ARTICLE IN PRESS

Physica B 405 (2010) 3501–3506

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

$The

(107741� Corr

E-m

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

First-principles calculations on third-order elastic constants and internalrelaxation for monolayer graphene$

Rui Wang �, Shaofeng Wang, Xiaozhi Wu, Xiao Liang

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 9 February 2010

Received in revised form

20 April 2010

Accepted 12 May 2010

Keywords:

First-principles calculations

Third-order elastic constants

Internal relaxation

Graphene

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

016/j.physb.2010.05.032

work is supported by the National Natural

96).

esponding author. Tel.: +86 13527528737.

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

a b s t r a c t

The method of homogeneous deformation is combined with first-principles total-energy calculations on

determining third-order elastic constants and internal relaxation for monolayer graphene. We employ

density functional theory (DFT) within generalized-gradient-approximation (GGA). The elastic

constants are obtained from a polynomial fitted to the calculations of strain–energy and strain–stress

relations. Our results agree well with recent calculations by DFT calculations, tight-binding atomistic

simulations, and experiments with an atomic force microscope. The internal relaxation displacement

has also been determined from ab initio calculations. The details of internal lattice relaxation by first

principles are basically consistent with the previous molecular dynamics (MD) simulation. But for tiny

deformation, there is an anomalous region in which the behavior of internal relaxation is backward

action. In addition, we have also demonstrated that the symmetry of the relationship between the

internal displacement and the infinitesimal stains can be satisfied.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Monolayer graphene, which is a true 2D material of sp2 bondedcarbon systems, has been intensely investigated on the horizon ofmaterials science and condensed-matter physics. Recently,graphene has been confirmed as the strongest material ever tobe measured being able to reversible elastic deformations inexcess of 20% [1]. The elastic response of the graphene must beconsidered nonlinear because the stress–strain response mustcurve over to a maximum point that defines the intrinsic breakingstress [1]. Moreover, numerical simulations for graphene sheetshave also demonstrated that the nonlinear elasticity shouldbe considered [2–6]. Besides, for noncentrosymmetric lattice,internal relaxation is very important to determine the elasticproperties [7] and electron band-structure [8].

In nonlinear elasticity theory, the second-order elasticconstants (SOECs) describe the linear elastic stress–strainresponse, including the propagation velocity of acoustic distur-bances along different crystallographic directions. Higher-orderelastic constants, such as third-order elastic constants (TOECs),reflect the nonlinear elasticity of the material, including changesin acoustic velocities due to finite-strain [9,10]. Therefore, bothTOECs and SOECs are very important parameters for modeling themechanical repones of lattice under high pressure. In previous

ll rights reserved.

Science Foundation of China

studies, the TOECs for a wide variety of materials have beendetermined from experiments [11]. First-principles calculationhas demonstrated its success in theoretical studies of elasticproperties on various materials.

More recently, several studies have studied the nonlinearin-plane elastic response and internal relaxation for monolayergraphene. Liu et al. [2] employed ab initio calculations ofideal strength and phonon instability by considering nonlinearelastic response. Cadelano et al. [4] combined continuumelasticity theory and tight-binding atomistic simulations tocalculate the corresponding moduli, and the density-functionaltheory has also used to calculate the nonlinear in-planeelastic properties by Wei et al. [3]. Zhou et al. [7] developed aMD approach to directly determine the internal displacementas a function of the macroscopic strain. In this paper, weperform first principles calculations of the linear and nonlinearelastic constants, and obtain the relationships betweenthe internal relaxation and the applied strain. The internalrelaxation displacement is consistent with the previous MDsimulation [7] in general. But for tiny deformation, it is foundthat the behavior of internal relaxation is quite different andabnormal.

The paper is organized as follows. In Section 2, we recall somebasic facts from nonlinear theory of elasticity, and give theindependent elastic constants. Section 3 contains description ofemployed computational methodology. The linear and nonlinearelastic moduli from first principles calculations are also presentedhere. Section 4 deals with the determining of relationshipbetween internal relaxation and the applied strain.

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R. Wang et al. / Physica B 405 (2010) 3501–35063502

2. Nonlinear elasticity theory applied to 2D graphene

Our method for calculating elastic constants mentionsfinite-strain continuum elasticity theory. Hence, we will recallsome basic facts from nonlinear theory of elasticity [12–14],and apply it on 2D graphene. Let us consider point x¼(x1,x2)which, after applying strain to a crystal in basal plane, movesto the position x

0

. After introducing the deformation Jacobianmatrix J,

Jij ¼@x0i@xj

, ð1Þ

we may define the Lagrangian strain g [14],

Zij ¼1

2

Xp

ðJpiJpj�dijÞ, ð2Þ

which is a convenient measure of deformation for an elastic body.Elastic constants are defined by expanding the internal energy

E as a Taylor series in Lagrangian strain g,

EðS0,fZijgÞ ¼ EðS0,0ÞþS0

2!

Xijkl

CijklZijZklþS0

3!

Xijklmn

CijklmnZijZklZmnþ � � � ,

ð3Þ

where E(S0,0) and S0 are the corresponding ground state internalenergy and the area of the unstrained graphene, respectively. Ifone uses a notation (xx-1, yy-2, xy-3), which is similar asVoigt notation in 3D solid, Eq. (3) can be rewritten as

EðS0,fZigÞ ¼ EðS0,0ÞþS0

2!

Xij ¼ 1,3

CijZiZjþS0

3!

Xijk ¼ 1,3

CijkZiZjZkþ � � � , ð4Þ

with the strain tensor becoming the following form:

g¼Z1

1

2Z3

1

2Z3 Z2

0BB@

1CCA, ð5Þ

where the Cij and Cijk denote here second- and third-order elasticconstants, respectively [14]. If we introduce Jij ¼ dijþeij andassume that eij is symmetric (rotation free) linear strain tensor,the definition of Zij (Eq. (2)) yields

Zij ¼ eijþ1

2

Xk

eikekj: ð6Þ

Substituting the above result to the expansion in Eq. (4) andleaving only terms up to second order with respect to componentsof eij recover the infinitesimal theory of elasticity.

Since the graphene is the 2D hexagonal lattice belonging topoint group D6h, one applies the symmetric transformation T andhas

Cijkl ¼Xghmn

TigTjhCghmnTkmTln, ð7Þ

Cijklpq ¼X

ghmnrs

TigTjhTkmCghmnrsTlnTprTqs: ð8Þ

The elastic constants are the expansion coefficients in the Taylorseries of Eq. (4) [14], i.e.,

Cij ¼1

S0

@2E

@Zi@Zj

�����Z ¼ 0

ðSOECÞ, ð9Þ

Cijk ¼1

S0

@3E

@Zi@Zj@Zk

�����Z ¼ 0

ðTOECÞ ð10Þ

so Cij and Cijk are the symmetric tensor for continuous functionEðS0,fZigÞ. The independent second-order elastic constants and the

relationships between the remaining non-zero components andthe independent ones are directly obtained from Eq. (7), as

fCijg ¼

C11 C12 0

C12 C11 0

0 01

2ðC11�C12Þ

0BBB@

1CCCA, ð11Þ

and there are two independent elastic constants, C11 and C12,which corresponds to in-plane isotropic linear elasticity for whichYoung’s modulus is E¼(C11

2�C12

2 )/C11 and Poisson’s ration¼ C12=C11 [7]. Also, from Eq. (8) we can directly obtain thenon-zero third-order elastic constants,

C112 ¼ C121 ¼ C211,

C133 ¼ C313 ¼ C331 ¼�13ðC111þC112�2C222Þ,

C233 ¼ C333 ¼ C332 ¼13ð2C111�C112�C222Þ,

C122 ¼ C212 ¼ C221 ¼13ðC111þ4C112�2C222Þ, ð12Þ

and there are three independent elastic constants, C111, C112 andC222.

Another fundamental quantity in the theory of finite deforma-tions is Lagrangian stress t

tij ¼1

S0

@E

@Zij

, ð13Þ

which can be expressed in terms of true stress tensor r using thefollowing formula:

t¼ detð J ÞJ�1rðJTÞ�1: ð14Þ

3. Determination of second- and third-elastic constants

3.1. Computational methodology

In the work presented here, we carry out first-principles total-energy calculations based on the density functional theory (DFT)level, using the Vienna ab initio simulation package (VASP 4.6)[15–17]. The Perdew–Wang [18] (PW91) exchange-correlationfunctional for the generalized-gradient-approximation (GGA) isused. A plane-wave basis set is employed within the frameworkof the projector augmented wave (PAW) method [19,20]. Vacuumspace of 20 A normal to graphene plane is used to avoidinteractions between two layers. Reciprocal space was representby Monkhorst-Pack special k-point scheme [21] with 15�15�1grid meshes. The structures are relaxed without any symmetryconstraints with a cut-off energy of 600 eV. To avoid anywrap-around errors a sufficiently large Fourier grid is used,including all wave-vectors up to twice the cut-off wavenumber.The equilibrium theoretical crystal structures for graphene aredetermined by minimizing the Hellmann–Feynman force on theatoms and the stress on the unit cell. The convergence of energyand force are set to 1.0�10�6 eV and 1.0�10�3 eV/A, respec-tively. The geometry optimization for the C–C bond length isaccurate to 0.001 A which is very reasonable to reach numericalaccuracy of the strain–energy and strain–stress, and our calcu-lated result 1.424 A is in good agreement with the experimentalvalue 1.419 A. To obtain the unit cell for the strained crystal, thedeformation Jacobian matrix J is applied to the undeformedgraphene primitive vectors ai to obtain the deformed latticevectors a

0

i,

a01a02

!¼

a1

a2

!J¼

a1

a2

!ð1þeÞ, ð15Þ

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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

−3000

−2500

−2000

−1500

−1000

−500

0

|ηmax

|

Ela

stic

con

stan

ts (N

m−1

)

C112

C111

C222

Fig. 1. Third-order elastic constants of graphene as a function of the maximum

stain jZmaxj including in polynomial fit.

Table 1Comparison of the calculated second- and third-order elastic constants for

monolayer graphene with previous calculations.

Present results Present results Previous calculations

Strain–energy Strain–stress

C11 342 343 347.25a, 353b, 358.1c

C12 64 65 54.99a, 58.2b, 60.4c

C111 �2832 �2834 �2817c, �2725d

C112 �390 �392 �337.1c, �591d

C222 �2684 �2686 �2693.3c, �2523d

All data are in units of N m�1.

a Ref. [7], MD simulation.b Ref. [23], from bulk graphite.c Ref. [3], ab initio calculations.d Ref. [4], Tight-binding atomistic simulation.

R. Wang et al. / Physica B 405 (2010) 3501–3506 3503

and the undeformed primitive vectors for graphene are defined as

a1

a2

!¼

a 0

1

2a

ffiffiffi3p

2a

0@

1A, ð16Þ

where a is the lattice for graphene. The deformation Jacobianmatrix J are determined from the Lagrangian strain g by invertingEq. (2). For a given g, the associated deformation Jacobian matrix Jare not unique, the various possible solutions differing from oneanother by a rigid rotation. The lack of a one-to-one relationshipbetween g and J is not a concern because the calculated energy isinvariant under rigid deformation [22]. For a given applieddeformation, relaxation of the crystal internal coordinates forthe deformed unit cell was performed to obtain the minimizedenergy for the strained lattice.

The calculated procedure is as follows. To obtain a solvablesystem for the TOECs, the number of applied strain tensors mustbe as large as the number of independent TOECs of graphene.Hence, we consider three sets of deformations parametrized by Z:

gA ¼ ðZ,0,0Þ,

gB ¼ ð0,Z,0Þ,

gC ¼ ðZ,Z,0Þ: ð17Þ

The corresponding strain–energy and strain–stress on deforma-tion parameter Z for considered types of deformation gA, gB andgC are expressed as

Energy

DEðgAÞ

S0¼

1

2C11Z2þ

1

6C111Z3,

DEðgBÞ

S0¼

1

2C11Z2þ

1

6ðC111þC133�C233ÞZ3,

DEðgCÞ

S0¼ ðC11þC12ÞZ2þ

4

3ðC111�C133�2C233ÞZ3: ð18Þ

Stress

t1ðgAÞ ¼ C11Zþ12C111Z2,

t2ðgBÞ ¼ C11Zþ12ðC111þC133�C233ÞZ2,

t1ðgCÞ ¼ ðC11þC12ÞZþðC111�23 C133�

43C233ÞZ2,

t2ðgCÞ ¼ ðC11þC12ÞZþðC111�13 C133�

53C233ÞZ2,

where DEðgÞ ¼ EðS0,gÞ�EðS0,0Þ. An important parameter in thesecalculations is the value of maximum strain parameter Zmax

included in the polynomial fit. In this work, the fitted coefficientsfor the second-order terms are very stable and are almostindependent of the range of fitting. However, the coefficients forthe third-order term are more sensitive to Zmax. To display thisfeature, several TOECs are plotted in Fig. 1 as a function of thejZmaxj in the fitting. The figure shows that the results from thepolynomial fit converge for maximum strain parameters Zmax

above a magnitude 0.04. In every case for gi (i¼A,B,C), Z is variedbetween �0.08 and 0.08 with step 0.005 to obtain accurateTOECs. For every deformed configuration, the positions of atomsoptimized and both energy and stress tensors are calculated onthe basis of quantum DFT formalism. In this way, for each type ofdistortion, dependencies of energy EðgÞ and stress tðgÞ on strainparameter g are obtained.

3.2. Results and discussion

The calculated results are listed in Table 1, which also providesour prediction for second- and third-order elastic constants andcompares them with previous calculations [3,4,7,23]. C11 value isobtained slightly different results from different fits (e.g.,C11¼344 and 340 N m�1 from coefficients in DEðgAÞ=S0 andDEðgBÞ=S0, respectively). In such cases, the average of allobtained values is given in Table 1. The VASP package calculatesthe strain–stress relation in 3D materials, so the elastic constantsobtained from strain–stress relation for 2D graphene must bemultiplied by the interlayer spacing 20 A. The strain–energyincluding the results of first principles calculations and the fittedcurves obtained from nonlinear and linear elasticity theory areshown in Fig. 2. The strain–energies with negative strains arealways larger than ones with positive strains (e.g., see Fig. 2(a),(b), and (c) for Lagrangian strains gA, gB, and gC , respectively), sothe values of third-order elastic constants Cijk is typically negative.The presence of the TOECs in strain–energy lead to a lessening ofstiffness at high tensile strains and an increasingly stiff responseat high compressive strains.

Next interesting issue is to examine for which range of strainsthe third-order effects really matter. In Fig. 2, we also compareenergy for the deformations with energy values obtained withinlinear and nonlinear elasticity theories. One can clearly see that

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

0.2

0.4

0.6

0.8

1

1.2

1.4

Lagrangian Strain η

Δ E

(ηA

)/S (J

m−2)

First principles resultsnonlinear elasticitylinear elasticity

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

0.5

1

1.5

2

2.5

3

3.5

Lagrangian Strain η

Δ E

(ηC

)/S (J

m−2

)

Finst principles resultsnonlinear elasticitylinear elasticity

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

0.2

0.4

0.6

0.8

1

1.2

1.4

Lagrangian Strain η

Δ E

(ηB

)/S (J

m−2)

First principles resultsnonlinear elasticitylinear elasticity

Fig. 2. Strain–energy relations for 2D graphene. Star points denote results of first principles calculations; solid and dashed lines indicate the curves obtained from

nonlinear and linear elasticity theory, respectively. (a), (b), and (c) describe Lagrangian strains gA , gB , and gC , respectively.

Table 2Comparison of the calculated linear elastic moduli for monolayer graphene with

the experimental values and previous calculations. Young’s moduli E is in units of

N m�1 and Poisson’s ratio n is dimensionless.

E n

This work 330.4 0.188

Ref. [1] 340750 –

Ref. [2] 350 0.186

Ref. [3] 348 0.169

Ref. [4] 312 0.31

Ref. [7] 338.5 0.158

Ref. [23] 343 0.165

Ref. [24] 384 0.227

Ref. [25] 345 0.149

Ref. [26] – 0.173

R. Wang et al. / Physica B 405 (2010) 3501–35063504

the linear approach is not sufficient for strains larger thanapproximately 3.0%. Besides, from Table 1, we obtain Young’smodulus and Poisson’s ratio for the monolayer graphene as330.4 N m�1 and 0.188, respectively. Young’s modulus agrees well

with experiment result 340750 N m�1 reported by Lee et al. [1].In Table 2, we list the values of calculated elastic moduli ofmonolayer graphene, accompanied by available experimentaldata [1] and previous theoretical values [2–4,7,23–26]. Incomparison with the previous first principles calculations [3],we employ GGA-PW91 exchange-correlation functional whichseems to be reasonable as well as the GGA-PBE functional forgraphene. The optimized C–C bond length is 1.424 A, which ismore close to the experimental value 1.419 A than previous result1.412 A obtained from GGA-PBE functional [3].

4. Internal relaxation displacements

In continuum elasticity theory, a macroscopically homoge-neous deformation is described by a constant deformationJacobian matrix J (Eq. (1)). Graphene is typically 2D compoundlattice and its unit cell contains two atoms which are denoted byA and B, respectively. Though the deformation for each sublatticecan fully described by the macroscopic strain, the displacementsof some atoms in the unit cell do not necessarily follow the same

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R. Wang et al. / Physica B 405 (2010) 3501–3506 3505

rule for compound lattice. When graphene undergoes a homo-geneous deformation to arrive the equilibrium-deformation state,the atom B does have an internal relaxation [7]. For example,applying a uniaxial strain ðe11a0, e22 ¼ 0 and e12 ¼ 0Þ to the unitcell and this procedure is consistent with applying Lagrangianstain gA, as shown in Fig. 3(a), would stretch the tilted bonds andchange the bond angles, while the length of the vertical bondwould remain unchanged by assuming homogeneousdeformation within the unit cell. The changes of bond angle andbond length would alter the overlap integral for s electrons, sothe internal equilibrium must be maintained by allowing verticaldisplacements of the atoms that lead to stretching or compressing

Fig. 3. Applying uniaxial strain ðe11 a0, e22 ¼ 0 and e12 ¼ 0Þ to graphene: (a) without in

internal relaxation, the two atoms in the unit cell have a relative vertical displacem

deformation is still considered homogeneous at the macroscopic sale and the overall s

−0.1 −0.05 0 0.05 0.1−4

−2

0

2x 10

−4

−0.1 −0.05 0 0.05 0.1−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

Strain ε

Strain ε

Inte

rnal

rela

xatio

n di

spla

cem

ents

(A)

Inte

rnal

rela

xatio

n di

spla

cem

ents

(A)

Fig. 4. The internal relaxation for graphene under applying four sets of strain: (a) denote

of atom B within the unit cell. (b) and (c) represent uniaxial strain in the x direction and

respectively, and there is only internal relaxation along the y-direction in both the cases

dominated in the x-direction in this case.

of the vertical bond, denoted by s, as illustrated in Fig. 3(b).Recently, Zhou and Huang [7] developed a MD approach todirectly determine the internal displacement as a function of themacroscopic strain. Here, the internal displacement is determinedby first principles calculations in which the B atom allow to relaxarbitrarily in-plane to get energy minimization.

We show the first principles results of internal displacementsthrough applying four types of deformation in Fig. 4. The internaldisplacement s is determined by comparing the position to itsinitial position upon homogeneous deformation. Fig. 4(a)represents the case of the equibiaxial deformation with themean strain, i.e., e11 ¼ e, e22 ¼ e and e12 ¼ 0, and the calculated

ternal relaxation, the atoms are only displaced in the horizontal direction; (b) with

ent denoted by s in addition to the horizontal displacement. Meanwhile, the

hape of the deformed unit cell is same for (a) and (b).

−0.1 −0.05 0 0.05 0.1−0.005

0

0.005

0.01

0.015

0.02

0.025

−0.1 −0.05 0 0.05 0.1−0.04

−0.02

0

0.02

0.04

τxτy

Strain ε

Strain ε

Inte

rnal

rela

xatio

n di

spla

cem

ents

(A)

Inte

rnal

rela

xatio

n di

spla

cem

ents

(A)

s the equibiaxial strain (i.e., e11 ¼ e, e22 ¼ e and e12 ¼ 0) without internal relaxation

(i.e., e11 ¼ e, e22 ¼ 0 and e12 ¼ 0Þ and the y-direction (i.e., e11 ¼ 0, e22 ¼ e and e12 ¼ 0Þ,

. (d) denotes shear mode (i.e., e11 ¼ 0, e22 ¼ 0 and e12 ¼ eÞ, the internal relaxation is

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R. Wang et al. / Physica B 405 (2010) 3501–35063506

results e show that internal displacements are less than 10�4 A, sothese can be negligible. In this case, B atom is always located atthe center of the unit cell and the strained lattice maintainsinternal equilibrium without any internal relaxation.

In Fig. 4(b) and (c), we apply the uniaxial deformation alongthe x-direction (i.e., e11 ¼ e, e22 ¼ 0 and e12 ¼ 0) and the y-direction (i.e., e11 ¼ 0, e22 ¼ e and e12 ¼ 0), respectively. Both thecases, the symmetry maintains the atom B always locating atmidpoint and has no internal relaxation along the x-direction inunit cell. The relationship between the internal displacements ty

and the strain parameter e is nonlinear and asymmetric forpositive and negative strain e. In small deformations (in the range�0:01oeo0:01Þ, it is interesting to find that the linearapproximation for relationship between the internal displace-ments along the y-direction and the strain is appropriate for bothapplying uniaxial strain along the x-direction and the y-direction.For the strain regions �0:08oe12o�0:01 and 0:01oe12o0:05the relationships between internal displacement and strainparameter e are consistent with the results obtained by Zhouand Huang using the MD simulation [7], but for the tiny strainregions �0:01oeo0:01, we obtain an opposite trends comparingwith ones obtained by Zhou and Huang [7].

For the shear mode (i.e., e11 ¼ 0, e22 ¼ 0 and e12 ¼ e), we find thatthe internal relaxation is dominated and nonlinear in the x-direction,while the magnitude of relaxation in the y-direction is less than 0.01 Afor �0:08oe12o0:08 in Fig. 4(d). The relationship between theinternal relaxation displacement and the shear strain e12 is symmetricwith respect to the positive and negative shear strains. For the strainregions both�0:08oe12o�0:01 and 0:01oe12o0:08, tx has linearrelationships with same scale coefficient �0.5035 A, which is goodagreement with the value �0.5779 A obtained by MD simulation [7].For tiny shear strains (in the range �0:01oe12o0:01Þ, the linearrelationship is also appropriate but has scale coefficient 0.1621 A,which has not been obtained by MD simulation [7].

Through the above discussions, we find that the internal relaxationfrom our first principles calculations is basically consistent with theprevious results obtained from the MD simulation [7] for strainregions�0:08oeo�0:01 and 0:01oeo0:05, but it is backward fortiny strain �0:01oeo0:01. The DFT results for the total energycombined with the strain-relaxation relation have enabled us todetermine internal displacement s. The comparison to the previousMD study (Ref. [7]) seems to shed some light on the puzzlingdiscrepancy on the predicted Young’s modulus of graphene by thefirst-principle and MD calculations. As discussed in Ref. [7], Young’smodulus predicted by MD would have compared closely to the first-principles calculations if the internal relaxation is turned off.However, with internal relaxation, the MD-predicted Young’smodulus is considerably lower. The present study (Fig. 4, in particular)seems to suggest that the internal relaxation is indeed negligibleunder infinitesimal strains (o0:01) based on the first-principlescalculations.

Since the linear relationships are appropriate for tiny strain,we can write internal relaxation displacement s as

ti ¼X

jk

Dijkejk: ð19Þ

Because stain tensor eij is symmetric tensor, the coefficientssatisfy

Dijk ¼Dikj, ð20Þ

and their values can be obtained by fitting the DFT results, asD111 ¼D122 ¼D212 ¼ 0,D211C�D222C0:10 A, and D112C0:08 A.The symmetry of the 2D hexagonal lattice requires

Dijk ¼Xmnl

TimTjnTklDmnl, ð21Þ

with symmetric operator T. We can directly obtainD111¼D122¼D212¼0 and D211 ¼�D222 ¼

ffiffiffi3p

D112 from Eq. (21).We must point out that the values of D122 and D222 is alwaysequivalent but with opposite sign in any strain range, because ofthe equibiaxial deformation without any internal relaxation. Thecoefficients Dijk obtained from the symmetry are in goodagreement with the first principles results.

5. Conclusions

We present first principles calculations on the nonlinear elasticproperties and internal relaxation for monolayer graphene. In ourmethod, second- and third-order elastic constants for grapheneare determined from applying three sets of homogeneous elasticdeformation. We have used two approaches involving eitherstrain–energy or strain–stress relations, obtaining consistentresults from both of them. The values of the third-order elasticconstants are in good agreement with the previous calculatedresults, and the linear elastic moduli is also consistent withprevious experimental and theoretic values. The linear approachis not sufficient for strains larger than approximately 3.0%.

Through applying homogeneous in-plane strain, we have alsodetermined the internal relaxation displacement. The resultsshows that nonlinear relationship between the internal relaxationdisplacement and the applied strain, which is basically consistentwith the previous MD simulation [7]. In tiny deformation(approximately �0:01oeo0:01Þ, the relationship between theinternal relaxation displacement and the applied strain is back-ward from the result obtained from MD simulation [7]. Theunified expression of linear relationships between the internalrelaxation displacement and the applied strain in the tinydeformation has been obtained. In addition, we have alsodemonstrated that the symmetry for the relationship betweenthe internal displacement and the tiny strain can be satisfied. Ourstudy seems to suggest that the internal relaxation is indeednegligible under infinitesimal strains ðo0:01Þ based on the first-principles calculations.

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