Structural models of increasing complexity for icosahedral boron carbide with compositions throughout the single-phase region from first principles

We perform first-principles calculations to investigate the phase stability of boron carbide, concentrating on the recently proposed alternative structural models composed not only of the regularly studied ${\mathrm{B}}_{11}{\mathrm{C}}^p$(CBC) and ${\mathrm{B}}_{12}$(CBC), but also of ${\mathrm{B}}_{12}$(CBCB) and ${\mathrm{B}}_{12}({\mathrm{B}}_4$). We find that a combination of the four structural motifs can result in low-energy electron precise configurations of boron carbide. Among several considered configurations within the composition range of ${\mathrm{B}}_{10.5}\mathrm{C}$ and ${\mathrm{B}}_4\mathrm{C}$, we identify in addition to the regularly studied ${\mathrm{B}}_{11}{\mathrm{C}}^p$(CBC) at the composition of ${\mathrm{B}}_4\mathrm{C}$ two low-energy configurations, resulting in a new view of the B-C convex hull. Those are $[{\mathrm{B}}_{12}$(CBC)${\mathrm{]}}_{0.67}{\left[{\mathrm{B}}_{12}\left({\mathrm{B}}_4\right)\right]}_{0.33}$ and $[{\mathrm{B}}_{12}$(CBC)${\mathrm{]}}_{0.67}[{\mathrm{B}}_{12}$(CBCB)${\mathrm{]}}_{0.33}$, corresponding to compositions of ${\mathrm{B}}_{10.5}\mathrm{C}$ and ${\mathrm{B}}_{6.67}\mathrm{C}$, respectively. As a consequence, ${\mathrm{B}}_{12}$(CBC) at the composition of ${\mathrm{B}}_{6.5}\mathrm{C}$, previously suggested in the literature as a stable configuration of boron carbide, is no longer part of the $\mathrm{B}-\mathrm{C}$ convex hull. By inspecting the electronic density of states as well as the elastic moduli, we find that the alternative models of boron carbide can provide a reasonably good description for electronic and elastic properties of the material in comparison with the experiments, highlighting the importance of considering ${\mathrm{B}}_{12}$(CBCB) and ${\mathrm{B}}_{12}({\mathrm{B}}_4$), together with the previously proposed ${\mathrm{B}}_{11}{\mathrm{C}}^p$(CBC) and ${\mathrm{B}}_{12}$(CBC), as the crucial ingredients for modeling boron carbide with compositions throughout the single-phase region.

Figure 1

Four structural motifs, used to model boron carbide in the present work, i.e., (a) ${\mathrm{B}}_{11}{\mathrm{C}}^p$(CBC), (b) ${\mathrm{B}}_{12}$(CBC), (c) ${\mathrm{B}}_{12}({\mathrm{B}}_4$), and (d) ${\mathrm{B}}_{12}$(CBCB). Instead of a linear chain, shown in (a), (b), and (d), the ${\mathrm{B}}_4$ unit in (c) forms a rhombic chain. Red and white spheres represent B and C atoms, respectively.

Figure 2

Density of states of (a) $12{\mathrm{B}}_{12}$(CBC), (b) $11{\mathrm{B}}_{12}$(CBC) $+ {\mathrm{B}}_{12}$(CBCB), and (c) $11{\mathrm{B}}_{12}$(CBC) $+{\mathrm{B}}_{12}({\mathrm{B}}_4$). The highest occupied state is located at 0 eV and indicated by the dashed line.

Figure 3

Formation energy $\mathrm{\Delta }{E}^{\mathrm{form}}$ of boron carbide, calculated with respect to $\alpha$ boron and diamond. A set of thick solid lines, connecting filled black circles at ${\mathrm{B}}_{10.5}\mathrm{C}, {\mathrm{B}}_{6.67}\mathrm{C}$, and ${\mathrm{B}}_4\mathrm{C}$, indicates the $\mathrm{B}-\mathrm{C}$ convex hull, derived from all of the configurations, listed in Table 1. A set of dashed lines, on the other hand, indicates the previously proposed convex hull, consisting of ${\mathrm{B}}_{6.5}\mathrm{C}$, given by ${\mathrm{B}}_{12}$(CBC) (open black circle) and ${\mathrm{B}}_4\mathrm{C}$.

Figure 4

GGA-PBE96-calculated electronic density of states of (a) ${\mathrm{B}}_{6.67}\mathrm{C}$ or $[{\mathrm{B}}_{12}$(CBC)${\mathrm{]}}_{0.67}[{\mathrm{B}}_{12}$(CBCB)${\mathrm{]}}_{0.33}$ and (b) ${\mathrm{B}}_{10.5}\mathrm{C}$ or $[{\mathrm{B}}_{12}$(CBC)${\mathrm{]}}_{0.67}{\left[{\mathrm{B}}_{12}\left({\mathrm{B}}_4\right)\right]}_{0.33}$. The highest occupied state is located at 0 eV. The black solid and red dashed lines denote, respectively, the density of states of the lowest-energy configurations, identified in Sec. 3b, and of that imitating the random alloy pattern, respectively.

Figure 5

Young's modulus of boron carbide, plotted as a function of carbon content (left $y$ axis). The experimentally measured Young's modulus (filled black circles) are taken from the work of Gieske et al. [53], while the theoretical values (filled and open red squares) are obtained from the calculations in the present and our previous work [32]. Filled blue triangles represent the experimental reduced Young's modulus of boron carbide at different carbon content (right $y$ axis), reported by Cheng et al. [60].

Figure 6

(a) Superatomic ground-state configuration of $[{\mathrm{B}}_{12}$(CBC)${\mathrm{]}}_{0.67}[{\mathrm{B}}_{12}$(CBCB)${\mathrm{]}}_{0.33}$, predicted from Monte Carlo simulations (the 22nd configuration, listed in Table 3), (b) superatomic configuration of $[{\mathrm{B}}_{12}$(CBC)${\mathrm{]}}_{0.67}[{\mathrm{B}}_{12}$(CBCB)${\mathrm{]}}_{0.33}$, imitating the SRO parameters of (a) up to the fourth coordination shell (the 23rd configuration, listed in Table 3), and (c) proposed primitive unit cells of the ground state of $[{\mathrm{B}}_{12}$(CBC)${\mathrm{]}}_{0.67}[{\mathrm{B}}_{12}$(CBCB)${\mathrm{]}}_{0.33}$, derived from (b) and represented by a supercell composed of $4\times 3\times 3$ rhombohedral primitive unit cells of boron carbide. The red and yellow spheres represent ${\mathrm{B}}_{12}$(CBC) and ${\mathrm{B}}_{12}$(CBCB) superatoms, respectively.