Origins of Dirac cone formation in AB3 and A3B (A, B = C, Si, and Ge) binary monolayers

Compared to the pure two-dimensional (2D) graphene and silicene, the binary 2D system silagraphenes, consisting of both C and Si atoms, possess more diverse electronic structures depending on their various chemical stoichiometry and arrangement pattern of binary components. By performing calculations with both density functional theory and a Tight-binding model, we elucidated the formation of Dirac cone (DC) band structures in SiC3 and Si3C as well as their analogous binary monolayers including SiGe3, Si3Ge, GeC3, and Ge3C. A “ring coupling” mechanism, referring to the couplings among the six ring atoms, was proposed to explain the origin of DCs in AB3 and A3B binary systems, based on which we discussed the methods tuning the SiC3 systems into self-doped systems. The first-principles quantum transport calculations by non-equilibrium Green’s function method combined with density functional theory showed that the electron conductance of SiC3 and Si3C lie between those of graphene and silicene, proportional to the carbon concentrations. Understanding the DC formation mechanism and electronic properties sheds light onto the design principles for novel Fermi Dirac systems used in nanoelectronic devices.

individual C atoms 28 . Recently the first-principles calculations predicted that SiC 7 silagraphene is a semiconductor with a direct band gap of 1.13 eV 29 . g-SiC 3 and g-Si 3 C are predicted to possess DC band structures 30,31 . Meanwhile, significant band gaps are opened and the band structures are topologically nontrivial after the introduction of spin-orbital coupling 30 .
DC featured band structures commonly lead to unique electronic properties. For example, the charge carrier mobility of graphene which possess DC band structure 32,33 can reach up to 10 7 cm 2 /(V s) 34 . Only a few 2D materials possess DCs. The pure 2D DC materials include graphyne 4,35 , square graphynes 36 , silicene 6,8 , germanene 8 , and borophene 37 . The binary 2D DC systems include g-SiC 3 , g-Si 3 C 30, 31 , t1-SiC, t2-SiC 25 , and silagraphye 38,39 . The modified 2D DC systems include 6(H 2 ), 14, 18 graphyne, 6 BN , 6, 12 graphyne 40 , janugraphene, chlorographene 41 , and hydrogenated and halogenated blue phosphorene 42 . The organic 2D DC systems include Mn 2 C 18 H 12 43 and Ni 2 C 24 S 6 H 12 44 . Despite many reports on 2D DC systems, fewer studies contribute to the origin of DC formation. Using a two bands model, Wang et al. summarized that the conditions of DC formation include specific symmetries, proper parameters, and a suitable Fermi level where there are only DC points and no other bands 45 . To understand the origin of DC of graphyne, it was clarified that the acetylenic linkages between vertexes atoms could be reduced to effective hopping terms whose combination decides the existence of DCs 46,47 . More recently, by performing calculations using both DFT and a tight binding (TB) model. We proposed "pair coupling" 25 and "triple coupling" 39 mechanisms to elucidate the origin of DC formation of t1-SiC and α-graphyne, showing different processes of DC formations.
In this work, by performing DF and TB calculations, we analyzed the formation process of band structures of g-SiC 3 and g-Si 3 C and elucidated the origin of DC formation by proposing a "ring coupling" mechanism referring to the couplings among the six same atoms forming a ring. On the basis of this mechanism, the conditions of the systems being self-doped were also discussed. Furthermore, we verified the "ring coupling" mechanism by studying analogous binary monolayers consisting of Ge and C as well as Ge and Si, showing DC featured band structures consistent with the results of Zhao et al. 30 . Finally, we calculated the electron transport properties of g-SiC 3 and g-Si 3 C using non-equilibrium Green's function method combined with density functional theory (NEGF-DFT), showing that the studied silagraphene exhibit electron conductance between silicene and graphene.

Results and Discussion
Atomic structures and stability of g-SiC 3 and g-Si 3 C. By geometry optimization using DFT calculation, the atomic structures of g-SiC 3 and g-Si 3 C (shown in Fig. 1) are acquired. They have planar forms with P6/ MMM symmetry. And they are both graphene-like but consisting of two elements, one of which forms 6-membered rings. The corresponding structure parameters and formation energies are listed in Table 1. For comparison, the results of graphene and silicene from our previous work 39 are also listed in Table 1.
To analyze the stability of the structures, we calculated two types of formation energy 39 . The first formation energy E f is defined as: indicating that the energy of g-SiC 3 or g-Si 3 C is higher than the ideal mixture of graphene and silicene with the same C/Si proportions as g-SiC 3 or g-Si 3 C.
We discussed the possibility of atomic segregation into the graphene and silicene nanoribbons with Si-C interfaces in section S2 of Supplementary Information.
To verify the structure stability, we carried out quantum molecular dynamics (MD) calculations at a canonical ensemble (NVT ensemble) at 600 K. The MD trajectories indicate that the atomic structures of g-SiC 3 and g-Si 3 C do not change significantly after 2.5 ps (See Figure S1 in Supplementary Information). Previous phonon calculations of g-SiC 3 and g-Si 3 C by Zhao et al. did not find modes with imaginary frequencies 30 . Ding et al. also verified the stability of g-SiC 3 by density-functional-based tight binding molecular dynamics simulations and phonon calculations 31 .
Band structures of g-SiC 3 and g-Si 3 C. In this work, the Brillouin zones of all the structures possess same models with hexagon shown in Fig. 1

(c).
Band structure of g-SiC 3 . The band structure of g-SiC 3 possesses DCs calculated by DFT as shown in  Table 1. For comparison, the electron/hole group velocities of graphene and silicene were also calculated and listed in Table 1. These values are the group velocities averaged over electrons and holes as well as different directions. The averaged group velocity of g-SiC 3 is lower than that of graphene but higher than that of silicene.
From the density of states (DOS) of g-SiC 3 ( Fig. 2(a)), the bands near Fermi energy mainly attribute to the p z orbitals of Si and C. So we constructed a TB model to reproduce the band structure by only considering the p z orbitals. For the sake of convenience, we translated properly the lattice of g-SiC 3 as Fig. 3, and labeled the vertex atoms A and B as well as the ring atoms 1-6. The TB Hamiltonian can be written as 30,39 : where E i is the onsite energy of the i-th atom, −t mn is the hopping energy between the n-th and m-th atom (only considering the nearest-neighboring atoms for simplicity), + a i and â i are creation and annihilation operators, respectively. The TB parameters are determined by fitting against DFT results 25 . The onsite energies of C and Si are E C = −1.090 eV and E Si = 2.459 eV, respectively. The hopping energies of C-C and C-Si are t C-C = 2.258 eV and t C-Si = 1.715 eV, respectively. The agreement between the TB and DFT results verifies the rationality of the TB model (See Fig. 2(a)).
To understand the origin of DC featured band structure of g-SiC 3 , we make the analysis based on a TB model as follows. For simplicity, we rewrite t C-Si as t, and t C-C as t C . where l or m are the atom labels for the six C ring atoms shown in Fig. 3, â 7 and + a 7 mean â 1 and + a 1 , respectively. The eigenfunctions are: ( 1, 2, , 6) (5) C j n l i j l C l n 1 6 3 where φ − C l n is the wave function of l-th C atom in the n-th cell. The conclusion that ϕ | 〉 − C j n is the eigenfunctions of Ĥ C can be verified as: The eigenvalues of ϕ − C 3 and ϕ − C 6 are There are four eigenvalues in total. Based on the wave functions ϕ | 〉 − C j n and the wave functions of Si atoms, we define the Bloch basis sets: 1, 2, , 6 (7) where ϕ − Si A n and ϕ − Si B n are the wavefunctions of Si atoms labeled A and B shown in Fig. 3. The eigenfunctions of this system are the linear superposition of these eight functions. With these eight functions as basis vectors in the order of The matrix H k ( ) of Hamilton operator Ĥ can be written as The diagonal elements are Refer to the off-diagonal elements, due to the couplings among six C ring atoms having been considered, ′ H jj ( ≠ ′ j j ) are all zero; and because the Si atoms at A and B are not neighbors, H AB is zero. So the non-zero elements of off-diagonal elements are only H jA and H jB as well as their conjugates. The non-zero elements are listed in Table 2 3 2 ( Fig. 1(a)). These deduction procedures are similar to the case of h-SiC 25 . From Table 2 Table 2. At the K point, some of the elements H jA and H jB are zeros. So we can divide the basis vectors in Eq. (10) into three groups so that the couplings at the K point only exist between the vectors from the same groups but not between the vectors from different groups: If the vector | ⟩ 1 k from the second group changes to −| ⟩ 1 k , the matrix of the second group is the conjugate of the matrix of the first group in total Brillouin zone. So after diagonalization of the first and second group respectively, three pairs of energy bands can be acquired, each pair of which are equal in total Brillouin zone, and the middle pair are located near the Fermi surface. Referring to the third group, because there is little coupling between 3 k and 6 k for any wave vector k, the two energy levels + E t 2 C C and − E t 2 C C remain unchanged in the total Brillouin zone when only considering the couplings within each group.
On the basis of analysis above, we divide the formation of DCs band structure into the following three steps conceptually to understand the origin of DCs band structure: (a) First, the couplings among the six C ring atoms generate six energy states, where two pairs of energy states are degenerated at , respectively, and the other two energy states are located at Including the two energy states of Si atoms degenerated at E Si , there are eight energy states in total. This process is shown in Fig. 4(a). (b) The eight energy states can be divided into three groups as mentioned above. And we only consider the intra-group couplings ignoring inter-group couplings in total Brillouin zone. Then after diagonalization, both of the first and the second group generate three bands making up three pairs of bands, among which, each pair are equal in total Brillouin zone, and the middle pair lie around the Fermi surface. As for the third group, two flat bands will be acquired due to little couplings between these two states. This process is shown in Fig. 4(b). (c) The inter-group couplings not considered above are included at this step. This makes the band gap to be generated except for the K points where no inter-group couplings exist. So the bands maintain touching at the K points and are separated in the other zones, resulting in the formation of DCs bands. This process is shown in Fig. 4(c).
On the basis of "ring coupling" mechanism, changing the TB parameters E C , E Si , t C-C , and t C-Si does not influence the formation of DC band structure. So this DC band structure is robust to change vertex element or ring element into other elements. This conclusion can be verified by the calculation of g-Si 3 C, g-GeC 3 , g-Ge 3 C, g-GeSi 3 , and g-Ge 3 Si later in this work. If the onsite energies of the two vertex atoms are not equal due to different types of vertex atoms, the three pairs of bands generated by the couplings within the first group and the second group  Conditions of g-SiC 3 -like systems possessing self-doped band structure. From the band structure of g-SiC 3 [ Fig. 2(a) corresponding to the energy levels C , E Si , and E Si , respectively. To acquire the values of VB and CB at the Γ point, the Hamiltonian matrix with the vectors in Eq. (14) as basis set is diagonalized at the Γ point with scanning E Si and other parameters remaining unchanged. The result is shownin Fig. 5(a). And for comparing the values of VB and CB at the Γ point with the DC point, we calculated the values of DC point with scanning E Si without changing the other parameters as shown in Fig. 5(a). We discuss the results as the follows.
(1) When E Si is near E C (−4.825 eV < E Si < 2.645 eV), the value of DC point is higher than the value of VB at the Γ and lower than the value of CB at Γ, so the DC point exists on the Fermi surface, and this system is a DC system. (2) When E Si is far from E C (E Si < −4.825 eV or E Si > 2.645 eV), the DC point deviated from the Fermi surface.  Decreasing the hopping energy between C and Si (t C-Si ) with hopping energy between C and C (t C-C ) as well as onsite energy of C (E C ) unchanged compared to g-SiC 3 . (c) Decreasing the hopping energy between C and C (t C-C ) with hopping energy between C and Si (t C-Si ) as well as the onsite energy of C (E C ) unchanged compared to g-SiC 3 .
Specially, when E Si < −4.825 eV, the value of CB at the Γ is lower than the value of DC point, leading to DC point higher than Fermi surface; while when E Si > 2.645 eV, the value of VB at the Γ is higher than the value of DC point, leading to DC point lower than Fermi surface.
(3) When E Si < −4.825 eV or E Si > 2.645 eV, but E Si is very close to −4.825 eV or 2.645 eV, the DC point deviated only slightly from the Fermi surface, leading to the formation of a self-doped system.
So, increasing the difference between the onsite energies of vertex atoms and ring atoms change the systems into self-doped systems. The calculations for g-GeC 3 and g-Ge 3 C later in this paper support this conclusion.
To examine the influence of hoping energy on the formation of self-doped systems, we changed the hoping energy − t C Si ( − t C C ), and performed the same calculations required in Fig. 5(a) to acquire Fig. 5(b) [Fig. 5(c)]. From Fig. 5 3.812] eV, in which the DC point is located on the Fermi level. So, decreasing − t C Si and increasing − t C C may change the system into self-doped system. Increasing (decreasing) bond length can mimic the decreasing (increasing) of hopping energy, so increasing the C-Si bond length and/or decreasing the C-C bond length may change g-SiC 3 into self-doped system, while increasing the C-C bond length and/or decreasing the C-Si bond length increase the difference between the value of Dirac point and the value of VB at the Γ point compared with the equilibrium system. We decreased (increased) the C-C bond length with 0.06 Å and increased (decreased) the C-Si bond length with 0.06 Å, keeping the lattice parameter unchanged; then calculated their band structures by DFT (Fig. 6). From Fig. 6, we found that: (1) When the C-C bond length is decreased by 0.06 Å and C-Si bond length is increased by 0.06 Å, with lattice parameter unchanged, the value of Dirac point (−0.121 eV) is lower than the value of VB at the Γ point (0.027 eV), forming a self-doped system [ Fig. 6(a)]. (2) When the C-C bond length is increased with 0.06 Å and C-Si bond length is decreased by 0.06 Å, with lattice parameter unchanged, the difference (0.242 eV) between the value of Dirac point (0.010 eV) and the value of VB at the Γ point (−0.232 eV) increases [ Fig. 6(b)] compared with the equilibrium system. (For the equilibrium system, the value of Dirac point is −0.027 eV, the value of valence at the Γ point is −0.067 eV, and the difference is 0.040 eV. [ Fig. 2(a)]) So the DF calculations support our TB analysis above, and we may tune the bond length of g-SiC 3 to change g-SiC 3 into a self-doped system by depositing the monolayers on appropriate substrates. Thermal vibrations around equilibrium atom positions are not expected to affect the self-doping behavior due to random features of bond length changes.
Band structure of g-Si 3 C. g-Si 3 C (Fig. 1(b)), possessing similar atomic structure as g-SiC 3 [ Fig. 1(a)], also displays DCs in band structure (Fig. 7) due to "ring coupling" mechanism referring to the couplings of six Si ring atoms. The group velocity of g-Si 3 C near Fermi surface is listed in Table 1 after averaged over electrons and holes as well as different directions. The electron/hole group velocity of g-Si 3 C is lower than that of graphene or g-SiC 3 and is similar to that of silicene. These results are related to the transport properties discussed later. Figure 8 shows the formation process of DCs band structure of g-Si 3 C similar to the formation process of DCs band structure of g-SiC 3 (Fig. 4). The TB parameters are obtained by fitting DFT results: the onsite energies of C and Si are E C = −2.113 eV and E Si = 0.428 eV, respectively; The hopping energies of Si-Si and C-Si are t Si-Si = 1.037 eV and t C-Si = 1.212 eV, respectively. Figure 8(a) shows differences opposite to Fig. 4(a): for Fig. 8(a) which is the "band structure" of g-Si 3 C, the band from vertex A (or B) lie out of the "other four bands" (the four energy levels from the coupling of the six same type atoms in a ring), while for Fig. 4(a) which is the "band structure" of g-SiC 3 , the band from vertex A (or B) lie among the "other four bands". This can be explained as follows: because the C-C coupling is stronger than the Si-Si coupling (t C-C > t Si-Si ), the differences between the highest band and the lowest band of the "other four band" for g-Si 3 C is smaller than g-SiC 3 , which results to the band from vertex A (or B) for g-Si 3 C laying out of the "other four bands".
Atomic structures and band structures of g-GeC 3 , g-Ge 3 C, g-GeSi 3 , and g-Ge 3 Si. Similar to g-SiC 3 and g-Si 3 C, substituting Si or C with Ge from the same main group in the periodic table, we constructed the binary models of g-GeC 3 , g-Ge 3 C, g-GeSi 3 , and g-Ge 3 Si. Their atomic structures optimized by DFT are shown in Fig. 9. Their atomic structure parameters and formation energy are listed in Table 3. For the purpose of comparison, we optimized the geometry structure of germanene. Due to silicene and germanene preferring to sp 3 hybridization, g-Ge 3 C, g-GeSi 3 , and g-Ge 3 Si are all buckled with non-planar structures. While, g-GeC 3 is a planar structure with all atoms in a plane because carbon prefer to sp 2 hybridization and this structure consists of more carbon atoms than g-Ge 3 C. These results are similar to the study of Zhao et al. except for the g-Ge 3 C (a planar structure in their studies) 30 . As shown in Tables 1 and 3, the formation energy E f of g-GeC 3 , g-Ge 3 C, g-GeSi 3 , and g-Ge 3 Si decreases gradually, which can be understood by the fact that the formation energy E f decreases in the order of graphene (9.23 eV), silicene (4.77 eV) and germanene (4.03 eV) and the energy E f of silicene is very close to germanene.
The g-GeC 3 , g-Ge 3 C, g-GeSi 3 , and g-Ge 3 Si systems also possess DCs band structures (Fig. 10) due to possessing similar atomic structures as g-SiC 3 and g-Si 3 C, as well as all elements belong to IV group. A small difference is that the Dirac points of g-GeC 3 and g-Ge 3 C deviate slightly from Fermi surface, leading to the formation of self-doped systems. Specifically, as for g-GeC 3 , because the value of VB at the Γ point is higher than the value of Dirac point, the DC point is lower than Fermi surface. According to the discussions above, it is understood that the difference between the onsite energies of Ge and C atom is larger than that between Si and C atom as well as the hopping energy between Ge and C atom is smaller than that between Si and C atom. And referring to g-Ge 3 C, the value of CB at the Γ point is lower than the Dirac point, so the Dirac point is higher than Fermi surface. Because the atomic structure of g-Ge 3 C is buckled, the p z orbitals may be coupled to the other orbitals, and the band near Fermi surface may include the other orbitals except for the p z orbitals, leading to the formation of self-doped system. From the band structure of planar g-Ge 3 C calculated by Zhao et al. 30 , there is a band from non-p z orbitals near Fermi surface, consistent with our analysis. These results agree with the studies of Zhao et al. 30 .
When the C atom of g-Si 3 C is substituted by B, N, Al, or P, the atomic model structures of XSi 3 (X = B, N, Al, or P) can be constructed. When the Si atom of g-SiC 3 is substituted by B, BC 3 can be constructed. According to the analyses above, these structures should also possess DC band structures. However, because the numbers of the valence electrons of these structures are different from g-SiC 3 or g-Si 3 C, the DCs of these structures are either above or under the Fermi surface. Previous studies support this discussion 48, 49 . Electron transport properties of g-SiC 3 and g-Si 3 C nanoribbons. For the potential nanoelectronic device applications, we calculated directly the electron transport properties of g-SiC 3 and g-Si 3 C nanoribbons.
The current density versus voltage curves were calculated and shown in Fig. 11 for the lead-molecule-lead junctions. Here we showed the current density, dividing the current by the surface area of electrode. The current density and voltage have nearly linear relationship over the bias voltages ranging from 0 to 2.0 V. We found that the current of g-SiC 3 is larger than g-Si 3 C, both of which are smaller than graphene but larger than silicenes in both bulked and planar forms. It is known that graphene has larger electron conductance than silicene 50 . The studied binary monolayers have conductance between graphene and silicene and the conductance increases as the C concentration increases ( Figure S5 in Supplementary Information). Table 4. lists the electron conductance of the systems under various bias voltages. These electron transport results are consistent with the electron/hole group velocities calculated from the band structures shown earlier.      Table 3. Bond lengths l (Å), lattice parameters a (Å), size of buckle d z (Å) and formation energies per atom [E f and ′ E f (eV)] of g-GeC 3 , g-Ge 3 C, g-GeSi 3 , g-Ge 3 Si, and germanene.

Conclusions
In this work we proposed a "ring coupling" mechanism to illustrate the formation of DCs of g-SiC 3 and g-Si 3 C as the examples of binary monolayers AB 3 and A 3 B (A, B = C, Si, and Ge): (1) the couplings of six C ring atoms form six new wave functions corresponding to four energy levels. The middle two energy levels are doubly degenerated, respectively. (2) The two wave functions of Si and the four wave functions corresponding to the middle two doubly degenerated levels are divided into two groups; each group contains one wave function of Si and two wave functions each from the two different doubly degenerated wave functions. The intra-group coupling of each group forms three bands, and there are six bands in total from these two groups. The six bands make up three pairs, and each pair are equal at the K point. The rest two of the six functions from the couplings of six C ring atoms form two flat bands (they are the third group). (3) After considering the inter-group couplings among the three groups, the gap is formed. However, there are no inter-group couplings at the K point where the bands remain contact, leading to the formation of DCs. Based on this "ring coupling" mechanism, the possible methods changing the g-SiC 3 -like monolayers into self-doped systems are discussed: (a) Increasing the difference between the onsite energies of ring atom and vertex atom, (b) decreasing the hopping energies between the ring atom and vertex atom. (c) increasing the hopping energy between the two ring atoms.
The "ring coupling" mechanism proposed in this work is applicable to 2D DC materials possessing ring patterns. We previously also studied other typical 2D structures. Specifically, we used the "pair coupling" mechanism to explain DC formation in 2D materials with paired atoms, e.g. t1-SiC 25 . Moreover, we proposed the "triple coupling" mechanism to understand DC formation in α-grahynes where triple atom-chains were coupled first 39 . The "ring coupling", "pair coupling", and "triple coupling" mechanisms share the similar methodology but account for various arrangement patterns in understanding the general mechanism of Dirac cone formation in 2D materials, thus they can be unified into a more general framework called "divide-and-couple", which can be applied to illustrate the origins of Dirac cone formation in other Fermi Dirac systems.
Method and computational details. In this work, the DFT calculations were carried out using the Vienna ab initio simulation package (VASP) 51,52 . The exchange-correlation function and pseudopotentials adopted the form of Perdew-Burke-Ernzerh (PBE) within a generalized gradient approximation (GGA) 53 and the projector augmented-wave (PAW) method 54 respectively. For binary 2D systems, we adopted 700 eV energy cutoff for the expansion of plane wave basis set and (7 × 7 × 1) for Monkhorst-Pack sampling, leading to convergence of 0.001 eV. For unitary 2D systems, the (17 × 17 × 1) Monkhorst-Pack grid was used. The SCF calculations converge to 5.0 × 10 −7 eV/atom, while the geometry optimizations converge to 5.0 × 10 −6 eV/atom using conjugated gradient method. The QMD calculations were carried out with a 700 eV energy cutoff, a (5 × 5 × 1) Monkhorst-Pack k-point sampling, and the SCF tolerance 1.25 × 10 −7 eV/atom. The vacuum region among layers is longer than 15 Å to avoid the influences among periodic images.
To evaluate the electron transport properties for their potential applications as electronic devices, we calculated the current-voltage (I-V) characteristics, electron transmission spectrum, and density of states of g-SiC 3 and g-Si 3 C, compared with graphene and silicene in both bulked and planar forms using ab initio modeling package nanodcal 55,56 . Figure S4(a) in the Supplementary Information illustrates the lead-molecule-lead junction with the semi-infinite Au lead. We first optimized the molecule-electrode distances using the DMol 3 program. The Perdew-Burke-Ernzerhof (PBE) functional under General Gradient Approximation (GGA) was adopted with double-ζ polarization basis set and DFT Semi-core pseudopotentials 57,58 . T = 300 K was used for the Fermi-Dirac distribution around Fermi level throughout the work of this section. Figure S4(b-f) shows the configurations of the optimized junctions.