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Periodic buckling patterns of graphene/hexagonal boron nitride heterostructure
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Nanotechnology Nanotechnology 25 (2014) 445401 (10pp)
doi:10.1088/0957-4484/25/44/445401
Periodic buckling patterns of graphene/ hexagonal boron nitride heterostructure Chenxi Zhang1, Jizhou Song2 and Qingda Yang1 1
Department of Mechanical and Aerospace Engineering, University of Miami, Coral Gables, FL 33146, USA 2 Department of Engineering Mechanics and Soft Matter Research Center, Zhejiang University, Hangzhou 310027, People’s Republic of China E-mail: [email protected] Received 13 July 2014, revised 9 September 2014 Accepted for publication 10 September 2014 Published 14 October 2014 Abstract
Graphene/hexagonal boron nitride (h-BN) heterostructure has showed great potential to improve the performance of a graphene device. A graphene on an h-BN substrate may buckle due to the thermal expansion mismatch between the graphene and h-BN. We used an energy method to investigate the periodic buckling patterns including one-dimensional, square checkerboard, hexagonal, equilateral triangular and herringbone mode in a graphene/h-BN heterostructure under equi-biaxial compression. The total energy, consisting of cohesive energy, graphene membrane energy and graphene bending energy, for each buckling pattern is obtained analytically. At a compression slightly larger than the critical strain, all buckling patterns have the same total energies, which suggests that any buckling pattern may occur. At a compression much larger than the critical strain, the herringbone mode has the lowest total energy by significantly reducing the membrane energy of graphene at the expense of a slight increase of the bending energy of graphene and cohesive energy. These results may serve as guidelines for strain engineering in graphene/h-BN heterostructures. Keywords: graphene, hexagonal boron nitride, buckling (Some figures may appear in colour only in the online journal) 1. Introduction
interaction of graphene with h-BN substrate, graphene/h-BN is also an ideal system to study the intrinsic physical properties of graphene. Although the electrical, magnetic and thermal properties of graphene/h-BN heterostructure have been extensively investigated [19, 25–34], its mechanical properties have been rarely explored thus far. Understanding graphene’s morphology is crucial in tailoring graphene’s properties for device applications. Scharfenberg et al [35] examined the competition between adhesive and bending energies for few-layer graphene on a microscalecorrugated metallic substrate. It was shown that the graphene undergoes a layer-thickness dependent snap-through transition between lying flat and conforming to the substrate. Lanza et al [36, 37] investigated the morphology and performance of graphene layers on as-grown and transferred substrates due to the compressive strain and surface roughness. They observed the important adhesion differences depending on the graphene thickness and the target substrate roughness. Folds, wrinkles
Since its discovery, graphene has attracted wide attentions for potential applications ranging from flexible and invisible displays, nanoelectronic components and nanosensors to energy conversion and storage devices due to its excellent electrical, thermal and mechanical properties [1–4]. Most graphene applications require a supporting substrate (e.g., SiC and SiO2) [5–10], which leads to a significant reduction in electron mobility due to the charged surface states and impurities, surface roughness and surface optical phonons [11–18]. Recently, hexagonal boron nitride (h-BN) has been used as an ideal substrate for graphene devices due to h-BN’s atomically flat structure with little dangling bonds and charge traps [19]. It has been shown that graphene/h-BN heterostructures exhibit less intrinsic doping, much higher electron mobility and improved on/off ratio than conventional graphene devices on SiO2 substrate [20–24]. Due to the reduced 0957-4484/14/445401+10$33.00
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© 2014 IOP Publishing Ltd Printed in the UK
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Figure 1. Schematic diagrams of the buckling patterns: (a) one-dimensional mode, (b) square checkerboard mode, (c) hexagonal mode, (d)
equilateral triangular mode, and (e) herringbone mode under equi-biaxial compression.
and cracks were also observed in the samples. Jiang et al [38] studied the interfacial sliding due to uniaxial tension and buckling due to uniaxial compression of monolayer graphene on a stretchable substrate. It was found that the critical compressive strain of buckling is around −0.7%. Yoon et al [39] realized an equi-biaxial compression in graphene on SiO2 due to their thermal expansion coefficient mismatch although the deformation and morphology of graphene were not well understood. Pan et al [40] studied biaxial compressive strain in graphene/h-BN heterostructures and found that the strain is spatially inhomogeneous with the occurrence of equilateral triangular and quadrilateral bubbles possibly due to the existence of defects. Although many researchers have studied the morphology of graphene on different substrates, some questions still remain unknown for an ideal graphene/hBN heterostructure under biaxial compression. For example, can the graphene/h-BN heterostructure buckle as the compression increases? What is the condition for buckling? What is the buckling mode? etc. These questions are very critical since the deformation of graphene is strongly tied to its electrical performance (e.g., electron mobility and band gap) [41–50]. It is shown that the film bonded to a substrate under equi-biaxial compression may buckle into several intriguing periodic buckling patterns such as one-dimensional, square checkerboard, hexagonal, equilateral triangular and herringbone modes as shown schematically in figure 1 [51]. Many researchers have investigated these periodic buckling modes. For example, Chen and Hutchinson [52] and Song et al [53] developed an energy method with the total energy consisting of the film energy and substrate energy to study one-dimensional, square checkerboard and herringbone mode under equi-biaxial compression and they
showed that the herringbone mode has the lowest energy in the buckled state while the one-dimensional mode has the greatest. Audoly and Boudaoud [54] explored further details of these modes including the transition from onedimensional mode to the herringbone mode as the compression is not equi-biaxial. Cai et al [51] established an analytical upper-bond method and performed a full numerical analysis to study the energies of all these periodic patterns in the buckled state. It is found that the square checkerboard mode has lowest energy in small strain state (slightly larger than the critical buckling strain) and the herringbone mode is dominant in relatively large strain state. They suggested that initial imperfections may play a role in mode selection. The objective of this paper is to use an energy method to investigate the buckling behavior of graphene/h-BN heterostructure under equi-biaxial compression. All five possible periodic patterns (i.e., one-dimensional, square checkerboard, hexagonal, equilateral triangular and herringbone modes) are considered. Different from the system of a film perfectly bonded to a substrate, where the total energy consists of the film energy and substrate energy, the graphene/h-BN heterostructure has film energy and cohesive energy due to the van der Waals interaction at the interface but not the substrate energy since the substrate could be assumed to be a rigid body [55, 56]. This paper is outlined as follows. An energy method is developed in section 2 to obtain the total energy of the buckled state of graphene/h-BN heterostructures. The analytical expressions of the energies for the five periodic buckling modes are presented in section 3. Section 4 describes the results and discussions. Conclusions are made in section 5. 2
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2. An energy method for buckling analysis
and ⎧ ∂ 2u2 ∂ 2u ∂ 2u1 ⎫ 1 1 ⎨ 2 + (1 − ν) 22 + (1 + ν) ⎬ 2 2 ∂x1∂x2 ⎭ ∂x1 ⎩ ∂x 2
We develop an energy method to determine the buckling geometry in this section. The total energy of the graphene/hBN heterostructure consists of the strain energy of graphene and cohesive energy at the interface (or graphene-substrate interaction energy) [55, 56]. Minimization of total energy yields the buckling geometry.
⎧ ⎡ ⎛ ∂ 2w ⎞ ⎤ ⎪ ⎛ ∂w ⎞ ⎛ ∂ 2w ⎞ 1 ⎨⎜ +⎪ ⎟ ⎢ ⎜ 2 ⎟ + (1 − ν) ⎜ 2 ⎟ ⎥ ⎝ ∂x1 ⎠ ⎥⎦ ⎩ ⎝ ∂x2 ⎠ ⎢⎣ ⎝ ∂x2 ⎠ 2 +
2.1. The strain energy density of graphene
∂u j ⎞ 1 ⎛ ∂w ⎞ ⎛ ∂w ⎞ 1 ⎛ ∂u i ⎜⎜ ⎟⎟ + ⎜ ⎟⎟ , + ⎟ ⎜⎜ 2 ⎝ ∂x j ∂x i ⎠ 2 ⎝ ∂x i ⎠ ⎝ ∂x j ⎠
⎪
(5)
⎪
Once u1 (x1, x 2 ) and u 2 (x1, x 2 ) are solved, membrane force and membrane strain can be easily obtained by equations (1) and (2). The membrane energy density is then given by
The graphene is subjected to an equi-biaxial initial strain 0 ε110 = ε22 = ε0 due to the thermal expansion coefficient mismatch between graphene and substrate, and it buckles once ε0 exceeds a critical value. The graphene is modeled as an elastic Von Karman plate with finite rotation [57]. The strain energy density of graphene consists of the membrane energy density and the bending energy density. The membrane strain εij (where i, j = 1, 2) is related to the in-plane displacements u1 (x1, x 2 ) and u 2 (x1, x 2 ) and outof-plane displacement w (x1, x 2 ) given by εij = εij0 +
⎛ ∂w ⎞ ⎛ ∂ 2w ⎞ ⎫ 1 ⎟ ⎬ = 0. (1 + ν) ⎜ ⎟⎜ 2 ⎝ ∂x1 ⎠ ⎝ ∂x1∂x2 ⎠ ⎭
Φm =
1 ( N11ε11 + 2N12 ε12 + N22 ε22 ). 2
(6)
The bending energy density only depends on the out-ofplane displacement w and is given by 2 ⎛ ∂ 2w ⎞ ⎛ ∂ 2w ⎞ ⎛ ∂ 2w ⎞2 ¯ 3 ⎡⎢ ⎛ ∂ 2w ⎞ Et ⎜ 2 ⎟ + ⎜ 2 ⎟ + 2ν ⎜ 2 ⎟ ⎜ 2 ⎟ Φb = 24 ⎢⎣ ⎝ ∂x1 ⎠ ⎝ ∂x1 ⎠ ⎝ ∂x2 ⎠ ⎝ ∂x 2 ⎠
⎛ ∂ 2w ⎞2 + 2(1 − ν) ⎜ ⎟ . ⎝ ∂x1∂x2 ⎠
(1)
where εij0 is the initial strain and i, j = 1,2. The Hooke’s law gives the membrane force in the graphene as
(7)
2.2. The cohesive energy density
¯ ⎡⎣ (1 − ν) εij + νεkk δ ij ⎤⎦ , Nij = Et
(
The interlayer bonding between graphene and h-BN is due to van der Waals interaction, which is usually represented by the Lennard-Jones 6–12 potential
(2)
)
where E¯ = E / 1 − ν 2 is the plan-strain modulus of graphene, ν is the Poisson's ratio of graphene, t is the thickness of graphene and δij is the Kronecker delta (i, j = 1, 2). The shear tractions at the graphene/h-BN interface are obtained from the force equilibrium as Ti =
∂Nij ∂x j
.
⎛ η12 η6 ⎞ V (r ) = 4ζ ⎜ 12 − ⎟ , r6 ⎠ ⎝r
(8)
where V (r ) is the energy between a pair of atoms of distance r , 6 2 η is the equilibrium distance between two atoms and ζ is the bond energy at the equilibrium distance. For a pair of carbon and nitrogen atoms, the bond energy and equilibrium distance are ζC − N = 0.004 068 eV and ηC − N = 0.3367 nm and ζC − B = 0.003 294 eV and ηC − B = 0.3411 nm for a pair of carbon and boron atoms [59]. To describe van der Waals interactions between graphene monolayer and h-BN substrate, graphene and h-BN are modeled as homogenous and infinitely large layer and solid 4 (as shown in figure 2), respectively, with ρC = 2 denoting
(3)
It has been shown that the shear stress at the graphene/hBN interface has a negligible effect on the buckling geometry [58]. The vanishing shear tractions, i.e., T1 = 0 and T2 = 0 , yields two governing equations for the in-plane displacements u1 (x1, x 2 ) and u 2 (x1, x 2 ) given the out-of-plane displacement w (x1, x 2 ),
3 3 lC
⎧ ∂ 2u1 ∂ 2u ∂ 2u2 ⎫ 1 1 ⎨ 2 + (1 − ν) 21 + (1 + ν) ⎬ 2 2 ∂x1∂x2 ⎭ ∂x 2 ⎩ ∂x1 ⎧ ⎡ ⎛ ∂ 2w ⎞ ⎤ ⎪ ⎛ ∂w ⎞ ⎛ ∂ 2w ⎞ 1 ⎢⎜ ⎜ 2 ⎟⎥ ⎟ + (1 − ) +⎨ ν ⎜ ⎟ 2 ⎪ ⎝ ∂x2 ⎠ ⎥⎦ ⎩ ⎝ ∂x1 ⎠ ⎢⎣ ⎝ ∂x1 ⎠ 2 ⎛ ∂w ⎞ ⎛ ∂ 2w ⎞ ⎫ 1 + (1 + ν) ⎜ ⎟ ⎬ = 0, ⎟⎜ 2 ⎝ ∂x2 ⎠ ⎝ ∂x1∂x2 ⎠ ⎭ ⎪ ⎪
the number of carbon atoms per unit area on graphene and 2 denoting the number of the nitrogen ρN = ρB = 2 3 3 l h − BN h h − BN
and boron atoms per unit volume in h-BN [60]. Here lC = 0.1408 nm and l h − BN = 0.1440 nm are the equilibrium bond length of graphene and h-BN [61], h h − BN = 0.3406 nm is the inter-layer distance in h-BN. Then we have ρC = 38.80 atoms nm−2 and ρN = ρB = 54.48 atoms nm−3. As shown in figure 2, the graphene is parallel to the h-BN with h denoting the equilibrium distance. The distance
(4)
3
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Figure 3. Schematic illustration of graphene subjected to the opening displacement w on a h-BN substrate.
Figure 2. Schematic diagram of graphene/h-BN heterostructure.
Equation (13) can be rewritten as between a carbon atom on graphene and a nitrogen (or boron) atom on h-BN is r = x 2 + z 2 , where z is the projected distance between two atoms. The cohesive energy per unit area is then given by Φ graphene/ h − BN = ΦC − N + ΦC − B,
⎞9 ⎛ 2 ⎞⎛ 1 Φc = (CC1 − N + CC1 − B ) ⎜ ⎟ ⎜ ⎟ ⎝ 15 ⎠ ⎝ 1 + w / h e ⎠ ⎛ ⎞3 1 − CC2 − N + CC2 − B ⎜ ⎟, ⎝ 1 + w /he ⎠
(
(9)
where ΦC − N and ΦC − B are the energies due to the interactions between carbon and nitrogen atoms, and between carbon and boron atoms, respectively, and they are obtained by −h
ΦC − N = 2πρC ρ N dA =
∫−∞
dx
∫0
(14)
where
∞
V (r ) z d z
⎛ 2η 9 η3 ⎞ 2 πρC ρN ζC − N ηC3−N ⎜⎜ C −9N − C −3N ⎟⎟ , 3 h ⎠ ⎝ 15h
)
(10)
CC1−N =
2 πρC ρN ζC − N ηC12−N h e9 , 3
(15)
CC2−N =
2 πρC ρN ζC − N ηC6−N h e3, 3
(16)
CC1−B =
2 πρC ρB ζC − B ηC12−B h e9 , 3
(17)
CC2−B =
2 πρC ρB ζC − B ηC6−B h e3, 3
(18)
and
and −h
ΦC − B = 2πρC ρB dA =
∫−∞
dx
∫0
∞
V (r ) z d z
⎛ 2η 9 η3 ⎞ 2 πρC ρB ζC − B ηC3−B ⎜⎜ C −9B − C −3 B ⎟⎟ . 3 h ⎠ ⎝ 15h
are all constants. For a small w /he , the cohesive energy density could be approximated by the Taylor series by ignoring the higher order terms, i.e.,
(11)
The equilibrium distance he is determined by minimizing the total cohesive energy, by
∂Φ graphene/ h − BN ∂h
⎡ ⎛ w ⎞1 ⎛ w ⎞2 ⎛ 2 ⎞ Φc ≈ (CC1−N + CC1−B ) ⎜ ⎟ ⎢ 1−9 ⎜ ⎟ + 45 ⎜ ⎟ ⎝ 15 ⎠ ⎢⎣ ⎝ he ⎠ ⎝ he ⎠
= 0, and is then given
⎛ 2 ζC − N η 12 + ζC − B η 12 ⎞1/6 C −N C −B ⎟ h e = ⎜⎜ ⋅ ⎟ , 6 6 5 ζ η ζ η + C −N C −N C −B C −B ⎠ ⎝
⎛ w ⎞3 ⎛ w ⎞4 ⎤ − 165 ⎜ ⎟ + 495 ⎜ ⎟ ⎥ − (CC2−N + CC2−B ) ⎝ he ⎠ ⎝ h e ⎠ ⎥⎦
(12)
⎡ ⎛ w ⎞1 ⎛ w ⎞2 ⎛ w ⎞3 ⎛ w ⎞4 ⎤ × ⎢ 1 − 3 ⎜ ⎟ + 6 ⎜ ⎟ − 10 ⎜ ⎟ + 15 ⎜ ⎟ ⎥ . ⎢⎣ ⎝ he ⎠ ⎝ he ⎠ ⎝ he ⎠ ⎝ h e ⎠ ⎥⎦ (19)
which is about 0.2908 nm. For the out-of-plane displacement w (x1, x 2 ) beyond the equilibrium distance he as shown in figure 3, the cohesive energy density can be similarly obtained as
This approximation is critical to derive an analytic expression for cohesive energy.
⎡ 2ηC9−N 2 Φc (w ) = πρC ρN ζC − N ηC3−N ⎢ ⎢⎣ 15 ( h e + w )9 3 ⎤ ⎥ + 2 πρ ρ ζC − B η 3 − C B C −B 3⎥ ( he + w) ⎦ 3 ⎡ ηC3−B ⎤⎥ 2ηC9−B ×⎢ − . ⎢⎣ 15 ( h e + w )9 ( h e + w )3 ⎥⎦
2.3. The total energy
ηC3−N
The total energy Utotal consists of the membrane energy Um and bending energy Ub in graphene and cohesive energy Uc , which can be obtained from the integration of the corresponding energy densities Φm , Φb , and Φc . Minimization of the total energy with respect to the buckling geometry yields solutions for different buckling patterns to be discussed in section 4.
(13)
4
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3. Energies of the buckling modes
obtained by solving equations (4) and (5) as
The total energies for one-dimensional mode, square checkerboard mode, hexagonal mode, equilateral triangular mode and herringbone mode are obtained in this section.
u1 =
1 ξ 2t 2k sin ( 2 kx1 + 2 kx2 ) 32 2 1 + ξ 2t 2k sin ( 2 kx1 − 2 kx2 ) 32 2 1 + ξ 2t 2k (1 − ν) sin ( 2 kx1) , 16 2
3.1. One-dimensional mode
The out-of-plane displacement of one-dimensional model is given by w = ξt cos (kx1) ,
u2 =
(20)
where k is the wave number along x1 direction, ξ is a nondimensional coefficient of amplitude, and t is the graphene thickness. Therefore, ξt denotes the amplitude. Substituting w into the governing equations [equations (4) and (5)] of inplane displacements u1 (x1, x 2 ) and u 2 (x1, x 2 ) yields u1 =
1 2 2 t ξ k sin (2kx1) , 8
(27)
1 ξ 2t 2k sin ( 2 kx1 + 2 kx2 ) 32 2 1 − ξ 2t 2k sin ( 2 kx1 − 2 kx2 ) 32 2 1 + ξ 2t 2k (1 − ν) sin ( 2 kx2 ) . 16 2
(28)
The membrane energy, bending energy and cohesive energy per unit area in square checkerboard mode are then given analytically by
(21)
and u2 = 0.
2 2π
(22)
∫
The membrane strain can be obtained by equation (1) as ε11 = ε0 + ξ 2t 2k 2 /4 and ε12 = ε21 = ε22 = 0 . The membrane energy, bending energy and cohesive energy per unit area in one-dimensional mode are then obtained analytically by 2π / k 1 Φm dx1 Um = 2π / k 0 ⎡ ¯ ⎢ (1 + ν) ε02 + 1 ξ 2t 2k 2 (1 + ν) ε0 = Et ⎣ 4 ⎤ 1 44 4 ξ t k ⎥, + ⎦ 32
∫
Ub =
Uc =
1 2π / k
∫0
1 2π / k
∫0
2π / k
Φ b dx1 =
E¯ ξ 2t 5k 4 , 48
Ub =
Uc = (24)
2 2π
1 2 2 8π / k
∫x =k0 ∫x 1
2 2π
∫x =k0 ∫x 1
2 2π k 2
=0
2 2π k 2
=0
Φ b dx1dx2 =
(29)
E¯ ξ 2t 5k 4 , (30) 96
Φc dx1dx2
2 ⎡ ⎛ 2 ⎞ 45 ⎛ ξt ⎞ = (CC1−N + CC1−B ) ⎜ ⎟ ⎢ 1 + ⎜ ⎟ ⎝ 15 ⎠ ⎢⎣ 4 ⎝ he ⎠
Φc dx1
4⎤ 4455 ⎛ ξt ⎞ ⎥ ⎜ ⎟ − CC2−N + CC2−B 64 ⎝ h e ⎠ ⎥⎦ 2 4⎤ ⎡ 3 ⎛ ξt ⎞ 135 ⎛ ξt ⎞ ⎥ × ⎢1 + ⎜ ⎟ + ⎜ ⎟ . ⎢⎣ 2 ⎝ he ⎠ 64 ⎝ h e ⎠ ⎥⎦
2 ⎡ ⎛ 2 ⎞ 45 ⎛ ξt ⎞ = (CC1−N + CC1−B ) ⎜ ⎟ ⎢ 1 + ⎜ ⎟ ⎝ 15 ⎠ ⎢⎣ 2 ⎝ he ⎠
+
+
(
) (31)
(25) 3.3. Hexagonal mode
The out-of-plane displacement of generalized hexagonal mode is given by [47]
3.2. Square checkerboard mode
The out-of-plane displacement of square checkerboard mode is given by [47] ⎛ 1 ⎞ ⎛ 1 ⎞ w = ξt cos ⎜ kx1⎟ cos ⎜ kx2 ⎟ . ⎝ 2 ⎠ ⎝ 2 ⎠
∫
1 2 2 8π / k
(23)
2π / k
4⎤ 1485 ⎛ ξt ⎞ ⎥ ⎜ ⎟ − (CC2−N + CC2−B ) 8 ⎝ h e ⎠ ⎥⎦ 4⎤ ⎡ ⎛ ξt ⎞2 45 ⎛ ξt ⎞ ⎥ × ⎢1 + 3⎜ ⎟ + ⎜ ⎟ . ⎢⎣ 8 ⎝ h e ⎠ ⎥⎦ ⎝ he ⎠
2 2π
1 k k Φm dx1dx2 Um = 2 2 x1 = 0 x2 = 0 8π / k ⎡ ¯ ⎢ (1 + ν) ε02 + 1 (1 + ν) ξ 2t 2k 2ε0 = Et ⎣ 8 ⎤ 1 (1 + ν)(3 − ν) ξ 4t 4k 4⎥ , + ⎦ 512
⎡ ⎛ 3 ⎞⎤ ⎛1 ⎞ w = ξt ⎢ cos ( kx1) + 2 cos ⎜ kx1⎟ cos ⎜ kx2 ⎟ ⎥ , ⎝2 ⎠ ⎢⎣ ⎝ 2 ⎠ ⎥⎦
(32)
(26) 1
3
where the wave numbers are 2 k along x1 direction and 2 k along x 2 direction. Solving the governing equations of in-
The in-plane displacements u1 (x1, x 2 ) and u 2 (x1, x 2 ) can be 5
Nanotechnology 25 (2014) 445401
C Zhang et al
plane displacement u1 (x1, x 2 ) and u 2 (x1, x 2 ) gives
u1 =
⎞ ⎛3 1 22 3 ξ t k (3 − ν) sin ⎜ kx1 + kx2 ⎟ 16 2 ⎠ ⎝2 ⎛3 ⎞ 1 22 3 + ξ t k (3 − ν) sin ⎜ kx1 − kx2 ⎟ 16 2 ⎝2 ⎠ 1 22 + ξ t k sin(kx1 + 3 kx2 ) 16 1 22 + ξ t k sin(kx1 − 3 kx2 ) 16 ⎞ ⎛1 1 22 3 + ξ t k (1 − 3ν) sin ⎜ kx1 + kx2 ⎟ 16 2 ⎠ ⎝2 ⎞ ⎛1 1 22 3 + ξ t k (1 − 3ν) sin ⎜ kx1 − kx2 ⎟ 16 2 ⎠ ⎝2 1 1 + ξ 2t 2k sin ( 2kx1) + ξ 2t 2k (1−3ν) sin ( kx1) , 8 8
u2 =
Uc =
(33)
∫
1 16π / 3 k 2 E¯ ξ 2t 5k 4 = , 16
Ub =
2
4π / k
4π / 3 k
1
2
=0
=0
Φc dx1dx2
(37)
The out-of-plane displacement of equilateral triangular mode is given by [47] ⎡ ⎛ 3 ⎞⎤ ⎛1 ⎞ w = ξt ⎢ − sin ( kx1) + 2 sin ⎜ kx1⎟ cos ⎜ kx2 ⎟ ⎥ , (38) ⎝2 ⎠ ⎢⎣ ⎝ 2 ⎠ ⎥⎦ 1
3
where the wave number is 2 k along x1 direction and 2 k along x 2 direction, which are same as those in hexagonal mode. The in-plane displacements are solved as u1 =
(34)
∫
∫x = 0 ∫x
2
3.4. Equilateral triangular mode
The membrane energy, bending energy and cohesive energy per unit area in hexagonal mode are obtained as 4π / k 4π / 3 k 1 Φm dx1dx2 16π 2 / 3 k 2 x1 = 0 x 2 = 0 ⎡ ¯ ⎢ (1 + ν) ε02 + 3 (1 + ν) ξ 2t 2k 2ε0 = Et ⎣ 4 ⎤ 3 (1 + ν)(11−5ν) ξ 4t 4k 4⎥ , + ⎦ 128
1
3 4⎤ 495 ⎛ ξt ⎞ 22 275 ⎛ ξt ⎞ ⎥ ⎜ ⎟ + ⎜ ⎟ 2 ⎝ he ⎠ 8 ⎝ h e ⎠ ⎥⎦ ⎡ ⎛ ξt ⎞2 ⎛ ξt ⎞3 − (CC2−N + CC2−B ) ⎢ 1 + 9 ⎜ ⎟ − 15 ⎜ ⎟ ⎢⎣ ⎝ he ⎠ ⎝ he ⎠ 4⎤ 675 ⎛ ξt ⎞ ⎥ + ⎜ ⎟ . 8 ⎝ h e ⎠ ⎥⎦
3 22 ξ t k sin(kx1 + 3 kx2 ) 16 3 22 − ξ t k sin(kx1 − 3 kx2 ) 16 ⎞ ⎛1 3 22 3 kx2 ⎟ + ξ t k (1 − 3ν) sin ⎜ kx1 + 16 2 ⎠ ⎝2 ⎛1 ⎞ 3 22 3 kx2 ⎟ − ξ t k (1 − 3ν) sin ⎜ kx1 − 2 16 ⎠ ⎝2
Um =
4π / 3 k
−
+
3 22 ξ t k (3 − ν) sin( 3 kx2 ). 24
4π / k
∫x = 0 ∫x
2 ⎡ ⎛ 2 ⎞ 135 ⎛ ξt ⎞ = (CC1−N + CC1−B ) ⎜ ⎟ ⎢ 1 + ⎜ ⎟ ⎝ 15 ⎠ ⎢⎣ 2 ⎝ he ⎠
⎛3 ⎞ 3 22 3 kx2 ⎟ ξ t k (3 − ν) sin ⎜ kx1 + 48 2 ⎝2 ⎠ ⎞ ⎛3 3 22 3 kx2 ⎟ − ξ t k (3 − ν) sin ⎜ kx1 − 48 2 ⎠ ⎝2
+
1 16π / 3 k 2 2
⎛3 ⎞ 3 1 22 ξ t k (3 − ν) sin ⎜ kx1 + kx2 ⎟ 16 2 ⎝2 ⎠ ⎛ ⎞ 1 22 3 3 + ξ t k (3 − ν) sin ⎜ kx1 − kx2 ⎟ 16 2 ⎝2 ⎠ 1 22 − ξ t k sin (kx1 + 3 kx2 ) 16 1 22 − ξ t k sin (kx1 − 3 kx2 ) 16 ⎞ ⎛1 1 22 3 − ξ t k (1−3ν) sin ⎜ kx1 + kx2 ⎟ 16 2 ⎠ ⎝2 ⎛1 ⎞ 1 22 3 − ξ t k (1−3ν) sin ⎜ kx1 − kx2 ⎟ 16 2 ⎝2 ⎠ 1 22 1 22 − ξ t k sin ( 2kx1) − ξ t k (1−3ν) sin ( kx1) , 8 8
(35)
u2 =
Φ b dx1dx2
⎛3 ⎞ 3 22 3 kx2 ⎟ ξ t k (3 − ν) sin ⎜ kx1 + 48 2 ⎝2 ⎠ ⎛3 ⎞ 3 22 3 kx2 ⎟ − ξ t k (3 − ν) sin ⎜ kx1 − 48 2 ⎝2 ⎠ −
(36)
6
3 22 ξ t k sin (kx1 + 16
3 kx2 )
(39)
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× cos ( 2k1x1 + k 2 x2 ) t 3ξ12 ξ2 k12 ⎡⎣ 8k14 + 2(2 − ν) k12 k 22 + k 24 ⎤⎦ + 4 (4k12 + k 22 )2
3 22 ξ t k sin (kx1 − 3 kx2 ) 16 ⎞ ⎛1 3 22 3 − ξ t k (1 − 3ν) sin ⎜ kx1 + kx2 ⎟ 16 2 ⎠ ⎝2 ⎛1 ⎞ 3 22 3 + ξ t k (1 − 3ν) sin ⎜ kx1 − kx2 ⎟ 16 2 ⎝2 ⎠ +
3 22 ξ t k (3 − ν) sin ( 3 kx2 ) . 24
+
× cos (2k1x1 − k 2 x2 ) ⎤ ⎡1 1 4 2 2 + ⎢ t 2ξ12 k1 − t ξ1 ξ2 k1 (k12 − νk 22 ) ⎥ ⎦ ⎣8 16 1 × sin (2k1x1) − t 3ξ12 ξ2 k12 cos (k 2 x2 ) , 2
(40)
The membrane energy, bending energy and cohesive energy per unit area in the equilateral triangular mode are obtained as 4π / k 4π / 3 k 1 Φm dx1dx2 16π 2 / 3 k 2 x1 = 0 x 2 = 0 ⎡ ¯ ⎢ (1 + ν) ε02 + 3 (1 + ν) ξ 2t 2k 2ε0 = Et ⎣ 4 ⎤ 3 (1 + ν)(11−5ν) ξ 4t 4k 4⎥ , + ⎦ 128
∫
Um =
4π / k
1 2 16π / 3 k 2
∫x = 0 ∫x = 0
1
(41)
− −
4π / 3 k
∫x = 0 ∫x = 0
Uc =
1 4 2 2 2 t ξ1 ξ2 k1 k 2 sin ( 2k1x1 + 2k 2 x2 ) 32 1 4 2 2 2 + t ξ1 ξ2 k1 k 2 sin ( 2k1x1 − 2k 2 x2 ) 32 t 3ξ 2 ξ2 k13 k 2 (4k12 − νk 22 ) cos ( 2k1x1 + k 2 x2 ) + 1 4 (4k12 + k 22 )2
u2 = −
∫
1 Ub = 2 16π / 3 k 2 E¯ ξ 2t 5k 4 = , 16
Φ b dx1dx2
1
t 3ξ12 ξ2 k13 k 2 (4k12 − νk 22 ) 4 (4k12 + k 22 )2
cos ( 2k1x1 − k 2 x2 )
1 4 2 2 k12 t ξ1 ξ2 (νk12 − k 22 ) sin ( 2k 2 x2 ). 16 k2
(46)
2
The membrane energy, bending energy and cohesive energy per unit area in the herringbone mode are obtained as
(42) 4π / k
(45)
4π / 3 k
Um =
Φc dx1dx2
2
2π / k 2
1
2
=0
Φm dx1dx2
{
(
)
(
(43)
)
(
(
3.5. Herringbone mode
The out-of-plane displacement of the herringbone mode is well approximated by [47] w = ξ1t ⎡⎣ cos ( k1x1) − k1ξ2 t sin ( k1x1) cos ( k 2 x2 ) ⎤⎦ .
2π / k1
∫x = 0 ∫x
⎡ ¯ (1 + ν) ε02 + ξ12 k12 ⎢ 1 (1 + ν) t 2 + 1 (1 + ν) t 4ξ22 = Et ⎣4 8 ⎡ 1 1 t 2ξ22 × k12 + k 22 ⎤⎦ε0 + ξ14 k14 t 4 ⎢ + ⎣ 32 32 ⎛ ⎞ k12 k 24 ⎜ 2 2 ⎟ 2 ×⎜ 1−ν + k 1 + νk 2 ⎟ 2 ⎜ ⎟ 4k12 + k 22 ⎝ ⎠ 1 4 4 2 2 1 νt ξ2 k1 k 2 + 3 − ν 2 t 4ξ24 + 64 256 (47) × k14 + k 24 ⎤⎦ ,
2 ⎡ ⎛ 2 ⎞ 135 ⎛ ξt ⎞ = (CC1−N + CC1−B ) ⎜ ⎟ ⎢ 1 + ⎜ ⎟ ⎝ 15 ⎠ ⎢⎣ 2 ⎝ he ⎠ 4⎤ 22275 ⎛ ξt ⎞ ⎥ + ⎜ ⎟ − (CC2−N + CC2−B ) 8 ⎝ h e ⎠ ⎥⎦ 4⎤ ⎡ ⎛ ξt ⎞2 675 ⎛ ξt ⎞ ⎥ × ⎢1 + 9⎜ ⎟ + ⎜ ⎟ . ⎢⎣ 8 ⎝ h e ⎠ ⎥⎦ ⎝ he ⎠
1 4π 2 / k1 k 2
)
) (
(
)
)
}
2π / k1 2π / k 2 1 Φ b dx1dx2 4π / k1 k 2 x1 = 0 x 2 = 0 1 ¯ 5 2 4 1 ¯ 7 2 2 2 2 = Et ξ1 k1 + Et ξ1 ξ2 k1 (k1 + k 22 )2 , 48 96
Ub =
(44)
2
∫
∫
The in-plane displacements are solved as Uc =
1 4 2 2 3 t ξ1 ξ2 k1 sin ( 2k1x1 + 2k 2 x2 ) 32 1 4 2 2 3 − t ξ1 ξ2 k1 sin ( 2k1x1 − 2k 2 x2 ) 32 t 3ξ12 ξ2 k12 ⎡⎣ 8k14 + 2(2 − ν) k12 k 22 + k 24 ⎤⎦ + 4 (4k12 + k 22 )2
u1 = −
1 2 4π / k1 k 2
2π / k1
2π / k 2
1
2
∫x = 0 ∫x
=0
Φc dx1dx2
⎛ 2 ⎞ = (CC1−N + CC1−B ) ⎜ ⎟ ⎝ 15 ⎠ 2 ⎡ ⎛1 ⎞ ⎛ ξ1t ⎞ 1 × ⎢ 1 + 45 ⎜ + t 2ξ22 k12 ⎟ ⎜ ⎟ ⎝2 ⎠ ⎝ he ⎠ ⎢⎣ 4 7
(48)
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Figure 4. (a) The amplitude coefficient and (b) wavelength of the buckled graphene on h-BN versus the compressive strain for onedimensional mode. 4⎤ ⎛3 3 9 4 4 4 ⎞ ⎛ ξ1t ⎞ ⎥ t ξ2 k1 ⎟ ⎜ ⎟ + 495 ⎜ + t 2ξ22 k12 + ⎝8 ⎠ ⎝ h e ⎠ ⎥⎦ 8 64 2 ⎡ ⎛ ⎞ ⎛ ξ1t ⎞ 1 − (CC2 − N + CC2 − B ) ⎢ 1 + 3 ⎜ 1 + t 2ξ22 k12 ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ he ⎠ ⎢⎣ 2 4⎤ ⎛3 3 9 4 4 4 ⎞ ⎛ ξ1t ⎞ ⎥ t ξ2 k1 ⎟ ⎜ ⎟ . + 15 ⎜ + t 2ξ22 k12 + ⎝8 ⎠ ⎝ h e ⎠ ⎥⎦ 8 64
¯ 5 4 ⎤ ⎡ ¯ ⎢ 1 t 2k 2 (1 + ν) ε0 + 1 ξ 2t 4k 4⎥ + Et k Et ⎦ ⎣2 8 24 ⎡ ⎛ ⎞2 ⎛ t ⎞4 ⎤ t + (CC1−N + CC1−B ) ⎢ 6 ⎜ ⎟ + 99ξ 2 ⎜ ⎟ ⎥ ⎢⎣ ⎝ h e ⎠ ⎝ h e ⎠ ⎥⎦ 4⎤ ⎡ ⎛ ⎞2 t 45 2 ⎛ t ⎞ ⎥ − (CC2−N + CC2−B ) ⎢ 6 ⎜ ⎟ + ξ ⎜ ⎟ = 0. ⎢⎣ ⎝ h e ⎠ 2 ⎝ h e ⎠ ⎥⎦
(49)
1 ¯ 3 b1 = 2 Et (1 + ν ),
Let The total energy Utotal for each mode is the summation of the corresponding membrane energy Um , bending energy Ub and cohesive energy Uc and it is a function of buckling geometry such as wave number k and amplitude ξ . Minimization of the total energy with respect to the buckling geometry (i.e., k and ξ ) yields the solutions.
b2 =
(
b3 = 6 CC1−N + CC1−B − CC2−N − CC2−B
)(
t he
)
ξ2 = −
(
and
)
b1k 2ε0 + b2 k 4 + b3 3b2 k 4 + b 4
1 ¯ 5 Et , 24
2
)
45 b 4 = ⎡⎣ 99 CC1−N + CC1−B − 2 CC2−N + CC2−B ⎤⎦ equation (51) then becomes
(
(51)
4
() t he
.
and
(52)
At the critical buckling state, ξ = 0 yields 4. Results and discussion ε0 = −
The graphene buckles once the equi-biaxial compressive strain exceeds a critical value, εc . It should be noted that the critical compressive strain, εc , and the corresponding wave number kc , can be solved analytically by minimizing the total energy for the one-dimensional mode, square checkerboard mode, hexagonal mode and equilateral triangular mode because only two parameters (i.e., k, ξ ) appear in the energy expression but not for the herringbone mode due to the involvement of too many parameters (i.e., k1, k 2, ξ1, ξ2 ). For example, for one-dimensional ∂U mode, the minimization of the total energy ∂total = 0 and k ∂Utotal ∂ξ
b1k 2
,
which reaches a minimum value of −
k2 =
b3 b2
(53) 2 b2 b3
when
b1
corresponding to the critical compressive strain
and critical wave number as
(
¯ 3 CC1−N + CC1−B − CC2−N − CC2−B 2 Et εc =
¯ e (1 + ν) Eth
⎡ 144 C 1 + C 1 − C 2 − C 2 C −N C −B C −N C −B kc = ⎢ 2 3 ⎢ ¯ he Et ⎣
(
= 0 give ¯ 5 3 ⎤ ⎡ ¯ ⎢ t 2k (1 + ν) ε0 + 1 ξ 2t 4k 3⎥ + Et k = 0. Et ⎦ ⎣ 4 6
b2 k 4 + b 3
),
) ⎤⎥
(54)
1/4
⎥ ⎦
.
(55)
For the solutions beyond the critical strain, no analytical solutions could be found and Matlab is used to find the solutions numerically. We take the representative values
(50)
8
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Figure 5. (a) The cohesive energy, (b) the bending energy, (c) the membrane energy and (d) the total energy per unit area versus the
compressive strain.
¯ = 2203.25 eV nm−2, Et ¯ 3 = 1.5 eV, and t = 0.34 nm, Et ν = 0.21 [62]. εc = 0.0055 and kc = 7.64 nm−1 from equations (54) and (55), and they agree well with the numerical solution as shown in figure 4. The predicted critical strain ∼0.55% under equi-biaxial compression for onedimensional mode agrees reasonable well with uniaxial experiments (∼0.7%) [38] although the loading condition is different. It is found that only when the compressive strain is larger than the critical compressive strain, εc , there exist a non-zero magnitude even on a perfectly flat substrate surface. As the compressive strain increases, the predicted amplitude increases while the wavelength decreases. Similar conclusions can also be obtained for other buckling patterns and we will focus on their energies in the following. Figure 5 shows the cohesive energy, bending energy, membrane energy and total energy for each buckling pattern. It is noted that the bending energy becomes non-zero only if the compressive strain is larger than the critical strain, which is quite same for all the buckling modes. When the compressive strain is slightly larger than the critical strain, the graphene may buckle to any buckling pattern since the
difference of total energy for the buckling patterns is negligible. In this case, the geometric imperfection may play a key role in selecting the buckling mode. As the compressive strain increases to be much larger than the critical strain, the equilateral triangular modes has the highest total energy and the herringbone mode has the lowest total energy due to the significantly reduction in membrane energy, and is therefore the energetically favorable mode. These results are very similar to the system of a thin film perfected bonded to a compliant substrate [47–50] and could serve as guidelines for the strain engineering of graphene.
5. Conclusions We have established an energy method to investigate the onedimensional, square checkerboard, hexagonal, equilateral triangular and herringbone buckling patterns in a graphene/hBN heterostructure under equi-biaxial compression. Total energy consisting of cohesive energy, graphene membrane energy and graphene bending energy is obtained analytically 9
Nanotechnology 25 (2014) 445401
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for each buckling pattern. It is found that the total energies are quite same for all buckling modes at a compression slightly larger than the critical strain while the herringbone mode has the lowest total energy at a compression much larger than the critical compression. As compared to other buckling patterns, the herringbone mode significantly reduces the membrane energy of graphene at the expense of slight increase of the graphene bending energy and cohesive energy.
[26] Sławińska J, Zasada I, Kosiński P and Klusek Z 2010 Phys. Rev. B 82 085431 [27] Ding X L, Sun H, Xie X M, Ren H C, Huang F Q and Jiang M H 2011 Phys. Rev. B 84 174417 [28] Jiang J W, Wang J S and Wang B S 2011 Appl. Phys. Lett. 99 043109 [29] Kaloni T P, Cheng Y C and Schwingenschlögl U 2012 J. Mater. Chem. 22 919–22 [30] Balu R, Zhong X L, Pandey R and Karna S P 2012 Appl. Phys. Lett. 100 052104 [31] Yelgel C and Srivastava G P 2012 Appl. Surf. Sci. 258 9339–42 [32] Wallbank J R, Patel A A, Mucha-Kruczynski M, Geim A K and Fal’ko V I 2013 Phys. Rev. B 87 245408 [33] Abergel D S L, Wallbank J R, Cheni X, Mucha-Kruczynski M and Fal’ko V I 2013 New J. Phys. 15 123009 [34] Park S, Park C and Kim G 2014 J. Chem. Phys. 140 134706 [35] Scharfenberg S, Mansukhani N, Chialvo C, Weaver R and Mason N 2012 Appl. Phys. Lett. 100 021910 [36] Lanza M et al 2013 J. Appl. Phys. 113 104301 [37] Lanza M, Wang Y, Sun H, Tong Y and Duan H 2014 Acta Mech. 225 1061–73 [38] Jiang T, Huang R and Zhu Y 2014 Adv. Funct. Mater. 24 396–402 [39] Yoon D, Son Y W and Cheong H 2011 Nano lett. 11 3227–31 [40] Pan W, Xiao J, Zhu J, Yu C, Zhang G, Ni Z, Watanabe K, Taniguchi T, Shi Y and Wang X 2012 Sci. Rep. 2 893 [41] Guinea F, Katsnelson M I and Vozmediano M A H 2008 Phys. Rev. B 77 075422 [42] Pereira V M and Castro Neto A H 2009 Phys. Rev. Lett. 103 046801 [43] Guinea F, Katsnelson M I and Geim A K 2010 Nat. Phys. 6 30–3 [44] Guinea F, Geim A K, Katsnelson M I and Novoselov K S 2010 Phys. Rev. B 81 035408 [45] Levy N, Burke S A, Meaker K L, Panlasigui M, Zettl A, Guinea F, Castro Neto A H and Crommie M F 2010 Science 329 544–7 [46] Low T, Guinea F and Katsnelson M I 2011 Phys. Rev. B 83 195436 [47] Jang W J, Kim H, Shin Y R, Wang M, Jang S K, Kim M, Lee S, Kim S W, Song Y J and Kahng S J 2014 Carbon 74 139–45 [48] Giovannetti G, Khomyakov P A, Brocks G, Kelly P J and Brink V 2007 Phys. Rev. B 76 073103 [49] Gazit D 2009 Phys. Rev. B 79 113411 [50] Bhattacharya A, Bhattacharya S and Das G P 2011 Phys. Rev. B 84 075454 [51] Cai S, Breid D, Crosby A J, Suo Z and Hutchinson J W 2011 J. Mech. Phys. Solids 59 1094–114 [52] Chen X and Hutchinson J W 2004 J. Appl. Mech. 71 597–603 [53] Song J, Jiang H, Choi W M, Khang D Y, Huang Y and Rogers J A 2008 J. Appl. Phys. 103 014303 [54] Audoly B and Boudaoud A 2008 J. Mech. Phys. Solids 56 2444–58 [55] Li T and Zhang Z 2010 J. Phys. D: Appl. Phys. 43 075303 [56] Aitken Z H and Huang R 2010 J. Appl. Phys. 107 123531 [57] Timoshenko S 1940 Theory of Plates and Shells (New York: McGraw-Hill) [58] Huang Z Y, Hong W and Suo Z 2005 J. Mech. Phys. Solids 53 2101–18 [59] Thamwattana N and Hill J M 2009 Physica B 404 3906–10 [60] Zhang C, Lou J and Song J 2014 J. Appl. Phys. 115 144308 [61] Altintas B, Parlak C, Bozkurt C and Eryiğit R 2011 Eur. Phys. J. B 79 301–12 [62] Kudin K N, Scuseria G E and Yakoboson B I 2001 Phys. Rev. B 64 235406
Acknowledgments JS acknowledges the supports from the National Natural Science Foundation of China (Grant Nos.11372272 and 11321202), the Fundamental Research Funds for the Central Universities (2014FZA4027), and the Thousand Young Talents Program of China. QDY acknowledges the funding support from the US ARO grant No. W911NF-13-1-0211.
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Periodic buckling patterns of graphene/hexagonal boron nitride heterostructure
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Nanotechnology Nanotechnology 25 (2014) 445401 (10pp)
doi:10.1088/0957-4484/25/44/445401
Periodic buckling patterns of graphene/ hexagonal boron nitride heterostructure Chenxi Zhang1, Jizhou Song2 and Qingda Yang1 1
Department of Mechanical and Aerospace Engineering, University of Miami, Coral Gables, FL 33146, USA 2 Department of Engineering Mechanics and Soft Matter Research Center, Zhejiang University, Hangzhou 310027, People’s Republic of China E-mail: [email protected] Received 13 July 2014, revised 9 September 2014 Accepted for publication 10 September 2014 Published 14 October 2014 Abstract
Graphene/hexagonal boron nitride (h-BN) heterostructure has showed great potential to improve the performance of a graphene device. A graphene on an h-BN substrate may buckle due to the thermal expansion mismatch between the graphene and h-BN. We used an energy method to investigate the periodic buckling patterns including one-dimensional, square checkerboard, hexagonal, equilateral triangular and herringbone mode in a graphene/h-BN heterostructure under equi-biaxial compression. The total energy, consisting of cohesive energy, graphene membrane energy and graphene bending energy, for each buckling pattern is obtained analytically. At a compression slightly larger than the critical strain, all buckling patterns have the same total energies, which suggests that any buckling pattern may occur. At a compression much larger than the critical strain, the herringbone mode has the lowest total energy by significantly reducing the membrane energy of graphene at the expense of a slight increase of the bending energy of graphene and cohesive energy. These results may serve as guidelines for strain engineering in graphene/h-BN heterostructures. Keywords: graphene, hexagonal boron nitride, buckling (Some figures may appear in colour only in the online journal) 1. Introduction
interaction of graphene with h-BN substrate, graphene/h-BN is also an ideal system to study the intrinsic physical properties of graphene. Although the electrical, magnetic and thermal properties of graphene/h-BN heterostructure have been extensively investigated [19, 25–34], its mechanical properties have been rarely explored thus far. Understanding graphene’s morphology is crucial in tailoring graphene’s properties for device applications. Scharfenberg et al [35] examined the competition between adhesive and bending energies for few-layer graphene on a microscalecorrugated metallic substrate. It was shown that the graphene undergoes a layer-thickness dependent snap-through transition between lying flat and conforming to the substrate. Lanza et al [36, 37] investigated the morphology and performance of graphene layers on as-grown and transferred substrates due to the compressive strain and surface roughness. They observed the important adhesion differences depending on the graphene thickness and the target substrate roughness. Folds, wrinkles
Since its discovery, graphene has attracted wide attentions for potential applications ranging from flexible and invisible displays, nanoelectronic components and nanosensors to energy conversion and storage devices due to its excellent electrical, thermal and mechanical properties [1–4]. Most graphene applications require a supporting substrate (e.g., SiC and SiO2) [5–10], which leads to a significant reduction in electron mobility due to the charged surface states and impurities, surface roughness and surface optical phonons [11–18]. Recently, hexagonal boron nitride (h-BN) has been used as an ideal substrate for graphene devices due to h-BN’s atomically flat structure with little dangling bonds and charge traps [19]. It has been shown that graphene/h-BN heterostructures exhibit less intrinsic doping, much higher electron mobility and improved on/off ratio than conventional graphene devices on SiO2 substrate [20–24]. Due to the reduced 0957-4484/14/445401+10$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
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Figure 1. Schematic diagrams of the buckling patterns: (a) one-dimensional mode, (b) square checkerboard mode, (c) hexagonal mode, (d)
equilateral triangular mode, and (e) herringbone mode under equi-biaxial compression.
and cracks were also observed in the samples. Jiang et al [38] studied the interfacial sliding due to uniaxial tension and buckling due to uniaxial compression of monolayer graphene on a stretchable substrate. It was found that the critical compressive strain of buckling is around −0.7%. Yoon et al [39] realized an equi-biaxial compression in graphene on SiO2 due to their thermal expansion coefficient mismatch although the deformation and morphology of graphene were not well understood. Pan et al [40] studied biaxial compressive strain in graphene/h-BN heterostructures and found that the strain is spatially inhomogeneous with the occurrence of equilateral triangular and quadrilateral bubbles possibly due to the existence of defects. Although many researchers have studied the morphology of graphene on different substrates, some questions still remain unknown for an ideal graphene/hBN heterostructure under biaxial compression. For example, can the graphene/h-BN heterostructure buckle as the compression increases? What is the condition for buckling? What is the buckling mode? etc. These questions are very critical since the deformation of graphene is strongly tied to its electrical performance (e.g., electron mobility and band gap) [41–50]. It is shown that the film bonded to a substrate under equi-biaxial compression may buckle into several intriguing periodic buckling patterns such as one-dimensional, square checkerboard, hexagonal, equilateral triangular and herringbone modes as shown schematically in figure 1 [51]. Many researchers have investigated these periodic buckling modes. For example, Chen and Hutchinson [52] and Song et al [53] developed an energy method with the total energy consisting of the film energy and substrate energy to study one-dimensional, square checkerboard and herringbone mode under equi-biaxial compression and they
showed that the herringbone mode has the lowest energy in the buckled state while the one-dimensional mode has the greatest. Audoly and Boudaoud [54] explored further details of these modes including the transition from onedimensional mode to the herringbone mode as the compression is not equi-biaxial. Cai et al [51] established an analytical upper-bond method and performed a full numerical analysis to study the energies of all these periodic patterns in the buckled state. It is found that the square checkerboard mode has lowest energy in small strain state (slightly larger than the critical buckling strain) and the herringbone mode is dominant in relatively large strain state. They suggested that initial imperfections may play a role in mode selection. The objective of this paper is to use an energy method to investigate the buckling behavior of graphene/h-BN heterostructure under equi-biaxial compression. All five possible periodic patterns (i.e., one-dimensional, square checkerboard, hexagonal, equilateral triangular and herringbone modes) are considered. Different from the system of a film perfectly bonded to a substrate, where the total energy consists of the film energy and substrate energy, the graphene/h-BN heterostructure has film energy and cohesive energy due to the van der Waals interaction at the interface but not the substrate energy since the substrate could be assumed to be a rigid body [55, 56]. This paper is outlined as follows. An energy method is developed in section 2 to obtain the total energy of the buckled state of graphene/h-BN heterostructures. The analytical expressions of the energies for the five periodic buckling modes are presented in section 3. Section 4 describes the results and discussions. Conclusions are made in section 5. 2
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2. An energy method for buckling analysis
and ⎧ ∂ 2u2 ∂ 2u ∂ 2u1 ⎫ 1 1 ⎨ 2 + (1 − ν) 22 + (1 + ν) ⎬ 2 2 ∂x1∂x2 ⎭ ∂x1 ⎩ ∂x 2
We develop an energy method to determine the buckling geometry in this section. The total energy of the graphene/hBN heterostructure consists of the strain energy of graphene and cohesive energy at the interface (or graphene-substrate interaction energy) [55, 56]. Minimization of total energy yields the buckling geometry.
⎧ ⎡ ⎛ ∂ 2w ⎞ ⎤ ⎪ ⎛ ∂w ⎞ ⎛ ∂ 2w ⎞ 1 ⎨⎜ +⎪ ⎟ ⎢ ⎜ 2 ⎟ + (1 − ν) ⎜ 2 ⎟ ⎥ ⎝ ∂x1 ⎠ ⎥⎦ ⎩ ⎝ ∂x2 ⎠ ⎢⎣ ⎝ ∂x2 ⎠ 2 +
2.1. The strain energy density of graphene
∂u j ⎞ 1 ⎛ ∂w ⎞ ⎛ ∂w ⎞ 1 ⎛ ∂u i ⎜⎜ ⎟⎟ + ⎜ ⎟⎟ , + ⎟ ⎜⎜ 2 ⎝ ∂x j ∂x i ⎠ 2 ⎝ ∂x i ⎠ ⎝ ∂x j ⎠
⎪
(5)
⎪
Once u1 (x1, x 2 ) and u 2 (x1, x 2 ) are solved, membrane force and membrane strain can be easily obtained by equations (1) and (2). The membrane energy density is then given by
The graphene is subjected to an equi-biaxial initial strain 0 ε110 = ε22 = ε0 due to the thermal expansion coefficient mismatch between graphene and substrate, and it buckles once ε0 exceeds a critical value. The graphene is modeled as an elastic Von Karman plate with finite rotation [57]. The strain energy density of graphene consists of the membrane energy density and the bending energy density. The membrane strain εij (where i, j = 1, 2) is related to the in-plane displacements u1 (x1, x 2 ) and u 2 (x1, x 2 ) and outof-plane displacement w (x1, x 2 ) given by εij = εij0 +
⎛ ∂w ⎞ ⎛ ∂ 2w ⎞ ⎫ 1 ⎟ ⎬ = 0. (1 + ν) ⎜ ⎟⎜ 2 ⎝ ∂x1 ⎠ ⎝ ∂x1∂x2 ⎠ ⎭
Φm =
1 ( N11ε11 + 2N12 ε12 + N22 ε22 ). 2
(6)
The bending energy density only depends on the out-ofplane displacement w and is given by 2 ⎛ ∂ 2w ⎞ ⎛ ∂ 2w ⎞ ⎛ ∂ 2w ⎞2 ¯ 3 ⎡⎢ ⎛ ∂ 2w ⎞ Et ⎜ 2 ⎟ + ⎜ 2 ⎟ + 2ν ⎜ 2 ⎟ ⎜ 2 ⎟ Φb = 24 ⎢⎣ ⎝ ∂x1 ⎠ ⎝ ∂x1 ⎠ ⎝ ∂x2 ⎠ ⎝ ∂x 2 ⎠
⎛ ∂ 2w ⎞2 + 2(1 − ν) ⎜ ⎟ . ⎝ ∂x1∂x2 ⎠
(1)
where εij0 is the initial strain and i, j = 1,2. The Hooke’s law gives the membrane force in the graphene as
(7)
2.2. The cohesive energy density
¯ ⎡⎣ (1 − ν) εij + νεkk δ ij ⎤⎦ , Nij = Et
(
The interlayer bonding between graphene and h-BN is due to van der Waals interaction, which is usually represented by the Lennard-Jones 6–12 potential
(2)
)
where E¯ = E / 1 − ν 2 is the plan-strain modulus of graphene, ν is the Poisson's ratio of graphene, t is the thickness of graphene and δij is the Kronecker delta (i, j = 1, 2). The shear tractions at the graphene/h-BN interface are obtained from the force equilibrium as Ti =
∂Nij ∂x j
.
⎛ η12 η6 ⎞ V (r ) = 4ζ ⎜ 12 − ⎟ , r6 ⎠ ⎝r
(8)
where V (r ) is the energy between a pair of atoms of distance r , 6 2 η is the equilibrium distance between two atoms and ζ is the bond energy at the equilibrium distance. For a pair of carbon and nitrogen atoms, the bond energy and equilibrium distance are ζC − N = 0.004 068 eV and ηC − N = 0.3367 nm and ζC − B = 0.003 294 eV and ηC − B = 0.3411 nm for a pair of carbon and boron atoms [59]. To describe van der Waals interactions between graphene monolayer and h-BN substrate, graphene and h-BN are modeled as homogenous and infinitely large layer and solid 4 (as shown in figure 2), respectively, with ρC = 2 denoting
(3)
It has been shown that the shear stress at the graphene/hBN interface has a negligible effect on the buckling geometry [58]. The vanishing shear tractions, i.e., T1 = 0 and T2 = 0 , yields two governing equations for the in-plane displacements u1 (x1, x 2 ) and u 2 (x1, x 2 ) given the out-of-plane displacement w (x1, x 2 ),
3 3 lC
⎧ ∂ 2u1 ∂ 2u ∂ 2u2 ⎫ 1 1 ⎨ 2 + (1 − ν) 21 + (1 + ν) ⎬ 2 2 ∂x1∂x2 ⎭ ∂x 2 ⎩ ∂x1 ⎧ ⎡ ⎛ ∂ 2w ⎞ ⎤ ⎪ ⎛ ∂w ⎞ ⎛ ∂ 2w ⎞ 1 ⎢⎜ ⎜ 2 ⎟⎥ ⎟ + (1 − ) +⎨ ν ⎜ ⎟ 2 ⎪ ⎝ ∂x2 ⎠ ⎥⎦ ⎩ ⎝ ∂x1 ⎠ ⎢⎣ ⎝ ∂x1 ⎠ 2 ⎛ ∂w ⎞ ⎛ ∂ 2w ⎞ ⎫ 1 + (1 + ν) ⎜ ⎟ ⎬ = 0, ⎟⎜ 2 ⎝ ∂x2 ⎠ ⎝ ∂x1∂x2 ⎠ ⎭ ⎪ ⎪
the number of carbon atoms per unit area on graphene and 2 denoting the number of the nitrogen ρN = ρB = 2 3 3 l h − BN h h − BN
and boron atoms per unit volume in h-BN [60]. Here lC = 0.1408 nm and l h − BN = 0.1440 nm are the equilibrium bond length of graphene and h-BN [61], h h − BN = 0.3406 nm is the inter-layer distance in h-BN. Then we have ρC = 38.80 atoms nm−2 and ρN = ρB = 54.48 atoms nm−3. As shown in figure 2, the graphene is parallel to the h-BN with h denoting the equilibrium distance. The distance
(4)
3
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Figure 3. Schematic illustration of graphene subjected to the opening displacement w on a h-BN substrate.
Figure 2. Schematic diagram of graphene/h-BN heterostructure.
Equation (13) can be rewritten as between a carbon atom on graphene and a nitrogen (or boron) atom on h-BN is r = x 2 + z 2 , where z is the projected distance between two atoms. The cohesive energy per unit area is then given by Φ graphene/ h − BN = ΦC − N + ΦC − B,
⎞9 ⎛ 2 ⎞⎛ 1 Φc = (CC1 − N + CC1 − B ) ⎜ ⎟ ⎜ ⎟ ⎝ 15 ⎠ ⎝ 1 + w / h e ⎠ ⎛ ⎞3 1 − CC2 − N + CC2 − B ⎜ ⎟, ⎝ 1 + w /he ⎠
(
(9)
where ΦC − N and ΦC − B are the energies due to the interactions between carbon and nitrogen atoms, and between carbon and boron atoms, respectively, and they are obtained by −h
ΦC − N = 2πρC ρ N dA =
∫−∞
dx
∫0
(14)
where
∞
V (r ) z d z
⎛ 2η 9 η3 ⎞ 2 πρC ρN ζC − N ηC3−N ⎜⎜ C −9N − C −3N ⎟⎟ , 3 h ⎠ ⎝ 15h
)
(10)
CC1−N =
2 πρC ρN ζC − N ηC12−N h e9 , 3
(15)
CC2−N =
2 πρC ρN ζC − N ηC6−N h e3, 3
(16)
CC1−B =
2 πρC ρB ζC − B ηC12−B h e9 , 3
(17)
CC2−B =
2 πρC ρB ζC − B ηC6−B h e3, 3
(18)
and
and −h
ΦC − B = 2πρC ρB dA =
∫−∞
dx
∫0
∞
V (r ) z d z
⎛ 2η 9 η3 ⎞ 2 πρC ρB ζC − B ηC3−B ⎜⎜ C −9B − C −3 B ⎟⎟ . 3 h ⎠ ⎝ 15h
are all constants. For a small w /he , the cohesive energy density could be approximated by the Taylor series by ignoring the higher order terms, i.e.,
(11)
The equilibrium distance he is determined by minimizing the total cohesive energy, by
∂Φ graphene/ h − BN ∂h
⎡ ⎛ w ⎞1 ⎛ w ⎞2 ⎛ 2 ⎞ Φc ≈ (CC1−N + CC1−B ) ⎜ ⎟ ⎢ 1−9 ⎜ ⎟ + 45 ⎜ ⎟ ⎝ 15 ⎠ ⎢⎣ ⎝ he ⎠ ⎝ he ⎠
= 0, and is then given
⎛ 2 ζC − N η 12 + ζC − B η 12 ⎞1/6 C −N C −B ⎟ h e = ⎜⎜ ⋅ ⎟ , 6 6 5 ζ η ζ η + C −N C −N C −B C −B ⎠ ⎝
⎛ w ⎞3 ⎛ w ⎞4 ⎤ − 165 ⎜ ⎟ + 495 ⎜ ⎟ ⎥ − (CC2−N + CC2−B ) ⎝ he ⎠ ⎝ h e ⎠ ⎥⎦
(12)
⎡ ⎛ w ⎞1 ⎛ w ⎞2 ⎛ w ⎞3 ⎛ w ⎞4 ⎤ × ⎢ 1 − 3 ⎜ ⎟ + 6 ⎜ ⎟ − 10 ⎜ ⎟ + 15 ⎜ ⎟ ⎥ . ⎢⎣ ⎝ he ⎠ ⎝ he ⎠ ⎝ he ⎠ ⎝ h e ⎠ ⎥⎦ (19)
which is about 0.2908 nm. For the out-of-plane displacement w (x1, x 2 ) beyond the equilibrium distance he as shown in figure 3, the cohesive energy density can be similarly obtained as
This approximation is critical to derive an analytic expression for cohesive energy.
⎡ 2ηC9−N 2 Φc (w ) = πρC ρN ζC − N ηC3−N ⎢ ⎢⎣ 15 ( h e + w )9 3 ⎤ ⎥ + 2 πρ ρ ζC − B η 3 − C B C −B 3⎥ ( he + w) ⎦ 3 ⎡ ηC3−B ⎤⎥ 2ηC9−B ×⎢ − . ⎢⎣ 15 ( h e + w )9 ( h e + w )3 ⎥⎦
2.3. The total energy
ηC3−N
The total energy Utotal consists of the membrane energy Um and bending energy Ub in graphene and cohesive energy Uc , which can be obtained from the integration of the corresponding energy densities Φm , Φb , and Φc . Minimization of the total energy with respect to the buckling geometry yields solutions for different buckling patterns to be discussed in section 4.
(13)
4
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3. Energies of the buckling modes
obtained by solving equations (4) and (5) as
The total energies for one-dimensional mode, square checkerboard mode, hexagonal mode, equilateral triangular mode and herringbone mode are obtained in this section.
u1 =
1 ξ 2t 2k sin ( 2 kx1 + 2 kx2 ) 32 2 1 + ξ 2t 2k sin ( 2 kx1 − 2 kx2 ) 32 2 1 + ξ 2t 2k (1 − ν) sin ( 2 kx1) , 16 2
3.1. One-dimensional mode
The out-of-plane displacement of one-dimensional model is given by w = ξt cos (kx1) ,
u2 =
(20)
where k is the wave number along x1 direction, ξ is a nondimensional coefficient of amplitude, and t is the graphene thickness. Therefore, ξt denotes the amplitude. Substituting w into the governing equations [equations (4) and (5)] of inplane displacements u1 (x1, x 2 ) and u 2 (x1, x 2 ) yields u1 =
1 2 2 t ξ k sin (2kx1) , 8
(27)
1 ξ 2t 2k sin ( 2 kx1 + 2 kx2 ) 32 2 1 − ξ 2t 2k sin ( 2 kx1 − 2 kx2 ) 32 2 1 + ξ 2t 2k (1 − ν) sin ( 2 kx2 ) . 16 2
(28)
The membrane energy, bending energy and cohesive energy per unit area in square checkerboard mode are then given analytically by
(21)
and u2 = 0.
2 2π
(22)
∫
The membrane strain can be obtained by equation (1) as ε11 = ε0 + ξ 2t 2k 2 /4 and ε12 = ε21 = ε22 = 0 . The membrane energy, bending energy and cohesive energy per unit area in one-dimensional mode are then obtained analytically by 2π / k 1 Φm dx1 Um = 2π / k 0 ⎡ ¯ ⎢ (1 + ν) ε02 + 1 ξ 2t 2k 2 (1 + ν) ε0 = Et ⎣ 4 ⎤ 1 44 4 ξ t k ⎥, + ⎦ 32
∫
Ub =
Uc =
1 2π / k
∫0
1 2π / k
∫0
2π / k
Φ b dx1 =
E¯ ξ 2t 5k 4 , 48
Ub =
Uc = (24)
2 2π
1 2 2 8π / k
∫x =k0 ∫x 1
2 2π
∫x =k0 ∫x 1
2 2π k 2
=0
2 2π k 2
=0
Φ b dx1dx2 =
(29)
E¯ ξ 2t 5k 4 , (30) 96
Φc dx1dx2
2 ⎡ ⎛ 2 ⎞ 45 ⎛ ξt ⎞ = (CC1−N + CC1−B ) ⎜ ⎟ ⎢ 1 + ⎜ ⎟ ⎝ 15 ⎠ ⎢⎣ 4 ⎝ he ⎠
Φc dx1
4⎤ 4455 ⎛ ξt ⎞ ⎥ ⎜ ⎟ − CC2−N + CC2−B 64 ⎝ h e ⎠ ⎥⎦ 2 4⎤ ⎡ 3 ⎛ ξt ⎞ 135 ⎛ ξt ⎞ ⎥ × ⎢1 + ⎜ ⎟ + ⎜ ⎟ . ⎢⎣ 2 ⎝ he ⎠ 64 ⎝ h e ⎠ ⎥⎦
2 ⎡ ⎛ 2 ⎞ 45 ⎛ ξt ⎞ = (CC1−N + CC1−B ) ⎜ ⎟ ⎢ 1 + ⎜ ⎟ ⎝ 15 ⎠ ⎢⎣ 2 ⎝ he ⎠
+
+
(
) (31)
(25) 3.3. Hexagonal mode
The out-of-plane displacement of generalized hexagonal mode is given by [47]
3.2. Square checkerboard mode
The out-of-plane displacement of square checkerboard mode is given by [47] ⎛ 1 ⎞ ⎛ 1 ⎞ w = ξt cos ⎜ kx1⎟ cos ⎜ kx2 ⎟ . ⎝ 2 ⎠ ⎝ 2 ⎠
∫
1 2 2 8π / k
(23)
2π / k
4⎤ 1485 ⎛ ξt ⎞ ⎥ ⎜ ⎟ − (CC2−N + CC2−B ) 8 ⎝ h e ⎠ ⎥⎦ 4⎤ ⎡ ⎛ ξt ⎞2 45 ⎛ ξt ⎞ ⎥ × ⎢1 + 3⎜ ⎟ + ⎜ ⎟ . ⎢⎣ 8 ⎝ h e ⎠ ⎥⎦ ⎝ he ⎠
2 2π
1 k k Φm dx1dx2 Um = 2 2 x1 = 0 x2 = 0 8π / k ⎡ ¯ ⎢ (1 + ν) ε02 + 1 (1 + ν) ξ 2t 2k 2ε0 = Et ⎣ 8 ⎤ 1 (1 + ν)(3 − ν) ξ 4t 4k 4⎥ , + ⎦ 512
⎡ ⎛ 3 ⎞⎤ ⎛1 ⎞ w = ξt ⎢ cos ( kx1) + 2 cos ⎜ kx1⎟ cos ⎜ kx2 ⎟ ⎥ , ⎝2 ⎠ ⎢⎣ ⎝ 2 ⎠ ⎥⎦
(32)
(26) 1
3
where the wave numbers are 2 k along x1 direction and 2 k along x 2 direction. Solving the governing equations of in-
The in-plane displacements u1 (x1, x 2 ) and u 2 (x1, x 2 ) can be 5
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plane displacement u1 (x1, x 2 ) and u 2 (x1, x 2 ) gives
u1 =
⎞ ⎛3 1 22 3 ξ t k (3 − ν) sin ⎜ kx1 + kx2 ⎟ 16 2 ⎠ ⎝2 ⎛3 ⎞ 1 22 3 + ξ t k (3 − ν) sin ⎜ kx1 − kx2 ⎟ 16 2 ⎝2 ⎠ 1 22 + ξ t k sin(kx1 + 3 kx2 ) 16 1 22 + ξ t k sin(kx1 − 3 kx2 ) 16 ⎞ ⎛1 1 22 3 + ξ t k (1 − 3ν) sin ⎜ kx1 + kx2 ⎟ 16 2 ⎠ ⎝2 ⎞ ⎛1 1 22 3 + ξ t k (1 − 3ν) sin ⎜ kx1 − kx2 ⎟ 16 2 ⎠ ⎝2 1 1 + ξ 2t 2k sin ( 2kx1) + ξ 2t 2k (1−3ν) sin ( kx1) , 8 8
u2 =
Uc =
(33)
∫
1 16π / 3 k 2 E¯ ξ 2t 5k 4 = , 16
Ub =
2
4π / k
4π / 3 k
1
2
=0
=0
Φc dx1dx2
(37)
The out-of-plane displacement of equilateral triangular mode is given by [47] ⎡ ⎛ 3 ⎞⎤ ⎛1 ⎞ w = ξt ⎢ − sin ( kx1) + 2 sin ⎜ kx1⎟ cos ⎜ kx2 ⎟ ⎥ , (38) ⎝2 ⎠ ⎢⎣ ⎝ 2 ⎠ ⎥⎦ 1
3
where the wave number is 2 k along x1 direction and 2 k along x 2 direction, which are same as those in hexagonal mode. The in-plane displacements are solved as u1 =
(34)
∫
∫x = 0 ∫x
2
3.4. Equilateral triangular mode
The membrane energy, bending energy and cohesive energy per unit area in hexagonal mode are obtained as 4π / k 4π / 3 k 1 Φm dx1dx2 16π 2 / 3 k 2 x1 = 0 x 2 = 0 ⎡ ¯ ⎢ (1 + ν) ε02 + 3 (1 + ν) ξ 2t 2k 2ε0 = Et ⎣ 4 ⎤ 3 (1 + ν)(11−5ν) ξ 4t 4k 4⎥ , + ⎦ 128
1
3 4⎤ 495 ⎛ ξt ⎞ 22 275 ⎛ ξt ⎞ ⎥ ⎜ ⎟ + ⎜ ⎟ 2 ⎝ he ⎠ 8 ⎝ h e ⎠ ⎥⎦ ⎡ ⎛ ξt ⎞2 ⎛ ξt ⎞3 − (CC2−N + CC2−B ) ⎢ 1 + 9 ⎜ ⎟ − 15 ⎜ ⎟ ⎢⎣ ⎝ he ⎠ ⎝ he ⎠ 4⎤ 675 ⎛ ξt ⎞ ⎥ + ⎜ ⎟ . 8 ⎝ h e ⎠ ⎥⎦
3 22 ξ t k sin(kx1 + 3 kx2 ) 16 3 22 − ξ t k sin(kx1 − 3 kx2 ) 16 ⎞ ⎛1 3 22 3 kx2 ⎟ + ξ t k (1 − 3ν) sin ⎜ kx1 + 16 2 ⎠ ⎝2 ⎛1 ⎞ 3 22 3 kx2 ⎟ − ξ t k (1 − 3ν) sin ⎜ kx1 − 2 16 ⎠ ⎝2
Um =
4π / 3 k
−
+
3 22 ξ t k (3 − ν) sin( 3 kx2 ). 24
4π / k
∫x = 0 ∫x
2 ⎡ ⎛ 2 ⎞ 135 ⎛ ξt ⎞ = (CC1−N + CC1−B ) ⎜ ⎟ ⎢ 1 + ⎜ ⎟ ⎝ 15 ⎠ ⎢⎣ 2 ⎝ he ⎠
⎛3 ⎞ 3 22 3 kx2 ⎟ ξ t k (3 − ν) sin ⎜ kx1 + 48 2 ⎝2 ⎠ ⎞ ⎛3 3 22 3 kx2 ⎟ − ξ t k (3 − ν) sin ⎜ kx1 − 48 2 ⎠ ⎝2
+
1 16π / 3 k 2 2
⎛3 ⎞ 3 1 22 ξ t k (3 − ν) sin ⎜ kx1 + kx2 ⎟ 16 2 ⎝2 ⎠ ⎛ ⎞ 1 22 3 3 + ξ t k (3 − ν) sin ⎜ kx1 − kx2 ⎟ 16 2 ⎝2 ⎠ 1 22 − ξ t k sin (kx1 + 3 kx2 ) 16 1 22 − ξ t k sin (kx1 − 3 kx2 ) 16 ⎞ ⎛1 1 22 3 − ξ t k (1−3ν) sin ⎜ kx1 + kx2 ⎟ 16 2 ⎠ ⎝2 ⎛1 ⎞ 1 22 3 − ξ t k (1−3ν) sin ⎜ kx1 − kx2 ⎟ 16 2 ⎝2 ⎠ 1 22 1 22 − ξ t k sin ( 2kx1) − ξ t k (1−3ν) sin ( kx1) , 8 8
(35)
u2 =
Φ b dx1dx2
⎛3 ⎞ 3 22 3 kx2 ⎟ ξ t k (3 − ν) sin ⎜ kx1 + 48 2 ⎝2 ⎠ ⎛3 ⎞ 3 22 3 kx2 ⎟ − ξ t k (3 − ν) sin ⎜ kx1 − 48 2 ⎝2 ⎠ −
(36)
6
3 22 ξ t k sin (kx1 + 16
3 kx2 )
(39)
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× cos ( 2k1x1 + k 2 x2 ) t 3ξ12 ξ2 k12 ⎡⎣ 8k14 + 2(2 − ν) k12 k 22 + k 24 ⎤⎦ + 4 (4k12 + k 22 )2
3 22 ξ t k sin (kx1 − 3 kx2 ) 16 ⎞ ⎛1 3 22 3 − ξ t k (1 − 3ν) sin ⎜ kx1 + kx2 ⎟ 16 2 ⎠ ⎝2 ⎛1 ⎞ 3 22 3 + ξ t k (1 − 3ν) sin ⎜ kx1 − kx2 ⎟ 16 2 ⎝2 ⎠ +
3 22 ξ t k (3 − ν) sin ( 3 kx2 ) . 24
+
× cos (2k1x1 − k 2 x2 ) ⎤ ⎡1 1 4 2 2 + ⎢ t 2ξ12 k1 − t ξ1 ξ2 k1 (k12 − νk 22 ) ⎥ ⎦ ⎣8 16 1 × sin (2k1x1) − t 3ξ12 ξ2 k12 cos (k 2 x2 ) , 2
(40)
The membrane energy, bending energy and cohesive energy per unit area in the equilateral triangular mode are obtained as 4π / k 4π / 3 k 1 Φm dx1dx2 16π 2 / 3 k 2 x1 = 0 x 2 = 0 ⎡ ¯ ⎢ (1 + ν) ε02 + 3 (1 + ν) ξ 2t 2k 2ε0 = Et ⎣ 4 ⎤ 3 (1 + ν)(11−5ν) ξ 4t 4k 4⎥ , + ⎦ 128
∫
Um =
4π / k
1 2 16π / 3 k 2
∫x = 0 ∫x = 0
1
(41)
− −
4π / 3 k
∫x = 0 ∫x = 0
Uc =
1 4 2 2 2 t ξ1 ξ2 k1 k 2 sin ( 2k1x1 + 2k 2 x2 ) 32 1 4 2 2 2 + t ξ1 ξ2 k1 k 2 sin ( 2k1x1 − 2k 2 x2 ) 32 t 3ξ 2 ξ2 k13 k 2 (4k12 − νk 22 ) cos ( 2k1x1 + k 2 x2 ) + 1 4 (4k12 + k 22 )2
u2 = −
∫
1 Ub = 2 16π / 3 k 2 E¯ ξ 2t 5k 4 = , 16
Φ b dx1dx2
1
t 3ξ12 ξ2 k13 k 2 (4k12 − νk 22 ) 4 (4k12 + k 22 )2
cos ( 2k1x1 − k 2 x2 )
1 4 2 2 k12 t ξ1 ξ2 (νk12 − k 22 ) sin ( 2k 2 x2 ). 16 k2
(46)
2
The membrane energy, bending energy and cohesive energy per unit area in the herringbone mode are obtained as
(42) 4π / k
(45)
4π / 3 k
Um =
Φc dx1dx2
2
2π / k 2
1
2
=0
Φm dx1dx2
{
(
)
(
(43)
)
(
(
3.5. Herringbone mode
The out-of-plane displacement of the herringbone mode is well approximated by [47] w = ξ1t ⎡⎣ cos ( k1x1) − k1ξ2 t sin ( k1x1) cos ( k 2 x2 ) ⎤⎦ .
2π / k1
∫x = 0 ∫x
⎡ ¯ (1 + ν) ε02 + ξ12 k12 ⎢ 1 (1 + ν) t 2 + 1 (1 + ν) t 4ξ22 = Et ⎣4 8 ⎡ 1 1 t 2ξ22 × k12 + k 22 ⎤⎦ε0 + ξ14 k14 t 4 ⎢ + ⎣ 32 32 ⎛ ⎞ k12 k 24 ⎜ 2 2 ⎟ 2 ×⎜ 1−ν + k 1 + νk 2 ⎟ 2 ⎜ ⎟ 4k12 + k 22 ⎝ ⎠ 1 4 4 2 2 1 νt ξ2 k1 k 2 + 3 − ν 2 t 4ξ24 + 64 256 (47) × k14 + k 24 ⎤⎦ ,
2 ⎡ ⎛ 2 ⎞ 135 ⎛ ξt ⎞ = (CC1−N + CC1−B ) ⎜ ⎟ ⎢ 1 + ⎜ ⎟ ⎝ 15 ⎠ ⎢⎣ 2 ⎝ he ⎠ 4⎤ 22275 ⎛ ξt ⎞ ⎥ + ⎜ ⎟ − (CC2−N + CC2−B ) 8 ⎝ h e ⎠ ⎥⎦ 4⎤ ⎡ ⎛ ξt ⎞2 675 ⎛ ξt ⎞ ⎥ × ⎢1 + 9⎜ ⎟ + ⎜ ⎟ . ⎢⎣ 8 ⎝ h e ⎠ ⎥⎦ ⎝ he ⎠
1 4π 2 / k1 k 2
)
) (
(
)
)
}
2π / k1 2π / k 2 1 Φ b dx1dx2 4π / k1 k 2 x1 = 0 x 2 = 0 1 ¯ 5 2 4 1 ¯ 7 2 2 2 2 = Et ξ1 k1 + Et ξ1 ξ2 k1 (k1 + k 22 )2 , 48 96
Ub =
(44)
2
∫
∫
The in-plane displacements are solved as Uc =
1 4 2 2 3 t ξ1 ξ2 k1 sin ( 2k1x1 + 2k 2 x2 ) 32 1 4 2 2 3 − t ξ1 ξ2 k1 sin ( 2k1x1 − 2k 2 x2 ) 32 t 3ξ12 ξ2 k12 ⎡⎣ 8k14 + 2(2 − ν) k12 k 22 + k 24 ⎤⎦ + 4 (4k12 + k 22 )2
u1 = −
1 2 4π / k1 k 2
2π / k1
2π / k 2
1
2
∫x = 0 ∫x
=0
Φc dx1dx2
⎛ 2 ⎞ = (CC1−N + CC1−B ) ⎜ ⎟ ⎝ 15 ⎠ 2 ⎡ ⎛1 ⎞ ⎛ ξ1t ⎞ 1 × ⎢ 1 + 45 ⎜ + t 2ξ22 k12 ⎟ ⎜ ⎟ ⎝2 ⎠ ⎝ he ⎠ ⎢⎣ 4 7
(48)
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Figure 4. (a) The amplitude coefficient and (b) wavelength of the buckled graphene on h-BN versus the compressive strain for onedimensional mode. 4⎤ ⎛3 3 9 4 4 4 ⎞ ⎛ ξ1t ⎞ ⎥ t ξ2 k1 ⎟ ⎜ ⎟ + 495 ⎜ + t 2ξ22 k12 + ⎝8 ⎠ ⎝ h e ⎠ ⎥⎦ 8 64 2 ⎡ ⎛ ⎞ ⎛ ξ1t ⎞ 1 − (CC2 − N + CC2 − B ) ⎢ 1 + 3 ⎜ 1 + t 2ξ22 k12 ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ he ⎠ ⎢⎣ 2 4⎤ ⎛3 3 9 4 4 4 ⎞ ⎛ ξ1t ⎞ ⎥ t ξ2 k1 ⎟ ⎜ ⎟ . + 15 ⎜ + t 2ξ22 k12 + ⎝8 ⎠ ⎝ h e ⎠ ⎥⎦ 8 64
¯ 5 4 ⎤ ⎡ ¯ ⎢ 1 t 2k 2 (1 + ν) ε0 + 1 ξ 2t 4k 4⎥ + Et k Et ⎦ ⎣2 8 24 ⎡ ⎛ ⎞2 ⎛ t ⎞4 ⎤ t + (CC1−N + CC1−B ) ⎢ 6 ⎜ ⎟ + 99ξ 2 ⎜ ⎟ ⎥ ⎢⎣ ⎝ h e ⎠ ⎝ h e ⎠ ⎥⎦ 4⎤ ⎡ ⎛ ⎞2 t 45 2 ⎛ t ⎞ ⎥ − (CC2−N + CC2−B ) ⎢ 6 ⎜ ⎟ + ξ ⎜ ⎟ = 0. ⎢⎣ ⎝ h e ⎠ 2 ⎝ h e ⎠ ⎥⎦
(49)
1 ¯ 3 b1 = 2 Et (1 + ν ),
Let The total energy Utotal for each mode is the summation of the corresponding membrane energy Um , bending energy Ub and cohesive energy Uc and it is a function of buckling geometry such as wave number k and amplitude ξ . Minimization of the total energy with respect to the buckling geometry (i.e., k and ξ ) yields the solutions.
b2 =
(
b3 = 6 CC1−N + CC1−B − CC2−N − CC2−B
)(
t he
)
ξ2 = −
(
and
)
b1k 2ε0 + b2 k 4 + b3 3b2 k 4 + b 4
1 ¯ 5 Et , 24
2
)
45 b 4 = ⎡⎣ 99 CC1−N + CC1−B − 2 CC2−N + CC2−B ⎤⎦ equation (51) then becomes
(
(51)
4
() t he
.
and
(52)
At the critical buckling state, ξ = 0 yields 4. Results and discussion ε0 = −
The graphene buckles once the equi-biaxial compressive strain exceeds a critical value, εc . It should be noted that the critical compressive strain, εc , and the corresponding wave number kc , can be solved analytically by minimizing the total energy for the one-dimensional mode, square checkerboard mode, hexagonal mode and equilateral triangular mode because only two parameters (i.e., k, ξ ) appear in the energy expression but not for the herringbone mode due to the involvement of too many parameters (i.e., k1, k 2, ξ1, ξ2 ). For example, for one-dimensional ∂U mode, the minimization of the total energy ∂total = 0 and k ∂Utotal ∂ξ
b1k 2
,
which reaches a minimum value of −
k2 =
b3 b2
(53) 2 b2 b3
when
b1
corresponding to the critical compressive strain
and critical wave number as
(
¯ 3 CC1−N + CC1−B − CC2−N − CC2−B 2 Et εc =
¯ e (1 + ν) Eth
⎡ 144 C 1 + C 1 − C 2 − C 2 C −N C −B C −N C −B kc = ⎢ 2 3 ⎢ ¯ he Et ⎣
(
= 0 give ¯ 5 3 ⎤ ⎡ ¯ ⎢ t 2k (1 + ν) ε0 + 1 ξ 2t 4k 3⎥ + Et k = 0. Et ⎦ ⎣ 4 6
b2 k 4 + b 3
),
) ⎤⎥
(54)
1/4
⎥ ⎦
.
(55)
For the solutions beyond the critical strain, no analytical solutions could be found and Matlab is used to find the solutions numerically. We take the representative values
(50)
8
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Figure 5. (a) The cohesive energy, (b) the bending energy, (c) the membrane energy and (d) the total energy per unit area versus the
compressive strain.
¯ = 2203.25 eV nm−2, Et ¯ 3 = 1.5 eV, and t = 0.34 nm, Et ν = 0.21 [62]. εc = 0.0055 and kc = 7.64 nm−1 from equations (54) and (55), and they agree well with the numerical solution as shown in figure 4. The predicted critical strain ∼0.55% under equi-biaxial compression for onedimensional mode agrees reasonable well with uniaxial experiments (∼0.7%) [38] although the loading condition is different. It is found that only when the compressive strain is larger than the critical compressive strain, εc , there exist a non-zero magnitude even on a perfectly flat substrate surface. As the compressive strain increases, the predicted amplitude increases while the wavelength decreases. Similar conclusions can also be obtained for other buckling patterns and we will focus on their energies in the following. Figure 5 shows the cohesive energy, bending energy, membrane energy and total energy for each buckling pattern. It is noted that the bending energy becomes non-zero only if the compressive strain is larger than the critical strain, which is quite same for all the buckling modes. When the compressive strain is slightly larger than the critical strain, the graphene may buckle to any buckling pattern since the
difference of total energy for the buckling patterns is negligible. In this case, the geometric imperfection may play a key role in selecting the buckling mode. As the compressive strain increases to be much larger than the critical strain, the equilateral triangular modes has the highest total energy and the herringbone mode has the lowest total energy due to the significantly reduction in membrane energy, and is therefore the energetically favorable mode. These results are very similar to the system of a thin film perfected bonded to a compliant substrate [47–50] and could serve as guidelines for the strain engineering of graphene.
5. Conclusions We have established an energy method to investigate the onedimensional, square checkerboard, hexagonal, equilateral triangular and herringbone buckling patterns in a graphene/hBN heterostructure under equi-biaxial compression. Total energy consisting of cohesive energy, graphene membrane energy and graphene bending energy is obtained analytically 9
Nanotechnology 25 (2014) 445401
C Zhang et al
for each buckling pattern. It is found that the total energies are quite same for all buckling modes at a compression slightly larger than the critical strain while the herringbone mode has the lowest total energy at a compression much larger than the critical compression. As compared to other buckling patterns, the herringbone mode significantly reduces the membrane energy of graphene at the expense of slight increase of the graphene bending energy and cohesive energy.
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Acknowledgments JS acknowledges the supports from the National Natural Science Foundation of China (Grant Nos.11372272 and 11321202), the Fundamental Research Funds for the Central Universities (2014FZA4027), and the Thousand Young Talents Program of China. QDY acknowledges the funding support from the US ARO grant No. W911NF-13-1-0211.
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