Theoretical study of the structure of boron carbide ${\mathrm{B}}_{13}{\mathrm{C}}_2$

We have resolved long-standing discrepancies between the theoretical and experimental crystal structures of boron carbide ${\mathrm{B}}_{13}{\mathrm{C}}_2$. Theoretical studies predict that ${\mathrm{B}}_{13}{\mathrm{C}}_2$ should be stoichiometric and have the highest symmetry of the boron carbides. Experimentally, ${\mathrm{B}}_{13}{\mathrm{C}}_2$ is a semiconductor and many defect states have been reported, particularly in the CBC chain. Reconciling the disordered states of the chain, the chemical composition, and the lowest-energy state is problematic. We have solved this problem by constructing a structural model where approximately three-quarters of the unit cells contain $\left({\mathrm{B}}_{11}\mathrm{C}\right)$(CBC) and one-quarter of them contain $\left({\mathrm{B}}_{12}\right)\left({\mathrm{B}}_4\right)$. This structural model explains many experimental results, such as the large thermal factors in x-ray diffraction and the broadening of the Raman spectra, without introducing unstable CBB chains. The model also solves the energy-gap problem. We show that there are many arrangements of these two types of unit cells, which are energetically almost degenerate. This demonstrates that boron carbides are well described by a geometrically frustrated system, similar to that proposed for $\beta$-rhombohedral boron.

Figure 1

Crystal structure of ${\mathrm{B}}_{12}{\mathrm{C}}_3$ and ${\mathrm{B}}_{13}{\mathrm{C}}_2$. Shaded spheres denote carbon atoms. The $p$ and $e$ sites in the icosahedron are shown.

Figure 2

LDA band and DOS structure of ${\mathrm{B}}_{13}{\mathrm{C}}_2$. The names of the Brillouin zone are given in Ref. [33]. The right-hand site of the figure shows an energy diagram of the gap states reported by Werheit [34].

Figure 3

Configurations of structural model I. Red circles represent C atoms and hexagons represent icosahedra.

Figure 4

Charge density of ${\mathrm{B}}_{12}\left({\mathrm{B}}_4\right)$. Red lines indicate the boundary of the primitive unit cell. The cut plane (blue) is shown in the right-hand figure, which is a top down view of the unit cell. The atoms are numbered.

Figure 5

Structural model II. Red circles represent C atoms and hexagons represent icosahedra.

Figure 6

LDA DOS of model IIg. The left-hand figure shows the total DOS, whereas the right-hand figure shows the partial DOS split into its ${\mathrm{B}}_4,{\mathrm{C}}_{\left(e\right)}$, and ${\mathrm{C}}_{\left(c\right)}$ components. The dashed line indicates the Fermi level.

Figure 7

LDA Raman spectra (red lines) of boron carbides with compositions of 20 and 13.3 at. % C. Experimental spectra reported by Tallant et al. [61] are shown as black lines. Insets: Eigenvectors for some important normal modes.