⇦Chinese Physics Letters, 2019, Vol. 36, No. 1, Article code 013101 Analysis of Transition Mechanism of Cubic Boron Nitride Single Crystals under High Pressure-High Temperature with Valence Electron Structure Calculation * Mei-Zhe Lv (吕美哲)1, Bin Xu (许斌)2**, Li-Chao Cai (蔡立超)1, Feng Jia (贾凤)2, Xing-Dong Yuan (袁兴栋)2 Affiliations 1School of Materials Science and Engineering, Shandong University, Jinan 250101 2School of Materials Science and Engineering, Shandong Jianzhu University, Jinan 250101 Received 13 September 2018, online 25 December 2018 *Supported by the National Natural Science Foundation of China under Grant No 51272139.
**Corresponding author. Email: xubin@sdjzu.edu.cn Citation Text: Lv M Z, Xu B, Cai L C, Jia F and Yuan X D et al 2019 Chin. Phys. Lett. 36 013101    Abstract The possibilities of hexagonal boron nitride (hBN) and lithium boron nitride (Li$_{3}$BN$_{2}$) transition into cubic boron nitride (cBN) under synthetic pressure 5.0 GPa and synthetic temperature 1700 K are analyzed with the use of the empirical electron theory of solids and molecules. The relative differences in electron density are calculated for dozens of bi-phase interfaces hBN/cBN, Li$_{3}$BN$_{2}$/cBN. These relative differences of hBN/cBN are in good agreement with the first order of approximation ($ < $10%), while those of Li$_{3}$BN$_{2}$/cBN are much greater than 10%. This analysis suggests that Li$_{3}$BN$_{2}$ is impossible to be intermediate phase but is a catalyst and cBN should be directly transformed by hBN. DOI:10.1088/0256-307X/36/1/013101 PACS:31.15.bu, 64.70.mf, 61.50.Ah, 31.15.bt © 2019 Chinese Physics Society Article Text At present, the static high pressure-high temperature (HPHT) catalytic method is most commonly used to synthesize cubic boron nitride (cBN) with hexagonal boron nitride (hBN) as starting materials and with lithium nitride as catalysts on an industrial scale.[1-3] The transition mechanism has been discussed in some studies and some models are proposed by considering the possibility of phase transition between similar structures or by presuming the transition process with some indirect experimental results.[4,5] The two main transition mechanisms are that cBN can be directly transformed from hBN and cBN can be precipitated from solvent.[6,7] Namely, in the Li$_{3}$N-hBN system, cBN can be transformed by hBN or be dissolved by Li$_{3}$BN$_{2}$. Thus, it is very desirable to study the relationships of hBN$\to$cBN, Li$_{3}$BN$_{2}$$\to$cBN+Li$_{3}$N to understand the phase transition mechanism of cBN. The empirical electron theory (EET) of molecules and solids[8] is established by Yu, based on Pauling's valence bond theory. The EET has successfully been applied to many aspects of materials research (e.g., phase diagrams, phase transition, hot working, physical properties).[9-11] However, the phase transition of cBN is rarely reported with the use of EET. The electron structures and electron densities of solid and molecules can be calculated by means of the bond length difference method (BLD) presented in EET. Meanwhile, the improved Thomas–Fermi–Dirac theory by Cheng (TFDC)[12] proposes that the boundary condition between atoms is simply the continuity of the electronic densities. Electron density is considered to be a bridge connecting EET and TFDC,[13] thus the two theories could be used to discuss the cBN transition mechanism by analyzing the continuity of electron density between bi-phase interfaces hBN/cBN and Li$_{3}$BN$_{2}$/cBN under the conditions of cBN synthesis. The experimental bond length $D_{\rm n\alpha}$ in lattice structures can be confirmed, knowing the atom sites and the lattice parameters. The fractional coordinates of atoms u and v in crystal structures can be expressed in ($x_{\rm u} y_{\rm u} z_{\rm u}$) and ($x_{\rm v} y_{\rm v} z_{\rm v}$), respectively. Due to the EET, the bonds ($D_{\rm n\alpha}>0.5$ nm) rarely have an effect on the results, and thus can be ignored. The value of $D_{\rm n\alpha}$ can be obtained by $$\begin{align} D_{\rm n\alpha}^{\rm u-v} =\,&[(x_{\rm u} -x_{\rm v})^{2}a^{2}+(y_{\rm u} -y_{\rm v})^{2}b^{2}\\ &+(z_{\rm u} -z_{\rm v})^{2}c^{2}]^{1/2}.~~ \tag {1} \end{align} $$ According to the main equations for the BLD method in EET, Eq. (1) can be expressed as $$\begin{align} D_{\rm nA}^{\rm u-v} =\,&R_{\rm u} (1)+R_{\rm v} (1)-\beta \log n_{\rm A},~~ \tag {2} \end{align} $$ $$\begin{align} D_{\rm n\alpha}^{\rm s-t} =\,&R_{s} (1)+R_{\rm t} (1)-\beta \log n_{\rm \alpha} {\rm ~~(\alpha =B\ldots N)},~~ \tag {3} \end{align} $$ $$\begin{align} \log \frac{n_{\rm \alpha}}{n_{\rm A}}=\,&\{(D_{\rm nA}^{\rm u-v} -D_{\rm n\alpha}^{\rm s-t})+(R_{s} (1)+R_{\rm t} (1))\\ &-(R_{\rm u} (1)+R_{\rm v} (1))\}/\beta,~~ \tag {4} \end{align} $$ where $n_{\rm \alpha}$ is the covalent electron pairs of the $\alpha$ bond, $R(1)$ is the bond radius which can be gained from the hybridization table in EET, and $\beta$ is a coefficient. The ratio of covalent electron distribution on bonds $\gamma_{\rm \alpha} =\frac{n_{\rm \alpha}}{n_{\rm A}}$ can be calculated by Eq. (4), which can be obtained by subtracting Eq. (2) from Eq. (3). The values of $n_{\rm A}$ and $n_{\rm \alpha}$ can be calculated by $$\begin{align} {\Sigma} n_{\rm c} ={\Sigma} I_{\rm \alpha} n_{\rm \alpha} =n_{\rm A} {\Sigma} I_{\rm \alpha} \gamma_{\rm \alpha},~~ \tag {5} \end{align} $$ where $n_{\rm c}$ is the total valence electron number of a crystal plane, which is equal to the total number of electrons distributed on all covalent bonds, $I_{\rm \alpha}$ represents equivalent bond number of bond ${\rm \alpha}$ on the plane, which means that the covalent bonds have the equivalent bond length and uniform environment. According to EET, the covalent electron density of a certain crystal plane can be calculated by $$\begin{align} \rho =\frac{{\Sigma} n_{\rm c}}{S}=\frac{{\Sigma} I_{\rm \alpha} n_{\rm \alpha}}{S},~~ \tag {6} \end{align} $$ where $S$ is the area of the crystal plane. The relative differences in electron density of an interface can be defined as follows: $$\begin{align} \Delta \rho =\frac{| {\rho_{1} -\rho_{_{2}}} |}{(\rho_{1} +\rho_{2})/2}\times 100\%.~~ \tag {7} \end{align} $$ The bond energy $E_{\rm a}$ of covalent bond formed by two atoms (u and v) in crystals is calculated by $$\begin{align} E_{\rm a} =\sqrt {b_{\rm u} b_{\rm v}} \cdot \sqrt {f_{\rm u} f_{\rm v}} \frac{n_{\rm \alpha}}{\bar{D_{\rm \alpha}}},~~ \tag {8} \end{align} $$ where $b$ is a coefficient, which represents the shielding factor of the electron to nuclear charge, whose value can be found in Ref. [8]. The value of $f$ represents the bond-forming ability of the atom hybrid orbital, which can be gained by $$\begin{align} f=\,&\sqrt \alpha +\sqrt {3\beta} +\sqrt {5\gamma},\\ \alpha =\,&\frac{n_{s}}{n_{\rm T}},~\beta =\frac{n_{p}}{n_{\rm T}},~\gamma =\frac{n_{d}}{n_{\rm T}},~~ \tag {9} \end{align} $$ where $n_{s}$, $n_{p}$, $n_{d}$ represent the valence electrons of $s$, $p$ and $d$, respectively, and $n_{\rm T}$ represents the total valence electron number, whose values are gained in Ref. [14]. The BLD method is excessively dependent on the accurate lattice parameters ($a$, $b$, $c$), which are influenced by the promotion of synthetic pressure and synthetic temperature. Different p-T areas of cBN growth in the Li$_{3}$N-hBN system have been proposed, which may be related to the purity and degree of order of hBN and the synthesis process. Considering the real synthesis condition and previous experimental results, 5.0 GPa and 1700 K are selected as the calculated pressure and calculated pressure, because stable cBN can exist in this region. In our previous work,[15] the calculations of lattice parameters have been performed using the Vienna ab-initio simulation package (VASP). The projector-augmented wave with generalized gradient approximation (PAW-PBE) was employed separately to calculate hBN, cBN and Li$_{3}$BN$_{2}$ at HPHT (5.0 GPa and 1700 K). A plane wave kinetic energy cut-off value of 700 eV was correspondingly used to ensure a total-energy convergence of 10$^{-5}$ eV/atom, and $9\times9\times3$, $9\times9\times9$, $9\times9\times8$ Monkhorst-pack $k$-points meshes were, respectively, employed to hBN, cBN and Li$_{3}$BN$_{2}$. The calculated results are in good agreement with the previous experimental studies, and this accrues the reliability of the calculation methods employed in this work for the lattice parameters of three structures. The lattice parameters of hBN are $a=0.24940$ nm, $c=0.60627$ nm, the lattice parameter of cBN is 0.36352 nm, and the lattice parameters of Li$_{3}$BN$_{2}$ are $a=0.44378$ nm, $c=0.50265$ nm at 5.0 GPa and 1700 K. The BLD method is efficient to determine the state group of every possible existent atom which must satisfy the difference between $D_{\rm n\alpha}$ and theoretical bonding length $\overline D_{\rm n\alpha}$ is lower than 0.005 nm ($\Delta D_{\rm n\alpha} =| {\overline D_{\rm n\alpha} -D_{\rm n\alpha}} | < 0.005$ nm) and this means that the condition for stable valence electron structure of a crystal is $\Delta D_{\rm n\alpha} < 0.005$ nm. Using the BLD method, their (hBN, cBN and Li$_{3}$BN$_{2}$) average valence electron structures of all qualified hybrid order could be obtained. The average valence electron structures of all qualified hybrid order are more accurate to analyze the properties and behavior of crystals.[9-11] The transition to cBN starts with displacement of atoms of the hBN or Li$_{3}$BN$_{2}$ phases which depends on their stability. The larger the structure stability, the greater the difficulty to break up the covalent bonds of structure units. The formation and breakup of covalent bonds of structure units can be characterized by the covalent electron pairs ($n_{\rm A}$) and the bond energy ($E_{\rm A}$) of the strongest covalent bond. For hBN and Li$_{3}$BN$_{2}$, the strongest covalent bonds are both B–N bonds. Their corresponding values of $n_{\rm A}$ (hBN and Li$_{3}$BN$_{2}$) are 1.20756, 2.32482. According to Eq. (8), $E_{\rm A}$ for hBN and Li$_{3}$BN$_{2}$ are calculated to be 166.28 kJ/mol and 345.05 kJ/mol, respectively. The results show that the values of $n_{\rm A}$ and $E_{\rm A}$ of Li$_{3}$BN$_{2}$ are much greater than those of hBN, which means that Li$_{3}$BN$_{2}$ is more stable than hBN under the condition of cBN transition. Because $n_{\rm A}$ and $E_{\rm A}$ are larger,[10,11] a greater driving force is needed to break up the bond of $D_{\rm nA}$, thus the transition of Li$_{3}$BN$_{2}$$\to$ cBN is more difficult than the reaction of hBN$\to$cBN.
Table 1. Covalent electron density of crystal planes in hBN at 5.0 GPa and 1700 K.
Crystal planes Bond $D_{\rm n\alpha}$ (nm) $n_{\rm \alpha}$ $I_{\rm \alpha}$ $\sum n_{\rm c}$ $S$ (nm$^{2}$) $\rho$ (nm$^{-2}$)
$(0001)$ A 0.14399 1.20756 6 7.80459 0.05387 144.88611
B 0.24940 0.05096 6
C 0.25147 0.03092 6
D 0.28798 0.01132 6
$(10\bar{1}0)$ B 0.24940 0.05096 2 0.19451 0.15121 1.28636
C 0.24940 0.03092 2
E 0.30314 0.00692 4
H 0.39254 0.00038 8
$(11\bar{2}0)$ A 0.14399 1.20756 4 4.95231 0.26190 18.90941
D 0.28798 0.01132 4
E 0.28798 0.00692 8
F 0.30314 0.00311 4
G 0.33560 0.00189 4
I 0.41812 0.00021 4
J 0.41812 0.00013 4
$(01\bar{1}2)$ B 0.24940 0.05096 2 0.16529 0.09842 1.67947
C 0.25147 0.03092 2
H 0.39254 0.00038 4
Table 2. Covalent electron density of crystal planes in cBN at 5.0 GPa and 1700 K.
Crystal planes Bond $D_{\rm n\alpha}$ (nm) $n_{\rm \alpha}$ $I_{\rm \alpha}$ $\sum n_{\rm c}$ $S$ (nm$^{2}$) $\rho$ (nm$^{-2}$)
(100)-B B 0.25705 0.04236 8 0.34959 0.13215 2.64547
E 0.36352 0.00134 8
(100)-N C 0.25705 0.02654 8 0.21901 0.13215 1.65734
F 0.36352 0.00084 8
(111)-B B 0.25705 0.04236 12 0.50943 0.11444 4.45144
H 0.44522 0.00009 12
(111)-N C 0.25705 0.02654 12 0.31915 0.11444 2.78874
I 0.44522 0.00006 12
(110) A 0.15741 0.84576 8 7.11424 0.18688 38.06774
B 0.25705 0.04236 4
C 0.25705 0.02654 4
D 0.30141 0.00793 8
E 0.36352 0.00134 4
F 0.36352 0.00084 4
I 0.44522 0.00006 8
cpl-36-1-013101-fig1.png
Fig. 1. (a) The main low-index planes of hBN crystals. (b)–(e) The distributions of atoms and covalent bonds in $(0001)$, $(10\bar{1}0)$, $(11\bar{2}0)$ and $(01\bar{1}2)$ planes of hBN, respectively.
cpl-36-1-013101-fig2.png
Fig. 2. (a) The main low-index planes of cBN crystals. (b)–(f) The distributions of atoms and covalent bonds in (100)-B, (100)-N, (110), (111)-B and (111)-N planes of cBN, respectively.
The lattice electron density can be obtained by Eq. (6), which is the ratio of the total lattice electron number and the total valence electron number in a cell. The three selected planes of Li$_{3}$BN$_{2}$ contain maximum the number of B atoms or N atoms. It is worth mentioning that as the (100) and (111) planes of cBN are polar, there are in total five structural models about cBN. Figures 1–3 show the distribution of atoms and covalent bonds in main low index planes of hBN, cBN and Li$_{3}$BN$_{2}$, respectively. Based on the calculation in Eqs. (6) and (7), the electron densities of different crystal planes in hBN, cBN and Li$_{3}$BN$_{2}$ crystals are listed in Tables 1–3, respectively, and Tables 4 and 5 show the relative differences in electron density of different bi-phase interfaces in hBN/cBN and Li$_{3}$BN$_{2}$/cBN, respectively. As listed in Table 4, the value about relative difference in electron density at hBN $(01\bar{1}2)$/cBN (100)-N is 1.33%, which is lower than 10%. According to the TFDC, the electron density of (100) plane in hBN is continuous with (100) plane in cBN at the first order of approximation ($ < $10%). The lower $\Delta \rho$ of two adjacent planes, the closer the covalent electron structure of interface, and the lower the driving force for transforming one structure into another. This indicates that driving force required for the direct transformation from hBN to cBN is low enough, and the differences of valence electron structure at hBN/cBN interface could induce direct transformation from hBN into cBN.
cpl-36-1-013101-fig3.png
Fig. 3. (a) The main low-index planes of Li$_{3}$BN$_{2}$ crystals. (b)–(d) The distributions of atoms and covalent bonds in (100), (001) and (110) planes of Li$_{3}$BN$_{2}$, respectively.
As listed in Table 5, the minimum relative difference in electron density of all bi-phase interfaces at Li$_{3}$BN$_{2}$/cBN is 49.47%, which does not satisfy the first order of approximation ($>$10%). This indicates that Li$_{3}$BN$_{2}$ cannot be dissolved into cBN as the intermediate phase under the condition of 5.0 GPa and 1700 K. The value of $\Delta \rho$ of Li$_{3}$BN$_{2}$/cBN interfaces is larger, and thus the change of atom size on interface is greater. Therefore, a larger driving force of phase transition is needed under the HPHT synthetic process. These results are consistent with the previous thermal results[16] that the Gibbs free energy of hBN$\to$cBN are negative, while those of Li$_{3}$BN$_{2}$$\to$cBN are positive at the synthetic temperature and synthetic pressure of cBN single crystals. Because of its similar structure with cBN and larger lattice parameters, Li$_{3}$BN$_{2}$ can be the matrix of cBN nucleation, and thus decreases the surface energy between hBN and cBN.[17] The lattice of Li$_{3}$BN$_{2}$ is composed of Li1, Li2 and NBN$^{-3}$ ions. The Li ions can attract outer electrons in N atoms from hBN and transfer them to B atoms under HPHT. The electron structures of B and N atoms have been changed from the $sp^{2}$ hybridized state into the $sp^{3}$ hybridized state.[16] The crystal defects trend to agglomerate at interfaces where $\Delta \rho$ is great,[14] hence, Li$_{3}$BN$_{2}$ can be adsorbed to the interior of cBN crystals, leading to the generation of internal defects in cBN crystals, which is beneficial for cBN gradual growth. Thus cBN single crystals are transformed from hBN directly with the catalysis of Li$_{3}$BN$_{2}$ at HPHT.
Table 3. Covalent electron density of common crystal planes in Li$_{3}$BN$_{2}$ at 5.0 GPa and 1700 K.
Crystal planes Bond $D_{\rm n\alpha}$ (nm) $n_{\rm \alpha}$ $I_{\rm \alpha}$ $\sum n_{\rm c}$ $S$ (nm$^{2}$) $\rho$ (nm$^{-2}$)
(001) A 0.13428 2.32482 4 13.40268 0.21701 61.76156
B 0.19512 1.02226 4
L 0.35453 0.00198 4
O 0.38288 0.00073 8
Y 0.46582 0.00005 4
Z1 0.46582 0.00005 8
(110) A 0.13428 2.32482 4 14.49779 0.34761 41.70731
B 0.19512 1.02226 4
D 0.26381 0.10256 8
H 0.29601 0.03877 4
I 0.32812 0.00433 8
J 0.32940 0.01222 8
R 0.42201 0.00019 8
(100) D 0.26381 0.10256 4 0.51130 0.24580 2.08020
E 0.26381 0.00035 4
F 0.26769 0.00933 8
G 0.26769 0.00296 8
W 0.45920 0.00001 2
X 0.46582 0.00048 2
Y 0.46582 0.00005 8
Table 4. Relative differences in electron density of hBN/cBN interfaces at 5.0 GPa and 1700 K. Here h represents hBN, and c represents cBN.
$\Delta \rho$ (%) ${\rm h}(0001)$ ${\rm h}(10\bar{1}0)$ ${\rm h}(11\bar{2}0)$ ${\rm h}(01\bar{1}2)$
c(100)-B 192.83 69.13 150.91 44.67
c(100)-N 195.48 25.20 167.77 1.33
c(111)-B 188.08 110.32 123.78 90.43
c(111)-N 192.45 73.73 148.59 49.65
c(110) 116.73 186.93 67.30 183.11
Table 5. Relative differences in electron density of Li$_{3}$BN$_{2}$/cBN interfaces at 5.0 GPa and 1700 K. Here c represents cBN, and L represents Li$_{3}$BN$_{2}$.
$\Delta \rho$ (%) L(001) L(110) L(100)
c(100)-N 90.27 190.61 189.69
c(100)-B 49.47 194.06 193.48
c(111)-B 126.62 184.44 182.95
c(111)-N 94.42 190.11 189.15
c(110) 189.77 97.12 89.67
By calculating $\Delta \rho$ of hBN/cBN and Li$_{3}$BN$_{2}$/cBN, it can be concluded that the reaction of hBN$\to$cBN could be performed, while the reaction of Li$_{3}$BN$_{2}$$\to$cBN is impossible under the condition of 5.0 GPa and 1700 K. The values of $n_{\rm A}$ and $E_{\rm A}$ of Li$_{3}$BN$_{2}$ are much larger than those of hBN, thus the greater driving force is needed for breaking up the bond of Li$_{3}$BN$_{2}$. These results show that cBN comes directly from the transformation of hBN but not from the decomposition of Li$_{3}$BN$_{2}$. References Auger electron spectroscopy analysis for growth interface of cubic boron nitride single crystals synthesized under high pressure and high temperatureCO2-laser-induced gas-phase synthesis of micron-sized diamond powders: recent results and future developmentsX-ray photoelectron spectroscopy study of cubic boron nitride single crystals grown under high pressure and high temperatureThe equilibrium phase boundary between hexagonal and cubic boron nitrideCharacterization of Growth Interface for Cubic Boron Nitride Synthesized by the Static High Temperature-High Pressure Catalytic MethodA density functional theory study on parameters fitting of ultra long armchair ( n , n ) single walled boron nitride nanotubesAnalysis of the effect of alloy elements on allotropic transformation in titanium alloys with the use of cohesive energyA gradient nanostructure generated in pure copper by platen friction sliding deformationCalculation of the cohesive energy of solids with the use of valence electron structure parametersBoundary conditions of electrons at the interfaceLattice Parameters of Hexagonal and Cubic Boron Nitrides at High Temperature and High PressureFracture morphology and XRD layered characterization of cBN CakeHigh- and low-temperature phases of lithium boron nitride, Li3BN2: Preparation, phase relation, crystal structure, and ionic conductivity
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