Figure 1

Schematics of the magnetic anisotropy for two extreme cases. (a) In $3d$ systems, where the crystalline electrostatic field (CEF) is large and spin-orbit interaction can be treated as a small perturbation, the electron orbitals are firmly coupled to the lattice by CEF and hybridization. The associated orbital moment (L), and through spin-orbit coupling the spin moment (S), are oriented along a well-defined direction, called the easy direction of magnetization. An applied magnetic field rotates S, but the charge density (depicted as the oval shape) remains fixed to the lattice. The difference in expectation value of L along different magnetization directions results in a weak magnetic anisotropy. (b) In $4f$ systems, as described by the standard model, the spin-orbit coupling is strong and the CEF is weak, so that the charge density rotates (oval shape) with the total angular momentum (J) which is aligned along the magnetization direction. The difference in energy associated with different orientations of the charge density results in a large magnetic anisotropy.Reuse & Permissions

Figure 2

Angular dependence of the Eu $M_{4,5}$ XMLD in $\mathrm{EuO}/\mathrm{Si}(001)$. (a) XA spectrum averaged over all angles $\varphi$. (b) XMLD spectra $I_0$ and $I_{45}$ as measured for $\varphi =0°$ and $\varphi =45°$. The experimental geometry is depicted in the inset of (a) indicating the linear polarization $\mathbf{E}$ (white arrow) at an angle $\varphi$ to the [100] axis (dashed line) and applied external fields $\mathbf{H}$ (gray arrows) at angles $\varphi$ and $\varphi +90°$, respectively [cf. Eq. (1)]. (c) Residual spectra $I_{\mathrm{XMLD}}^{\mathrm{RES}}$ for $0°\le \varphi \le 45°$. These are the observed XMLD spectra minus the averaged XMLD according to $I_{\mathrm{XMLD}}^{\mathrm{RES}}=I_{\mathrm{XMLD}}(\varphi )-\frac{1}{2}(I_0+I_{45})$. Symbols indicate the experimental data and (red) lines give the modeled angular dependence using the experimental $I_0$ and $I_{45}$ spectra.Reuse & Permissions

Figure 3

Comparison of experimental Eu $M_{4,5}$ XA and XMLD spectra obtained from $\mathrm{EuO}/\mathrm{Si}(001)$ (black dots) with results of atomic multiplet calculations (green lines) for $V_0^4=175\mathrm{meV}$. (a) XA spectrum. (b) Fundamental XMLD spectra $I_0$ (upper spectrum) and $I_{45}$ (lower spectrum). (c) Difference spectrum $I_0-I_{45}$.Reuse & Permissions