Three-dimensional acetylenic modified graphene for high-performance optoelectronics and topological materials

Seeking carbon phases with versatile properties is one of the fundamental goals in physics, chemistry, and materials science. Here, based on the first-principles calculations, a family of three-dimensional (3D) graphene networks with abundant and fabulous electronic properties, including rarely reported dipole-allowed truly direct band gap semiconductors with suitable band gaps (1.07–1.87 eV) as optoelectronic/photovoltaic materials and topological nodal-ring semimetals, are proposed through stitching different graphene layers with acetylenic linkages. Remarkably, the optical absorption coefficients in some of those semiconducting carbon allotropes express possibly the highest performance among all of the semiconducting carbon phases known to date. On the other hand, the topological states in those topological nodal-ring semimetals are protected by the time-reversal and spatial symmetry and present nodal rings and nodal helical loops topological patterns. Those newly revealed carbon phases possess low formation energies and excellent thermodynamic stabilities; thus, they not only host a great potential in the application of optoelectronics, photovoltaics, and quantum topological materials etc., but also can be utilized as catalysis, molecule sieves or Li-ion anode materials and so on. Moreover, the approach used here to design novel carbon allotropes may also give more enlightenments to create various carbon phases with different applications.


INTRODUCTION
Carbon is one of the central and essential elements in the current scientific research and industrial applications due to its novel properties, i.e., superhardness, exceptional toughness, and the highest thermal conductivity [1][2][3][4][5][6][7] . Although elemental carbon exists in two natural allotropes, i.e., diamond and graphite, there are also many novel carbon phases that have been manufactured in experiments over the past decades, such as fullerenes, carbon nanotubes, graphene, and so on 8,9 . Of them, graphene has a very central and vital role in that fullerenes can be viewed as wrappedup graphene and carbon nanotubes can also be made of graphene by rolling up along specific direction. Moreover, lots of methods have been proposed to obtain novel carbon allotropes based on graphene. For instance, a new family of two-dimensional (2D) carbon materials, graphynes, has been revealed by replacing different single bonds in graphene with acetylenic bonds 10 . Using graphene nanoribbons as building blocks, a series of 3D graphene networks have been obtained and intensively studied in various fields [11][12][13][14] . Therefore, it occurs to us whether or not there is another possibility to design novel carbon polymorphs from graphene.
It should be emphasized that we are not just curious about how many esthetic carbon phases there are, more importantly, looking back in the last decades, the discovery and invention of new forms of carbon have always opened doors to new science and technology [15][16][17][18] . Such as the fullerenes, carbon nanotubes and graphene have been immensely studied from catalysis, energy storage, optoelectronics, photovoltaics, organic electronics, etc. 19,20 . One of the most important topics in developing carbon materials is to replace silicon-based electronics with carbon-based electronics 21 , which need carbon phases with appropriate direct band gap and well electron transition of optical absorption. It is well-known that semiconductors are classified into direct and indirect band gap semiconductors, respectively, depending on whether their valence band maximum (VBM) and conduction band minimum (CBM) occur at the same wavevector in reciprocal space. However, only when the optical transition selection rules are satisfied by the symmetries of wavefunctions of VBM, CBM, and transition dipole operator, the semiconductors can be viewed as dipole-allowed truly direct band gap semiconductors 22 , which are really pursued by us. Unfortunately, diamond is unsuitable due to its wide indirect band gap (~5.5 eV). Despite many efforts being paid to search high-performance semiconducting carbon allotropes, such as recent experimentally synthesized T-Carbon 23,24 or theoretically proposed CKL 25,26 and CYKL 27 , etc., they are not dipole-allowed truly direct band gap semiconductors and may encounter challenges in chemical synthesis because of their high total energies or structural complexities. Thus, it is highly desired to seek a high-performance semiconducting carbon phase with great potential of being realized.
On the other hand, studying physics in carbon materials also created a new era in condensed matter physics, i.e., topological physics in solid states [28][29][30] , as evidenced by the Dirac fermion and quantum spin Hall effect in graphene 31 . Although topological insulator was first proposed in graphene, its negligible spinorbital coupling (SOC) is not suitable for inducing observable topological effects at ambient condition. Therefore, the study of a topological insulator is now mainly concentrated on heavy element compounds with strong SOC that can induce more prominent topological effects. However, carbon, as a light element, is more preferable for spawning topological semimetal since the SOC is not mandatory in topological semimetal [32][33][34][35][36] . In fact, there are lots of topological semimetal states, such as topological nodal line semimetal [37][38][39] , topological nodal surface semimetal [40][41][42] , or new topological fermions 43,44 , have been proposed in carbon materials theoretically. Unfortunately, we notice that none of those reported topological semimetals in carbon materials have been realized in experiments, for which we think the reason is partial that those proposed carbon structures have relatively high total energies and lack any available structural precursors that can be used to realize the proposed carbon structures. Thus, searching for topological semimetal in carbon materials from the perspective of easy synthesis is still challenging and desirable.
In this work, we propose a new strategy to tailor graphene through stitching graphene layers with acetylenic linkage, as shown in Fig. 1a, b, which here is termed as 3D acetylenic modified graphene (3D-AMG). According to our first-principles calculations, all of the 3D-AMGs demonstrate excellent stabilities, as confirmed by their total energies, phonon spectra, and ab initio molecular dynamics (AIMD). Prominently, the family of 3D-AMGs present very abundant and fabulous electronic properties, some of which are dipole-allowed truly direct band gap semiconductors with band gaps in the range of 1.0-1.5 eV, which are located in the optimal band gap value range for photovoltaic application, and possess probably the highest optical absorption strength amongst all of the semiconducting carbon allotropes known to date, especially the band gap of 3D-AMG-7 is 1.19 eV that is very close to the optimal value of 1.34 eV for solar cell material; on the other hand, some 3D-AMGs are topological nodal-ring semimetals. Our work discovers some semiconducting carbon allotropes with excellent optical absorption strength and some topological nodalring semimetals, meanwhile the approach used here to design novel carbon allotropes through connecting graphene with acetylenic linkage may also give more thoughts to create various carbon phases with different applications.

RESULTS
The structure information and the stabilities of 3D-AMGs As shown in Fig. 1a, b, the graphene layers (sp 2 -hybridized atoms colored with blue and denoted as C2) are stitched vertically by acetylenic linkage (sp-hybridized atoms colored with red and denoted as C1) going through a carbon dimer (sp 3 -hybridized atoms colored with cyan and denoted as C3) along the armchair direction of graphene, so the family of 3D-AMG-n is obtained, where n denotes the width of the armchair nanoribbons in between two acetylenic connected carbon dimer (i.e., the cyan atoms). Thus Fig. 1a, b are 3D-AMG-4 and 3D-AMG-5, respectively. There are two space groups in 3D-AMG-n depending on n being odd (P2/M) or even (C2/M). As two representatives, we tabulate the basic structural information of 3D-AMG-4 and 3D-AMG-5 in Table 1 together with other carbon allotropes such as the CKL, CYKL, diamond, graphite, and T-carbon for comparison, and those of the rest members of 3D-AMG-n can be found in supplementary Table 1.
One would expect 3D-AMGs should express good stability since they are constructed from graphene. It is actually true. The calculated total energies of 3D-AMG-4 and 3D-AMG-5 are −8.86 eV and −8.91 per atom, respectively, which arẽ 0.05~1.00 eV/atom lower than those of the previously reported CKL (−8.81 eV/atom), CYKL (−8.24 eV/atom), and recently realized T-carbon (−7.91 eV/atom). Moreover, the total energies decrease as the increase of n in 3D-AMG-n (see supplementary Table 1), such as the total energy of 3D-AMG-10 is −9.03 eV/atom, which almost approaches that of diamond (−9.09 eV/atom). On the other hand, the phonon spectra of 3D-AMG-4 and 3D-AMG-5 have no soft mode throughout the entire Brillion zone (BZ), implying they are also dynamically stable (see Fig. 2a, b). Their AIMD simulations manifest that they can be stable under at least 1000 K (see Fig.  2c-f), showing very excellent thermodynamic stability as well.
The electronic and optical properties of 3D-AMGs Now, we proceed to investigate the electronic properties of 3D-AMGs. It has been known that the main drawback of graphene to impede its application in semiconductor devices is the absence of band gap. Focusing on the band gap's engineering of graphene, there are plenty of methods that have been proposed, such as cutting into graphene nanoribbons or surface functionalization 45 . The acetylenic modification here on the surface of graphene can be treated as the extension of the method of surface functionalization, but just of extending to three dimension by connecting different graphene layer with a functional group (i.e., the acetylenic linkage). Based on our first-principles calculations, these 3D-AMGs demonstrate very abundant and fascinating electronic properties. Altering the value of n in 3D-AMGs, indirect band gap, direct band gap semiconductors, and topological nodal-ring semimetals can be found. Here, we are mainly interested in the direct band gap semiconductors and topological nodal-ring semimetals, thus they are discussed elaborately below as exemplified with 3D-AMG-4 and 3D-AMG-5, respectively. The results of the rest members of 3D-AMGs can be found in Supplementary Information (SI).
The band structure of 3D-AMG-4 is shown in Fig. 3a. One can clearly observe that the VBM and the CBM locate at the same M point, indicating that it is a direct band gap semiconductor. Although 3D-AMG-4 is a direct band gap semiconductor, it is known that in a crystal with inversion symmetry there are likely parity-forbidden transitions induced between conduction and valence band edges, which can in turn seriously affect its optical absorption properties 46 . The space group of 3D-AMG-4 is C2/M containing the inversion symmetry, therefore more information is needed to analyze its optical performance. According to our firstprinciples calculations, the VBM and CBM of 3D-AMG-4 are mainly derived from the p orbitals of C2 carbon atoms (see Fig. 3b) and have respective odd and even parities (denoting with "−" and "+" in Fig. 3a), which means 3D-AMG-4 is a direct band gap semiconductor exhibiting no parity-forbidden (even to even or odd to odd) transitions.
The transition probabilities between the topmost valence and the lowest conduction bands can be revealed by the sum of the squares of the dipole transition matrix elements (TME), P 2 , at various k points 46 . The calculated TME of 3D-AMG-4 is shown in Fig. 3c. One can observe that the high transition probability at M point, implying its potential strong optical absorption strength.  Fig. 7), which make them quite suitable for solar cell materials, according to the Shockley-Queisser's theory 47 that the optimal band gap for solar cell materials should straddle between 1.0 and 1.5 eV.
The topological properties of 3D-AMGs Another interesting feature of 3D-AMG-n family is that some of them are topological nodal-ring semimetals as exemplified by 3D-AMG-5 here. The band structure of 3D-AMG-5 is shown Fig. 4a, in which the Dirac point crossed by the valence and conduction bands along B-A can be clearly observed. We also confirm the results by the HSE calculation (see supplementary Fig. 6b). According to our calculations, the bands near the Fermi level are mainly contributed from the p x orbitals of C2 atoms, as presented in the charge densities and PDOS (see the inset of Fig.  4a, b). Based on those analysis, we develop a tight-binding (TB) model to describe the hopping integrals in 3D-AMG-5, and the TB band structure fits very well with the density functional theory (DFT) results around the Fermi level (more details can be found in SI).
3D-AMG-5 has the little point group of C 2 along the B-A path, and the two bands involved in the crossing belong to different irreducible representations of A and B (see the inset in Fig. 4a), implying that the crossed point is protected by crystalline symmetry. In addition, the system preserves the inversion (P) and time-reversal (T) symmetries. For such PT-symmetric system without SOC, the crossed 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.  4a). This nodal ring is protected by a quantized one-dimensional Table 1. Crystal system, space group (SG), lattice parameters a/b/c (Å) and α/β/γ (°), mass density ρ (g/cm 3 ), band gap E g (eV), and total energy E t (eV/ atom) of 3D-AMG-4, 3D-AMG-5, diamond, CKL, CYKL, and T-carbon obtained from our work in comparison with those in literature.   Y. Gao et al. winding number N ¼ 1 π H l Tr A k ½ Á dk where A k is the Berry connection at point k for the occupied bands, and the integration path l is around a loop encircling the nodal ring. This winding number is essentially the Berry phase in a unit of π. Our calculation confirms that N = 1 for the nodal ring, indicating its topological character.
One significant consequence of topological nodal-ring semimetal is the presence of "drumhead-like" surface states. Here, we study the surface spectrum (Fig. 4c, d) of 3D-AMG-5 on the (001) surface. The result confirms that the "drumhead-like" surface states do exist and they are nestled outside the projected nodal ring in Fig. 4d (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. For 3D-AMG-2 and 3D-AMG-8, also possess nodal rings, but the patterns of the nodal rings and surface states are different (see supplementary Fig. 8). In particular, 3D-AMG-2 has two helical nodal loops in the first BZ.
We would like to point out the electronic properties of 3D-AMGs are mainly contributed by the atoms in the armchair nanoribbons (i.e., the C2 atoms). This implies that the electronic properties of 3D-AMGs are intimately connected with those of armchair nanoribbons. As analyzed by Son et al. 48 , the band gaps of the armchair nanoribbons with a width equal to 3p + 2 (where p is a positive integer) are much smaller than those of the others, owing to the quantum confinement effect; Likewise, although local details of the armchair nanoribbons are perturbed by the inserted acetylenic linkages, the whole tendency of electronic properties of 3D-AMGs are partially consistent with those of the armchair nanoribbons. Specially speaking, 3D-AMG-n with n = 3p+2 are semimetals, and the others are semiconductors.

DISCUSSION
Before closing, we would like to give some remarks about our work. The Shockley-Queisser limit 47 suggests that the theoretically maximum solar converting efficiency 33.7% of a single-junction solar cell occurs for a semiconductor with a band gap of 1.34 eV, as evidenced by silicon 49 , GaAs 50 , CuIn x Ga 1-x Se 2 51 , and hybrid organic-inorganic perovskite compounds (CH 3 NH 3 PbX 3 , X=I, Br, and Cl) 52 . Nevertheless, these materials proposed so far have various limitations. For example, silicon is an indirect-band gap (1.1 eV) semiconductor and its optical absorption is limited by the requirement of phonon assistance. As, Cd, and Pb are toxic; In is a  The effective masses are given in units of the free electron mass.
Y. Gao et al. rare element; and CH 3 NH 3 PbX 3 is not stable 53 . In comparison with these well-known materials, the 3D-AMGs may be better candidates, since they are non-toxic, carbon is cheap and abundant, and most importantly, their band gaps are adjustable.
To guide the experiment, we simulate X-ray diffraction peaks of 3D-AMGs and contrast them with those of representative carbon allotropes (see supplementary Fig. 10). Taking 3D-AMG-4 as an example, we also suggest a possible experimental scheme for its synthesis (see supplementary Fig. 11). One may start from the graphene, and functionalize the graphene with bromide 54 . Then, stacking the multilayer brominated graphene sheets and inserting acetylene molecules, and finally, deacidification will lead to the formation of 3D-AMG-4. Each step in this process is either already achieved or has a high chance to be achieved because of the existence of similar reactions. Given its energetic and dynamic stabilities and in view of the rapid progress in experimental techniques, which have realized a number of carbon structures in recent decades, we expect 3D-AMGs could also be realized in the near future.
Last but not least, it is worthily noted that 3D-AMGs would have several other advantages and promising applications. For instance, the large specific surface and space can enhance reversible storage space and storage capacity for lithium and sodium ions. Moreover, 3D-AMGs have the graphene-like basic unit, so they may inherit the excellent properties of graphene, such as superhigh conductivity and super-high thermal conductivity.
In this work, we propose a new strategy to tailor the properties of graphene through stitching different graphene layers with acetylenic linkages. According to the different connected ways, a new family of 3D carbon networks is obtained, namely 3D-AMG-n here. 3D-AMGs show very excellent stability in that they are constructed from graphene. More strikingly, 3D-AMGs are not trivial carbon phases, actually they manifest many fascinating properties. For instance, 3D-AMG-3/4/7/10 are dipole-allowed truly direct band gap semiconductors with band gaps ranging from 1.07 eV to 1.87 eV and possess very outstanding optical absorption strength in comparison with other semiconducting carbon allotropes, which makes them quite suitable for optoelectronic or photovoltaic applications. On the other side, 3D-AMG-2/5/8 are topological semimetals, which is a hot topic in the study of topological materials. Moreover, 3D-AMGs have large specific surface and space, making them also preferable as catalysis, molecule sieves, and Li-ion anode materials, etc.

First-principles calculations
The first-principles electronic structure calculations were based on the DFT with the Perdew-Burke-Ernzerhof (PBE) 55 approximation to the exchangecorrelation energy. The core-valence interactions were described by the projector augmented wave (PAW) 56 method, as implemented in the VASP package 57,58 . The kinetic energy cutoff of 550 eV was adopted for the plane-wave basis. The atomic positions were fully relaxed by the conjugate gradient method; the energy and force convergence criteria were set to be 10 −6 eV and 10 −3 eV/Å, respectively. The k-point meshes 7 × 7 × 7 and 9 × 7 × 11 were used for the BZ integration of 3D-AMG-4 and 3D-AMG-5, respectively. The band gaps of semiconductors were also calculated by using the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional 59 . The frequency-dependent dielectric matrix was calculated using the method described by Gajdos et al. within PAW potentials 60 . In order to study the dynamical stability, we used a finite displacement approach as implemented in the PHONOPY package 61 to calculate the phonon dispersions. The thermal stability was investigated with the AIMD simulations in a canonical ensemble with a Nose-Hoover thermostat 62,63 . The band crossing characteristics were analyzed with the WannierTools 64 package according to the TB model constructed via the Wannier90 code 65 .

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.