Electronic and Structural Origin of Ultraincompressibility of $5d$ Transition-Metal Diborides $M{\mathrm{B}}_2$ ($M=\mathrm{W}$, Re, Os)

First-principles theory was used to investigate the roles of bond topology and covalency in the phase stability and elastic strength of $5d$ transition-metal diborides, focusing on elements ($M=\mathrm{W}$, Re, Os) that have among the lowest compressibilities of all metals. Among the phases studied, the ${\mathrm{ReB}}_2$-type structure exhibits the largest incompressibility ($c$ axis), comparable to that of diamond. This ${\mathrm{ReB}}_2$ structure is predicted to be the ground-state phase for ${\mathrm{WB}}_2$ and a pressure-induced phase (above 2.5 GPa) for ${\mathrm{OsB}}_2$. Both strong covalency and a zigzag topology of interconnected bonds underlie these ultraincompressibilities. Interestingly, the Vickers hardness of ${\mathrm{WB}}_2$ is estimated to be similar to that of superhard ${\mathrm{ReB}}_2$.

Figure 1

Crystal structures of (a) ${\mathrm{ReB}}_2$-type (Hex-I), (b) ${\mathrm{RuB}}_2$-type (Orth), and (c) ${\mathrm{AlB}}_2$-type (Hex-II). There is a similarity between Hex-I and Orth structures, as emphasized by the shaded area in (a) and (b). The larger (open) and smaller (filled) spheres indicate transition-metal and B atoms, respectively, in each outlined unit cell.

Figure 2

Heats-of-formation ($\Delta H_f$) at zero pressure of $M{\mathrm{B}}_2$ ($M=\mathrm{Hf}$, Ta, W, Re, Os, Ir) in ${\mathrm{ReB}}_2$-type (Hex-I), ${\mathrm{AlB}}_2$-type (Hex-II), and ${\mathrm{RuB}}_2$-type (Orth) structures (a), enthalpies [$\Delta H(P)$] of different structures of ${\mathrm{WB}}_2$ as a function of pressure relative to that of the Hex-II phase (b), and electronic density-of-states of $M{\mathrm{B}}_2$ [from (c) to (e)] in Hex-II, Hex-I, and Orth structures, respectively. In (c)–(e), the energy zero is chosen as the Fermi energy ($E_F$) of ${\mathrm{ReB}}_2$ and the corresponding Fermi levels of other $M{\mathrm{B}}_2$ are represented by dashed lines.

Figure 3

Calculated LDA elastic constants of the ${\mathrm{ReB}}_2$-type (Hex-I), ${\mathrm{RuB}}_2$-type (Orth), and ${\mathrm{AlB}}_2$-type (Hex-II) structures for the $M{\mathrm{B}}_2$ compounds ($M=\mathrm{Hf}$, Ta, W, Re, Os, Ir). Not shown in the plot are the ${\mathrm{C}}_{12}$ and ${\mathrm{C}}_{13}$ of these three structures and the ${\mathrm{C}}_{23}$ of the Orth structure; these elastic constants have values in the range of 200–300 GPa for W, Re, and Os diborides.

Figure 4

Contours of electron localization function (ELF) of ${\mathrm{ReB}}_2$ on: (a) the (110) plane of the ${\mathrm{ReB}}_2$-type (Hex-I) structure, (b) the plane corresponding to the shaded area of Fig. 1b of the ${\mathrm{RuB}}_2$-type (Orth) structure, and (c) the (110) plane of the ${\mathrm{AlB}}_2$-type (Hex-II) structure. The zigzag chains in (a) and (b) are drawn to emphasize the alternating directionality of the network of strong interconnected B-B, Re-B bonds.