Strain Effect on the Electronic Properties of Single Layer and Bilayer

Article pubs.acs.org/JPCC

Strain Effect on the Electronic Properties of Single Layer and Bilayer Graphene Jen-Hsien Wong,† Bi-Ru Wu,*,‡ and Ming-Fa Lin*,† †

Department of Physics, National Cheng Kung University, Tainan, Taiwan Department of Natural Science, Center for General Education, Chang Gung University, Taoyuan, Taiwan

‡

ABSTRACT: This paper investigates strain effects on the electronic properties of single-layer and bilayer graphene using a first-principles method. The deformation significantly alters energy dispersion, band overlap, band gap, and the band edges of graphenes. Fermi velocity behaves both linearly and nonlinearly with the strains, depending on the types of deformation and the direction of the Fermi velocity. In bilayer graphene, the uniaxial strain enhances the band overlap by 2 orders of magnitude. A semimetal−insulator transition occurs when bilayer graphene is under a compressive uniaxial strain along the zigzag chain direction. These strain-dependent results are useful for acquiring the intralayer and interlayer atomic relations or Slonczewski−Weiss−McClure parameters. The intralayer coupling γ0 under the H-strain and interlayer couplings γ1, γ3, and γ4 under the P-strain decrease dramatically as the strain increases. Nevertheless, interlayer couplings vary more slowly with the H-strain than the P-strain.

1. INTRODUCTION Graphene has attracted considerable attention and has been the object of extensive study since its first successful synthesis in 2004.1−16 Because of its hexagonal symmetry, graphene possesses many fascinating properties, such as an extremely high mobility of the charge carriers,2,6 an anomalous quantum Hall effect,2,3,7 and ambipolar transport phenomena.1,7−10 Graphene offers great flexibility for device design: it can be cut or patterned into any shape by using lithographic techniques.17,18 It is thus a promising material for future electronic devices. In particular, the electronic structure in the low-energy region determines the electronic and the transport properties of graphene. To further increase its applications, various studies have attempted to modulate the electronic structure of graphene reversibly, using mechanical deformation, the external electric field, or the magnetic field. Mechanical deformation can tune the metallic carbon nanotube so that it becomes semiconductive.19−21 Recent discoveries show that graphene grown on a flexible substrate can stretch reversibly under a strain of up to 18%, whereas between 18 and 30% it can only stretch irreversibly.18 These results suggest that modulating the electronic structure of graphene by strain is feasible. However, investigations of the band modulation of graphene focus mainly on the effects of external fields and impurities. Few studies based on the first-principles method have been considered for the strain effect on graphene, and the results are controversial.22,23 Effects of mechanically deforming graphene remain to be explored in detail. This study investigates the effect of the electronic properties of single layer and bilayer graphenes caused by the homogeneous and uniaxial strains. © 2012 American Chemical Society

Single-layer graphene (SLG) consists of carbon atoms arranged in a honeycomb shape. The hexagonal symmetry yields a pair of linear energy bands intersecting at the Fermi level (EF). The intersections (known as the Dirac points) occur at high-symmetry points K and K′ in the first Brillouin zone (FBZ). The holes and electrons of graphene near the Dirac point behave like massless Dirac fermions.2,11 The carriers have an extremely high Fermi velocity of approximately 106 m/ s.24−27 While adding a layer of graphene, the bilayer graphene exhibits distinct low-energy electronic dispersion with the SLG because of interlayer interaction. Bilayer graphene (BLG) consists of two graphene sheets overlaid in the Bernal type stacking (Figures 1a and 1b). It shares many of the interesting properties of SLG but presents a richer band structure.12−16 An external perpendicular electric field can achieve BLG band gap tuning.13−15 The second layer’s appearance changes the linear dispersion of SLG into the massive parabolic dispersion of BLG, which remains gapless. Two pairs of parabolic bands are in the low-energy region. The pair of parabolic bands near the Fermi level overlaps each other and crosses at two points. The overlap of the conduction and valence bands indicates a domain where electrons and holes coexist. Because of the coexistence of holes and electrons, the charge carrier transport can change from electrons to holes or vice versa using gate voltage.1,7 This band overlap (δE) is proportional to the carrier concentration, and charge carriers dominate ambipolar transport.1 This paper presents a systematic study of the electronic structure modulation of deformed SLG and BLG by using a Received: January 25, 2012 Revised: March 18, 2012 Published: March 21, 2012 8271

dx.doi.org/10.1021/jp300840k | J. Phys. Chem. C 2012, 116, 8271−8277

The Journal of Physical Chemistry C

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avoid interactions between graphenes belonging to neighboring supercells, the thickness of the supercell in the z-direction was tested and a value of 15 Å used in the calculations. Figures 1a and 1b show the BLG structure. All atoms are relaxed until the Hellmann−Feynman force is smaller than 0.005 eV/Å. After geometric optimization, an in-plane lattice constant of 2.447 Å and an interlayer distance of 3.280 Å were obtained for BLG. Researchers considered a two-dimensional homogeneous strain and three types of uniaxial strains for the graphenes (Figures 1c-1e). The homogeneous strain (εH) is defined as εH = (a-a0)/a0, where a (a0) is the deformed (undeformed) lattice constant of the graphene. The homogeneous strain (H-strain) is parallel to the plane of graphene and retains the hexagonal symmetry (Figure 1c). εu = (L-L0)/L0 describes the uniaxial strain, where L and L0 correspond to the deformed and undeformed unit lengths of graphene in a specific direction, u = P, A, and Z for the P-, A-, and Z-strain, respectively. Uniaxial strains include one perpendicular deformation (P-strain) and two types of parallel deformations, denoted as A-strain and Zstrain. The P-strain was only relevant for BLG; its deformation is perpendicular to the plane of BLG, and its influence modifies the interlayer distance. Similar to the H-strain, BLG under the P-strain retains its hexagonal symmetry (Figure 1c). The Astrain resulted in deformation along the armchair chain direction and parallel to the C−C bonds (Figure 1d), whereas the deformation of the Z-strain was evident along the zigzag chain direction and perpendicular to the C−C bonds (Figure 1e). Positive and negative values of strains represent tensile and compressive strains, respectively. These calculations consider tensile and compressive strains of up to 36% and 24%, respectively. Based on the tight-binding model,28,29 SWMc parameters include the intralayer atomic interactions of graphene γ0 and the interlayer atomic interactions of BLG γ1, γ3, and γ4 (Figure 1b). The SWMc Hamiltonian for SLG and BLG are given in eq 1 and eq 2, respectively

Figure 1. Structure and the FBZ of single layer and bilayer graphenes. The four types of strains on BLG shown in (a), where the arrows denote the deformation direction of the compressive P-strain and the tensile H-, A-, and Z-strains. SWMc parameters are defined in (b). The unit cell and the FBZ of SLG and BLG under the H- and the P-strain is plotted in (c), and a tensile A- and a tensile Z-strain are depicted in (c) and (d), respectively.

⎡ 0 γ f⎤ 0 ⎥ HSLG = ⎢ ⎢⎣ γ0f * Δ ⎥⎦

first-principles method. There are four types of deformation: one homogeneous and three uniaxial strains. The character of the band structure was strongly dependent on the type of deformation. Strain effects are more significant for the low energy electronic properties, especially the band edge states of BLG. The tight binding Slonczewski−Weiss−McClure (SWMc) parameters28,29 are useful to describe the intralayer and interlayer atomic interaction. To explore the interactions of intralayer and interlayer atoms under deformations, the study also investigates SWMc parameters of SLG under the H-strain and of BLG under the H- and the P-strains.

⎡ Δ ⎢ ⎢ γ f* 0 HBLG = ⎢ ⎢ γ ⎢ 1 ⎢ −γ f ⎣ 4

(1)

−γ4f * ⎤ ⎥ 0 −γ4f * γ3f ⎥ ⎥ −γ4f Δ γ0f * ⎥ ⎥ ⎥ γ3f * γ0f 0 ⎦

(2)

f = eikxa / 3 + 2e−ikxa /2 3 cos(k ya /2)

(3)

γ0f

γ1

In these equations, a is the lattice constant of graphenes, and Δ denotes the on-site energy difference between the A and B sites. The Δ is assumed to be zero because the discrepancy of the site energies is tiny. The Fermi velocity of SLG is vF = √3γ0a/2ℏ.

2. COMPUTATIONAL DETAILS Calculations were performed based on density-functional theory30 within the local density approximation.31 The projector-augmented wave potential interpreted electron-ion interactions as implemented in the ab initio VASP code.32−35 Plane waves expanded the wave functions with the energy cutoff at 550 eV. The Monkhorst-Pack scheme was applied for k-point sampling in the FBZ. A 21 × 21 × 1 k-mesh in the FBZ was used for the geometry optimization, whereas a 45 × 45 × 1 k-mesh was used for the electronic structure calculations. To

3. RESULTS AND DISCUSSION All graphenes have a hexagonal structure and the FBZ of the unstrained graphenes is an equilateral hexagon, in which the K and K′ points are at the corners of the hexagon. Graphenes retain their hexagonal symmetry under an H-strain or under a P-strain, thus preserving the hexagonal FBZ (Figure 1c). 8272



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However, an A-strain and a Z-strain break the hexagonal symmetry of graphenes, and so does the FBZ. Figure 1d shows the deformed structure and the FBZ for graphenes under the influence of an A-strain, and Figure 1e displays the deformation of graphene under a Z-strain. 3.1. Electronic Properties of SLG under Strains. Because SLG retains hexagonal symmetry under an H-strain, the band structure of strained SLG preserves the character of the bands without the strain (Figure 2a). This includes a pair of

conduction (valence) band remains at the R point. The subband spacing at the R point increases with the magnitude of the compressive (tensile) A-strain. The sub-band spacing opened by a tensile A-strain is also larger than by a compressive strain. However, the behavior of the energy dispersion near the K and the R points reverse for SLG under a Z-strain (Figures 2d and 2e). 3.2. The Fermi Velocity for SLG under Strains. The definition of a Fermi velocity is vF = (1/ℏ)(dE/dk)|k=kF, where E and kF is energy and the Fermi momentum, respectively. The linear energy dispersion around the Dirac point (Figure 2) reveals the existence of the massless fermion in SLG. The Fermi velocity of the massless fermion is proportional to the slope of the linear energy dispersion. The strains dramatically change Fermi velocity vF of the holes and electrons in SLG because they strongly affect the interaction of carbon atoms. The vF varies linearly with the H-strain, and it changes either linearly or nonlinearly with the A- and the Z-strains. The Fermi velocity in the direction of the K (R) to Γ points is v1F, whereas v2F is the Fermi velocity in the direction of the K (R) to M (S) points (Figure 3 inset). The v1F and v2F both decrease linearly

Figure 2. Band structure of SLG in the low energy region under an Hstrain (a) and the uniaxial strains (b)-(e). (b) and (c) show the band structure around the K and the R points under the A-strain. (d) and (e) display the band structure near the K and the R point under the Zstrain, respectively.

Figure 3. Fermi velocity of SLG near the K and the R points in the FBZ. The Fermi velocity v1F and v2F denote the velocity in the direction of the K (R) to Γ points and the K (R) to M (S) points, respectively. The solid lines and the dashed lines represent v1F and v2F, respectively. The circles, squares, and triangles denote Fermi velocities of SLG under the H-, the A- and the Z-strains, respectively.

linear bands crossing at the Fermi level, the position of the Dirac points and a slight asymmetry of the two linear bands. The asymmetry of the two linear bands is enhanced by a compressive H-strain. The tensile H-strain decreases the slope of the linear band, whereas the compressive one increases it. On the other hand, when SLG is under the influence of an Astrain (Figure 1d) or a Z-strain (Figure 1e), the uniaxial strains break the hexagonal symmetry of SLG. The linear bands change their shape, and the Dirac point either shifts or vanishes. The anisotropy of the two energy dispersion curves is more significant under the uniaxial strains. However, the energy states near the Fermi level still exist around the K point and the R point in the FBZ. Under an A-strain, the bands near the K point intersecting at the Fermi level remain, but the Dirac point shifts away from the K point along the K to Γ path (KΓ path). The shift caused by a tensile A-strain is greater than that caused by the compressive strain (Figure 2b). With an increase in both tensile and compressive A-strains, the shifting range of the Dirac point also increases. In Figure 2c, a gap or sub-band spacing at the R point opens because of the A-strain; thus, the Dirac point vanishes. The local minimum (maximum) of the

as the H-strain increases. Because the slightly asymmetric behavior of the two energy dispersion curves leads the small anisotropy of the Fermi velocity. The calculated Fermi velocities of SLG without strain v1F0 (8.13 × 105 m/s) and v2F0 (8.18 × 105 m/s) agree with the previous studies well.11 The tensile H-strain reduces the anisotropy of the vF and the Dirac cone, whereas the compressive one enhances them. When SLG is under an A-strain (Figure 3), the Fermi velocity v1F near the K point increases linearly with the A-strain (−10% < εA < 10%), whereas both the compressive and the tensile A-strains raise the v2F. Oppositely, the behavior of the v1F and v2F for SLG under a Z-strain reverse. v1F varies linearly with the Z-strains as the strain is greater than −12%. vF can be obtained experimentally from the measured cyclotron resonances.24−27 Experiments can also verify the strain effect on vF. 3.3. Electronic Properties of BLG under Strains. Band structure of unstrained BLG contains two pairs of parabolic bands in the low-energy region (Figure 4a). The two bands near the Fermi level are inner bands, whereas the other pair is named the outer bands. The outer bands are 0.8 eV apart. The 8273



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circumstances, band structure reverts to that of SLG, i.e., the original parabolic bands become linear bands again because the BLG is perceptible as two isolated SLGs. The interlayer interaction of BLG was strongly affected by the P-strain. For a deformed material, the vertical deformation will bring out a lateral deformation. However, the induced lateral deformation of BLG under the P-strain is tiny, and it is homogeneous even though the P-strain is up to 50%. Similar to SLG, the uniaxial nature of the A- and Z-strains alters the low-energy dispersion of BLG significantly. A subband spacing or gap opens at the K point or R point; the band overlap changes dramatically or vanishes. The eye-shaped structure still exists at the Fermi level (Figure 5) near the K

Figure 4. Band structure of BLG around the K point in the FBZ under the H-strain (a)-(b) and under the P-strain (c)-(d). The inset in (b) is the band overlap of BLG, and the two intersections C1 and C2 are denoted. The band overlap δE and the δEcp are defined in (d).

inner bands intersect with the Fermi level and cross each other at two points C1 and C2 (Figure 4b inset). C1 is located on the Fermi level and at the K point, whereas C2 lies above the Fermi level along the KΓ path. The two intersections C1 and C2 have a tiny energy difference δEcp of 0.76 meV. The inner bands have a small overlap δE of around 2.6 meV (Figure 4d). The band overlap region resembles the shape of an eye. Due to this overlap, BLG possesses holes and electrons exhibiting ambipolar properties. Because the band structure in the lowenergy region plays a crucial role in graphene transport phenomena, this study focuses on the band structure near the Fermi level. The BLG preserves hexagonal symmetry under an H-strain or under a P-strain; therefore, its band structure retains its character without strain. The tensile H-strain reduces the curvature of the parabolic bands (Figure 4a), whereas the compressive one increases it. The eye-shaped band overlap is enlarged under a tensile H-strain but shrunk under a compressive strain (Figure 4b). C1 remains at the K point, whereas C2 lies away from (but close to) the K point under a tensile (compressive) H-strain. On the other hand, the outer bands move closer to the Fermi level under a tensile P-strain but shift away from it under a compressive strain (Figure 4c). C1 remains at the K point on the Fermi level for BLG under a P-strain (Figure 4d). When the compressive P-strain is greater than 26%, C1 shifts away from the K point. Nevertheless, C2 is always above the Fermi level, independent of strain magnitude. As the compressive P-strain increases, C2 moves further away from the K point along the KΓ path. In contrast to the H-strain, the eye-shaped band overlap region is enlarged under a compressive P-strain but shrunk under a tensile P-strain. When BLG is under the influence of a tensile P-strain, the eye-shaped region reduces and the band overlap finally vanishes, resulting in a large separation of the two graphene sheets. Under these

Figure 5. Band structure of BLG around the K and R points in the FBZ under the A-strains.

point for BLG under a tensile A-strain. A local maximum in the lowest conduction band and a local minimum in the highest valence band emerge at the K point. The band overlap region enlarges and moves away from the K point along the KΓ path as tensile strain increases. In the bands near the R point, the two intersections C1 and C2 vanish for BLG under the tensile A-strain. Thus, a gap opens at the R point and increases along with the tensile A-strain increasing. When BLG is under a compressive A-strain, the intersection C1 is vanished and a subband spacing opens at the K point. The other intersection C2 moves away from the K point along the KΓ path, shifting downward and away from the Fermi level as the magnitude of the compressive A-strain increases. On the contrary, C2 vanishes and a sub-band spacing opens near the R point, while C1 moves away from the R point along the RS path. C1 shifts upward and away from the Fermi level as the magnitude of the compressive A-strain increases. The eye-shaped structure thus vanishes when BLG is under the influence of a compressive A-strain. However, BLG creates no band gap under an A-strain. The density of states (DOSs) of BLG under the A- and the Z-strains are shown in the bottom of Figures 5 and 6. It reveals that the energy states in the higher energy region are also affected by the strains. The variation of the 8274



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Figure 7. Band gap varies with the compressive Z-strain.

Figure 6. Band structure of BLG around the K and R points in the FBZ under the Z-strains.

higher energy states relates to the optical properties. We will further discuss the strain effect on optical properties elsewhere. Different from SLG, the influences on BLG from a Z-strain and an A-strain are distinct. The eye-shaped structure vanishes and reappears under the influence of a Z-strain. A band gap is also generated. A gap is created at the K point (Figure. 6) and grows with the tensile Z-strain. The gap at the K point is on the Fermi level at first but shifts below the Fermi level as tensile Zstrain surpasses 10%. For the bands near the R point, the eyeshaped band overlap is replaced by two staggered parabolic bands when the tensile Z-strain is lower than 8%. Nevertheless, a tiny indirect band gap of 4.2 meV around the R point exists for BLG under a tensile Z-strain of 8%. The minimum of the conduction band and the maximum of the valence band lie along the RS path. The eye-shaped structure reappears as the tensile Z-strain is between 10 and 18%. On the other side, a gap also opens at the K point by the compressive Z-strain; this gap increases with the magnitude of strain. BLG develops a direct band gap near the R point along the RΓ path. This band gap grows to 50 meV as the magnitude of the compressive Z-strain increases to 12%, after which the band gap decreases with a further continuous supply in strain (Figure 7). The band gap vanishes at a compressive Z-strain of −18% because of the sliding movement of the two graphene layers. 3.4. The Band Overlaps of BLG under Strains. The band overlaps δE and the energy difference between the two intersections δEcp change significantly when BLG is under the influence of the homogeneous and uniaxial strains. The δE and the δEcp increase monotonically from 0.9 and 0.1 meV to 7.5 and 2.9 meV, respectively (Figure 8a), as the H-strain increases from −24% to 24%. Figure 8b is the δE and the δEcp for BLG under the uniaxial P-strain. The δE and the δEcp increase dramatically with the magnitude of the compressive P-strain. The δE and the δEcp for BLG under the compressive P-strain are 10× that without strain. This indicates that the number of charge carriers grows as BLG is under the influence of a compressive P-strain. On the other side, the tensile A-strain also affects the δE and the δEcp intensely. When the tensile A-strain

Figure 8. The band overlap δE and the δEcp under the H-strain, the Pstrain, and the A-strain are illustrated in (a), (b), and (c), respectively.

is increased to 24%, the δE and the δEcp reach 242 and 137 meV, respectively (Figure 8c). Under this circumstance, the magnitude of the δE is several hundred times that without strain. It can be estimated that the carrier concentration of BLG can grow with 2 orders of magnitude under the tensile A-strain. However, the carrier concentration of BLG is still small. 3.5. The SWMc Parameters of Graphenes under Strains. Previous sections have discussed the effect on the electronic properties of graphenes caused by the homogeneous and uniaxial strains. These strain influences are primarily from changing intralayer and interlayer atomic interactions. These strain-dependent results are useful in acquiring intralayer and interlayer atomic relations or Slonczewski−Weiss−McClure parameters. Strain-dependent SWMc parameters are useful for further investigation into other basic physical properties. According to the tight-binding model the strengths of the intralayer interaction and the interlayer interaction can be parametrized by the transfer integral γ0 and γ1, γ3, γ4 (Figure 1b), respectively. 8275



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properties of graphenes. Furthermore, the predicted properties are experimentally quantifiable. For example, Fermi velocity can be measured from cyclotron resonances; the sub-band spacing and energy gap can be examined by angle-resolved photoemission spectroscopy.36−38 SLG under strains retains its linear dispersion and gapless character, but the Fermi velocity changes dramatically. The character of the electronic structure of the undeformed SLG remains under the H-strain. The Dirac point shifts away from the K point along the KΓ path for SLG under an A-strain, while a sub-band spacing opens at the R point. This reverses for SLG under a Z-strain. For the influence of the H- and the P-strains, BLG energy dispersion retains the character of an undeformed graphene. The tensile H-strain enlarges the band overlap, while the compressive one reduces it. On the contrary, band overlap reduces for BLG under tensile P-strain but enhances strongly under compressive P-strain. Moreover, tensile A-strain dramatically increases the band overlap of BLG around the K point and creates a sub-band spacing at the R point. The compressive A-strain opens a sub-band spacing at the K and R points. The strain shifts C2 (C1) away from the K (R) point along the KΓ path and also shifts it downward (upward) away from the Fermi level near the K (R) point. In particular, a semimetal−insulator transition exists for BLG under a compressive Z-strain. Compressive Z-strain creates a small direct gap and increases to 50 meV near the R point as the strain reaches to −12%.

For SLG, the Fermi velocity of the massless Dirac fermion can be expressed as vF = √3γ0a/2ℏ, based on the tight-binding model. The value of γ 0 (in eV) decreases rapidly γ0 = 2.54e−2.55εH as the H-strain increases. This is because strain weakens the intralayer interaction of carbon atoms. The results suggest the interaction of the nearest carbon atoms of SLG decreases exponentially as the H-strain increases. For the H-strain in a small range of −5−5%, the γ0 varies linearly with the small deformation. Because the intralayer interaction of BLG resembles that of SLG, this study focuses on the interlayer interaction of BLG. The interlayer interaction of BLG varies slowly under the Hstrain, because the H-strain mainly changes the intralayer interaction rather than the interlayer interaction. The transfer integrals γ1, γ3, and γ4 vary sublinearly with the H-strain. The relation of the interlayer transfer integrals with the H-strain (εH) in the range of −20−20% find expression in the following γ1 = −0.637εH 2 + 0.466εH + 0.393

(4)

γ3 = −1.26εH 2 − 0.093εH + 0.392

(5)

γ4 = −0.621εH 2 + 0.107εH + 0.167

(6)

where the unit of γs′ are in eV. When the magnitude of both the tensile and the compressive H-strains are less than 8%, 6%, and 5% for γ1, γ3, and γ4, respectively, the γs′ vary linearly with the small deformation. Opposite to the H-strain, the P-strain strongly affects the interlayer interaction but has nearly no influence on the intralayer interaction. The interlayer transfer integrals change dramatically with the compressive P-strain. The relation of the interlayer transfer integrals with the P-strain (εP) in the range of −20−20% finds expression as the following γ1 = 0.394e−5.56εP

(7)

γ3 = 0.380e−4.48εP

(8)

γ4 = 0.163e−4.58εP

(9)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (B.-R.W.), [email protected]. edu.tw (M.-F.L.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the Physics division of NCTS (South), NCHC and the financial support of the Nation Science Council of Taiwan under Grant No. NSC 98-2112-M004-003-MY3 and NSC 98-2112-M-006-013-MY4. B.R.W. also acknowledges the support of Chang Gung University under Grant No. UMRPD590071.

The interaction of the on top atom (A1 and A2 in Figure 1b) belonging to different layer γ1 increases exponentially as the magnitude of the compressive P-strain increases, i.e., decreasing the interlayer distance of BLG. However, the interlayer interaction of the atoms on different sites (B1 and B2, A1 and B2), γ3 and γ4, are less sensitive to the P-strain than γ1. The strain effect on the interlayer interaction of BLG under a Pstrain is larger than that under an H-strain; therefore, the linear response range of the P-strain is smaller than that of the Hstrain. The γs′ vary linearly with the small P-strain, when the magnitude of deformation is less than 2%.

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REFERENCES

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4. CONCLUSIONS This study investigates the low-energy electronic structure of deformed SLG and BLG using a first-principles method and considers one homogeneous strain and three uniaxial strains. For the influence of deformations, the electronic properties of the graphenes change dramatically, such as energy dispersion, the sub-band anisotropic, Fermi velocity, Fermi momentum, sub-band spacing, band overlap, and intralayer and interlayer atomic interactions. It also obtains the tight-binding SWMc parameters. This is useful for investigation of other physical 8276



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Strain Effect on the Electronic Properties of Single Layer and Bilayer

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