Exchange interactions in paramagnetic amorphous and disordered crystalline $\mathrm{CrN}$-based systems

We present a first principles supercell methodology for the calculation of exchange interactions of magnetic materials with arbitrary degrees of structural and chemical disorder in their high temperature paramagnetic state. It is based on a projection of the total magnetic energy of the system onto local pair clusters, allowing the interactions to vary independently as a response to their local environments. We demonstrate our method by deriving the distance dependent exchange interactions in vibrating crystalline CrN, a Ti${}_{0.5}$Cr${}_{0.5}$N solid solution as well as in amorphous CrN. Our method reveals strong local environment effects in all three systems. In the amorphous case we use the full set of exchange interactions in a search for the noncollinear magnetic ground state.

Figure 1

Calculated magnetic exchange interactions on the first two Cr-Cr coordination shells of cubic CrN as a function of the interatomic distance. The values obtained from geometries derived with DLM-MD as well as idealized displacements of the atoms are shown. ${\mathrm{R}}_0^{\alpha }$ is the equilibrium pair distance of the $\alpha$th coordination shell ($\alpha =1,2)$. The inset illustrates the application of Eqs. (3) and (4).

Figure 2

Magnetic exchange interactions between Cr-Cr pairs in a Ti${}_{0.5}$Cr${}_{0.5}$N alloy as a function of the interatomic distances (top panel) and number of Ti nearest neighbors to the Cr-Cr pairs (lower panel). In the top panel the Ti rich environments are highlighted by open symbols and the values in pure CrN are included for comparison. In the lower panel, a linear regression trend line is provided as a guide for the eye.

Figure 3

Cr-Cr radial distribution function for the amorphous CrN model (a-CrN) compared to crystalline cubic CrN (c-CrN) (top panel) and the magnetic exchange interactions (lower panel) in the amorphous cell obtained with MDCA and MCW approaches, respectively. The inset in the lower panel is a zoom in of the short distance interactions comparing the MDCA results with the same interactions obtained with MCW.