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DOI: 10.1039/C5RA03422A (Communication) RSC Adv., 2015, 5, 43810-43814

Ab initio strain engineering of graphene: opening bandgaps up to 1 eV

Nicolas Kerszberg† and Phanish Suryanarayana†*
College of Engineering, Georgia Institute of Technology, USA. E-mail: phanish.suryanarayana@ce.gatech.edu; Tel: +1-404-894-2773

Received 25th February 2015 , Accepted 5th May 2015

First published on 7th May 2015

Abstract

We employ electronic structure calculations based on Density Functional Theory (DFT) to strain engineer graphene's bandgap. Specifically, working in the finite deformation setting, we traverse the three-dimensional in-plane strain space to determine states capable of opening significant bandgaps in graphene. We find that biaxial strains comprising of tension in the zigzag direction and compression in the armchair direction are particularly effective at tuning graphene's electronic properties, with resulting bandgaps of up to 1 eV. Notably, we ascertain that a 11% strain in the zigzag direction in combination with −20% in the armchair direction produces a bandgap of approximately 1 eV. We also establish that uniaxial and isotropic biaxial strains of up to ±20% are incapable of opening bandgaps, while shear strains of ±20% can introduce bandgaps of around 0.4 eV.

Graphene is a two-dimensional allotrope of carbon consisting of atoms arranged in a honeycomb lattice.1,2 Graphene's great potential for technological applications arises from its fascinating mechanical,3,4 electronic,5,6 optical7,8 and thermal9,10 properties. Among the notable electronic features are an extremely high charge mobility11,12 and unusual quantum Hall effect.13,14 However, the prospect of graphene being used in electronic devices like transistors is severely hindered by the absence of a bandgap.15,16 This has motivated a number of efforts to engineer a bandgap in graphene, including epitaxial growth,17,18 quantum confinement,19,20 application of electric fields in bilayered graphene,21,22 and utilization of strains.23–28 Amongst these, strain engineering is a particularly attractive option because of graphene's ability to sustain large strains of up to 25–27% before failure.3,29 This provides the motivation for the current work.

Gui et al.23 employed Density Functional Theory (DFT)30,31 calculations to conclude that while isotropic biaxial strains are unable to open a bandgap, uniaxial strains of 12.2% in the armchair direction can produce a gap of 0.486 eV. Using first principles calculations, Ni et al.24 initially surmised that bandgaps of 0.3 eV can be opened with less than 1% uniaxial strain. However, upon closer examination, they realized that uniaxial strains much larger than 20% are required to open a bandgap. This prediction is in agreement with tight-binding simulations of Pereira et al.26 and DFT calculations of Choi et al.25 The initial misinterpretation of the results was attributed to the shift of the Dirac point—point of vanishing density of states where the valence and conduction bands touch conically—from the high symmetry point in the Brillouin zone.

Cocco et al.27 utilized the tight-binding method to predict that bandgaps of up to 0.9 eV can be obtained in graphene by applying uniaxial strains in combination with shear strains. However, Naumov and Bratkovsky28 concluded that homogeneous in-plane strains are incapable of opening bandgaps below the breaking point. Additionally, they demonstrated that it is possible to introduce bandgaps by applying a sine-like inhomogeneous deformation in any direction other than armchair, with the zigzag direction being the most effective. Recently, Gui et al.32 showed using DFT calculations that localized strains in the zigzag direction can open bandgaps in the range of 0.1 to 1.0 eV. In this work, we demonstrate using DFT calculations that it is indeed possible to open a bandgap in graphene using homogeneous in-plane deformations. In fact, there exist certain regions of the strain space where relatively large bandgaps of up to 1 eV can be opened.

The honeycomb structure of graphene can be viewed as a triangular lattice with a two atom basis.5 We sketch this hexagonal configuration in Fig. 1 along with its lattice vectors

 
image file: c5ra03422a-t1.tif(1)
where a0 denotes the carbon–carbon bond length. The resulting unit cell has two atoms with fractional coordinates of image file: c5ra03422a-t2.tif. The corresponding reciprocal lattice is also honeycomb with lattice vectors
 
image file: c5ra03422a-t3.tif(2)
The first Brillouin zone is as shown in Fig. 1, with the area enclosed by Γ–K–M–Γ defining the irreducible region. The point Γ has coordinates {0, 0}, and denotes the center of the Brillouin zone. The high symmetry points K, M, K′, M′, K′′, M′′ and K′′′ have fractional coordinates image file: c5ra03422a-t4.tif respectively. The unstressed graphene lattice, i.e. bond length a0 minimizes the energy, is chosen as the reference configuration with respect to which all the deformations and strains are defined.


image file: c5ra03422a-f1.tif
Fig. 1 Honeycomb lattice on the left and its first Brillouin zone on the right. The x and y axes correspond to the zigzag and armchair directions respectively. The lattice vectors are denoted by a1 and a2. The reciprocal lattice vectors are represented by b1 and b2. The parallelogram formed by a1 and a2 represents the unit cell.

Since graphene is capable of accommodating very large strains, utilizing finite-deformation theory33 is essential for performing an accurate analysis. To this end, we introduce the deformation gradient

 
F = ∇f(x), (3)
where f(x) denotes the homogeneous in-plane deformation of the unstressed graphene lattice. Here, x ∈ [Doublestruck R]2 represents the position of any carbon atom in the reference configuration. The second-order Lagrangian strain tensor can be expressed in terms of F as
 
image file: c5ra03422a-t5.tif(4)
where I ∈ [Doublestruck R]2×2 is the identity matrix and [scr E, script letter E]yx = [scr E, script letter E]xy. The eigenvalues and eigenvectors of E are referred to as the principal strains and directions respectively.

In this work, we will consider homogeneous deformations of graphene's honeycomb lattice, i.e. F is independent of x. The lattice vectors of such a strained system are

 
a′1 = Fa1, a′2 = Fa2, (5)
and the corresponding reciprocal lattice vectors can be shown to take the form
 
b′1 = F−Tb1, b′2 = F−Tb2. (6)
For a generalized state of strain, the symmetry of the system reduces from 6/mmm hexagonal to 2/m monoclinic. As a result, the irreducible Brillouin zone is now given by the area enclosed within Γ–K–K′–K′′–K′′′–Γ. However, the fractional coordinates of atoms in real space as well as all points in reciprocal space are independent of the applied deformation.

On homogeneously deforming graphene's lattice as described above, the atoms in the strained unit cell experience a net force. If these forces are equilibrated, though the triangular sub-lattices still possess the prescribed uniform state of strain, the overall honeycomb lattice becomes non-uniformly strained. Since we are interested in the effect of homogeneous deformations on graphene's bandgap, we do not relax the positions of the atoms in the simulations described below. This is representative of practical scenarios where the movement of the atoms is constrained, e.g. when substrates are utilized to impart the desired strain state.24,34,35 In such situations, we also anticipate a significant reduction in graphene's susceptibility to buckling under compressive strains.

In the above setting, we perform Density Functional Theory (DFT)30,31 based electronic structure simulations using the software ABINIT.36,37 For a given strain tensor E, we calculate the deformation gradient F using the relation (eqn (4))

 
image file: c5ra03422a-t6.tif(7)
which is evaluated in practice through eigen decomposition of the matrix (2E + I). Unless specified otherwise, we choose the local density approximation (LDA)31 with the Perdew–Zunger38 parameterization of the correlation energy calculated by Ceperley–Alder.39 Further, we employ the Troullier and Martins40 norm-conserving nonlocal pseudopotential. The electronic ground-state is determined using a plane-wave energy cutoff of 34 Hartree, a 20 × 20 × 1 Monkhorst–Pack k-point grid41 and a supercell such that the spacing between the graphene layer and its periodic images is 30 Bohr. With this choice of parameters, the energy is found to be converged to within 0.002 eV per atom.

In typical ab initio calculations, once the ground-state is established, the bandgap is determined by evaluating the electronic band structure along certain high-symmetry directions.42 For the strained graphene lattice, this corresponds to the Γ–K–K′–K′′–K′′′–Γ circuit. However, in the presence of strains, particularly those containing shear components, such an approach can result in spurious predictions.26,28 In view of this, we calculate the band structure at 100[thin space (1/6-em)]000 k-points that are randomly generated within the irreducible Brillouin zone. The estimate for the bandgap so determined is further improved by zooming into the identified region of interest and utilizing an additional 10[thin space (1/6-em)]000 random k-points. We have found the bandgaps so determined to be accurate within 0.01 eV.

We now employ the DFT simulations described above to ascertain the influence of homogeneous in-plane deformations on the bandgap of graphene. As the first step, we determine the reference configuration by relaxing the dimensions of the unit cell while maintaining its geometry. We find that the carbon–carbon bond length of a0 = 2.66 Bohr minimizes the energy, with the resulting stresses having magnitude smaller than 0.005 GPa. This value of a0 is in good agreement with previous such predictions43,44 as well as experimental observations.45,46 We define all future deformations and strains with respect to this equilibrium honeycomb lattice, which we have found to be semi-metallic with a zero bandgap.

First, we study the effect of the most fundamental strains—uniaxial ([scr E, script letter E]xx = [scr E, script letter E] or [scr E, script letter E]yy = [scr E, script letter E]), isotropic biaxial ([scr E, script letter E]xx = [scr E, script letter E]yy = [scr E, script letter E]) and shear ([scr E, script letter E]xy = [scr E, script letter E]yx = [scr E, script letter E])—on the bandgap of graphene. In Table 1, we present the bandgaps obtained by varying [scr E, script letter E] in steps of 0.05 within the interval −0.20 to 0.20. It is clear that both uniaxial and isotropic biaxial strains are unable to open a bandgap of any significance. Shear strains fare only marginally better, with a noticeable bandgap only appearing at [scr E, script letter E] = ±0.20. Specifically, an indirect bandgap of Δ = 0.401 eV opens up between the k-points M and M′′. The inability of uniaxial, isotropic biaxial and shear strains to open a bandgap is in agreement with previous conclusions.25,26,28 The difference in the prediction for 20% shear can be attributed to the use of finite deformation theory here compared to infinitesimal theories employed previously. Overall, we find that uniaxial and isotropic biaxial strains are by themselves unsuitable for opening bandgaps in graphene.

Table 1 Graphene's bandgap (Δ) in eV for uniaxial, isotropic biaxial and shear strains. The subscripts of Z and A correspond to zigzag and armchair directions respectively. Omitted strain components are implied to be zero
[scr E, script letter E] UniaxialZ UniaxialA Biaxial Shear
[scr E, script letter E]xx = [scr E, script letter E] [scr E, script letter E]yy = [scr E, script letter E] [scr E, script letter E]xx = [scr E, script letter E]yy = [scr E, script letter E] [scr E, script letter E]xy = [scr E, script letter E]yx = [scr E, script letter E]
−0.20 0.005 0.005 0.023 0.401
−0.15 0.012 0.003 0.013 0.006
−0.10 0.006 0.008 0.002 0.012
−0.05 0.010 0.014 0.003 0.007
0.00 0.000 0.000 0.000 0.000
0.05 0.014 0.008 0.010 0.003
0.10 0.007 0.001 0.012 0.012
0.15 0.008 0.014 0.010 0.007
0.20 0.007 0.006 0.001 0.401

Next, we traverse the complete three-dimensional strain space in search for homogeneous deformations capable of opening significant bandgaps in graphene. Specifically, we consider all possible combinations of [scr E, script letter E]xx, [scr E, script letter E]xy = [scr E, script letter E]yx and [scr E, script letter E]yy in steps of 0.05 within the range of −0.20 to 0.20. In Table 2, we list all the strain states (excluding those related by symmetry) that introduce a bandgap of Δ > 0.4 eV and that have principal strains of magnitude less than 0.20. Interestingly, we observe that all the strain tensors possess one tensile and one compressive principal strain, with the associated principal directions being close to zigzag and armchair respectively. This is indeed highlighted by the fact that the largest bandgap of Δ = 0.923 eV occurs for the biaxial strain state having [scr E, script letter E]xx = 0.10, [scr E, script letter E]yy = −0.20. From these observations, we surmise that biaxial strains with tension along the zigzag direction and compression along the armchair direction are the ideal candidates for opening large bandgaps in graphene.

Table 2 Strain states with bandgap Δ > 0.4 eV discovered during the traversal of the three-dimensional strain space. All bandgaps are direct and occur at the listed k-point
[scr E, script letter E]xx [scr E, script letter E]xy [scr E, script letter E]yx [scr E, script letter E]yy Δ (eV) k-point
−0.15 −0.10 −0.10 0.10 0.411 M
−0.15 −0.10 −0.10 0.15 0.605 M
−0.10 −0.15 −0.15 0.05 0.544 M
−0.05 −0.15 −0.15 0.05 0.434 M
0.10 0.00 0.00 −0.20 0.923 M′

In view of the above finding, we now consider biaxial states of strain with tension and compression along the zigzag and armchair directions respectively. Specifically, we discretize the two-dimensional strain space bounded by 0 ≤ [scr E, script letter E]xx ≤ 0.20 and −0.20 ≤ [scr E, script letter E]yy ≤ 0 with a step size of 0.01 in each direction. We plot a contour of the bandgaps so obtained in Fig. 2, wherein all the nonzero bandgaps are direct and appear at the k-point M′. It is clear that there is a significant portion of the strain space where noticeably large bandgaps are opened. In fact, the largest bandgap of Δ = 1.009 eV appears for the strain state: [scr E, script letter E]xx = 0.11, [scr E, script letter E]yy = −0.20. We plot the corresponding band structure diagram in Fig. 3. It can be deduced that on the application of the aforementioned strains, the Dirac points originally positioned at K′ and K′′ merge at the k-point M′. In order to determine the effect of the aforementioned biaxial strains on the nature of the bonding, we plot the electron density contours for the undeformed and deformed unit cells in Fig. 4. We observe that there is a significant increase in the electron density between the two atoms. This highly preferential bonding in the armchair direction could provide a possible explanation for the appearance of relatively large bandgaps.


image file: c5ra03422a-f2.tif
Fig. 2 Bandgap (Δ) contours for biaxial strains comprising of compression and tension in the armchair and zigzag directions respectively.

image file: c5ra03422a-f3.tif
Fig. 3 Band structure plot for the strain state: [scr E, script letter E]xx = 0.11, [scr E, script letter E]yy = −0.20. The bandgap Δ = 1.009 eV.

image file: c5ra03422a-f4.tif
Fig. 4 Electron density contours in the unit cell, with the white circles denoting the atoms. Left: undeformed graphene. Right: homogeneously deformed graphene with strain state of [scr E, script letter E]xx = 0.11, [scr E, script letter E]yy = −0.20.

Finally, we verify that the salient predictions of this work are not a consequence of the two major approximations associated with DFT simulations, i.e. the pseudopotential and the exchange–correlation energy. We choose the strain state for which the largest bandgap of Δ = 1.009 eV was obtained (i.e. [scr E, script letter E]xx = 0.11, [scr E, script letter E]yy = −0.20) as the representative example. On replacing the LDA functional with the Perdew–Burke–Ernzerhof Generalized Gradient Functional (GGA),47 we find the predicted bandgap to be Δ = 0.775 eV. When the Goedecker–Teter–Hutter pseudopotential48 is used in partnership with the LDA, we obtain Δ = 0.782 eV. It can therefore be inferred that the results presented in this work are an accurate representation of the electromechanical response of graphene.

In this work, we have demonstrated through ab initio calculations that electronic properties of graphene can be effectively tuned through the judicious application of homogeneous in-plane deformations. In particular, we have shown that biaxial strains with tension in the zigzag direction and compression in the armchair direction are ideally suited for this purpose, with a notable portion of this strain space opening significant bandgaps. In fact, we have found that a gap of around 1 eV can be opened using a combination of 11% strain in the zigzag direction and −20% in the armchair direction. Overall, we conclude that homogeneous strains represent a powerful tool for controlling the electronic properties of graphene, and possibly other two-dimensional materials. A promising approach for achieving this in practice appears to be through substrate induced deformations,35 making it a worthy subject for future research.

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Footnote

† These authors contributed equally to this work.
This journal is © The Royal Society of Chemistry 2015
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