Hubbard interactions in magnetic monolayers: FePS$_3$ and CrI$_3$*

Hubbard-corrected density-functional theory has proven to be successful in addressing self-interaction errors in 3D magnetic materials. However, the effectiveness of this approach for 2D magnetic materials has been much less explored. Here, we use PBEsol+$U$ and its extensions PBEsol+$U$+$V$ to investigate the electronic, structural, and vibrational properties of 2D antiferromagnetic FePS$_3$ and ferromagnetic CrI$_3$. The Hubbard parameters (onsite $U$ and intersite $V$) are computed self-consistently using density-functional perturbation theory (DFPT) [PRB 98, 085127 (2018)], thus fully from first-principles. We show that for FePS$_3$ Hubbard corrections are crucial for obtaining the experimentally observed insulating state with the correct crystal symmetry, with Hubbard-corrected vibrational frequencies in good agreement with Raman experiments. Finally, we discuss CrI$_3$, and the requirements it elicits in correcting the relative positions of the majority and minority conduction bands via a spin-resolved $U$ [PRB 98, 235157 (2018)].