Hardness Descriptor Derived from Symbolic Regression

Hard and superhard materials play a vital role in numerous industrial applications necessary for sustainable development. However, discovering new materials with high hardness is challenging due to the complexity of this multiscale property and its and its intricate relationship with the atomic properties of the material. Here, we introduce a low-dimensional physical descriptor for Vickers hardness derived from a symbolic-regression artificial intelligence approach to data analysis. This descriptor is a mathematical combination of materials' properties that can be evaluated much more easily than hardness itself through the atomistic simulations, therefore suitable for a high-throughput screening. The artificial intelligence model was developed and trained using the experimental hardness values and high-throughput screening was performed on 635 compounds, including binary, ternary, and quaternary transition-metal borides, carbides, nitrides, carbonitrides, carboborides, and boronitrides to identify the optimal superhard material. The proposed descriptor is a physically interpretable analytic formula that provides insight into the multiscale relationship between atomic structure (micro) and hardness (macro). We discovered that hardness is proportional to the Voigt-averaged bulk modulus and inversely proportional to the Poisson's ratio and Reuss-averaged shear modulus. Results of high-throughput search suggest the enhancement of material hardness through mixing with harder, yet metastable structures (e.g., metastable VN, TaN, ReN$_2$, Cr$_3$N$_4$, and ZrB$_6$, all of them exhibit high hardness).