Quasimolecular electronic structure of the spin-liquid candidate ${\mathrm{Ba}}_3{\mathrm{InIr}}_2{\mathrm{O}}_9$

The mixed-valent iridate ${\mathrm{Ba}}_3{\mathrm{InIr}}_2{\mathrm{O}}_9$ has been discussed as a promising candidate for quantum spin-liquid behavior. The compound exhibits ${\mathrm{Ir}}^{4.5+}$ ions in face-sharing ${\mathrm{IrO}}_6$ octahedra forming ${\mathrm{Ir}}_2{\mathrm{O}}_9$ dimers with three $t_{2g}$ holes per dimer. Our results establish ${\mathrm{Ba}}_3{\mathrm{InIr}}_2{\mathrm{O}}_9$ as a cluster Mott insulator. Strong intradimer hopping delocalizes the three $t_{2g}$ holes in quasimolecular dimer states while interdimer charge fluctuations are suppressed by Coulomb repulsion. The magnetism of ${\mathrm{Ba}}_3{\mathrm{InIr}}_2{\mathrm{O}}_9$ emerges from spin-orbit entangled quasimolecular moments with yet unexplored interactions, opening up a new route to unconventional magnetic properties of $5d$ compounds. Using single-crystal x-ray diffraction we find the monoclinic space group $C2/c$ already at room temperature. Dielectric spectroscopy shows insulating behavior. Resonant inelastic x-ray scattering reveals a rich excitation spectrum below 1.5 eV with a sinusoidal dynamical structure factor that unambiguously demonstrates the quasimolecular character of the electronic states. Below 0.3 eV, we observe a series of excitations. According to exact diagonalization calculations, such low-energy excitations reflect the proximity of ${\mathrm{Ba}}_3{\mathrm{InIr}}_2{\mathrm{O}}_9$ to a hopping-induced phase transition based on the condensation of a quasimolecular spin-orbit exciton. The dimer ground state roughly hosts two holes in a bonding $j=\frac{1}{2}$ orbital and the third hole in a bonding $j=\frac{3}{2}$ orbital.

Figure 1

Left: Sketch of the crystal structure of ${\mathrm{Ba}}_3{\mathrm{InIr}}_2{\mathrm{O}}_9$ with triangular layers of well-separated ${\mathrm{Ir}}_2{\mathrm{O}}_9$ units. Bold blue lines connect Ir ions within a dimer. Olive/gray/red: Ba/In/O. Right: Permittivity ${\varepsilon }^{\prime }$ (black) and dielectric loss ${\varepsilon }^{\prime \prime }$ (red) at 20 K in the microwave range. The small ${\varepsilon }^{\prime \prime }$ demonstrates the insulating character of ${\mathrm{Ba}}_3{\mathrm{InIr}}_2{\mathrm{O}}_9$.

Figure 2

Quasimolecular orbitals based on local single-hole states with $j=\frac{1}{2}$ and $\frac{3}{2}$. In the hole picture, bonding (antibonding) orbitals are depicted in red (blue). In this $j$ basis, hopping is block-diagonal for $t_{e_g^{\pi }}=-t_{a_{1g}}$ (see main text). For three holes, increasing hopping yields a phase transition from $j_{\dim }=\frac{1}{2}$ (left) to $j_{\dim }=\frac{3}{2}$ (right) [38]. In the latter state, all three holes occupy bonding orbitals. In ${\mathrm{Ba}}_3{\mathrm{InIr}}_2{\mathrm{O}}_9$, the hopping parameters deviate from the idealized case $t_{e_g^{\pi }}=-t_{a_{1g}}$ depicted here, but this sketch still provides a good starting point for an intuitive understanding of the electronic structure.

Figure 3

High-resolution RIXS spectra of ${\mathrm{Ba}}_3{\mathrm{InIr}}_2{\mathrm{O}}_9$ at 10 K for $\mathbf{q}$ = (0.3 0 $2.8n$) r.l.u. with integer $n$ and $2.8\times 2\pi /c\approx \pi /d=Q_d$, where $d$ denotes the intradimer Ir-Ir distance. Left: The intra-$t_{2g}$ excitations below about 1.5 eV show a pronounced even/odd behavior with respect to $n$. The intensity is modulated with the period $2Q_d=2\pi /d=5.56\left(2\right)\times 2\pi /c$; cf. Fig. 5. We use a finite $h=0.3$ to avoid fulfilling the Bragg condition and the concomitant enhancement of the elastic line. Right: Excitations involving $e_g^{\sigma }$ orbitals are observed above 2.5 eV.

Figure 4

RIXS spectra of ${\mathrm{Ba}}_3{\mathrm{InIr}}_2{\mathrm{O}}_9$ at the high-symmetry points $\mathrm{\Gamma }, M$, and $K$ for two different values of $q_c$, the component of the transferred momentum parallel to the $c$ axis. For $\mathrm{\Gamma }$, the data were measured with a small offset in $\mathbf{q}$ to avoid strong elastic scattering. We find a striking insensitivity to the in-plane components of the transferred momentum. Within the experimental energy resolution, the RIXS features do not show any considerable dispersion. The largest change of up to 10 meV is found for the peak at 0.45 eV (see inset). The data are dominated by the strong dependence of the intensity on $q_c$, very similar to the behavior shown in Fig. 3. The integer values $l=17$ and 20 were chosen since they are close to extrema of the intensity modulation at $6Q_d$ and $7Q_d$; see Fig. 5. Compared to Fig. 3, the larger overall intensity stems from the difference in $h$; see Fig. 5.

Figure 5

RIXS intensity $I\left(\mathbf{q}\right)$ of ${\mathrm{Ba}}_3{\mathrm{InIr}}_2{\mathrm{O}}_9$ at 10 K. (a) Data for $\mathbf{q}$ = ($-0.3$ 0 $l$) r.l.u. (top axis) reveal a sinusoidal intensity modulation that unambiguously demonstrates the quasimolecular dimer character of the electronic structure. The bottom axis emphasizes the period $2Q_d=2\pi /d=5.56\times 2\pi /c$ which is incommensurate with the Brillouin zone. Solid lines show damped sinusoidal fits; see Eq. (1). Green symbols: Low-resolution data, $\mathrm{\Delta }E=0.36\mathrm{eV}$, covering the range 0.62–0.98 eV. Integration time was 10 s per $\mathbf{q}$ point. Brown symbols: Data integrated from 0.08 to 0.21 eV with the high-resolution setup, $\mathrm{\Delta }E=25$ meV, and 20 s integration time per $\mathbf{q}$ point. The advantage of the low-resolution setup is the enhanced signal-to-noise ratio. Data were normalized by the incident flux. (b) As a function of $h$, the RIXS intensity, integrated from 0.3 to 1.16 eV, shows monotonic behavior. Data were normalized to the value at $h=0$. The solid red line depicts the expected behavior due to self-absorption.

Figure 6

Comparison of RIXS spectra of ${\mathrm{Ba}}_3{\mathrm{InIr}}_2{\mathrm{O}}_9$ (top) and ${\mathrm{Ba}}_3{\mathrm{CeIr}}_2{\mathrm{O}}_9$ (bottom). The Ce compound has two holes per dimer, i.e., no partially filled quasimolecular orbitals, and exhibits a clear onset of RIXS intensity at about 0.5 eV. The different Ir-Ir distance $d_{\mathrm{Ce}}=c_{\mathrm{Ce}}/5.83$ gives rise to a slightly different period $2Q_{\mathrm{Ce}}$ of the interference pattern.

Figure 7

Low-energy RIXS spectra of ${\mathrm{Ba}}_3{\mathrm{InIr}}_2{\mathrm{O}}_9$ at 10 K. (a) Zoom-in on two data sets shown in Fig. 3, measured with 10 meV step size for $q_c$ values on extrema of the interference pattern; see Fig. 5. Integration time was 20 s per detector image. (b) Data collected with $2\theta ={90}^{\circ }$ to suppress the elastic line. With the given incident energy, this fixes $\left|\mathbf{q}\right|$. With this restriction, we can reach $q_c=18.5\times 2\pi /c\approx 6.7Q_d$, reasonably close to the interference maximum at $7Q_d$, and $\mathbf{q}$ = ($-2.66$ 1/3 17) r.l.u. with $q_c \approx 6Q_d$, an interference minimum. For a quantitative comparison, one has to take into account that the intensity at $6.7Q_d$ (dark red symbols) is roughly 10% smaller than at $7Q_d$; see Fig. 5. Self-absorption has been corrected. Dashed lines with small symbols show an estimate of the inelastic contribution obtained by subtracting the already suppressed elastic line. Dark blue (dark red): 5 meV (10 meV) step size and 40 s (30 s) integration time.

Figure 8

Oscillator fit (solid lines) of RIXS data averaged over the spectra measured at about $\mathrm{\Gamma }, M$, and $K$; cf. Fig. 4. We distinguish $l\approx 20$ and 17 which roughly correspond to the extrema of the interference pattern at $q_c=7Q_d$ and $6Q_d$; cf. Fig. 5. The elastic line has been subtracted. Parameters are given in Table 2. Vertical bars denote the excitation energies found in the fit.

Figure 9

Excitation energies of the many-body states calculated by exact diagonalization for $U=1.5\mathrm{eV}$. Left: Dependence on $\zeta$ for $t={\mathrm{\Delta }}_{\mathrm{CF}}=0$ and $J_H=0.33\mathrm{eV}$. For $\zeta =0.43\mathrm{eV}$, the lowest excitation energies of about 0.4 eV and 0.7 eV agree with the dominant characteristic RIXS features of $t_{2g}^4$ iridates [59, 60, 61] and with the spin-orbit exciton of $t_{2g}^5$ iridates at $\frac{3}{2}\zeta$. Right: Effect of hopping in the simplified case $t=t_{a_{1g}}=-t_{e_g^{\pi }}$ for $\zeta =0.43\mathrm{eV}$ and $J_H={\mathrm{\Delta }}_{\mathrm{CF}}=0$, where hopping is block-diagonal in the $j$ basis. Notably, several excitation energies are independent of $t$. Middle: Excitation energies as a function of hopping $t_{a_{1g}}$ for the realistic parameter set with $\zeta =0.43\mathrm{eV}, J_H=0.33\mathrm{eV}, {\mathrm{\Delta }}_{\mathrm{CF}}=0.1\mathrm{eV}$, and $t_{e_g^{\pi }}/t_{a_{1g}}$= $-0.6$. For these parameters, the ground state changes at about 0.52 eV, and good agreement with the experimental peak energies (symbols) is obtained for $t_{a_{1g}}=0.53$ eV. The weak RIXS feature at 0.23 eV (open blue symbol) is tentatively attributed to Ir/In site disorder; see Sec. 3f.

Figure 10

Exact diagonalization results for RIXS spectra at (0 0 $n{Q}_d$). Top and middle panels show data on opposite sides of the phase transition for $t_{a_{1g}}=0.52\mathrm{eV}$ and 0.53 eV, respectively. All other parameters are identical, i.e., $\zeta =0.43\mathrm{eV}, U=1.5\mathrm{eV}, J_H=0.33\mathrm{eV}, {\mathrm{\Delta }}_{\mathrm{CF}}=0.1\mathrm{eV}$, and $t_{e_g^{\pi }}/t_{a_{1g}}=-0.6$, as used in the middle panel of Fig. 9. Bottom: Result for larger $t_{a_{1g}}=0.62\mathrm{eV}$ and $U=3.0\mathrm{eV}$. To take into account the finite In-Ir site disorder, all spectra contain the small contribution of individual $t_{2g}^4$ sites depicted by dashed lines in the top panel (see main text), using the parameters given above.