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Site dependency of the magnetism for Mn adsorption on MgO/Ag(001): a combined DFT$+U$ and STM investigation

Theoretical and experimental investigation of the electronic and magnetic structure of transition metal atoms on an insulating interface with a metallic substrate at low temperatures is quite challenging. In this paper, we show a density functional theory plus Hubbard $U$ based protocol to study an Mn adatom on three symmetrically allowed absorption sites, namely on O, hollow (between two Mg and two O) and on Mg on MgO(001). We added a thick enough(bulk-like) metallic Ag slab beneath MgO for faithful replication of scanning tunneling microscopy (STM) experiments. Our study reveals that to determine the stable most binding site, we need to obtain a Hubbard $U$ value from the density functional theory(DFT) calculations, and our results show agreement with STM experiments. Our calculated Hubbard $U$ values for the three adatom sites are different. When Mn sits on O, it retains 2.4 $μ_B$ spin moment, close to its atomic spin moment. However, when Mn sits on other adatom sites, the spin moment decreases. Using the atom projected density of states, we find Mn on O atom shows very narrow crystal field splitting among \textit{d} orbitals, whereas on other adatom sites Mn \textit{d} orbitals show significant splitting. We calculate charge and spin densities and show vertical and horizontal propagation of charge and spin density varies widely between sites. Mn on Mg top shows an unusual feature, that Mn pushes Mg below, to make a coplanar of Mn geometry with the four oxygen atoms. In addition, we explained the reason for spin leaking down to the Ag layers. Mn on MgO/Ag does not show any spin-flip behavior in the STM, an unusual phenomenon, and we used our first principles-based approach to explain this unique observation.