Ab initio elastic tensor of cubic Ti0.5Al0.5N 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) Ti0.5Al0.5N 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 (C-ij) and Zener's type elastic anisotropy (A) of Ti0.5Al0.5N. 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.