- Email: [email protected]
h-BN with different interlayer distances using DFT
Superlattices and Microstructures 72 (2014) 230–237
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Structural and elastic properties of hybrid bilayer graphene/h-BN with different interlayer distances using DFT R. Ansari ⇑, S. Malakpour, S. Ajori Department of Mechanical Engineering, University of Guilan, P.O. Box 3756 Rasht, Iran
a r t i c l e
i n f o
Article history: Received 14 February 2014 Received in revised form 8 April 2014 Accepted 22 April 2014 Available online 1 May 2014 Keywords: Hybrid graphene/hexagonal boron nitride Elastic properties Density functional theory
a b s t r a c t Importance of synthesizing graphene-substrate hybrid structure to open a band gap in graphene and apply them in novel nanoelectronic devices is undeniable. Graphene/hexagonal boron-nitride (h-BN) hybrid bilayer is an important type of these structures. The synthesized h-BN/graphene is found to have interesting electrical properties which is very sensitive to the change of the interlayer distance. This has encourages researchers to tune the energy and band gap of such structures. A change in the interlayer distance can also alter the mechanical properties, considerably, due to the variation of interaction energies. The current study is aimed to characterize the mechanical properties variation with interlayer distance change for h-BN/graphene hybrid bilayer structure. To this end, density functional theory calculations are employed within the generalized gradient approximation (GGA) framework. The results demonstrate that there are different possible equilibrium interlayer distances between layers related to two types of layer configuration, i.e. AA and AB. It is found that increasing the interlayer distance causes reduction of Young’s modulus. Also, Young’s modulus of hybrid structure is approximately between those of graphene/graphene and h-BN/h-BN bilayer structures and also lower than pristine monolayer graphene and graphite. Unlike the pure bilayer structures, Poisson’s ratio of hybrid bilayer structure is found to be higher than those of pristine monolayer graphene and h-BN nanosheets. Ó 2014 Elsevier Ltd. All rights reserved.
⇑ Corresponding author. Tel./fax: +98 131 6690276. E-mail address: [email protected] (R. Ansari). http://dx.doi.org/10.1016/j.spmi.2014.04.017 0749-6036/Ó 2014 Elsevier Ltd. All rights reserved.
R. Ansari et al. / Superlattices and Microstructures 72 (2014) 230–237
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1. Introduction Since 2004 [1], two-dimensional nanomaterials have been the focus of researchers all over the world. Graphene, as a two-dimensional allotrope of carbon with sp2 structure, due to having unique mechanical, electrical and physical properties, is so promising for novel applications in nanodevices, especially next generation of nanoelectronics. Graphene is a zero-gap semiconductor owing to the similarity of two carbon sublattices, and hence its applications in modern electronic devices is restricted. In this regard, many efforts have been made to open a band gap in graphene based on different methods like molecular adsorption [2], defects [3], hydrogenation [4–6], application of external electric field [7] and developing graphenesubstrate hybrid structures [8,9]. Among these approaches, developing graphenesubstrate hybrid structures, are found to be more straightforward to implement. Moreover, if a weak interaction is formed between graphene and the substrate, the intrinsic mechanical and physical properties of graphene do not change remarkably which is considered as an important advantage [10–17]. The band gap of graphene is strongly affected by its substrate [10,14,15]. Hexagonal boron-nitride (h-BN) nanosheet is an inorganic counterpart of graphene in which each pair of carbon–carbon bond is replaced by boron-nitrogen one [18,19]. Due to this specific structure, using h-BN as a substrate can alter the band gap of graphene by introducing the inequivalency in the graphene lattice [20–24]. There are different theoretical and experimental studies [25–32] in which the electronic properties of hybrid graphene/BN systems have been investigated.
Fig. 1. schematic representation of AA and AB types of bilayer structures (a) graphene/h-BN, (b) h-BN/h-BN, and (c) graphene/ graphene.
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R. Ansari et al. / Superlattices and Microstructures 72 (2014) 230–237 Table 1 Equilibrium interlayer distance for AA and AB type of bilayer structures. Bilayer nanostructure
Equilibrium interlayer distances (Å)
Graphene/h-BN (AA) Graphene/h-BN (AB1) Graphene/h-BN (AB2) h-BN/h-BN (AA) h-BN/h-BN (AB) Graphene/graphene (AA) Graphene/graphene (AB)
3.767, 3.928, 4.065, 4.41, 4.425, 4.675 3.796, 3.858, 3.912, 4.08, 4.214, 4.44 3.876, 3.914, 4.044, 4.105, 4.36, 4.596, 5.038 3.79, 3.96, 4.014, 4.03, 4.14, 4.42 3.57, 3.66, 3.76, 3.83, 4.08, 4.22 3.956, 4.43, 5.4, 6.78 5.4
Fig. 2. variation of energy with equilibrium interlayer distance for (a) AA and (b) AB type of bilayer h-BN/h-BN, (c) AA, (d) AB1, (e) AB2 type of bilayer graphene/h-BN and (f) AA type of graphene/graphene.
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Fig. 3. variation of Young’ modulus with equilibrium interlayer distance for (a) AA and (b) AB type of bilayer h-BN/h-BN, (c) AA, (d) AB1, (e) AB2 type of bilayer graphene/h-BN and (f) AA type of graphene/graphene.
The fascinating electrical properties of h-BN/graphene are very sensitive to the interlayer distance which results in the energy gap of such structures to change up to 0.55 eV [39]. The variation of the interlayer distance also can alter the mechanical properties. To the best of author’s knowledge, the elastic properties of hybrid bilayer graphene/h-BN structures including Young’s modulus and Poisson’s ratio for different equilibrium interlayer distances have not been taken into consideration
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Table 2 the values of A and B constants of Eq. (1). Bilayer nanostructure
A (TPa/Å)
B (TPa)
Graphene/h-BN (AA) Graphene/h-BN (AB1) Graphene/h-BN (AB2) h-BN/h-BN (AA) h-BN/h-BN (AB) Graphene/graphene (AA)
0.1839 0.1803 0.1552 0.1732 0.1995 0.1271
1.5 1.483 1.38 1.405 1.506 1.341
Fig. 4. variation of Poisson’s ratio with equilibrium interlayer distance for (a) AA and (b) AB type of bilayer h-BN/h-BN, (c) AA, (d) AB1, (e) AB2 type of bilayer graphene/h-BN and (f) AA type of graphene/graphene.
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so far. Hence, the current study aims to determine the aforementioned elastic properties of hybrid bilayer graphene/h-BN structure with different equilibrium interlayer distances using density functional theory calculations. Also, the results generated herein are compared with those related to bilayer graphene/graphene and h-BN/h-BN structures.
2. Methodology and models In this study, density functional theory (DFT) calculations are employed to determine the elastic properties of nanostructures. DFT calculations are implemented through the Quantum-Espresso code [33] employing the exchange correlation of Perdew–Burke–Ernzerhof (PBE) within the GGA framework [34,35] to compute the energy of system during applying the corresponding loads. Since the results are not affected by changing the size of unit cell, the smallest hexagonal unit cell of structures is selected in the calculations. Also, Brillouin zone integration is taken with a Monkhorst–Pack [36] k-point mesh of 12 12 1 and the value of cut-off energy for plane wave expansion is considered to be 80 Ry. Three basic models of hybrid bilayer structures including graphene/h-BN, graphene/ graphene, and h-BN/h-BN are considered in AA and AB types as presented in Fig. 1. As revealed in this figure, in AA type, hexagonal part of two layers are located above each other completely. Besides, in AB types, atoms on hexagonal structure of two layers locate decussately in which in the case of hybrid bilayer graphene/h-BN, AB type is divided into two categories i.e., boron on carbon (AB1) and nitrogen on carbon (AB2). The uniaxial tension is applied to the unit cell of bilayer structures and energy of structures is computed in 2% to 2% strain in the harmonic elastic rage and finally utilizing the second derivation of energy with respect to applied strain, Young’s modulus is calculated. Furthermore, Poisson’s ratio is determined by using eaxial as the uniaxial strain (eaxial ¼ Daa ; a denotes the first lattice constant) and transverse strain (etrans ¼ Dbb ; b denotes the second lattice constant), as the ratio of the transverse strain to the axial strain.
3. Results and discussion In order to compute the equilibrium distance between layers of bilayer structures, different distances are chosen and the system is allowed to reach its equilibrium state first and then new equilibrium positions are determined to apply tension load and compute aforementioned properties. From the initial optimizations, it is observed that the number of equilibrium positions of two layers and corresponding distances are different. For example, h-BN/h-BN bilayer structure has 6 equilibrium distances in both AA and AB types which is generally different from ones with hybrid bilayer graphene/h-BN structures and graphene/graphene as outlined in Table 1. To demonstrate the variation of equilibrium energy with equilibrium interlayer distance, Fig. 2(a–f) is presented. According to this figure, except graphene/graphene bilayer structures, the equilibrium energies of h-BN/h-BN and graphene/h-BN structures reduce by increasing the distance between layers in a homographic-like trend whereas in AB type of hybrid bilayer graphene/h-BN, however, the equilibrium energy approximately remains constant as interlayer distance increases. Considering the graphene/graphene bilayer structure, AA type reveals totally different trend with a minimum value in which increasing the equilibrium distance results in enlarging the energy. DFT calculations also demonstrate that there is only one equilibrium distance for AB type of graphene/graphene bilayer structure equal to 5.4 Å. It should be noted that the change in the value of equilibrium energy is negligible which can be observed from Fig. 2. Comparing the stiffness of different bilayer structures at the minimum value of equilibrium interlayer distance, it is observed that the highest value of Young’s modulus belongs to AB type graphene/graphene bilayer structure with the value of 0.89 TPa which is less than the values of pristine monolayer graphene sheet and graphite [37,38] and minimum one is for AA type of this structure. It further is observed that Young’s modulus of h-BN/h-BN bilayer structures, is about 0.76 and 0.79 TPa for AA and AB types, respectively, which are near to Young’s modulus of pristine monolayer h-BN sheet [37]. Comparing these results with those of hybrid bilayer structures which are equal to 0.81 for AA and AB1 types and 0.79 for AB2 type, demonstrate that the values of Young’s modulus of hybrid
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bilayer structures are between the values of pure bilayer structures of the constituent layers of hybrid structures. In order to demonstrate the variation of Young’s modulus with equilibrium interlayer distance, Fig. 3(a–f) is given. In accordance with this figure, it is observed that Young’s modulus decreases by increasing the equilibrium bilayer distances in a linear trend as follows:
Y ¼ Ad þ B
ð1Þ
In this equation, Y is Young’s modulus in TPa, d denotes the equilibrium interlayer distance in Å, and A TPa and B (TPa) are constants. By fitting a straight line on the data sets of each type of presented Å bilayer models, A and B constants are determined and presented in Table 2. As observed from this table, the variation of Young’s modulus with equilibrium bilayer distance is more pronounced in the case of AB type of h-BN/h-BN bilayer structure. Moreover, it is shown that AA type of graphene/ graphene bilayer structure has the minimum slope. Besides, it is illustrated that the maximum reduction in Young’s modulus belongs to AA type of graphene/graphene bilayer structure with 45% of reduction and the minimum value is for h-BN/graphene hybrid bilayer structure with the value of 15%. Considering Poisson’s ratio, Fig. 4(a–f) is demonstrated. As observed, although a specific trend cannot be seen generally in variation of Poisson’s ratio with equilibrium interlayer distances, there is a particular value of interlayer distance in which Poisson’s ratio rises remarkably, unlike the AA type of pure graphene and h-BN bilayer structures in which noticeable drop are demonstrated. In addition, the results demonstrate that the value of Poisson’s ratio of bilayer pure graphene and h-BN structure are less than those of pristine monolayer graphene and h-BN sheets [37]. In contrast, the minimum value of Poisson’s ratio of hybrid structure of bilayer graphene is considerably higher than those of pristine monolayer graphene and h-BN nanosheets. 4. Conclusion In this investigation, the elastic properties of hybrid bilayer Graphene/h-BN structure were calculated employing density functional theory calculations within the GGA framework. The calculations revealed different possible equilibrium interlayer distances between layers of hybrid graphene/hBN and pure graphene/graphene and h-BN/h-BN bilayer structures for two types of AA and AB layer positioning. Young’s modulus was shown to be reduced by increasing the interlayer distance. Moreover, the value of Young’s modulus of hybrid structure was observed to be between those of graphene/graphene and h-BN/h-BN and also lower than those of pristine monolayer graphene and graphite. Additionally, in contrast to pure graphene and h-BN bilayer structures, the value of Poisson’s ratio of hybrid bilayer structure was observed to be higher than those of pristine monolayer graphene and h-BN. The electrical properties of such layered structures can be modified by changing the interlayer distance which led to considerable alteration of the mechanical properties to be characterized The results of this investigation seem to be useful for application of these bilayer structures in nanoelectromechanical systems (NEMS). References [1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A. Firsov, Electric field effect in atomically thin carbon films, Science 306 (2004) 666–669. [2] J. Berashevich, T. Chakraborty, Tunable bandgap and magnetic ordering by adsorption of molecules on graphene, Phys. Rev. B 80 (2009) 033404. [3] F. Banhart, J. Kotakoshi, A.V. Krasheninnikov, Structural defects in graphene, ACS Nano 5 (2011) 26–41. [4] D.C. Elias, R.R. Nair, T.M.G. Mohiuddin, S.V. Morozov, P. Blake, M.P. Halsall, A.C. Ferrari, D.W. Boukhvalov, M.I. Katsnelson, A.K. Geim, K.S. Novoselov, Control of graphene’s properties by reversible hydrogenation: evidence for graphane, Science 323 (2009) 610–613. [5] H.J. Xiang, E.J. Kan, S.H. Wei, X.G. Gong, M.H. Whangbo, Thermodynamically stable single-side hydrogenated graphene, Phys. Rev. B 82 (2010) 165425. [6] H.J. Xiang, E.J. Kan, S.H. Wei, M.H. Whangbo, J.L. Yang, ‘‘Narrow’’ graphene nanoribbons made easier by partial hydrogenation, NanoLetter 9 (2009) 4025–4030. [7] F. Xia, D.B. Farmer, Y.M. Lin, P. Avouris, Graphene field-effect transistors with high on/off current ratio and large transport band gap at room temperature, NanoLetter 10 (2010) 715–718.
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Jacobsen, Nonlocal screening of plasmons in graphene by semiconducting and metallic substrates: first-principles calculations, Phys. Rev. Lett. 106 (2011) 146803. [14] Y. Fan, M. Zhao, Z. Wang, X. Zhang, H. Zhang, Tunable electronic structures of graphene/boron nitride heterobilayers, Appl. Phys. Lett. 98 (2011) 083103. [15] L. Liu, Z. Shen, Bandgap engineering of graphene: a density functional theory study, Appl. Phys. Lett. 95 (2009) 252104. [16] C.R. Dean, A.F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K.L. Shepard, J. Hone, Boron nitride substrates for high-quality graphene electronics, Nat. Nanotechnol. 5 (2010) 722–726. [17] L. Liu, Y.P. Feng, Z.X. Shen, Structural and electronic properties of h-BN, Phys. Rev. B 68 (2003) 104102. [18] N. Alem, R. Erni, C. Kisielowski, M.D. Rossell, W. Gannett, A. Gannett, Atomically thin hexagonal boron nitride probed by ultrahigh-resolution transmission electron microscopy, Phys. Rev. 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Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Structural and elastic properties of hybrid bilayer graphene/h-BN with different interlayer distances using DFT R. Ansari ⇑, S. Malakpour, S. Ajori Department of Mechanical Engineering, University of Guilan, P.O. Box 3756 Rasht, Iran
a r t i c l e
i n f o
Article history: Received 14 February 2014 Received in revised form 8 April 2014 Accepted 22 April 2014 Available online 1 May 2014 Keywords: Hybrid graphene/hexagonal boron nitride Elastic properties Density functional theory
a b s t r a c t Importance of synthesizing graphene-substrate hybrid structure to open a band gap in graphene and apply them in novel nanoelectronic devices is undeniable. Graphene/hexagonal boron-nitride (h-BN) hybrid bilayer is an important type of these structures. The synthesized h-BN/graphene is found to have interesting electrical properties which is very sensitive to the change of the interlayer distance. This has encourages researchers to tune the energy and band gap of such structures. A change in the interlayer distance can also alter the mechanical properties, considerably, due to the variation of interaction energies. The current study is aimed to characterize the mechanical properties variation with interlayer distance change for h-BN/graphene hybrid bilayer structure. To this end, density functional theory calculations are employed within the generalized gradient approximation (GGA) framework. The results demonstrate that there are different possible equilibrium interlayer distances between layers related to two types of layer configuration, i.e. AA and AB. It is found that increasing the interlayer distance causes reduction of Young’s modulus. Also, Young’s modulus of hybrid structure is approximately between those of graphene/graphene and h-BN/h-BN bilayer structures and also lower than pristine monolayer graphene and graphite. Unlike the pure bilayer structures, Poisson’s ratio of hybrid bilayer structure is found to be higher than those of pristine monolayer graphene and h-BN nanosheets. Ó 2014 Elsevier Ltd. All rights reserved.
⇑ Corresponding author. Tel./fax: +98 131 6690276. E-mail address: [email protected] (R. Ansari). http://dx.doi.org/10.1016/j.spmi.2014.04.017 0749-6036/Ó 2014 Elsevier Ltd. All rights reserved.
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231
1. Introduction Since 2004 [1], two-dimensional nanomaterials have been the focus of researchers all over the world. Graphene, as a two-dimensional allotrope of carbon with sp2 structure, due to having unique mechanical, electrical and physical properties, is so promising for novel applications in nanodevices, especially next generation of nanoelectronics. Graphene is a zero-gap semiconductor owing to the similarity of two carbon sublattices, and hence its applications in modern electronic devices is restricted. In this regard, many efforts have been made to open a band gap in graphene based on different methods like molecular adsorption [2], defects [3], hydrogenation [4–6], application of external electric field [7] and developing graphenesubstrate hybrid structures [8,9]. Among these approaches, developing graphenesubstrate hybrid structures, are found to be more straightforward to implement. Moreover, if a weak interaction is formed between graphene and the substrate, the intrinsic mechanical and physical properties of graphene do not change remarkably which is considered as an important advantage [10–17]. The band gap of graphene is strongly affected by its substrate [10,14,15]. Hexagonal boron-nitride (h-BN) nanosheet is an inorganic counterpart of graphene in which each pair of carbon–carbon bond is replaced by boron-nitrogen one [18,19]. Due to this specific structure, using h-BN as a substrate can alter the band gap of graphene by introducing the inequivalency in the graphene lattice [20–24]. There are different theoretical and experimental studies [25–32] in which the electronic properties of hybrid graphene/BN systems have been investigated.
Fig. 1. schematic representation of AA and AB types of bilayer structures (a) graphene/h-BN, (b) h-BN/h-BN, and (c) graphene/ graphene.
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R. Ansari et al. / Superlattices and Microstructures 72 (2014) 230–237 Table 1 Equilibrium interlayer distance for AA and AB type of bilayer structures. Bilayer nanostructure
Equilibrium interlayer distances (Å)
Graphene/h-BN (AA) Graphene/h-BN (AB1) Graphene/h-BN (AB2) h-BN/h-BN (AA) h-BN/h-BN (AB) Graphene/graphene (AA) Graphene/graphene (AB)
3.767, 3.928, 4.065, 4.41, 4.425, 4.675 3.796, 3.858, 3.912, 4.08, 4.214, 4.44 3.876, 3.914, 4.044, 4.105, 4.36, 4.596, 5.038 3.79, 3.96, 4.014, 4.03, 4.14, 4.42 3.57, 3.66, 3.76, 3.83, 4.08, 4.22 3.956, 4.43, 5.4, 6.78 5.4
Fig. 2. variation of energy with equilibrium interlayer distance for (a) AA and (b) AB type of bilayer h-BN/h-BN, (c) AA, (d) AB1, (e) AB2 type of bilayer graphene/h-BN and (f) AA type of graphene/graphene.
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Fig. 3. variation of Young’ modulus with equilibrium interlayer distance for (a) AA and (b) AB type of bilayer h-BN/h-BN, (c) AA, (d) AB1, (e) AB2 type of bilayer graphene/h-BN and (f) AA type of graphene/graphene.
The fascinating electrical properties of h-BN/graphene are very sensitive to the interlayer distance which results in the energy gap of such structures to change up to 0.55 eV [39]. The variation of the interlayer distance also can alter the mechanical properties. To the best of author’s knowledge, the elastic properties of hybrid bilayer graphene/h-BN structures including Young’s modulus and Poisson’s ratio for different equilibrium interlayer distances have not been taken into consideration
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Table 2 the values of A and B constants of Eq. (1). Bilayer nanostructure
A (TPa/Å)
B (TPa)
Graphene/h-BN (AA) Graphene/h-BN (AB1) Graphene/h-BN (AB2) h-BN/h-BN (AA) h-BN/h-BN (AB) Graphene/graphene (AA)
0.1839 0.1803 0.1552 0.1732 0.1995 0.1271
1.5 1.483 1.38 1.405 1.506 1.341
Fig. 4. variation of Poisson’s ratio with equilibrium interlayer distance for (a) AA and (b) AB type of bilayer h-BN/h-BN, (c) AA, (d) AB1, (e) AB2 type of bilayer graphene/h-BN and (f) AA type of graphene/graphene.
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so far. Hence, the current study aims to determine the aforementioned elastic properties of hybrid bilayer graphene/h-BN structure with different equilibrium interlayer distances using density functional theory calculations. Also, the results generated herein are compared with those related to bilayer graphene/graphene and h-BN/h-BN structures.
2. Methodology and models In this study, density functional theory (DFT) calculations are employed to determine the elastic properties of nanostructures. DFT calculations are implemented through the Quantum-Espresso code [33] employing the exchange correlation of Perdew–Burke–Ernzerhof (PBE) within the GGA framework [34,35] to compute the energy of system during applying the corresponding loads. Since the results are not affected by changing the size of unit cell, the smallest hexagonal unit cell of structures is selected in the calculations. Also, Brillouin zone integration is taken with a Monkhorst–Pack [36] k-point mesh of 12 12 1 and the value of cut-off energy for plane wave expansion is considered to be 80 Ry. Three basic models of hybrid bilayer structures including graphene/h-BN, graphene/ graphene, and h-BN/h-BN are considered in AA and AB types as presented in Fig. 1. As revealed in this figure, in AA type, hexagonal part of two layers are located above each other completely. Besides, in AB types, atoms on hexagonal structure of two layers locate decussately in which in the case of hybrid bilayer graphene/h-BN, AB type is divided into two categories i.e., boron on carbon (AB1) and nitrogen on carbon (AB2). The uniaxial tension is applied to the unit cell of bilayer structures and energy of structures is computed in 2% to 2% strain in the harmonic elastic rage and finally utilizing the second derivation of energy with respect to applied strain, Young’s modulus is calculated. Furthermore, Poisson’s ratio is determined by using eaxial as the uniaxial strain (eaxial ¼ Daa ; a denotes the first lattice constant) and transverse strain (etrans ¼ Dbb ; b denotes the second lattice constant), as the ratio of the transverse strain to the axial strain.
3. Results and discussion In order to compute the equilibrium distance between layers of bilayer structures, different distances are chosen and the system is allowed to reach its equilibrium state first and then new equilibrium positions are determined to apply tension load and compute aforementioned properties. From the initial optimizations, it is observed that the number of equilibrium positions of two layers and corresponding distances are different. For example, h-BN/h-BN bilayer structure has 6 equilibrium distances in both AA and AB types which is generally different from ones with hybrid bilayer graphene/h-BN structures and graphene/graphene as outlined in Table 1. To demonstrate the variation of equilibrium energy with equilibrium interlayer distance, Fig. 2(a–f) is presented. According to this figure, except graphene/graphene bilayer structures, the equilibrium energies of h-BN/h-BN and graphene/h-BN structures reduce by increasing the distance between layers in a homographic-like trend whereas in AB type of hybrid bilayer graphene/h-BN, however, the equilibrium energy approximately remains constant as interlayer distance increases. Considering the graphene/graphene bilayer structure, AA type reveals totally different trend with a minimum value in which increasing the equilibrium distance results in enlarging the energy. DFT calculations also demonstrate that there is only one equilibrium distance for AB type of graphene/graphene bilayer structure equal to 5.4 Å. It should be noted that the change in the value of equilibrium energy is negligible which can be observed from Fig. 2. Comparing the stiffness of different bilayer structures at the minimum value of equilibrium interlayer distance, it is observed that the highest value of Young’s modulus belongs to AB type graphene/graphene bilayer structure with the value of 0.89 TPa which is less than the values of pristine monolayer graphene sheet and graphite [37,38] and minimum one is for AA type of this structure. It further is observed that Young’s modulus of h-BN/h-BN bilayer structures, is about 0.76 and 0.79 TPa for AA and AB types, respectively, which are near to Young’s modulus of pristine monolayer h-BN sheet [37]. Comparing these results with those of hybrid bilayer structures which are equal to 0.81 for AA and AB1 types and 0.79 for AB2 type, demonstrate that the values of Young’s modulus of hybrid
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bilayer structures are between the values of pure bilayer structures of the constituent layers of hybrid structures. In order to demonstrate the variation of Young’s modulus with equilibrium interlayer distance, Fig. 3(a–f) is given. In accordance with this figure, it is observed that Young’s modulus decreases by increasing the equilibrium bilayer distances in a linear trend as follows:
Y ¼ Ad þ B
ð1Þ
In this equation, Y is Young’s modulus in TPa, d denotes the equilibrium interlayer distance in Å, and A TPa and B (TPa) are constants. By fitting a straight line on the data sets of each type of presented Å bilayer models, A and B constants are determined and presented in Table 2. As observed from this table, the variation of Young’s modulus with equilibrium bilayer distance is more pronounced in the case of AB type of h-BN/h-BN bilayer structure. Moreover, it is shown that AA type of graphene/ graphene bilayer structure has the minimum slope. Besides, it is illustrated that the maximum reduction in Young’s modulus belongs to AA type of graphene/graphene bilayer structure with 45% of reduction and the minimum value is for h-BN/graphene hybrid bilayer structure with the value of 15%. Considering Poisson’s ratio, Fig. 4(a–f) is demonstrated. As observed, although a specific trend cannot be seen generally in variation of Poisson’s ratio with equilibrium interlayer distances, there is a particular value of interlayer distance in which Poisson’s ratio rises remarkably, unlike the AA type of pure graphene and h-BN bilayer structures in which noticeable drop are demonstrated. In addition, the results demonstrate that the value of Poisson’s ratio of bilayer pure graphene and h-BN structure are less than those of pristine monolayer graphene and h-BN sheets [37]. In contrast, the minimum value of Poisson’s ratio of hybrid structure of bilayer graphene is considerably higher than those of pristine monolayer graphene and h-BN nanosheets. 4. Conclusion In this investigation, the elastic properties of hybrid bilayer Graphene/h-BN structure were calculated employing density functional theory calculations within the GGA framework. The calculations revealed different possible equilibrium interlayer distances between layers of hybrid graphene/hBN and pure graphene/graphene and h-BN/h-BN bilayer structures for two types of AA and AB layer positioning. Young’s modulus was shown to be reduced by increasing the interlayer distance. Moreover, the value of Young’s modulus of hybrid structure was observed to be between those of graphene/graphene and h-BN/h-BN and also lower than those of pristine monolayer graphene and graphite. Additionally, in contrast to pure graphene and h-BN bilayer structures, the value of Poisson’s ratio of hybrid bilayer structure was observed to be higher than those of pristine monolayer graphene and h-BN. The electrical properties of such layered structures can be modified by changing the interlayer distance which led to considerable alteration of the mechanical properties to be characterized The results of this investigation seem to be useful for application of these bilayer structures in nanoelectromechanical systems (NEMS). References [1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A. Firsov, Electric field effect in atomically thin carbon films, Science 306 (2004) 666–669. [2] J. Berashevich, T. Chakraborty, Tunable bandgap and magnetic ordering by adsorption of molecules on graphene, Phys. Rev. B 80 (2009) 033404. [3] F. Banhart, J. Kotakoshi, A.V. Krasheninnikov, Structural defects in graphene, ACS Nano 5 (2011) 26–41. [4] D.C. Elias, R.R. Nair, T.M.G. Mohiuddin, S.V. Morozov, P. Blake, M.P. Halsall, A.C. Ferrari, D.W. Boukhvalov, M.I. Katsnelson, A.K. Geim, K.S. Novoselov, Control of graphene’s properties by reversible hydrogenation: evidence for graphane, Science 323 (2009) 610–613. [5] H.J. Xiang, E.J. Kan, S.H. Wei, X.G. Gong, M.H. Whangbo, Thermodynamically stable single-side hydrogenated graphene, Phys. Rev. B 82 (2010) 165425. [6] H.J. Xiang, E.J. Kan, S.H. Wei, M.H. Whangbo, J.L. Yang, ‘‘Narrow’’ graphene nanoribbons made easier by partial hydrogenation, NanoLetter 9 (2009) 4025–4030. [7] F. Xia, D.B. Farmer, Y.M. Lin, P. Avouris, Graphene field-effect transistors with high on/off current ratio and large transport band gap at room temperature, NanoLetter 10 (2010) 715–718.
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