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Graphyne and its family: Recent theoretical advances Jun Kang, Zhongming Wei, and Jingbo Li ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.8b03338 • Publication Date (Web): 17 Apr 2018 Downloaded from http://pubs.acs.org on April 17, 2018

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ACS Applied Materials & Interfaces

Graphyne and Its Family: Recent Theoretical Advances Jun Kang,†,‡ Zhongming Wei,† and Jingbo Li∗,† State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences & College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100083, China, and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States E-mail: [email protected]

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To whom correspondence should be addressed State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences & College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100083, China ‡ Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States †

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Abstract Graphyne and its family are new carbon allotropes in 2D form with both sp and sp2 hybridization. Recently the graphyne with different structures have attracted great attentions from both experimental and theoretical communities, especially since the first successful synthesis of graphdiyne, which is a typical member of the graphyne family. In this review, recent theoretical progresses in the research of the graphyne family are summarized. More specifically, we systematically introduce the structural, mechanical, band, electronic transport, and thermal properties of graphyne and its family, as well as their possible applications, such as gas separation, water desalination and purification, anode material for ion battery, H2 storage, and catalysis application. Several related theoretical methods are also reviewed. The co-existence of sp and sp2 hybridization and the unique atom arrangement of the graphyne family members bring many novel properties and make them promising materials for many potential applications.

Keywords: Graphyne, Graphdiyne, theoretical calculation, electronic properties, applications.

Introduction There are various hybridization states (sp, sp2 , sp3 ) of carbon, which allow diverse covalent bonding between carbon atoms and result in numerous carbon allotropes. 1 The two most stable natural carbon allotropes are graphite and diamond, which have sp2 and sp3 hybridization characters, respectively. During the several past decades, continuous efforts have been made to search for new carbon allotropes. Some of the breakthroughs include the synthesis of fullerenes, 2 carbon nanotubes, 3 and graphene. 4 In 1987, Baughman et al. theoretically proposed graphyne and the graphyne-family members (GFMs), 5 which are new forms of carbon with one-atom-thickness and consist of sp and sp2 carbon atoms. These structures can be constructed by either partially or completely replacing the C-C bonds in graphene with acetylenic groups -C≡C-. Since the prediction of the graphyne family, there 2


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are several pioneer works focusing on the electronic properties and the synthesis of building blocks for the GFMs. 6–10 In 2010, through a cross-coupling reaction on Cu surface, Li et al. realized the first synthesis of large area films of graphdiyne, which is a member of the graphyne family. 11 Since then, the GFMs have attracted great attentions because of their excellent structural, mechanical, electronic, and thermal properties. 12–14 New strategies for synthesizing graphdiyne structures have been explored. 15 Preparation of graphdiyne films with different thickness was realized on ZnO nanorod arrays. 16 Very recently, Eco-friendly method of synthesizing graphdiyne nanosheets at a interface between gas and liquid or between liquid and liquid was developed. 17 Graphdiyne has also been made into different morphologies, such as nanotubes, 18 nanowires, 19 and nanowalls. 20,21 Moreover, the applications of graphdiyne in ion batteries, 22,23 catalysis, 24,25 water purification, 26,27 and solar cells 28,29 have been experimentally demonstrated. The exciting progresses in experiment have stimulated many theoretical researches regarding the fundamental properties and possible applications of the GFMs. In this review, recent theoretical progresses in the research of GFMs are summarized and discussed. More specifically, we will introduce the structural, mechanical, band, electronic transport, and thermal properties of the GFMs, as well as their possible applications.

Computational approaches Before reviewing the studies on GFMs, we first briefly introduce several widely used theoretical approaches for the simulation of different properties of GFMs. Density Functional Theory. Density Functional Theory (DFT) reveals that the electron density of a many-electron system can solely determine the properties of this system, and the total energy is minimized by the correct ground state electron density. 30 In DFT, the many-body problem is transformed into a single-particle Kohn-Sham equation, by attributing all the contributions of many-body effects to the exchange-correlation energy term

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Exc . 31 DFT is exact in principle, but the actual form of Exc is unknown, hence approximate functionals for Exc are usually used. It is assumed by the local density approximation (LDA) that the Exc functional only depends on the value of local electronic density. 31,32 The generalized gradient approximation (GGA) takes both the electron density and its gradient into account. 33 LDA and GGA are employed extensively to investigate the structural, mechanical, electronic, and magnetic properties of materials. Usually they give acceptable results. However, one of the major problems of LDA and GGA is that both of them severely underestimate the band gap, resulted from the missing of derivative discontinuity of total energy at integer particle numbers. The hybrid functionals such as HSE 34 improve the total energy estimation by mixing non-local Hartree-Fock exchange with LDA or GGA energy. They usually give much better band gaps than LDA or GGA does, but the computational cost is significantly larger. GW approximation. Up to date, the most suitable approach to study electronic quasiparticle excitations is the many-body perturbation theory based on the one-body Green’s function. 35 In this approach, the energy dependent and non-local self-energy term Σxc includes all non-classical many-body effects. The GW method approximates the Σxc using its first-order expansion with respect to the dynamically screened Coulomb interaction W and the Green’s function G. 35,36 W and G are often calculated based on the eigen states of a reference single-particle Hamiltonian, and the quasi-particle energies are calculated as a first-order correction to the single-particle eigen energies. So far, the GW method has been successfully applied to the calculation of quasi-particle band structure properties for a wide class of materials. However, it also suffers from convergence issues and unfavorable scaling of the computational cost regarding to the system size. Semi-Empirical Tight-Binding Method. Tight-binding method is primarily used for band structure calculation of a material. It uses atomic orbitals as basis to expand the single-electron wave functions of the system. The Hamiltonian matrix elements between these atomic orbitals are treated as adjustable parameters 37 and fitted to the results of

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experiments or first-principles calculations, and the eigen values and eigen states are then calcualted by diagonalizing the Hamiltonian matrix. Despite its simplicity, tight-binding model can give good qualitative results with much low computational cost compared to DFT calculations. A major problem of the tight-binding method is that the fitted parameters are highly system-dependent, thus the transferability is poor. Non-Equilibrium Green’s Function Method. The Non-Equilibrium Green’s Function (NEGF) formalism is a popular method to calculate the electron or phonon transport properties of extended systems. 38,39 In this approach, the simulated system is constructed by three parts. Two semi-infinite leads serve as the electron or heat baths, and they are connected by a central conductor region. The transmission of electron or phonon is calculated based on the Green’s function for the center region, and the self-energy of the leads which describes the lead-center interaction. The main advantage of NEGF is that the quantum mechanical effects such as tunneling and diffraction are preserved, which allows a highly accurate description of nanoscale devices. However, for large devices, the NEGF method can be computationally expensive.

Structure of graphyne and its family The structure of graphyne can be viewed as a 2D network of hexagonal carbon rings (sp2 hybridized) connected by acetylenic linkages (sp hybridized), as presented in Fig. 1(a). To distinguish from other graphyne family members, this structure is often referred as γgraphyne in many literatures. However, here we follow the original notation in the Baughman paper 5 and mention it as graphyne. It has a hexagonal symmetry similar to that of graphene. The length of the acetylenic linkages can be different, leading to the graphyne-n structures, in which n indicates the number of -C≡C- bonds in the linkage (Fig. 1(a)). Graphdiyne (graphyne-2) is the first experimentally accessible member in the graphyne family. 11 Compared with graphene, the graphyne-n structures show weaker stability, because inserting

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Figure 1: (a) Illustration of the graphyne-n structures. Reprinted with permission from Ref. 40 Copyright 2013 American Chemical Society.(b) Structures of α-graphyne, β-graphyne, and 6,6,12-graphyne. Reprinted from Ref., 5 with the permission of AIP Publishing.

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the acetylenic linkages into the carbon network reduces the cohesive energy. 5 Among the graphyne-n structures, graphyne is predicted to be most stable, and the cohesive energy decreases as the length of acetylenic linkage increases. 5 The lattice constant an of graphyne-n has the expression of: an = agraphyne + (n − 1)∆aacetylene , where agraphyne is the lattice constant of graphyne, and ∆aacetylene is the length of the acetylenic linkage in the structure. 41 The optimized value of agraphyne was found to be in the range of 6.86-6.90 Å, 5,6,42,43 and ∆aacetylene has a value of ∼2.58 Å. 6,40 The coexistence of sp and sp2 hybridization in graphyne-n results in different C-C bonding types. In graphyne, there are three types of C-C bonds: the C(sp2 )-C(sp2 ) bond with a length of 1.43 Å, the C(sp2 )-C(sp) bond with a length of 1.40 Å, and the C(sp)≡C(sp) bond with a length of 1.23 Å. 6 In graphyne-n with n larger than 1, besides the above three types of bonds, there is another C(sp)-C(sp) bond that connects two C(sp)≡C(sp) bonds, and its length is 1.33 Å. 6 Besides the graphyne-n structures, Baughman et al. also proposed that sp and sp2 carbon atoms can form 2D networks with different arrangements, thus the graphyne family members can have many other structures. These structures are indicated as α,β,γ-graphyne. 5 Here α and β are the numbers of C atoms in the smallest and next smallest rings which are connected by acetylenic linkage, whereas γ is the number of C atoms in a third ring connected to the β through a acetylenic linkage. With different α, β, and γ, the structure can have different ratio of sp and sp2 carbons, as well as different symmetry. In Fig. 1(b) the structures of three α,β,γ-graphynes are given, namely 18,18,18-graphyne, 12,12,12-graphyne, and 6,6,12-grapyne. The symmetries of the former two structures are hexagonal, whereas 6,6,12graphyne possesses rectangular symmetry. The 18,18,18-graphyne and 12,12,12-graphyne are often referred as α-graphyne and β-graphyne, 44 and we adopt such notation in the following discussions.

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Mechanical properties For 2D materials as the GFMs, there are two important elastic parameters, namely the inplane stiffness C and Poisson’s ratio ν. C is calculated by (1/S0 )(∂ 2 E/∂2 ), where S0 , E, and stand for the equilibrium area, the total energy, and the applied uniaxial strain, respectively. 45 It describes the tendency of a 2D material to deform along an axis. ν is defined by −trans /axial , which is the ratio of the transverse strain to the applied axial strain. Literature reported C and ν for different GFMs are summarized in Tab. 1. The elastic parameters are almost isotropic for structures with hexagonal symmetry like graphyne, but are anisotropic for 6,6,12-graphyne which has rectangular symmetry. Compared with graphene, GFMs have much smaller in-plane stiffness. This can be explained by two arguments. 42,46 First, each C atom in graphene has a coordination number NC of 3, whereas the acetylenic linkages in GFMs reduce the NC . As a result, GFMs have fewer number of bonds than graphene does. Second, the in-plane atomic mass density of the GFMs are smaller than that of graphene. Moreover, a longer acetylenic linkage will reduce the NC and the atomic density, leading to smaller in-plane stiffness. Buehler et al. proposed a linear scaling laws for the stiffness of the graphyne-n structures as: 41 Cn = C1 (a1 /an ), where Cn and an are the in-plane stiffness and lattice constant for graphyne-n. Fonseca et al. showed that, in general, the in-plane stiffness of the GFMs grows proportionally to the power of the density ρ, namely C ∼ ρp . 47 The Poisson’s ratios for the GFMs, ranging from 0.39 to 0.87, are all larger than that of graphene. Note for 2D case, the upper bound of Poisson’s ratio can be 1, instead of the 0.5 for 3D case. 48 A Poisson’s ratio of 1 indicates that the area of the 2D sheet will not change under uniaxial strain. Interestingly, the Poisson’s ratio for α-graphyne can be as high as 0.87. Therefore, α-graphyne has a strong ability to conserve its area under uniaxial strain. Because of the -C≡C- linkages, the GFMs exhibit significantly reduced ultimate strength compared with graphene, as listed in Tab. 1. Nevertheless, the ultimate strength of the GFMs is much higher than that of common polymer-based membranes. Along the y (zigzag) direction, the ultimate strength of GFMs is higher than that along the x (armchair) direction. 8


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Table 1: Literature reported in-plane stiffness C, Poisson’s ratio ν, ultimate strength σ, and ultimate strain along the x and y directions in Fig. 1 for different GFMs. Values reported without the direction information are included in the x columns. Graphyne

Graphdiyne

α-graphyne

β-graphyne

6, 6, 12-graphyne Graphene

Cx (N/m) 166, 42 170.4, 49 166.3, 40 162.1, 46 150, 52 170.2, 50 164, 51 163.0 47 123.1, 40 150.2, 41 100, 52 118.6, 47 121.8 53 39.9, 50 24, 54 21.98, 55 22.48, 56 42.8 47 87.1, 50 83, 57 77, 54 73.07, 55 93.6 47 117.3, 50 121.1 47 347.0, 40 348, 58 333 50 341.09 55

Cy (N/m) 224.0, 49 169.2, 50 162.5, 51 159.6 47

νx 0.417, 42 0.416, 40 46 0.429, 0.42, 51 0.39 47

νy 0.42, 51 0.38 47

σx (N/m) 17.84, 46 16.68, 50 14.34, 49 14.44 52 10.71, 41 9.54 52

σy (N/m) 18.83, 46 21.16, 50 31.97, 49 20.47 52 13.54, 41 20.84 52

x (%) 20, 46 11.2, 50 8.19, 49 11.2 52 6.3, 41 10.9 52

y (%) 20, 46 14.8, 50 13.24, 49 17.7 52 8.0, 41 20.8 52

185.2, 41 117.5 47

0.446, 40 0.40, 47 0.453 53

0.40 47

40.2, 50 42.4 47

0.863, 54 0.87, 55 0.874, 56 0.72 47

0.72 47

10.88 50

12.18 50

15.6 50

17.8 50

87.4, 50 92.1 47

0.49, 57 0.647, 54 0.67, 55 0.52 47

0.51 47

12.75 50

15.50 50

13.0 50

16.2 50

149.1, 50 152.1 47

0.39 47 0.164, 40 0.169, 58 0.18 55

0.49 47

13.09 50 34.71, 50 30.15 59

20.64 50 41.94, 50 35.85 59

11.6 50 0.134, 50 0.13 59

14.7 50 0.191, 50 0.2 59

The ultimate strength is in proportional relationship with the percentage of the linkages in the structures, and a linear scaling law is predicted for graphyne-n. 41 By analyzing the fracture process, it is revealed that the the origin of the crack, which initiates the fracture of the whole sheet, first occurs in the acetylenic linkages due to the breaking of the weak single bonds. 50 The GFMs also show considerable ultimate strains in the order of 10%, therefore, they can maintain the structure under a large strain. The mechanical parameters indicate that the GFMs are mechanically robust, and can be ideal candidates for membrane materials superior to polymer-based membranes.

Band properties Band structures The band structures of the GFMs have been thoroughly investigated. The graphyne-n structures like graphyne and graphdiyne possess a nonzero band gap, which is absent in graphene. At the LDA or GGA level, the calculated band gap of the graphyne-n structures is around 0.5 eV, 6,42,60 and only shows weak dependence on n. 6,40 With methods that include self-interaction correction, such as HSE or GW, the calculated band gap increases to ∼1

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eV, 40,42,60,61 which is comparable to that of Si. One advantage of graphyne-n over Si is that its band gap is direct, thus allows higher light absorption efficiency, as well as higher radiative recombination rate. In Fig. 2(a), the band structure of graphyne is presented. The conduction band minimum (CBM) and valence band maximum (VBM) are located at the M point of hexagonal Brillouin zone. By analyzing the projected band structures for sp2 carbon (C1) and sp carbon (C2), one can divide the bands of graphyne into different parts as shown in Fig. 2(b): the deep-lying σ/σ ∗ bands, the wide pz π/π ∗ bands at the band edges, and the narrow px − py π/π ∗ bands located inside the pz π/π ∗ bands. 42 The origin of the px − py π/π ∗ bands is the sp-sp interaction, and these bands do not exist in graphene. Such a band structure leads to anisotropic optical property of graphyne. In the energy range of 0 to 8 eV, the optical adsorption can be strong for in-plane polarized light, but quite weak for light with out-of-plane polarization. 42 The band structure of graphdiyne is quite similar to that of graphyne, as shown in Fig. 2(c). However, the direct gap of graphdiyne is at the Γ point rather than the M point. More generally, for graphyne-n structures, if n is odd, the band gap is at M, whereas if n is even, the band gap locates at Γ. 6,40 More interestingly, first-principles calculations suggest that many other GFMs can have Dirac-cone-like feature, such as α-graphyne, β-graphyne, and 6,6,12-grapyne. 44,62 Their band structures are presented in Fig. 2(d)-(f). The Dirac cones of α-graphyne are located at the high-symmetry K and K’ points, whereas those of β-graphyne and 6,6,12-graphyne are at low-symmetry points. Moreover, the 6,6,12-graphyne is self-doped because the two Dirac points therein are located below and above the Fermi level, respectively. 44 These findings demonstrated that Dirac cone is not a unique property of graphene. Also, Dirac cone can occur at low-symmetry points, and does not require hexagonal lattice symmetry. To better understand the electronic structures, as well as the existence/absence of Dirac cones, in different GFMs, several works have proposed tight-binding models to describe the pz π/π ∗ bands. 62,64–66 The basic idea of these models is that the effect of acetylenic linkages -C≡C- is equivalent to a renormalized direct hopping term between vertex atoms

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Figure 2: (a) Band structure of graphyne calculated by GGA (lines) and HSE (circles). Reprinted with permission from Ref. 42 Copyright 2011 American Chemical Society.(b) Projected band structures for two C atoms in graphyne. One is sp2 hybridized (C1) and the other is sp hybridized (C2). Open circles indicate the character of the px and py states, and solid circles indicate the pz character. The distribution of different bonding types in the band structure is also shown. Reprinted with permission from Ref. 42 Copyright 2011 American Chemical Society. (c) Band structure of graphdiyne. Reprinted with permission from Ref. 63 Copyright 2011 American Chemical Society.(d)-(f) Band structure for α-graphyne (d), β-graphyne (e), and 6,6,12-graphyne (f). Reprinted with permission from Ref. 44 Copyright 2012 by the American Physical Society. (g) Effective direct hopping between the two vertex carbon atoms connected by an acetylenic linkage. Reprinted with permission from Ref. 62 Copyright 2012 by the American Physical Society.

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(Fig. 2(g)). As a result, a GFM can be considered as a hexagonal lattice with the same topology as graphene, and a unified tight-binding model can be applied to different GFMs. The magnitudes and combinations of the renormalized hopping terms determine whether Dirac cones exist in the system. Using such a model, new graphyne-based structures that possess Dirac cones are also predicted. One is the 14,14,14-graphyne which shows a rhombic symmetry, and it is equivalent to graphene deformed along the armchair direction. Another one is the 14,14,18-graphyne which has rectangular symmetry, and exhibits anisotropy linear band dispersion with slopes of 5.0 eV/Å along x and 4.1 eV/Å along y. 65 Besides the pz band model, a more complicated tight-binding model including the σ orbitals and spin-orbit coupling (SOC) was proposed, and it was found that the Dirac cone features can be affected by the SOC. 67 A nontrivial gap in both α- and β-graphyne can be opened by intrinsic SOC. The Rashba SOC leads to the splitting of each Dirac cone in α-graphyne into four, and can be used to open or close a trivial gap in β-graphyne.

Band modulation For flexibility in device design and application, it is highly desirable that the band structure of materials can be tuned as needed. Lots of efforts have been devoted to exploring effective methods for band modulation of the GFMs. The most investigated strategies include: structural engineering, strain tuning, doping, and applying electric field. Through structural engineering, it is possible to make the 2D sheets of GFMs into 1D nanoribbons (NRs) or nanotubes (NTs), or 0D cages. It is also possible to stack the GFMs on top of other 2D materials, forming the van der Waals heterostructures. Due to the quamtum confinement, edge, and interface effects, these structural modifications can greatly change the band structure of GFMs. Gao et al. 68 studied the band structures of graphyne and graphdiyne NRs with armchair and zigzag edges (Fig. 3(a)). All these NRs are semiconductors, and their band gaps decrease with increasing NR widths due to reduced quantum confinement, as seen in Fig. 3(b). For graphdiyne, the quantum confinement effect in the 12


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armchair ribbon is found to be weaker than that in the zigzag ribbon, possible because of the smaller band dispersion along the Γ-M path than that along the Γ-K path in 2D graphdiyne sheet. 69 In armchair α-graphyne NRs, it is found that the band gap shows oscillation with width, similar to the case of armchair graphene NR. 70 In addition to quantum confinement, edge states also bring new features in the NRs. Dispersionless subbands and antiparallel edge magnetic ordering are proposed for zigzag NRs of graphyne, α-graphyne, β-graphyne, and 6,6,12-graphyne. 70,71 The edge magnetism in these NRs allows spin transport, as will be discussed later. The electronic properties of GFM-based NTs were investigated by several works. 72–74 α-graphyne NTs were predicted to have similar band structures as the usual graphene NTs. 72,73 The band gaps of zigzag α-graphyne NTs show an oscillatory dependence on the diameter, whereas the armchair α-graphyne NTs have very small or even zero band gaps. Tight-binding calculations suggested that armchair β-graphyne NTs are metallic whereas zigzag ones can be either metallic or semiconducting, and the band gaps of graphyne NTs are independent of diameter. 72 However, DFT calculations predicted that all β-graphyne NTs exhibit quite small band gaps which do not depend on tube size, and the band gaps of graphyne NTs decrease as tube size increases. 73 The disagreement between tight-binding and DFT calculations may come from the fact that structure relaxation is included in the later but not in the former. Zero-dimensional graphyne cages, namely fullerenynes, were studied by Zhang et al. 75 The GGA calculated band gaps of different fullerenynes depend on the size and symmetry, and vary in the range of 0.50-1.44 eV. Formation of van der Waals heterostructures can also change the band structure of GFMs. For instance, Sun et al. showed that graphyne/WSe2 heterostructure exhibits an reduced indirect band gap compared with the two constitute monolayers, and spatial separation of electron-hole pairs is proposed in the heterostructure. 76 Strain modulation is an effective way to modify the band structure of nanomaterials, since reduced dimension makes them more easily to be deformed compared with bulk materials. It is predicted that with uniform biaxial tensile strain, the band gap of graphyne increases. 42

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When the applied strain varies from 0 to 0.10, the band gap increases from 0.46 eV (0.96 eV) to 0.9 eV (1.5 eV) according to GGA (HSE) calculation. The increasing is almost linear, with a slope of 4.15 eV (5.13 eV) from GGA (HSE). When strain increases from 0.10 to 0.15, the band gap increases from 0.9 eV (1.5 eV) to 1.3 eV (2.0 eV) according to GGA (HSE). The linear relationship is maintained, but the slope is twice larger, with a value of 8.10 eV (10.14 eV) from GGA (HSE). With compressive strain, as strain value increases, the band gap shows linear reduction. For compressive strain from 0 to 0.1, the band gap decreases from 0.46 eV (0.96 eV) to 0.16 eV (0.6 eV) according to GGA (HSE), and the slope is 2.97 eV (3.71 eV) for GGA (HSE). The origin of the gap variation is explained by the width change of pz π/π ∗ bands due to C-C distance increase or decrease. 42 In graphdiyne, similar trend is observed. 53 A more systematic investigation of the strain effect on graphyne-n structures is carried out in Ref. 40 . The findings therein are that although biaxial tensile strain increases the band gap, uniaxial tensile strain results in a reduced band gap, as presented in Fig. 3(c). For example, it is seen that, a 10 % uniaxial strain along the zigzag or armchair directions decreased the band gap of graphyne by 0.16 eV or 0.29 eV, respectively. Uniaxial tensile breaks the symmetry and introduces two inequivalent valleys in the band structures of graphyne-n. The band edge states in these two valleys have different characters, and response differently when the uniaxial tensile strain increases. In one of the valleys the band gap will increase, but in the other one the gap will decrease, thus the overall band gap decreases. The movement of Dirac points in α- and β-graphyne with rotating uniaxial and shear strains are studied in Ref. 77 . The movement is found to be circular in α-graphyne, but elliptical in β-graphyne with the center displaced from the origin. It is also reported that a tensile strain of 6.3% on 6,6,12-graphyne causes merging of Dirac cones, and a semi-metal to semiconductor transition occurs when the strain exceeds 6.3%. 78 Doping is another method to change the electronic properties of materials. The effects of B and N substitutional doping in graphyne, α-graphyne, and β-graphyne are investigated by several works. 79–83 In these structures, N-doping is proposed to be more favored than

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Figure 3: (a) Illustration of the armchair and zigzag graphyne NRs. Reprinted from Ref., 68 with the permission of AIP Publishing.(b) Band gap of graphyne and graphdiyne NRs with different width. Reprinted from Ref., 68 with the permission of AIP Publishing. (c) Band gap variation of graphyne and graphdiyne sheets under biaxial strain (H-strain) and uniaxial strain along the armchair (A-strain) and zigzag (Z-strain) directions. Reprinted with permission from Ref. 40 Copyright 2013 American Chemical Society. (d) Band gap of armchair and zigzag graphdiyne NRs under transverse electric field. 69 Copyright IOP Publishing. Reproduced with permission. All rights reserved.

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B-doping. 80,83 The preferred substitution site for the N-doping is the sp C site, while for B-doping the sp2 C site is preferred, except for the case of α-graphyne. 80,83 Depending on the dopant concentration, the doping may lead to semiconductor-to-metal transitions, to band gap opening, or to the formation of new Dirac points. 80,81 Moreover, when the dopant is on the C(sp) site, magnetism may appear in graphyne based structures. 80,81 Isoelectronic B/N co-doping of graphdiyne and α-graphyne is discussed in Refs. 79 and, 82 respectively. For graphdiyne, B/N co-doping does not change the direct gap feature. As the portion of BN component increases, the band gap shows gradual increase at the beginning and then it increases abruptly. For α-graphyne, B/N co-doping introduces a band gap, and the gap oscillates periodically as the distance between the doped B and N atoms increases. Using DFT+U approach, Sun et al. 84 calculated the electronic structures of single transition-metal atom absorbed graphyne and graphdiyne. According to their study, V-, Mn-, Co-adsorbed graphyne, and Co-doped graphdiyne have spin-polarized half-semiconductor character, with 100% spin-polarization at the HOMO state. When Cr/Mn/Fe/Co is adsorbed on graphdiyne, and Cr/Fe is adsorbed on graphyne, the system becomes metallic with non-zero net magnetism. Ni adsorption does not introduce magnetism, but reduces the band gap compared with that of un-doped graphyne and graphdiyne. Lastly, several studies have discussed the possibility of modulating the band structure of GFMs using electric field. Electric field induced giant Stark effect is predicted in graphdiyne NRs. 69 As a result, the band gap decreases with increasing field strength, which finally leads to a semiconductor-metal transition, as shown in Fig. 3(d). In addition, the rate of band gap decreasing shows linear dependency on the ribbon width. Another study showed that applying a transverse electric field can realize half-metallicity in zigzag α-graphyne NRs. 70 Lu et al. found that the band gaps of the bilayer and trilayer graphdiyne decrease as the vertically applied electric field increases, independent of the stacking configuration. 85 In bilayer α-graphyne with a doubled Dirac-cone feature, a vertically applied electric field can open up a band gap with a rate of 0.3 eÅ. 86

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Electronic transport properties Intrinsic carrier mobility Table 2: Calculated hole µh and electron mobility µe for different GFMs from Ref. 87 The x and y directions are indicated in Fig. 1. Reprinted with permission from Ref., 87 Copyright 2013 American Chemical Society direction α-graphyne x y β-graphyne x y 6,6,12-graphyne x y graphdiyne x y graphene x y

µh (104 cm2 V−1 s−1 ) µe 104 cm2 V−1 s−1 2.960 2.716 3.316 3.327 0.856 0.798 1.076 0.892 12.29 24.48 42.92 54.10 1.91 17.22 1.97 20.81 35.12 32.02 32.17 33.89

Carrier mobility is an important parameter for the application of materials in electronic devices. Fast response of carriers to an external field can be expected in high-mobility materials, which is feasible for high-speed field effect devices. Graphene has been shown to possess extremely high carrier mobility in the order of 105 cm2 V−1 s−1 . 87 Theoretical calculations suggest that the graphyne family could have excellent intrinsic carrier mobility comparable to that of graphene. The mobility of several GFMs have been studied by Shuai et al. using the Boltzmann transport equation with the relaxation time approximation and deformation potential theory. 87,88 In their method, it is assumed that the dominant mechanism of the scattering of carriers at low energy is the electron-acoustic phonon coupling, so the obtained mobility can be considered as a upper limit. In practical, for structures with delocalized electronic states, this approximation works quite well, and gives a mobility of 3×105 cm2 V−1 s−1 for graphene. 87 The results in Ref. 87 are summarized in Tab. 2. It is seen that all the structures exhibit considerable carrier mobility in the order of 104 -105 cm2 V−1 s−1 . The mobility of 6,6,12-graphyne shows obvious anisotropy, resulted from its

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rectangular symmetry. Interestingly, 6,6,12-graphyne has higher carrier mobilities than those of graphene along one direction, which are 5.41×105 cm2 V−1 s−1 and 4.29×105 cm2 V−1 s−1 for electrons and holes, respectively. The origin of the high mobility in 6,6,12-graphyne can be understood by its weak electron-phonon coupling and the inequivalence of the two Dirac cones in its Brillouin zone. 87 Another notable finding is that the electron mobility in graphdiyne is also comparable to graphene. The mobility in 1D graphdiyne NRs is discussed in Refs. 63 and. 89 Depending on the width, orientation and carrier type, the mobility in graphdiyne NRs ranges from 103 cm2 V−1 s−1 to 106 cm2 V−1 s−1 . For graphdiyne NRs with the same orientation, the mobility is higher if the ribbon width is larger. Orientation dependence is also observed. With similar width, the mobility for armchair NRs is higher than that for zigzag NRs. Moreover, the electron is found to have higher mobility than the hole mobility. Graphdiyne and its NRs are semiconductors, so it is much easier to achieve high on/off ratio in these structures than in graphene. Thus, graphdiyne and its NRs could have potential application in high-speed field effect transistor.

Quantum and spin transport As mentioned above, the GFMs can be either semiconductors or semi-metals with Dirac cones. In addition, it is possible to introduce magnetism via structure engineering or doping. Therefore, there could be many interesting quantum and spin transport properties in the GFMs, which can be utilized for novel nanoelectronic devices. NEGF is the most widely used method for studying the quantum transport in the GFMs. Yang et al. studied the tunneling transport behavior of a zigzag graphyne NR tuned by gate and drain voltages, and proposed a tunneling FET based on it. 90 Once the drain bias becomes larger than the band gap, the NR turns from the off-state to on-state, and the on/off ratio can reach 103 . Further modulation on the tunneling current can be achieved by gate bias, and a larger gate bias leads to a larger current. Zeng et al. discovered directional anisotropy in the quantum conductance of 6,6,12-graphyne sheet. 91 When the bias is smaller than 0.4 V, the tunneling 18


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current along the armchair direction is higher than that along the zigzag direction, whereas the trend reverses when the bias is larger than 0.4 V. This anisotropy can be understood by the different shapes of the two Dirac cones in 6,6,12-graphyne due to the rectangular symmetry. Besides, negative differential resistance in several types of graphyne-based NRs have also been reported. 91,92 Yue et al. demonstrated symmetry-dependent transport properties in zigzag α-graphyne NRs (ZαGYNRs), 93 as shown in Fig. 4(a) and 4(b). They found that the current in symmetric ZαGYNRs under finite bias voltages is very small, due to the forbidden transport between the π and π ∗ bands which have opposite parity. Asymmetric ZαGYNRs exhibit linear current-voltage relationships. Their π and π ∗ bands have no parity thus the transport is always allowed. Based on such symmetry matching of the electrodes band structures, the bipolar spin-filtering effect in the symmetric ZαGYNRs is demonstrated. Current with almost complete spin-polarization and large magnetoresistance can be generated and tuned by the direction of bias voltage and/or spin-polarization configuration of the electrodes. 93 Spin Seebeck effect is found in the ZαGYNRs, 94,95 which can result in thermal-driven net spin current. Zhai et al. also showed that B/N doping may induce a large magnetoresistance behavior in the asymmetric ZαGYNRs. 95 Besides ZαGYNRs, spin-filtering and half-metallic properties can be achieved in graphyne NRs or 6,6,12-graphyne NRs through transition-metal-doping 96–98 or control of edge structures. 99 Edge states in zigzag graphdiyne nanoribbons are expected to induce magnetic-controlled switching effect. 100 The rich quantum and spin transport properties allow the applications of the GFMs in many nanoelectronic devices such as FET, spintronic logic-device, dual spin filter diode, molecule signal converter, and spin caloritronics device.

Thermal properties Thermal conductivity of graphyne-n sheets with different n number has been examined by several studies, 101–105 with different theories including the Green-Kubo formula on top

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Figure 4: (a) The Γ-point wave functions of π and π ∗ bands in 4-ZαGNR, and the electrode band alignments and transmission spectrum for a 4-ZαGNR with a 0.3 V bias. (b) The same as (a) but for a 5-ZαGYNRs. Reprinted with permission from Ref. 93 Copyright 2012 by the American Physical Society. of equilibrium molecular dynamics, nonequilibrium molecular dynamics, NEGF, and Boltzmann transport equation. Despite the different methods used, the conclusions of these works are similar. First, compared with graphene, the heat transport in graphyne-n is much more difficult, as indicated by their low thermal conductivity. For example, the reported thermal conductivity for graphyne at room temperature is in the range of 18-82 W/m-K. 101–105 In contrast, the thermal conductivity of graphene is much higher (∼103 W/m-K). 103,104 Second, the thermal conductivity of graphyne-n decreases rapidly as the length of the acetylenic linkage increases. According to Ref., 104 at room temperature, the thermal conductivity decreases from 64 W/m-K in graphyne to 8 W/m-K in graphyne-10, as shown in Fig. 5(a). The origin of such intrinsically low thermal conductivity is studied by phonon normal mode analysis. 102,104 As seen in Fig. 5(b), the linkage atoms have much larger contribution to most of low-frequency phonon modes, which dominate the heat transport, than the atoms in the hexagonal ring. Thus, the atoms on the linkage and the ring show vibrational mismatch. As a result, the energy transfer between the linkage and the neighboring ring is not efficient, leading to the low thermal conductivity. Moreover, the vibrational mismatch is enhanced when the length of acetylenic linkage increases, as indicated by the decreased phonon vibra-

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tional density overlap between the linkage and ring. Thus the thermal conductivity is lower when the linkage is longer. Nevertheless, Chen. et al. also showed that in graphyne there are some low-frequency (close to 0 THz) coherent phonons which have similar contributions from the linkage and ring atoms. When temperature is extremely low (

Graphyne and its family: Recent theoretical advances - ACS Applied

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