Interplay between breathing-mode distortions and magnetic order in rare-earth nickelates from ab initio magnetic models

We use density-functional theory calculations to explore the magnetic properties of perovskite rare-earth nickelates $\mathcal{R}\mathrm{Ni}{\mathrm{O}}₃$ by constructing microscopic magnetic models containing all relevant exchange interactions via Wannierization and Green's function techniques. These models elucidate the mechanism behind the formation of antiferromagnetic order with the experimentally observed propagation vector, and explain the reason previous DFT plus Hubbard $U$ calculations favored ferromagnetic order. We perform calculations of magnetic moments and exchange-coupling parameters for different amplitudes of the ${R}₁⁺$ breathing-mode distortion, which results in expanded and compressed $\mathrm{Ni}{\mathrm{O}}₆$ octahedra. Our analysis shows that the strong competition between nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic couplings determine the magnetic ordering. The inclusion of spin-orbit coupling demonstrates that the magnetic anisotropy is very small, while the magnetic moments of the short bond nickel atoms tend to zero similar to the collinear case. Finally, we show that nickelates with larger rare-earth ions display overall stronger exchange couplings, resulting in a more stable antiferromagnetic phase. Our results provide a clear picture of the trends of the magnetic order across the nickelate series and give insights into the coupling between magnetic order and structural distortions.