Based on the idea that the hardness of covalent crystal is intrinsic and equivalent to the sum of the resistance to the indenter of each bond per unit area, a semiempirical method for the evaluation of hardness of multicomponent crystals is presented. Applied to $\beta \mathrm{\text{−}}{\mathrm{B}\mathrm{C}}_2\mathrm{N}$ crystal, the predicted value of hardness is in good agreement with the experimental value. It is found that bond density or electronic density, bond length, and degree of covalent bonding are three determinative factors for the hardness of a polar covalent crystal. Our method offers the advantage of applicability to a broad class of materials and initializes a link between macroscopic property and electronic structure from first principles calculation.

It has been widely accepted that planar boron structures, composed of triangular and hexagonal motifs are the most stable two-dimensional (2D) phases and likely precursors for boron nanostructures. Here we predict, based on an ab initio evolutionary structure search, a novel 2D boron structure with nonzero thickness, which is considerably, by $50\mathrm{meV}/\mathrm{atom}$, lower in energy than the recently proposed $\alpha$-sheet structure and its analogues. In particular, this phase is identified for the first time to have a distorted Dirac cone, after graphene and silicene the third elemental material with massless Dirac fermions. The buckling and coupling between the two sublattices not only enhance the energetic stability, but also are the key factors for the emergence of the distorted Dirac cone.