Encoding the Electronic Density of States for Machine Learning of Mechanical Properties in Transition Metal Carbides

Abstract

Transition metal carbides feature high hardness and phase stability, but suffer from inherent brittleness. The properties of these carbides can be tuned by combining different carbides in solid solutions, giving rise to multinary or compositionally complex carbides. But the vast combinatorial space of these carbides present a new challenge for materials design. To tackle this problem, we used ab initio calculations to screen the mechanical properties of the multinary carbides of the 9 IVb-VIb metals and Al. The chosen compunds encompass all equimolar combinations of solid solutions with 1 to 5 metals, amounting to 637 individual compositions. The simulation cells were set up as 2x2x2 supercells of the TiC prototype (SG Fm-3m) with 64 atoms, the metals were distributed on the metal-sublattice by the special quasi-random structure algorithm. The cells were then relaxed using VASP, and the mechanical properties were calculated from the equilibrium structure with the stress-strain method. This high-throughput exploration still requires significant computational expense, we therefore use machine-learning (ML) algorithms trained on the ab initio data to predict the properties of the full desired phase space from a small representative subset. The efficacy of these ML models depends strongly on expressive co-variates. Here, we explore the impact of the electronic density of states (DoS) as co-variate to improve the prediction efficacy. We have calculated the DoS from the relaxed structures in the range from -10 to +5 eV around the Fermi level. To implement the DoS in the ML algorithm efficiently, we extracted the salient information by two procedures. First, we fitted the DoS with a simple mathematical function, describing the DoS with a double triangle fit. One of the triangle functions is fitted to the states forming the covalent bonds in the region around -10 to -4 eV, the second triangle to the metallic states around the Fermi-level. We thus express the DoS with 6 variables: the position, height, and width of the triangles. The second route represents an integrative binning of the DoS in certain regions. Here, we found that bins in the regions from -2 to 0 and 0 to +2 eV offer the best prediction improvement. With both methods we can improve the absolute R2 prediction scores of mechanical properties like bulk, shear, Young’s modulus, and Cauchy pressure by up to 30%, thus allowing a decent property prediction from a very small subset of the phase space.