Mechanical and electronic properties of graphene

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Computational Materials Science 153 (2018) 64–72

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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

Mechanical and electronic properties of graphene nanomesh heterojunctions a

a

b

Ji Zhang , Weixiang Zhang , Tarek Ragab , Cemal Basaran a b

a,⁎

T

University at Buffalo, SUNY, Buffalo, NY 14260, United States Arkansas State University, State University, AR 72467, United States

A R T I C LE I N FO

A B S T R A C T

Keywords: Heterojunction Graphene Nano Ribbon Graphene Nanomesh GNR Mechanics Band structure Midgap States GNM

It is well known that introducing periodic holes into graphene can be used to obtain semiconducting graphene nanomeshes (GNM). Using Molecular Dynamics (MD) simulations as well as semi-empirical Extended Hückel (EH) method, the mechanical and electronic properties of GNM heterojunction are studied. In this study the mechanical and electronic properties of graphene nanoribbons with different hole sizes and different hole shapes (circular, square and equilateral triangle) were studied. Midgap states were observed near the Fermi level, which are also affected by the geometries of the holes. Dependence of the properties on the density (ρ) was also investigated for each geometry. It has been found that GNM is significantly brittle compared to pristine graphene nanoribbons. It is observed that the relationship between the hole shape and size and the band gap is different for armchair and zigzag chirality.

1. Introduction Graphene is single layer of carbon atoms that arranged in a hexagonal honeycomb lattice. It is a material that has a great potential to revolutionize electronics due to its excellent mechanical properties with high failure strength that is in the order of 100 GPa [1,2] and high flexibility [3,4]. Also it has extraordinary thermal properties (about (4.84 ± 0.44) × 103 to (5.30 ± 0.48) × 103 W/mK) [5–7] and extremely high electron mobility (about 200,000 cm2/Vs) [8]. Recent development of highly flexible electronics [9] and the ever growing requirement on the current density capacity of materials make graphene a good candidate to replace traditional metallic and semiconductor materials, especially in power electronic devices where very high electrical current is essential [10]. A graphene sheet intrinsically has no energy bandgap [11] thus introducing a band gap into this material is essential in semiconductor applications [12–15]. The methods of introducing an energy band gap include cutting graphene sheets into graphene nanoribbons (GNR) [16–19], application of either a tensile or shear strain on the lattice [17,20–25], processing graphene sheet into graphene nanomesh (GNM) by introducing periodic nano holes [26–33], hydrogenating graphene with a certain pattern [34,35], and growth of graphene on various substrates [3,4]. Using the above method realized the application of graphene as semiconductor, but they may alter the properties and behavior of graphene [36–40]. The purpose of this study is to verify the viability of the graphene nanomesh in high current density power electronics applications, thus it is important to understand the effect the ⁎

Corresponding author. E-mail address: cjb@buffalo.edu (C. Basaran).

https://doi.org/10.1016/j.commatsci.2018.06.026 Received 11 April 2018; Received in revised form 12 June 2018; Accepted 18 June 2018 0927-0256/ © 2018 Published by Elsevier B.V.

nano hole patterns on its energy dispersion relations. Graphene heterojunctions, where metallic and semiconducting regions are connected [41–44], is an essential component for nanoelectronics. The metallic regions can be obtained by fabricating GNR with specific widths such that there is no energy bandgap [19] or using an graphene sheet. The semiconducting regions can be obtained by introducing nano holes [27]. It has been proven that the band gap can be tuned by the hole dimensions and the geometry of the hole [26,29,45]. However, introducing these periodic holes may largely degrade the material strength and ductility. Under high electrical current density, there will be electron migration forces acting on the lattice which is generated by the electron scattering with phonons, impurity and edges [46,47]. Under high electrical current density, current crowding around these nano holes will lead to an increased electron wind forces [48] leading to disintegration of the graphene nanoribbons [49]. In order to better understand the influence of periodic nano holes on the current capacity of graphene, it is necessary to study the fracture strength of GNM under uniaxial tensile loading [12,49–54]. Previous studies [55,56] on mechanical behavior of GNM only focused on the response or mechanical properties of one unit lattice where the interaction of the metallic and semiconducting regions in the heterojunctions are ignored. Hu, Wyant [55] studied the mechanical behavior and fracture of graphene nanomesh with circular pores, in which the simulation was performed to hexagonal lattice. Comparison has been made between the results of Hu, Wyant [55] and this study and they are discusses further in the next section. Although some study has been done, the simulation directly

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performed to the graphene heterojunctions is still necessary. In graphene heterojunctions, the metallic regions and semiconducting regions act together thus only studying the mechanical behavior of a unit lattice is not sufficient when assessing the viability of GNM heterojunctions. In this study, the mechanical properties of GNM heterojunctions with circular, square, and equilateral triangular holes under uniaxial tension are studied using Molecular Dynamics (MD) simulations because of the good performance and low computational cost [1,2,12,49,54,57–64]. Moreover, for every hole geometry studied mechanically, the energy dispersion relation is studied to quantify the effect of the hole geometry and hole size on the induced band gap. In the literature previous studies focuse on the band structure of GNM. For example, Ouyang et al. [65] have studied the electronic structure and chemical modification of GNM using spin-polarized density functional theory (DFT). They found that the energy band gap of GNM is sensitive to the hole shape and size, in which GNM with hexagonal holes is typical semiconductor and GNM with triangular holes present semiconducting behavior with some localized states. Pedersen et al. [66] have studied the electronic properties of GNM using finite-element method, Tight Binding (TB) scheme as well as DFT. They show that using all three methods, band gap of a few hundred meVs is opened in GNM and the presence of carbon vacancies along the hole edges induce midgap bands. Yu et al. [67] studied the influence of edge imperfections on the transport behavior of GNM using DFT. They observed that edge imperfection of the GNM nano holes would induce localized edge state which contributes to the metallic conducting behavior of the GNM; by decorating the hole edge with Oxygen-containing group, the Fermi level will shift to the valence band and make the GNM p-doped. Chandratre and Sharma [32] showed that by introducing holes of the right geometry graphene can be tuned into piezoelectric material. Previous studies have shed light on the development of GNM based nano-devices. It is worth mentioning that all these researchers used hexagonal or rhombus shape supercell for GNM energy band gap calculations, which the same reciprocal lattice as graphene was used and give rise to A-B-A stacking of the nano holes. In this study, however, rectangle supercell was used, which lead to A-A-A stacking of the nano hole and an orthorhombic Brillouin zone. In addition, both armchair and zigzag supercell were also considered.

Fig. 1. Simulated GNM with (a) circular holes, (b) square holes, and (c) triangular holes.

2. Molecular dynamics simulations Graphene Nanoribbons (GNRs) can be metallic and semiconducting depending on the chirality and the width of the GNR. For instance, armchair GNR can be both metallic and semiconducting depending on the number of atoms along the width of the GNR, while zigzag GNR is always metallic [3]. In this paper, both armchair and zigzag GNR were studied. The semiconducting region in the middle was produced through the introduction of the holes (see Fig. 1). Three unit cells at each end of the nanomesh which is enclosed by the blue1 rectangles were used to apply a prescribed uniaxial tensile displacement as shown in Fig. 2. The prescribed displacement is applied at constant a speed of 0.25 Å/picosecond until complete fracture of the GNM. The width of the simulated graphene heterojunctions is 10.0 nm and the overall length is 25.0 nm. The metallic region is 4.5 nm long on each side and the semiconducting region is 16.0 nm. Periodic boundary conditions along the width direction were applied for the unit cell in the energy bandgap calculations. In the MD simulations periodic boundary condition along the width direction was also applied. As shown in Fig. 3, the result shows that using periodic boundary condition along the width direction of the GNM has very little effect on the stress-strain diagram compared to the simulations using shrink-wrapped boundary conditions. The dimensions of the circular, square and triangular holes

Fig. 2. GNM under uniaxial tension.

introduced in the semiconducting region are summarized in Table 1 as multiples of the graphene lattice constant a which is 2.46 Å. Strain rate and loading scheme may have an influence on the stressstrain behavior [68–71]. To consider this factor, two types of loading schemes are applied with the same average speed which is compatible with previously recommended values in the literature [72]. In the first one, GNM was first subjected to one displacement increment of 0.0125 Å and then fully relaxed for 50 time steps until thermodynamic equilibrium is reached before the next displacement increment. In the second loading scheme, GNM was subjected to a uniaxial tension with

1 For interpretation of color in Fig. 2, the reader is referred to the web version of this article.

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Fig. 4. Stress-strain diagram of armchair GNMs with circular nano holes with different hole sizes (Diameter in graphene lattice constant a).

Fig. 3. Stress-strain diagram comparison for Armchair GNM with circular holes of 8a in diameters using periodic and shrink-wrapped boundary conditions (a = 3 a0 ≈ 2.46 Å ). Table 1 Nano hole geometries and dimensions. Hole geometry

Circular

Square

Equilateral triangular

Hole dimension as multiples of the lattice constant a (a = 2.46 Å)

Diameter 2 4 6 8

Side length 2 4 6 8

Side length 3 4 5 6 7 8 9

constant speed of 0.25 Å/picosecond. Simulations were performed in the NVT ensemble at a constant temperature of 300 K using LAMMPS [73]. The time step was 0.5 fs which is less than 10% of the vibration period of a carbon atom [74]. Before the application of the initial loading, the system was left to relax for 20,000 time steps to reach a thermodynamically equilibrium state. Virial stress [75] is calculated for each atom in the GNM and the details of the MD simulation methodology can be found in the supplementary material of this paper.

Fig. 5. Stress-strain diagram of armchair GNMs with square nano holes with different hole sizes (Edge length in graphene lattice constant a).

3. Mechanical properties of the heterojunction The stress-strain diagrams of the pristine GNR and semiconducting GNMs with circular, square and equilateral triangular holes obtained from the MD simulations are plotted in Figs. 4–7, respectively. In each figure, stress-strain diagram is plotted for each case until complete fracture. The size of the circular holes is denoted by the diameter in graphene lattice constant a (a = 3 a0 ≈ 2.46Å ), where a is the bond length of graphene, and the size of square holes and triangular holes are denoted by the edge length. The ultimate strength and the corresponding failure strain are plotted against the density (ρ) in Figs. 7 and 8, respectively. The density is defined as the ratio between the area of the semiconducting region and the area of the same region in the pristine graphene ribbon. From the figures, it can be observed that increasing the hole size (decreasing density) will lead to a monotonic decrease in the ultimate failure stress and failure strain regardless of the geometry of the nano holes. According to the simulation results, each hole geometry has different sensitivity to strength and fracture strain as shown in Figs. 7 and

Fig. 6. Stress-strain diagram of armchair GNMs with equilateral triangular nano holes with different hole sizes (Edge length in graphene lattice constant a). 66

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Fig. 9. Comparison with the fracture strength of armchair GNM of Hu, Wyant [55].

Fig. 7. Fracture strength of armchair GNM vs. density (Density of 1 represents pristine geometry).

Fig. 10. Comparison with the fracture strain of armchair GNM of Hu, Wyant [55].

Fig. 8. Fracture strain of GNM vs. density (Density of 1 represents pristine geometry).

8, respectively. From Fig. 7, it is clear that under the same density, square shape holes will degrade the ultimate strength less than the circular holes and the triangular holes. In order to understand the reason for this different stress and strain sensitivity, the stress contour of the GNM of different geometries were calculated and plotted in Fig. 11. In Fig. 11, it is shown that the stress contour of GNM with square holes act as multiple parallel GNRs where the atomic stresses are quite uniformly distributed with no significant stress concentration. In comparison, the stress contours in GNMs with circular and triangular holes experienced larger stress concentration around the holes. GNMs with triangular holes have higher stress concentration than the GNMs with circular holes. Thus we can attribute the stress concentration as the main factor affecting the geometry sensitivity to decreasing density. Also, the results on circular holes has been compared with Hu, Wyant [55] regarding the fracture strength and strain. As shown in Fig. 9, the fracture strengths of GNMs versus the density in this study are pretty comparable with the results of Hu, Wyant [55] even though the stacking of nano pores are different. A-A-A stacking geometry is simulated in this study and in Hu, Wyant [55] hexagonal lattice is simulated hence A-B-A stacking type was investigated. As compared to the similar results in the fracture strength between these two study, the

Fig. 11. Stress contour of GNMs of different hole geometries under tensile load.

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Fig. 12. Stress-strain diagram of zigzag GNMs with circular nano holes of different sizes (Diameter in graphene lattice constant a).

Fig. 14. Stress-strain diagram of zigzag GNM with equilateral triangular nano holes of different sizes (Edge length in graphene lattice constant a).

fracture strains which are compared in Fig. 10 are comparable in the magnitude but has different changing trend with respect to the density. GNMs in Hu, Wyant [55] are more ductile when the density becomes small but this is not the case in the result of this study- the decreasing of fracture strain is monotonic when the density decreases. In order to compare the behavior of zigzag GNM versus the behavior of armchair GNM and quantify the difference in their behavior, the stress-strain diagrams for zigzag GNMs are calculated and plotted in Figs. 12, 13 and 14 for circular, square and triangular holes, respectively. As in armchair GNM, it is observed that the fracture strength decreases as the density decreases. By comparing the fracture strength and failure strain versus density in Figs. 15 and 16, respectively, it can be observed that GNMs with square holes are again the least sensitive to decreasing of density. In Figs. 17 and 18, the results of zigzag GNM with circular pores have been compared with Hu, Wyant [55]. Similar to the armchair GNM, the strengths of the zigzag GNM in this study are comparable to the result in Hu, Wyant [55]. When the density in two study are similar, the strength of the GNMs are also close, which means in the zigzag direction, the strength of GNM is not sensitive to the nano hole arrangement. The comparison of the fracture strain in two cases in Fig. 18

Fig. 15. Strengths of zigzag GNMs vs. density.

Fig. 16. Fracture strain of zigzag GNM vs. density. Fig. 13. Stress-strain diagram of zigzag GNMs with square nano holes of different sizes (Edge length in graphene lattice constant a). 68

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Fig. 17. Comparison of the fracture strength of zigzag GNM of Hu, Wyant [55].

Fig. 19. Fracture behaviors of (a) armchair GNM and (b) zigzag GNM.

expanded in a basis set of local atomic orbitals (LCAOs):

φnlm (→ r ) = Rnl (r ) Ylm ( r )̂

(1)

where Ylm is a spherical harmonic and Rnl is a superposition of Slater orbitals and written as following:

Rnl (r ) =

r n−1−l [C1 (2n1)2n + 1e−n1 r + C2 (2n2)2n + 1e−n2 r ] (2n)!

(2)

where n1, n2, C1 and C2 are parameters must be defined for the valence orbitals of each element. In this section, the three types of nano holes studied in the previous section were examined, namely circular, square, and equilateral triangular holes. Similar to the mechanical section of the paper, for each type of hole four diameter or edge-length sizes were examined, namely 2a, 4a, 6a, and 8a. In order to stabilize the GNM structure, the carbon atoms on the hole-edges were all hydrogen-passivated, which is a common practice in engineering graphene. Before calculating the band structure, each geometry was optimized till a residual force of 0.001 eV/Å was obtained. In order to ensure numerical accuracy of the EH calculation, a 10 × 10 × 1 MonkhorstPack (MP) k-point grid was used, and 100 Hartree was selected as the mesh density cut-off energy. In our calculations, spin-polarization was not considered as we mianly studied the variation of band gap with the size and the geometry of nano holes. Spin-polarization calculation would lift degeneracy for certain half-filled localized states [65], but it will not change the band gap of GNM. First we examine the pristine case of the supercells and their corresponding Brillouin zone lattice as shown in Fig. 20. It is noticed in Fig. 20(a) and (b) that instead of having a two-dimensional supercell structure, our supercells have a thickness of 0.335 nm in the out of plane thickness direction, which is a typical value of the thickness for monolayer graphene. These three-dimensional supercells have threedimensional Brillouin zone lattices as shown in Fig. 20(c), in which kB and kC are corresponding to the B and C direction in the supercell structure. However, since graphene is considered as a two-dimensional material, the wavevector k for calculating band structure should be confined in the in-plane directions, and therefore we set the Brillouin zone path for band structure calculation to be Γ-Y-A-Z-Γ which is

Fig. 18. Comparison of the fracture strain of zigzag GNM of Hu, Wyant [55].

shows the disparity of the fracture strain under two lattice geometries. The fracture strain in Hu, Wyant [55] has its minimum value when the density is near 0.89, but in this study the fracture strains generally has lower value when the density is lower. Aside from degrading the material, nano holes in graphene also decrease the fracture strain. As shown in Fig. 16, failure strain of the zigzag GNR decrease by around 50% or more due to the presence of the holes. As a result, GNM is more brittle compared to pristine GNR. In addition, Fig. 19 shows a sample of the fracture mechanism in armchair GNM versus the fracture mechanism zigzag GNM which was consistent in all simulated geometries and sizes. Fracture sections are perpendicular to the loading direction in armchair, while for zigzag GNMs the fractured sections are more serrated. This difference in fracture mechanism is quite similar to previous work in the literature [2]. 4. Band structure and electronic properties of the heterojunctions In order to calculate the energy band structure of GNM, the ATKVNL package was used [76–78]. In order to reach a balance between the computational accuracy and efficiency, the semi-empirical Extended Hückel (EH) method was utilized. Proven to be equally accurate compared to the first principle DFT method but much less time-consuming, the EH method is ideal to tackle graphite-type materials [79]. Within the EH method the electronic structure of the system is 69

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Fig. 20. Supercell of the pristine graphene nanoribbon, top and side views. a: armchair, b: zigzag, c: the corresponding Brillouin zone lattice.

Fig. 21. Typical geometries of graphene nanomesh supercells with holes with size of 4a and the corresponding band structure of (a: Armchair circular, b: Zigzag circular, c: Armchair square, d: Zigzag square, e: Armchair triangular, and f: Zigzag triangular).

corresponding to the coordinates of (0,0,0), (0,0.5,0), (0,0.5,0.5), (0,0,0.5) and (0,0,0) in the k space, respectively. By analyzing the band structure of the GNM as shown in Fig. 21, it is observed that a bandgap of a few hundred meVs is opened in an armchair supercell with circular and triangular nano holes, making the structure semi-conducting. In the square holes case, however, there is no band gap. The conduction band (CB) and valence band (VB) intersect in between Γ and Y and the structure is semi-metallic. In a zigzag supercell, a small band gap of a few meVs is opened at the Γ point for the circular hole case. For the square hole case, the CB and VB are crossed at a k point between Z and Γ, making the structure semi-metallic again. In the triangular hole case a band gap of a few hundred meVs is opened, resembling the triangular case in armchair supercell. It is also observed in Fig. 21(a), (b), (e), and (f) that one or more midgap states exist near the Fermi level, which is due to the unequal number of the A-type and B-type atoms in the supercell. The number of the midgap states are simply |NA-NB| (NA and NB are the number of A atoms and B atoms) [65] and is counted for all geometries as shown in Table 2. The existence of these midgap states is believed to have

Table 2 Number of midgap states vs. nano hole and supercell geometry.

Circular Square Triangular a b

Armchair supercell

Zigzag supercell

1 0 (n − 1)a

1 1b (n − 1)a

n is the number of graphene lattice constant a along the nano hole edge. Except for 4a case which has 0 midgap state.

contribution to the metallic transport behavior in low bias field. [67] In order to further examine the effect of the nano hole size on the energy band structure of the GNM, the relationship between band gap and the density is plotted in Fig. 22. It is found that in both armchair and zigzag supercells, as the triangular hole size increase (materials density decrease) the band gap of the GNM increase from about 0.6 eV to 1.3 eV. This is smaller than the result in previous work [65] in which they use hexagonal supercell and obtain band gap ranges from 1.5 eV to 2.3 eV as the edge length increase. We believe the smaller value of the

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Fig. 22. Band Gap of GNMs vs. Density. (a) armchair supercell, (b) zigzag supercell.

behavior in low bias.

band gap is due to the special A-A-A patterning of the nano holes. The “A” here denotes certain lay-out fashion of one row of nano holes, and A-A-A patterning simply means the patterning fashion of each row of nano holes is same. This is different from the commonly used A-B-A patterning with hexagonal unit cell where there is a misalignment between row A and row B. In the square hole cases, there is no band gap in armchair GNM supercell for all square hole sizes. However, finite band gap is opened in zigzag supercell and the only exception is the 4a case in which the CB and the VB intersect at a k point between Z and Γ. The intersection of the CB and VB is due to the lack of midgap states in the GNM with hole size of 4a. While in the other cases, the existence of the midgap states prevent the state near the Fermi level to be occupied by the CB and VB thus open a finite band gap. For zigzag GNM supercell with circular holes, the band gap increases as the nano hole diameter increases. However, in armchair case, the band gap decreases as the nano hole diameter increases. It is noticed that when the nano hole diameter increases to 8a, there is no band gap. Such abrupt change of the band gap is never discussed in the literature and the underlying mechanism is yet to be explored.

Acknowledgements This work is sponsored by the U.S. Navy, Office of Naval Research, Advanced Electrical Power Systems Program under the direction of Capt. Lynn Peterson. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.commatsci.2018.06. 026. References [1] J. Zhang, T. Ragab, C. Basaran, Influence of vacancy defects on the damage mechanics of graphene nanoribbons, Int. J. Damage Mech. 26 (1) (2017) 28–48. [2] J. Zhang, T. Ragab, C. Basaran, The effects of vacancy defect on the fracture behaviors of zigzag graphene nanoribbons, Int. J. Damage Mech. 26 (4) (2017) 608–630. [3] S. Bae, et al., Roll-to-roll production of 30-inch graphene films for transparent electrodes, Nat. Nanotechnol. 5 (8) (2010) 574–578. [4] K.S. Kim, et al., Large-scale pattern growth of graphene films for stretchable transparent electrodes, Nature 457 (7230) (2009) 706–710. [5] A.A. Balandin, Thermal properties of graphene and nanostructured carbon materials, Nat. Mater. 10 (8) (2011) 569–581. [6] A.A. Balandin, et al., Superior thermal conductivity of single-layer graphene, Nano Lett. 8 (3) (2008) 902–907. [7] Y. Chu, C. Basaran, Review of Joule heating in graphene nano-ribbon, Thermal and Thermomechanical Phenomena in Electronic Systems (ITherm), 2012 13th IEEE Intersociety Conference on, IEEE, 2012. [8] K.I. Bolotin, et al., Ultrahigh electron mobility in suspended graphene, Solid State Commun. 146 (9–10) (2008) 351–355. [9] C. Dagdeviren, et al., Recent progress in flexible and stretchable piezoelectric devices for mechanical energy harvesting, sensing and actuation, Ext. Mech. Lett. 9 (2016) 269–281. [10] R. Murali, et al., Breakdown current density of graphene nanoribbons, Appl. Phys. Lett. 94 (24) (2009) 243114. [11] M. Fujita, et al., Peculiar localized state at zigzag graphite edge, J. Phys. Soc. Jpn. 65 (7) (1996) 1920–1923. [12] Y. Chu, et al., Strained phonon–phonon scattering in carbon nanotubes, Comput. Mater. Sci. 112 (2016) 87–91. [13] P. Gautreau, et al., Phonon dispersion and quantization tuning of strained carbon nanotubes for flexible electronics, J. Appl. Phys. 115 (24) (2014) 243702–243707. [14] Y. Chu, P. Gautreau, C. Basaran, Parity conservation in electron-phonon scattering in zigzag graphene nanoribbon, Appl. Phys. Lett. 105 (11) (2014) 113112. [15] Y. Chu, T. Ragab, C. Basaran, Temperature dependence of Joule heating in zigzag graphene nanoribbon, Carbon 89 (2015) 169–175. [16] M.Y. Han, et al., Energy band-gap engineering of graphene nanoribbons, Phys. Rev. Lett. 98 (20) (2007) 206805. [17] Y. Lu, J. Guo, Band gap of strained graphene nanoribbons, Nano Res. 3 (3) (2010)

5. Conclusions The ultimate strength and ductility of graphene nanomesh (GNM) is significantly decreased after introducing the periodic holes. GNMs with square holes tend to be less sensitive to decreasing density. The stress concentration in square holes is less than the circular and triangular holes-GNM. Failure strain is reduced by ∼50% or more after introducing holes. As a result, GNM is much more brittle compared to pristine GNR. Based on this study, more care is required in using GNM in the high current density electronics applications since the material ductility and strength are largely degraded. Moreover, the energy band structure of graphene nanomesh is sensitive to the shape and size of the nano holes. Triangular holes in both armchair and zigzag supercell open finite band gap which increase as the hole size increases. The situations for circular hole are different for armchair and zigzag cases in that the band gap increases as the hole size increases in armchair supercell and band gap decreases in zigzag supercell. For square holes, there is no band gap in armchair supercell, while in zigzag supercell a finite band gap is opened which increases as the hole size increases. Midgap states are found near the Fermi level for some geometries which have unequal number of A-type and B-type atoms. The existence of midgap states would prevent the intersecting of the CB and VB and is believed to contribute to the metallic transport 71

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