Symmetry analysis of magneto-optical effects: The case of x-ray diffraction and x-ray absorption at the transition metal $L_{2,3}$ edge

A general symmetry analysis of the optical conductivity or scattering tensor is presented and used to rewrite the conductivity tensor as a sum of linear independent spectra multiplied by simple functions depending on the local magnetization direction. This allows one to describe the full, magnetization directional dependent, magneto-optical response of a system in arbitrary symmetry by only a few linear independent fundamental spectral functions. Using this formalism, we discuss the azimuthal dependence of the resonant x-ray diffracted intensity on magnetic Bragg reflections. We present several numerical examples at the transition metal $L_{2,3}$ edge. From these numerical calculations, we can conclude that for realistic parameters several fundamental spectra, not present in spherical symmetry, become important and should not be neglected. Deviations from the standard analysis in spherical symmetry become large in cases where orbital order coexists with magnetic order, even if the orbital order is at a different $\mathit{q}$ vector. In the extreme case of the layered cuprates, one finds that one is not sensitive to the projection of the magnetic moments onto the plane of the ordered $d_{x^2-y^2}$ hole. Not including the correct crystal symmetry can lead to incorrectly determined magnetic orientations and structures.

Figure 1

(Color online) Scattering tensor of (top) ${\text{Mn}}^{2+}$ and (bottom) ${\text{Ni}}^{2+}$ as a function of the cubic crystal-field splitting $10Dq$. Left column shows the $a_{1g}$ component, middle column the $t_{2g}$ and $e_g$ component and right column the $t_{1u}$ component. All graphs are on the same intensity scale. For ${\text{Ni}}^{2+}$ with $10Dq=0$ two graphs are included, once starting from a spherical ground state (F) and once starting from the cubic ground state $\left({\text{A}}_2\right)$. Calculations done on an ionic model without spin-orbit coupling on the $d$ shell.

Figure 2

(Color online) Top: an artificially created super lattice of six layers of NiO alternated by one layer of MnO stacked in the [111] cubic crystal direction. Bottom: RXD intensity of the first magnetic Bragg reflection as a function of the azimuthal angle $\varphi$ and resonant energy. In the left column calculations are done for the scattering tensor in cubic symmetry. In the middle column spherical symmetry has been created by taking $F^{\left(1\right)}$ proportional to the average of $F_{xy}$, $F_{zy}$ and $F_{xz}$ as calculated in cubic symmetry. The calculations are done for incoming $\sigma$, $\pi$, linear $45°$ rotated $\left(\sigma +\pi \right)$, and circular $\left(\sigma +ı\pi \right)$ polarized light.

Figure 3

(Color online) Fundamental spectra of the elastic scattering tensor for ${\text{Ni}}^{2+}$ (top panel), ${\text{Mn}}^{3+}$ (middle panel), and ${\text{Cu}}^{3+}$ (bottom panel) as a function of tetragonal crystal-field splitting $\Delta e_g$ defined as the energy difference between the $d_{z^2}$ and $d_{x^2-y^2}$ orbital. $\left(\Delta t_{2g}=1/4\Delta e_g\right)$. $\Delta e_g>0$ stands for a tetragonal contracted system, i.e., the $d_{z^2}$ orbital higher in energy than the $d_{x^2-y^2}$ orbital.

Figure 4

(Color online) Left: crystal structure of ${\text{NaCu}}_2{\text{O}}_2$ including a possible local magnetization direction, which is used for the RXD calculations. Right: RXD calculations of the scattered intensity as a function of the azimuthal angle, for incoming $\sigma$, $\pi$, linear $45°$ rotated $\left(\sigma +\pi \right)$, and circular $\left(\sigma +ı\pi \right)$ polarized light. Whereby the scattering tensor once has been assumed to have tetragonal symmetry and once has been assumed to have spherical symmetry.