Prediction of a new ground state of superhard compound B₆O at ambient conditions

Abstract

Boron suboxide B₆O, the hardest known oxide, has an Rm crystal structure (α-B₆O) that can be described as an oxygen-stuffed structure of α-boron, or, equivalently, as a cubic close packing of B₁₂ icosahedra with two oxygen atoms occupying all octahedral voids in it. Here we show a new ground state of this compound at ambient conditions, Cmcm-B₆O (β-B₆O), which in all quantum-mechanical treatments that we tested comes out to be slightly but consistently more stable. Increasing pressure and temperature further stabilizes it with respect to the known α-B₆O structure. β-B₆O also has a slightly higher hardness and may be synthesized using different experimental protocols. We suggest that β-B₆O is present in mixture with α-B₆O, and its presence accounts for previously unexplained bands in the experimental Raman spectrum.

Introduction

Superhard materials are used in many applications, from cutting, grinding and drilling tools to wear-resistant coatings^1,2,3. However, most superhard materials⁴, such as diamond⁵ and cubic-BN⁶, are synthesized at high pressure, which makes them expensive, but some (boron allotropes, B₆O, B₄C) are thermodynamically stable already at ambient conditions. The hardness of α-B₆O⁷ was reported to be in the range 30–45 GPa^8,9, making it the hardest known oxide^9,10,11.

Objects with icosahedral symmetry (I_h) bear a special fascination because of incompatibility of fivefold symmetry with crystalline periodicity. The discovery of multiply-twinned particles B₆O, an icosahedral packing of B₁₂ icosahedra with I_h symmetry, had aroused interest⁷. Here we report the prediction of a new phase of B₆O, with space group Cmcm, which we name β-B₆O. This structure is energetically almost degenerate with α-B₆O (and slightly more stable), is predicted to have a higher hardness, and actually corresponds to twinned α-B₆O structure.

Results

Discovery of β-B₆O at ambient conditions

Our variable-composition evolutionary searches expectedly find B₂O₃ and B₆O to be the only stable compounds in the B-O system. Interestingly, there are also several compounds very close to stability - B₂O₇ (this is a 2D-form of B₂O₃ intercalated with oxygen molecules) and oxygen-deficient versions of B₆O with B₆O-like structures and compositions between B and B₆O. To our surprise, Cmcm-B₆O (β-B₆O, see Table 1 for structural parameters), instead of the well-known Rm-B₆O (α-B₆O)^7,12,13,14, turned out to be the most stable structure at ambient pressure, as shown in Fig. 1; phonon calculations have confirmed its dynamical stability. Transmission electron microscopy recently confirmed its existence¹⁵. Structural parameters and some of the physical properties of β-B₆O are shown in Table 1, in comparison with α-B₆O and two related forms of pure boron.

Table 1 Calculated structural parameters, hardness, elastic moduli and band gap of α-B, Cmcm-B, α-B₆O, and Cmcm-B₆O phases at ambient pressure.

Figure 1: Stability of Cmcm-B₆O.

(a) Convex hull of the B-O system at ambient pressure. The solid (hollow) points represent the stable (metastable) structures. (b) Enthalpy difference between β-B₆O and α-B₆O, including zero-point energy.

In order to further compare the stability of β-B₆O and α-B₆O, we calculated their enthalpies as a function of pressure, as shown in Fig. 1b. We found that the enthalpy of β-B₆O is lower than that of α-B₆O at ambient pressure, but the energy difference is only about 1.8 meV/formula within the GGA (and almost degenerate within the HSE06 hybrid functional). As pressure increases, β-B₆O becomes progressively more favorable than α-B₆O, indicating that β-B₆O might be more easily synthesized under pressure. The enthalpies of the two structures are so close that it makes us think: will the two structures coexist? what is their relationship? how to synthesize β-B₆O? In order to answer these questions, we perform a detailed comparison of their structure, Raman spectra and phonon densities of states (PDOS).

Comparison of crystal structures of α-B₆O and β-B₆O

β-B₆O structure has hexagonal close packing of B₁₂ icosahedra (ABAB… stacking), while α-B₆O is based on the cubic close packing (ABCABC… stacking) of B₁₂ icosahedra, as shown in Fig. 2d,c. As is the case of hcp and fcc metals, twinning of α-B₆O can produce local β-B₆O stackings. It may also be possible to obtain β-B₆O-like stacking faults by deformation of α-B₆O, through plane slips. Most properties of these two phases are very similar: e.g. predicted volume per formula (V(α-B₆O = 51.71 ų/formula; V(Cmcm-B₆O) = 51.69 ų/formula), hardness (Hv (α-B₆O) = 38 GPa; Hv (Cmcm-B₆O) = 39 GPa), elastic moduli (Table 1), DFT band gaps (α-B₆O has a 1.85 eV direct band gap, while Cmcm-B₆O has a 1.81 eV indirect band gap).

Figure 2

Crystal structures of (a) α-B, (b) Cmcm-B, (c) α-B₆O, (d) Cmcm-B₆O, and their local structures, (e,f) B₁₂ icosahedra. Green (large) and red (small) spheres denote B and O atoms, respectively.

Another interesting aspect is that if we remove the oxygen atoms from α-B₆O and Cmcm-B₆O, they turn into α-B¹⁶ and Cmcm-B, respectively (Fig. 2a,b). α-B and Cmcm-B are energetically nearly degenerate structures of boron at low pressure¹⁷, while Cmcm-B is a newly predicted structure^18,19. As shown in Fig. 2b, displacements of layers I and II can transform this structure into α-B, and vice versa; Cmcm-B₆O and α-B₆O have a similar relationship (Fig. 2d). Furthermore, we found that the conversion of Cmcm-B₆O and α-B₆O will cause a deflection of B-O bond by 36°, as shown in their local structure (Fig. 2e,f).

However, it should be pointed out that it is not easy to change the stacking in covalent systems, though some examples are known – lonsdaleite (metastable “hexagonal diamond”) is formed in shocked cubic diamond. To obtain multiple polytypes, methods like physical vapor transport (PVT), also known as seeded sublimation growth, can be used: e.g., different polytypes of SiC were obtained using the PVT method²⁰.

Comparison of Raman spectra

As mentioned above, β-B₆O and α-B₆O are energetically almost degenerate at zero pressure, structurally related and can coexist. To test the latter, we calculated their Raman spectra and compared with the experimental results^12,13,21. In Fig. 3, the topmost curve is the Raman spectrum reported by Solozhenko et al.¹²; below it are the Raman frequencies reported by Werheit and Kuhlmann²¹ and marked by vertical bars (|). The two curves below are our Raman spectra of β-B₆O and α-B₆O. The experimental data correspond to normal isotopic abundance, and so do our calculations. Correspondingly, the atomic mass of boron in the calculations adopt the weighted average value, i.e. 10.811, which is based on the isotopic abundance of boron. As one could expect, the Raman spectra of β-B₆O and α-B₆O are similar and match perfectly with the experimental spectra. For example, the Raman modes at 374, 499, 541, 737, 785, 833, 1088, 1119, 1141 cm⁻¹ are consistent with Solozhenko’s results¹². However, there are four Raman modes which are unique for β-B₆O. The first two are computed to be at 516 and 533 cm⁻¹ (marked by letter A in Fig. 3), and Werheit and Kuhlmann indeed observed these two modes at 519 and 534 cm⁻¹. The third mode is predicted to be at 630 cm⁻¹ (marked with letter B), and Werheit and Kuhlmann have indeed observed a Raman-active phonon at 627 cm⁻¹. We note that α-B₆O does not have Raman-active modes between 570 and 700 cm⁻¹, thus the one observed by Werheit and Kuhlmann at 627 cm⁻¹ cannot come from α-B₆O, but β-B₆O. The fourth mode has the theoretical Raman frequency of 896 cm⁻¹ (marked with letter C), while α-B₆O has no Raman-active modes between 850 and 1000 cm^-1. Thus, this mode is also unique to the β-B₆O structure, and again seen in experiments: Solozhenko et al. observed Raman active phonon at 889 cm⁻¹, and Werheit and Kuhlmann observed a Raman-active phonon at 902 (909) cm⁻¹. Moreover, Wang et al. also observed similar Raman spectra in their B₆O samples (see Fig. 1 in ref. 13). This analysis clearly shows that experimental samples contain β-B₆O.

Figure 3

Raman spectra of B₆O experimental spectrum of Solozhenko et al.¹², Raman mode frequencies experimentally observed by Werheit and Kuhlmann²¹, and our calculated Raman spectra of Cmcm-B₆O and α-B₆O.

Comparison of PDOSs and Gibbs free energy

Comparing phonon densities of states (PDOSs) of α-B₆O and β-B₆O at ambient pressure (Fig. 4), we once again see a great degree of similarity. In order to further confirm the stability of β-B₆O, we have calculated the Gibbs free energy of β-B₆O and α-B₆O as a function of temperature, shown in Supplementary Fig. S2. We conclude that β-B₆O remains more stable than α-B₆O also when temperature is taken into account – and is even slightly stabilized by thermal effects.

Figure 4: Phonon density of states of α-B₆O phase and Cmcm-B₆O phase at ambient pressure.

For clarity, the PDOS of α-B₆O was shifted.

Having found that β-B₆O should be more stable than α-B₆O, and demonstrated that the two phases actually coexist in experimental samples, we ask a question: why is the synthetic compound mostly α-B₆O, instead of the more stable β-B₆O? While there can be no definitive answer at this point, we suggest that this may be because of the use of α-rhombohedral-B (Rm)⁹ or β-rhombohedral-B (Rm)²² as a starting material (together with B₂O₃) for synthesis of rhombohedral-B₆O. Hubert et al.²³ used amorphous-B as the starting material, but they got not only B₆O but also B₆O twinning particles and some amorphous phases. And they observed a stacking of ABAB… around the planar defect, which is the same as β-B₆O (We speculate it probably is β-B₆O). To obtain β-orthorhombic-B₆O, one would need to crystallize it from the melt (preferably at high pressure, to increase its thermodynamic advantage over α-B₆O), or use other phases of boron as precursors. Using Cmcm-B would be ideal, but this phase (just 11 meV/atom higher in energy than α-B¹⁹) remains hypothetical, though likely to be eventually synthesized. Alternatively, the PVT method²⁰ could be used.

In summary, to our big surprise, ab initio structure prediction calculations discovered a new ground state for the widely studied superhard compound B₆O – our predicted β-B₆O is more stable than experimentally known α-B₆O. The two phases are polytypes and have nearly the same densities (β-B₆O is slightly denser), energies (slightly lower for β-B₆O), band gaps (slightly smaller and indirect, rather than direct, for β-B₆O), hardnesses (β-B₆O is slightly harder) and phonon densities of states, but have important differences in Raman spectra. By comparing calculated and experimental Raman spectra, we demonstrated that the experimental samples are actually a mixture of α-B₆O and β-B₆O. The discovery of β-B₆O opens up new possibilities, in view of its greater stability, hardness and indirect band gap. Our findings also indicate possibilities of tuning the properties of B₆O by obtaining phase-pure samples (probably not obtained to date), and the possibility of metastable oxygen-deficient compounds based on α-B₆O or β-B₆O - these can be obtained at high temperatures (where disordered oxygen vacancies will stabilize the structure) and low chemical potentials of oxygen.

Methods

We used the ab initio evolutionary algorithm USPEX^24,25,26,27 to search for thermodynamically stable B-O compounds and their structures at ambient pressure. This methodology has shown its predictive power in many studies (e.g., ref. 17 and 27, 28, 29). All structures were relaxed; structure relaxations and total energy calculations were done using density functional theory (DFT) within the generalized gradient approximation (GGA)³⁰ as implemented in the VASP code³¹, with the projector-augmented wave method³². We used plane-wave kinetic energy cutoff of 600 eV, and sampled the Brillouin zone with uniform Γ-centered meshes of is 2π*0.07 Å⁻¹ resolution within structure search, and 2π*0.04 Å⁻¹ for subsequent highly precise relaxations and properties calculations. In order to confirm the relative stability of α-B₆O and β-B₆O, we used local density approximation (LDA)³³ and HSE06 hybrid functional³⁴. Phonon spectra was computed by PHONOPY³⁵ and VASP, and Raman spectra were calculated using the Fonari-Stauffer method³⁶. Hardness was calculated with Chen model³⁷ and Lyakhov-Oganov model³⁸. Elastic moduli were computed using Voigt-Reus-Hill averaging³⁹.