Observation of spin-orbit excitations and Hund's multiplets in ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$

We use Ru $L_3$-edge (2838.5 eV) resonant inelastic x-ray scattering (RIXS) to quantify the electronic structure of ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$, a layered $4d$-electron compound that exhibits a correlation-driven metal-insulator transition and unconventional antiferromagnetism. We observe a series of Ru intraionic transitions whose energies and intensities are well described by model calculations. In particular, we find a $J=0\to 2$ spin-orbit excitation at 320 meV, as well as Hund's-rule driven $S=1\to 0$ spin-state transitions at 750 and 1000 meV. The energy of these three features uniquely determines the spin-orbit coupling, tetragonal crystal-field energy, and Hund's rule interaction. The parameters inferred from the RIXS spectra are in excellent agreement with the picture of excitonic magnetism that has been devised to explain the collective modes of the antiferromagnetic state. $L_3$-edge RIXS of Ru compounds and other $4d$-electron materials thus enables direct measurements of interactions parameters that are essential for realistic model calculations.

Figure 1

(a) Geometry of the RIXS experiment. The incoming and outgoing photon beams subtend a fixed angle of ${90}^{\circ }$. By varying the angle $\theta$ between the incoming beam and the ${\mathrm{RuO}}_2$ planes of ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$, the incoming photon polarization ($\pi$) can be changed from $E//c$ ($\theta =0^{\circ }$) to $E//ab$ ($\theta ={90}^{\circ }$). (b) X-ray absorption spectrum (XAS) collected at the Ru $L_3$ edge of ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$. The red triangles ($E_1$ and $E_2$) represent excitations into the empty Ru $4d{t}_{2g}$ and $e_g$ orbitals, respectively. (c) Color map of the incident-energy dependence of the RIXS spectrum across the Ru $L_3$ edge. The vertical white dashed lines show the resonance energies ($E_1$ and $E_2$) of features $A/C$ and $B/D$. All data were collected at room temperature.

Figure 2

(a) RIXS spectrum of ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$ collected for incident energy $E_1=2838.5\mathrm{eV}$. The sample was kept at a temperature of 12 K while varying the incident angle $\theta$. Each spectrum was shifted vertically for clarity. (b) Fitted RIXS spectrum taken at $\theta ={60}^{\circ }$. The vertical white bars are a result of multiplet calculations (see text). (c) Intensity of the ${\mathrm{A}}_2$ (${\mathrm{A}}_3$ and ${\mathrm{A}}_4$) feature as a function of $\theta$. The data were normalized to the value at $\theta =9^{\circ }$ (${45}^{\circ }$); solid curves represent theory results. (d) The $t_{2g}^4$ multiplet energy levels as a function of $\mathrm{\Delta }/\xi$. The SOC was fixed at $\xi =0.13\mathrm{eV}$. Hund's coupling separates the multiplet into levels with different spin S and orbital angular momentum L. The shaded region below 0.4 eV is magnified on the right hand side for the detailed structure of the $\mathrm{S}=1$ states. For large values of $\mathrm{\Delta }$, the lowest three levels ($J=0$ ground state, and $J_z=\pm 1$ doublet at ${\mathrm{E}}_{\mathrm{so}}$) form an effective $\tilde{S}=1$ system hosting excitonic magnetic order [22, 23]. (e) Spin-orbit excitation energy $E_{\mathrm{so}}$ (see inset) as a function of $J_{\text{H}}/\xi$ at different values of the tetragonal crystal field $\mathrm{\Delta }/\xi$. The limiting case of $J_{\text{H}}/\xi =0$ ($\infty$) corresponds to the so-called $jj$-coupling ($LS$-coupling) scheme. The location of ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$ with $J_{\text{H}}/\xi \simeq 2.6$ and $\mathrm{\Delta }/\xi \simeq 2$ is indicated.

Figure 3

The calculated RIXS spectra including magnetic excitations $A_1$, spin-orbital excitations $A_2, A_3$, and $A_4$ within the $t_{2g}$ multiplets, and $t_{2g}\to e_g$ transitions $B_1$ and $B_2$. Vertical bars represent the energy and intensities of the transitions, and their Lorentzian-broadened profiles result in the dashed curves. Different widths 0.14 and 0.6 eV are used for the transitions within $t_{2g}$ and $e_g$ sectors, respectively. The scattering momenta $[H,H]$ and corresponding $\theta$ values are indicated.

Figure 4

(a) RIXS spectrum of a carbon tape demonstrating the resolving power at 2.839 keV. (b) Additional results for the low-energy RIXS fit. Peak positions of $A_2, A_3$, and $A_4$ as a function of the angle $\theta$. The data points are all found within an energy of $320\pm 15\mathrm{meV}, 750\pm 15\mathrm{meV}$, and $1000\pm 15\mathrm{meV}$ (dashed horizontal lines).