The electronic structure and magnetic properties of a single Fe adatom on a CuN surface have been studied using density functional theory in the local spin density approximation (LSDA), the LSDA+U approach and the local density approximation plus dynamical mean-field theory (LDA+DMFT). The impurity problem in LDA+DMFT is solved through exact diagonalization and in the Hubbard-I approximation. The comparison of the one-particle spectral functions obtained from LSDA, LSDA+U and LDA+DMFT show the importance of dynamical correlations for the electronic structure of this system. Most importantly, we focused on the magnetic anisotropy and found that neither LSDA, nor LSDA+U can explain the measured, high values of the axial and transverse anisotropy parameters. Instead, the spin excitation energies obtained from our LDA+DMFT approach with exact diagonalization agree significantly better with experimental data. This affirms the importance of treating fluctuating magnetic moments through a realistic many-body treatment when describing this class of nano-magnetic systems. Moreover, it facilitates insight to the role of the hybridization with surrounding orbitals.
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