The Analysis of the Transformation Mechanism of cBN Crystals with the First-Principle Calculation

To clarify the synthesis mechanism of cubic boron nitride (cBN) with catalysts at high temperature and high pressure, we calculate the surface energy of the main phases in the Li-N-B synthesis system using the first-principle method. Based on the density functional theory, the surface energy of low-index surfaces of cBN, hexagonal boron nitride (hBN), and lithium boron nitride (Li3BN2) at the cBN synthetic temperature of 1700K and synthetic pressure of 5.0GPa is calculated. +e surface energy of the main low-index surfaces of cBN is σ (111)> σ (001)> σ (110), that of hBN is σ (1010)> σ (1120)> σ (0001), and that of Li3BN2 is σ (100)> σ (110)> σ (001). +e energy orders of the main low-index surfaces were well contrary to the corresponding orders of the valence electron density of the low-index surfaces of cBN, hBN, and Li3BN2, which were calculated by the empirical electron theory (EET) of solids and molecules. +e result shows that the calculation results in this paper are well consistent with the previous results of the EET theory and support for the results of the “direct transformation of hBN to cBN under the catalysis of Li3BN2” obtained by the EET theory.


Introduction
Cubic boron nitride (cBN) single crystals have a high hardness value, high melting point, high thermal conductivity, wide energy gap, and low dielectric constant, which make them highly attractive as promising materials [1][2][3][4]. At present, cBN single crystals are synthesized using hexagonal boron nitride (hBN) as raw materials and lithium nitride (Li 3 N) as catalysts by the static high-temperature and highpressure (HPHT) catalytic method [5][6][7]. However, the present cBN single crystals still cannot meet the demand for advanced product development.
e clear transformation mechanism of cBN crystals is significant to determine the quality of cBN crystals at HPHT. e research on the material surface and material catalysis mainly adopts the methods of characterization and theoretical research.
In our previous experimental study, the main phases are hBN, cBN, and Li 3 BN 2 in the interface layers [8]. e empirical electron theory (EET) of solids and molecules has been successfully applied to the analysis of phase diagrams, phase transformation, and natural properties [9][10][11]. In the previous study, the transformation possibilities from hBN to cBN and from Li 3 BN 2 to cBN were discussed with the valence electron structure [9]. However, the first-principle calculations based on the density functional theory became the most successful method and were applied widely in material calculations [12,13]. e most stable surface of cBN crystals has been reported in detail using the CASTEP code [14,15]. Meanwhile, the analysis about properties of (110) surfaces of cBN crystals has been discussed systematically [16]. e characters and properties of (111) and (100) of cBN crystals have also few reports in the previous studies [17,18].
However, the information of surfaces for hBN and Li 3 BN 2 phases is barely referred in the previous studies. us, it is undoubtedly necessary to analyze the surface stability of three phases systematically, which helps understand the transformation mechanism of cBN crystals.

Materials and Methods
All calculations in the study were performed in the framework of the density functional theory using the VASP code (Vienna ab initio simulation program). e surface energies of lower-index surface planes on the three main phases, hBN, cBN, and Li 3 BN 2 , were calculated systematically [8]. An energy cutoff of 550 eV for the plane-wave basis was sufficient to obtain converged structural properties. Brillouin zone sampling was set with the 9 × 9 × 1 k-point Monkhorst Pack grid meshes for all surfaces and with 9 × 9 × 3, 9 × 9 × 9, and 9 × 9 × 8 k-points for hBN, cBN, and Li 3 BN 2 bulks, respectively. For Perdew-Burke-Ernzerhof (PBE) calculations, the lattice constants used were as follows: a � 2.505Å and c � 6.601Å for hBN, a � 3.615Å for cBN, and a � 4.643Å and c � 5.259Å for Li 3 BN 2 [14,15].
After structural relaxation, the surface energies (σ) [16,17] of cBN, hBN, and Li 3 BN 2 crystals were calculated as follows: where A is the area of the surface unit cell, E slab is the total energy of the slab after relaxation, and E bulk is the total energy of the crystals after relaxation. e main lower-index surfaces (100) (100) and (111), as shown in Figure 2, i.e., including (110), B-and N-terminated (100), and B-and N-terminated (111) surfaces. ese surface energies (σ) [19,20] of cBN single crystals can be calculated as follows: where N B and N N are the number of B and N atoms in the surface slab, respectively; μ bluk is the total energy of an atom in a bulk crystal at T � 0 K, and μ slab N is the chemical potential of nitrogen. Figure 4 shows the surface energies of different cBN singlecrystal models as a function of the N chemical potential in its allowed range. e calculated (110) surface energy was 0.175 eV/Å 2 in this work, which is in good agreement with the calculation results in [14][15][16]. Figure 4 shows that all lower surface energies of cBN single crystals are well consistent with those reported in previous theoretical studies [15]. e accuracy of the calculated cBN surface energy results ensures the validity of the calculation method used in the present study. For further analysis, the surface energies of cBN (111) and (100) are defined by the average surface energy of their B-terminated and N-terminated surfaces. erefore, the main lower-index surface energies of cBN, hBN, and Li 3 BN 2 at the cBN synthetic temperature of 1700 K and synthetic pressure of 5.0 GPa are obtained by the firstprinciple method. e main average surface energies of the lower-index surfaces (100), (111), and (110) at 1700 K and 5.0 GPa for cBN are 0.475 eV/Å 2 , 0.436 eV/Å 2 , and 0.25 eV/Å 2 , respectively. us, the order of the average lower-index surface energy of cBN at 1700 K and 5.0 GPa is σ (100) > σ (111) > σ (110). According to the previous EET study [9], valence electron density of surface (110) is greater than that of surfaces (111) and (100) at 1700 K and 5.0 GPa, so surface (110) of cBN is the most stable under this condition. Furthermore, the order of the valence electron density of cBN is ρ (110) > ρ (111) > ρ (100) [6], which is completely contrary to the corresponding order of the surface energy of cBN. e higher the valence electron density, the smaller the surface energy and the more stable the surface. is means that the result of the cBN surface energy calculated by the first-principle theory well verifies the conclusion of the valence electron density calculated by the EET theory [9]. e surface energies of the main lower-index surfaces  for Li 3 BN 2 are 0.82 eV/Å 2 , 0.33 eV/Å 2 , and 0.25 eV/Å 2 , respectively. us, the order of the surface energy of Li 3 BN 2 is σ (100) > σ 110) > σ (001). Meanwhile, the order of the valence electron density of hBN and Li 3 BN 2 at 1700 K and 5.0 GPa is ρ (0001) > ρ (1120) > ρ (1010) and ρ (001) > ρ (110) > ρ (100), respectively [9]. It could be concluded that the order of surface energies of hBN and Li 3 BN 2 is also completely contrary to the corresponding order of the valence electron density of hBN and Li 3 BN 2 . us, the surface energies of hBN and Li 3 BN 2 with the first-principle theory are well consistent with those results of the valence electron density by the EET theory. It could be concluded that the consistency between the first-principle theory for the surface energy and the EET theory for the valence electron density further confirms the credibility of the calculation results.

Discussion
Previous studies proved that the transformation possibilities from hBN to cBN are more than those from Li 3 BN 2 to cBN at the cBN synthetic temperature of 1700 K and synthetic pressure of 5.0 GPa, and Li 3 BN 2 could be the real catalyst in the synthesis process of cBN crystals [9,[21][22][23]. e surface energy of the catalyst could directly influence the nucleation rate. Among all values of the surface energy obtained so far, the surface energy of the (100) surface of Li 3 BN 2 is the highest, i.e., 0.82 eV/Å 2 , which is much greater than the other surface energies reported in this work. us, the nucleation of cBN crystals can easily occur around the (100) surface of Li 3 BN 2 ; therefore, the catalyst material Li 3 BN 2 using the (100) surface as the exposed surfaces could be used as substrates to induce effectively the growth of cBN single crystals.

Conclusions
e order of the main lower-index surface energies of cBN is σ (100) > σ (111) > σ (110); the order of the main lower-index surface energies of hBN is σ (1010) > σ (1120) > σ (0001); and the order of the main lower-index surface energies of Li 3 BN 2 is σ (100) > σ (110) > σ (001). e orders of the surface energy of cBN, hBN, and Li 3 BN 2 are all completely contrary to their corresponding orders of the valence electron density. Hence, the surface energies with the firstprinciple theory are well consistent with those results of the valence electron density by the EET. Comparing all surface energies in three phases (hBN, cBN, and Li 3 BN 2 ), the (100) surface of Li 3 BN 2 has the highest surface energy; therefore, it can be used as the substrate to induce effectively the growth of cBN crystals.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.

Authors' Contributions
Lichao Cai contributed to data curation and wrote the original draft; Bin Xu was involved in funding acquisition and supervised the study; Meizhe Lv and Xiaohong Fan investigated the study; Xiaohong Fan contributed to methodology; and Meizhe Lv contributed to writing, reviewing, and editing.