The modern synthesis of superhard and, especially, ultrahard phases is a fascinating area of research that could lead to the design of new, industrially important materials. Computational methods built within the well-established quantum mechanics framework of density functional theory (DFT) play an important role in the search for these advanced materials and the prediction of their properties. The close relationship between the physical properties of carbon and boron nitride has led to particular interest in the B-C-N ternary system, characterized by the small radii of the elements, resulting in short interatomic distances and reduced volumes-the parameters being 'recipes' for very high hardness in three-dimensional structures. The purpose of this review is to provide a brief outline of recent developments and problems in predicting novel ultrahard carbon allotropes as well as binary and ternary compounds of the B-C-N system with particular emphasis on the analysis of the models used to evaluate the hardness of the theoretically predicted structures.
Keywords: B–C–N system; DFT; crystal structure; elastic moduli; hardness.