Magnetic anisotropy of $4f$ atoms on a WSe$_2$ monolayer: a DFT+U study

Inspired by recent advancements in the field of single-atom magnets, particularly those involving rare-earth (RE) elements, we present a theoretical exploration employing DFT+$U$ calculations to investigate the magnetic properties of selected $4f$ atoms, specifically Eu, Gd and Ho, on a monolayer of the transition-metal dichalcogenide WSe$_2$ in the 1H-phase. This study comparatively examines RE with diverse $4f$ orbital fillings and valence chemistry, aiming to understand how different coverage densities atop WSe$_2$ affect the magnetocrystalline anisotropy. We observe that RE elements lacking $5d$ occupation in the atomic limit exhibit larger magnetic anisotropy energies at high densities, while those with outer $5d$ electrons show larger anisotropies in dilute configurations. Additionally, even half-filled $4f$ shell atoms with small orbital magnetic moments can generate substantial energy barriers for magnetization rotation due to prominent orbital hybridizations with WSe$_2$. Open $4f$ shell atoms further enhance anisotropy barriers through spin-orbit coupling effects. This aspect is crucial for the experimental realization of stable magnetic information units.