Stability and Strength of Transition-Metal Tetraborides and Triborides

Using density functional theory, we show that the long-believed transition-metal tetraborides ($T{\mathrm{B}}_4$) of tungsten and molybdenum are in fact triborides ($T{\mathrm{B}}_3$). This finding is supported by thermodynamic, mechanical, and phonon instabilities of $T{\mathrm{B}}_4$, and it challenges the previously proposed origin of superhardness of these compounds and the predictability of the generally used hardness model. Theoretical calculations for the newly identified stable $T{\mathrm{B}}_3$ structure correctly reproduce their structural and mechanical properties, as well as the experimental x-ray diffraction pattern. However, the relatively low shear moduli and strengths suggest that $T{\mathrm{B}}_3$ cannot be intrinsically stronger than $c$-BN. The origin of the lattice instability of $T{\mathrm{B}}_3$ under large shear strain that occurs at the atomic level during plastic deformation can be attributed to valence charge depletion between boron and metal atoms, which enables easy sliding of boron layers between the metal ones.

Figure 1

Bond structures at equilibrium for (a) ${\mathrm{WB}}_4$ and (b) ${\mathrm{WB}}_3$. The isosurface maps of the ELF correspond to $0.0006\mathrm{electrons}/{\mathrm{bohr}}^3$; the large blue and small red spheres represent W and B atoms, respectively. Maps of the ELF on the ($10\overline{1}0$) plane for (c) ${\mathrm{WB}}_4$ and (d) ${\mathrm{WB}}_3$. Calculated phonon dispersion curves for (e) ${\mathrm{WB}}_4$ and (f) ${\mathrm{WB}}_3$. The phonon density of states for (g) ${\mathrm{WB}}_4$ and (h) ${\mathrm{WB}}_3$. The numbers close to W and B atoms in (a) and (b) are the corresponding Bader charges.

Figure 2

Total and partial electronic density of states of (a) ${\mathrm{WB}}_4$, (b) ${\mathrm{WB}}_3$, (c) ${\mathrm{MoB}}_4$, and (d) ${\mathrm{MoB}}_3$. The vertical dashed lines indicate the Fermi levels.

Figure 3

Simulated XRD curves for (a) ${\mathrm{WB}}_4$ and (b) ${\mathrm{WB}}_3$. (c) Experimental XRD pattern for ${\mathrm{WB}}_x$ reproduced from Ref. 17.

Figure 4

Stress-strain relationships for (a) ${\mathrm{WB}}_3$ and (b) ${\mathrm{MoB}}_3$. Curves with solid symbols indicate tension deformations, and curves with open symbols denote shear deformations. The isosurfaces of deformed valence charge density difference of ${\mathrm{WB}}_3$ at a shear strain of (c) $\gamma =0.2044$ (before) and (d) $\gamma =0.2531$ (after lattice instability) in the $(0001)⟨10\overline{1}0⟩$ slip system. The isosurfaces of valence charge density difference correspond to $\pm 0.016\mathrm{electrons}/{\mathrm{bohr}}^3$; large and small spheres represent tungsten and boron atoms, respectively. The arrows highlight the charge depletion states.