Physica E 43 (2011) 914–918

Contents lists available at ScienceDirect

Physica E

1386-94

doi:10.1

$The

(110743

Researc� Corr

E-m

journal homepage: www.elsevier.com/locate/physe

Nonlinear elasticity of monolayer zinc oxide honeycomb structures:A first-principles study$

Rui Wang �, Shaofeng Wang, Xiaozhi Wu, Shaorong Li, Lili Liu

Institute for Structure and Function and Department of Physics, Chongqing University, Chongqing 400044, People’s Republic of China

a r t i c l e i n f o

Article history:

Received 14 September 2010

Received in revised form

19 October 2010

Accepted 5 November 2010Available online 13 November 2010

77/$ - see front matter & 2010 Elsevier B.V. A

016/j.physe.2010.11.013

work is supported by the National Natural

13) and Project No. CDJXS11102211 supp

h Funds for the Central Universities of China.

esponding author.

ail address: rcwang@cqu.edu.cn (R. Wang).

a b s t r a c t

We present the second- and third-order elastic constants and discuss the nonlinear elasticity for

monolayer zinc oxide (ZnO) with honeycomb structure. Density functional theory (DFT) within

generalized-gradient-approximation (GGA) combining with the method of homogeneous deformation

is employed. The predictions for the elastic constants are obtained from the nonlinear least-squares

polynomial fit to the calculated strain-energy relations from first-principles total-energy calculations.

In comparison with the linear approach, the nonlinear effects really matter for strain larger than

approximately 3.0%. We discuss how internal relaxation acts on the elastic properties, and internal

relaxation displacements for the corresponding applied strain are obtained. Our results show that internal

relaxation is important for the values of elastic constants, and especially influence the third-order elastic

constants. Finally, we discuss force–displacement behavior and the breaking strength of monolayer ZnO

within a framework of nonlinear stress–strain relationship. Monolayer ZnO exhibits very high ductility, in

our study exceeding 20% ductility in tension, and the elastic response will exhibit highly nonlinear while

the third-order effects really matter.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Monolayer graphene, which is a true two-dimensional (2D)material of honeycomb structure with sp2 bonded carbon systems,has been intensely investigated on the horizon of materials scienceand condensed-matter physics due to its wide range of unusualproperties [1–3]. Normally, monolayer graphene is a semimetalwith a band profile having linear dispersion and zero gap, and itselectrons and holes behave like a massless Dirac fermion. Recently,it was predicted that Si and Ge, and even binary compoundsof Group IV elements and III–V compounds can form 2D stablemonolayer honeycomb structures based on state-of-the-art first-principles calculations [4,5]. ZnO has been the subject of variousresearches due to its wide band gap of � 3:3 eV and large excitonbinding energy of 60 meV resulting from photo-electronic applica-tions [6,7]. In earlier studies, Claeyssens and Kulkarni et al. haveindicated that graphitic ZnO exists through investigating the ZnOand its nanowires [8,9], and two-monolayer-thick ZnO(0 0 0 1)films grown on Ag(1 1 1) had been reported in experiments [10].More recently, Topsakal et al. discussed the stability of 2D ZnO, itsnanoribbons and flakes that are analyzed by phonon frequency, aswell as by finite temperature ab initio molecular-dynamics

ll rights reserved.

Science Foundation of China

orted by the Fundamental

calculations, and meanwhile the electric, magnetic and mechanicproperties of 2D monolayer and bilayer ZnO in honeycombstructure and its armchair and zigzag nanoribbons were alsoinvestigated by first-principles calculations [11].

Graphene has been confirmed to be the strongest material evermeasured being able to reverse elastic deformations in excess of 20%and the elastic response of the graphene must be considered non-linear [12–14]. ZnO is a 2D planar material of honeycomb lattice aswell as graphene, so applying finite deformation and nonlinearelasticity are also very important for practical applications. Inaddition, recent works indicated the effect of deformation on theelectronic properties for band-gap engineering [15,16]. In the lineartheory of elasticity, the infinitesimal deformations are assumed, andthe second-order elastic constants (SOECs) are sufficient to describethe elastic stress–strain response and wave propagation in solids[17]. In nonlinear elastic theory, high-order elastic constants, such asthird-order elastic constants (TOECs), as well as SOECs play animportant role [18–21]. To understand the physical properties ofthese compounds and provide significant information with respect toapplications of 2D monolayer ZnO, it is necessary to study thenonlinear elasticity. In this paper, the investigation of the nonlinearelasticity of monolayer ZnO compound in honeycomb structures isperformed by using first-principles calculations combined with themethod of applying series of finite strains, used to successfullycalculate TOECs and even high-order elastic constants [14,22–24].Besides, for noncentrosymmetric lattice, internal relaxation is veryimportant to determine the elastic properties [25] and the electronband-structure [26]. So the internal relaxation displacements of the

R. Wang et al. / Physica E 43 (2011) 914–918 915

atoms of sublattice O has also been determined until the deformedunit cell which yields the minimum total energy. We also discusshow the internal relaxation influences the values of elastic constants.In addition, the breaking strength of monolayer ZnO has beeninvestigated in the framework of the calculated SOECs and TOECs.

2. Computational method

We have performed first-principles total-energy calculationsbased on the density functional theory (DFT) level, using the Viennaab initio simulation package (VASP 4.6) developed at the Institut furMaterialphysik of Universitat Wien [27–29]. The exchange-correla-tion potential was approximated by generalized-gradient-approxi-mation(GGA) using Perdew–Burke–Ernzerhof (PBE) functional[30,31]. A plane-wave basis set is employed within the frameworkof the projector augmented wave (PAW) method [32,33]. Thestructure of monolayer ZnO is optimized using periodically repeatingsupercell having hexagonal lattice in 2D and a vacuum space of 20 Anormal to the ZnO plane is used to avoid interactions between twolayers. Reciprocal space was represented by Monkhorst–Pack specialk-point scheme [34] with 23�23�1 grid meshes. The structures arerelaxed without any symmetry constraints with a cutoff energy of500 eV. The equilibrium theoretical crystal structures are determinedby minimizing the Hellmann–Feynman force on the atoms and thestress on the unit cell. The convergence of energy and force are set to1.0�10�6 eV and 1.0�10�3 eV/A, respectively. The optimizedstructure for the Zn–O bond length is accurate to 0.001 A which is

1.89 1.892 1.894 1.896 1.898 1.9 1.902

−8.5658

−8.5657

−8.5656

−8.5655

−8.5654

−8.5653

−8.5652

−8.5651

−8.565

−8.5649

−8.5648

The length of Zn−O bond (A)

Ene

rgy

(eV

/uni

t cel

l)

Fig. 1. The energy of per unit cell as a function of the length of Zn–O bond. The

calculated length for honeycomb ZnO is 1.896 A. which is determined from the

corresponding minimum value of the internal energy.

yx

a’2

a’1

a2

a1

Fig. 2. Zinc oxide lattice under uniaxial strain in zigzag direction. Dashed line and solid lin

coincides with x-axis and armchair direction coincides with y-axis. Solid circles and dashed

O is denoted by s.

very reasonable to reach numerical accuracy of the strain-energy, andour calculated result of 1.896 A is in good agreement with theprevious first-principles calculations of 1.895 A [11]. The calculatedequilibrium lattice is shown in Fig. 1.

Our method for calculating elastic constants mentions thefinite-strain continuum elasticity theory. The relation betweenthe elastic constants and the density of strain energy C for 2D zincoxide can be written as

CðgÞ ¼1

2!CijklZijZklþ

1

3!CijklmnZijZklZmnþ . . . , ð1Þ

where Zij is Lagrangian strain tensor [21] and summation conven-tion is employed for repeating indices; herein C denotes eachhigher-order elastic constants that were defined by Brugger [18] as

Cijklmn... ¼@3C

@Zij@Zkl@Zmn � � �

�����Z ¼ 0

i,j,k, . . . ¼ 1,2: ð2Þ

After applying a macroscopically homogeneous deformation, thedeformed crystal can be described by the deformation tensorJij ¼ @xi=@aj which relates the initial and final coordinates xi and ai,respectively, and the Lagrangian strain tensor Zij ¼ 1=2ðJikJjk�dijÞ.Since the monolayer ZnO is the 2D hexagonal lattice, one applies thesymmetric transformation T and has

Cijklpq... ¼ TigTjhTkm � � �Cghmnrs...TlnTprTqs � � � : ð3Þ

In this paper, we are interested in the second-order elastic constants(SOEC), Cijkl, and third-order elastic constants (TOEC), Cijklmn. If we usea notation (xx-1, yy-2, xy-3), the Lagrangian strain tensor g linksthe notation by

g¼Z1

12Z3

12Z3 Z2

!, ð4Þ

and there are two independent SOECs (C11, C12) and three TOECs(C111, C112, C222). In addition, the relationships between the remainingnon-zero SOECs and the independent ones are as

fCijg ¼

C11 C12 0

C12 C11 0

0 0 12ðC11�C12Þ

0B@

1CA, ð5Þ

and the remaining non-zero TOECs are,

C112 ¼ C121 ¼ C211,

C133 ¼ C313 ¼ C331 ¼�14ð3C222�2C111�C112Þ,

C233 ¼ C323 ¼ C332 ¼14ð2C111�C112�C222Þ,

C122 ¼ C212 ¼ C221 ¼ C111þC112�C222: ð6Þ

To obtain a solvable system for the TOECs, the number of appliedstrain tensors must be as large as the number of independent TOECs.Hence, we consider three sets of deformations parameterized by x. InFig. 2, we show the zinc oxide unit cell with lattice vectors ai (i¼ 1,2)

O

Zn

x: Zigzagy: Armchair

τ

a’2

a2

a’1a1

e denote the undeformed lattice and deformed lattice, respectively. Zigzag direction

circles respectively, represent, sublattices O and Zn. Internal relaxation of sublattice

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.080

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Lagrangian strain ξ

Ψ(η

zig) (

Jm−2

)

DFT results

nonlinear elasticity

linear elasticity

Fig. 3. Strain-energy relations of monolayer ZnO for particular deformation gzig .

Star points denote results of first principles results; solid and dashed lines indicate

the curves obtained from nonlinear and linear elasticity theory, respectively.

R. Wang et al. / Physica E 43 (2011) 914–918916

and aiu in the undeformed and deformed configurations, respectively.The lattice vectors of the deformed zinc oxide crystal have aiu¼ Jijaj.We consider three sets of deformation:

gzig ¼x 0

0 0

� �, garm ¼

0 0

0 x

!, geq ¼

x 0

0 x

!, ð7Þ

and represent the uniaxial strain in the zigzag direction, uniaxialstrain in the armchair direction, and equibiaxial strain, respectively.The corresponding strain-energy on deformation parameter x forconsidered types of deformation are expressed as

CðgzigÞ ¼12 C11x

2þ1

6C111x3,

CðgarmÞ ¼12 C11x

2þ1

6C222x3,

CðgeqÞ ¼ ðC11þC12Þx2þ1

3ð2C111þ3C112�C222Þx3: ð8Þ

In addition, to implement the different deformation modes in ourcalculation, we must have the deformation tensor Jij, which isdetermined from the Lagrangian strain as

Jij ¼ dijþZij�12ZikZkjþ

1

2ZikZklZljþ � � � ð9Þ

For a given x, in general, J is not unique but this is not a problem sincethe Lagrange strain brings rotational invariance.

In continuum elasticity theory, a macroscopically homogeneousdeformation is described by a constant deformation gradient tensorJij. Zinc oxide honeycomb structure is typically 2D compoundlattice and its unit cell contains two atoms, Zn and O. The strains areimposed by specifying the positions of atoms of sublattice Zn on theedges of the unit cell. The positions of the atoms of sublattice O inthe plane are not constrained, and do perform internal relaxationuntil the deformed unit cell yields the minimum total energy forthe imposed strain state of the unit cell (see Fig. 2). In the followingsection, we will obtain the SOECs and TOECs by using first-principles calculations and discuss how internal relaxation influ-ences the values of elastic constants.

−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.080

0.1

0.2

0.3

0.4

0.5

Lagrangian strain ξ

Ψ(η

zig) (

Jm−2

) Without internal relaxation

With internal relaxation

Fig. 4. Strain-energy relationship with internal relaxation (square points) compares

with that without internal relaxation (circle points).

3. Results and discussions

The outputs of the fitting procedure are reported in Table 1where the full set of TOECs of monolayer is shown. In order todiscuss the internal relaxation influencing the values of elasticconstants, we provide our prediction for second- and third-orderelastic constants in both cases of the atoms of sublattice Orelaxation and unrelaxation. The strain-energy relationship foruniaxial strain gzig including the results of first-principles calcula-tions and the fitted curves obtained from nonlinear and linearelasticity theory in the case of internal relaxation are shown inFig. 3. It is worth noting that the 2D ZnO with Lagrangian strains upto 8.0%, including the terms up to third-order in energy expansionsufficed to obtain good agreement with our ab initio results. Thestrain-energies with negative strains are always larger than theones with positive strains, so the values of third-order elasticconstants CIJK is typically negative. The presence of the third-ordersin strain-energy lead to a lessening of stiffness at high tensilestrains and an increasingly stiff response at high compressive

Table 1The calculated second- and third-order elastic constants for monolayer ZnO in both relax

nonlinear elastic modulus D. All data are in units of Nm�1.

C11 C12 C111

Within internal relaxation 87.5 58.3 �511.3

Without internal relaxation 110.6 35.2 �1218.2

strains. In Fig. 3, we also compare the energy values obtained fromlinear with those obtained from nonlinear elasticity theories. Onecan clearly see that the linear approach is not sufficient for strainslarger than approximately 3.0%, and the third-order effects dom-inate the properties of solids in this region.

Next we focus on discussing the effects of internal relaxationinfluencing the values of SOECs and TOECs. In Fig. 4, we present thestrain-energy relationship within internal relaxation and thatwithout internal relaxation for uniaxial strain gzig . We can see thatthe values of strain energy within internal relaxation is alwayslower than the cases without internal relaxation, and this isreasonable. If the small strain regions are (�0:01oxo0:01), the

ed and nonrelaxed cases are listed. We also give the Young’s moduli E and effective

C112 C222 E D

�112.4 �435.5 48.7 �116.7

�275.7 �1196.3 99.4 �451.1

−0.1 0 0.1−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Strain

Inte

rnal

rela

xatio

n di

spla

cem

ents

(A)

−0.1 0 0.1−1.2

−1

−0.8

−0.6

−0.4

−0.2

0x 10−4

Strain−0.1 0 0.1

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Strain

τx

τy

Fig. 5. The internal relaxation displacements for monolayer ZnO under applying strain: (a) and (b) represent uniaxial zigzag strain gzig and armchair strain garm , respectively,

and there is only internal relaxation along the y-direction in both the cases. (c) denotes the equibiaxial strain geq without internal relaxation of atom O within the unit cell.

R. Wang et al. / Physica E 43 (2011) 914–918 917

values of energy are basically equivalent in both the cases withinand without internal relaxation. Our study seems to suggest thatthe internal relaxation is indeed negligible under infinitesimalstrains based on the first-principles calculations. In the lineartheory of elasticity, the infinitesimal deformations are assumed, sothe internal relaxation influences the values of SOECs slightly. Onthe contrary, when applying finite deformation to the materials, thenonlinear elasticity must be considered and the internal relaxationwill influence the values of the TOECs strongly. As shown in Table 1,the values of the SOECs and TOECs are in agreement with the abovediscussions and this is the physical quality. The internal relaxationdisplacements of the atoms of sublattice O s, which is defined bycomparing the position to its initial position upon homogeneousdeformation, are determined by the first-principles calculationsand are shown in Fig. 5. Fig .5(a) and (b) denote the cases of uniaxialzigzag strain gzig and armchair strain garm, respectively. In both thecases, the symmetry maintains the atom O always locating atmidpoint and has no internal relaxation along the x-direction inunit cell. Fig. 5(c) represents the case of the equibiaxial deformationgeq and the calculated results show that internal displacements areless than 10�4 A, so these can be negligible. In this case, O atom isalways located at the center of the unit cell and the strained latticemaintains internal equilibrium without any internal relaxation.

If the stress–strain response of the materials curve to a max-imum point that defines the intrinsic breaking stress, the elasticresponse must be considered nonlinear and the force–displace-ment behavior is interpreted within a framework of nonlinearstress–strain relationship. The intrinsic strength of monolayergraphene has been discussed in experiments by using an atomicforce microscope [12] and in theory by using the tight-bindingatomistic simulations [13]. The stability of 2D monolayer ZnO hasbeen demonstrated by using calculation of phonon-dispersioncurves as well as ab initio finite-temperature molecular-dynamics

calculations [11]. ZnO has 2D hexagonal lattice forming a planarhoneycomb structure as well as graphene. The intrinsic breakingstress of 2D ZnO can be described as a phenomenological nonlinearscalar relation between the applied stress s and the uniaxialLagrangian strain e [12],

s¼ EeþDe2, ð10Þ

where E¼ ðC211�C2

12Þ=C11 is the Young modulus, and D is an effectivenonlinear elastic modulus of the 2D ZnO sheet and is determinedfrom components of both the SOECs and TOECs as [13]

D¼ 32ð1�nÞ½ð1þnÞ

2L1þð1�nÞ2L2�, ð11Þ

with the Poisson ratio n¼ C12=C11 and

L1 ¼14 ðC222�C112Þ, L2 ¼

112ð2C111�C222þ3C112Þ: ð12Þ

By inserting the elastic constants of Table 1 into Eq. (11), we obtainthe nonlinear elastic moduli D as also shown in Table 1. Themaximum of the elastic stress–strain response defines the intrinsicstress, which for this functional form is sint ¼�E2=4D at the straineint ¼�E=2D. The value of breaking strengthsint for monolayer ZnOis approximated to 5:1 Nm�1 at the maximum strain e¼ 20:8% inthe case of internal relaxation. In comparison with the monolayergraphene, the value of breaking strength is approximated to41:8 Nm�1 at the strain e¼ 24:6% [12]. The results show that thegraphene is much stronger than monolayer ZnO. Meanwhile, 2Dmonolayer ZnO has very high ductility due to its large breakingstrain in tension. The elastic response will exhibit highly nonlinearwhile the third-order effects really matter. In contrast to 2Dhoneycomb structures of carbon, the calculated length of Zn–Obond is significantly larger than that of C–C due to fact that theatom Zn has a larger radius than that of C, so the strength of sp2 ofZnO is smaller than that of graphene.

R. Wang et al. / Physica E 43 (2011) 914–918918

4. Conclusions

We present the first-principles total-energy calculations ondiscussing the nonlinear elastic properties for monolayer zinc oxidehoneycomb structure within generalized-gradient-approximation(GGA) combining with the method of homogeneous elastic deforma-tion. From the nonlinear fitting, we obtained the predictions for SOECsand TOECs from the coefficients of the fitted polynomials of theinternal energy-strain functions. In comparison with the linearapproach, the nonlinear effects is sufficient for strain being largerthan approximately 3.0%. Zinc oxide honeycomb structure is typically2D compound lattice and its unit cell contains two atoms Zn and O,so we discuss how internal relaxation acts on the elastic properties.Our results show that internal relaxation is important for the values ofelastic constants and strongly influence the TOECs especially. Besides,internal relaxation displacements for the corresponding strain arealso determined. Finally, we discuss force–displacement behaviorand the breaking strength of monolayer ZnO within a frameworkof nonlinear stress–strain relationship. Our results show that thegraphene is much stronger than monolayer ZnO. Meanwhile, 2Dmonolayer ZnO exhibits very high ductility, in our study exceeding20% ductility in tension. The elastic response will exhibit highlynonlinear while the third-order effects really matter.

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