RIXS interferometry and the role of disorder in the quantum magnet ${\mathrm{Ba}}_3{\mathrm{Ti}}_{3-x}{\mathrm{Ir}}_x{\mathrm{O}}_9$

Motivated by several claims of spin-orbit-driven spin-liquid physics in hexagonal ${\mathrm{Ba}}_3{\mathrm{Ti}}_{3-x}{\mathrm{Ir}}_x{\mathrm{O}}_9$ hosting ${\mathrm{Ir}}_2{\mathrm{O}}_9$ dimers, we report on resonant inelastic x-ray scattering (RIXS) at the Ir $L_3$ edge for different $x$. We demonstrate that magnetism in ${\mathrm{Ba}}_3{\mathrm{Ti}}_{3-x}{\mathrm{Ir}}_x{\mathrm{O}}_9$ is governed by an unconventional realization of strong disorder, where cation disorder affects the character of the local moments. RIXS interferometry, studying the RIXS intensity over a broad range of transferred momentum $\mathbf{q}$, is ideally suited to assign different excitations to different Ir sites. We find pronounced Ir-Ti site mixing. Both ions are distributed over two crystallographically inequivalent sites, giving rise to a coexistence of quasimolecular singlet states on ${\mathrm{Ir}}_2{\mathrm{O}}_9$ dimers and spin-orbit-entangled $j=1/2$ moments of $5d^5{\mathrm{Ir}}^{4+}$ ions. RIXS reveals different kinds of strong magnetic couplings for different bonding geometries, highlighting the role of cation disorder for the suppression of long-range magnetic order in this family of compounds.

Figure 1

Left: Hexagonal crystal structure of ${\mathrm{Ba}}_3{\mathrm{Ti}}_{3-x}{\mathrm{Ir}}_x{\mathrm{O}}_9$. In the hypothetical absence of Ir-Ti site mixing for $x=2$, Ti (Ir) ions occupy the sites with Wyckoff position $2a$ ($4f$). Hence layers of ${\mathrm{TiO}}_6$ octahedra (light blue) are sandwiched between layers of ${\mathrm{Ir}}_2{\mathrm{O}}_9$ dimers (light green), the latter being built from face-sharing ${\mathrm{IrO}}_6$ octahedra. The real compound shows Ir-Ti site mixing, i.e., Ir ions also occupy $2a$ sites (dark green) and Ti ions also can be found on $4f$ sites (dark blue), the latter giving rise to ${\mathrm{TiIrO}}_9$ units instead of dimers. Right: Local structure around an Ir defect ion on a $2a$ site (dark green). It is connected to six neighbors on $4f$ sites in corner-sharing geometry with ${180}^{\circ }$ bonds (thick lines). In this example, five of the six neighbors belong to ${\mathrm{Ir}}_2{\mathrm{O}}_9$ dimers, but there is a single ${\mathrm{Ir}}^{2a}-{\mathrm{Ir}}^{4f}$ pair (thick black line) where the Ir ion on the $4f$ site is part of an ${\mathrm{IrTiO}}_9$ unit.

Figure 2

Magnetic susceptibility $\chi \left(T\right)$ of ${\mathrm{Ba}}_3{\mathrm{Ti}}_{3-x}{\mathrm{Ir}}_x{\mathrm{O}}_9$ for $x=0.5$ and 1.5, measured on the same samples as studied in RIXS. The inset displays $1/\chi$ vs $T$. Solid lines are Curie-Weiss fits according to Eq. (1) with fixed ${\mu }_{\mathrm{eff}}=1.73$ ${\mu }_{\mathrm{B}}$.

Figure 3

RIXS resonance map of ${\mathrm{Ba}}_3{\mathrm{Ti}}_{1.2}{\mathrm{Ir}}_{1.8}{\mathrm{O}}_9$ measured at 300 K for transferred momentum $\mathbf{q}$ = (0  0  18.5) r.l.u. The upper part depicts a three-dimensional (3D) plot of the RIXS intensity as a function of both incident energy $E_{\mathrm{in}}$ and energy loss. The lower part shows the same data as a color plot, where the energies are easier to identify. The dominant RIXS peak at 3 eV energy loss corresponds to excitations to $e_g^{\sigma }$ orbitals which are resonantly enhanced for an incident energy of about 11.218 keV. The intra-$t_{2g}$ excitations below about 1.2 eV are maximized for an incident energy of 11.215 keV. The thick gray line is the elastic line at zero loss.

Figure 4

(a)–(d) RIXS spectra of ${\mathrm{Ba}}_3{\mathrm{Ti}}_{3-x}{\mathrm{Ir}}_x{\mathrm{O}}_9$ at $T$ = 20 K, showing the coexistence of single-site and dimer excitations at least for $x\ge 0.5$. For each $x$, we compare data for different $l$. For $x$ = 0.3 and 1.8 we measured on the (001) surface with $h$ = $k$ = 0 and large $l$, while the data for $x$ = 0.5 and 1.5 were collected on the (100) surface with large $h$ and smaller $l$. For small $x$, the two-peak structure of the spin-orbit exciton dominates, a mark of single-site $j$ = 1/2 moments. For large $x$, the broad features above 0.2 eV correspond to quasimolecular excitations on ${\mathrm{Ir}}_2$ dimers, which is reflected in the sinusoidal $q_c$ dependence of the intensity with a period $2Q_d$ = $2\pi /d$; cf. Fig. 6. Compared with an intensity minimum at $m·2Q_d$ with integer $m$, the data in (d) correspond to $0Q_d, 0.3Q_d, 0.6Q_d$, and $1.0Q_d$. The arrows in (b) and (c) denote the integration ranges used in Fig. 6.

Figure 5

RIXS spectra of ${\mathrm{Ba}}_3{\mathrm{Ti}}_{1.2}{\mathrm{Ir}}_{1.8}{\mathrm{O}}_9$ for $0\le h\le 2$ at $T$ = 20 K. For the dimer excitations above 0.2 eV, the different intensities in (a) and (b) reflect the ${sin}^2(q_{c}d/2)$ modulation. With $Q_d$ = $\pi /d$, the value $l$ = 16.8 (19.6) corresponds to $q_c \approx 6.3Q_d$ ($7.3Q_d$), which is close to a minimum (maximum); see Figs. 4 and 8. Only the weak 0.15 eV peak exhibits a pronounced $h$ dependence. It corresponds to a magnetic excitation that involves an Ir ion on a $2a$ site, giving rise to an intensity modulation along $h, k$, and $l$; see Sec. 5c. For constant $h$ and $k$, it shows a period of $2Q_{2a}\approx 2.3Q_d$. In contrast to the dimer features above 0.2 eV, its intensity for $h$ = $k$ = 0 is enhanced for $l$ = 16.8 but suppressed for $l$ = 19.6.

Figure 6

RIXS interference patterns as a function of $l$ for $x$ = 0.5 (a) and 1.5 (b). Measuring on a (100) surface with $c$ in the scattering plane allows us to cover a large range of $l$ that includes $l$ = 0. Data were corrected for self-absorption [51] (see Appendix pp1) and integrated over the indicated energy ranges. The local, single-site character of the spin-orbit exciton yields roughly constant intensity around 0.57 eV (orange). In contrast, the ${sin}^2(q_{c}d/2)$ modulation of the 0.35 eV peak (dark blue) with period $2Q_d$ = $2\pi /d$ demonstrates two-site interference as sketched in the inset of (a) and hence a quasimolecular dimer character. The larger period observed for integration below 0.25 eV for $x$ = 1.5 [green in (b)] unveils a different microscopic origin related to Ir ions on $2a$ sites; see Sec. 5c. For each point, the integration time was 20 s (orange) and 60 s (blue) in (a) and 30 s in (b).

Figure 7

RIXS interference patterns for $x$ = 1.5. The 0.95 eV dimer peak shows ${cos}^2(q_{c}d/2)$ behavior (red) [cf. Fig. 4], while a ${sin}^2(q_{c}d/2)$ modulation is observed for integration from 0.275 to 0.45 eV, in agreement with Fig. 6. Data were collected on a (100) surface with $c$ (nearly) perpendicular to the scattering plane, i.e., rotated by ${90}^{\circ }$ with respect to the geometry used in Fig. 6. A finite range of $l$ can be covered by tilting the sample. Solid lines are fits according to $I\left(l\right)=a_0{sin}^2(\pi l/2Q_d)+c_0$ and equivalently for the ${cos}^2$ curve, using the period $2Q_d$ = $5.34\times 2\pi /c$ determined from the data in Fig. 6.

Figure 8

Modulated RIXS intensity for $x$ = 1.8 for large $l$. In contrast to the $\mathbf{q}$ scans with a fixed energy window depicted in Figs. 6 and 7, the data (symbols) were obtained from a series of RIXS spectra, measured up to 1.5 eV, by integrating the intensity of the dominant peak from 0.275 to 0.45 eV (blue) and from 0.65 to 0.80 eV (light blue). Particularly large values of $l$ are reached by measuring on a (001) surface. Solid lines are fits that yield the period $2Q_d$ = $(5.33\pm 0.05)\times 2\pi /c$. With the knowledge of the ${sin}^2(q_{c}d/2)$ character from Fig. 6, the large $l$ value allows us to determine the period with high accuracy.

Figure 9

RIXS interference patterns of ${\mathrm{Ba}}_3{\mathrm{Ti}}_{1.5}{\mathrm{Ir}}_{1.5}{\mathrm{O}}_9$ as a function of $h$. The dominant dimer peak at 0.35 eV, integrated from 0.275 to 0.45 eV (blue), does not depend on $h$ since the dimer axis is parallel to $c$; cf. Eq. (3) and Fig. 5. In contrast, the 0.15 eV peak (dark green) shows a modulation with two different periods along $h$. The solid line depicts the total fit [cf. Eq. (12)], while the light green lines show the two contributions with periods 3 and 3/2 in $h$. Accordingly, the 0.15 eV peak corresponds to a magnetic excitation for a pair of Ir ions located on neighboring $2a$ and $4f$ sites. Inset: projection of the ${\mathrm{Ir}}^{2a}$-O-${\mathrm{Ir}}^{4f}$ bonds onto the $ab$ plane, illustrating the two values of $|\mathbf{q}·{\mathbf{r}}_i|$ for $\mathbf{q}$ = ($h$  0  0) r.l.u.; cf. Eqs. (7, 8, 9). The numbers $i\in \{1,2,3\}$ denote the six bonds $\pm {\mathbf{r}}_i$ for a given $2a$ site. The ${\mathrm{Ir}}^{2a}{\mathrm{O}}_6$ octahedra in the neighboring layers are rotated by $\pi$ around $c$, giving rise to the identical projection onto the $ab$ plane.

Figure 10

Effect of self-absorption on the RIXS data. Bottom panel: RIXS intensity of ${\mathrm{Ba}}_3{\mathrm{Ti}}_{1.5}{\mathrm{Ir}}_{1.5}{\mathrm{O}}_9$ at $T$ = 20 K as a function of $l$. The data were integrated over the indicated energy windows. For comparison, Fig. 6 shows the same data after self-absorption correction, i.e., divided by the factor $f_{sa}$ shown in the top panel; cf. Eq. (A1).

Figure 11

RIXS spectra of ${\mathrm{Ba}}_3{\mathrm{Ti}}_{1.2}{\mathrm{Ir}}_{1.8}{\mathrm{O}}_9$ at $T$ = 20 K. The data of Fig. 4 are plotted on a logarithmic intensity scale to include the entire elastic line. Additionally, open symbols depict the data for (0  0  21.3) multiplied by 1/7.