Predicted phase diagram of boron-carbon-nitrogen

Noting the structural relationships between phases of carbon and boron carbide with phases of boron nitride and boron subnitride, we investigate their mutual solubilities using a combination of first-principles total energies supplemented with statistical mechanics to address finite temperatures. Thus we predict the solid-state phase diagram of boron-carbon-nitrogen (B-C-N). Owing to the large energy costs of substitution, we find that the mutual solubilities of the ultrahard materials diamond and cubic boron nitride are negligible, and the same for the quasi-two-dimensional materials graphite and hexagonal boron nitride. In contrast, we find a continuous range of solubility connecting boron carbide to boron subnitride at elevated temperatures. An electron-precise ternary compound ${\mathrm{B}}_{13}\mathrm{CN}$ consisting of ${\mathrm{B}}_{12}$ icosahedra with NBC chains is found to be stable at all temperatures up to melting. It exhibits an order-disorder transition in the orientation of NBC chains at approximately $T=500$ K. We also propose that the recently discovered binary ${\mathrm{B}}_{13}{\mathrm{N}}_2$ actually has composition ${\mathrm{B}}_{12.67}{\mathrm{N}}_2$.

Figure 1

Cutaway view of idealized boron carbide structure. The primitive cell is a rhombohedron with a ${\mathrm{B}}_{12}$ icosahedron on each vertex and a three-atom CBC chain along the rhombohedral axis. Intericosahedral bonds along rhombohedron edges connect boron atoms at polar sites. Boron atoms at equatorial sites bond to the chain carbon atoms. Icosahedra at two vertices have been removed for visual clarity.

Figure 2

Boron-rich corner of the B-C-N ternary phase diagram. Black dots show stable structures lying on vertices of the convex hull. Solid lines connecting black dots are convex hull edges. The green circle marks ${\mathrm{B}}_{12.67}{\mathrm{N}}_2$ (i.e., ${\mathrm{B}}_{12}·{\mathrm{NB}}_{0.67}\mathrm{N}$), which lies above the convex hull. Squares indicate ternary structures with enthalpies above the convex hull by $0<\mathrm{\Delta }\mathrm{\Delta }H<20$ meV/atom (blue) and $20<\mathrm{\Delta }\mathrm{\Delta }H$ meV/atom (red). Inset shows the convex hull of the complete ternary.

Figure 3

Electron donation by chains. Electrons are color coded to match the atoms that donated them. The green electrons at top and bottom are donated by equatorial borons on icosahedra (not shown) that are bonded to the chain. Atomic radii and bond lengths are qualitatively accurate. (a) CBC chain donates +1, (b) unbonded N-N chain donates +0, (c) NBN chain donates +3, (d) bonded PP chain donates +2, and (e) NBC chain donates +2.

Figure 4

Enthalpies of B-C showing the ${\mathrm{B}}_{13}{\mathrm{C}}_2$ and ${\mathrm{B}}_4\mathrm{C}$ structures on the convex hull and a scatter plot of higher enthalpy structures. Inset shows the region ${\mathrm{B}}_{13}{\mathrm{C}}_2$ to ${\mathrm{B}}_4\mathrm{C}$ with structures restricted to CBC chains and a single choice of polar carbon site. $\mathrm{\Delta }\mathrm{\Delta }H$ is enthalpy relative to the tie line joining competing compounds. The red line is the parabolic approximation to $\langle E\rangle$ at $T=2000$ K.

Figure 5

Convex hull, enthalpy of BN binary at $P=0$ and 5 Gpa. Vertices of the convex hull are highlighted as solid dots. The inset shows $\mathrm{\Delta }\mathrm{\Delta }\mathrm{H}$ for rhombohedral structures with stoichiometry ${\mathrm{B}}_{13}{\mathrm{N}}_2, {\mathrm{B}}_{12.67}{\mathrm{N}}_2$, and ${\mathrm{B}}_{12}{\mathrm{N}}_2$, from left to right, respectively.

Figure 6

Heat capacity $C$ and susceptibility ${\chi }_z$ of ${\mathrm{B}}_{12}·\mathrm{NBC}$, evaluated in a $2\times 2\times 2$ supercell.

Figure 7

Relative enthalpies $\mathrm{\Delta }\mathrm{\Delta }H$ along line ${\mathrm{B}}_{12}·\mathrm{NBC}-{\mathrm{B}}_{38}{\mathrm{N}}_6$. Points are calculated supercell structures; stars are reference (convex hull) states. Curve is $\mathrm{\Delta }\mathrm{\Delta }H$ derived from Eq. (2) at $T=2000$ K.

Figure 8

(a) Free-energy curves over narrow temperature range with two miscibility ranges. Orange dashed line shows the convex hull for $T=880$ K. Inset shows free-energy curves over broad temperature range from low to complete miscibility. (b) Vertical section with temperature-dependent solubility region shaded.

Figure 9

Solubility range (shown as blue) at $T=300$ K (top left), $T=1000$ K (top right), $T=1500$ K (bottom left), and $T=2500$ K (bottom right).