Theory of the ferromagnetism in TiCrN solid solutions

First-principles calculations are used to investigate the magnetic properties of Ti$_{1-x}$Cr$_{x}$N solid solutions. We show that the magnetic interactions between Cr spins that favour antiferromagnetism in CrN is changed upon alloying with TiN leading to the appearance of ferromagnetism in the system at approximately $x \leq 0.50$ in agreement with experimental reports. Furthermore we suggest that this effect originates in an electron density redistribution from Ti to Cr that decreases the polarization of Cr d-states with t$_{2g}$ symmetry while it increases the polarization of Cr d-states with e$_g$ symmetry, both changes working in favour of ferromagnetism.