Mechanical and Electronic Properties of Graphyne and Its Family

Article pubs.acs.org/JPCC

Mechanical and Electronic Properties of Graphyne and Its Family under Elastic Strain: Theoretical Predictions Qu Yue,† Shengli Chang,† Jun Kang,‡ Shiqiao Qin,† and Jingbo Li*,‡ †

College of Science, National University of Defense Technology, Changsha 410073, Hunan Province, China State Key Laboratory for Superlattice and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China

‡

ABSTRACT: Using the first-principles calculations, we investigate the mechanical and electronic properties of graphyne and its family under strain. It is found that the inplane stiffness decreases with increasing the number of acetylenic linkages, which can be characterized by a simple scaling law. The band gap of the graphyne family is found to be modified by applying strain through various approaches. While homogeneous tensile strain leads to an increase in the band gap, the homogeneous compressive strain as well as uniaxial tensile and compressive strains within the imposed range induce a reduction in it. Both graphyne and graphyne-3 under different tensile strains possess direct gaps at either M or S point of Brillouin zone, whereas the band gaps of graphdiyne and graphyne-4 are always direct and located at the Γ point, irrespective of strain types. Our study suggests a potential direction for fabrication of novel strain-tunable nanoelectronic and optoelectronic devices.

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gaps change with ribbon widths.20,21 The work of Kang et al.22 indicated that the band gaps of graphdiyne nanoribbons can also be modulated by transverse electric fields. Bu et al.23 demonstrated that the C hexagons or C chains in the graphdiyne structure can be replaced with corresponding BN hexagons or BN units and found the band gap of doped graphdiyne varying with different BN component concentration. Similarly, Zhou et al.24 studied the BCN analog of graphyne with BN hexagons joined by C chains and found its band gap can be tuned by introducing different lengths of C chains. However, on the level of practical application, it should be noted that the above approaches which involve the fabrication of nanoribbons and precise control of doping position in the nanoscale are still challenging and may encounter fundamental difficulties. As an alternative, strain engineering is considered as a simple and promising way to modify properties of materials. It has been demonstrated both experimentally and theoretically that the band gap and optical properties of graphene can be controlled and tuned by applying strain.25−28 In the case of the graphyne family, although previous works mainly focus on their mechanical properties under strain,29−36 few works report the strain effects on their electronic behaviors. Although the response of band gap of graphyne to the homogeneous biaxial strain has been recently calculated,29 other strains including

INTRODUCTION Two dimensional nanosheet materials are currently a focus of intense research due to their potential application in nextgeneration nanodevices.1 Among these, graphene has been the most extensively studied because of its high mobility2 and rich physics.3,4 However, its applications are hindered by the absence of band gap in pristine graphene. As an alternative material, graphyne, which has a native band gap, has attracted much attention.5−7 Its geometric structure can be simply regarded as replacing a portion of the carbon−carbon sp2 bonds in graphene by acetylenic linkages. Because the number of acetylenic linkages in graphyne can be variable, it brings in a family including graphyne, graphdiyne, graphyne-3, etc. To date, works on synthesis of the substructures of graphyne and its family have been conducted.8−10 Recently, Li et al.11,12 successfully fabricated the graphdiyne film on the surface of Cu foil via a cross-coupling reaction. Motivated by the advances in synthesis, theoretical investigations are also performed on their structural and electronic properties, which suggest that the graphyne family is promising material for applications in nanoelectronics13,14 and optoelectronics,15 as well as hydrogen storage16,17 and purification.18,19 For allowing flexibility in nanodevice design and application, a tunable band gap of material is mostly required. For graphyne and its family members, different strategies have been proposed to meet this demand. One potential approach is to pattern the sheet structure into nanoribbons. It is shown that all of the graphyne and graphdiyne nanoribbons with armchair and zigzag edges exhibit semiconducting behavior and their band © XXXX American Chemical Society

Received: March 1, 2013 Revised: June 19, 2013

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dx.doi.org/10.1021/jp4021189 | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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Figure 1. (a) Geometric structure of graphyne sheet where the primitive cell is denoted by yellow parallelogram. r⃗1 and r⃗2 symbolize the in-plane lattice vectors, which are defined as (−a/2,√3a/2,0) and (a/2,√3a/2,0), respectively. The green and blue arrows indicate the deformation directions of Z-strain and A-strain. (b) Schematic illustration of the units of the graphyne family which are constructed by acetylene linkages between hexagons. Brillouin zone with high-symmetry points labeled under (c) H-strain, (d) A-strain, and (e) Z-strain.

Numerical calculation is implemented by the Vienna ab initio simulation package (VASP).42,43 Figure 1a shows the atomic configuration of graphyne sheet, in which hexagonal carbon rings are joined together by single acetylenic linkages, forming the new carbon network containing both sp and sp2 carbon atoms. r1⃗ and r2⃗ represent the in-plane lattice vectors. Analogous to graphyne, graphdiyne contains two acetylenic linkages between the nearest neighboring hexagons, whereas graphyne-m corresponds to the structure with three or more acetylenic linkages. In our calculations, we consider graphyne, graphdiyne, graphyne-3, and graphyne-4 sheets as a representative set (Figure 1b). The strained graphyne and other sheets are obtained via applying different strains, incluing a homogeneous biaxial strain (H-strain) and two types of uniaxial strains (A-strain and Zstrain), which are realized by stretching or compressing the sheets in the x and/or y directions. For the H-strain case, same magnitude of strain is applied in both x and y directions and the hexagonal symmetry is hence preserved (Figure 1c). Its strain value is defined as εH = (a′ − a)/a × 100%, where a and a′ are the lattice constants of the systems containing n acetylenic linkages before and after deforming, respectively. The A-strain corresponds to structure deformation in the x direction (along armchair chain direction) and parallel to the acetylenic linkages, whereas Z-strain represents deformation in the y direction (along zigzag chain direction) and perpendicular to the acetylenic linkages, Consequently the hexagonal symmetry is disturbed in the two cases (Figures 1d,e). Value of uniaxial strain is defined as εA(Z) = (u′ − u)/u × 100%, where εA and εZ are strain values corresponding to A-strain and Z-strain, u and u′ are undeformed and deformed unit lengths in a specific direction, respectively.

uniaxial tensile and compressive strains along different directions have not been considered yet. Moreover, how the band gaps of other family members besides graphyne would behave under these strains have not been examined either. In order to address these issues, a systematic investigation of strain effects on the electronic properties of graphyne family would be highly desirable. Apart from that, the elastic constants of the graphyne family are still worth studying from DFT method. In this work, we present our first-principles study of the elastic and electronic properties of graphyne and its family under strain. We show that the in-plane stiffness decreases with the addition of acetylenic linkages. While the band gap of the graphyne family can be increased by the homogeneous tensile strain, homogeneous compressive strain as well as uniaxial tensile and compressive strains tend to reduce the gap.

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MODEL AND METHODOLOGY First-principles calculations are performed on the basis of density-functional theory (DFT) using projector-augmented wave (PAW) potentials.37 The exchange-correlation interactions are treated by the generalized gradient approximation (GGA) with Perdew−Burke−Ernzerhof (PBE) functional.38 Because the GGA-PBE generally underestimates the band gap, the Heyd−Scuseria−Ernzerhof (HSE06) hybrid functional39,40 is also adopted for band structure calculations in some specific cases. A plane-wave basis set with cutoff energy of 400 eV is employed. The Brillouin zone is sampled using a 7 × 7 × 1 Monkhorst-Pack grid,41 which is tested to give converged results for all of the properties. The sheet is treated within supercell using the periodic boundary condition, and vacuum layer of 12 Å is chosen to prevent the interaction between adjacent sheets. The sheet is fully relaxed by using the conjugate gradient method until the maximum Hellmann− Feynman forces acting on each atom is less than 0.02 eV/Å. B



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Table 1. Lattice Constant a, Fitting Parameters p1 and p2, InPlane Stiffness C, and Poisson’ Ratio ν of Graphyne Sheet and Its Familya

a

system

a (Å)

p1 (eV)

p2 (eV)

ν

C (N/m)

graphyne graphdiyne graphyne-3 graphyne-4 graphene

6.89 9.46 12.04 14.60 2.47

257.9 371.8 492.6 621.3 58.8

214.4 331.8 429.4 536.3 19.3

0.416 0.446 0.436 0.432 0.164

166.3 123.1 101.8 87.7 347.0

The results of graphene are also presented for reference.

Figure 4. (a) Wave functions for Mc, Mv, Sc, and Sv in unstrained graphyne. Red (blue) distribution represents positive (negative) value. The isosurface level is taken as 0.007 e/Å3. Orbital energies of Mc, Mv, Sc, and Sv under (b) H-strain, (c) A-strain, and (d) Z-strain. The vacuum level is taken as zero reference in the calculation of orbital energy. Figure 2. In-plane stiffness C as a function of number of acetylenic linkages n. Corresponding fit is drawn with dashed line. Δ in inset denotes the bond length of a single acetylenic linkage. C value of graphene is also plotted for comparison.

equilibrium area of the system, the in-plane stiffness is expressed by C = (1/S0)(∂2ES/∂ε2), where ES is the strain energy calculated by subtracting the total energy of equilibrium system from the total energy of strained system, ε is uniaxial strain. The Poisson’s ratio ν, which represents the ratio of the transverse strain to the axial strain is equal to ν = −εtrans/εaxial. These elastic parameters are determined by using the method based on strain energy calculation in the harmonic elastic deformation range.44 Specifically, the unit lengths of both lattice vectors r1⃗ and r2⃗ along armchair direction (zigzag direction) change with uniaxial A-strain εA (Z-strain εZ) ranging from −2% to +2% in an increment of 1%. At each (εA, εZ) point the system is fully reoptimized and the corresponding total energy is obtained. Afterward, the data is fitted by ES = p1εA2 + p1εZ2 + p2εAεZ. The in-plane stiffness and Poisson’s ratio then can be calculated by C = (2p1 − p22/2p1)/S0 and ν = p2/2p1.

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RESULTS AND DISCUSSION Elastic Properties. The structure optimization is performed first. The calculated lattice constants of graphyne, graphdiyne, graphyne-3 and graphyne-4 by GGA-PBE functional are 6.89, 9.46, 12.04, and 14.60 Å, respectively, which are consistent with previous results obtained from the local density approximation.6 For graphyne family, its elastic properties can be characterized by two independent parameters, namely, Young’s modulus Y and Poisson’s ratio ν. Since the ambiguous for thickness of the sheet system poses difficulty in the definition of equilibrium volume when calculating the Young’s modulus, one can use the in-plane stiffness C instead.44,45 Defining S0 as the

Figure 3. (a) Band structure of unstrained graphyne. Solid line and filled circle represent the GGA-PBE and HSE06 calculated results. Mv (Sv) and Mc (Sc) symbolize the highest valence state and lowest conduction state at M-point (S-point), respectively. Fermi level is set at zero. Variation of band gap versus strain value calculated from (b) GGA-PBE and (c) HSE06 functionals. Open triangle, square and circle symbols correspond to the direct band gap under H-strain, A-strain, and Z-strain, respectively. The edges of valence and conduction bands are also labeled in the form of VBMCBM. C



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Figure 5. (a) Band structure of unstrained graphdiyne calculated with GGA-PBE (solid line) and HSE06 (filled circle) functionals. Γv1 and Γv2 represent the highest (doubly degenerate) valence states, and Γc1 and Γc2 correspond to lowest (doubly degenerate) conduction state at Γ-point. Fermi level is set at zero. Variation of band gap versus strain value calculated with (b) GGA-PBE (c) HSE06 functionals. Open triangle, square, and circle symbols correspond to the direct band gap under H-strain, A-strain, and Z-strain, respectively.

Figure 6. (a) Wave functions for Γc1, Γv1, Γc2, and Γv2 in unstrained graphdiyne. Red (blue) distribution represents positive (negative) value. The isosurface level is taken as 0.007 e/Å3. Orbital energies of Γc1, Γv1, Γc2, and Γv2 under (b) H-strain, (c) A-strain, and (d) Z-strain. The vacuum level is taken as zero reference.

The fitting parameters p1 and p2 and calculated elastic constants C and ν are listed in Table 1. The in-plane stiffness C of graphyne, graphdiyne, graphyne-3, and graphyne-4 are 166, 123, 102, and 88 N/m, respectively. Also the Poisson’s ratios are 0.416, 0.446, 0.436, and 0.432. The variation of ν with graphyne family is seen to be small. By means of this method, the stiffness of graphene is found to be 347 N/m, agreeing well with the experimental value of 340 ± 50 N/m,46 and the calculated stiffness of the monolayer MoS2 is 123 N/m,47 close to both the earlier calculated value of 145.82 N/m48 and experiment value of 180 ± 60 N/m,49 which confirms the reliability of the method. The calculated C agrees well with the reported results of 16629 and 162 N/m30 for graphyne and 122 N/m31 for graphdiyne. On the basis of molecular dynamics (MD) approaches, Cranford et al.34 recently calculated the stiffness of graphyne family with ReaxFF potential and found them to be 170−224, 150−185, 117−153, and 119−145 N/m from graphyne to graphyne-4, respectively (the above values are converted from original data in units of GPa using the thickness of 3.20 Å assumed in ref 33). It can be seen that the MD results are somewhat larger than our DFT determined values. As

Figure 7. (a) Band structure of unstrained graphyne-3. Fermi level is set at zero. (b) Variation of band gap as a function of strain value. Open (filled) triangle, square and circle symbols correspond to the direct (indirect) band gap under H-strain, A-strain, and Z-strain, respectively. All of the results are calculated with GGA-PBE functional. (c and d) The same captions as panels a and b but for the case of graphyne-4.

graphyne is about one-half the stiffness of graphene, it is concluded that the former is mechanically much softer than the latter. Figure 2 gives the relationship between in-plane stiffness of graphyne family and the number of acetylenic linkages n that they contains (n ≥ 1). It is shown that the stiffness gradually degrades with n increasing from 1 to 4 (from graphyne to graphyne-4). Remarkably, the result can be fitted using a simple scaling law,34 Cn = C1(a1/an), by which the stiffness of system beyond graphyne-4 can be predicted. Herein, an and Cn specifically refer to the lattice constant and in-plane stiffness of the family member having n acetylenic linkages; that is, a1 D



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Table 2. Summary of Calculated Band Gaps (in eV) for Graphyne Sheet and Its Family under Different Strains H-strain

A-strain

Z-strain

functional

system

0%

−2%

+10%

−2%

+10%

−2%

+10%

GGA-PBE

graphyne graphdiyne graphyne-3 graphyne-4 graphyne graphdiyne

0.46 0.48 0.56 0.54 0.94 0.89

0.40 0.41 0.49 0.46 0.87 0.80

0.88 0.94 1.07 1.11 1.47 1.53

0.40 0.39 0.50 0.44 0.87 0.78

0.17 0.31 0.43 0.45 0.56 0.69

0.37 0.39 0.47 0.45 0.83 0.78

0.30 0.21 0.47 0.36 0.71 0.56

HSE06

10%. Both the VBM and CBM retain at M-point for −2% ≤ εA < 0 but shift to S-point for 0 ≤ εA ≤ 10%. Upon application of Z-strain, the band gap is reduced to 0.37 eV at εZ = −2% and 0.30 eV at εZ = 10%. In this case, graphyne has direct gap at Mpoint for 0 ≤ εZ ≤ 10% and at S-point for −2% ≤ εZ < 0. Through the calculations we see that unlike graphene, graphyne are sensitive to almost all types of strain and its band gap can be modified and controlled by various straining approaches. Especially, when tensile H-strain and tensile A-strain are applied, the gap of graphyne can be altered in the range from 0.17 to 0.88 eV with the direct gap feature. For comparison, the band gap of strained graphyne calculated by HSE06 is also presented in Figure 3c. The gaps under H-strain, A-strain, and Z-strain are 1.47, 0.56, and 0.71 eV at strain value of 10%, respectively, and are found to be 0.87, 0.87, and 0.83 eV at strain value of −2%. Although HSE06 seems to predict a rather large gap than GGA-PBE at a given strain, the variation trend of gap versus strain is almost the same for them. Thus, one can expect that there should be no obvious difference between the electronic properties calculated from the two functionals. The following results are obtained based on PBE functional if not specified. To reveal the underlying mechanism for the band gap variation in strained graphyne, we investigate the orbital energy fluctuations of lowest unoccupied molecular orbital (LUMO) and highest occupied molecular orbital (HOMO) at M- and Spoint, where CBM or VBM most appear under different strain loadings. In particular, the LUMO and HOMO at M-point (Spoint) are denoted as Mc (Sc) and Mv (Sv), respectively. Figure 4a shows the wave functions for the four states in strain-free graphyne. Mv (Sv) exhibits antibonding character between carbon hexagons and acetylenic linkages, whereas Mc (Sc) exhibits bonding character between carbon hexagons and acetylenic linkages. Then, from Figures 4b−d, it can be seen that the gap variation upon straining arises from the orbital energy shift of the four states. In Figure 4b, when tensile Hstrain is applied to graphyne, both orbital energies of Mc and Mv decrease with increasing strain value. Since the decreasing trend of the former curve is less steep than that of the latter curve, it results in a lager band gap relative to the equilibrium value. On the contrary, when increasing compressive H-strain, both orbital energies of Mc and Mv increase with the latter raising faster, consequently narrowing the band gap. This phenomena also can be simply understood from the characters of wave functions for Mv and Mc. As Mv has more nodes than Mc in either x or y direction, its orbital energy will be prone to change under straining.13 When graphyne is subjected to the Astrain, the situation becomes different, its band gap depends on the energy difference not only between Mv and Mc but also between Sv and Sc, due to the breaking of hexagonal symmetry (Figure 4c). Under tensile A-strain, while orbital energy of Sv decreases slower than that of Mv with strain increasing, orbital

and C1 refer to those of graphyne. Since an = a1+(n − 1)Δ, Δ being the bond length of single acetylenic linkage (Δ ≅ 2.57 Å in our calculations), Cn can then be given by Cn = C1/(1 + (n − 1)(Δ/a1)) = 166.3/(1 + 0.373(n − 1)). Electronic Properties of Graphyne under Strain. Figure 3a gives the band structure of strain-free graphyne. On the GGA-PBE level, it is found that graphyne is a semiconductor with a direct band gap of 0.46 eV at M-point (S-point) in the hexagonal Brillouin zone, agreeing well with reported values of 0.4717,24 and 0.52 eV.6 The states near band edges mostly arise from the overlap of carbon 2pz orbitals. Also, the effective masses of carries are found to be extraordinarily anisotropic, for example, effective masses of holes in valence band (mv*) are 0.083m0 along S-R (M-K) direction and 0.216m0 along S-Γ (MΓ) direction, whereas effective masses of electrons in conduction band (mc*) are 0.080m0 along S-R (M-K) direction and 0.202m0 along S-Γ (M-Γ) direction, m0 is the electron mass. Note that the effective masses are obtained using m* = ℏ2/(∂2E/∂k2) and the k points are very close to S point (less than 0.04 Å−1). The above results are in good agreement with earlier works.6,29 On the other hand, as GGA usually underestimates the band gap relative to the experimental value, the HSE06 hybrid functional is also employed to calculate the band structure of graphyne, which has been proven to be able to give more accurate gap value. Compared to GGA-PBE result, HSE06 reveals a larger gap of 0.94 eV. Nonetheless, it seems the band structures obtained from the two functionals share similar characteristics, as shown in Figure 3a. We now study the behavior of graphyne under strain by using GGA-PBE. Since the fracture strain values of graphyne, gradiyne, graphyne-3 and graphyne-4 are 11.2, 10.9, 10.9, and 10.8% for uniaxial strain along armchair chain direction and 17.7, 20.8, 22.3, and 22.4% for strain along zigzag chain direction,32 respectively, we constrain the tensile strain within the range from 0 to 10% in our calculations. To remain the planar structure of graphyne family, only the small compressive strain ranging from −2% to 0 is imposed, beyond which the systems start to distort with out-of-plane structure. For both tensile and compressive strains within the loaded range, the strain energy increases monotonously as strain increases, indicating that deformation of graphyne family is elastic. The strained systems can return to their original geometries once the strain is removed. The gap variation of graphyne as a function of strain is summarized in Figure 3b. In the case of Hstrain, the band gap increases with the increase of tensile strain but decreases under compressive strain. The gap value ranges from 0.40 to 0.88 eV for −2% ≤ εH ≤ 10%. Meanwhile, the valence band maximum (VBM) and conduction band minimum (CBM) remain at M-point and thus the direct gap feature is not disturbed. By contrast, in the presence of A-strain, the gap is reduced to 0.40 eV at εA = −2% and 0.17 eV at εA = E



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becomes higher than that of Γv1, orbital energy of Γc2 is slightly lower than that of Γc1, giving rise to smaller gap between Γv2 and Γc2. The above analysis can also be applied to the Z-strain case when gap of graphdiyne decreases with VBM and CBM at Γv2 and Γc1 under tensile Z-strain and at Γv1 and Γc1 under compressive Z-strain, respectively, as shown in Figure 6d. Electronic Properties of Graphyne-3 and Graphyne-4 under Strain. We next examine the strain effects on the electronic properties of graphyne-3 and graphyne-4 sheets. The band gaps of graphyne-3 and graphyne-4 without strain are 0.56 eV at M-point and 0.54 eV at Γ-points, respectively. Their band structures are presented in parts a and b of Figure 7. It is interesting to find that the band gaps of these sheets exhibit the following rule: the band gaps will occur at M-point for sheets having odd number of acetylenic linkages and at Γ-point for sheets having even number of acetylenic linkages. Therefore, one may expect that the gap variations of graphyne-3 and graphyne-4 under strain would be similar to those of graphyne and graphdiyne, respectively. The results in parts b and d of Figure 7 certainly verify this expectation. That being said, the band gap gradually increases with increasing tensile H-strain and decreases under compressive H-strain and other uniaxial tensile and compressive strains. Nevertheless, it seems the gaps of graphyne-3 and graphyne-4 become less sensitive to the uniaxial tensile strain, comparing to the graphyne and graphdiyne cases. Graphyne-3 also undergoes a direct-toindirect transition under compressive A-strain with VBM shifting to M-point and CBM locating at S-point (in the case of compressive Z-strain, VBM shifts to S-point and CBM locates at M-point). By contrast, H-strain still have strong ability to tune the gaps of graphyne-3 and graphyne-4. For instance, when the H-strain increases from −2% to +10%, the band gap of graphyne-3 increases from 0.49 to 1.07 eV with VBM and CBM remaining at M-point. Similarly, gap of graphyne-4 increases from 0.46 to 1.11 eV with VBM and CBM at Γ-point under the same condition. For the purpose of comparison, the gap values of the four studied systems under H-, A-, and Z-strain are summarized in Table 2. Shear strain or combined shear-uniaxial strain may be other effective ways to tune their band gaps, which, however, is beyond the scope of this paper.

energy of Sc decreases faster than that of Mc, which leads to smaller direct gap at S-point and larger direct gap at M-point. Instead, for the compressive A-strain case, Mv is pushed up and Mc is slightly pushed down, narrowing the direct gap at the Mpoint and enlarging the gap at the S-point simultaneously. The orbital energy variations of Sc, Mc, Mv, and Sv under Z-strain are generally similar to the corresponding ones of Mc, Sc, Sv, and Mv in the A-strain case, as shown in Figure 4d, therefore the same analysis can be conducted. Electronic Properties of Graphdiyne under Strain. On the basis of both GGA-PBE and HSE06 functionals, the straininduced modulation effect on the electronic properties of graphdiyne is considered. Band structure of unstrained graphdiyne is given in Figure 5a. GGA-PBE reveals a direct band gap of 0.48 eV at Γ-point, in good agreement with earlier reported values of 0.536 and 0.46 eV.13,14 The effective masses are isotropic, m*v and m*c are 0.095m0 and 0.091m0, respectively. Also, HSE06 gives a lager band gap of 0.89 eV, which is close to the reported value of 1.10 eV from the GW many-body theory.15 The relations between band gap and strain for PBE and HSE06 are summarized in Figure 5b,c, respectively. Because the gap variations obtained from the two functionals exhibit same trend, we only focus on the GGA-PBE results hereafter. Similar to graphyne, the gap of graphdiyne obeys a descending trend under compressive H-strain and decreases to 0.41 eV at εH = −2%, while it follows a ascending trend under tensile H-strain and increases to 0.94 eV at εH = 10%. In the case of A-strain, the gap monotonically decreases to 0.39 eV at εA = −2% and 0.31 eV at εA = 10%. The gap under Z-strain decreases to 0.39 eV at εZ = −2% and 0.21 eV at εZ = 10%. While tensile A-strain has a stronger ability than tensile Z-strain to modulate the gap of graphyne, the gap of graphdiyne is prone to decrease under tensile Z-strain instead. Different from graphyne, it is also interesting to find that the band gap of strained graphdiyne is always located at Γ-point regardless of the types of strain loading, which makes graphdiyne a potential candidate for fabricating strain-tunable optoelectronic device. Similarly, we plot the orbital energies of LUMO and HOMO at Γ-point versus strain to illustrate how the band gap of graphdiyne is modulated. Because the conduction band and the valence band of unstrained graphdiyne are quasi-degenerate along the Γ-K direction, both the LUMO and HOMO at Γpoint consist of doubly degenerate states, denoted as Γc1, Γc2, Γv1, and Γv2, respectively (also labeled in Figure 5a). The wave functions for the four states are presented in Figure 6a. It can be seen that Γc1 and Γc2 exhibit bonding character between carbon hexagons and diacetylenic linkages, whereas Γv1 and Γv2 display antibonding character between carbon hexagons and diacetylenic linkages. When graphdiyne is subjected to H-strain, the hexagonal symmetry remains and orbital energy of Γc1 (Γv1) is still identical to that of Γc2 (Γv2). Since orbital energy of Γv1 is more prone to change relative to that of Γc1, the band gap becomes larger under tensile tension and conversely smaller under compressive tension. In the case of uniaxially strained grapdiyne, the hexagonal symmetry is broken and both the LUMO and HOMO become nondegenerate, as a result, the orbital energy of Γc1 (Γv1) is no longer equal to that of Γc2 (Γv2). In Figure 6c, when tensile A-strain increases, orbital energies of Γv1 and Γv2 tend to decrease at different decreasing rates, orbital energy of Γv1 becomes higher over Γv2. As orbital energy of Γc2 becomes lower than that of Γc1 at the same time, the band gap is thereby reduced with VBM and CBM at Γv1 and Γc2. Under compressive A-strain, while orbital energy of Γv2

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CONCLUSIONS In summary, first-principles calculations are carried out to reveal the elastic constants and strain-tunable band gaps in graphyne and its family. The in-plane stiffness is found to degrade gradually with increasing the number of acetylenic linkages, namely, 166, 123, 102, and 88 N/m from graphyne to graphyne-4, whereas the Poisson’s ratio varies by a small amount among them. The relation between in-plane stiffness and number of acetylenic linkages can be characterized by a simple scaling law. Upon application of strain, the band gaps of graphyne and its family can be modified and controlled through different loading types. The gaps are found to steadily increase under homogeneous tensile strain but decrease when uniaxial tensile or compressive strains as well as homogeneous compressive strain are applied. Moreover, while the band gaps of graphyne and graphyne-3 are direct and located at either M or S point depending on the types of applied tensile strains, graphdiyne and graphyne-4 always maintain the direct gap at Γ point under these strains. The variations in their band structures are attributed to the shift of energy states near the Fermi level under strains. Our findings suggest graphyne and its F



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family as promising materials for application in strain-tunable nanoelectronics and optoelectronics.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +86-010-82304982. Fax: +86010-82304982. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS J.L. gratefully acknowledges financial support from the National Science Fund for Distinguished Young Scholar (Grant No. 60925016). This work is supported by the National Natural Science Foundation of China (NSFC; Grant No. 11104347 and No. 11104349) and Advanced Research Foundation of National University of Defense Technology (Grant No. JC02-19).

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Mechanical and Electronic Properties of Graphyne and Its Family

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