Ab initio elastic tensor of cubic Ti${}_{0.5}$Al${}_{0.5}$N alloys: Dependence of elastic constants on size and shape of the supercell model and their convergence

In this study we discuss the performance of the special quasirandom structure (SQS) method in predicting the elastic properties of B1 (rocksalt) Ti${}_{0.5}$Al${}_{0.5}$N alloy. We use a symmetry-based projection technique, which gives the closest cubic approximate of the elastic tensor and allows us to align the SQSs of different shapes and sizes for a comparison in modeling elastic tensors. We show that the derived closest cubic approximate of the elastic tensor converges faster with respect to SQS size than the elastic tensor itself. That establishes a less demanding computational strategy to achieve convergence for the elastic constants. We determine the cubic elastic constants $\left(C_{ij}\right)$ and Zener's type elastic anisotropy $\left(A\right)$ of Ti${}_{0.5}$Al${}_{0.5}$N. Optimal supercells, which capture accurately both the configurational disorder and cubic symmetry of elastic tensor, result in $C_{11}=447$ GPa, $C_{12}=158$ GPa, and $C_{44}=203$ GPa with 3$\%$ of error and $A=1.40$ with 6$\%$ of error. In addition, we establish the general importance of selecting proper SQS with symmetry arguments to reliably model elasticity of alloys. We suggest the calculation of nine elastic tensor elements: $C_{11}$, $C_{22}$, $C_{33}$, $C_{12}$, $C_{13}$, $C_{23}$, $C_{44}$, $C_{55}$, and $C_{66}$, to analyze the performance of SQSs and predict elastic constants of cubic alloys. The described methodology is general enough to be extended for alloys with other symmetry at arbitrary composition.

Figure 1

The calculated nine projected cubic elastic constants of Ti${}_{0.5}$Al${}_{0.5}$N together with the derived cubic-averaged elastic constants.

Figure 2

Calculated euclidian distance deviations $\left|\right|C-C^{\text{cub}}\left|\right|/\left|\right|C\left|\right|$ obtained in the 21-dimensional space, see Eqs. (2) and (5).

Figure 3

Comparison of the calculated elastic tensor elements with the projected principal cubic elastic constants in Ti${}_{0.5}$Al${}_{0.5}$N.

Figure 4

The calculated projected cubic elastic constants of Ti${}_{0.5}$Al${}_{0.5}$N relative to the values obtained for the $(4\times 4\times 4)$ SQS model.

Figure 5

Zener's elastic anisotropy values in Ti${}_{0.5}$Al${}_{0.5}$N for each structural model considered in this study. The horizontal solid line shows the value of $\overline{A}$ obtained for the $(4\times 4\times 3)$ SQS.