Effect of Spin-Orbit Coupling on the Magnetic Properties of Materials: Results

Abstract

This contribution concerning the effect of spin–orbit coupling on the magnetic properties of materials is divided into two sections. In the first section we review the method based on the density functional theory (DFT) within the local density approximation (LDA) used to compute the electronic structure, the magnetic anisotropy, the x-ray absorption spectra, and the x-ray magnetic circular dichroism. We give the major approximations used to derive the Kohn–Sham equations with or without the Hubbard interaction for correlated orbitals. We give also a brief introduction to the generalized gradient approximation (GGA). We then provide a solution of the latter equations using the full-potential linear augmented plane wave (FLAPW) basis set and discuss the so-called LDA+U method, where the Hubbard U is included for localized orbitals. We show how the relativistic effects, such as the spin–orbit coupling, can be introduced into band structure calculations and show their effect on magnetism, i.e., magnetic anisotropy energy (MAE), magnetooptical properties, and x-ray magnetic circular dichroism (XMCD). Then we show a brief derivation of the force theorem for the calculation of the magnetic anisotropy as well as a description of its application to the MAE calculations and show the details of the calculation of the XMCD matrix elements in the electric dipole approximation. The second section of this contribution includes some applications of the method to the computation of the electronic, magnetic, and spectroscopic properties of spintronics materials. In particular, we investigate the electronic structure and x-ray magnetic circular dichroism (XMCD) of Sr₂FeMoO₆ (SFMO for short) and other useful ferromagnetic half-metals with 100% spin polarization, materials useful for spin injection. In particular, we show that the spin–orbit coupling reduces the spin polarization, while the intra-site electronic correlations tend to increase it. For example, SFMO is found to be a half-metallic ferrimagnet with a gap in the spin-up channel. The calculated spin magnetic moments on iron and Mo sites confirm the ferromagnetic ordering and settle the controversy existing between the earlier experimental works. The orbital magnetism at the Fe and Mo sites agrees quite well with the recent experimental XMCD measurements. The computed L_2,3 XMCD at the Fe and the Mo sites compares fairly well with the experiment. The XMCD sum rule computed spin and orbital magnetic moments are in good agreement with the values obtained from the direct self-consistent calculations. In the last application, we focus on the GGA+U treatment of the electronic and magnetic structure of Gd and Gd-related compounds, such as GdN and GdFe₂. We compare the calculated density of states to the experimental photoemission and inverse photoemission spectra (XPS and BIS) and determine the Fermi surface with and without the Hubbard U and spin–orbit coupling. The GGA+U is found to be the most appropriate for treating the 4f Gd electrons. We have investigated the bulk properties and calculated the XMCD spectra at the L_2,3 edges at the Gd site of GdN. The agreement of the calculated spectra with experiment is the indication of the relevance of the XMCD formalism within the one-electron picture. The results also show that the ground-state electronic structure of GdN is that of a half-metal. Finally our computational method is used to determine the magnetic anisotropy aspect of Gd and its compounds GdN and GdFe₂. Using force theorem, we have calculated the MAE of Gd, GdN, and GdFe₂ for different directions of the magnetization. Indeed, owing to the nil spin–orbit interaction of the 4f half-filled shell, the force theorem is expected to be efficient for Gd and Gd compounds’ MAE calculations. This theorem allows a considerable computational effort gain since the spin–orbit coupling could be calculated only for one self-consistent iteration. Once again, the GGA+U method is found to be the most adequate approach for the force theorem calculations of the Gd MAE. The GGA and GGA-core model treatments of the 4f states have led to a wrong MAE. It turns out that the electronic properties and the magnetic properties of 4f systems are tightly related, and the 4f electrons play a crucial role in the computed magnetic anisotropy. Although the Gd MAE is found to be similar to that of a typical 3d transition metal like hcp Co, the GdN and GdFe₂ cubic crystal MAEs are found to be different from that of a pure 3d cubic material like fcc Ni.