Comparative study on the nonlinear properties of bilayer graphene and silicene under tension

Accepted Manuscript Comparative study on the nonlinear properties of bilayer graphene and silicene under tension Peng Xu, Zhongyuan Yu, Chuanghua Yang, Pengfei Lu, Yumin Liu, Han Ye, Tao Gao PII: DOI: Reference:

S0749-6036(14)00330-9 http://dx.doi.org/10.1016/j.spmi.2014.08.022 YSPMI 3399

To appear in:

Superlattices and Microstructures

Received Date: Revised Date: Accepted Date:

15 July 2014 22 August 2014 23 August 2014

Please cite this article as: P. Xu, Z. Yu, C. Yang, P. Lu, Y. Liu, H. Ye, T. Gao, Comparative study on the nonlinear properties of bilayer graphene and silicene under tension, Superlattices and Microstructures (2014), doi: http:// dx.doi.org/10.1016/j.spmi.2014.08.022

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Comparative study on the nonlinear properties of bilayer graphene and silicene under tension Peng Xu1, Zhongyuan Yu1*, Chuanghua Yang1, Pengfei Lu1, Yumin Liu1, Han Ye1, Tao Gao2 1

State Key Laboratory of Information Photonics and Optical Communications,

Beijing University of Posts and Telecommunications (BUPT), P.O. Box 72 Beijing 100876, China 2

Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China.

Abstract Atomic structures and nonlinear properties of single layer graphene (SLG), bilayer graphene (BLG), single layer silicene (SLS), and bilayer silicene (BLS) under equiaxial tension and uniaxial tensions along armchair and zigzag directions have been investigated comparatively using first-principles calculations. First, we have calculated the dependences of atomic structures (bond length, interlayer distance, and buckling height) of BLG and BLS on strain under three types of tensions. There exists the weak Van der Waals interaction between two layers of BLG and the interlayer distance is not variable with strain for three types of tensions. However, the interlayer of BLS is the covalent bond interaction, and the distance decreases with the increasing strain for three types of tensions. The continuum description of elastic response is formulated by expanding the elastic strain energy density in a Taylor series in strain truncated after the third-order term. The in-plane second- and third-order elastic constants of BLG and BLS have been obtained by fitting to the strain energy density versus Lagrangian strain relationships. The results show the in-plane stiffnesses of BLG and BLS become slightly larger than those of their single layer counterparts. In spite of the interlayer Si-Si covalent bond between two layers of BLS, its stiffness is still much less than BLG and SLG. Poisson’s ratios of BLG and BLS basically maintain unchanged compared to their single layer counterparts. Keywords: Bilayer graphene; Bilayer silicene; Nonlinear elastic properties;

*

Corresponding author Address: P.O.Box 72, Xitucheng Road No.10, Beijing, 100876, China.

Tel:+86-10-61198062. E-mail: [email protected].

First-principles calculations.

Introduction Graphene is a two-dimension (2D) honeycomb monolayer of carbon atoms, known as the strongest 2D material ever measured by experiments[1]. Graphene has always been a research hot spot, since it was prepared by mechanical exfoliation method in 2004[2]. The non-trivial properties of graphene make it possess great research value and lots of promising applications in numerous fields, such as semiconductor devices and biomedical[3]. Silicene is the silicon counterpart of graphene. Many properties of graphene and silicene hold highly comparability and relevance since silicon is adjacent to carbon belonging to the IV family elements. There have been a variety of methods to produce graphene sheet[4-6]. In recent years, silicene sheets have been synthesized on the silver (111) and (110) substrates[7, 8]

. Fleurence and his collaborators has also grown silicene on Si substrate with ZrB2

buffer layer using UHV chemical vapor deposition method[9]. But graphene and silicene are gapless, which hinders their applications in microelectronics and optoelectronics. Therefore, some researchers[10,

11]

have managed to make bilayer

graphene (BLG) to solve its gap issue. Recently, Ping Wu et al. have proposed BLG growth via a penetration mechanism[12]. Lin and Avouris[13] have proved that BLG devices may have lower noise levels than graphene devices due to their special electronic structure. Like graphene, the bilayer as well as multilayer silicene also exhibits distinctive physical properties[14-16]. Two studies[17, 18]also have demonstrated that the stable structure of BLG and bilayer silicene (BLS) are the AB stacking structure based on the phonon calculation and structure optimization. The electronic properties of BLG and BLS have also been reported[19, 20]. The mechanical properties are crucial for designing and manufacturing devices with 2D nanomaterials in practice. Strain engineering is also a common and effective approach to tailor the electronic properties of materials and the performance of the devices. Vitor et al. have showed strain-induced anisotropy and local deformations can tailor the transport characteristics of graphene devices[21]. Ruthet al.[22] have

demonstrated the stiffness and strength of single layer silicene (SLS) are weaker, but its bending rigidity is stronger compared to the single layer graphene (SLG). Qin et al.[23] have performed the first-principles calculations to study the mechanical and electronic properties of silicene under large equiaxial strain. Nonlinear elastic properties of graphene and silicene have been investigated[24, 25]. Wang and Zhang have investigated the elastic behavior of BLG under different in-plane loadings[26]. They have proposed that the elastic response of BLG is sensitive to chirality and loading direction. Zhang et al.[27] have investigated the mechanical properties of BLG with twist or grain boundaries based on density functional theory (DFT) calculations. However, nonlinear elastic properties of BLG and BLS have not been reported. BLG with respective to graphene has Van der Waals interaction between the two layers. Nevertheless, BLS has covalent bond interaction between the two layers. These differences will inevitably affect mechanical properties of BLG and BLS including their nonlinear elastic properties. Therefore, a comparative study on nonlinear elastic properties of BLG, BLS, SLG, and SLS is necessary and significant. In this work, we focus on the calculations of nonlinear elastic constants, Young’s modulus and the Poisson’s ratio of BLG, BLS, SLG and SLS using first-principles. Three types of tensions are adopted. They are uniaxial tensions along the armchair (AC) and zigzag (ZZ) directions and equiaxial (EQ) tension. We perform a series of DFT calculations to get the strained energy density of four materials under these tensions. Nonlinear elastic response is established by expanding strained energy density in a Taylor series with respect to Lagrangian strain. The independent secondand third-order elastic constants are defined by fitting the continuum model to the strained energy density versus Lagrangian strain results from DFT calculations.

Density functional calculations The strain energy density versus strain relationships of SLG, SLS, BLG, and BLS under the desired deformation configurations is characterized via DFT calculations. The electrons explicitly include in the calculations are the 2s22p2, and 3s23p2 electrons for C and Si atoms, respectively. The core electrons are replaced by

thee proj p jecttor auugm men ntedd-wavee (P PAW W)[28,

29]

annd pseeuddo-ppoteentiial appprooachh. Thhe

exxchaangge-ccorrrelaationn pot p tenttialss iis app a proxxim mateed by loocall-deensity appproxiimaationn (L LDA A)[228]. For F r all sttudiies herre, thee heeighht of o out-o o of-pplanne axiis inn unnitee ceell is i m maiintaaineed at a 166 Å too ellim minaate thee innteractiionns. The T e k-po k ointt mesh m h deenssitiees for f (2× ×1)) unnit celll coorreespoondd to 366×188×1 1 Gam G mmaa ceenteeredd grrid.. Thhe kin k neticc-ennerggy cuttofff off thee planne waavee foor all a calc c culaatioon is seet at a 600 6 0 eV V. In thhe opttim mizaationn proc p cesss, a tottal eneergyy coonveergencce crite c erioon of o 1.0× 1 ×100-6eV V per p atom is i appl a lied d. Inn orrderr too rellax as muuch as

FIG G. 1. Atom A mic strructuure of bilaayerr graaphenee (BLG G) annd (bila ( ayerr sillicene) BL LS. Thee bllackk annd yellow w sppherres repr r resent C annd S Si attomss, reespeectivvelyy: (aa) toop view v w of BL LG. (b) ( side s e vieew of o BLG B G. ( side s e vieew of BLS B S. The T axe a s laabeleed by b the t blac b ck arro a ws dennote thee AC C (c)) topp viiew of BLS. (d) and ZZ Z ddirecctioons, resppectiveely. h1 and a h2 indi i icatee thhe dista d ancees betw weenn tw wo laayerrs oof BLG B G annd BL LS, resppecttiveely. t deenottes the t buccklinng heig h ght of o BLS B S. (22×1)) unnit cells c s fraameed by grreenn linne in i panel (a) and (cc) are a uused d too peerfoorm thee strrain eneergyy deensiity vverssus Laggran ngiaan strai s in unde u er unniaxiial ttenssion.

possible watching the residual forces, the max convergence threshold for the maximum force is specified as -1.0×10-5 eV/Å per atom. The calculations are performed at zero temperature. The optimized structure parameters of SLG, SLS, BLG and BLS are summarized in Table 1. The results are good agreement with the previous theoretical and experimental values[30-33]. Fig.1 shows the top and side views of BLG and BLS atomic unit cell in the undeformed configurations. The strain energy density versus strain relationships calculations for both AC and ZZ uniaxial tensions are performed through the (2×1) unit cell indicated in Fig.1.(a) and Fig.1.(c). For the uniaxial tension, a series of incremental tensions are applied on the (2×1) unit cell in the AC or ZZ direction, then keep another direction constant, and relax the system to minimum energy. For the BLG and BLS, there exist additional internal relative displacements beyond the affine displacements under uniaxial tension. Therefore, the Hellmannn-Feynman forces have Table 1.The optimized structural parameters of SLG, BLG, SLS and BLS. The lattice constant a0, buckling height t, nearest-neighbor C-C (Si-Si) distance d, and interlayer spacing h. The previous calculated results are also present here for comparisons. a0 (Å)

t(Å)

d(Å)

SLG

2.446, 2.46a

0

1.412, 1.42a

BLG

2.446, 2.45b,

0

1.412

2.46c

a

h(Å)

3.324, 3.33b, 3.40c

SLS

3.826, 3.826d

0.437, 0.438d

2.252, 2.251d

BLS

3.806

0.658

2.294

2.482

Reference [31]

b

Reference [32]

c

Reference [33]

d

Reference.[30]

been relaxed to zero at each tension. The equiaxial tension is imposed by uniformly

pulling the primitive cell. The DFT calculations validate there is no internal relative displacement within the primitive cells when they undergo equiaxial tension. This is consistent with the graphene’s results in other studies[24, 29].

The description of third-order nonlinear continuum elastic constants The undeformed reference configurations are shown in Fig.1.(a) and Fig.1.(c), with lattice vectors a1 and a2. When a macroscopically homogeneous deformation (deformation gradient tensor F) is applied, the lattice vectors of the deformed BLG and BLS are ai′ = Fai (with i=1, 2). The Lagrangian strain is defined as η = 12 (FTF - I) , where I is the identity tensor. Here we define strain energy Es = (Etot − E0), where Etot is the total energy of the strained system, E0 is the total energy of the strain-free system. The strain energy density has the functional form Φ =Φ(η) and is defined as Es/V, where V is the volume of the undeformed supercell. Nonlinear elastic constitutive behavior is established by expanding Φ in a Taylor series in terms of powers of Lagrangian strain η as

Φ(η ) =

1 1 1 Cijklηijηkl + Cijklmnηijηklηmn + Cijklmnopηijηklηmnηop 2! 3! 4!

1 + Cijklmnopqrηijηklηmnηopηqr + ..., 5!

(1)

where Cijkl, Cijklmn, Cijklmnop, and Cijklmnopqr correspond to the second-, third-, fourth-, and fifth-order elastic constants, respectively; the summation convention is adopted for repeating indices and summation for lower case indices runs from 1 to 3. In this work, we discuss the nonlinear elastic behavior of SLG, BLG, SLS, and BLS within the framework of second-order elastic constant (SOEC), Cijkl, and third-order elastic constants (TOEC), Cijklmn. We used conventional Voigt notation for subscripts: 11 → 1, 22 → 2, 33 → 3, 23 → 4, 31 → 5, and 12 → 6. Please note that for strain η4 = 2η23, η5 = 2η31, η6 = 2η12. The summation convention for upper case subscripts runs from 1 to 6. The Eq.(1) can

be rewritten as: Φ=

1 1 1 1 CIJη Iη J + CIJKη Iη Jη K + CIJKLη Iη Jη Kη L + CIJKLMη Iη Jη Kη Lη M +… , 2! 3! 4! 5!

(2)

In this work, we assume that the contribution of bending to the strain energy density is negligible for all the tensions as compared to the in-plane strain contribution. This assumption is reasonable since the radius of curvature of out-of-plane deformation is significantly larger than the in-plane inter-atomic distance. Then we only consider the in-plane components for these kinds of structures. Similarly, the components of the SOEC and TOEC tensors can be simplified based on the symmetries of the graphene-like atomic lattice point group D6h which consists of a six-fold rotational axis and six mirror planes as formulated in Ref.[23]. Previous study[24]has shown there are two and three independent nonzero in-plane components for the SOEC and TOEC tensors, respectively. The five independent elastic constants of four materials are determined by a least-squares fitting to strain energy density versus Lagrangian strain results from first-principles calculations. Three relationships between strain energy density and Lagrangian strain are necessary and sufficient because there are three independent TOECs. We obtain the relationships by simulating the following deformation states: uniaxial tensions in the AC and ZZ directions, and EQ tension. For uniaxial tension in the ZZ direction, the strain tensor is ⎛0

0 ⎞

ηijZZ = ⎜ ⎟, ⎝ 0 η ZZ ⎠

(3)

where ηZZ is the amount of strain in ZZ direction. For a given strain tensor, the associated deformation gradient tensor is not unique. The various possible solutions differ from one to another by a rigid rotation. Here the lack of a one-to-one map relationship between the strain tensor and deformation gradient tensor is not concern since the calculated energy is invariant under rigid deformation. One of the corresponding deformation gradient tensor FZZ for uniaxial strain in the ZZ direction is selected as

0 ⎞ ⎛1 FZ Z = ⎜ ⎟, ⎝ 0 ε ZZ + 1⎠

(4)

where εZZ is the nominal strain along the ZZ direction. The strain energy density versus Lagrangian strain relationships for ZZ tension is

1 1 2 3 . Φ(ηZZ ) = C11ηZZ + C111ηZZ 2 6

(5)

For uniaxial tension along the AC direction, the strain tensor is, ⎛0

0 ⎞

ηijAC = ⎜ ⎟, ⎝ 0 η AC ⎠

(6)

One of the corresponding deformation gradient tensor FAC for AC tension is 0 ⎞ ⎛1 FAC = ⎜ ⎟, ⎝ 0 ε AC + 1⎠

(7)

where εAC is the nominal strain along the AC direction. The strain energy density versus Lagrangian strain relationship is

1 1 2 3 . Φ(η AC ) = C11η AC + C222η AC 2 6

(8)

For EQ tension, ηAC = ηZZ= ηEQ, the strain tensor is

0 ⎞ ⎛η EQ ηijEQ = ⎜ ⎟, ⎝ 0 η EQ ⎠

(9)

The corresponding deformation gradient tensor FEQ for EQ tension is

0 ⎞ ⎛ ε EQ + 1 FEQ = ⎜ , ε EQ + 1⎟⎠ ⎝ 0

(10)

where εEQ is the nominal strain under EQ tension. The strain energy density versus Lagrangian strain relationship is

1 2 3 . Φ(ηEQ ) = (C11 + C12 )ηEQ + (2C111 − C222 + 3C112 )ηEQ 3

(11)

From the definition of Lagrangian strain η, we can obtain

1 2

η = ε + ε 2, where ε represents the nominal strain εAC, εZZ and εEQ, respectively.

(12)

FIG G.2 2. Evvoluutionn off geomeetriees of o BLG (lefft) unde u er armcchaiir (A AC), ziggzagg (Z ZZ) andd equuiaxxial (EQ Q) tennsio ons: (a)) Boond lenngth hs of o d1; (b) ( Bonnd leng l gthss off d2; (cc) Dista D ancee beetweeen tw wo laayerrs h. h Evvolu utionn off geoomeetriees off BL LS (rig ( ht) und der A AC,, ZZ Z annd EQ E tensi t ionss: (dd) Bond B d lenngthhs of o d1; (ee) Boond lenggthss off d2; (f) Disstannce bbetw weenn tw wo layers hh.

FIG G.3. Evvoluutionn off thee bu uckling heiight t off BL LS uundeer A AC, ZZ, annd EQ E teensiionss.

FIG G.4 4.Ploots of strai s in ener e rgy dennsityy veersu us Lagr L ranggiann strrain rellatioonshhipss off (a)) BL LG andd (bb) BL LS und u der AC, A , ZZ Z, and a EQ tennsioons. Sollid ppoinnts dennote thee coorressponndinng DFT D T reesults; soli s d linnes indi i icatee thee cuurvees ob btaiinedd froom the t thir t rd-orrderr poolynnomiial fittin f ng.

R ultts and Resu a d disc d cusssess In I ourr sttudyy, thhe moost eneergeeticcallyy fa favoorabble strructturee is set ass thhe sstraain--free strructturee. A All thee attom ms aare allooweed fulll frreeddom m of moti m ion unnderr th he tens t sionns. Thhe evvoluutioons of atoomicc sttruccturres of BL LG (lefft) andd BLS B (riightt) und u er the t AC C, ZZ Z andd EQ Q teenssionns are a shoown n inn the Fig.2 F 2. The T e boondd d1 annd d2 of o BLG B G and a BL LS hav h ve been b n inddicatedd inn F Fig.1(aa) an nd Figg.1((c), resspeectivvelyy. Whe W en thee AC C ttenssion n iss apppliied,, the boond d1 of B BLG G is i para p alleel to t the t strretcchinng forrce.. And A thhe bon b nd lenngthh off d1 moono otonnicaallyy inncreeasees witth thee sttrainn, as a illuustrrateed in i Fig F g.2.((a). Whe W en the t ZZ Z tennsioon iis appl a liedd, thhe bon b nd d1 oof BLG B G iss peerpeenddicuular to thee diirecction n of teensiion andd

the change of bond d1 length is very small. When the EQ tension is applied on BLG, the lengths of bond d1 and d2 are equal and linearly increase with the increasing strain, as shown in Fig.2.(a) and Fig.2.(b). It is the weak Van der Waals interaction between the layers of BLG. So we can see from Fig.2.(c) that the distance between two layers of graphene is hardly affected by tensions and maintains at about 3.324 Å. Observing the Fig.2.(d) and Fig.2.(e), the variations of bond d1 and d2 of BLS with the AC and ZZ tensile strain are similar to those of BLG. However, observe the Fig.1.(d) and find that the Si-Si covalent bonds exist in the interlayer of BLS. From the Fig.2.(f), we can note that the dependences of interlayer heights of BLS on three types of tensile tensions are more sensitive compared to BLG. The curves of interlayer height versus two types of uniaxial strain exhibit monotonous decreases and they are almost identical at the small strain (linear response area). The interlayer height h of BLS under EQ tension increases with the increasing strain, and then decreases with the increasing strain (blue angles in the Fig.2.(f)). Detailed reasons will be presented in the following. Evolution of the buckling height t of BLS under AC, ZZ, and EQ tensions is shown in Fig.3. All the three curves decrease with the increasing strain. In small tension, the variations for AC and ZZ tensions are nearly identical. The variation of buckling height with EQ tension is more sensitive with respect to the other two uniaxial tensions. In other word, buckling height of BLS under EQ tension more quickly decreases with increasing strain. One side, the decrease of buckling height of BLS under EQ tension makes its surfaces flat and further makes the interlayer distance greater. On the other side, EQ tension affects the covalent bonds between two layers of BLS and renders the distance smaller. So the joint effects of two factors cause the interlayer distance of BLS increases and then decreases with the increasing strain. At small (large) tension, the effect of the former factor is stronger (weaker) than that of the latter factor. So the interlayer distance of BLS increases with the increasing strain at the small tension. On the contrary, it decrease with the increasing strain at the large tension. All five elastic constants appear in the strain energy density versus Lagrangian strain constitutive relationships for the three special cases collectively (Eq.(5), Eq.(10),

Eq.(11)). Thus, the values of the elastic constants can be determined by fitting to the strain energy density versus Lagrangian strain relationships as calculated from first-principles calculations. The results of DFT simulations are shown in Fig.4 where the strain energy density of BLG (Fig.4.(a)) and BLS (Fig.4.(b)) are plotted as functions of the Lagrangian strain. Solid points denote corresponding DFT results. Solid lines indicate the curves obtained from the third-order polynomial fitting, respectively. Calculated five independent elastic constants, the Young modulus, and the in-plane Poisson’s ratio for the four materials are summarized in Table 2. For the SLG and SLS, our results are good agreement with previous calculations[24, 25]. The differences between our results and the results in Ref.[24, 25] attribute to adoptive functionals. Generally, the GGA functional used by them always underestimates the bind energy and overestimates the bond length. The LDA functional adopted by us is just the opposite: it overestimates the bind energy and underestimate the bond length. In addition, the bond angle, lattice constants, elastic constants and so on obtained by LDA and GGA functionals are slightly different. But they are basically consistent and do not seriously affect the research of material properties. It is interesting to note that the TOECs are all negative, which ensures that the elastic stiffness softens with strain up to ideal strain. The in-plane SEOCs and TEOCs (C11, C12, C111, C112 and C222) of BLG and BLS are larger than those of SLG and SLS, respectively. In other word, the in-plane stiffnesses of BLG and BLS are larger compared to their single layer counterparts. These changes attribute to the interlayer interactions in BLG (Van der Waals) and BLS (covalent bond). Furthermore, from the SOEC in Table 2, the in-plane Young modulus E and Poisson’s ratio ν can be obtained from the following 2 2 relationships: E = (C11 − C12 ) / C11 and ν =C12 / C11 .We have E = 334.77 N/m, ν = 0.18

N/m for SLG, E = 419.77 N/m, ν = 0.18 N/m for BLG, E = 63.3 N/m, ν = 0.32 N/m for SLS, and E=68.25 N/m, ν=0.33 N/m for BLS for small strain. The Young’s modulus E of BLG is 25% larger than SLG. Similarly, the one of BLS is 7% larger than SLS. The Poisson’s ratios ν of SLG and BLG maintain at 0.18. The Poisson’s ratios ν of SLS and BLS are 0.32 and 0.33, respectively.

Under large hydrostatic pressure, it is useful to describe the nonlinear elastic properties of materials by the pressure dependent elastic constants C ij . For most applications, it is sufficient to only consider the linear terms in the external Table 2. Independent components for the SOEC and TOEC tensor components, in-plane Young’s modulus E and Poisson’s ratio v of SLG, BLG, SLS and BLS.

SLG

C11

C12

C111

C112

C222

E

(N/m)

(N/m)

(N/m)

(N/m)

(N/m)

(N/m)

346.2

62.9

-2151.6

-265.2

-2083.8

334.77

a

a

-3089.7

a

a

a

340.8

a

b

b

b

b

352.0

b

60.4

BLG

434.0

78.6

SLS

70.6 71.3c

BLS

76.68

358.1

a

62.6

-453.8

-2928.1

ν

0.178a

-2693.3

-2691.0

-371.4

-2653.2

419.77

0.18

22.7

-400.5

-13.5

-313.7

63.3

0.32

c

c

-397.6

-14.1

c

c

-318.9

63.8

c

-511.98

-77.38

-438.3

68.25

23.2

25.42

-337.1

348

C12′

′ C 22

5.91

0.65

5.09

5.98

0.72

5.18

4.45

0.15

3.37

5.77

0.75

4.29

0.18

b

-2817

C11′

0.169

b

0.325c 0.33

Reference.[34]

b

Reference.[24]

c

Reference.[25]

hydrostatic pressure. When pressure is applied, the pressure dependent second-order elastic constants ( C11 , C 22 , C12 ) and corresponding derivatives with pressure

′ , C12′ ) can be obtained from C11, C22 , C12 ,C111, C112 , C222 , E andν as: ( C11′ , C 22 1 −ν C11 = C11 − ( C111 + C112 ) P， E 1 −ν C 11′ = − ( C 111 + C 112 ) ， E 1 −ν C 22 = C11 − C 222 P， E 1 −ν ′ = − C 222 C 22 , E

1 −ν C12 =C12 − C112 P， E 1 −ν C12′ = − C112 . E

(13) (14) (15) (16) (17) (18)

Results for Cij′ are also shown in Table 2. The values of Cij′ for BLS and BLG get larger compared to the SLG and SLS. Cij′ can be measured by the experiments, but

there has been no relevant report so far.

Conclusions In summary, we have studied the nonlinear elastic properties of SLG, BLG, SLS, and BLS under AC, ZZ, and EQ tensions based on first-principles DFT calculations. The study on the atomic structure has shown that the distance between two layers of BLG almost maintains unchanged under three types of tensions. But the interlayer distance of the BLS monotonically decreases with the increase of these strains. The differences of two variations are caused by the different types of interactions between the interlayers. The SOECs and TOECs of BLG and BLS have been obtained by combining continuum elasticity theory and the homogeneous deformation method. Calculated results have demonstrated that in-plane stiffnesses of BLG and BLS become slightly larger than their respective monolayer counterparts. We have also calculated Young’s modulus and Poisson’sratio (E = 419.77 Nm-1, ν = 0.18 for BLG and E = 68.25 Nm-1, ν = 0.33 for BLS). The Young’s modulus of BLG and BLS are larger than those of their respective single layer counterparts. The Poisson’s ratios of BLG and BLS almost keep constant compared to their respective single layer counterparts. Our work can contribute to understand the nonlinear elastic properties of the BLS and BLG materials and calculated data can be applied to finite element analyses models for their applications at large scale.

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Highlights

1. There is Van der Waals interaction between two layers of BLG 2. There is covalent bond interaction between two layers of BLS 3. stiffnesses of BLG and BLS become slightly larger. 4. second-and third-order elastic constants of BLG and BLS have been obtained