Grazynes: Carbon-Based Two-Dimensional Composites with Anisotropic Properties

A new family of two-dimensional carbon allotropes is presented, based on graphene stripes linked to each other by acetylenic connections. The large amount of allowed connectivities demands a family name for them: grazynes. The present study reports the energetic, structural, elastic, and electronic physicochemical properties of a set of simple grazynes by means of density-functional-theory-based calculations, suggesting also possible synthetic routes. The main results conclude that these are exotic yet stable materials, stiffer than graphene in the acetylenic direction, highly anisotropic, and with the presence of Dirac points in the reciprocal space along the graphene stripes direction resistant to strain, regardless of its direction. Thus, grazynes infer directionality in electron conductivity and resilience to the material stretching/compression, quite important, for instance, in the nanoelectronics applicability point of view.


INTRODUCTION
The last decade has witnessed the bloom of C-based two-dimensional (2D) materials, 1 mostly leaded by graphene, 2 and posteriorly followed by other, more exotic 2D carbon allotropes, e.g., graphynes, 3 graphane, 4 and graphone, to name a few. 5 The applied research on them has even opened the gate towards other few-layered non C-based materials. 6 Focusing on the 2D Carbon allotropes group, they have become richer in their types and diversity, and the forecasted structures experimentally synthesized and/or isolated, spurred by theoretical predictions of their appealing properties; see e.g.
the corroborated larger electrical conductivity of graphynes -2D periodic arrangements of C atoms displaying sp and sp 2 orbital hybridizations-compared to graphene, 7 as previously predicted by computational simulations. 3 From the nanoelectronics applicability point of view, it would be highly desirable to infer directionality in the electron conductivity of such 2D materials, a feature that can be externally inducible, but that can be intrinsically/structurally prompted as well, as does happen in anisotropic graphyne structures. 3 Directionality may also affect other properties, such as compressibility, and this correspondence is de facto both ways, as compression/expansion of 2D lattices may affect the network structure and, in turn, its electronic structure, 8 a critical point in many 2D C-based Covalent Organic Frameworks (COFs). 9,10 Therefore, 2D carbon allotrope materials with inherent anisotropic properties are technologically desirable so as to control and/or exploit the elastic/electronic properties at will on a given preferred direction. Here we present a new family of 2D carbon allotropes that accomplish this task, to which we refer to as grazynes in the following. These allotropes can be also inherently tagged as graphynes, given that they contain both sp 2 and sp C atoms, although, at variance with graphynes, their directional display is inherently determined by the atomic graphenic building blocks, regarded as graphene stripes, are concurrently aligned on a given direction, and transversally interlinked by acetylenic building blocks. The vast variety of graphene stripe widths and acetylenic lengths as well as the unit cell periodicity implies a rich palette of building blocks connectivity, allowing for such materials tailoring at will.  Figure   1 shows the three grazyne structures studied in the present work (i.e., [1], [1]{0}grazyne, [2], [1]{0}-grazyne and [3], [1]{0}-grazyne). As can be seen in Figure 1, and for clarity purposes, when the acetylenic bridges occupy all the possible linking positions, the {0} index can be omitted. Moreover, Figure 2 shows two grazyne structures, one with different alternating graphenic stripes (i.e., [1,3], [1]{0}-grazyne) and another where acetylenic linkers do not occupy all the possible linking points (i.e., [1,3], [2]{2}-grazyne).

COMPUTATIONAL DETAILS
Here we have analyzed, by means of first-principles calculations, the three simplest grazyne structures (i.e., [1], [1]-grazyne, [2],[1]-grazyne and [3],[1]-grazyne, see Figure   1) with different graphene stripe width. To this end Density Functional Theory (DFT) calculations have been performed, as done previously in the past for graphynes and COFs, 3,9,10 exploiting the systems periodicity by imposing periodic boundary conditions. Two ab initio codes have been employed, the Vienna Ab initio Simulation Package (VASP) 11 and the Fritz-Haber Institute Ab Initio Molecular Simulation (FHI-AIMS) Package. 12 explicitly treated, 11 whereas core electrons are described by the Projector Augmented Wave (PAW) method. 13 The kinetic energy cutoff for the plane-wave expansion of the valence electron density was set to 850, 1200, and 1050 eV for [1], [1]-, [2],[1]-, and [3], [1]-grazynes, respectively. These values are large enough to obtain energies converged below 1 meV per unit cell in all cases. Long-range dispersion corrections were included through the pairwise D2 force field as developed by Grimme, 14 even though previous studies highlight the minimum impact that other corrections, such as D3, can have on 2D C-based materials, with variations of cell parameters below 0.0001 Å, and changes in energy below 1 meV, 15,16 although one has to keep in mind that the use of one or another correction can sensibly influence the stacking of such sheets, or their interaction with other material surfaces. 16 A vacuum layer of 20 Å was added perpendicular to the materials layer direction to prevent any interaction between adjacent grazyne layers. All optimizations were carried out spin polarized, given that open-shell configurations are energetically reachable in carbon-based 2D Covalent Organic Radical Frameworks (CORFs), 9,17 although these spin-polarized optimizations revealed either non magnetic moments or negligible magnetic moments in the studied cases, and so, all posterior single-point calculations were performed non spin-polarized to reduce the computational cost.
The systems were fully relaxed in the grazyne plane with an electronic energy convergence criterion of 10 -5 eV per unit cell and using a conjugated gradient method for the atomic relaxations until the Hellmann-Feynman forces acting on atoms became smaller than 0.01 eV·Å -1 . Moreover, the cell vectors containing the 2D structure were allowed to relax to accommodate the material. After the simultaneous optimization of atomic positions and cell vectors, the slight modifications of the vacuum region were reset to 20 Å, and ensuring afterwards that both the electronic and atomic convergence criteria were met by carrying out a single-point calculation. Test calculations with a stronger optimization criterion (i.e., of 0.001 eV·Å -1 ) revealed negligible changes of only meV. The minimum character of the optimized structures was evaluated by vibrational frequency analyses of 0.03 Å, showing in all cases that the optimized structures correspond to true minima in the potential energy surface. Further than that, atomic out-of-plane displacement of carbon atoms and further relaxations were also tested, although the final structures back relaxed to the perfectly planar structures.
Finally, layer rumplings and other collective deformations were applied and posteriorly relaxed, including graphenic or acetylenic rumplings, or both in opposite directions, similar to the observed deformations of graphyne upon molecular adsorption upon. 18 In all cases the same back relaxation to the perfectly planar optimal structures was observed, confirming the stability of the structures.
During the electronic structure calculations, the reciprocal spaces were sampled using optimal G-centered Monkhorst respectively, and a 750 eV cutoff energy yielded converged cohesive energies but cell parameter variations of ±0.002 Å and spurious residual magnetic moments of 0.0342 , 0.0308, and -0.1231 µ B , respectively. The grazyne structures have been described within the Generalized Gradient Approximation (GGA) for the exchange-correlation functional, using for this purpose the functional developed by Perdew, Burke, and Ernzerhof (PBE), 20 a functional proven to be accurate enough in the description of carbon-based materials. 3,9 As far as strains along the grazyne sheet directions were concerned, the elastic properties were derived from energy calculations performed on the same unit cells depicted in Figure 1, where the a and b cell parameters correspond to those located on x-and y-directions, respectively.
In order to obtain the systems band structures, the geometrical structures were optimized again using the seamless parallel FHI-AIMS package, 12 where the electron density is described with a basis set of Numerical Atomic Orbitals (NAO). The PBE0 hybrid 21 functional has been used including an all electron description and taking into account scalar-relativistic effects at the Zero Order Regular Approximation (ZORA) level. 22 A Tier-1 basis set together with light grid options have been chosen, which is of quality similar or even higher than Dunning 23 GTO aug-cc-pVDZ. 24  reciprocal length slopes. 9 In the case of band structures calculations, a tighter electronic energy convergence criterion was applied (10 -7 eV). The grazynes energetic stabilities were analyzed by computing the cohesive energies once the in-plane cell parameters were optimized. These cohesive energies per C atom, "#$ , were computed according to Equation (1), where % is the total energy of the grazyne unit cell, & is the total energy of a carbon atom in its vacuum ground electronic state (i.e., in its triplet 3 P ground state), and n the number of carbon atoms per unit cell. According to this notation, the more positive the E coh , the higher the stability. (2).
These elastic constants can be deduced from DFT calculations through Equation (3), which relates the total energy of the system and the applied strain within the harmonic approximation, where is the energy of the system when the strain is applied, ; is the energy of the unstrained structure, and = and > are the applied strains along the parallel and perpendicular directions of the graphene stripes of grazynes, respectively. .
Following the procedure described in a previous work of some of us 15 , the inplane stiffness of the material along x and y directions are defined as Equation (4), where ; is the unstretched area of the cell. Similarly, the Poisson's ratio of the material along x and y directions are defined as Equation (5), respectively.
As mentioned above, these equations are only valid for linear elastic regimes up to the yield strength, which in the here explored grazynes corresponds to strains in the ±2% range, and by that, contemplating both stretched and compressed situations. At larger strains the grazynes behave in a highly anisotropic fashion, and for these cases a nonlinear elastic analysis has been done up to the Ultimate Tensile Strength (UTS) following the continuum perspective procedure as outlined by Wei and coworkers 25 and detailed in Section 3.2.
Finally, outstanding electronic properties such as an extremely high carrier mobility and charge transport arise from the presence of Dirac points in a 2D material electronic structure, as happens in graphene. Near any Dirac point the electronic energy is linear with respect to the vector displacements in the reciprocal space, so the 2D band structure in this region can be adjusted by Equation (6). In the case of grazynes, Dirac points have been located (see below), and in order to compare their conductivity capacity among them, and, also with respect graphene and graphynes, their Fermi velocities have been evaluated.  Table 1 and Table 2, respectively. The simplest grazyne structure, the [1], [1]-grazyne is equivalent to the previously studied undistorted squarographene structure. 27 The present relative stability with respect graphene per C atom, dE, is here computed to be 0.49 eV atom -1 , which is very close to the previously reported value of 0.54 eV atom -1 obtained through Density Functional based Tight Binding (DF-TB) simulations. Further than that, the closer energetic relative stability found here for grazynes, being in between graphene and graphynes, is also in accordance to previous findings. 27 As expected, here we found that, the higher the n value, the closer to graphene stability is the E coh . Notice how, ultimately, graphene is only 0.27 eV per C atom more stable than [3],[1]-grazyne.

Energetic and Structural Properties.
It is worth to stress out that other carbon allotropes like graphynes present markedly lower cohesive energies (e.g., 6.93, 7.01, and 7.21 eV atom -1 for a-, b-, and ggraphyne, respectively). These values reinforce the higher stability of [1], [1]-grazyne with respect to graphynes, and consequently, their feasibility and stability when structures. 28 Thus, their synthesis becomes only a problem of organic synthesis, although a feasible procedure could well start from iodine or bromine capped graphene stripes, 29 where there is a plethora of procedures to achieve them, either through bottom-up or top-down strategies. 30 Such halide-capped graphene stripes can be linked to acetylenic building blocks through Sonogashira cross-coupling reactions, 28 see Figure   5. Actually, such a procedure can be step-wise controlled by protective trimethylsilylacetylene (TMS) endings applied either on the graphene stripes or the acetylenic linkages, which offer diverse interesting routes for knitting them. 31 However, there are other possible synthetic routes. As an example, it is worth highlighting the similarities with the bottom-up synthesis of nanoporous graphene, as it resembles the here presented grazynes, but with sp 2 type of links in between the graphene stripes. 32 Briefly, there, a precursor reactant, namely the diphenyl-10,10'-dibromo-9,9'bianthracene (DP-DBBA), was used to create graphene ribbons (stripes) through Ullmann coupling and cyclohydrogenation. The resulting graphene stripes were actually transversally linked directly through dehydrogenative cross coupling. All in all, the above successful synthesis allows stating that, although the grazynes synthesis may appear challenging, seems at hand, nevertheless.
Leaving the possible synthetic routes apart, and focusing on the grazynes geometric structure, as far as the rectangular unit cell parameters (a,b)

Elastic Properties.
For the above commented structures, we analyzed in detail the elastic constants in the linear regime according to Equation (3), thus applying = (in x-direction) and > (in y-direction) strains in the range ±2% to the cells shown in Figure 4. To do so the lattice constants were changed with 0.5% steps along both directions (i.e., = = ∆ / ; and > = ∆ / ; , with ; and ; being the cell parameters corresponding to the minimum energy structure, Figure 4). Thus, a total of 81 DFT points were used to fit the previous = = , > expression (see Table 3). In isotropic 2D materials, the elastic constants along x and y directions are identical due to the symmetry of the lattice. Our calculations show anisotropy in the Poisson's ratio and the in-plane stiffness along x-(parallel direction) and y-(perpendicular direction). This is expected given the presence of acetylenic linkages that break the structural symmetry.
The obtained values of c 1 , c 2 , and c 3 , along with the minimum E 0 and the exposed surface, S 0 , are shown in Table 3. With the adjusted parameters, shown in Table 3, the Poisson's ratio and the inplane stiffness can be obtained, according to Equations (4) and (5), respectively, in a straightforward fashion, and these have been gained and gathered in Consequently, their nonlinear elastic constants were evaluated using the fourth order continuum description of the nonlinear elasticity theory, 25  with an upper-case summation subscript running from 1 to 6. Since the deformed state of grazynes has no bending contribution only in-plane stress and strain components can be considered (i.e., components with subscripts 1 and/or 2).
From VASP calculations, the true Cauchy stress ( ) is obtained at different strains. Then, the stress is converted to the second P-K stress Σ through the deformation tensor ( ) by means of, where is the deformation tensor determinant.
The nonlinear elastic constants were evaluated by performing a least-squares fitting of the stress-strain data to Equations (11)- (15). Different magnitudes of uni-and biaxial strains in the x-and y-directions were applied on the grazyne cells (see Figure 1 for Moreover, increasing the stripes width enables the material resist higher stresses in both directions. Table 5 lists the elastic coefficients for the analyzed grazyne structures.
Obviously, the second order elastic constants obtained here from stress-strain data are equivalent to those listed in Table 3 that were obtained in the harmonic analysis of strain energy although considering that @ = @ A @@ , A = @ A AA , and C = @A .    Figure 6 with Equations (11)- (15

Electronic Properties.
Regarding the electronic properties of the studied grazynes, these have been obtained following the k-space paths defined in Figure 3. The   The systems feature linear band dispersion in the proximity of the Dirac cones, as characteristic of graphene and graphynes. As commented, in two cases they are located across the X→S or Y→G paths, which are placed along the graphene stripe direction. Indeed, no Dirac cones are featured along the graphyne direction, with just a succinct approach of bands near the X point. Therefore, it seems, as the ballistic transport is achievable along the graphene stripes channels, but inhibited perpendicular to it, pointing for anisotropy in the charge carrier mobility, with fundamental implications in oriented nanoelectronic devices.
One may wonder whether strain affects the grazynes band structures. To this end, the electronic structure has been evaluated for [1], [1]-grazyne when compressed/stretched in the elastic region by ±2% along the graphenic (x) and acetylenic (y) directions, also considering the uniaxial and biaxial deformations. Figure   9 shows the results of the original band structure with no deformations, plus the eight possibilities of compressions/stretching.
From the inspection of Figure 9 several conclusions can be withdrawn: First, the overall electronic structures are marginally affected by the strain, with only some changes in the bands energetic dispersion. More importantly is that Dirac cones prevail with strain, regardless of the strain direction, the simultaneous strains along different axes, even opposite strain directions on different axes. This implies that such Dirac cones are resilient to such in-plane deformations. If any, only some minor changes are detected. The above-commented band at X approaches the E F when compressed along the graphenic direction, and separates when stretched, with marginal effect when the strain is applied in the acetylenic direction.
Given the above, the Fermi velocities in the proximity of the Dirac cones have been calculated using Equation (6) on the unstrained geometry, and employing five points located in the range of ±0.04 Å -1 in the fitting procedure. The Fermi velocity values, listed in Table 6, are slightly smaller than the Fermi velocities, calculated likewise, for graphene and graphynes. However, as the width of the graphene stripes increases (e.g., n = 1, 2, 3, …), the Fermi velocities become closer to that of graphene.
In any case, grazynes can be still regarded as materials with directional ballistic transport. Another aspect of interest is that graphenic stripes are actually isolated, as happens on H-capped graphene nanoribbons reported in the literature, which present edge effects that perturb their electronic structure, even more when non-capped. 41 In 29 this sense, the grazyne formation would be a way of displaying a semimetal character on very thin graphene stripes, without losing too much conductivity in the process.  3) Dirac cones, and their location in k-space (in parentheses, see Figure 3). Values for graphene and a-, and b-graphyne are shown for comparison, yet previously obtained at DFT PBE level.

CONCLUSIONS
Altogether, here we present a new family of 2D carbon allotropes based on graphene stripes knitted to each other by acetylenic linkages. Given the large amount of allowed connectivities we name them as grazynes. The results on the structural, elastic, and electronic properties of a set of simple grazynes by means of DFT calculations, including consistent calculations within GGA but also considering hybrid functionals, allow concluding that such grazynes are exotic yet stable materials, which could be experimentally achieved, being a priori more stable than already synthesized graphynes. In general terms, they are stiffer than graphene in the acetylenic direction, and present Dirac points in the reciprocal space along the graphene stripes, resilient to strain, and featuring Fermi velocities comparable to the latter.