The magnetic anisotropy energy (MAE) of hcp Gd was calculated from first principles using the full-potential linear muffin-tin orbital method. It was found that the principal contributions to the MAE are the dipole-dipole interaction between the localized $4f$ spins and the spin-orbit interaction of the valence band states. The dipole contribution has the form $\frac{1}{2}{K}_1^d(1-\cos 2\theta )$, where $\theta$ is the angle between the magnetization direction and the $c$ axis. The contribution of the spin-orbit interaction is shown to arise from the polarization of the conduction band that becomes exchange split due to exchange interaction with the localized $4f$ electrons. We argue that this leads to significant contributions from higher order anisotropy constants. An imposed reduced $4f$ moment leads to a repopulation of the electronic states at the Fermi level and a reduced exchange splitting of the valence states, which we demonstrate leads to a modification of the MAE. This modification is in qualitative agreement with the observed temperature dependence of the MAE. In addition, the dependence of the MAE on the $c∕a$ ratio has been studied.

• Received 12 April 2005

DOI:https://doi.org/10.1103/PhysRevB.72.094423