Localised Magnetism in 2D Electrides

In this work, we investigate intrinsic magnetic properties of monolayer electrides LaBr$_{2}$ and La$_{2}$Br$_{5}$, where excess electrons do not reside at any atomic orbital and act as anions located at interstitial regions. Having demonstrated that conventional first-principles approaches are incapable of treating such non-atomic magnetic orbitals largely underestimating insulating band gaps, we construct effective electronic models in the basis of Wannier functions associated with the anionic states to unveil the microscopic mechanism underlying magnetism in these systems. Being confined at zero-dimensional cavities in the crystal structure, the anionic electrons will be shown to reveal an exotic duality of strong localisation like in $d$- and $f$-electron systems and large spatial extension inherent to delocalised atomic orbitals. While the former tends to stabilise a Mott-insulating state with localised magnetic moments, the latter results in direct exchange between neighbouring anionic electrons that dominates over antiferromagnetic superexchange interactions. On the basis of the derived spin models, we argue that any long-range magnetic order is prohibited in LaBr$_{2}$ by Mermin-Wagner theorem, while intersite anisotropy stabilises weakly coupled ferromagnetic chains along the monoclinic $\mathbf{b}$ axis in La$_{2}$Br$_{5}$. Our study shows that electride materials combining peculiar features of both localised and delocalised atomic states constitute a unique class of strongly correlated materials.