Magnetic anisotropy in Gd, GdN, andGdFe2tuned by the energy of gadolinium4fstates

Abstract

The tiny magnetocrystalline anisotropy energies (MAEs) of bulk Gd, GdN, and ${\text{GdFe}}_{2}$ have been calculated by means of the force theorem in conjunction with the full-potential linear augmented plane-wave (FLAPW) method. The generalized gradient correction including the Hubbard interaction $U$ $(\text{GGA}+U)$ produced the best possible agreement with the experimental MAE strength compared to either the generalized gradient approximation (GGA), where the $4f$ electrons are treated as valence states, or the GGA core, where they are treated as core electrons. Indeed, the magnetic anisotropy is three times that of the $\text{GGA}+U$ if the $4f$ orbitals are prevented to hybridize correctly with the other orbitals like in the GGA-core calculation and 1 order of magnitude if they hybridize too much, like in the GGA calculation. The $\text{GGA}+U$ results can be explained in terms of orbital moment anisotropy using Bruno's model showing that the MAE is due to the orbital magnetic-moment anisotropy. In addition, because the $4f$ states of Gd are half filled their orbital moment and spin-orbit coupling are zero; the Gd MAE is tuned by the spin-orbit coupling of $5d$ states rather than by that of the $4f$ states like in other rare-earth systems, such as Tb or Dy. Nevertheless, the strength of MAE is found to depend on the energy position of the $4f$ states. The MAE of Gd is therefore much similar to that of a transition metal rather than that of a typical rare-earth metal such as Tb or Dy. It is not surprising that Gd shows a calculated easy axis along the (0001) direction like hcp cobalt. All converged calculations within the GGA, the GGA core, or the $\text{GGA}+U$ methods show that the magnetization is along the $c$ axis, in disagreement with an experiment and a recent calculation which show that the easy axis makes an angle of $20\ifmmode^\circ\else\textdegree\fi{}$ with the hcp $c$ direction. Based on the present calculations, the disagreement with experiment might be due to possible presence of symmetry-breaking imperfections, such as defect states or impurities, and cannot be explained using bulk MAE calculations. As for the MAE of GdN and ${\text{GdFe}}_{2}$ compounds, crystallizing in, respectively, cubic rocksalt and Laves phase structures, despite the qualitative agreement with Bruno's model, their interpretation is much more complex. Indeed, their predicted magnetization easy axis is along one of the symmetry equivalent (100), (010), and (001) directions rather than the (111) direction of fcc nickel, and their MAEs are much smaller than that of Gd. The removal of N from the GdN structure without changing the lattice parameter re-established the easy axis along the (111) direction as expected, showing that the easy axis of GdN is a consequence of $\text{Gd}\text{ }5d$ and $\text{N}\text{ }2p$ hybridizations.