Theoretical Study of Novel B2CO Compounds with High Hardness

doi:10.21203/rs.3.rs-659960/v1

Two B₂CO phases (oP8’ with sp²-sp³ hybridization coexist and mP16 with unitary sp³ 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 B₂CO phases with high symmetry and composed of sp³ hybridization, their band gaps all increases monotonically; for B₂CO phases composed by sp³-sp² hybridization coexist or with low symmetry like mP16, their band gaps increased first 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.

Ceramics

stability analysis

pressure

formation enthalpy

property

As an important role in modern precision machining such as drilling, cutting, grinding, etc, hard materials have important scientific research significance and industrial application value, therefore attracting continuous attention. [1–3] 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.. [4–8] Since the synthesis of B-C-O compounds in large volume press (LVP) high pressure experiments, [9–11] ternary B-C-O compounds have entered the field of scientific research. Extensive research has been conducted around the mining of B-C-O compounds with high hard properties.

B₂CO compounds, as the simplest ternary B-C-O compounds of isoelectronic with diamond (IED), have become the research hotspot of B-C-O compounds. In 2011, two B₂CO phases (tP4-B₂CO and tI16-B₂CO) were theoretically designed as potential superhard materials through the particle swarm optimization method for crystal structure prediction. tP4-B₂CO and tI16-B₂CO stabilizes in diamond-like structures with typically strong covalent sp³ B-C and B-O bonds. The calculated Vickers hardness of B₂CO is ∼50 GPa. The research opens up a possibility to quest superhard materials within the ternary B-C-O system. [12] Thereafter, a lonsdaleite-like orthorhombic structure B₂CO (oP8-B₂CO) 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-B₂CO. The strong sp³ covalent B-C and B‐O bonds benefit oP8-B₂CO 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-B₂CO and tI16-B₂CO share diamond-like structures, [12] oP8-B₂CO has lonsdaleite-like structure [13]), two orthorhombic B₂CO phases (oP16-B₂CO and oC16-B₂CO) with confirmed 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 first-principles calculations, oP16- and oC16-B₂CO are both superhard materials with indirect band gaps.

Even the structure design has achieved great success, the most stable structure of B₂CO in the ground state remains a mystery until a novel orthorhombic structure oI16-B₂CO was predicted as the B₂CO thermodynamic ground-state phase.[16] Different from these previously proposed B₂CO phases formed by covalent sp³ bonds, oI16-B₂CO is formed by covalent sp²-sp³ B-C and B-O bonds. Theoretical research reveals that oI16-B₂CO has indirect semiconduct nature, common hard nature, but extended ductility.[16] Inspired by the first reported sp² and sp³ hybridization bonds coexist in B₂CO structure, a new metastable tetragonal phase for B₂CO (tP16-B₂CO) with sp²-sp³ coexisted was explored.[17] With the energy lower than those of previously proposed candidates, except oI16-B₂CO, tP16-B₂CO was identified as the thermodynamic metastable phase for B₂CO compound. The calculation of mechanical properties indicates its hard properties with high hardness of 23.19 GPa and anisotropy as the max stress along [001] far larger than that along [100]. [17]

The increased C ratio will lead to a stacking of C in the structure, forming shorter sp³ C-C bonds, so B₂CxO (x ≥ 2) might have greater hardness values than B₂CO does. [12] Inspired by this notion, three potential ultra-incompressible and thermodynamically stable B₂CxO (x = 2, 3, 5) phases were introduced via systematic particle swarm optimization algorithm structure search.[18] By evaluating the trends of the crystal configuration, electronic structure, and mechanical properties as a function of the C concentration, it is found that the high carbon concentration benefits the formation of the sp³ C-C covalent bonds and leads to the enhanced elastic moduli, hardness and ideal strengths in these B₂CxO compounds.

There is also significant progress on B-C-O compounds of non-IED. In 2016, a body-centered tetragonal structure B₄CO₄ 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 B₄CO₄, and the weakest ideal shear strength of 27.5 GPa demonstrates that the B₄CO₄ compound is indeed a hard material. [20] Also two low-enthalpy metastable compounds B₆C₂O₅ and B₂CO₂ with high pressure stability to 20 GPa have been discovered.[19] The hardness of B₆C₂O₅ and B₂CO₂ are 29.6 GPa and 41.7 GPa, respectively. [21] Two years later, a tetragonal B-C-O compound (tI12-B₆C₄O₂ with I4m2 symmetric structure) was reported. [22] And the research about the mechanical properties reveals that tI12-B₆C₄O₂ 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 sp²-sp³ coexisting and low space group (SG) B₂CO structures hinders the further exploration of the relationship between B₂CO phase and properties. In this paper, we focus on B₂CO 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 B₂CO compounds have been systematically studied, and the regulatory effect of pressure on properties has also been further investigated.

The potential structures of B₂CO compounds have been generated through evolutionary simulation methods CALYPSO [23–25] and USPEX [26–28] with fixed 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 field 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 finite 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, 0.5)→T(-0.5, 0, 0.5)→Y(-0.5, 0, 0)→S(-0.5, 0.5, 0)→X(0, 0.5, 0)→U(0, 0.5, 0.5)→R(-0.5, 0.5, 0.5) and Z(0, 0, 0.5)→G(0, 0, 0)→Y(0, 0.5, 0)→A(0.5, 0.5, 0)→B(0.5, 0, 0)→D(0.5, 0, 0.5)→E(0.5, 0.5, 0.5)→C(0, 0.5, 0.5), respectively.

3.1 Optimized structure

Two newly discovered B₂CO structures have been proposed and their crystal structures schematic at atmospheric pressure are shown in Fig. 1. For the first phase with SG Pmn21 is primitive orthorhombic structure containing 8 atoms per unit cell. In order to distinguish from the previously proposed oP8-B₂CO, here denoted it as oP8’-B₂CO (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 sp² hybridization and form BC3 triangular configuration, however for B2 atoms, they connect with three O atoms and one C atom by sp³ 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 sp² hybridization. In other word, there are sp²-sp³ hybridization forms coexist in oP8’, which is quite different from oP8, which is just sp³ hybridization. There are holes in the oP8’ structure that run along the X-axis, result in its low density of 2.813 g/cm³.

For the second B₂CO phase, it belongs to the primitive monoclinic with the SG of P21/c, and the unit cell contains 16 atoms, hence denoted as mP16-B₂CO (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/cm³) and Atomic Wyckoff positions for two newly B₂CO phases.

oP8ʹ				mP16 (β=88.6°)
ρ (g/cm³)	a (Å)	b (Å)	c (Å)	ρ (g/cm³)	a (Å)	b (Å)	c (Å)
2.813	2.617	5.804	3.859	3.052	4.509	5.246	4.567
AWP	x	y	z	AWP	x	y	z
B1	0.5	0.049	0.118	B1	0.820	0.354	0.410
B2	0.5	0.620	0.303	B2	0.662	0.625	0.915
C	0.5	0.876	0.443	C	0.304	0.869	0.401
O	0.5	0.437	0.598	O	0.186	0.137	0.910

3.2 Stability analysis

The elastic stabilities of oP8’- and mP16-B₂CO at AP are evaluated by the calculation of the independent elastic constants C_ijs, as presented in Table 2.

For oP8’, it belongs to orthorhombic system, Born criteria are listed in Eq. (1) [34]:

For mP16-B₂CO with Laue class as 2/m, belong to monoclinic system, it has 13 independent elastic constants. The generic necessary and sufficient criterion that all eigenvalues of elastic matrix be positive is easy to check with simple linear algebra routines. [34]

Table 2 The elastic matrices of two newly B₂CO phases at AP, the numerical unit is unified with GPa.

711.6	39.2	68.3				483.5	211.8	132.1		-3.0
……	338.2	120.4				……	409.5	69.9		-10.3
……	……	195.4				……	……	621.1		52.1
			177.7						212.7		1.2
				184.4		……	……	……		222.1
					147.1				……		188.4
oP8’-B₂CO						mP16-B₂CO

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_ijs satisfy the criteria above, indicating oP8’-B₂CO and mP16-B₂CO 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 B₂CO 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 B₂CO 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 B₂CO under high pressure are studied. As displayed in Table S1, the elastic constant C_ijs of oP8’ and mP16 B₂CO under high pressure all satisfied 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 B₂CO have high pressure stability.

3.3 Potential synthesis reaction

For providing guidance of experimental synthesis, it’s of great significance and extremely urgent to deploying the potential synthesis of novel IED B-C-O phases. Based on the common reactants such as boron trioxide B₂O₃, boron-rich oxides B₆O, graphite C, and α-boron B, construct two different reaction paths, then balanced the synthesis equations of these novel IED B₂CxO 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, for B₂CO constituted by sp³ hybridization, no matter what kind of synthesis path, the critical pressure of formation enthalpy from positive to negative is the minimum for tI16 and the maximum for mP16. And the critical pressures of sp³ hybridization B₂CO fall into two categories: mP16 and oC16 with high critical pressures, others with low critical pressures. For B₂CO contained sp³ and sp² hybridization displaying in 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 B₂O₃, B₆O, C and B. Compared with tP16, oP8’ has significantly smaller critical pressure, which indicates that oP8’ has the advantage of synthetic conditions. As exhibited in Fig. 3c, for B₂C_XO with similar structure type, with the increase of carbon content, the critical pressure gradually decreases in either synthesis path. In other words, the increase of carbon is conducive to the synthesis of B₂C_XO (X=2, 3, 5).

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 B₂CO, 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. S2b, for path 1 of B₂C_XO (X=2, 3, 5), their critical pressures are 29.2 GPa, 20.0 GPa and 14.9 GPa, respectively. In other words, with the increase of carbon content of B₂C_XO (X=2, 3, 5), the critical pressure of formation enthalpy gradually decreases from positive to negative. For the path 2 of B₂C₃O and B₂C₅O, their critical pressures are 27.4 GPa and 19.3 GPa, both are 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 benefit to reduce the reaction barrier, hence synergistic high temperature and HP experiments in LVP maybe a potentially efficient 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.

3.4 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 densification during HP compression. As exhibited in Fig. 4, for these B₂CO 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 sp³ 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 sp² and sp³ hybridization coexist phases, all have large pressure induced volume change. Detailed research has revealed that the B₂CO phases with sp³ hybrid connection have lower volume compression ratio than that of sp² and sp³ hybrid coexist. This is due to the existence of holes in the sp² and sp³ hybridization coexisting phases, which results in their weaker resistance to deformation than the dense phases formed by pure sp³ hybridization.

In addition, the volume per molecular formula of B₂C_XO (X=2, 3, 5) as a function of pressure was studied in detail, and shown in Fig. S3a. For B₂C_XO (X=2, 3, 5), their volumes vary with pressure between AP to 100 GPa high pressure as follows: B₂C₂O 6.388 Å³/20.034 %; B₂C₃O 7.154 Å³/19.144 %; B₂C₅O 8.855 Å³/18.213 %. That is to say, with the increase of carbon content, their volumes shrinkage gradually increase, while under the effect of volume size at AP, their volume compression rate decrease.

In fact, the phonon vibration frequency is affected by the strength of the chemical bond. The max phonon vibration frequency is an indicator of the strength of the chemical bond. Based on the phonon scattering of oP8’ and mP16 B₂CO under different pressures, it is found that their max phonon vibration frequencies are 40.162 THz/AP, 41.547 THz/10 GPa, 45.483 THz/50 GPa, 47.512 THz/100 GPa for oP8’ and 39.411 THz/AP, 40.754 THz/10 GPa, 44.691 THz/50 GPa, 48.765 THz/100 GPa for mP16. With the increase of pressure, the maximum vibration frequency increases, indicating that pressure will induce the increase of bond energy, which may be due to the decrease of atomic spacing under the action of pressure. Furthermore, zero point vibrational energy (ZPVE) is the energy that a quantum will vibrate under the zero point of absolute temperature. The amplitude of zero point vibration increases with increasing temperature, and the lighter the atom (that is, the smaller the element number), the more pronounced the zero point vibration. Based on the phonon scattering spectrum and its DOS, ZPVE can be calculated based on is the phonon density of states, is Planck constant. ZPVE per B₂CO chemical formula of oP8’ are 0.541 eV/AP, 0.560 eV/10 GPa, 0.614 eV/50 GPa, 0.666 eV/100 GPa. And ZPVE (eV) per formula of mP16 are 0.522 /AP, 0.547 /10 GPa, 0.615 /50 GPa, 0.676 /100 GPa. The researched relationship of ZPVE and pressure has revealed that there is also a positive correlation between pressure and ZPVE of oP8’ and mP16.

3.5 Electrical property

Based on GGA-PBE functional, the electronic band structures along the symmetry points of the Brillouin zone of all known B₂CO phases at AP are investigated and exhibited in Fig. 5, thereinto a~f are the B₂CO phases formed by sp³ hybridized, and g~i are the B₂CO phases constituted by sp²-sp³ hybridized.

For tP4 and tI16, there is no band crossing Fermi level, and the forbidden bands are determined by valence band max VBM (trefoil) and conduction band min CBM (heart). As the VBM and CBM locates at different symmetry points, both tP4 and tI16 are indirect bandgap semiconductors with gaps as 1.666 eV and 2.971 eV, respectively. For three orthorhombic phases oP8, oP16 and oC16, they all have semiconductor properties with the indirect band gap as 3.522 eV, 3.244 eV and 3.461 eV, respectively. For sp²-sp³ hybridized phases oI16 and tP16, they both have semiconductor properties with an indirect band gap as 3.658 eV and 1.880 eV, respectively. Among all the B₂CO, mP16 has the widest band gap as 4.681 eV, oP8’ has the narrowest band gap as 1.607 eV. The proposed two phases greatly expand the band-gap properties of B₂CO as semiconductor materials and enrich the potential applications of B₂CO in the electronic industry field.

Here the electrical properties of the carbon-rich type B₂C_XO (X=2, 3, 5) at atmospheric pressure were revealed based on PBE. As displayed in Fig. 6, as the VBM (trefoil) and CBM (heart) of B₂C₂O and B₂C₃O locates at the same symmetry points G, both B₂C₂O and B₂C₃O are direct semiconductors, which is consistent with the conclusions of Zhang et al [18]. As for B₂C₅O, the VBM and CBM locates at symmetry points G and M, indicating that B₂C₅O is an indirect semiconductor, which is also in well agreement with previous studies [18]. The band gap values of B₂C_XO (X=2, 3, 5) are 1.888 eV, 2.485 eV and 2.280 eV, respectively.

The relationship of band gap values and pressure of these IED B-C-O phases are researched. As shown in Fig. 7, there are two variation trends of band gaps with pressure. For these B₂CO phases with high symmetry and composed of pure sp³ hybridization, such as tP4, tI16, oP8, oP16 and oC16, their band gaps all increases monotonically as the pressure increases. In the pressure range of 100 GPa, their band gaps increased 0.166 eV/9.96%, 0.187 eV/6.29%, 0.639 eV/18.14%, 0.252 eV/7.77%, and 0.515 eV/15.04%, respectively. As for these B₂CO phases composed by sp³-sp² hybridization coexist (oI16, oP8’ and tP16) or with low symmetry like mP16, their band gaps increased first and then decreased with the increase of pressure. In different phases, the inflection point of rising first and falling second is different. For mP16 and oP8’, the inflection point is at 10 GPa, and their max band gaps are 4.706 eV and 1.643 eV, respectively. As for oI16 and tP16, their inflection points are where the pressure is higher, and the max gap values are 4.807 eV and 2.662 eV, respectively.

Here the relationship between the band gap of the carbon-rich system B₂C_XO (X=2, 3, 5) and the pressure has been investigated. As displayed in Fig. S3, in the pressure range of ambient pressure up to high pressure as 100 GPa, the band gap value of B₂C₃O undergo near invariance and then slow decline, among the band gap between ambient pressure and 30 GPa varies by only several meV, then the gap drops about 30 meV within 30 GPa to 100 GPa. For B₂C₂O, its gap increases at first and then decrease, the max gap value 2.416eV appears in 70~80 GPa. The gap of B₂C₂O goes through sustained decline with pressure increasing, and the 100 GPa pressure result in the gap decreasing amplitude as 0.323 eV/17.11%.

3.6 Mechanical property

Here the mechanical properties as bulk modulus B and shear modulus G of two newly predicted B₂CO phases and their relationship with pressure are detailed studied. The independent elastic constant C_ijs 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 sp³ hybridization, which is denser than oP8' formed by the coexistence of sp² and sp³ 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 fitting 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 R²=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 fitting 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 R²=0.993, and G of mP16 has the relationship as G=193.3+3.281P-0.037P²+1.715×10^-4P³, R²=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-0.078P²+3.481×10^-4P³ with R²=0.998, and oP8’: E=337.4+5.271P with R²=0.992.

As reflect 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 first point at AP, the pressure P (10 GPa~100 GPa) to Poisson's ratio of mP16 satisfies the following relationship as: μ=0.190+8.549×10^-4P with R²=0.997.

The hardness is extensively adopted to evaluate the mechanical properties of condensed matter. Here, the Vickers hardness (Hv) of two newly discovered B₂CO phases were calculated based on Eq. (4). [1] The directional dependence of material’s elastic property can be quantified 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 satisfied as Hv=34.2-0.0756P with R²=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 first 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 first 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^-3P with R²=0.998.

3.7 Simulated tensile deformation

The material’s ideal strength, which is defined 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 bond-breaking processes. Here the tensile deformation of oP8’ with high symmetry has been studied in detail.

Fig. 9a presents the simulated tensile stress–strain relationships along the specific directions including [100], [010] and [001] for oP8’. It can be seen that tI16-B₈C₆O₂ has the max tensile stress of 64.7 GPa in the [100] direction with the strain at 0.21, with the strain exceed 0.24 in [100] direction, oP8’ will failure. For [010] and [001] direction, overall tensile stress increases synchronously with the strain, the max tensile stress and max strain appear simultaneously. Once the stress or strain exceeds the maximum ([010]: about 37.2 GPa /0.16; [001]: about 12.0 GPa /0.13), oP8’ will be destroyed. As shown in Fig. 9b, with the appearance of structural strains in different directions, the band gap of oP8’ will mutate, then as the strain of [010] direction increases, the band gap continues to decrease. As the strain of [100] direction increases, the band gap continues to increase to 2.455 eV until the strain reach 0.17, and then decreases with strain increase. For [001] direction, the gap decreases during the strain up to 0.08 and then increase with strain increase.

Through the candidate structures generated by evolutionary simulation methods and structural stability screening by density functional theory, two B₂CO phases (oP8’ with sp²-sp³ hybridization coexist and mP16 with unitary sp³ 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 specific 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 B₂CO phases with high symmetry and composed of sp³ hybridization, their band gaps all increase monotonically; for B₂CO phases composed by sp³-sp² hybridization coexist or with low symmetry like mP16, their band gaps increase first 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.

IsoElectronic with Diamond: IED; ambient pressure: AP; high pressure: HP;

space group: SG; Perdew-Burke-Ernzerhof: PBE; density of states: DOS;

diamond anvil cell: DAC; large volume press: LVP.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Science Foundation of China (Grant no. 12064013), the Natural Science Foundation of Jiangxi Province (Grant no. 20202BAB214010), the Open Funds of the State Key Laboratory of Metastable Materials Science and Technology (Grant no. 201906), Ganzhou Science and Technology Project (Grant no. 202060), and the Program of Qingjiang Excellent Young Talents, Jiangxi University of Science and Technology.

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Theoretical Study of Novel B₂CO Compounds with High Hardness

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Abstract

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1 Introduction

2 Computational Methods And Details

3 Results And Discussion

4 Conclusions

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References

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