Finite temperature magnetic interactions from first principles

Density functional theory has demonstrated remarkable predictive power in calculating magnetic properties at zero temperature. At finite temperatures, thermally excited phonons may affect magnetism. Efficient ab-initio methods to calculate the temperature dependence of magnetic exchange interactions are still lacking despite the importance of room temperature magnetism for applications. Exchange is controlled by an interplay between metal-ligand hybridization, Hubbard repulsion, band gap, interatomic distances and bond angles, all of which change with temperature. Here we present a method to calculate the exchange interactions at finite temperatures from first principles using only two supercell calculations and quantify these mechanisms. Changes in bond angles and the band gap are identified as a primary factors. In NiO with 180-degree bonds exchange decreases with temperature, while in Cr$_2$O$_3$ with the bond angles away from 180 degrees the exchange increases by 10% at room temperature.