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Carbon 117 (2017) 462e472

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Carbon

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Strain gradient polarization in graphene

S.I. Kundalwal a, 1, S.A. Meguid a, *, G.J. Weng b

a Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario M5S 3G8,Canadab Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ 08903, USA

a r t i c l e i n f o

Article history:Received 26 September 2016Received in revised form3 March 2017Accepted 4 March 2017Available online 10 March 2017

Keywords:GrapheneFlexoelectricityPiezoelectricityPolarizationDefectsDensity functional theory

* Corresponding author.E-mail addresses: kundalwal@iiti.ac.in (S.I. Kunda

ca (S.A. Meguid).1 Currently Assistant Professor in the Department

Indian Institute of Technology Indore, Indore 453 552

http://dx.doi.org/10.1016/j.carbon.2017.03.0130008-6223/© 2017 Elsevier Ltd. All rights reserved.

a b s t r a c t

Flexoelectricity phenomenon is the response of electric polarization to an applied strain gradient and isdeveloped as a consequence of crystal symmetry in all materials. In this study, we show that the presenceof strain gradient in non-piezoelectric graphene sheet does not only affect the ionic positions, but alsothe asymmetric redistribution of the electron density, which induce strong polarization in the graphenesheet. Using quantum mechanics calculations, the resulting axial and normal piezoelectric coefficients ofthe graphene sheet were determined using two loading conditions: (i) a graphene sheet containing non-centrosymmetric pores subjected to an axial load, and (ii) a pristine graphene sheet subjected to abending moment. Particular emphases were placed on the role of edge and corner states of pores arisingdue to the functionalization. We also investigated the electronic structure of graphene sheet underdifferent in-plane strain distributions using quantum mechanics calculations and tight-bindingapproach. The findings of our work reveal that the respective axial and normal electromechanical cou-plings in graphene can be engineered by changing the size of non-centrosymmetric pores and radii ofcurvature. Our fundamental study highlights the possibility of using graphene sheets in nano-electromechanical systems as sensors or actuators.

© 2017 Elsevier Ltd. All rights reserved.

1. Introduction

Recent advances in nanoscale technologies have renewed in-terest in flexoelectricity. Large strain gradients present at thenanoscale level may lead to strong electromechanical coupling.Piezoelectricity - electrical polarization induced by a uniform strain(or vice-versa) - is the most widely known and exploited forms ofelectromechanical coupling that exists in non-centrosymmetriccrystals. In centrosymmetric crystals, the presence of the centerof inversion results in the absence of bulk piezoelectric properties.In contrast to piezoelectricity, flexoelectricity can be found in anycrystalline material, regardless of the atomic bonding configuration[1]. The symmetry breaking at surfaces and interfaces in nonpolarmaterials would allow new forms of electromechanical coupling,such as surface piezoelectricity and flexoelectricity, which cannotbe induced in bulk materials. The first step towards a theoretical

lwal), meguid@mie.utoronto.

of Mechanical Engineering,, India.

understanding of flexoelectricity was due to Kogan [2]. It is worthmentioning that the term “flexoelectricity” for crystalline materialswas coined by a similar phenomenon in liquid crystals [3]. Piezo-electric materials are commonly usedwhere precise and repeatablecontrolled motion is required, such as atomic force microscopyprobes, smart structures [4e6], piezotronic devices [7], sensors andactuators, and nanogenerators [8]. The conventional piezoelectricceramics are heavy, brittle, and some pose significant environ-mental risk due to the inclusion of high lead content [9]. Piezo-electric polymers, on the other hand, are light and environmentallybenign, but typically display considerably smaller piezoelectricresponse in terms of actuation. Piezoelectric single-layer hexagonalboron nitride is a promising nanomaterial for nano-electromechanical system (NEMS), but its synthesis is much harderwhich has been a major issue for more than a decade.

The successful synthesis of 2D single layer graphene sheet [10]has attracted considerable attention from both academia and in-dustry. This is due to its unique scale-dependent electronic, me-chanical, and thermal properties [11e14]. It is widely considered tobe one of the most attractive materials of the twenty-first century.Remarkably, high electron mobility and other striking features,such as anomalous quantum hall effect, pseudomagnetic field, and

S.I. Kundalwal et al. / Carbon 117 (2017) 462e472 463

spin transport [15], have made graphene an intensely studiedmaterial from both basic science viewpoint as well as advancedapplications in micro- and nano-electronics [7], spintronic devices,electron lenses, energy storage systems [16], and gas separationmembranes [17]. In view of its unique 2D structure and electricalproperties with zero bandgap, graphene is most suitable for NEMSapplications.

The flexoelectric effect in carbon nanotubes was observed byWhite et al. [18] for the first time in 1993. In their study, the bondsymmetry breaking due to curvature was visible in the electronicproperties of carbon nanotubes (CNTs). A homogeneous me-chanical deformation of graphene cannot induce polarization dueto the symmetry of its lattice. However, second-order electronicflexoelectric effect in graphene can be induced by the presence ofa strain gradient. The strain gradient changes the ionic positionsas well as leads to asymmetric redistribution of the electrondensity. Contrary to 3D systems, graphene are also able to sustainlarge strains up to 25% [13] and, thus can exhibit exceptional formsof electromechanical coupling [19]. Zeinalipour-Yazdiand andChristofides [20] determined the size-dependent Young'smodulus and CeC binding energy in graphene nanoribbons viaelectron first principles computations. They reported linearitybetween the CeC binding energy and Young's modulus, whichwas explained by the decrease in the molecular polarizability dueto deformations within the proportional limit of the graphenenanoribbons. Dumitric�a et al. [21] investigated the normal polar-ization induced by bending of graphite shells, which microscop-ically occurs because of a shift in sp2 hybridization at each atomicsite. Using density functional theory (DFT) calculations for bentnanographitic ribbons made of up to 400 atoms, Kalinin andMeunier [22] predicted the electromechanical coupling, inagreement with the prediction of linear theory for flexoelectriccoupling in quantum systems. Note that the stability of planarcarbon materials is higher than their curved nanostructures (e.g.Fullerenes and CNTs) and this is associated to the existence of theenergy requirements to bend a graphene sheet. Based on averagebond-dissociation-enthalpies, Zeinalipour-Yazdi and Loizidou[23] applied physical sphere-in-contact models to elucidate thecap geometry of single walled CNT with sub-nanometer width(i.e., (3,3), (4,4) and (5,5)-SWCNTs). Their DFT calculations revealthat these new carbon geometries have similar stability to carbonfound in other ball-capped zigzag and arm-chair nanotubes (i.e.,(5,5), (6,6), (9,0) and (10,0)). Recently, experimental and theo-retical investigations reported that creating nanoscale non-centrosymmetric pores in graphene nanoribbons would lead to

Fig. 1. The unit cells of (a) armchair graphene sheet, and (b) zigzag graphene sheet. a1, a2

apparent piezoelectric behavior [24,25]. These exceptional andhighly desirable electromechanical properties have motivated ourinterest in this work. To the best of the authors' knowledge, theaxial and normal flexoelectricity coefficients of graphene have notyet been reported. In this study, we first examined the bandstructures of graphene sheet subjected to different in-plane straindistributions using quantum mechanics calculations and tight-binding (TB) approach. Then, we focused our attention to deter-mining the induced piezoelectric coefficients of graphene withand without non-centrosymmetric pores of different sizes usingquantum mechanics calculations. The outcome of this workshould lead to a greater insight into strain-induced electric po-larization in graphene sheets that would allow exploring theessence of their strong piezoelectric response through differentloadings.

2. TB modeling of electronic structure of graphene

Graphene has unique electronic properties evolving from itshexagonal honeycomb lattice structure, which makes electrons ingraphene behave as massless relativistic fermions that satisfy theDirac equation [26]. In-plane strain distributions in graphene sheetsignificantly modify its band structure around the Fermi level,which break the inversion symmetry [27]. Theoretically, both theTB model and ab initio approaches have been widely adopted toinvestigate the effect of strain on the band structure of grapheneand graphene nanoribbons [28,29]. In this research, we first studythe electronic structure of graphene sheet under different in-planestrain distributions using quantum mechanics calculations andHückel TB model [29,30]. If the inversion symmetry breaking isconfirmed with particular planer strain distribution case, then theflexoelectric effect in graphene sheet with non-centrosymmetricpore is examined.

We define x-direction to be along the unstrained graphene axisand y as the transverse direction, as shown in Fig. 1.

The following changes occur in the x-component of the coor-dinate of the ith atom when the graphene sheet is subjected touniaxial or shear strains:

xi/ð1þ εAÞxi

xi/ð1þ εzÞxi

9=; Uniaxial strains (1a)

and a3 are the bond vectors. (A colour version of this figure can be viewed online.)

Fig. 2. (a) Graphene system with symmetrical strain distribution, (b) armchair graphene with asymmetrical strain distribution, and (c) zigzag graphene with asymmetrical straindistribution. Corresponding primitive cells in black solid line, reciprocal lattices in green dashed line, BZs in red dashed line, and irreducible BZs in sky blue color are shown belowthe deformed graphene systems. G, K, M, R, and S are the high symmetrical points; and Lx and Ly are the half diagonal lengths of the primitive cells. (A colour version of this figurecan be viewed online.)

S.I. Kundalwal et al. / Carbon 117 (2017) 462e472464

xi/xi þ gyi Shear strain (1b)

where εA and εz are the respective uniaxial strains along armchairand zigzag directions, and g is the shear strain. The uniform strainin graphene sheet can be written as a strain tensor:

ε ¼�εA g0 εz

�(2)

In the deformed graphene sheet, real space vectors are

r ¼ ðIþ εÞr0 (3)

in which I ¼ ðdijÞ2�2 and subscript “0” denotes the undeformedstate of graphene sheet.

The resulting Brillouin zone (BZ) for the reciprocal vector k inthe deformed space is no longer a regular hexagon [28]. By intro-ducing a new quantity

k0 ¼ ðIþ εÞTk (4)

The quantity k0can be regarded as the undeformed reciprocal

vector. The BZ in the k0space is restored to hexagonal because

kr ¼ kðIþ εÞr0 ¼ ðIþ εÞTkr0 ¼ k0r0 (5)

This approach facilitates analysis of variations in electronicstates near Fermi point kF with strains and the Hückel TB

Hamiltonian becomes

HðkÞ ¼X

i¼1;2;3

ti expðikaiÞ ¼X

i¼1;2;3

tiexpðik0ai0Þ (6)

with Harrison hopping parameter ti ¼ t0ða0=aiÞ2 [31]. Thisparameter incorporates the effect of strain into the variation of ti.Under the application of symmetric strains i.e., t1 ¼ t2 ¼ t3 (seeFig. 2a), the Fermi point kF can be determined by solvingEðkFÞ ¼ jHðkFÞj ¼ 0. On the other hand, t1, t2, and t3 are asymmetricin case of uniaxial strains, as shown Fig. 2b and (c), and the Fermipoints deviate from K in the k0 space.

Under the linear approximation, the deviation Dk0F ¼ k0

F � k0K is

determined to be [30]:

Dk0Fxa0 ¼ Ct

hεy

hð1þ yÞcos3qþ gsin3q

i(7a)

Dk0Fya0 ¼ �Ct

�εyð1þ yÞsin3qþ gcos3q

�(7b)

where Dk0Fx and Dk

0Fy are the respective components of Dk

0F; q is the

angle between the x-axis and the zigzag direction, εy is the uniaxialstrain along the y-direction, y is Poisson's ratio, a0 is the equivalentbond length and Ct is given by

Ct ¼ � a2t

dtda

����a¼a0

(8)

S.I. Kundalwal et al. / Carbon 117 (2017) 462e472 465

The deviation of K0is opposite to that of K in the k

0space. The

dispersion relation of the deformed graphene is given by expandingE(k) near the Fermi points:

E�k

0� ¼ ±32t0a0

���k0 � k0F

��� (9)

Thus, the effect of small strain is to move the energy cone in thek

0space, as demonstrated in Fig. 3.Combining Eqs. (7) and (9), we can obtain the following re-

lations for the pseudo gap at K and K0points in the k

0space:

Egap�k

0V

�¼ 3t0Stð1þ yAÞεA (10a)

Egap�k

0V

�¼ 3t0Stð1þ yzÞεz (10b)

3. Quantum mechanics calculations

3.1. Density functional theory

DFT is an attractive method when compared with conventionalwave function based methods, because the electronic wave func-tion of a system with N-electrons depends on 3 N spatial co-ordinates, while the electron density depends only on threecoordinates (x, y and z). In DFT, the total electronic energy EðrÞ isthe sum of the kinetic energy of the electrons TðrÞ, the energy of theelectrons in the external field VðrÞ, and the electron-electroninteraction energy Ve�eðrÞ; viz.,

EðrÞ ¼ TðrÞ þ VðrÞ þ Ve�eðrÞ (11)

In 1964, Hohenberg and Kohn proved [32] that the energy andelectronic properties of atoms and molecules can be determined bythe electron density rðrÞ ¼ rðx; y; zÞ. Using their theory, the prop-erties of a many-electron system can be characterized using func-tionals, i.e. the spatially dependent electron density. The Kohn-Sham equation in Hartree atomic units [33] is given by"� V2

2þ wextðrÞ þ wHartree½r�ðrÞ þ wXC½r�ðrÞ

#jiðrÞ ¼ jiðrÞ (12)

where the first term represents the kinetic energy of electrons, the

Fig. 3. Schematic illustration of shift of energy cone away from K points in k0space. (A

colour version of this figure can be viewed online.)

second is the external potential, and the third is the Hartree po-tential that describes the classical electrostatic repulsion.betweenthe electrons. The XC potential wXC½r� is defined by

wXC½r�ðrÞ ¼dEXC½r�drðrÞ (13)

where EXC½r� is the XC energy functional associated with the elec-tronic density r defined by

rðrÞ ¼XOccupied

i

jjiðrÞj2 (14)

where the sum runs over the occupied states.

3.2. Band gap and flexoelectric effect in graphene

In this section, all electron quantum mechanical calculationsbased on DFT were developed using Quantum ESPRESSO softwarepackage [34] and ultrasoft pseudopotentials. We considered bothsymmetric and asymmetric in-plane strain distributions in gra-phene sheets (see Fig. 2) and calculated their band gaps. In oursimulations, we keep one of εA, εz and g at a fixed value, while theother two are relaxed to achieve the lowest total energy. The edgesin graphene sheet were passivated using hydrogen because theyboth result in dangling bonds that lead to extended reconstructions[35]. First, the initial configuration of graphene sheet was preparedand then relaxed under the application of zero and finite strainconditions. Once the graphene system was fully relaxed, electronicband structure calculations were carried out to ensure the semi-conducting nature of its geometry. The quantum mechanics cal-culations were performed in an unstrained reference state andunder various levels of strain up to 3% in the directions as shown inFig. 2. The geometry of graphene was optimized until the residualforce per atom becomes less than 0.04 eV/Å and then the electronicground statewave functions were collected using amesh of 45� 45Monkhorst-Pack k-points for k-point sampling in the BZ. The en-ergy was considered to be converged once the change in the totalenergy of the system between subsequent steps was less than0.1 meV. Poisson ratios εA and εz in the directions parallel to orperpendicular to CeC bonds were calculated during each level ofstrain using their respective definitions; viz.,

εA ¼ DLx=LxDLy

Ly

and εz ¼DLy

Ly

DLx=Lx(15)

The constitutive relation for the polarization vector induced dueto the flexoelectricity effect may be written as [2]:

Pi ¼ eijkεjk þ f ijklvεklvxj

(16)

where eijk and fijkl are the respective piezoelectric and flexoelectrictensors; εjk and vεkl

vxjare the elastic strain and strain gradient,

respectively. Note that the first term in the right-hand side of Eq.(16) is the well-known piezoelectric effect.

Graphene sheets with non-centrosymmetric pores act as adielectric material and under the application of axial stress it showsflexoelectricity effect [24]. This is attributed to the fact that thedifferences in material properties at the interfaces result in thepresence of strain gradients under the application of an axial stress,which eventually induces the necessary polarization. Note thatcertain symmetry rules for the pores should be followed to inducepolarization in the graphene sheet. Curvature-induced polarizationis another way to induce flexoelectric effect in the graphene. As a

S.I. Kundalwal et al. / Carbon 117 (2017) 462e472466

result of bending of graphene, Coulomb repulsion inside a cavityincreases with curvature and it leads to a redistribution of the p-orbitals. This results into an electronic charge transfer from theconcave to the convex region, and induces the normal atomicdipole at each atomic site [21]. We use both the loading conditionsto obtain the induced axial and normal piezoelectricity coefficientsthrough quantummechanics calculations. In the following sections,we will investigate how these loading conditions induce flex-oelectricity phenomena in graphene systems. In addition, we willalso consider different pores geometries of varying sizes to inves-tigate their effect on any induced flexoelectric effects in graphenesheets.

3.3. Calculation of induced axial piezoelectric coefficient

We performed DFT calculations as those described in Section 2.2to obtain the apparent piezoelectric behavior from a graphenesheet containing non-centrosymmetric trapezoidal shaped pore(see Fig. 4).

The edges and porosity in a graphene sheet were passivatedusing hydrogen and the electronic ground state wave functionswere collected using a mesh of 1 � 6 � 6 Monkhorst-Pack k-pointsfor k-point sampling in the BZ. Spacing of 0.1�A�1 k-point was used

Fig. 4. Passivated armchair graphene sheet with trapezoidal pore subjected to an axialstress: (a) atom vacancy concentration of 4% and (b) atom vacancy concentration of19.5%. (A colour version of this figure can be viewed online.)

during the relaxation of the graphene sheet. The Berry-phasemethod was employed to compute the piezoelectric properties ofgraphene, which are directly related to the polarization differencesbetween its strained and unstrained states. In the modern theory ofpolarization, the spontaneous electric polarization in periodicsolids is obtained from the geometric phase (i.e., Berry phase)accumulated by the occupied electronic states as one introduces apotential that adiabatically connects an unpolarized and a polarizedstates of system [36]. For graphene sheet, the Berry phase andhence the polarization is controlled by the periodic boundaryconditions of the electronic wave functions. The average inducedpolarization was obtained for a strained graphene using thefollowing Berry phase formulation (discretized on a dense k-pointgrid along the direction of the polarization):

e11 ¼ eo þ dεþ O�ε3�

(17)

where eo is the pre-existing polarization due to surface effectspresent in the unstrained state. In our simulations, we ensured theapplication of very small values of strain and the term (Oðε3Þ)exhibiting nonlinear effects was neglected.

3.4. Calculation of induced normal piezoelectric coefficient

In this section, we determine the normal polarization inducedby the bending of a graphene sheet. Note that there is no dipolemoment across the flat graphene sheet due to the symmetry of p-orbitals (see Fig. 5a). By bending the graphene sheet, we canintroduce asymmetry in the p-orbital overlap (see Fig. 5b).

Therefore, Coulomb repulsion inside the cavity increases withcurvature and leads to a redistribution (rehybridization) of the p-orbitals from the sp2 of graphene to something intermediate be-tween sp2 and sp3 [21]. This results into an electronic chargetransfer from the concave to the convex region. Fig. 5b demon-strates that the carbon atoms in the bent graphene sheet are not inthe tangential plane and the three si e bonds (i ¼ 1; 2; 3) are tilteddown with respect to that tangential plane. In that configuration,the si e bonds are forced to lie along the internuclear axes and wecan use orbital orthogonality to solve the p-orbital hybridizationand direction. Due to the geometrical tilting of s-bonds, p- and s-orbital mixing occurs which we call it p-hybrids. As shown inFig. 5b, the angle between the s and p hybrids has the respectivevalues of 1090 and 900 for sp3 and sp2 hybridization. Such p- and s-orbital mixing is also reported to occur in case of conjugatedpolymers such as polyenes, polyacenes, and polyynes [37]. Redis-tribution of ions and charges occur upon bending of graphene sheetwhich in turn results in the formation of a net dipole momentacross the graphene. Such dipole moment (m) was obtained byfitting the energy curve E (ε) as a function of electric field (0, 0, εz)by

E ¼ Eo þ mεz þ e33ε2z þ O

�ε3z

�(18)

where Eo is the pre-existing energy present in a flat graphene state.

4. Results and discussions

We first investigated the electronic structure of a graphenesheet subjected to different in-plane strain distributions usingquantum mechanics calculations and TB approach. We consideredboth armchair and zigzag graphene sheets. In case of zigzag gra-phene sheets, edge magnetism may be an issue, but we did notconsider magnetic edge states in the current study because they donot exist in real passivated graphene systems. Kunstmann et al. [38]

Fig. 5. Graphene sheets in which p-orbitals are (a) symmetric, and (b) asymmetric. (A colour version of this figure can be viewed online.)

S.I. Kundalwal et al. / Carbon 117 (2017) 462e472 467

critically discussed the stability of edge states and magnetism inzigzag edge graphene nanoribbons and showed that there are atleast three very natural mechanisms. They defined these threestates as edge reconstruction, edge passivation, and edge closur-edwhich dramatically reduce the effect of edge states in zigzagedge graphene nanoribbons or even totally eliminate them.Therefore, edge magnetism can be neglected in case of passivatedgraphene sheets. The anisotropic nature of the graphene sheet wasobserved by the varied Poisson's ratios under asymmetrically largestrains for armchair and zigzag graphene sheets, as shown in Fig. 6.This is attributed to the significant changes of the band-gapstructure, the electron redistribution due to the induced largestrains and the variation of angles of two adjacent CeC bonds of agraphene sheet. The angles of CeC bonds change when a graphenesheet is subjected to uniaxial strains along the armchair and thezigzag directions, which eventually influence Poisson's ratios. OurDFT results concerning the anisotropic behavior of the Poisson'sratio for the armchair and the zigzag graphene sheets under largestrain qualitatively agrees well with the trend observed in the

Fig. 6. Poisson's ratio of armchair and zigzag graphene sheets. (A colour version of thisfigure can be viewed online.)

existing DFT and molecular dynamics studies conducted inRefs. [27,39]. We can observe that the Poisson's ratio is almostconstant under small strains up to 1.2%, but decays differently un-der large strains. Our results also indicate that the graphene sheetbehaves in an isotropic manner under small strains (<1:8%)because the 2D hexagonal symmetry of pure C lattice ensures itsisotropic elastic properties.

Fig. 7 demonstrates the variations of Egap for armchair andzigzag graphene sheets obtained from DFT and TB models forvarious strains up to 3%. In the TB modeling, we adopted t0 ¼ 2:67eV and Ct ¼ 1:29 [29,30] and used the Poisson's ratio determinedfrom DFT calculations (see Fig. 6). Fig. 7 clearly shows that asym-metrical strain distributions in graphene sheets result in theopening of the band gaps at the Fermi level and these band gapsincrease as the strain increases. For the first case (see Fig. 2a), wefound that the band gap of graphene does not open irrespective ofthe magnitude of symmetrical strain distribution and it remainsalways a semiconductor. This finding is consistent with a previouslyreported results on graphene [27]. It may also be observed from

Fig. 7. Variations of Egap for armchair and zigzag graphene sheets subjected to uniaxialstrain. (A colour version of this figure can be viewed online.)

Fig. 8. Passivated armchair graphene sheet with trapezoidal pore subjected to an axialstress: (a) atom vacancy concentration of 4.5%, and (b) atom vacancy concentration of20%. (A colour version of this figure can be viewed online.)

S.I. Kundalwal et al. / Carbon 117 (2017) 462e472468

Fig. 7 that the values of Egap for armchair and zigzag graphenesheets are almost the same and that negligible differences attrib-uted to different Poisson's ratio (when graphene is strained in onedirection, it will shrink in the perpendicular direction) leads to afurther shift the Fermi points. Fig. 7 clearly shows good agreementbetween the TB modeling results and DFT calculations, validatingour simulations and enabling us to determine uniaxial straingradient induced polarization in graphene sheets with non-centrosymmetric trapezoidal shaped pore.

To further validate our quantum mechanics simulations, theexperimentally determined dipole moments of carbonyl sulfideand aniline were considered. DFT calculations for carbonyl sulfideand aniline were performed using a generalized gradient approxi-mation of Perdew-Burke-Ernzerhof [40]. Molecular systems ofcarbonyl sulfide and aniline were modeled by the ultrasoft pseu-dopotential method using the PWscf code from the QuantumESPRESSO software package [34]. Orbitals were expanded in aplane-wave basis set up to a kinetic energy cutoff of 400 eV;3000 eV for the charge density cutoff. Initial configurations ofcarbonyl sulfide and aniline structures were prepared and relaxed.Then, BZ integrations were performed with Gaussin-smearing [41]and Monkhorst-Pack k-point technique using a smearing param-eter of 0.04 eV and a 4 � 4 � 4 mesh for both structures. Table 1summarises the outcome of this comparison. The two sets of re-sults predicted are in good agreement and validate our simulations.

4.1. Uniaxial strain induced piezoelectric effect

In this section, we report results concerned with uniaxial straininduced polarization in armchair and zigzag graphene sheetscontaining non-centrosymmetric trapezoidal and triangular pores.An additional aspect of pore edge states by hydrogen termination isrelated to the current study because such states may alter theflexoelectric phenomena in graphene sheets. Figs. 4 and 8demonstrate passivated armchair graphene sheets with edge andcorner (red circled) states. Therefore, we focus on examining theeffect of functionalization on strain induced polarization, consid-ering not only the edge but also the corner states. Note that noadditional force acts on the carbon atoms due to HeH repulsionalong the zigzag direction. However, inhomogeneous force acts onthe carbon atoms along the armchair direction, which causes latticerelaxation and, hence, interatomic bonding energy weakening[44,45]. It may be observed from Figs. 4 and 8 that the passivatedarmchair graphene sheets under uniaxial strain always have trap-ezoidal pore with a modified zigzag edge. However, repulsioninduced by hydrogen termination at the corner of a pore (redcircled), which we call corner strain effect, may result. Accordingly,two cases were considered: two armchair graphene sheets with4.5% and 20% atom vacancy concentrations (r) forming non-centrosymmetric trapezoidal pores with and without cornerstrain effect. The predicted strain gradient induced polarization inthese graphene sheets as a function of strain is shown in Fig. 9. Thisfigure clearly depicts that the graphene sheet with non-centrosymmetric trapezoidal shaped pore subjected to axialstress exhibits the required polarization, since the net averagepolarization is nonzero. The strain gradient induced polarization

Table 1Comparisons of molecular electric dipole moments.

Structure Experimental Present calculations

Carbonyl Sulfide 0.71521 ± 0.00020 Da [42] 0.715Aniline 1.129 ± 0.005 D (Oþ state) [43]

1.018 ± 0.005 D (O� state) [43]1.1251.014

a 1D ¼ 3:33564� 10�30 Cm.

increases with the trapezoidal pore size. This is attributed to thestrong dielectric behavior of a graphene sheet with larger pores. Itmay also be observed from these figures that the corner strain ef-fect marginally affects the strain gradient induced polarization incase of armchair graphene sheet containing small pores. As ex-pected, the reverse is true in case of armchair graphene sheetcontaining big pores. Fig. 9(a) and (b) depict linear piezoelectricresponses. In the case where corner strain effects are not present,the respective axial piezoelectric coefficients were determinedfrom these linear least square fits to be 0.027 C=m2 and 0.12 C=m2;while in the case involving corner strain effects, the correspondingvalues are determined to be 0.023 C=m2 and 0.115 C=m2. The fullpiezoelectric tensor for graphene sheet can be determined on thebasis of symmetry of hexagonal 6m2 class using the relations: e12 ¼�e11 and e26 ¼ �e11.

As mentioned earlier, certain symmetry rules for the poresshould be followed to induce polarization in the graphene sheet.Therefore, we endeavoured to establish whether other pore ge-ometries induce strain polarization in a graphene sheet. For such aninvestigation, we considered a triangular pore in a graphene sheet,as shown in Fig. 10. This figure clearly indicates that an armchairgraphene sheet with a triangular pore is more sensitive to cornerstrain effects. Note that the armchair graphene sheet containing a

Fig. 9. Strain induced average polarization in armchair graphene sheets containing non-centrosymmetric trapezoidal pore with and without corner strain effects. (A colour versionof this figure can be viewed online.)

S.I. Kundalwal et al. / Carbon 117 (2017) 462e472 469

triangular pore without corner strain effects represent the earlierdiscussed case of armchair graphene sheet containing trapezoidalpore (see Fig. 4a). We compared these two cases of armchair gra-phene sheets with 4.8% and 4% atom vacancy concentrationsforming non-centrosymmetric triangular pores with and withoutcorner strain effect. The predicted strain gradient induced polari-zation in these graphene sheets as a function of strain is shown inFig. 11. We can clearly observe from this figure that the inducedpolarization increases with the applied strain. It may also beobserved that corner strain effects significantly affect the straingradient induced polarization even if the vacancy concentration ismaximum in the case of a triangular pore. Note that corner straineffects become less pronounced in case of a larger pore size, as wehave seen in the earlier case of a trapezoidal pore. The respectiveaxial piezoelectric coefficients were determined from linear leastsquare fits to be 0.0216 C=m2 and 0.27 C=m2 with and withoutcorner strain effects. It is important to note from Figs. 9 and 11 thatthe corner strain effects can be systematically controlled in case ofsmall pores in armchair sheets by adopting various symmetry rulesand can be neglected in case of larger pores.

In the previous sets of results, the strain induced polarizationhas been investigated by applying the tensile load along thearmchair direction of graphene sheets. However, the application of

Fig. 10. Passivated armchair graphene sheet with triangular pore subjected to an axialstress (4.8% atom vacancy concentration). (A colour version of this figure can be viewedonline.)

the tensile load along the zigzag direction of graphene sheet mayaffect the strain induced polarization in it because of the ever-present existence of HeH repulsion along the armchair edges of apore (elliptically marked region in blue color) and corner straineffects (red circled), as shown in Fig. 12. Fig. 13 shows the predictedstrain gradient induced polarization in zigzag graphene sheets as afunction of the applied strain. A similar trend of results is obtainedfor the values of e11 as has already been observed in case ofarmchair graphene sheets but in case of zigzag graphene sheet, thestrain gradient induced polarization is much less than that found incase of former with less vacancy concentration. For instance, thevalues of axial piezoelectric coefficients are 0.031 C=m2 and 0.034C=m2 for the zigzag graphene sheets with 7.5% and 8.2% atom va-cancy concentrations, respectively.

4.2. Curvature induced piezoelectric effect

Next, we investigated the curvature-induced polarization intothe rolled graphene sheet by determining the magnitude of theinduced normal atomic dipole at each atomic site. Fig. 14 shows the

Fig. 11. Strain induced average polarization in armchair graphene sheets containingnon-centrosymmetric triangular pore with and without corner strain effect. (A colourversion of this figure can be viewed online.)

Fig. 12. Passivated zigzag graphene sheet with trapezoidal pore subjected to an axial stress: (a) atom vacancy concentration of 7.5%, and (b) atom vacancy concentration of 8.2%. (Acolour version of this figure can be viewed online.)

S.I. Kundalwal et al. / Carbon 117 (2017) 462e472470

dipole moment per atom for four different radii of curvature. Asexpected, the magnitude of the dipole moment per atom decreasesas the radius of curvature increases. Note that the graphene layercan sustain large deformations as witnessed by their ability to formnear perfect carbon nanotube. Some studies suggest that curvatureinduced charge redistribution is presents in carbon nanotubewhich is non-existent in flat graphene sheet [21]. A build-in elec-trostatic field perpendicular to CNTs surface exists due to its curvedgeometry. The surface electric field is believed to cause anti-localization, whose signature is seen in the reported positivemagnetoresistance measurements in CNTs under low magneticfield. It is worth noting that the piezoelectric coefficients are givenper unit volume. Therefore, a quantitative determination of thepiezoelectric properties of the 2D graphene sheet to the 3Dequivalent graphene layer requires specification of the interlayerspacing and packing, representing the electron distribution aroundit. We assumed 2D graphene sheet as an equivalent 3D graphene

Fig. 13. Strain induced average polarization in zigzag graphene sheets containing non-centrosymmetric trapezoidal pore with the ever-present corner and edge strain effects.(A colour version of this figure can be viewed online.)

layer using an assumed interlayer separation of 3.4 Å as its thick-ness [46e49]. Once the dipole moment per atom for different radiiof curvature of graphene is obtained, we can determine the dipoledensity (i.e., average curvature-induced polarization) using thefollowing relation:

e33 ¼ dipole moments of n number of atomsvolume occupied by n number of atoms

(19)

Note that CeC bond length was taken as 1.41 Å while computingthe volume of an equivalent 3D graphene layer. Fig. 15 demon-strates the normal polarization in a graphene sheet for fourdifferent radii of curvature as a function of number of atoms (n) init. We considered a square shaped graphene made of n atoms toinvestigate the effect of its different radii of curvature. In general,the obtained values of normal piezoelectric constants are greaterthan Gallium nitride (0.73 C=m2) [50], a-Quartz (0.18 C=m2) [51],and piezoelectric polymer poly(vinylidene fluoride) (0.168 C=m2)

Fig. 14. Evolution of dipole moment per atom as a function of radius of curvature (R) ofgraphene. (A colour version of this figure can be viewed online.)

Fig. 15. Curvature induced normal piezoelectric coefficient (e33) as a function ofnumber of atoms (n) in graphene. (A colour version of this figure can be viewedonline.)

S.I. Kundalwal et al. / Carbon 117 (2017) 462e472 471

[52], which are all commonly used as piezoelectric materials.Alternatively, the piezoelectric constants computed from differentdimensional system can be expressed as a total dipole per stoi-chiometric unit. The full piezoelectric tensor for graphene sheet canbe determined on the basis of symmetry of hexagonal 6m2 classusing the relations: e31 ¼ �e33 and e15 ¼ �e33.

5. Conclusions

In this study, we report the induced piezoelectricity in graphenesheets through interplay between different non-centrosymmetricpores with edge and corner states, curvature, and flexoelectricityphenomena in the presence of strain gradients using quantummechanics calculations. We have also calculated the electronicstructure of armchair and zigzag graphene sheets under plannerasymmetrical strain distributions using quantum mechanics cal-culations and tight-binding approach, which result in the openingof band gaps at the Fermi level. Our results suggest that a graphenesheet with non-centrosymmetric trapezoidal-shaped pores andbent graphene show strong electromechanical coupling whensubjected to large strain gradients. We also studied the effects offunctionalization, considering not only the edge state but also thecorner state. We found that zigzag graphene sheets provide weakerflexoelectric effects in comparison with armchair graphene sheetsfor the same vacancy concentration. This is due to the ever-presentexistence of HeH repulsion along the armchair edge of a pore andcorner strain effects. In the case of armchair graphene sheets, thecorner strain effects can be systematically controlled, or eveneliminated by adopting various symmetry rules and can beneglected in cases involving larger pores. The current study offersdifferent pathways to alter the axial and normal electromechanicalcoupling in graphene sheets through changing the size of the non-centrosymmetric pores and the radius of curvature of these sheets.In particular, the latter case provides a strong electromechanicalcoupling without creating pores in the graphene sheet and possiblysacrificing its elastic properties and mechanical strength.

Acknowledgements

This work was supported by the Natural Sciences and Engi-neering Research Council of Canada (NSERC) (BAF - 143101) and aBanting Postdoctoral Fellowship awarded to the first author. The

authors wish to acknowledge the anonymous reviewers for theirconstructive remarks and suggestions.

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