Abstract
Currently, the available algorithms for unbiased structure searches are primarily atombased, where atoms are manipulated as the elementary units and energy is used as the target function without any restrictions on the bonding of atoms. In fact, in many cases such as nanostructureassembled materials, the structural units are nanoclusters. We report a study of a bondingrestricted structure search method based on the particle swarm optimization (PSO) for finding the stable structures of twodimensional (2D) materials containing dispersed C_{2} dimers rather than individual C atoms. The C_{2} dimer can be considered as a prototype of nanoclusters. Taking SiC, BC and TiC systems as test cases, our method combined with density functional theory and phonon calculations uncover new ground state geometrical structures for SiC_{2}, Si_{2}C_{2}, BC_{2}, B_{2}C_{2}, TiC_{2} and Ti_{2}C_{2} sheets and their lowlying energy allotropes, as well as their electronic structures. Equally important, this method can be applied to other complex systems even containing f elements and other molecular dimers such as S_{2}, N_{2}, B_{2} and Si_{2}, where the complex orbital orientations require extensive search for finding the optimal orientations to maximize the bonding with the dimers, predicting new 2D materials beyond MXenes (a family of transition metal carbides or nitrides) and dichalcogenide monolayers.
Introduction
Determining the geometrical structures is essential to design new materials. The traditional Xray diffraction technique for determining the crystal structure requires high quality samples and is incapable of dealing with materials under extreme conditions such as high pressure, strong electric or magnetic fields. On the other hand, determining a groundstate geometry based on chemical and physical intuitions is difficult due to the complex potential energy surface especially for materials with large unit cells or complicated structural units. Hence many structure prediction methods based on quantum mechanical simulations combined with global optimization algorithms have been developed, such as simulated annealing^{1}, genetic algorithm^{2,3} and basin hopping^{4}. However, in these currently available methods, energy is used as the target function and there are no restrictions on the bonding of atoms and atoms are usually manipulated as the elementary units. Thus, they are unsuitable for doing structure search for nanoclusterassembled materials where the structural units are nanoclusters, which are not directly bonded to each other but through linkers in most cases. Therefore, it is highly desirable to develop new search algorithms for doing such bondingrestricted structure search. An earlier work reported a constrained evolutionary search algorithm that enables prediction of polymeric crystal^{5}. However, algorithms for motifbased prediction of low dimensional materials are not yet reported.
In this study, we report a dimerbased bondingrestricted search method for finding stable 2D structures with a restriction of no direct bonding between the dimers. C_{2} is a wellknown pseudohalogen unit with an electron affinity of 3.4 eV^{6} and is the basic structural unit of diverse carbon compounds. It has received increasing attention due to its rich physical and chemical properties. R. Hoffmann^{7} recently made a comment on C_{2}: “C_{2} is one of my favorite molecules, for this diatomic epitomizes many of the fundamental questions of chemistry”. Shaik and coworkers^{8} demonstrated that an unusual quadruple bond may exist in C_{2}, which is strong enough to be considered as a typical chemical bond. In addition, dimerization of carbon is not only the first stage toward final crystallization, but also plays a vital role in the growth of many carbon related materials^{9,10}. For instance, a C_{2} dimer on a Cu surface was identified as the dominant feeding species for graphene growth due to its lower diffusion and attachment barrier as compared to the carbon monomer and trimer^{11,12}. Although the C_{2} dimer is quite chemically active and inclined to aggregate, it naturally occurs in the carbon vapor of interstellar medium and electric arcs. Vapor from a 16V carbon arc was found to contain 28 wt% diatomic carbon^{13}. As a result, in molecular dynamics (MD) simulations of growth of carbon–based materials, such as graphene, carbon nanotube and C_{60}, C_{2} dimers are usually adopted as the initial carbon source^{14,15,16}.
Furthermore, the C_{2} dimer is the building block of metalalkynide complexes^{17}, alkynide complexes^{18}, metallocarbohedrenes clusters M_{m}C_{n}^{19,20} (usually termed as metcars) and ternary metal carbides (LiAgC_{2}, KAgC_{2}, CsAgC_{2} and NaPdC_{2})^{21}. In recently studied silicon carbides (Si_{x}C_{y}), the dimerization of C in the SiC monolayer reintroduces the Dirac cone into the honeycomb lattice^{22}, while SiC sheets without C_{2} dimer are wideband gap semiconductors^{23,24,25,26}. In SiC_{2} silagraphene^{27}, each silicon atom binds to four C_{2} units in a flat plane, resulting in a metallic sheet. When C_{2} dimers are embedded in a 2D porphyrin sheet, the system becomes highly active for oxygen reduction reaction^{28}. In addition, it was demonstrated that the formation of C_{2} dimers plays an important role in stabilizing metcar clusters^{29}.
Due to the C_{2} dimer’s unique properties as well as its importance in forming numerous carbonbased threedimensional materials and zerodimensional nanostructures, we focus this study on developing a dimerbased global search algorithm to design 2D materials containing dispersed C_{2} dimers to retain its intrinsic properties. We then apply this method to identify the ground state geometries of the 2D SiC_{2}, BC_{2} and TiC_{2} systems containing C_{2} dimers as test cases.
Results and Discussion
To test our method, we applied it to three systems where C_{2} dimers are linked with Si, B and Ti atoms respectively. New ground state geometries and new lowlying isomers are identified.
2D SiC_{2} system
Recently, SiC_{2} and Si_{2}C_{2} sheets with dispersed carbon dimers have received some attention. The studies were carried out using a priori geometric structures to investigate the properties of 2D SiC sheets^{22,27,30}. However, a complete understanding of SiC_{2} phases is still lacking and it is unclear whether the studied structures are the ground states under the condition of discrete C_{2} dimers. Thus we have conducted a comprehensive structure search to identify the ground state geometry and the lowlying energy isomers of the SiC_{2} system.
The most stable structure of Si_{2}C_{2} and three lowlying energy isomers of SiC_{2} have been identified using our C_{2} dimerbased global search. Because the lowest energy geometry of Si_{2}C_{2} is found to be identical to a recently reported structure^{22}, we only concentrate on the three isomers of the SiC_{2} sheet, as shown in Fig. 1. In these structures all carbon atoms are pairwisely bonded with each other and covalently bonded with Si atoms. They all are comprised of fourfold coordinated Si with threefold coordinated C atoms. The structure, shown in Fig. 1(a), is composed of pure pentagons, thus it is named pentaSiC_{2}. The other two isomers shown in Fig. 1(b,c) consist of 4, 5 and 6rings and 4 and 6rings, respectively, labeled as 456SiC_{2} and 46SiC_{2} accordingly. SiC and CC bond lengths are about 1.90 Å and 1.36 Å respectively. The side views show the sandwichlike structures with the fourfold coordinated Si atoms in the middle sandwiched between C_{2} dimers. When we consider the Si sublattice and C_{2} dimer (treating C_{2} as a structural unit) sublattice separately, it turned out that the major difference among these three isomers is the orientation of C_{2} dimers, namely the dimers are parallel to each other in 46SiC_{2}, while they are perpendicular to each other in pentaSiC_{2} (In Supplementary Information, we shortly discuss the kinetic barriers connecting different isomers). It is obvious that the different arrangement of C_{2} dimers remarkably alters the geometrical structures, resulting in the different electronic properties of these isomers as demonstrated in the following paragraph.
Total energy calculations reveal that all three isomers are energetically more favorable than the previously proposed SiC_{2}silagraphene^{27} with planar tetracoordinate silicon atoms. PentaSiC_{2}, 456SiC_{2} and 46SiC_{2} are 0.63, 0.57 and 0.51 eV/formula unit (f.u.) lower in energy than that of SiC_{2}silagraphene, suggesting that the SiC_{2} sheet is more likely to adopt a buckled structure with the partially sp^{3} hybridized Si and the sp^{2} hybridized C. PentaSiC_{2} is found to be the lowest energy configuration among the three isomers with the SiC and CC bond lengths of 1.90 Å and 1.36 Å respectively, which is identical to that reported recently by Bezanilla et al.^{30} and as 456SiC_{2} lies only 0.06 eV higher in energy than pentaSiC_{2}, the two structures can be considered as energetically degenerate. The relative energy of 46SiC_{2} with respect to pentaSiC_{2} is 0.12 eV. The dynamical stability of the predicted lowlying energy isomers of SiC_{2} is verified by calculating their phonon dispersions. The results plotted in Fig. S1 (see Supplementary Information) show that the three structures are all dynamically stable since their vibration modes are real in their entire Brillouin zones, respectively.
To understand the electronic properties of the three isomers, their band structures are calculated using the HSE06 hybrid functional. Furthermore, a more detailed investigation of the contribution from each atomic orbital to the band structure is carried out using the colormapped bands. The calculated results are presented in Fig. 1, which shows that pentaSiC_{2}, 456SiC_{2} are large band gap compounds with band gaps of 3.1 eV, 2.7 eV, while the bandgap of 46SiC_{2} is 0.55 eV. We note that 46SiC_{2} is a direct band gap semiconductor and its band gap is much smaller than that of pentaSiC_{2} and 456SiC_{2}. While the others are indirect band gap semiconductors because the valence band maximum (VBM) and the conduction band minimum (CBM) are not at the same point in their Brillouin Zones. The localized distribution of p_{z} orbitals of the C_{2} dimer, above and below the Fermi level, is disclosed in the colored band structure (red dot curves in decomposed band structure). Further analyses of the chemical bonding of the C_{2} dimer reveals that the localized distribution originates from the bonding and antibonding p_{z} orbital of C_{2} dimers. Due to the unique geometrical structure, the alternation of partially sp^{3} hybridized Si and slightly distorted sp^{2} C prevents the p_{z} orbitals from forming a delocalized π bond. This could explain, to some degree, why these p_{z} composed bands are less dispersive than those in graphene or other planar silicon carbon sheets.
To understand the origin of the different electronic structures exhibited in the three isomers, the detailed structural configurations of the local C and Si atoms in pentaSiC_{2} and 46SiC_{2} are sketched in Fig. S3 (see Supplementary Information). PentaSiC_{2} has higher symmetry subjecting it to the least geometrical distortion among the three isomers. 46SiC_{2}’s parallel arrangement of the C_{2} dimer imposes a larger geometrical distortion; the angle of C with its neighboring Si is 92.9 °, while that in pentaSiC_{2} is 109.6 °. The Si atoms in 46SiC_{2} also adopt the distorted hybridization as compared to that in pentaSiC_{2}. Besides, the decomposed band structures (Fig. 1) show that the highest and the second highest occupied bands near the M point in 46SiC_{2} are exclusively from the SiC σ bonds. The larger geometrical distortion of SiC σ bonds in 46SiC_{2} leads to a stronger interaction between the relevant orbitals, resulting in the highly dispersive bands and a smaller band gap.
2D BC_{2} system
For the second test system, we study the 2D BC_{2} system due to boron’s rich chemistry: for example its electron deficiency and versatile bonding ability. In fact, nanostructures composed of boron and carbon atoms have attracted growing interest both theoretically and experimentally. For instance, a BC_{3} honeycomb structure with high crystalline quality was identified by electron diffraction^{31}. A B_{2}C sheet with planar tetracoordinated carbon (ptC) moiety was theoretically predicted^{32}. A thorough study of BC compounds with different stoichiometry was carried out by performing a PSO search^{33}. However, a study on BC sheets comprised of discrete C_{2} dimers has not been reported. Due to its versatility in forming various chemical bonds and the differing chemical nature of B from Si, the bonding between B and the C_{2} dimer is expected to be quite different from that of Si and C_{2}, leading to BC_{2} systems with different structural and electronic properties.
Using our dimerbased search algorithm, three lowlying energy isomers of B_{2}C_{2} and one stable structure of BC_{2} are obtained (Fig. 2). Geometry optimizations and total energy calculations suggest that the structure shown in Fig. 2(a1) has the lowest energy of the three isomers of B_{2}C_{2} with a total energy of 0.04 eV, 0.13 eV lower than that of the structures shown in Fig. 2(a2,a3), respectively. To examine the dynamical stability of these structures, we calculated their phonon dispersions. The results are plotted in Fig. S4. The absence of imaginary modes in the entire Brillouin zone for each of the structures confirms that they are all dynamically stable. However, we note that these structures are energetically metastable as compared to the BC planar monolayers predicted by Luo et al. (B_{2}C_{2}a is 0.23 eV/f.u. higher in energy than the most stable BC sheet).
In order to gain more insight into the chemical bonding in the BC_{2} sheets, we plotted an electron localization function (ELF)^{34} isosurface (with a relatively large isovalue of 0.75 to reflect the σ bonds between the atoms) in Fig. 3. According to the ELF diagrams, carbon in B_{2}C_{2}a and B_{2}C_{2}b adopt sp^{3} hybridization and each carbon atom is covalently bonded to three adjacent B and one C atoms. B atoms take a slightly distorted sp^{2} configuration. The CC bond length in Fig. 2(a1,2) are 1.51 Å (close to that of a CC single bond). While in B_{2}C_{2}c, both the B and C atoms have sp^{2} hybridization. B_{2}C_{2}c adopts a planar honeycomb structure similar to graphene^{35}, however, the total energy of B_{2}C_{2}c is 0.13 eV/f.u. higher than that of B_{2}C_{2}a. Multicenter bonds are pretty common in boronrelated materials^{36,37} and contribute a lot to their stability. However, no such bonds are found in B_{2}C_{2}a, B_{2}C_{2}b and B_{2}C_{2}c, which might explain the physical origin of their higher energy.
For the BC_{2} sheet, only one stable structure without imaginary modes was discovered (Figs 2(b) and S4(b)). In particular, all the BC_{2} structures isostructural to those of SiC_{2} (namely pentaBC_{2} or 46BC_{2}) are unstable, reflecting the differing chemical nature of B and Si. In this highly buckled BC_{2} sheet (the thickness is 1.53 Å), all C atoms are strongly covalently bonded together; the CC bond length is 1.31 Å. The calculated electron localization function (ELF) shows that there are sp^{3} hybridized B and nearly planar tetracoordinate B atoms. In our case, the bonding type of planar tetracoordinated B is closer to the case of planar tetracoordinated Si^{27} and C^{32}.
Our electronic band structure calculations (Fig. 2) show that the ground state structures of B_{2}C_{2} and BC_{2} with the discrete C_{2} dimers have indirect band gaps of 2.9, 2.8 and 4.1 eV at the HSE06 level for B_{2}C_{2}a, B_{2}C_{2}b and BC_{2}, respectively, which is in contrast to the metallic BC sheet^{33} where C is not in C_{2} form. The reason is, once we restrict the carbon atoms to be pairwisely bonded together, the most stable structures (B_{2}C_{2}a and BC_{2}) discovered by using our dimerbased search are buckled. The separation of C_{2} dimers by boron atoms and the buckling nature of the sheets would destroy the delocalization of p_{z} π electrons of boron and carbon (the main reason for the metallicity of many planar stable BC compounds). In the B_{2}C_{2}c isomer, the planar geometry, as well as its delocalized π electrons is preserved. Geometrically speaking, B_{2}C_{2}c is similar to honeycomb SiC^{22}. Since boron has few valence electrons it is impossible to fill all the electronic states below the Dirac cone giving rise to metallicity.
2D TiC_{2} system
To study the interaction between the C_{2} dimer and transition metal elements, we have systematically studied TiC_{2} sheets for the following reasons: (1) both Ti and Si possess four valence electrons, thus it is intriguing to look at the difference between TiC_{2} and SiC_{2} sheets and Ti_{2}C_{2} and Si_{2}C_{2} sheets. (2) Titanium and carbon have been found to form various kinds of stable compounds ranging from 0D clusters^{19,38}, to 2D sheets^{39,40} and to 3D bulk materials. Among them, in metcar clusters all carbon atoms exist in the form of a C_{2} dimer^{19}. Therefore, it might be possible to find a family of 2D crystals that consists of transition metal atoms and C_{2} dimers. (3) Recently, a group of 2D layered early transition metal carbides termed as MXenes, including Ti_{3}C_{2} and Ti_{2}C, were experimentally synthesized by exfoliation from their bulk MAX phases^{41,42,43}. However, in MXenes sheets, C is in atomic form rather than a C_{2} dimer. Therefore it is necessary to study the possibility of forming stable 2D titanium carbides containing C atoms that are all in C_{2} dimer form.
We concentrated on the 2D titanium carbide sheets with the stoichiometry of TiC_{2} and Ti_{2}C_{2} where C is in the form of a C_{2} dimer. Recently, a TiC_{2} sheet was predicted to have good performance as anode material for a lithium ion battery^{40}. However, the geometrical structure was artificially designed. Thus, it is easy to miss some lowlying isomers. In fact, we found, using our dimerbased algorithm, that among the isomers of TiC_{2}, there is a one that is lower in energy by 0.13 eV/f.u. than the recently reported structure^{40}. In this lowest energy structure, as shown in Fig. 4(a), due to the different orientations of C_{2} dimers, the total thickness of 1.97 Å is less than the 2.02 Å of the reported structure^{40}. One can see when going from Si to Ti, although both have four valence electrons, the complex 3d orbitals in Ti require a delicate arrangement of C_{2} dimers to match with the d orbital orientations for maximizing the interactions. The stability of the TiC_{2} sheet can be traced back to the most stable like Ti_{8}C_{12} cluster^{44}, where all carbon dimers bind to Ti in endon configuration (EOC) and sideon configuration (SOC), which strengthens the bonding between Ti and C and stabilizes the Ti_{8}C_{12} cluster. In our 2D TiC_{2} sheet EOC and SOC are apparently well preserved. The CC bond length is 1.33 Å, which is slightly longer than that of a dispersed C_{2} molecule (1.31 Å) due to the partial occupation of π_{g} orbital of C_{2} in TiC_{2}. The bond length of TiC in EOC and SOC modes is 2.05 Å and 2.19 Å, respectively, close to 2.1 Å in Ti_{2}C MXene phase.
We then increased the atomic ratio of Ti from TiC_{2} to Ti_{2}C_{2} to study the effect of Ti concentration on the geometry and properties of the TiC sheets. The structure shown in Fig. 4(b) is found to have the lowest energy among the isomers of Ti_{2}C_{2} under the requirement of discrete C_{2} dimers and it lies lower in energy by 0.25 eV/f.u than the tTiC sheet^{39}. This suggests that when the same atomic ratio of Ti to C exists, C_{2} is energetically more favorable than atomic C to bind with Ti in forming a sheet. The CC bond length is 1.48 Å, close to that of CC single bond but longer than that in a TiC_{2} sheet. This is because the hybridization between the C_{2}π_{g} and Ti3d is stronger in Ti_{2}C_{2}, resulting in more π_{g} states being occupied, thus leading to the elongated CC bond.
The calculated band structures and partial DOS are also plotted in Fig. 4, which clearly show that the TiC_{2} sheet is metallic, while the Ti_{2}C_{2} sheet, like the tTiC sheet, is an indirect narrow gap semiconductor with a gap of 0.1 eV at the HSE06 level. It is known that the ground state electronic configuration of an dispersed C_{2} is (1σ_{g})^{2}(1σ_{u})^{2}(2σ_{g})^{2}(2σ_{u})^{2}(1π_{u})^{4} with the higher energy states (3σ_{g})(1π_{g})(3σ_{u}) unoccupied and the (2σ_{g})^{2}(2σ_{u})^{2}(1π_{u})^{4}(3σ_{g}) having much lower energy as compared to the orbitals of Ti3d. Therefore, a weak interaction between the deep level orbitals of C_{2} and Ti3d is expected. Thus, the electronic states near the Fermi level are mainly from the interplay between C_{2}π_{g} and Ti3d orbitals.
For the structure of tTiC^{39} a strong electronic resonance between C2p and Ti3d is obvious. Due to the strong hybridization between C2p and Ti3d and the low buckled geometry of tTiC, the bonding states of C2p and Ti3d are separated from the unoccupied Ti3d states forming a band gap. TiC_{2} is geometrically analogous to tTiC with carbon atoms replaced with discrete C_{2} dimers. However, replacing C with C_{2} leads to drastic change in their electronic structures, namely tTiC is semiconducting with a strong electronic resonance, while TiC_{2} is metallic. This is because the π_{g} orbitals of C_{2} hybridize with the Ti3d orbitals, forming the energy bands crossing the Fermi level, thus, resulting in metallicity of TiC_{2}. For the Ti_{2}C_{2} sheet, the electronic states near the Fermi level are also derived from C_{2} π_{g} and Ti 3d orbitals similar to TiC_{2}, yet the interaction between C_{2} π_{g} and Ti 3d orbitals is stronger as compared to that in TiC_{2} due to the higher Ti concentration, which reduces the electronic states near the Fermi level and thus results in the small band gap as shown in Fig. 4(b).
Based on the above discussions, the following intriguing points can be concluded: (1) compared with the sandwiched structure of the TiC_{2} sheet, the Ti atoms are more exposed in the Ti_{2}C_{2} sheet and hence have higher chemical reactivity; (2) Although having a higher atomic ratio of Ti, the Ti_{2}C_{2} sheet is semiconducting with a band gap of 0.1 eV, while the TiC_{2} sheet is metallic; (3) For the tTiC^{39} and Ti_{2}C_{2} sheets, although they have the same composition ratio, when going from atomic C to C_{2} dimer, the band gap is reduced from 0.2 eV to 0.1 eV, showing promising applications of the Ti_{2}C_{2} sheet in infrared lightrelated technology.
Summary
In this study, we developed a dimerbased bondingrestricted structure search method to find the ground state structures of the 2D materials containing C_{2} dimers with a restriction of no direct bonded dimers and then applied this method to three systems composed of discrete C_{2} dimers that are coordinated with Si, B and Ti atoms, respectively, as test cases. For the SiC_{2} sheet, three energetically nearly degenerate allotropes with very different electronic structures are identified, which also differ from the previously proposed SiC_{2}silagraphene^{27}, showing that tuning the orientation of the C_{2} dimers not only can modulate the size of the band gap, but also can induce the transition from an indirect gap to a direct gap. For the BC system, the ground state structures of both BC_{2} and B_{2}C_{2} sheets are determined. Due to the different chemical nature of B and Si, all the possible structures of BC_{2} isostructural to those of SiC_{2} are unstable. In the ground state configuration of BC_{2}, each C_{2} dimer is bonded with three B atoms forming a 2D sheet with a thickness is 1.53 Å and a band gap of 4.1 eV. When increasing the number of atomic B to form B_{2}C_{2}, each C_{2} dimer is bonded to six B atoms having the maximum number of BC σ bonds and reducing the band gap to 2.9 eV. Unlike the metallic BC sheet^{33} where C atoms do not form any dimers, both the BC_{2} and B_{2}C_{2} sheets identified here are semiconducting. The emergence of band gap is attributed to the buckled structures, which prevents the delocalization of p_{z} orbitals. For the TiC system, a new structure of the TiC_{2} sheet with a lower energy than the metallic TiC_{2} sheet reported recently^{40} was found. A Ti_{2}C_{2} sheet was also found to be energetically more stable than the previously proposed tTiC sheet^{39}, suggesting it is more favorable for Ti to bind with C_{2} dimers in forming 2D structures. Unlike the metallic TiC_{2} monolayer^{40}, the Ti_{2}C_{2} sheet is an indirect band gap semiconductor. In addition, compared with the structures composed of C_{2} dimers and nonmetallic elements (Si, B), both the stable TiC_{2} and Ti_{2}C_{2} sheets adopt relatively more compact structures due to the complex orientations of d orbitals, which require a more extensive search to find the optimal orientation of the C_{2} dimers. These cases show that our searching algorithm is efficient and indispensable to design new 2D materials beyond the atombased ones.
Methods
Dimerbased search algorithm
Particle Swarm Optimization (PSO), one of the most popular swarm intelligence algorithms, was originally developed by Kennedy and Eberhart in 1995^{45}. The PSO algorithm utilizes the collective intelligence of the whole population rather than a single individual, it is quite efficient in handling various optimization problems. Recently, the PSO algorithm has been successfully applied to crystal structure prediction^{46,47,48}, another intractable optimization problem. Just like in the natural world where collective intelligence facilitates the locating of food or optimal habitats for insects and fishes, PSO implementation in CALYPSO was proven to be efficient in predicting the most stable structures for undiscovered materials^{48}.
Due to the great success of PSO in predicting crystal structures, we have studied a dimerbased structure search program based on the PSO algorithm, where additional constraints and augmented algorithm are added. The main framework of our dimerbased structure search method is illustrated by the flowchart as shown in Fig. 5.
Before outlining the details of the steps given in the flowchart, we will explain the treatment of the C_{2} dimer in our constrained search method. The requirements are that all carbon atoms exist in the form of a C_{2} dimer and each dimer must be separated from the others. A C_{2} dimer can be treated naturally as a “pseudoatom”. However, general treatments for ordinary atoms are not sufficient for “pseudo atoms”, as an ordinary atom is treated as an isotropic sphere, while a C_{2} dimer is considered a pseudo atom exhibiting structural anisotropy along the molecular axis. Therefore, to fully represent a C_{2} dimer, extra indices are imperative. As shown in Fig. 6, besides the three coordinates, x, y and z, the dimeric orientation angles φ and θ are required. Therefore, in our augmented PSO algorithm, the coordinates needed are r_{i} = (x_{i}, y_{i}, z_{i}, φ_{i}, θ_{i}), where x_{i}, y_{i} and z_{i} are used to denote the barycenter, φ_{i} and θ_{i} are used to represent the orientation of the dimer. Besides, during the PSObased structure searches, the bond length of C_{2} dimer is allowed to vary within the length of CC single and triple bond. Accordingly, the PSO algorithm is modified to accommodate these changes.
There are four main steps enumerated in the flowchart, some operative techniques are directly adapted from previous works^{46,48}. First of all, most of the trial structures are partially randomly generated under symmetry restrictions^{46,49,50} or evolved via the PSO algorithm^{48,51}. Fully randomly generated structures are also used. The reason is in 2D space there are 17 plane groups overall belonging to four different crystallographic systems (oblique, rectangular, square and hexagonal); one of the groups is selected each time to create a new trial structure. The lattice shape of the new structure is determined by the selected space group and its area. Generally, the positions of the atoms or the barycenter of C_{2} dimer are allocated on Wyckoff positions for the chosen plane group. An extra step is needed to generate bond length and orientation for each dimer randomly or partially restricted, which is different from atombased structure prediction methods. All these steps are sketched in Fig. 7. In order to predict 2D materials with finite thickness, the coordinate z_{i} of atoms and dimers are assigned randomly within a thickness Δ as shown in Fig. 7(d). However, because the number of 2D plane groups is finite, the generating method for a symmetry constrained structure is prone to produce similar structures. In this case totally randomly generated structures are needed to guarantee the diversity of structures.
Another source of the new trial structures is the PSO evolution which is a populationbased algorithm. The population in our case is defined as a group of structures. For generation t, we define a population P_{t} and every structure in generation t is represented as a particle P_{i}^{t}. The evolution of structures is controlled by PSO algorithm:
where i and t are indexes of particle (structure) and generation, represents the structural information of a certain structure including the orientation of dimers and the coordinates of atoms, is the velocity or rate of structural change of certain structure i. The velocity of structure i is calculated as follows:
where ω is the inertia weight and controls the momentum of the particle, is the optimized structure of particle i and is the structure with the lowest energy obtained so far for generation t, r_{1} and r_{2} are two random numbers uniformly distributed in range [0, 1], c_{1} and c_{2} are set to be 2^{47,52}. The definition of velocity fully reflects the idea of collective intelligence. The trajectory of each particle is biased, to a certain degree, toward the best structure of whole population and itself at the same time.
The three structure generation approaches work synergistically and serve different purposes during the structure search. The partially random structure generation with symmetry constraints help reduce the degree of freedom and narrow the search space. The fully random structure generation warrants the structural diversity and the PSObased structure evolution provides an effective way to explore the configuration space and locate the thermodynamically favorable structures.
It is important to note here that due to the nonequivalent coordinates x_{i}, y_{i}, z_{i} and φ_{i}, θ_{i} in terms of their magnitudes and range, C_{2} dimers and ordinary atoms must be manipulated separately. In addition, the newly generated trial structures must satisfy the condition that all C_{2} dimers are disconnected from each other, thus the structures with dimer clustering are eliminated. Next, similarities between the different structures are examined. Each structure is characterized by a set of predetermined fingerprinting functions^{48,52} and the distance between two structures are calculated based on the difference of their fingerprinting functions. If a newly produced structure is similar to the previous one, the new structure is discarded. Finally, local structure optimization is performed to drive the structure to the local minimum on multidimensional energy space, which also provides physical and rational structures for future PSO evolution.
First principles calculation
Combined with our bondingrestricted algorithm, first principles calculations within the framework of density functional theory (DFT) are performed for local geometry optimizations during structure search. Calculations are mainly performed by using the Vienna Ab initio Simulation Package (VASP)^{53}. The PerdewBurkeErnzerhof (PBE) functional^{54} is used to incorporate the exchangecorrelation interaction. The HSE06 hybrid functional^{55,56} is used for high accuracy of electronic structure calculations. The projector augmented wave (PAW)^{57} method is used to treat the interactions between ion cores and valance electrons. Plane waves are used to expand the valance electron (2s^{2}2p^{1} for B, 2s^{2}2p^{2} for C, 3s^{2}3p^{2} for Si and 3d^{3}4s^{1} for Ti) wave functions. During the massive structure searching stage, in order to reduce the workload, plane waves with a kinetic energy cutoff of 350 eV and MonkhorstPack scheme with a sparse grid density (2π × 0.04 Å^{−1}) are adopted. To compare the relative stability of different candidate structures, we used the same kinetic energy cutoff, kpoint grid density and exchangecorrelation functionals to perform geometry optimizations and total energy calculations. Plane waves with a kinetic energy cutoff of 450 eV and MonkhorstPack scheme^{58} with a grid density of 2π × 0.02 Å^{−1} are used to optimize the structures and calculate their electronic properties. The convergence criteria for total energy and forces are set to be 10^{−4} eV and 10^{−3} eV/Å, respectively. The supercell approach is used to calculate force constants. A vacuum space of 20 Å in the perpendicular direction is added to avoid the interaction between periodic images. Phonon properties are calculated using the finite displacement method as implemented in the Phonopy package^{59,60}.
Additional Information
How to cite this article: Zhang, C. et al. Bondingrestricted structure search for novel 2D materials with dispersed C_{2} dimers. Sci. Rep. 6, 29531; doi: 10.1038/srep29531 (2016).
References
Kirkpatrick, S., Gelatt, C. D. & Vecchi, M. P. Optimization by Simulated Annealing. Science 220, 671–680 (1983).
Abraham, N. & Probert, M. A periodic genetic algorithm with realspace representation for crystal structure and polymorph prediction. Phys. Rev. B 73, 224104 (2006).
Trimarchi, G. & Zunger, A. Global spacegroup optimization problem: Finding the stablest crystal structure without constraints. Phys. Rev. B 75, 104113 (2007).
Wales, D. J. & Doye, J. P. K. Global Optimization by BasinHopping and the Lowest Energy Structures of LennardJones Clusters Containing up to 110 Atoms. J. Phys. Chem. A 101, 5111–5116 (1997).
Zhu, Q., Sharma, V., Oganov, A. R. & Ramprasad, R. Predicting polymeric crystal structures by evolutionary algorithms. J. Chem. Phys. 141, 154102 (2014).
Cederbaum, L., Domcke, W. & Von Niessen, W. Manybody calculation of electron affinities: C2 and a prediction for P2 . J. Phys. B 10, 2963 (1977).
Shaik, S., Rzepa, H. S. & Hoffmann, R. One molecule, two atoms, three views, four bonds? Angew. Chem. Int. Ed. 52, 3020–3033 (2013).
Shaik, S. et al. Quadruple bonding in C2 and analogous eightvalence electron species. Nat. Chem. 4, 195–200 (2012).
Wang, Q., Ng, M.F., Yang, S.W., Yang, Y. & Chen, Y. The mechanism of singlewalled carbon nanotube growth and chirality selection induced by carbon atom and dimer addition. ACS Nano 4, 939–946 (2010).
Sternberg, M., Zapol, P. & Curtiss, L. A. Carbon dimers on the diamond (100) surface: Growth and nucleation. Phys. Rev. B 68, 205330 (2003).
Wu, P. et al. Carbon Dimers as the Dominant Feeding Species in Epitaxial Growth and Morphological Phase Transition of Graphene on Different Cu Substrates. Phys. Rev. Lett. 114, 216102 (2015).
Riikonen, S., Krasheninnikov, A., Halonen, L. & Nieminen, R. The Role of Stable and mobile carbon adspecies in copperpromoted graphene Growth. J. Phys. Chem. C. 116, 5802–5809 (2012).
Skell, P. S. & Plonka, J. H. Chemistry of the singlet and triplet C2 molecules. Mechanism of acetylene formation from reaction with acetone and acetaldehyde. J. Am. Chem. Soc. 92, 5620–5624 (1970).
Artyukhov, V. I., Liu, Y. & Yakobson, B. I. Equilibrium at the edge and atomistic mechanisms of graphene growth. Proc. Natl. Acad. Sci. 109, 15136–15140 (2012).
Artyukhov, V. I., Penev, E. S. & Yakobson, B. I. Why nanotubes grow chiral. Nat. Commun. 5, 4892 (2014).
Irle, S., Zheng, G., Elstner, M. & Morokuma, K. From C2 molecules to selfassembled fullerenes in quantum chemical molecular dynamics. Nano Lett. 3, 1657–1664 (2003).
Long, N. J. & Williams, C. K. Metal Alkynyl σ Complexes: Synthesis and Materials. Angew. Chem. Int. Ed. 42, 2586–2617 (2003).
Yam, V. W.W. Molecular Design of Transition Metal Alkynyl Complexes as Building Blocks for Luminescent MetalBased Materials: Structural and Photophysical Aspects. Accounts. Chem. Res. 35, 555–563 (2002).
Guo, B., Kerns, K. & Castleman, A. Jr. Ti8C12^{+}MetalloCarbohedrenes: A New Class of Molecular Clusters? Science 255, 1411 (1992).
Rohmer, M.M., Bénard, M. & Poblet, J.M. Structure, reactivity and growth pathways of metallocarbohedrenes M8C12 and transition metal/carbon clusters and nanocrystals: A challenge to computational chemistry. Chem. Rev. 100, 495–542 (2000).
Buschbeck, R., Low, P. J. & Lang, H. Homoleptic transition metal acetylides. Coordin. Chem. Rev. 255, 241–272 (2011).
Qin, X. et al. Origin of Dirac Cones in SiC Silagraphene: A Combined Density Functional and TightBinding Study. J. Phys. Chem. Lett. 6, 1333–1339 (2015).
Sun, X.H. et al. Formation of silicon carbide nanotubes and nanowires via reaction of silicon (from disproportionation of silicon monoxide) with carbon nanotubes. J. Am. Chem. Soc. 124, 14464–14471 (2002).
Bekaroglu, E., Topsakal, M., Cahangirov, S. & Ciraci, S. Firstprinciples study of defects and adatoms in silicon carbide honeycomb structures. Phys. Rev. B 81, 075433 (2010).
Zhao, K., Zhao, M., Wang, Z. & Fan, Y. Tightbinding model for the electronic structures of SiC and BN nanoribbons. Physica. E 43, 440–445 (2010).
Gori, P., Pulci, O., Marsili, M. & Bechstedt, F. Sidedependent electron escape from grapheneand graphanelike SiC layers. Appl. Phys. Lett. 100, 043110 (2012).
Li, Y., Li, F., Zhou, Z. & Chen, Z. SiC2 silagraphene and its onedimensional derivatives: where planar tetracoordinate silicon happens. J. Am. Chem. Soc. 133, 900–908 (2010).
Li, Y. et al. A new C = C embedded porphyrin sheet with superior oxygen reduction performance. Nano. Res. 8, 2901–2912 (2015).
Joswig, J.O. & Springborg, M. The influence of C2 dimers on the stability of TimCn metcar clusters. J. Chem. Phys. 129, 134311–134311 (2008).
LopezBezanilla, A. & Littlewood, P. B. σ–πBand Inversion in a Novel TwoDimensional Material. J. Phys. Chem. C. 119, 19469–19474 (2015).
Kouvetakis, J., Kaner, R. B., Sattler, M. L. & Bartlett, N. A novel graphitelike material of composition BC3 and nitrogencarbon graphites. J. Chem. Soc., Chem. Commun. 1758–1759 (1986).
Wu, X., Pei, Y. & Zeng, X. C. B2C graphene, nanotubes and nanoribbons. Nano Lett. 9, 1577–1582 (2009).
Luo, X. et al. Predicting twodimensional boron–carbon compounds by the global optimization method. J. Am. Chem. Soc. 133, 16285–16290 (2011).
Silvi, B. & Savin, A. Classification of chemical bonds based on topological analysis of electron localization functions. Nature 371, 683–686 (1994).
Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004).
Tang, H. & IsmailBeigi, S. Novel precursors for boron nanotubes: the competition of twocenter and threecenter bonding in boron sheets. Phys. Rev. Lett. 99, 115501 (2007).
Galeev, T. R. et al. Deciphering the mystery of hexagon holes in an allboron graphene αsheet. Phys. Chem. Chem. Phys. 13, 11575–11578 (2011).
Wei, S. & Guo, B. Metallocarbohedrenes: Formation of multicage structures. Science 256, 818–820 (1992).
Zhang, Z., Liu, X., Yakobson, B. I. & Guo, W. Twodimensional tetragonal TiC monolayer sheet and nanoribbons. J. Am. Chem. Soc. 134, 19326–19329 (2012).
Zhao, T., Zhang, S., Guo, Y. & Wang, Q. TiC2: a new twodimensional sheet beyond MXenes. Nanoscale 8, 233–242 (2016).
Naguib, M., Mochalin, V. N., Barsoum, M. W. & Gogotsi, Y. 25^{th} Anniversary Article: MXenes: A New Family of TwoDimensional Materials. Adv. Mater. 26, 992–1005 (2014).
Naguib, M. et al. Twodimensional transition metal carbides. ACS Nano 6, 1322–1331 (2012).
Naguib, M. et al. TwoDimensional Nanocrystals: TwoDimensional Nanocrystals Produced by Exfoliation of Ti3AlC2 . Adv. Mater. 23, 4207–4207 (2011).
Berkdemir, C., Castleman, A. & Sofo, J. O. Metalsubstituted Ti8C12 metallocarbohedrynes: toward less reactive clusters as building blocks of clusterassembled materials. Phys. Chem. Chem. Phys. 14, 9642–9653 (2012).
Kennedy, J. In Encyclopedia of Machine Learning 760–766 (Springer, 2010).
Wang, Y. & Ma, Y. Perspective: Crystal structure prediction at high pressures. J. Chem. Phys. 140, 040901 (2014).
Wang, Y., Lv, J., Zhu, L. & Ma, Y. CALYPSO: A method for crystal structure prediction. Comput. Phys. Commun. 183, 2063–2070 (2012).
Wang, Y., Lv, J., Zhu, L. & Ma, Y. Crystal structure prediction via particleswarm optimization. Phys. Rev. B 82, 094116 (2010).
Wu, S. et al. An adaptive genetic algorithm for crystal structure prediction. J. Phys. Condens. Matter 26, 035402 (2014).
Liu, Z.L. MUSE: Multialgorithm collaborative crystal structure prediction. Comput. Phys. Commun. 185, 1893–1900 (2014).
Call, S. T., Zubarev, D. Y. & Boldyrev, A. I. Global minimum structure searches via particle swarm optimization. J. Comput. Chem. 28, 1177–1186 (2007).
Gao, B., Shao, X., Lv, J., Wang, Y. & Ma, Y. Structure Prediction of Atoms Adsorbed on TwoDimensional Layer Materials: Method and Applications. J. Phys. Chem. C. 119, 20111 (2015).
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169 (1996).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
Heyd, J., Scuseria, G. E. & Ernzerhof, M. Hybrid functionals based on a screened Coulomb potential. J. Chem. Phys. 118, 8207–8215 (2003).
Heyd, J., Scuseria, G. E. & Ernzerhof, M. Erratum: “Hybrid functionals based on a screened Coulomb potential”. J. Chem. Phys. 124, 219906 (2006).
Blöchl, P. E. Projector augmentedwave method. Phys. Rev. B 50, 17953 (1994).
Monkhorst, H. J. & Pack, J. D. Special points for Brillouinzone integrations. Phys. Rev. B 13, 5188–5192 (1976).
Togo, A. & Tanaka, I. First principles phonon calculations in materials science. Scripta Mater. 108, 1–5 (2015).
Togo, A., Oba, F. & Tanaka, I. Firstprinciples calculations of the ferroelastic transition between rutiletype and CaCl2type SiO2 at high pressures. Phys. Rev. B 78, 134106 (2008).
Acknowledgements
This work is partially supported by grants from the National Natural Science Foundation of China (NSFC51471004), the National Grand Fundamental Research 973 Program of China (Grant No. 2012CB921404) and the Doctoral Program of Higher Education of China (20130001110033). The calculations were carried out at the National Supercomputer Center in Guangzhou, China.
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Q.W. designed the project; C.Z. performed the calculations; C.Z., S.Z. and Q.W. analyzed the results and wrote the manuscript.
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Zhang, C., Zhang, S. & Wang, Q. Bondingrestricted structure search for novel 2D materials with dispersed C_{2} dimers. Sci Rep 6, 29531 (2016). https://doi.org/10.1038/srep29531
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DOI: https://doi.org/10.1038/srep29531
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