Grapheayne: a class of low-energy carbon allotropes with diverse optoelectronic and topological properties

A series of carbon allotropes with novel optoelectronic and rich topological properties is predicted by systematic first-principles calculations. These fascinating carbon allotropes can be derived by inserting acetylenic linkages (-C$\equiv$C-) into graphite, hence they are termed as grapheaynes. Grapheaynes possess two different space groups, $P$2/$m$ or $C$2/$m$, and contain simultaneously the $sp$, $sp^2$, and $sp^3$ chemical bonds. They have formation energies lower than the already experimentally synthesized graphdiyne and other theoretically predicted carbon allotropes with acetylenic linkages. Particularly, when the width $n$ of grapheayne-$n$ exceeds 15, its cohesive energy is lower than that of diamond, and approaches that of graphite with increasing $n$. Remarkably, we find that some grapheaynes behave as semiconductors with direct narrow band gaps and own the highest absorption coefficients among all known semiconducting carbon allotropes, while some others are topological semimetals with nodal lines. Especially, some grapheaynes can be engineered with tunable direct band gaps in the range of 1.07-1.87 eV and have ideal properties for photovoltaic applications. Our work not only uncovers the unique atomic arrangement and prominent properties of the grapheayne family, but also offers a treasury that provides promising materials for catalyst, energy storage, molecular sieves, solar cell, and electronic devices.


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Depending on the width of the nanoribbon, this gives a family of new 3D carbon allotropes, which can be termed as grapheaynes. By first-principles calculations, we show that the grapheaynes enjoy excellent stability. They are energetically more stable than the experimentally synthesized graphdiyne and other proposed carbon allotropes with acetylenic linkages, and they are thermally stable up to 1000 K. Following the design philosophy, the geometric confinement of the nanoribbons and the sp-sp 2 -sp 3 hybrid bonding endow the grapheayne family with a variety of electronic band features. We find that if the width of the nanoribbon n satisfies = 3 + 2 ( is an integer), then the system is a topological nodal loop semimetal; otherwise, it is a semiconductor. Interestingly, several semiconducting members of the family (with n = 3, 7, and 10) possess direct band gaps in the range of 1.0-1.5 eV, which are comparable to that of silicon (1.1 eV) and are very close to the optimal band gap of 1.34 eV for solar cell absorber materials. This suggests a great potential of grapheaynes for optoelectronic applications, and our calculation confirms that they have strong optical absorption response superior than other carbon allotropes as well as most optoelectronic materials. In addition, as an advantage of 3D carbon network structures, the grapheaynes are also promising for energy storage and molecular sieve applications. We also discuss possible experimental routes for synthesizing and characterizing these new carbon allotrope materials.

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Å), the C2-C2 bonds (1.34-1.46 Å), the C2-C3 bonds (1.52-1.55 Å), and the C3-C3 bonds (1.55-1.63 Å). Due to the insertion of the −C ≡ C − linkages, the interlayer spacing between graphene-like sheets increases from 3.35 Å for graphite to 4.12-5.00 Å for the grapheaynes. Meanwhile, the mass density of grapheayne-n decreases from 2.26 g/cm 3 to 1.70 g/cm 3  Stability is an important issue when proposing a new carbon allotrope. By analyzing the calculated cohesive energies (E coh ), one finds that grapheaynes have very low formation energies compared with other systems containing the −C ≡ C − linkages. For example, it is more stable than graphyne (-8.58 eV per C atom) which has been predicted to be the most stable among the 2D graphyne family, and is also more stable than carboneyane (-8.27 eV per C atom) which is another metastable 3D carbon allotrope with sp-sp 2 -sp 3 chemical bonds. We also calculated the cohesive energies (E coh ) as a function of width n for grapheayne-n. The result is plotted in Fig.   2. In the figure, the black dotted line represents the energy of diamond and the lower bound of the energy axis represents the cohesive energy of graphite. One can observe that when the width of grapheayne-n exceeds 15, the system is energetically lower than diamond, and its E coh approaches graphite (Fig. 2). We have also calculated the total energy against volume curves for fourteen typical carbon allotropes, including grapheayne-n (n=1-10), carboneyane, T-carbon, diamond, and graphite (as shown in 6 / 24 Supplementary Fig. S1). Among them, graphite has the lowest energy, followed by diamond, while most grapheaynes approach diamond. This indicates that the grapheaynes are energetically metastable. It is worth noting that with increasing negative pressure, the grapheaynes become more and more energetically favorable than T-carbon, diamond, and even graphite (see Fig. S1).
To inspect whether grapheaynes are dynamically stable or not, we calculated their phonon spectra. The results for grapheayne-4 and 5 are shown in Figs. 3a and 3b, respectively (others are shown in Supplementary Figs. S2 and S3). One can see that there is no imaginary phonon mode in the whole BZ, which confirms that grapheaynes are dynamically stable. We have also checked the thermal stability of the grapheaynes by performing the ab initio molecular dynamics (AIMD) simulations at finite temperatures. For grapheayne-4 (grapheayne-5), we find that they can retain their integrity at 1000 K (1200 K) ( Fig. 3c-3f), indicating its excellent stability at high temperature.
Electronic, optical, and topological properties. The grapheayne family exhibits rich electronic properties depending on the width of the nanoribbon component. We find that grapheayne-n is a topological nodal-line semimetal (indicated by the red box in  Figure 4c shows that the grapheayne-3, 4, 7, and 10 have effective masses of electrons and holes comparable to some well-known electronic materials, such as GaN, ZnO, Si, and carbon kagome lattice (CKL) 28 . Especially, due to the graphene-like nanoribbon unit in grapheaynes, the effective masses of grapheayne-4 and grapheayne-7 in the ribbon plane (i.e., ∥ and ℎℎ ∥ ) are very small, which may lead to promising potential application in electronic devices.
Besides grapheayne-4, there are also a series of grapheaynes with direct and narrow band gaps (Table I) eV, in order to maximize energy conversion efficiency. Therefore, our result implies that grapheaynes with tunable direct band gaps could be a good candidate for photovoltaic applications.
To gain better understanding of the band structure, in Fig. 4b, we plot the partial density of states (PDOS) of grapheayne-4. We note that the p orbital of C2 atoms dominate the bands near the band gap (the red-colored bands in Fig. 4a). In contrast, the p orbitals of C1 atoms occupy the lower valence bands (the blue-colored bands in around the crossing point reveals that the p x orbital of C2 atoms form  bands (see the inset in Fig. 5a), which is very similar to the origin of Dirac cone of graphene, whose  bands are attributed to the p z orbitals. Therefore, we can use a simple tight-binding (TB) model only consisting of C2 atoms to capture the essential physics around the Fermi level for grapheayne-5 (refer to the Supplementary Information with Fig. S9).
According to the crystal symmetry, grapheayne-5 has the little point group of 2 along the B-A path, and the two bands involved in the crossing belong to different irreducible representations of A and B along the path (see the inset in Fig. 5a). As a result, the band crossing is protected by symmetry. In addition, the system preserves the inversion (P) and time reversal (T) symmetries. For such PT-symmetric system without SOC, the crossing point cannot be an isolated single point. Indeed, a careful inspection reveals that the band crossing points form a continuous nodal ring in the whole BZ (see the inset in Fig. 5a). This nodal ring is protected by a quantized one-dimensional winding number where is the Berry connection at point k for the occupied bands, and the integration path l is around a loop encircling the ring. This winding number is essentially the Berry phase in unit of π. Our calculation confirms that = 1 for the nodal ring.
One important characteristic of a topological nodal-ring semimetal is the presence of "drumhead-like" surface states. Here, we study the surface spectrum (Figs. 5c and 5d) of grapheayne-5 on the (001) surface. The result confirms that the "drumhead-like" surface states exist and they are nestled outside the projected nodal ring in Fig. 5d (the red dashed lines). These "drumhead-like" surface states are slightly below the Fermi level; hence they should be easily detected by angle-resolved photoemission spectroscopy (ARPES). For grapheayne-2 and grapheayne-8, they also possess nodal rings, and the patterns of the nodal ring and surface states are different (see Supplementary Fig. S6). In particular, grapheayne-2 has two helical nodal loops in the first BZ. Like grapheayne-5, their nodal lines are also protected by PT symmetry with a nontrivial winding number.

Discussion
In this work, we have proposed a strategy for engineering the band gap in carbon, namely, by assembling graphene-like nanoribbons with acetylenic linkages, which combines the quantum confinement effect and hybrid bonding character to achieve a strong band structure modulation. According to above strategy, we design a family of novel stable carbon allotropesthe grapheaynes, with fascinating electronic properties.
We have shown that the semiconducting grapheayne members can have direct eV) semiconductor and its optical absorption is limited by the requirement of phonon assistance. As, Cd, and Pb are toxic; In is a rare element; and CH 3 NH 3 PbX 3 is not stable 37 . In contrast with these well-known materials, the grapheaynes may be better candidates, since they are non-toxic, consist of carbon which is cheap and abundant, and most importantly, their band gaps might be adjusted.
To facilitate experimental observation, we simulate the the X-ray diffraction (XRD) patterns of the grapheaynes along with several other carbon allotropes. The results are given in Supplementary Fig. S10, which can be used as structural indicators in future experiments. Taking grapheayne-4 as an example, we also suggest a possible experimental scheme for synthesis ( Supplementary Fig. S11). One may start from the graphene, and functionalize the graphene with bromide. Then, stacking the multilayer brominated graphene sheets and inserting acetylene molecules and finally deacidification will lead to the formation of grapheayne