Origin of Magnetic Anisotropy of Gd Metal

Using first-principles theory, we have calculated the energy of Gd as a function of spin direction, $\theta$, between the $c$ and $a$ axes and found good agreement with experiment for both the total magnetic anisotropy energy and its angular dependence. The calculated low temperature direction of the magnetic moment lies at an angle of $20°$ to the $c$ axis. The calculated magnetic anisotropy energy of Gd metal is due to a unique mechanism involving a contribution of $7.5\mu \mathrm{e}\mathrm{V}$ from the classical dipole-dipole interaction between spins plus a contribution of $16\mu \mathrm{e}\mathrm{V}$ due to the spin-orbit interaction of the conduction electrons. The $4f$ spin polarizes the conduction electrons via exchange interaction, which transfers the magnetic anisotropy of the conduction electrons to the $4f$ spin.

Figure 1

Calculated energy of Gd metal as a function of angle from the $c$ axis. The full line is the experimental $E_A$ from data in Ref. 5, Eq. (4). The open circles are the calculated total MAE results for the electronic structure plus dipole contribution. The dashed line is the dipole contribution. The dotted line is a fit to the calculated results using $-11.568(\cos 2\theta -1)+3.765(\cos 4\theta -1)$.

Figure 2

Calculated $E_A$ (band) of Gd metal for different $k$ points in the $k_x\mathrm{\text{−}}{k}_y$ plane ($k_z=0$) (top left). The central figure reports contributions from all states with energy $ϵ$ ($E_F-1.714\mathrm{e}\mathrm{V}\le ϵ\le E_F$), while the top-right figure refers to contributions from states with energy $-\infty \le ϵ\le E_F-1.714\mathrm{e}\mathrm{V}$. $E_F$ indicates the Fermi energy. The color scale (blue, green, brown, yellow) represents an increasing MAE going from $\le -5000\mu \mathrm{e}\mathrm{V}/\mathrm{\text{atom}}$ (blue) to $\ge +5000\mu \mathrm{e}\mathrm{V}/\mathrm{\text{atom}}$ (yellow). The transition from negative to positive contributions occurs between green and brown coloration.