Magnetic anisotropy and orbital ordering in ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$

We review the magnetic and orbital ordered states in ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$ by performing resonant elastic x-ray scattering (REXS) at the Ru ${\mathrm{L}}_{2,3}$ edges. In principle, the point symmetry at Ru sites does not constrain the direction of the magnetic moment below $T_N$. However early measurements reported the ordered moment entirely along the $\vec{b}$ orthorhombic axis. Taking advantage of the large resonant enhancement of the magnetic scattering close to the Ru ${\mathrm{L}}_2$ and ${\mathrm{L}}_3$ absorption edges, we monitored the azimuthal, thermal, and energy dependence of the REXS intensity and find that a canting ($m_c\simeq 0.1m_b$) along the $\vec{c}$-orthorhombic axis is present. No signal was found for $m_a$ despite this component also being allowed by symmetry. Such findings are interpreted by a microscopic model Hamiltonian and pose new constraints on the parameters describing the model. Using the same technique we reviewed the accepted orbital ordering picture. We detected no symmetry breaking associated with the signal increase at the “so-called” orbital ordering temperature ($\simeq 260$ K). We did not find any changes of the orbital pattern even through the antiferromagnetic transition, suggesting that, if any, only a complex rearrangement of the orbitals, not directly measurable using linearly polarized light, can take place.

Figure 1

Crystal structure of ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$. Gray spheres indicate the canted octahedral Ru sites, light blue spheres are calcium, and red are oxygen. Blue arrows indicate the magnetic moment on each Ru site. The local $xyz$ frame around ${\mathrm{Ru}}_1$ is highlighted in the inset.

Figure 2

Temperature variation of refined structure parameters from single crystal XRD measurements. (a) Lattice parameters, with the $c$ axis on a separate scale. (b) Octahedral distortions angles, as defined in Fig. 3. Panels (c) and (d) show the change in Ru-O bond distance and bond angles, respectively. Oxygen ions are labeled as in Fig. 3.

Figure 3

${\mathrm{RuO}}_6$ octahedra labeling the oxygen atoms and bond angles for Fig. 2 and for Table 2.

Figure 4

Rocking curve of reflections (013) and (103) at 2.967 keV, above and below $T_N$. Crosstalk between the polarization channels has been removed in each case and intensities are corrected for self-absorption.

Figure 5

Temperature dependence of the (013) reflection, showing the absence of any practical change at $T_N$ in the $\sigma \sigma$ channel. This implies that the corresponding orbital OP is unchanged at $T_N$.

Figure 6

Azimuthal scan of the (110) reflection, at 10 K, that can be described only by the quadrupole term $Q_{bc}$, in both $\sigma \sigma$ and $\sigma \pi$ channels, as shown by the ab initio FDMNES calculations represented here as blue and red continuous lines. This points to the absence of any magnetic component $m_a$, whose theoretical contribution from Eq. (C8) is also reproduced here, as a black continuous line.

Figure 7

Temperature dependence of several reflections in the insulating region. (inset) Magnetic reflections associated to $m_b$ highlighted in logarithmic scale, with a comparison (see main text for details) to FDMNES simulations for the (013). Reflections are measured on multiple samples using an area detector, without polarization analysis. Reflections (200), (100), (110), and (003) were measured at 2.967 keV, whereas (013) and (103) were measured at 2.838 keV.

Figure 8

Variation in intensity of (100) resonant reflections with azimuthal angle, well described by Eq. (C5) at all temperatures.

Figure 9

In-plane (${\mathbf{S}}_{ab}$) and out-of-plane (${\mathbf{S}}_c$) Ru-Ru spin correlations evaluated in the ground state of the Ru-O-Ru cluster as a function of the Ru-O-Ru bond angle $\varphi$ at different values of the crystal field potential for two ratio of the Hund and Coulomb interaction, i.e., (a) $J_H/U=0.175$ and (b) $J_H/U=0.25$.

Figure 10

Dependence of resonant intensity at several reflections at the Ru $L_3$ absorption edge. Each measurement is made at the specified azimuth using an area detector, summing both $\sigma \sigma$ and $\sigma \pi$ channels and corrected for self-absorption. The absorption cross section, $\mu$, is also shown in (a), determined from the fluorescence. From it, the approximate position of $t_{2g}$ and $e_g$ orbitals is highlighted by vertical lines. The log scale inset in (c) indicates the remaining resonant intensity above the magnetic transition on the (013) reflection.

Figure 11

Dependence of resonant intensity at several reflections at the Ru $L_2$ absorption edge. Each measurement is made at the specified azimuth using an area detector, summing both $\sigma \sigma$ and $\sigma \pi$ channels and corrected for self-absorption. The absorption cross section, $\mu$, is also shown in (a), determined from the fluorescence. From it, the approximate position of $t_{2g}$ and $e_g$ orbitals is highlighted by vertical lines. The log scale inset in (c) indicates the remaining resonant intensity above the magnetic transition on the (013) reflection.

Figure 12

Variation in intensity of (003) resonant reflection with azimuthal angle in $\sigma \pi$ rotated channel. The azimuthal reference is along the (010), meaning that the angle is zero when the $b$ axis is within the scattering plane of the diffractometer.

Figure 13

Variation in intensity around the azimuth of the (013) off-specular reflection. Both the as-measured and absorption corrected intensities are shown, highlighting the significance of the correction. The corrected intensity is very close to the signal expected from a pure magnetic moment along the $m_b$ direction.

Figure 14

Calculations of the resonant scattering arising solely from the effect of the atomic structure, as calculated by the FDMNES code. The $\sigma \pi$ resonant spectra of the three measured reflections is shown as light-to-dark lines for low-to-high temperatures, respectively, with the calculated intensity at the $t_{2g}$ and $e_g$ energies given in the inset.