes the deformation in the primitive unit cell. If the strain is symmetric then the band property of the system does not change but for asymmetrical strain , the band property of the system changes due to symmetry breaking. When an asymmetrical strain parallel to C-C bond is applied, it causes a deformation in the primitive unit cell. This deformation is taken as a change in angle between the primitive unit vectors. Here the strain is applied upto 12.2 % and it is seen that with the increase in strain the angle between the primitive unit vectors is reduced by following a 3 degree polynomial with respect to Lx and Ly(where Lx and Ly are in nanometer). The equation of is
(3) This value of is then put in equation (2) to calculate the band gap under different strain distribution .
It is seen that up to Ly =0.2396 nm band gap of graphene increases then the bandgap begin to fall . For this region the assumption of is different and it is,
(4) In case asymmetrical applied strain perpendicular to C-C bond , up to 7.3 % strain the angle between the primitive unit vectors is increased by
following a 2 degree polynomial with respect Lx and Ly. The equation of is,
Now up to Lx = 0.1323 nm band gap of graphene increases and then the bandgap begins to fall. For this region the assumption of is,
Asymmetrical strain distribution results in the opening of the bandgap between the maximum of the valance band and the minimum of the conduction band in graphene. When an asymmetrical strain parallel to carbon-carbon bond is applied, Ly increases. Then for the system in order to come back to its lowest energy, Lx decreases during the structural relaxation. Due to change of Lx and Ly, the angle between the primitive unit vectors decreases and causes the symmetry breaking. This angular change is taken as the parameter of deformed primitive cell to calculate the electronic structure of graphene. For example, for Ly = 0.2196, 0.2236, 0.2396, and 0.2436 nm the corresponding optimized values of Lx are Lx= 0.1228, 0.1224, 0.1217 and 0.1216 nm. Then from our proposed model the corresponding angle between the primitive unit vectors are =59.47o, 58.91o, 54.79o and 57.75o. The corresponding electronic structure or band diagrams are shown in fig.4 with the extended view at K point
Fig.4 Extended view of bandgap opening for (a) Ly=0.2196 nm and Lx=0.1228 nm (b) Ly=0.2236 nm and Lx=0.1224 nm (c) Ly=0.2396 nm and Lx=0.1217 nm (d) Ly=0.2436 nm and Lx=0.1216 nm. Similar behavior is obtained in the
graphene system, when asymmetrical strain perpendicular to carbon-carbon bond is applied. In this case for example for Lx =0.1268, 0.1292, 0.1353 nm the corresponding optimized Ly are Ly=0.2126, 0.2120 and 0.2105 nm and the corresponding deformed angle are = 60.52o, 61.05oand 60.38o. The opening of bandgap corresponding to these deformed angle are shown in fig.5
FIG.4 EXTENDED VIEW OF BANDGAP OPENING FOR (A) LX=0.1268 NM AND LY= 0.2126 NM (B) LX= 0.1292 NM AND LY=0.2120 NM (C) LY=0.1353 NM AND LX= 0.2105 NM . These results revels that the zero bandgap or semi-metallic behavior of graphene sheet gets modified or a bandgap is opened when asymmetrical strain is applied to it. Now the question is what is the reason behind this? We know that planner graphene consists of strong bonds and delocalized pz electrons. Here orbitals are formed by overlapping the pz orbitals of the carbon atoms in the hexagonal lattice. These and bands touches each other at the K point causing zero bandgap.
When the strain is applied the carbon-carbon bonds of graphene get elongated. Due to this elongation of carbon-carbon bonds, the electron clouds get localized on the corresponding carbon atoms. Therefore a restriction is imposed on movement of the electrons which causes the opening the bandgap at the K point. In this way, under strained condition, graphene loses its semimetal characteristics and turns into a direct bandgap semiconductor.
IV.COMPARISON WITH ESTABLISHED METHOD:
In this study, we have calculated the band gap of graphene under the application of asymmetrical strain by an analytical approach. The calculated value is found to be in great agreement with the measured value obtained by first principle calculation . It is found that in case of asymmetrical strain distribution parallel and perpendicular to C-C bond, the percentage of error is more or less 4 % for most of the data which is shown
In this study an analytical model has been proposed for investigating the energy dispersion of strained graphene under the distribution of the planner strain in parallel and perpendicular to carbon-carbon bond. Using the proposed model the energy dispersion for different planner strain has been calculated. From our study it is seen that for relaxed or symmetrically strained graphene, the band gap of the system is zero. When asymmetrical strain parallel or perpendicular to carbon-carbon bond is applied then a finite bandgap is opened. And it is also seen that tuning the percentage strain, the band gap can also be tuned. Here it is also seen that comparing the perpendicular strain, parallel strain parallel to the carbon -carbon bond more induces more bandgap. Finally we have compared our results with the results obtained from the established method and found good agreement with around 96 % accuracy. References
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