Theoretical Study of Novel B2CO Compounds with High Hardness

Two B 2 CO phases (oP8’ with sp 2 -sp 3 hybridization coexist and mP16 with unitary sp 3 hybridization) were discovered via structure searching and stability analysis. The study of formation enthalpy reveals that high pressure (HP) technology performed maybe an important method for synthesis. Among all isoelectronic with diamond (IED) B-C-O phases, oP8’ has the smallest gap and mP16 has the widest gap. With pressure increasing, for B 2 CO phases with high symmetry and composed of sp 3 hybridization, their band gaps all increases monotonically; for B 2 CO phases composed by sp 3 -sp 2 hybridization coexist or with low symmetry like mP16, their band gaps increased rst and then decreased. oP8’ and mP16 both have large mechanical modulus and they are typical materials with high hardness. Pressure has a positive correlation with its mechanical modulus.


Introduction
As an important role in modern precision machining such as drilling, cutting, grinding, etc, hard materials have important scienti c research signi cance and industrial application value, therefore attracting continuous attention. [1][2][3] Materials composed of light elements (boron, carbon, nitrogen, oxygen) have the characteristics of light weight, strong adhesion and high hardness, such as diamond, cubic boron nitride and boron carbide, boron suboxide, etc., so they have the potential to become superhard materials and have been a research hotspot for a long time.. [4][5][6][7][8] Since the synthesis of B-C-O compounds in large volume press (LVP) high pressure experiments, [9][10][11]  theoretically designed as potential superhard materials through the particle swarm optimization method for crystal structure prediction. tP4-B 2 CO and tI16-B 2 CO stabilizes in diamond-like structures with typically strong covalent sp 3 B-C and B-O bonds. The calculated Vickers hardness of B 2 CO is ∼50 GPa. The research opens up a possibility to quest superhard materials within the ternary B-C-O system. [12] Thereafter, a lonsdaleitelike orthorhombic structure B 2 CO (oP8-B 2 CO) with mechanical, dynamical and thermodynamic stabilities have been predicted theoretically by the crystal structure prediction package, and the results indicate that pressure may promote the synthesis of oP8-B 2 CO. The strong sp 3 covalent B-C and B-O bonds bene t oP8-B 2 CO to possess high hardness of 47.70 GPa. [13] Whereafter the structural, electronic, and mechanical anisotropy properties of oP8-, tI16-, and tP4-B2CO under pressure are systematically theoretical studied. [14] The electronic band structures indicate that they are all indirect and wide semiconductor materials, and their band gaps all increase with the increase of pressure.
Considering the important effect of structure on performance, and also encouraged by the structural characteristic of newly predicted B-C-O (both tP4-B 2 CO and tI16-B 2 CO share diamond-like structures, [12] oP8-B 2 CO has lonsdaleite-like structure [13]), two orthorhombic B 2 CO phases (oP16-B 2 CO and oC16-B 2 CO) with con rmed stability have been predicted via manual construction, [15] and due to the structural genetic, they have similar structures to Cco-C8 and Bct-C4, respectively. Based on rst-principles calculations, oP16-and oC16-B 2 CO are both superhard materials with indirect band gaps.
Even the structure design has achieved great success, the most stable structure of B 2 CO in the ground state remains a mystery until a novel orthorhombic structure oI16-B 2 CO was predicted as the B 2  The increased C ratio will lead to a stacking of C in the structure, forming shorter sp 3 C-C bonds, so B 2 CxO (x ≥ 2) might have greater hardness values than B 2 CO does. [12] Inspired by this notion, three potential ultraincompressible and thermodynamically stable B 2 CxO (x = 2, 3, 5) phases were introduced via systematic particle swarm optimization algorithm structure search.
[18] By evaluating the trends of the crystal con guration, electronic structure, and mechanical properties as a function of the C concentration, it is found that the high carbon concentration bene ts the formation of the sp 3 C-C covalent bonds and leads to the enhanced elastic moduli, hardness and ideal strengths in these B 2 CxO compounds.
There is also signi cant progress on B-C-O compounds of non-IED. In 2016, a body-centered tetragonal structure B 4 CO 4 with high hardness was predicted by ab initio variable-composition evolutionary simulations, which is thermodynamically stable at high pressure above 23 GPa. [19] Based on the calculation of the ideal tensile and shear strengths along the principal crystal directions, the obtained results indicate that the shear mode along (001)[100] slip system dominates the plastic deformation of B 4 CO 4 , and the weakest ideal shear strength of 27.5 GPa demonstrates that the B 4 CO 4 compound is indeed a hard material. [20] Also two lowenthalpy metastable compounds B 6 C 2 O 5 and B 2 CO 2 with high pressure stability to 20 GPa have been discovered. [19] The hardness of B 6 C 2 O 5 and B 2 CO 2 are 29.6 GPa and 41.7 GPa, respectively. [21] Two years later, a tetragonal B-C-O compound (tI12-B 6 C 4 O 2 with I4m2 symmetric structure) was reported. [22] And the research about the mechanical properties reveals that tI12-B 6 C 4 O 2 possesses hard nature with hardness of 21.9 GPa. [22] Previous theoretical studies have made good progress in predicting high-performance structures, however, due to the lack of new B-C-O compounds, the study on the properties of B-C-O compounds, especially the consistency of sp 2 -sp 3 coexisting and low space group (SG) B 2 CO structures hinders the further exploration of the relationship between B 2 CO phase and properties. In this paper, we focus on B 2 CO compounds. Undergo rigorous structural stability analysis including mechanical and dynamical stabilities, two novel phases were discovered (oP8' and mP16). Together with these previously reported structures, and the optimize reactants and high pressure conditions were obtained based on the formation enthalpies calculation, which will provide guidance for high pressure experimental synthesis. Based on density function theory, the mechanical and electronic properties of all B 2 CO compounds have been systematically studied, and the regulatory effect of pressure on properties has also been further investigated.

Computational Methods And Details
The potential structures of B 2 CO compounds have been generated through evolutionary simulation methods CALYPSO [23][24][25] and USPEX [26-28] with xed components. The following full relaxation geometry optimizations were performed in CASTEP models [29]. The gradient-corrected functionals was adopted Perdew-Burke-Ernzerhof (PBE) form. [30] The ultrasoft pseudopotentials were adopted with energy cutoff as 380 eV and the tolerance of electronic self-consistent eld is 5.0 ⋅ 10 − 7 eV/atom, and the K-Points of oP8' and mP16 are 10⋅4⋅7 and 6⋅5⋅5 at atmospheric pressure. [31] During the geometry optimizations, [32] iterations were continued until energy change didn't beyond 5⋅10 − 3 meV per atom; force tensor on atoms was less than 0.1eV/nm; displacement on atom no more than 5×10 − 5 nm and stress didn't exceed 20 MPa. The phonon calculation was implemented via the nite displacement method. [33] The elastic constants were calculated with 9 patterns generated for each strain and max strain amplitude as 0.009.
The unit cells were adopted within the following research, and the symmetry points and their path for the Brillouin zone of oP8' and mP16 are G(0, 0, 0)→Z(0, 0, 0.  Fig. 1. For the rst phase with SG Pmn21 is primitive orthorhombic structure containing 8 atoms per unit cell. In order to distinguish from the previously proposed oP8-B 2 CO, here denoted it as oP8'-B 2 CO (oP8' for short). As shown in Fig. 1a, there are two classes of B atoms (B1 and B2) that are not equivalent in oP8', the Y coordinates of B1 atom are 0.049 or 0.951, which almost located at the cell boundary, while the Y coordinates of B2 atom are 0.380 or 0.620. For B1 atoms, they just bind with three C atoms by sp 2 hybridization and form BC3 triangular con guration, however for B2 atoms, they connect with three O atoms and one C atom by sp 3 hybridization and form tetrahedron. The C atoms all have 4 coordination with four neighboring atoms as three B1 atoms and one B2 atom, while for O atoms all bonding with three B2 atoms by sp 2 hybridization. In other word, there are sp 2 -sp 3 hybridization forms coexist in oP8', which is quite different from oP8, which is just sp 3 hybridization. There are holes in the oP8' structure that run along the X-axis, result in its low density of 2.813 g/cm 3 .
For the second B 2 CO phase, it belongs to the primitive monoclinic with the SG of P21/c, and the unit cell contains 16 atoms, hence denoted as mP16-B 2 CO (mP16 for short). There are also two kinds of B atoms (B1 and B2) that not equivalent in mP16, the X coordinates of B1 atom is 0.180 or 0.820, and the X coordinates of B2 atom are 0.338 or 0.662. Different from the oP8' structure, all the atoms in the mP16 structure are quadruple coordination relations, thereinto all O atoms are bonded to two B1 atoms and two B2 atoms, all C atoms are also connected with two B1 atoms and two B2 atoms, all of B atoms are bonded to two O atoms and two C atoms. Hence there is no any case about C atom directly connected with O atoms. At the same time, from the Z-axis, the structure is similar to a six-cell honeycomb, while the composition of B-C-O six membered ring is diverse and the six atoms are not in the same plane. Other detail structural information about the two new structures is exhibited in Table 1. Table 1 The cell parameters a, b, c (Å), ρ (g/cm 3 ) and Atomic Wyckoff positions for two newly B 2 CO phases.

Stability analysis
The elastic stabilities of oP8'-and mP16-B 2 CO at AP are evaluated by the calculation of the independent elastic constants C ij s, as presented in Table 2.
For oP8', it belongs to orthorhombic system, Born criteria are listed in Eq. (1) [34]: For mP16-B 2 CO with Laue class as 2/m, belong to monoclinic system, it has 13 independent elastic constants. The generic necessary and su cient criterion that all eigenvalues of elastic matrix be positive is easy to check with simple linear algebra routines. [34]  The elastic matrices are symmetric about the main diagonal, and these symmetric nonzero data are represented by apostrophe symbol "……", the blank cell represents data that is 0. There is no doubt that from the above elastic matrix in Table 2, the C ij s satisfy the criteria above, indicating oP8'-B 2 CO and mP16-B 2 CO are elastic stable at AP.
The imaginary frequency will lead to crystal distort, indicates the dynamical instability. Phonon dispersion spectrum and the related phonon density of states (DOS) of two newly predicted B 2 CO phases at AP are researched and plotted in Fig. 2. There are no negative phonon modes in entire Brillouin zone of their unit cells, suggesting their dynamically stable.
The stability of oP8' and mP16 B 2 CO were also intensively researched at high pressure. Here, taking 10 GPa, 50 GPa and 100 GPa as examples, the elastic constant and phonon scattering of oP8' and mP16 B 2 CO under high pressure are studied. As displayed in Table S1, the elastic constant C ij s of oP8' and mP16 B 2 CO under high pressure all satis ed the Born criterion of elastic stability. And the phonon scattering spectra and their density of states in Fig. S1 all indicate that there are no virtual frequency. That is to say, in the range of pressures that we studied, both oP8' and mP16 B 2 CO have high pressure stability.

Potential synthesis reaction
For providing guidance of experimental synthesis, it's of great signi cance and extremely urgent to deploying the potential synthesis of novel IED B-C-O phases. Based on the common reactants such as boron trioxide B 2 O 3 , boron-rich oxides B 6 O, graphite C, and α-boron B, construct two different reaction paths, then balanced the synthesis equations of these novel IED B 2 CxO phases, and therewith the relationships between formation enthalpy ΔH f and pressure re revealed according the following Eq. (2)~(3). The ΔH f1 and ΔH f2 represent the formation enthalpy via path 1 and path 2, respectively. Fig. 3 summarizes the relationships between formation enthalpy (normalized on a per formula) and pressure for all IED B-C-O compounds. The enthalpy of formation is negative, indicating the likelihood that the target IED B-C-O compounds will be synthesized by the selected reaction, and more negative the formation enthalpy, more reaction driven energy, more possible to be synthesized. Hence the critical pressure for formation enthalpy reaching zero is the key to determine the HP synthesis.
As shown in Fig. 3a, Fig. 3b, oI16 is not a pressure driven structure due to the formation enthalpy increases gradually with the increase of pressure and changes from negative to positive, hence it is unlikely to be synthesized by high pressure of common reactants such as B 2 O 3 , B 6 O, C and B. Compared with tP16, oP8' has signi cantly smaller critical pressure, which indicates that oP8' has the advantage of synthetic conditions. As exhibited in Except that oI16 is not a pressure-driven structure, the rest are all pressure-driven structures, and the critical pressure is within 100 GPa, which can be synthesized by high pressure experiment with diamond anvil cell (DAC). Some of these IED B-C-O phases have critical pressures of less than 30 GPa and may be synthesized by high pressure experiments in LVP. Hence, the formation enthalpies as a function of pressure at 0 to 30 GPa with sampling intervals of 2 GPa has been studied detailed. As shown in Fig. S2a, for the path 1 of B 2 CO, their critical pressures are between 12~20 GPa, and following the sequence as tI16 < tP4 < oP16 < oP8 < oP8'. The path 2 critical pressures of tI16 and tP4 are in the range of 27~30 GPa, which has obvious disadvantages in terms of synthetic pressure than path 1. As exhibited in Fig higher than that of path1. As the working pressure and temperature of LVP can reach tens of GPa and thousands of degrees centigrade, and the high temperature will bene t to reduce the reaction barrier, hence synergistic high temperature and HP experiments in LVP maybe a potentially e cient synthesis technology for the target B-C-O product at a lower pressure than the theoretical critical pressure value at zero temperature. Of course, special technologies need to be combined, such as rapid pressure relief, rapid cooling and so on, which will help to preserve these IED B-C-O phases.

High pressure deformation
It is well known that pressure is an important parameter to regulate the state of matter. In general, solid materials undergo a continuous densi cation during HP compression. As exhibited in Fig. 4, for these B 2 CO phases we studied, during the pressure from AP to HP as 100 GPa, the volumes per molecular formula of these phases decrease monotonically. For the six pure sp 3 hybridization connected phases, their volume compressions are basically similar, the largest volume compression appears on mP16, while the minimum volume compression appears on tP4, the difference between the two is very small. For three sp 2 and sp 3 hybridization coexist phases, all have large pressure induced volume change. Detailed research has revealed that the B 2 CO phases with sp 3 hybrid connection have lower volume compression ratio than that of sp 2 and sp 3 hybrid coexist. This is due to the existence of holes in the sp 2 and sp 3 hybridization coexisting phases, which results in their weaker resistance to deformation than the dense phases formed by pure sp 3 hybridization.
In addition, the volume per molecular formula of B 2 C X O (X=2, 3, 5) as a function of pressure was studied in detail, and shown in Fig. S3a

Mechanical property
Here the mechanical properties as bulk modulus B and shear modulus G of two newly predicted B 2 CO phases and their relationship with pressure are detailed studied. The independent elastic constant C ij s of oP8' at different pressure are listed in Table S2, and the elastic matrix of mP16 are displayed in Table S3. Because mP16 is composed of pure sp 3 hybridization, which is denser than oP8' formed by the coexistence of sp 2 and sp 3 hybridization. Generally, B indicates the ability to resist volume deformation by loading pressure, [35] and can be calculated by independent elastic parameter.
[36] Hence mP16 has larger elasticity moduli than oP8' across the whole studied pressure range, as shown in Fig. 8a. The pressure has a positive correlation to the bulk modulus B of oP8' and mP16, therein the linear tting shows that the bulk modulus B of mP16 has a perfect linear relationship with pressure P (AP to 100 GPa) as B=268.3+3.323P with R 2 =0.998.
Generally, shear modulus G can be acquired by independent elastic parameter, [36] represents the ability to resist deformation upon shear stress. [35] Due to the structural differences, the shear modulus G of oP8' is smaller than that of mP16 at AP. As exhibited in Fig. 8b, the shear modulus G of oP8' and mP16 is also positively affected by pressure. When the pressure is above 80 GPa, the shear modulus G of oP8' will be larger than that of mP16. The data tting results show that the shear modulus G of oP8' has a perfect linear relationship with pressure P (AP to 100 GPa) as G=144.9+2.073P with R 2 =0.993, and G of mP16 has the relationship as G=193.3+3.281P-0.037P 2 +1.715×10 -4 P 3 , R 2 =0.996.
The Young modulus E represents the rigidity of the condensed material, the greater E is, the less likely to deform. Based on the relationship for B, G and E as E=9BG/(3B+G), [36] the Young modulus E of oP8' and mP16 are also detailed studied. As displayed in Fig. 8c, mP16 has higher E than oP8' at AP. However, with the increase of pressure, the Young's modulus E of oP8' increases rapidly, which is nearly the same as that of mP16 at 80 GPa, and then exceeds that of mP16 with the increase of pressure. The positive correlation between pressure P (AP to 100 GPa) and Young's modulus E can be expressed as mP16: E=464.2+7.839P-0.078P 2 +3.481×10 -4 P 3 with R 2 =0.998, and oP8': E=337.4+5.271P with R 2 =0.992.
As re ect the brittleness or toughness of materials, Poisson's ratio μ is also an important mechanical property of materials and can be calculated based on μ=0.5(3B-2G)/(3B+G).
[36] We also studied in detail the effect of pressure on the Poisson's ratio μ of oP8' and mP16. As demonstrated in Fig. 8d, they both have value of μ smaller than 0.333, which is the critical value for a criterion of ductility or brittleness. The results of Poisson's ratio show that both of them are brittle materials. Ignored the rst point at AP, the pressure P (10 GPa~100 GPa) to Poisson's ratio of mP16 satis es the following relationship as: μ=0.190+8.549×10 -4 P with R 2 =0.997.
The hardness is extensively adopted to evaluate the mechanical properties of condensed matter. Here, the Vickers hardness (Hv) of two newly discovered B 2 CO phases were calculated based on Eq. (4). [1] The directional dependence of material's elastic property can be quanti ed as anisotropy. The mechanical behavior of materials such as plastic deformation, elastic instability and crack behavior are dominantly affected by the elastic anisotropy. Hence, it's essential for us to systematically investigate their elastic anisotropy for potential industrial applications. The universal elastic anisotropy index A u [37] is employed to evaluate the integrated anisotropy as Eq. (5).
We also studied in detail the effect of pressure on their hardness and elastic anisotropy index A u . As displayed in Fig. 8e, both oP8' and mP16 have high hardness than 20 GPa in all the pressure range we studied, suggesting that oP8' and mP16 are hard materials. The hardness of mP16 increases during the pressure from AP reaches to 20 GPa, and then the hardness continues to decrease in the process of increasing the pressure to 100 GPa. The relationship between hardness Hv and pressure P (30 GPa ~ 100 GPa) in the descending stage is satis ed as Hv=34.2-0.0756P with R 2 =0.998. For oP8', its hardness starts with a small increase and then a rapid increase and ends with a near constant change during the process from chamber pressure to 100 GPa. The value of A u for isotropic crystal is 0. Any departure from 0 indicates the anisotropic degree which considers the contributions of both the bulk and shear moduli. As exhibited in Fig. 8f, with the increase of pressure, the anisotropy of oP8' experienced a process of rst decreasing, then nearly unchanged, and then decreasing again. For mP16, it has the degree of anisotropy lower than oP8'. The elastic anisotropy index A u of mP16 rst decrease and then increase with pressure increase from AP to high pressure as 100 GPa. In the pressure range from 20 to 100 GPa, the A u of mP16 has a relationship with pressure P as A u =0.071+2.79×10 -3 P with R 2 =0.998.

Simulated tensile deformation
The material's ideal strength, which is de ned as the critical stress when a perfect crystal lost its elastic stability, equal to the upper limit for material strength. The atomistic mechanism for structural deformation and failure models can be clearly interpretated with the investigation of strain-stress relations and bondbreaking processes. Here the tensile deformation of oP8' with high symmetry has been studied in detail.

Conclusions
Through the candidate structures generated by evolutionary simulation methods and structural stability screening by density functional theory, two B 2 CO phases (oP8' with sp 2 -sp 3 hybridization coexist and mP16 with unitary sp 3 hybridization) have been discovered. The research of the phonon scattering spectra and the independent elastic constants verify their stability. The formation enthalpies and the relationship with pressure of all known IED B-C-O phases have been systematically studied, and the negative formation enthalpies at speci c pressures indicate that HP technology performed in DAC or LVP maybe an important method for synthesizing all IED B-C-O phases. Among all IED B-C-O phases, oP8' has the smallest gap and mP16 has the widest gap. With pressure increasing, for B 2 CO phases with high symmetry and composed of sp 3 hybridization, their band gaps all increase monotonically; for B 2 CO phases composed by sp 3 -sp 2 hybridization coexist or with low symmetry like mP16, their band gaps increase rst and then decrease. The study of mechanical properties shows that oP8' and mP16 both have large mechanical modulus and they are typical materials with high hardness. Pressure has a positive correlation with its mechanical modulus. The simulated tensile process also reveales differences in the ultimate stress and strain carried by oP8' in different directions and the resulting changes in electrical properties. The maximum tensile stress and strain are in the direction of [100], which are 64.7 GPa and 0.24, respectively.

Con icts of interest
There are no con icts to declare.  Phonon dispersion spectrum and phonon DOS at AP of (a) oP8'-and (b) mP16-B2CO.

Figure 3
Page 16/20 Calculated formation enthalpies ΔHf via pressure of all known IED B2CxO, a~c represents sp3 hybridization in B2CO, sp2-sp3 hybridization coexist in B2CO and B2CXO, respectively. Fig. d is the schematic diagram of DAC.

Figure 4
The relationship of pressure and volume per formula of all B2CO phases at AP to 100 GPa.
Page 17/20    a: Calculated band gaps as a function of pressure for all B2CO phases based on GGA-PBE with pressure range 0~100 GPa. b: Calculated band gaps as a function of pressure for oP8' B2CO based on GGA-PBE with pressure range 0~30 GPa. c is graphical description.