An isoelectronic and asymmetrical compound of diamond, B₂O, was synthesized by Endo et al. through the chemical reaction of BP with oxygen^[49] under the condition of 2.0–6.0 GPa and 800–1350 °C. The measured Vickers hardness is in a range from 33.5 GPa to 40.5 GPa, which is close to the criterion of superhard materials. From the measured x-ray diffraction data, Endo et al. proposed a possible crystal structure consisting of diamond lattice sites with planes of O alternating with two planes of B along the [111] crystallographic direction. Subsequently, Grumbach et al.^[50] theoretically proposed that the 111 structure be segregated into layers of BO and B and then proposed a 100 structure with stacking the B and O planes in the [100] direction which is energetically more stable than the 111 structure. However, the x-ray diffraction (XRD) peaks of the 111 and 100 structures could not match the experimental data.

Using ab initio evolutionary method^[51–53] for crystal structure prediction, we have investigated the candidate crystal structures of the synthesized superhard B₂O.^[54] We obtained an intriguing monoclinic C2/m structure which is most energetically favorable. The C2/m structure contains four B₂O formula units in one unit cell, in which both B and O atoms occupy the Wyckoff 4i (z1, 0, z2) sites. Interestingly, B atoms possess coordination numbers ranging from 4 to 7. A novel tetragonal P4₂mc contains both sp³- and sp²-hybridized boron atoms, in which vertical flat 6-fold rings are linked together by sp³-hybridized B atoms along c axis. The inspection of the crystal structure of hexagonal P-62m along c axis reveals that it can be viewed as distorted graphite with one B atom shifting to the middle of the two flat layers and forming six-coordinated bonds. All O atoms in C2/m, P4₂mc, and P-62m structures are three-coordinated (sp²). The monoclinic C2/m of B₂O is an excellent potential superhard material with a simulated hardness of 66.7 GPa. However, the XRD of C2/m cannot fit to the experimental XRD data^[49] of the synthesized sample.

Fig. 2.

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	Fig. 2. Crystal structures of P-4m2 B2O,[54] I-42d B2CO,[55] R3m-2u BC2N,[56] and Imm2 B3NO.[57]

From the simulated XRD patterns, the most likely structure for the synthesized superhard B₂O is the P-4m2 (Fig. 2) or even a mixture of P-4m2 and P4₂mc, which are not diamond-like. The simulated hardness values of P4₂mc and P-4m2 are 40.5 and 44.8 GPa, respectively, in accordance with the experimental values of 33.5–40.5 GPa.^[54]

Several B–C–O materials have successfully been synthesized at high pressure and high temperature, such as B₆C_1.1O_0.33 to B₆C_1.28O_0.31^[58] and B(C, O)_0.1555,^[59] which demonstrate that superhard B–C–O compounds can form typical sp³ covalent hybridization, which contain B–C, B–O, C–O, and C–C bonds. Therefore, B–C–O materials can be expected to be good candidates for designing new superhard materials. B2CO has 16 valence electrons per formula unit (4 valence electrons per atom) and is thus isoelectronic with diamond.

Using the CALYPSO method^[60–62] for predicting the crystal structure, we design a potential superhard ternary material, B₂CO, which is an isoelectronic example in the B–C–O system.^[55] We predict two energetically competitive diamond-like structures with space groups of P-4m2 (tP4) and I-42d (tI16), see Fig. 2. The P-4m2-1u structure is isostructural to t-B₂CN.^[12,63,64] It is found that B atoms in both structures occupy four diagonal positions; while O atoms sit in the vertexes and two opposite surface positions in the tP4 structure, and sit on half of the vertexes and surface positions in the tI16 structure. Both structures adopt the same sp³ bonding environment: each B is tetragonally bonded with two C and two O atoms, and C and O atoms are tetragonally bonded with four B atoms, forming B–C and B–O covalent bonds in the diamondlike lattices. The results of total energy calculations at ambient pressure show that tI16 structure is 43 meV/f.u. lower than that of tP4. The dynamical and mechanical stabilities of tI16 and tP4 are examined by the calculations of phonon dispersions and elastic constants, respectively. The two structures exhibit similar mechanical properties, such as high bulk modulus (approximately 31 GPa), high shear modulus (approximately 260 GPa), high hardness (approximately 50 GPa), and low Poisson’s ratio (approximately 0.17).

In 1994, Nakano et al.^[11] experimentally observed the formation of c-BC₂N from graphite BC₂N at 7.7 GPa and approximately 2000–2400 K. Later, Knittle et al.^[8] prepared a c-BC_xN (x = 0.9–3.0) solid solution by using even higher static pressure of 30 GPa and laser heating, starting from either the microstalline BCN, or the mechanical mixture of graphitic carbon and boron nitride. The resulting BC₂N has a bulk modulus of 355 GPa. Zhao et al.^[6] have successfully synthesized millimeter-sized bulk samples of c-BC₂N with the H_v about 62 GPa. Using shock-wave compression, Komatsu et al.^[10] obtained c-BC₂N with a bulk modulus of 401 GPa, which is larger than that of c-BN. Under high pressure above 18 GPa and high temperature over 2200 K, Solozhenko et al.^[9] experimentally reported that the bulk modulus 282 GPa of synthesized c-BC₂N is lower than that of c-BN but the measured hardness of 76 GPa is much higher than that of c-BN single crystal and slightly lower than that of diamond. By all the evidence, it is clear that c-BC₂N is a superhard material. Diamond-like BC₂N (c-BC₂N) has gained extensive attention since it has been expected to be thermally and chemically more stable than diamond, and harder than c-BN.

On the theoretical side, much effort has also been made to uncover the crystal structure of the superhard BC₂N.^[65–77] As is well known, the stable crystal structure has the lowest Gibbs free energy under the given conditions. Based on this principle, Sun et al.^[70] proposed seven possible structures (struc-m, m = 1–7) with different atomic configurations in the eight-atom zinc-blende (ZB) cubic unit cell. They proposed that the struc-1 with Pmm2 space group having the lowest energy is the best candidate. While this structure has different appellations in the literature such as BC₂N-1,^[66] β -BC₂N,^[67] II-BC₂N,^[68] and HD1,^[68] in fact, they are identical. Sun et al.^[70] further predicted a chalcopyrite structure (cp-BC₂N) with the tetragonal symmetry (space group, I-42d). The calculated hardness of cp-BC₂N is 72.2 GPa very close to the experimental value. More recently, Luo et al.^[78] suggested that the body-centered BC₂N phase (bc6-BC₂N) has lower density than ZB BC₂N, wurtzite BC₂N and cp-BC₂N, and they have estimated that the hardness of bc6-BC₂N is over 60 GPa. Among the above structures, struc-1 possesses the lowest free energy. Very recently, Zhou et al.^[72] predicted a z-BC₂N structure, which is is comprised of the sixteen-atom supercell of diamond. The simulated XRD spectrum agrees well with the experimental data, and importantly the total energy of z-BC₂N is lower than that in struc-1. Luo et al.^[79] further predicted three structures of wurzite-type BC₂N, among which BC₂N-w3 also has lower total energy than struc-1. Very recently, Chen et al.^[74] proposed a series of short period (C₂)_n(BN)_n (111) superlattices (denoted as BC₂N_n×n, with n = 1,2,3,…). The authors suggested that these superlattices have the lowest energies in those of the previously proposed structures. However, the intense debate is still going on.^[76,77] Also, it is worth noting that the crystal structure of superhard c-BC₂N should be unique instead of a structural series.

To clarify the intensive debate on the superhard phase of c-BC₂N, we have extensively explored the crystal structure of BC₂N by using ab initio evolutionary method.^[56] We have predicted three polytypic structural families: orthorhombic Pmm2-nu, hexagonal P3m1-nu, and rhombohedral R3m-nu, where n refers to that the structure contains n BC₂N units per primitive cell (n = 1, 2, 4). Through total energy calculations, it is found that R3m structural series possesses the lowest enthalpy. The structural stability of R3m structural series has been confirmed by the phonon calculations. Analyses of the simulated x-ray diffraction pattern and energy-loss near-edge spectroscopy suggest that our predicted R3m-2u [Fig. 2] is the best candidate phase for the observed superhard BC₂N. We have also demonstrated that the previously proposed high density and low density forms might be identical and their x-ray diffraction patterns could be reasonably understood by the single phase of R3m-2u. The calculated electronic structures show that R3m-2u BC₂N is a wide gap semi-conductor with an indirect DFT band gap of 3.8 eV. The calculated bulk modulus of R3m-2u BC₂N is 396 GPa, which is in good accordance with the experimental value of 401 GPa by Komatsu et al.,^[10] but much higher than the measured value of 282 GPa by Solozhenko et al.^[9] The estimated theoretical Vichers hardness of R3m-2u BC₂N is 62 GPa, in agreement with the experimental value of 62–76 GPa.^[6,9]

Previous studies demonstrated that diamond-like superhard c-BN and B₂O compounds can form typical sp³ covalent network bonds.^[54] It is reasonable to explore whether those materials with mixed B–N and B–O bonds may possess high hardness values. We have performed superhard-driven search by using an unbiased structure search method based on CALYPSO method^[57] in a ternary B–N–O system, B₃NO, which is also an isoelectronic compound with diamond.^[80] Crystal structure is critical to understand the hardness and relevant physical properties of B₃NO. We have predicted various structures for B₃NO under 0–100 GPa.^[57] It is in contrast to the traditional ground-state prediction method where the total energy is solely used as the fittness function. We choose hardness as the fittness function in combination with the first-principles method to obtain the hardness versus energy map by seeking a balance between hardness and energy for a better mechanical description of given chemical systems. We predicted a variety of structures with high hardness values and relatively low energies.^[80]

A variety of newly structures of B₃NO compound contain short, strong, and three-dimensional covalent bonds, which are responsible for the predicted superior mechanical properties.^[57] Two energetically stable B₃NO structures with space groups of Imm2 (oI20) and Pmn2₁(oP20) are uncovered. The oI20 structure (Fig. 2) is the most stable structure which possesses irregular 6-member rings, while oP20 is formed by 4-numbered, 6-numbered, and 8-numbered rings. To understand the electronic properties, we perform calculations on the electronic band structures and partial densities of states of oI20 and oP20. The calculated results reveal the semiconductor features of oI20 and oP20 structures with indirect band gaps of 0.87 eV and 0.12 eV, respectively. The electronic states near the Fermi level are mainly contributed by B-2p and a few N-2p, and O-2p states. The partial densities of states for B-2p, N-2p, and O-2p are very similar in an energy range from −12 eV to −4 eV, indicating the significant hybridizations between these orbitals and strong covalent interactions in both the B–N and B–O bonds. The calculated electron localization function (ELF) shows that the O atoms in oI20 and oP20 structures are threefold coordinated with three near-neighbor B atoms and a lone pair of electrons. The results show that oI20 and oP20 structure may have a broad prospect in industrial applications, such as solar battery, and luminous materials. The two structures exhibit high bulk moduli of 326–332 GPa, significantly higher than that of B₂O (240 GPa),^[54] indicating the structures are difficult to compress near their respective equilibrium volumes. The further hardness calculations show the two structures are likely to become superhard materials with Vickers hardness of 45.9 GPa (oI20) and 47.4 GPa (oP20), respectively, exceeding the criterion of superhard materials (40 GPa).

Eproperties are of considerable demand for the creation of multifunctional device.^[81] Since the discovery of superconductivity in boron-doped diamond (4 K),^[82] the superconducting behaviors in “covalent metals” (dominated by strong covalent bonds, rather than the metallic bonding) have also been an exotic subject.^[83] It is suggested that the strongly directional covalent characteristics in metallic compounds lead to large phonon frequencies and electron–phonon coupling potential, which contribute to the increase of T_c.^[84] Thus, metallic B–C–N compound with exclusively covalent bond can thus be expected to simultaneously possess superhard and superconducting properties. Metallicity of ternary B–C–N compounds can be evaluated by the following simple rule: pZ_V(B) + mZ_V(C) + lZ_V(N) ≠ 4n.^[85] The values p, m, l, and n are integers, and Z_V(B), Z_V(C), and Z_V(N) are the atomic valence states (2s and 2p) for B, C, and N, respectively. Recently, diamond-like B₂CN^[12,63] and BC_x,^[13,21] crystals have been synthesized under the condition of high-pressure and high-temperature (1500–2500 K), which are suggested to be conductors.

We have extensively investigated the crystal structures of B₂CN by using ab initio CALYPSO algorithm in a large pressure range of 0–100 GPa.^[64] Five competitive structures with clear tetrahedrally sp³ hybridization are predicted: tetragonal P-4m2 (1 f.u. /cell), hexagonal P3m1 (1 f.u./cell), rhombohedral R3m (1 f.u./cell), and orthorhombic Pmm2 (2 f.u./cell), and Pmma (2 f.u./cell). P-4m2, Pmm2, and Pmma can be viewed as a periodic sandwich-like structure formed by stacking CBNB, BNBBCCBN, and CCBNBBNB along the ⟨100⟩ direction of diamond structure, respectively. Instead, the stacking sequences of P3m1 and R3m follows the ⟨111⟩ direction. With fixing the numbers of 1 and 2 formula units in the primitive cell, it is found that the energetically best structures are the diamond-[100] structures: t-B₂CN and Pmma. The C–C and B–N chemical bonds are more favorable than B–B, B–C, and C–N for forming the diamond-like structure with sp³-hybridized, thus we define C–C and B–N bonds as stable bonds and the other relatively weaker bonds as less stable bonds. In the two most stable structures (Pmma [Fig. 3] and Pmm2), it is interesting to note that the ratio of the stable chemical bonds to less stable bonds reaches 5:3, which is the largest in the five structures. Further increasing the formula units (3 or 4 f.u./cell) in the simulation cell, we find that the produced structures tend to separate into bulk diamond + c-BN + boron layers. We find that all the B2CN structures have positive formation energies, indicating that these structures tend to phase decomposition. The calculated large bulk modulus (318–333 GPa) and high hardness (56–58 GPa) reveal that they are ultra-incompressible and superhard materials. Therefore, the synthesized B₂CN can be expected to be applied to the high-pressure devices for investigating the electric properties of various materials under extreme conditions.

Fig. 3.

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	Fig. 3. Crystal structures of Pmma B2CN,[64] Pmma-a BC3,[86] P3m1 BC5,[87] and P-4m2 BC7.[88]

Due to the electron deficiency of sp³ hybridized B₂CN in comparison with that of diamond, the five diamond-like B₂CN structures show metallic with hole nature. Further electron-phonon coupling calculations predict that all the five phases are superconductors. The estimated T_c for diamond-[111] structures (R3m and P-3m1) reaches a very high value, 44–53 K, in contrast to the smaller value (2–9 K) in diamond-[100] structures. The large difference in superconductivity of B₂CN is mainly related to the relatively low frequency (< 20 THz). In view of the excellent properties of diamond-like B₂CN, it is necessary to pay much attention to the experimental synthesis and conductivity measurements.

Boron-doped diamond has been widely explored as better oxidation and ferrous resistant materials to cover the shortage of diamond and expand the applications to electric devices.^[89,90] Electrical measurements demonstrate that Boron-doped diamond becomes metallic and even turns into a superconductor with a transition temperature T_c of 4 K at a higher doping level (2%–3%).^[82] Further experimental and theoretical studies indicate a T_c increases with increasing dopant concentrations.^[91–93] However, it remains a major challenge to introduce such a large amount of B into diamond in experiments, and the maximum B concentration has been limited up to approximately 5%.^[93] An alternative route to achieving high B content is to directly synthesize the B–C compounds. Recently, Solozhenko et al.^[13] successfully synthesized a diamond-like B–C phase (c-BC₅) with a high B content ever achieved (∼ 16.7%). A subsequent hexagonal structure constructed from a six-atom supercell of diamond for c-BC₅ was proposed^[94–96] and a large T_c (45 K) was predicted.^[97] The obtained c-BC₅ sample has been measured to possess a large bulk modulus (335 GPa), unusually high fracture toughness (9.5 MPa·m^1/2), and high thermal stability (up to 1900 K). Remarkably, the indentation experiment has demonstrated an extremely high hardness (71 GPa), harder than cubic BN.^[13]

We have uncovered four other low-energy structures: tetragonal I-4m2 (1 f.u./cell), orthorhombic Imm2 (1 f.u./cell), hexagonal P-3m1 (2 f.u./cell), and orthorhombic Pmma (2 f.u./cell).^[98] It is found that except for Imm2, other structures are built up by the sandwich-like layers with different stacking sequences. Specifically, I-4m2 and Pmma structures having the atomic packing of ABCABC··· along the [100] crystallographic direction of diamond can be viewed as diamond-[100] structure. Instead, P3m1 and P-3m1 structures are diamond-[111] structures as the atomic packing follows the [111] direction. Subsequently, a new P3m1_2u [Fig. 3] with slightly lower formation energy was theoretically proposed for the candidate structures for the synthesized BC₅.^[87] All the predicted structures are tetrahedrally bonded with clear sp³ hybridization. Further increasing the formula units (3 or 4 f.u./cell) in the simulation cell, the produced structures tend to decompose into bulk diamond + boron-carbon layers. This fact is understandable in view of the metastable nature of the synthesized BC₅ compound supported also by the calculated formation energy.

These diamond-like BC₅ possess similar structural and mechanical properties. The calculated bulk modulus is in a range from 373 GPa to 398 GPa, in good accordance with the experimental value of 335 GPa.^[98] On the basis of the microscopic hardness model,^[99,100] the Vickers hardness of Pmma structure is estimated to be 61–74 GPa, in satisfactory agreement with the experimental data (71 GPa). It was previously found that a small metallic component has a strong negative effect on hardness and thus the correction of metallic bonding is necessary to account for the experimental hardness for electron conductors. The underlying mechanism is that those electrons delocalized to contribute to the conduction should be excluded from the hardness calculation. However, for the hole conductors, the major carriers are holes and the valence electrons are mainly localized to form covalent bonds. It is thus unnecessary to include the metallic correction in the hardness calculation. Indeed, if we force to include the correction of metallic bonding, the simulated hardness decreases by about 15 GPa, which is largely deviated from the experimental data.

Using CALYPSO methodology for crystal structure prediction of the potential superhard materials of BC₇, we find four potential superhard structures of tetragonal P-4m2, hexagonal P3m1, orthorhombic Pmm2, and hexagonal R3m, which are energetically much superior to the previously proposed P-43m structure.^[88] All the predicted structures are tetrahedrally bonded with sp³ hybridization. The energetically best structure is the tetragonal P-4m2 (Fig. 3). The calculated Vickers hardness and B₀ of the lowest energy P-4m2 polymorph reaching very high values (75.2 GPa and 377 GPa) indicate that diamond-like BC₇ is a superhard and ultra-incompressible material. Further phonon and elastic constant calculations imply that these structures are all mechanically stable.

Encouragingly, diamond-like c-BC₃ with high B content is also successfully synthesized under the condition of high pressure and high temperature^[21] by using the precursor materials. Numerous structural models have been theoretically proposed to understand the structural nature of diamondlike c-BC₃.^[86,87,101] However, none of these proposed sp³ structural models is consistent with the experimental observation of cubic diamond structure. A lack of accurate structural determination for these materials has severely impeded further study of their properties. We report a novel high symmetric phase of low-energy cubic BC₃ (I-43m symmetry and 64 atoms per unit cell, d-BC₃) consisting of unique eight B–B bond distributions identified by using an unbiased structure search method based on particle-swarm optimization algorithms in combination with density functional theory calculations.^[102] At ambient pressure, the optimized lattice parameter of d-BC₃ is 7.330 Å (Fig. 4), with boron occupying Wyckoff 8c (0.89711, 0.89711, 0.10289) and 8c (0.24273, 0.24273, 0.75727), and carbon occupying 12e (0.26755, 0.0, 0.0), 12d (0.0, 0.5, 0.75), and 24g (0.61792, 0.87580, 0.87580). A Bader charge analysis reveals that the charge density at the B–B bond critical point is 0.463 electron charges per ų with a Laplacian value of −1.14, indicating the covalent nature of the B–B bond in d-BC₃. As shown in Fig. 5, the strong covalent B–B bonding can also be shown in the two-dimensional ELF on the (110) planes passing through the B–B bonds as shown in Fig. 5. The proposed d-BC₃ becomes stable structure with exclusively sp³ hybridization above 41.3 GPa, in good agreement with the reported synthesis pressure of 39 GPa.^[21] The simulated x-ray diffraction pattern and Raman modes of this d-BC₃ phase are in excellent agreement with the experimental data.^[21]

Fig. 4.

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	Fig. 4. Crystal structures of d-BC3.

Fig. 5.

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	Fig. 5. The two-dimensional ELF on the (110) planes passing through the B–B bonds.

To deal with the metastable material with a positive formation enthalpy, it is interesting to investigate whether the newly predicted ordered d-BC₃ is more energetically favorable than diamondlike disordered structure. We have used the special quasirandom structure (SQS) approach^[103] to randomly distribute B and C in the diamond lattice with different n-atoms (n = 16, 32, 64, and 128). The results show that the enthalpies of all the quasirandom diamond-like structures are higher than those of d-BC₃, indicating that the current d-BC₃ is clearly much more stable than the disordered phase at zero temperature. Since the d-BC₃ was synthesized under the condition of high-pressure and high-temperature, the temperature effect on energy should be taken into account. We have calculated the Gibbs free energies with the consideration of the contribution of vibrational and configurational entropy for d-BC₃ and the 128-atom quasirandom BC₃ structure. In any case, our calculations show that with or without the vibrational and configuration entropy contributions, the Gibbs free energy of predicted d-BC₃ are more energetically favorable than that of quasirandom structure below 1166 K (Fig. 6), indicating that our predicted d-BC₃ structure is more favorable for the experimentally recovered sample at atmospheric temperature than the disordered phase.

Fig. 6.

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	Fig. 6. Gibbs free energy per atom of quasirandom 128-atom BC3 structure relative to d-BC3 at 40 GPa.

The simulated hardness of BC₃ reaches 62 GPa. We have performed first-principles calculations of the stress–strain relations, which provide an insight into the local bond deformation and breaking mechanisms that determine the incipient plasticity in a crystal.^[104–106] Its deformation modes under tensile and x shear strains show intriguing bond elongation and sequential bond-breaking processes that lead to remarkable extended ductility and elastic response. The [111] tensile stress decreases very gradually past the peak and the peak-to-valley drop extends over a wide range of tensile strain from 0.08 to 0.18, and the system undergoes a second elastic response regime from tensile strain 0.18 to 0.31. This behavior is highly unusual for a superhard material, and it is in stark contrast to the results for diamond and c-BN where the stress drops precipitously past the peak. The ratio between shear strength and tensile strength (53.0/52.5), which is nearly unity, is the lowest in superhard covalent materials, indicating the superior ductility of d-BC₃ to those of top superhard materials. The results provide a crucial insight into the structural assignment for BC₃ and clarify the actual existence of cubic symmetric phase for other boron carbide.