PHYSICAL REVIEW B 99, 054105 (2019)

Intrinsic bending flexoelectric constants in two-dimensional materials

Xiaoying Zhuang,1,2,* Bo He,2 Brahmanandam Javvaji,2 and Harold S. Park3,†1College of Civil Engineering, Tongji University, 1239 Siping Road, 200092 Shanghai, China

2Institute of Continuum Mechanics, Leibniz Universität Hannover, Appelstr. 11, 30167 Hannover, Germany3Department of Mechanical Engineering, Boston University, Boston, Massachusetts 02215, USA

(Received 11 November 2018; revised manuscript received 28 December 2018; published 21 February 2019)

Flexoelectricity is a form of electromechanical coupling that has recently emerged because, unlike piezoelec-tricity, it is theoretically possible in any dielectric material. Two-dimensional (2D) materials have also garneredsignificant interest because of their unusual electromechanical properties and high flexibility, but the intrinsicflexoelectric properties of these materials remain unresolved. In this work, using atomistic modeling accountingfor charge-dipole interactions, we report the intrinsic flexoelectric constants for a range of two-dimensionalmaterials, including graphene allotropes, nitrides, graphene analogs of group-IV elements, and the transitionmetal dichalcogenides (TMDCs). We accomplish this through a proposed mechanical bending scheme thateliminates the piezoelectric contribution to the total polarization, which enables us to directly measure theflexoelectric constants. While flat 2D materials like graphene have low flexoelectric constants due to weak π -σinteractions, buckling is found to increase the flexoelectric constants in monolayer group-IV elements. Finally,due to significantly enhanced charge transfer coupled with structural asymmetry due to bending, the TMDCs arefound to have the largest flexoelectric constants, including MoS2 having a flexoelectric constant ten times largerthan graphene.

DOI: 10.1103/PhysRevB.99.054105

I. INTRODUCTION

Piezoelectricity is perhaps the best-known mechanism ofconverting mechanical deformation into electrical energy, andhas been widely used in engineering practice [1,2]. Whilepiezoelectricity is well-established, other types of electrome-chanical coupling, such as flexoelectricity, have recently at-tracted significant interest [3–7]. One reason for this is thatpiezoelectricity is limited to materials with noncentrosym-metric crystal structures. In contrast, in flexoelectricity thepolarization is not only related to the strain as in piezoelec-tricity, but also the strain gradient. Thus, for flexoelectricity,the polarization P induced due to mechanical deformation isgiven as [8]

Pα = dαβγ �βγ + μαβγ δ ∂�γ δ

∂xβ, (1)

where dαβγ is the piezoelectric coefficient, �βγ is the strain,μαβγ δ is the flexoelectric coefficient, ∂�

γ δ

∂xβis the strain gradient

and α, β, γ , and δ represent the directional components of thecoordinate system.

Because flexoelectricity is dependent on the gradient ofstrain it can, in principle, occur in any dielectric material.Furthermore, significant potential for flexoelectricity emergesas the dimensions of materials reduce to the nanometer scaledue to the ability to produce larger strain gradients for smallsize scales. However, studies of electromechanical coupling

*zhuang@ikm.uni-hannover.de†parkhs@bu.edu

in nanomaterials, and specifically two-dimensional (2D) ma-terials such as graphene and molybdenum disulfide (MoS2)have largely focused on their piezoelectric properties [9–22];we also note a recent review article summarizing the vari-ous simulation and experimental methods for characterizingpiezoelectricity in 2D materials [23].

In contrast to the extensive study of piezoelectricity innanomaterials, relatively few studies on flexoelectricity havebeen performed. Majdoub and co-workers reported an en-hancement of flexoelectricity in nanoscale barium titaniumoxide (BTO) [24]. Surface effects on flexoelectricity in BTOnanobelts were investigated using core-shell potentials [25].In addition, several preliminary studies on flexoelectricity in2D materials have recently been carried out using densityfunctional theory (DFT) calculations, theoretical analyses orexperiments. For instance, a linear relationship was foundbetween induced dipole moment and bending curvature ingraphene using DFT calculations [6]. A theoretical analy-sis [26] of flexoelectricity in carbon nanostructures (nan-otubes, fullerenes, and nanocones) confirmed the dependenceof flexoelectric atomic dipole moments on local curvature.Others have patterned graphene to generate strain gradientsand enhance the electromechanical coupling and polarization[15,16,20]. Furthermore, a recent experimental study [11]provided evidence that monolayer MoS2 exhibits an out-of-plane flexoelectric response using the piezoresponse force mi-croscopy. However, one key issue in calculating or measuringthe flexoelectric constants of 2D materials is that it has beendifficult to isolate the relative contributions of piezoelectricityand flexoelectricity to the resulting polarization [11]. As aresult, the intrinsic flexoelectric properties of 2D materials re-main unresolved, and furthermore the mechanisms controlling

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ZHUANG, HE, JAVVAJI, AND PARK PHYSICAL REVIEW B 99, 054105 (2019)

NX

XC

( d)( b) ( c)( a)

XS

FIG. 1. Top and side view of the studied materials: (a) grapheneallotropes; (b) nitrides XN, X = B, Al, Ga; (c) graphene analoguesof group-IV elements X , X = Si, Ge, Sn; and (d) transition metaldichalcogenides XS2, X = Cr, Mo, W. For (a)–(c), h refers to thebuckling height, while in (d), h1 and h2 refer to intralayer distances.

the intrinsic flexoelectric properties of different 2D materialsare also unresolved.

In this work, we develop a classical charge-dipole (CD)atomistic model that couples with classical molecular dy-namics (MD) simulations to calculate the intrinsic bendingflexoelectric constants of the four different 2D material groupsshown in Fig. 1: graphene allotropes, nitrides, grapheneanalogues of group-IV elements, and transition metaldichalcogenides (TMDCs). Specifically, we propose and val-idate a mechanical bending formulation that eliminates thepiezoelectric contribution to the polarization in Eq. (1), thusenabling us to directly calculate the intrinsic flexoelectricconstants. By comparing these different classes of 2D materi-als, we investigate and elucidate the effects of charge-dipoleinteractions, out of plane buckling in monolayers, intralayerbuckling asymmetry, and charge transfer on the flexoelectricresponse of 2D materials.

II. SIMULATION METHOD

In this work, we performed MD simulations by utilizing aCD model in conjunction with bonded interactions to deter-

TABLE II. Piezoelectric coefficients (C/m2) for BN and MoS2.

Material Calculated Reported

BN 0.163 0.390a; 0.417b

MoS2 0.646 0.564c; 0.453d

aReference [69]; bRef. [14]; cRef. [14]; dRef. [70].

mine the atomic configurations, as well as the point chargesqi and dipole moments pi associated with each atom i. Thebonded interactions were modeled using well-known poten-tials, i.e., adaptive intermolecular reactive empirical bondorder (AIREBO), Tersoff, and Stillinger Weber (see Table Ifor references to all potentials), while the point charges anddipole moments were calculated using the well-known CD po-tentials [27–29] (further details are given in Appendix A). Thepotential parameters for the CD model were determined usingDFT calculations (more details are given in Appendix B), andwere validated through calculation of piezoelectric constantsfor boron nitride and MoS2, which as shown in Table II inAppendix C are in good agreement with previous studies. Allsimulations were performed using the open-source MD sim-ulation code large-scale atomic/molecular massively parallelsimulator (LAMMPS) [30].

The MD simulations were performed using the unit celldimensions (a, b, c, and h) for each material given in Table I,along with the potential functions employed to estimate thebonded interactions. A fixed unit cell size of 80 × 80 Å [31]was adopted for all simulations to estimate the flexoelectriccoefficients. The flexoelectric constants were determined byfirst prescribing the following displacement field to the atomicsystem

uz = K x2

2, (2)

TABLE I. Calculation details for each material. The unitcell dimensions a, b, c, and h are given in angstroms. The bonding interactionsare modeled using different types of “short-range potentials.” αDFTtotal and α

CALtotal are the polarizability estimates from DFT and calculated using

Eq. (B6) in Å3. Calculated in angstroms, RA and RB are the CD potential parameter for atom types A and B in a given unit cell, respectively.

material a b c h short-range potential αDFTtotal αCALtotal RA RB

C1 2.46a 4.26a 3.5b 0.0 AIREBOc 2.49 2.78 0.64 0.64C2 4.87a 8.84a 3.5 0.0 AIREBOc 2.46 2.72 0.64 0.64C3 5.70a 7.56a 3.5 0.0 AIREBOc 2.45 2.77 0.64 0.64

BN 2.50d 2.50d 3.33d 0.0 Tersoff e 2.85 2.84 0.76 0.35AlN 3.13f 3.13f 3.39f 0.0 Tersoff f 19.79 19.97 1.04 0.48GaN 3.21g 3.21g 3.63g 0.0 Tersoff h 15.80 16.38 1.05 0.48

Si 3.82i 6.62i 2.41j 0.44i Tersoff k 20.62 20.92 1.37 1.37Ge 3.97i 6.87i 3.20l 0.65i Tersoff m 13.43 13.14 1.27 1.27Sn 4.67n 8.09n 3.30o 0.89n Tersoff o 15.25 15.86 1.52 1.52

MoS2 3.16p 3.16p 12.29q 1.58 SWr 12.32 12.35 0.69 1.04WS2 3.18s 3.18s 12.16s 1.56 SWt 15.36 15.38 0.70 1.09CrS2 3.04q 3.04q 14.41q 1.45 SWt 10.86 10.87 0.75 1.00

aReference [32]; bRef. [50]; cRef. [51]; dRef. [52]; eRef. [53]; fRef. [54]; gRef. [55]; hRef. [56]; iRef. [57]; jRef. [58]; kRef. [59]; lRef. [60];mRef. [61]; nRef. [62]; oRef. [63]; pRef. [64]; qRef. [65]; rRef. [66]; sRef. [67]; tRef. [68].

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Initi

al c

ongu

ratio

n

Def

orm

ed c

ongu

ratio

n

FIG. 2. Schematic illustration of the geometry and loading con-dition for 2D material system.

where x represents the atom coordinate in the x direction,K represents the inverse of curvature (strain gradient) of thebending plane, and where the prescribed mechanical defor-mation is shown in Fig. 2. Once the bending deformation isprescribed, the edge region atoms are held fixed while the inte-rior atoms are allowed to relax to energy minimizing positionsusing the conjugate-gradient algorithm, after which the pointcharges and dipole moments are found for each atom. Fromthe MD simulations, we establish the relationship betweenpolarization and strain gradient as follows. The strain gradientfrom Eq. (2) is

∂εxz

∂x= 1

2

∂2uz

∂x2= 1

2K (3)

where εxz is the strain in the x direction from the applieddeformation in the z direction. Substituting Eq. (3) in Eq. (1)and assuming that the imposed mechanical deformation inEq. (2) removes the piezoelectric contribution, we obtain

Pz = 12μzxzxK, (4)

where μzxzx is the out-of-plane or bending flexoelectric co-efficient and Pz is the out-of-plane polarization. We willverify the assumption of the removal of the piezoelectriccontribution through the prescribed bending deformation inthe next section.

III. RESULTS AND DISCUSSION

We use the simulation procedure described previously inSec. II to study the flexoelectric properties of four groups of2D materials: graphene allotropes (C1, C2, and C3), nitrides(BN, AlN, and GaN), graphene analogues of group-IV ele-ments (Si, Ge, and Sn) and TMDC monolayers (MoS2, WS2,and CrS2). C1 corresponds to pristine graphene, while C2and C3 represent graphene with Stone-Wales defects whichreplace some hexagons by pentagons and heptagons withdifferent periodicity, respectively [32]. BN, AlN, and GaNare the nitrogen-based hexagonal monolayers with boron,aluminum, and gallium, respectively. Silicene (Si), germanene(Ge), and stanene (Sn) are the group-IV 2D graphene analogs.However, the vertical distance between the atoms or buckling

height (h) in the unit cell is nonzero when compared to thegraphene allotropes and nitride material groups [see Figs. 1(a)and 1(c)]. The TMDCs possess three sublayers or intralayerswhere element X (center layer) forms bonds with two Satoms in the top and bottom layers. The layers are verticallyseparated by the intralayer heights h1 and h2, as shown inFig. 1(d).

We first demonstrate that the proposed bending schemeeliminates the piezoelectric contribution to the total polariza-tion, such that we can focus on the resulting intrinsic flex-oelectric properties of the different 2D material groups. Theapplied deformation [using Eq. (2)] results in strain (�xz ) andstrain gradient ( ∂�

xz

∂x ) along the xz direction, and a polarizationalong the z direction, where we use MoS2 as an example asit has the most complex 2D structure of the 2D materials weconsider. We calculate the local atomic strain for each atomi using the local deformation gradient F, which involves theinitial and deformed atomic coordinates. The local atomicstrain tensor for atom i (�i) is [33]

�i = 12 [(Fi )TFi − I], (5)where I is the identity matrix.

Figure 3(a) represents the atomic configuration of MoS2system colored with the xz component of strain, which is

calculated from Eq. (5) at a given curvature (K = 0.01 Å−1).The variation of strain �xz along the x direction is plottedin Fig. 3(b), where the strain was found by dividing theatomic system into several equal width bins and averaging thestrain in each bin. A linear variation in �xz is observed fromFig. 3(b). This demonstrates that the induced deformationis symmetric and the resulting polarization due to strain iscanceled out. Therefore the total strain �xz is zero (sum overall the bins), which eliminates the piezoelectric contributionto the polarization in Eq. (1) and supports the assumptionmade in obtaining Eq. (4), i.e., for the prescribed bendingdeformation, the out-of-plane polarization is only dependenton the strain gradient. Furthermore, symmetry analysis on thepiezoelectric tensor show that dzxz is zero for a point groupsymmetry associated with the 2D material sets [34].

The mechanical bending deformation that is imposedserves to strictly to zero out the out-of-plane piezoelectriccontribution to total polarization. However, it is important tonote that an in-plane polarization is generated due to the out-of-plane bending. Furthermore, the in-plane polarization fromout-of-plane bending may receive a contribution from in-planepiezoelectricity. We further discuss the in-plane polarizationin Sec. III D and Appendix E.

A. Mechanisms of inducing polarization in 2D materials

Our analysis of the mechanisms governing the flexoelectricconstants for the 2D materials depends on understanding,within the framework of the utilized CD model, the variouscontributions to the dipole moments that are induced fromthe prescribed bending deformation. Specifically, the dipolemoment pi on atom i depends on its polarizability and thepresence of a local electric field, which consists of threeparts: the electric field at position ri due to neighboring (i)dipoles p j , (ii) charges q j , and (iii) from the externally appliedelectric fields Eext. Because the only external stimulus is the

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-40 -20 20 40 0x ( )

-40 -30 -20 -10 0 10 20 30 40-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2MoS

2

linear

x ( )

-0.2

0.0

0.2

(b)

(a)

FIG. 3. (a) Atomic configuration colored with strain �xz in x

direction for MoS2 sheet when the strain gradient K = 0.01Å−1;the large spheres represent Mo atoms, and small spheres representS atoms. (b) Binwise distribution of strain �xz along x axis, circlesrepresent the calculated average strain �xz at location x and solid lineis linear fitting to the calculated data.

prescribed bending deformation, Eext = 0 and the governingequation for the dipole moments [Eq. (B1)] becomes

Tp−pii pi −N∑

j,i �= jTp−pi j p j =

N∑j,i �= j

Tq−pi j q j, (6)

where Tp−pi j and Tq−pi j are the polarizability tensors. These

two tensors represent dipole-dipole and charge-dipole inter-actions, respectively, which can also be interpreted as ac-counting for σ -σ and σ -π electron interactions, respectively[28,29,35], and can be written as [29]

Tq−pi j =1

4π�0

ri jr3i j

≈ 14π�0

ri jr3i j

×[

erf

(ri j√2R

)−

√2

π

ri jR

exp

(− r

2i j

2R2

)], (7)

Tp−pi j =1

4π�0

3ri j ⊗ ri j − r2i jIr5i j

×[

erf

(ri j√2R

)−

√2

π

ri jR

exp

(− r

2i j

2R2

)]

− 14π�0

√2

π

ri j ⊗ ri jr2i j

1

R3exp

(− r

2i j

2R2

). (8)

From Eqs. (7) and (8), the interatomic distance (ri j ) and R(factor related to polarizability) are identified as the importantfactors in defining the dipole moment of atoms via the polar-izability tensors.

B. Flat 2D monolayers

We first consider the simplest 2D structures, flat grapheneand BN monolayers. To aid in the analysis, we rewrite Eq. (6)for only the pzi component, which is

T p−p,zzii pzi = E p,zi + Eq,zi , (9)

where E p,zi =∑N

j,i �= j{T p−p,xzi j pxj + T p−p,yzi j pyj + T p−p,zzi j pzj}and Eq,zi =

∑Nj,i �= j T

q−p,zi j q j are the electric fields on

atom i due to neighboring dipoles, charges and associatedpolarizability components.

For the undeformed graphene sheet, the out of plane dipolemoment pzi is zero due to the flat nature of the monolayer.However, once graphene is bent, the π -σ interactions in-crease, leading to a nonzero pzi . Specifically, for deformed

graphene with bending curvature 0.002 Å−1

, the measuredcontributions of E p,zi and E

q,zi to the total electric field on

atom i are 93.45% and 6.55%, respectively. As graphene isbent further, these contributions change to 93.27% and 6.73%,

respectively, when K = 0.01 Å−1. The increased importanceof Eq,zi with increasing bending implies an increasing im-portance of π -σ interactions on the total electric field anddipole moment on atom i. This can also be interpreted throughpyramidalization [36–38], in which sp2 bonding converts tosp3 bonding. In this process, the valence electrons of eachcarbon atom develop bonding interactions with the neigh-boring atoms due to the bond bending involved symmetryreduction, which allows mixing between π and σ electrons,leading to π -σ interactions [39]. This interaction modifiesthe charge state of the carbon atom as well as the locallygenerated electric fields, which is captured by the CD modelin the form of the charge-induced electric fields Eq,z. Overall,these increased π -σ interactions result in the flexoelectriccoefficient for graphene being found as μgr = 0.00286 nC/m,which is found from linear fitting of the polarization as afunction of bending curvature in Fig. 4.

In case of BN, the contribution of Eq,z also increases from1.86% to 1.99% when bending curvature increases from 0.002

to 0.01 Å−1

, though the overall contribution of Eq,z to thetotal electric field is smaller than for graphene. This suggeststhat the π -σ interactions in BN are weaker than in graphene,which may be related to the difference in the tendency of pyra-midalization between B and N atoms. Specifically, B atomsprefer the sp2 hybridization while N atoms are more likely to

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INTRINSIC BENDING FLEXOELECTRIC CONSTANTS IN … PHYSICAL REVIEW B 99, 054105 (2019)

0 2 4 6 80

5

10

15

Graphene C1

BN

Silicene

MoS2

FIG. 4. Polarization Pz vs strain gradient K for graphene, BN,silicene, and MoS2. Markers indicate the simulation data and solidlines indicate the linear fitting.

achieve sp3 hybridization or pyramidalization [40,41]. Thus,even though the polarizability of BN is similar to graphene(see αCALtotal values in Table I), the flexoelectric constant of BNof 0.00026 nC/m is ten times smaller than graphene due tothe smaller Eq,z contribution in BN.

The graphene allotropes C2 and C3 show similar flexo-electric coefficients to defect-free monolayer graphene (C1),as shown in Fig. 5. Though C2 and C3 contain differentarrangements of defects, the sp2 hybridization is unchanged,which induces nearly equal charges and dipole moments foratoms in C2 and C3 under deformation. As a result, theflexoelectric coefficients are nearly constant for this materialgroup. In the case of the nitride group, AlN and GaN are foundto have larger flexoelectric constants than BN as shown inFig. 5, though still significantly smaller than graphene. Thisis due to a corresponding increase in the contribution of Eq,z,from 1.99% for BN to 2.25% for AlN to 6.85% for GaN for acurvature of 0.01 Å

−1.

FIG. 5. Bending flexoelectric coefficient for 2D materials.

X

Y

X

Y

AB

C

DE

F

2.42

2.41

2.26

2.75

2.42

2.42

2.42

2.42

2.42

2.42

2.42

2.42

2.42

2.42

2.42

2.42

(a) undeformed con guration

(b) deformed con guration

FIG. 6. (a) Undeformed and (b) deformed atomic configurationof MoS2 sheet. Red colored atoms are used to explain the changesin bond length. Dashed lines represent the Mo layer and dash-dottedlines indicate the S layers. Atoms X and Y in (a) possess bond lengthsof 2.42 Å with neighbor sulfur atoms. The bond length betweenatoms Y-A and Y-D is 2.42 Å and 2.41 Å, respectively. Atom Y hasbond length of 2.26 Å with atoms B and C. The bond length betweenatoms Y-E and Y-F is 2.76 Å. Only left portion of the atomic systemwas shown here.

C. Buckled 2D monolayers

As seen in Fig. 4, the induced polarization for flat 2Dmaterials is much smaller than is seen in silicene. From astructural point of view, silicene and graphene differ in thatthe atomic polarizability of silicene is larger, and also that itexists in a buckled configuration as compared to graphene (seeh values in Table I). Therefore we performed simulations toexamine the effects of both of these factors on the inducedpolarization in silicene. We first performed a bending test forsilicene in which the buckling height was kept to zero, inorder to understand the effect of buckling on the polarization.To do so, we simply imposed the bending deformation onsilicene without allowing any subsequent relaxation of theatomic positions. The variation of polarization for flat sil-icene and silicene is plotted in Fig. 8. From the numericalfitting, the flexoelectric coefficients for flat silicene μsi-flatand silicene μsi are identified as 0.00634 and 0.00728 nC/m,respectively. Noting that the graphene flexoelectric coefficientis μgr = 0.00286 nC/m, the ratio of μsi-flat/μgr is 2.217, whichis close to the ratio of their atomic polarizability parameters(Rsi/Rgr = 2.141 from Table I). From this, it is clear that theatomic polarizability increases the induced polarization andthus flexoelectric constants. The ratio of μsi/μgr is 2.545,which is about 15% higher than 2.217. This increase inpolarization of about 15% between silicene and flat silicenecan therefore be ascribed to the buckled structure of silicene.

Further understanding can be drawn from the contributionsof the electric fields from dipole-dipole and charge-dipoleinteractions. The contributions from E p,zi and E

q,zi to the

total electric field are estimated for flat silicene and silicenewhen the bending curvature is 0.008 Å

−1. The numerical

values for flat silicene are 91.89% and 8.10%, respectively,which are similar to that of graphene. Therefore the increaseddipole moment and flexoelectric coefficient for flat silicene is

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ZHUANG, HE, JAVVAJI, AND PARK PHYSICAL REVIEW B 99, 054105 (2019)

Init

ial

con

gura

tion

Def

orm

ed c

ongu

rati

on

(a)

0 0.01 0.02 0.03-0.02

-0.01

0

0.01

0.02

0.03

MoS2BN

(b)

FIG. 7. (a) Loading scheme for estimating piezoelectric coefficient and (b) polarization Px vs strain �xx for BN and MoS2 material systems.

primarily due to its larger atomic polarizability (Rsi/Rgr =2.141) compared to graphene.

For buckled silicene, the contributions from E p,zi and Eq,zi

to the total electric field are 76.89% and 23.10%, respectively.Comparing to flat silicene, there is an increase in Eq,zi anddecrease in E p,zi for silicene. Thus the CD model predictsthat π -σ interactions are dramatically enhanced in buckledsilicene as compared to flat silicene, which is in agreementwith recent DFT studies by Podsiadły-Paszkowska et al. [42],who found that it is easier to achieve sp3 bonding in buckledsilicene. Such changes in hybridization (pyramidalization)lead to significant charge modulations and induce large dipolemoments. The difference in the numerical contribution of Eq,z

to the total electric field in flat versus buckled silicene of15% is identical to the observed difference in magnitude ofthe flexoelectric coefficients. This demonstrates that bucklingin the atomic structure of 2D materials can induce increasedpolarization, and thus flexoelectric constants.

Germanene and stanene also have flexoelectric constantsthat are larger than graphene and BN as shown in Fig. 5,though lower than silicene. This is due to a combinationof lower polarizability of these materials as compared to

0 2 4 6 8

0

1

2

3at silicene

silicene

FIG. 8. Polarization Pz vs given strain gradient K for silicene andflat silicene.

silicene (see αCALtotal values in Table I), and due to reduced Eq,zcontributions of 20.91% and 18.45%, respectively, indicatingweaker π -σ interactions in these buckled structures ascompared to silicene.

D. TMDCs

As shown in Fig. 4, the polarization under bending issignificantly higher in MoS2 than the other 2D materials.Interestingly, for MoS2, the contributions from the dipole andcharge-induced electric fields in the z direction are 15.23%and 84.76%, respectively, where the contribution of Eq,z issignificantly higher than for the previously discussed 2Dmaterials.

As we now elaborate, the mechanism enabling the largepolarization, and thus large flexoelectric constant in MoS2, isdifferent from the other 2D materials. As shown in Fig. 6(a),MoS2 is a trilayer 2D materials in which each central Moatom bonds with the S atoms in the layers above and be-low. The thickness of this sheet is defined as the sum ofthe vertical separation between these layers. The imposedbending deformation causes the top and bottom S layers todeform differently with respect to the central Mo atom. For theinitial (flat) configuration in Fig. 6(a), the central Mo atoms,labeled as X and Y, are located 2.42 Å away from both theneighboring top and bottom layer S atoms. This initial atomicconfiguration also induces nonzero dipole moments to eachatom since the z component of ri j is nonzero. An equal andopposite dipole moment is observed for the top and bottom Satoms due to the equidistant separation with the central Moatoms, whereas no dipole moment is found on the Mo atomsdue to symmetry.

However, after bending, there are significant changes inbond length, as shown in Fig. 6(b). The bond lengths betweenatom X and its nearest S atom neighbors are unchangedeven after deformation; the bond lengths Y-A and Y-D aremeasured as 2.42 and 2.41 Å. In contrast, significant changesin bond length result for other nearest S neighbors, where acompression in the Y-B and Y-C bond lengths is identified(2.42 to 2.26 Å) in Fig. 6(b), and where an elongation ofthe Y-E and Y-F bond lengths (2.42 to 2.75 Å) is seen. Theidentified differences in bond lengths break the symmetryseen in undeformed MoS2 in Fig. 6(a), which leads to nonzerodipole moments, and increases the Eq,z contribution to thetotal electric field as compared to buckled silicene.

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0

2

4

6

-10

-8

-6

-4

-2

0

2

Graphene C1

BN

SiliceneMoS

2

Graphene C1

BN

SiliceneMoS

2

(a) (b)

0 2 4 6 8 0 2 4 6 8

FIG. 9. The variation of polarization (a) Px and (b) Py with bending curvature (K ) for graphene, BN, silicene, and MoS2 materials. Theinset in (a) and (b) represent the polarization variation for materials other than MoS2.

Interestingly, the polarizability of silicene is significantlylarger than MoS2, i.e., RMoS2/Rsi is about 0.5, according toTable I. This indicates that MoS2 has a significantly higherpolarization and flexoelectric constant than buckled silicenefor other reasons, starting with the enhanced π -σ interactions.Furthermore, a recent DFT study on the bonding charac-teristics and charge transfer in MoS2 [43] found that theS atoms share their electrons with the Mo atoms, whichresults in the transfer of electrons back to the Mo atoms.This charge transfer, coupled with the bond length asymmetrydue to bending, results in a large Eq,z, and thus large dipolemoments.

The flexoelectric coefficients for other members of TMDCgroup are smaller than MoS2 as shown in Fig. 5, wherethe flexoelectric coefficient of WS2 is three times smallerthan MoS2, and where CrS2 has an even smaller value. Wefound that the bond length asymmetry between the layersafter bending is highest for MoS2 and decreases for WS2 andCrS2, and also that the local difference in radius of curvaturefor MoS2, WS2, and CrS2 materials is 49%, 40%, and 26%,respectively, both of which lead to a decreasing contributionof Eq,z for WS2 and CrS2. The DFT study of Pike et al. [43]also found a smaller Born effective charge for WS2 comparedto MoS2, which supports the observation of lower Eq,z forWS2 compared to MoS2.

We also calculated the in-plane flexoelectric constants forall the 2D materials, as summarized in Appendix E. However,we focus on the in-plane flexoelectric constant for MoS2as they are larger than the out-of-plane constants for theother 2D materials. From Figs. 9 and 4, we observe that thepolarization Py is about an order of magnitude higher thanPz for MoS2, whereas the electric field Eq,y is less than Eq,z

for MoS2 (Eq,z/Eq,y = 7). This is because of cancellations inthe induced dipole moments in calculating the polarization.Specifically, the dipole moments py have the same sign for allS atoms, whereas pz has a different sign for the top and bottomplanes of S atoms, which induces cancellation of polarizationin the z direction making Pz smaller than Py. Thus, while Py

is higher than Pz, the flexoelectric coefficient μyxzx is less thanμzxzx due to its correlation with Eq. Overall, the enhancedπ -σ interactions and bond length asymmetry also leads to

strong in-plane electromechanical coupling, and an in-planeflexoelectric constant of μyxzx = 0.00962 nC/m.

An interesting observation from Fig. 5 is that the flexo-electric constants of graphene (C1) and CrS2 are nearly equal.Though CrS2 exhibits higher atomic polarizability and bondlength asymmetry, the final dipolar polarization is similar tographene. This is because there is a relatively low asymmetryin dipole moment between S atoms in the top and bottomlayers in CrS2, which results in some cancellation of theinduced polarization, leading to a flexoelectric constant thatis similar to graphene. However, for WS2 and MoS2, the in-creased asymmetry between layers avoids the dipole momentcancellation to achieve larger flexoelectric coefficients.

We find that MoS2 has an intrinsic bending flexoelectricconstant of 0.032 nC/m. This value is about ten times largerthan found in graphene, and about 3–5 times larger than seenin the buckled monolayers. We can compare our computedvalue with one extracted from the recent experimental studyon the electromechanical properties of MoS2 reported byBrennan and coworkers [11]. In that work, the out-of-planepiezoelectric coefficient (d ) of MoS2 using piezoresponseforce microscopy was measured to be 1.03 pm/V [11]. Thatwork also established a relationship between the flexoelectricconstant (μ) and piezoelectric (d ) coefficient under the as-sumption of small length scales and linear electric field [11] as

μ = dY t2, (10)

where Y is the elastic modulus of MoS2 and t is the monolayerthickness of MoS2. With Y = 270 GPa and t = 0.65 nm, μis about 0.091 nC/m, which is significantly higher than ourcalculated value of 0.032 nC/m. This difference is due tothe usage of elastic modulus in Eq. (10), where the usage ofthe bending modulus may be more appropriate. The bendingmodulus of MoS2 was previously found to be about 9.61 eV or65.01 GPa [44]. Using this bending modulus, the flexoelectriccoefficient from Eq. (10) with bending modulus gives a valueof 0.021 nC/m. This value is close compared to the calculatedvalue of 0.032 nC/m from this work and demonstrates thatthe flexoelectric constant for MoS2 estimated using the

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-30 -20 -10 0 10 20 30-1.5

-1

-0.5

0

0.5

1

1.5

-30 -20 -10 0 10 20 300

2

4

6

(a) (b)

FIG. 10. Binwise distribution of polarization (a) Px and (b) Py along x axis for BN at a curvature of K = 0.08 nm−1.atomistic CD model is in good agreement with experimentalmeasurements.

IV. CONCLUSION

In this work, we used classical atomistic simulations ac-counting for charge-dipole interactions to study the bend-ing flexoelectric constants for four groups of 2D materi-als: graphene allotropes, nitrides, graphene analog mono-layer group-IV elements, and TMDs. Our proposed bendingsimulations enabled us to directly estimate the flexoelectricconstants by eliminating the piezoelectric contribution to thepolarization. In doing so, we were able to analyze the mech-anisms underpinning the calculated flexoelectric constantsby interpreting them through the electric fields generatedfrom dipole-dipole (σ -σ bonding) and charge-dipole (π -σbonding) interactions. While the charge-dipole interactionsincrease with bending curvature, their relative weakness in theflat monolayers (graphene, h-BN) lead to lower flexoelectricconstants for these materials. In contrast, we found that buck-ling, which occurs in the monolayer group-IV elements, leadto > 10% increases in flexoelectric constant. Finally, due tosignificantly enhanced charge transfer coupled with structuralasymmetry due to bending, the TMDCs are found to havethe largest flexoelectric constants, including MoS2 having aflexoelectric constant ten times larger than graphene.

ACKNOWLEDGMENTS

The authors would like to acknowledge the support ofNSFC (11772234), the Fundamental Research Funds for theCentral Universities and ERC Starting Grant (802205, X.Z.).

APPENDIX A: CHARGE-DIPOLE POTENTIAL MODEL

The charge-dipole potential model was first proposed byOlson et al. [27]. This model assumes that atom i in a systemis associated with a net point charge qi and a dipole momentpi. This model has been further developed to overcome the nu-merical divergence under point charge approximation [28,29].The total electrostatic energy (ECD) for an N atom system is

given as

ECD = 12

N∑i

N∑j,i �= j

qiTq−q

i j q j −N∑i

N∑j,i �= j

qiTq−pi j p j

− 12

N∑i

N∑j,i �= j

piTp−pi j p j +

1

2

N∑i

qiTq−q

ii qi

+ 12

N∑i

piTp−pii pi −

N∑i

qiχi −N∑i

piEext(ri ), (A1)

where χi is the electron affinity of atom i. T q−q, Tq−p,and Tp−p represent charge-charge, charge-dipole, and dipole-dipole interaction coefficients, respectively. The first threeterms in Eq. (A1) describe the mutual interaction of atomiccharges and dipoles among different atoms. The fourth andfifth terms in Eq. (A1) represent the energy required to createa charge and dipole on atom i. The sixth term represents thenucleus-to-electron interaction energy. The last term representthe energy due to external electric fields. T q−q represents theCoulombic interaction between atomic charges, which areseparated by a distance ri j . This coefficient diverges underthe point charge approximation when accounting for the self-charge term in Eq. (A1). In order to avoid the divergence of theself-energy term, the point charge approximation is modifiedinto atoms with Gaussian distributed charges [28,29]. Thecharge distribution for atom i at position r is

ρi(r) = qiπ3/2R3

exp

(−|r − ri|

R2

), (A2)

where ri is position vector of atom i. R is equal to√R2A,i + R2B, j/

√2, where RA,i represents the width of Gaus-

sian distribution for atom index i with type A. RB, j representthe Gaussian distributed charge width for atom type B andwith index j. For further details about charge-dipole potentialrefer to Refs. [28,29] and references therein. The parameter Rcan be estimated from the atomic polarizability. The method-ological details are given in next section.

For a given atomic configuration, the charge q and dipolemoment p for each atom can be found from the minimizationof total electrostatic energy Eq. (A1). The energy minimiza-

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tion with respect to qi yields

T q−qii qi +N∑

j,i �= jT q−qi j q j −

N∑j,i �= j

Tq−pi j p j = χi. (A3)

The energy minimization with respect to pi is

Tp−pii pi −N∑

j,i �= jTp−pi j p j −

N∑j,i �= j

Tq−pi j q j = Eext(ri ). (A4)

From the numerical solution of Eqs. (A3) and (A4), the chargeand dipole moment are known for the atomic configuration.The electrostatic force and energy calculated from the chargeand dipole moment of each atom are supplied to the atomicdynamical equation of motion in addition to the strong shortrange interactions. Further details on the numerical implemen-tation of charge-dipole potential can be found in recent workby the authors [20].

From the known values of the dipole moment, polarizationfor the unit cell is defined as the sum of dipole moments ofatoms present in that unit cell divided by the volume of theunit cell. The polarization of the mth unit cell (Pm) is

Pm = 1Vm

(n∑

i=1pi

), (A5)

where n is the number of basis atoms present in unit cell m,Vm is the volume of the unit cell, and the total polarization isthe average among all unit cells in the system.

APPENDIX B: ESTIMATION OF CHARGE-DIPOLEPOTENTIAL PARAMETER R

Consider Eq. (A4) for dipole moments, which isrewritten as

Tp−pii pi −N∑

j,i �= jTp−pi j p j =

N∑j,i �= j

Tq−pi j q j + Eext(ri ). (B1)

This represents that the dipole moment of an atom is definedby three different parts: electric field at position ri due toneighboring (i) dipoles p j [the left-hand-side second term inEq. (B1)]; (ii) charges q j [the right-hand-side first term inEq. (B1)], and (iii) from the externally applied electric fields.The diagonal coefficient Tp−pii is known as the inverse ofatomic polarizability tensor (α). The mathematical expressionfor Tp−pii is given under the CD potential approximations is[28,29]

Tp−pii =1

4π�0

√2

3√

πR3= 1

αi, (B2)

where �0 is the dielectric permitivity of vacuum. The CDparameter R is related to the polarizability α. For an Natomic system, Eq. (B1) modifies into a matrix-vector system,which is

Ap = E, (B3)

where

A =

⎡⎢⎢⎢⎣

α−11 Tp−p12 · · · Tp−p1N

Tp−p21 α−12 · · · Tp−p2N

· · · · · ·Tp−pN1 T

p−pN2 · · · α−1N

⎤⎥⎥⎥⎦ (B4)

and p and E represent the vector of dipole moments andassociated external electric field of each atom, respectively. Inorder to estimate the polarizability, assuming that the dipolesare experiencing a uniform electric field (E) (which includesboth external fields and charge related fields) [45,46], the totaldipole moment (ptotal) of atomic system is written as

ptotal = αtotalE, (B5)where αtotal is the total polarizability of the atomic system,which is expressed as

αtotal =N∑i

N∑j

Bi j, (B6)

where Bi j is the components of matrix A−1. Equation (B6)represents that, in order to estimate R, αtotal has to be known.The polarizability can be calculated from the changes inelectronic wave functions. We have used the function polarin GAUSSIAN [47] software to estimate αtotal, where detailsabout computing polarizability in DFT calculations are foundelsewhere [48,49].

In order to estimate the R value for graphene, DFT sim-ulations are performed for different sized graphene systems.The isotropic polarizability values from DFT (αDFTtotal ) for thesesystems are noted. With the atomic coordinate informationand assuming R between 0.1 to 1.0 Å, the total polarizabilityis calculated (αCALtotal ) from Eq. (B6). The atoms present ingraphene unit cell are named as A and B. Since both arecarbon atoms, it is assumed that RA is equal to RB becauseof the identical electron negativities of these atoms in the unitcell. For graphene samples with size greater than 1 nm showthat for R = 0.64 Å, αCALtotal is identical to αDFTtotal . The estimatedR value for graphene is in close agreement with the estimatebased on fullerene structure [29]. The size based studies arecarried for BN. RA defines the parameter for N atom and RB forthe B atom. The electron negativities suggest B atom shouldhave a low Gaussian distribution of electronic density whencompared to N atom. During the estimation of αCALtotal usingEq. (B6), it is assumed that RA is greater than RB and theestimate matches with the αDFTtotal at 0.76 and 0.35 for N and B,respectively. Similar type of studies are performed to estimatethe CD parameter for other materials. The calculated totalpolarizability from DFT and derived estimate from Eq. (B6)and the parameter R are tabulated in Table I.

APPENDIX C: VALIDATION OF CHARGE-DIPOLEMODEL

In this section, the CD model parameters are validated bycalculating the piezoelectric coefficients of BN and MoS2. A80-Å square sheet of BN and MoS2 is subjected to in-planestretching by displacing (ux ) the left and right ends of thesheet, shown in Fig. 7(a). The deformed atomic configurations

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TABLE III. Anisotropic flexoelectric coefficients [μzxzx, μyxzx (ay1), μxxzx (ax1)] given in nC/m. a0 has units of C/m

2, while a2 has units of C.

Material μzxzx ay1(μyxzx ) ax1(μ

xxzx ) ay0 ax0 a

y2 a

x2

Graphene 0.00286 1.53 × 10−7 5.18 × 10−6 −7.40 × 10−7 −1.61 × 10−8 −1.72 × 10−4 1.97 × 10−4Silicene-flat 0.00634 1.04 × 10−5 1.91 × 10−5 1.69 × 10−5 5.35 × 10−8 −0.00180 0.00072BN 0.00026 0.00146 3.06 × 10−5 −5.76 × 10−6 3.04 × 10−7 0.26296 −1.96 × 10−4Silicene 0.00728 0.0027 2.94 × 10−5 −6.43 × 10−5 8.03 × 10−7 −0.0164 0.00152MoS2 0.03194 0.00962 0.00164 −0.00039 0.00060 −4.5484 −0.0512

are energy minimized using the conjugate gradient scheme,after which the charge and dipole moments are obtainedfrom Eqs. (A3) and (A4) of CD model. A linear variation isobserved between total polarization and strain, as shown inFig. 7(b). The piezoelectric coefficient (slope of this variation)for BN and MoS2 are in good agreement with the reportedDFT estimations (see Table II). This validates the CD parame-ters derived from DFT and the prediction of electromechanicalbehavior.

APPENDIX D: FLEXOELECTRICITY INUNSTABLE STRUCTURE

The developed simulation scheme with CD model is ap-plied to unstable 2D structure of silicene. For unstable silicene(flat silicene), the buckling height is assumed as zero tounderstand the effect of buckling height on the flexoelectricpolarization. The polarization response with the strain gradi-ent is given in Fig. 8. The response of silicene was also addedfor comparison purpose. The slope of silicene and flat siliceneare differ only by 15%, which is directly connected with theabsence of buckling height.

APPENDIX E: IN-PLANE FLEXOELECTRICPOLARIZATION

The variation of in-plane polarization Px and Py show aquadratic dependence with bending curvature K as shown inFig. 9, which is similar to earlier reports for BN [71–73].

Py = dyzx�zx + μyxzx ∂�zx

∂x= ay0 +

1

2ay1K +

1

4ay2K

2, (E1)

Px = dxzx�zx + μxxzx ∂�zx

∂x= ax0 +

1

2ax1K +

1

4ax2K

2. (E2)

In the above equations, ax1 and ay1 have units of C/m, which

are those of flexoelectric constants, while a0 and a2 have unitsof C/m2 and C, respectively. Taking a1 as the flexoelectriccoefficient, the numerical values for graphene, BN, silicene,and MoS2 are tabulated in Table III along with the bendingflexoelectric coefficients. We note that because the mechanicalbending we imposed to generate the out-of-plane flexoelectricconstants only generate a constant strain in the x directionfrom applied deformation in the z direction, the in-planepolarization that is generated also results in contributions toin-plane piezoelectricity.

From Table III, the in-plane flexoelectric coefficients(μyxzx and μxxzx ) are significantly smaller than the out-of-plane coefficient (μzxzx ) of graphene under bending deforma-tion. The corresponding in-plane polarizations Px and Py arealso lower than the out of plane polarization Pz. This implies

that the in-plane π -σ interactions for graphene generate rel-atively small in-plane dipole moments. A symmetry analysiscan be used to show that μyxzx and μxxzx are zero [74], whichimplies that the graphene system is isotropic [75]. Similarly,lower in-plane flexoelectric coefficients are observed for flatsilicene (see Table III).

In the case of anisotropic BN [75], the in-plane coeffi-cient (μyxzx ) is nearly five times larger than the out-of-planecoefficient (μzxzx ), as shown in Table III, while Figs. 9(b)and 4 show that the polarization Py is higher than Pz. Inaddition, the charge-dipole coupling induced electric fieldEq,y is greater than Eq,z, and that the ratio of Eq,y/Eq,z is about4.6, which is similar to the ratio between the in-plane and outof plane flexoelectric coefficients. First principles calculationsfor a corrugated BN sheet [76] provide significant in-planepolarization, which is related to π and σ chemical bond shiftsdue to the out-of-plane atomic displacements. The very smalldifference in out of plane displacements of the B and N atoms[77,78] leads to relatively small out-of-plane dipole moments,and also suggests that in-plane π -σ interactions are strongerwhich makes μyxzx is higher than μzxzx.

The flexoelectric coefficient μzxzx is higher than μyxzx forbuckled silicene. The corresponding polarization Pz is greaterthan Py, as observed from Figs. 4 and 9(b), and the out of planeelectric field is larger than the in-plane electric fields, whichimplies that the π -σ coupling is stronger out-of-plane thanin-plane. It is also noted that the atomic buckling in silicenesignificantly enhances the in-plane flexoelectric coefficientcompared to flat silicene (see values of μyxzx for siliceneand silicene-flat materials in Table III). Because the bucklingheight of silicene is significantly larger than seen in BN, largerdipole moments and thus larger in-plane and out-of-planeflexoelectric constants are predicted for buckled silicene ascompared to BN.

In BN, silicene, and MoS2, it is observed from Fig. 9for BN that the in-plane polarization Py is higher than thein-plane polarization Px. In the present study, the x direction isconsidered as the armchair configuration while the y directionrepresents the zigzag configuration, which means that polar-ization in the zigzag direction is higher than in the armchairdirection, which was previously observed for BN [76]. Tofurther confirm this, we rotated the atomic system to changethe x direction to zigzag and the y direction to armchair andrepeated the bending test. There is no change in polarizationPz and the corresponding flexoelectric coefficients. Whencoming to in-plane polarizations, we found px (zigzag) ishigher than py (armchair). We also observed that px is linearand py is parabolic [see Figs. 10(a) and 10(b)], which is alsothe same form for the strain fields that were observed in thearmchair and zigzag directions. This implies that the local

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atomic configuration strongly impacts the deformation, andthus the induced polarization, which was also observed insilicene and MoS2. The observation of anisotropic in-planepolarization due to bending is similar to the earlier findingsreviewed by Ahmadpoor and Sharma [75].

In summary, we find that monatomic unit cells, such asgraphene and flat silicene, do not exhibit spatial variationsin the out of plane displacements due to bending, whereasMoS2, buckled silicene and BN do exhibit, to varying degrees,spatial variations in the out of plane displacements due to

bending. As a result, the in-plane charge-dipole interactionsare week for graphene and flat silicene, resulting in lowin-plane flexoelectric constants. In contrast, MoS2 exhibitssignificant structural asymmetry under bending, which en-hances the in-plane π -σ coupling, with a similar effect seenin buckled silicene. While BN does not exhibit some spatialvariation in the out of plane displacements, the out of planedisplacements are relatively small, and as such, the in-planeflexoelectric constants are smaller than buckled silicene andMoS2.

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