Predicting hardness of covalent/ionic solid solution from first-principles theory

We introduce a hardness formula for the multicomponent covalent and ionic solid solutions. This expression is tested on nitride spinel materials A(3)N(4) (A=C,Si,Ge) and applied to titanium nitrogen carbide (TiN1-xCx with 0 <= x <= 1), off-stoichiometric transition-metal nitride (TiN1-x and VN1-x with x <= 0.25), and B-doped semiconductors (C1-xBx, Si1-xBx, and Ge1-xBx with x <= 0.1). In all cases, the theoretical hardness is in good agreement with experiments.