Electronic structure and elasticity of the Ta-W solid solution

TL;DR

This work tackles how electronic structure governs the composition-dependent elasticity of Ta$_{1-x}$W$_x$ by combining density functional theory with DOS, band structure, crystal orbital Hamilton population (COHP), and wavefunction analysis within a rigid-band framework. It identifies a pronounced change in the Fermi-level DOS near $VEC=5.5$, coinciding with a slope change in the shear modulus, and links this to covalent bonding and anti-bonding trends of ${T_{2g}}$ orbitals that shape $C_{11}$, $C_{12}$, and $C_{44}$ and thus the Pugh ratio $P=K/G$. The study demonstrates that occupancy of ${T_{2g}}$ bonds as $VEC$ increases drives the observed elastic anomalies, with NN bonding strengthening and NNN anti-bonding diminishing around $VEC=5.5$, and shows that the rigid-band model captures these tendencies for Ta–W and qualitatively for other group V–VI binaries. This electronic-structure–driven picture provides a mechanistic link between bonding characteristics and ductility, offering a predictive framework for ductility in related transition-metal alloys.

Abstract

The brittleness or ductility of metals has long been attributed to their elastic constants, with high Poisson ratio, or equivalently high Pugh ratio, favoring greater ductility. Growing evidence links ductility with their electronic structure. Consequently, it is desirable to understand how the electronic structure affects the elastic constants. Here, we examine the Ta-W binary alloy system, which evolves from ductile character at Ta-rich compositions to brittleness at high W. We show that a change in slope of the composition-dependent shear modulus near the equiatomic composition coincides with an abrupt change in the Fermi level density of states. We relate the behaviors of the elastic constants to the characters of occupied electronic orbitals close to the Fermi level. Finally, we consider additional alloy systems from groups V and VI and show that qualitatively similar behavior occurs more broadly.

Figures (9)

• Figure 1: Cubic elastic constants of Ta-W at various compositions. The Pugh ratio $P=K/G$ is shown in the inset.
• Figure 2: (a) Total and partial densities of states of TaW (assuming Pearson type cP2, Strukturbericht B2) with Fermi energy shifted to 0; inset marks Fermi levels corresponding to valence electron counts for elemental Ta, equiatomic TaW, and elemental W.
• Figure 3: (top) Electronic band structure of TaW.cP2. Plotting point marker sizes indicate the projections onto atomic orbitals. (bottom) Fermi surface of flat band (left) and three merged bands (right).
• Figure 4: Selected crystal orbital Hamilton populations between the Ta and W atom of TaW.cP2. Energies corresponding to VEC=5.0, 5.5 and 6.0 are marked as in Fig. \ref{['fig:bands']}. Solid curves are for nearest neighbor (NN) bonds, while dashed curves are for next nearest-neighbors (NNN).
• Figure 5: (top): $\Gamma$-point ${d_{xz}}$ orbital of TaW.cP2 at energy $E=0.12$ eV. The horizontal [110] axis and vertical [001] axis are labeled. The Ta atom position is marked in magenta and the W atom in turquoise. The nearest neighbor interatomic bond lying along the [111] direction is marked in gold, while the next nearest-neighbor lying along the [001] axis at $x=0$ is marked in blue. (middle): Partial charge in range $E\in(-0.474,0.12)$. The isosurface is at $5\times 10^{-3}$ electrons per Å$^{3}$. (bottom): Bonds and their associated forces.