Vibronic effect on resonant inelastic x-ray scattering in cubic iridium hexahalides

Strong spin-orbit coupling gives rise to rich quantum phenomena in $4d/5d$ transition metal compounds [1, 2, 3]. One of the most investigated phenomena is the Kitaev spin liquid phase in Mott-insulating honeycomb compounds [4]. The candidate honeycomb materials consist of edge-sharing octahedra containing heavy $d^{5}$ ions. The spin-orbit coupling of the $d^{5}$ ion in the octahedral environment makes the local quantum states spin-orbit entangled $j=1/2$ type and the exchange interaction between the $d^{5}$ sites bond-dependent (Kitaev-type anisotropy). The Kitaev exchange interaction appears in other lattices such as triangular and face centered cubic (fcc) lattices [4], which could induce various magnetic phases [5, 6].

Motivated by the theoretical predictions, much experimental effort has been made to search Kitaev and related systems, while realizing pure $j=1/2$ states in materials is far from trivial. Contrary to the assumption of a perfect octahedral environment in theories, candidate materials exhibit symmetry lowering, resulting in the hybridization of the $j=1/2$ and the excited $j=3/2$ states.

Recently, K₂IrCl₆ was reported as the first iridium compound with pure $j=1/2$ states on metal sites [7]. The neutron powder diffraction data show that the compound is cubic down to 0.3 K [8]. The $g$-tensor measured by electron paramagnetic resonance is consistent with the $g$ for $j=1/2$ in an octahedral environment, suggesting no symmetry lowering [9].

Nevertheless, spectroscopic data look controversial to the cubic structure of K₂IrCl₆. Within the octahedral environment, the excited states of a single Ir site are four-fold degenerate $j=3/2$ multiplet. In contrast, the resonant inelastic x-ray scattering (RIXS) measurement of the $j=3/2$ states shows a splitting of $\approx 50$ meV down to low temperature (10 K) [8]. The RIXS spectra of cubic K₂IrBr₆ (in high-temperature phase) and Cs₂IrCl₆ resemble those of K₂IrCl₆ too [8, 10, 11]. The Raman scattering spectra of K₂IrCl₆ also show a similar splitting in the same range of energy [12]. As the origin of the splitting, non-cubic deformation of each octahedron and crystal-field splitting by rotational modes of the octahedra were discussed, whereas both interpretations require symmetry lowering of the system. Moreover, the latter leads to the splitting of only a few meV according to the previous experiments and calculations [10].

We recently proved ab initio that the vibronic excitations induce the splittings of peaks in the RIXS spectra of a similar compound, K₂RuCl₆ [13, 14]. In this compound, the dynamic Jahn-Teller (JT) effect develops in the excited spin-orbit multiplet states and determines the shape and temperature evolution of the RIXS spectra. The dynamic JT effect is characterized by the superpositions of many electron-lattice configurations with JT-deformed structures, and hence, structural anisotropy does not appear in K₂RuCl₆. Since the vibronic coupling tends to be stronger in $5d$ systems than in $4d$ systems [15], the dynamic JT effect would give nonnegligible changes in the RIXS spectra of the Ir compounds.

In this work, we examine the presence of the dynamic JT effect and reveal its impact on the RIXS spectra of the Ir⁴⁺ compounds. We calculate the vibronic states of the dynamic JT model and, based on them, the RIXS spectra. We confirm the validity of the derived interaction parameters by a series of post Hartree-Fock calculations.

Here we explain the theoretical framework for the quantum states and the RIXS spectra of a single Ir site in the cubic K₂Ir$X_{6}$ ($X=$ Cl, Br). We discuss only the phenomena in a single Ir octahedron because the energy scale of the intersite exchange interaction (about 1 meV) [7, 9, 10] is several tens times smaller than the splitting (about 35-60 meV) in the RIXS spectra [8, 10].

The quantum states of each $5d^{5}$ site are determined by the interplay of the ligand field, spin-orbit, and vibronic interactions. We consider the interactions in the order because their energy scales decrease from several eV by about one order of magnitude. The octahedral ligand field splits the $5d$ orbitals into $e_{g}$ doublet and $t_{2g}$ triplet. The $t_{2g}$ orbitals are lower in energy and have five electrons (one hole). Hereafter, we use the hole picture. Within the $t_{2g}$ orbitals, the spin-orbit coupling is unquenched [16]:

where $\tilde{\bm{l}}$ is the effective $\tilde{l}=1$ orbital operator, $\hat{\bm{s}}$ is the spin operator, and $\lambda$ $(>0)$ is the spin-orbit coupling parameter. The $t_{2g}$ hole couples to the JT active $E_{g}$ vibrations too: the JT model including the harmonic oscillator Hamiltonian for the $E_{g}$ modes is [17]

Here $\hat{q}_{\gamma}$ are the dimensionless normal coordinates for the $E_{g}$ modes ($\gamma=u,v$, where $u$ and $v$ transform as $z^{2}$ and $x^{2}-y^{2}$, respectively), $\hat{p}_{\gamma}$ the conjugate momenta of $\hat{q}_{\gamma}$, $\omega$ the frequency of the $E_{g}$ modes, $g$ the dimensionless vibronic coupling parameter, and $\hat{P}_{\gamma}$ ($\gamma=yz,zx,xy$) is the projection operator into the $t_{2g}\gamma$ orbital. We ignore the vibronic coupling to the $T_{2g}$ modes because it is much weaker than that to the $E_{g}$ modes.

The spin-orbit interaction splits the $t_{2g}^{1}$ hole configurations into effective $j=1/2$ and $j=3/2$ multiplet states. These states belong to the eigen energies of $\hat{H}_{\text{SO}}$, $-\frac{\lambda}{2}$ and $\lambda$, respectively. Using the spin-orbit multiplet states, the vibronic coupling part of $\hat{H}_{\text{JT}}$ becomes

The basis of the matrix is in the order of $j=\frac{1}{2}$ ($m_{j}=-\frac{1}{2},\frac{1}{2}$) and $j=\frac{3}{2}$ ($m_{j}=-\frac{3}{2},-\frac{1}{2},\frac{1}{2},\frac{3}{2}$). Eq. (3) indicates the presence of the JT effect in the $j=3/2$ multiplet states and the pseudo JT effect between the $j=1/2$ and $j=3/2$ spin-orbit multiplets.

The energy eigenstates of $5d^{5}$ sites take the form of spin-orbit and lattice entanglement. Since the vibronic interaction, Eq. (3), hybridizes the spin-orbit multiplet states via the quantized normal coordinates, the energy eigenstates (vibronic states) of $\hat{H}_{\text{SO}}+\hat{H}_{\text{JT}}$ are

where $|\chi_{jm_{j};\nu}\rangle$ include the information of the lattice degrees of freedom. The vibronic states reduce to the direct products of the spin-orbit multiplet and vibrational states in the limit of $g\rightarrow 0$.

Now, we turn to the Ir $L_{3}$ RIXS spectra of the vibronic system. The $L_{3}$ RIXS is a photon-in photon-out process involving electron transitions from the $2p_{3/2}$ states to the $t_{2g}$ ($5d$) orbitals, and then, from the $t_{2g}$ orbitals to the $2p_{3/2}$ within an Ir ion. Assuming that the initial and final states of the system are of vibronic type, Eq. (4), the cross-section for the $L_{3}$ RIXS process in a single Ir center is [14]

by using the Kramers-Heisenberg formula [18, 19] and the dipole and fast collision approximations [20, 21]. In Eq. (5), $\nu$ ($\nu^{\prime}$) indicate the initial (final) vibronic states, $\rho_{\nu}$ the canonical distribution for the Ir center, $\hat{d}_{\alpha}$ the $\alpha$ ($=x,y,z$) component of the electric dipole operator, $\hat{P}_{\text{eh}}$ the projection operator into the electron-hole states with $(2p_{3/2})^{3}(t_{2g})^{6}$ electron configurations, and $\hslash\omega_{i}$ ($\hslash\omega_{f}$) and $\bm{\epsilon}_{\bm{k}\lambda}$ ($\bm{\epsilon}_{\bm{k}^{\prime}\lambda^{\prime}})$ are, respectively, the energy and the polarization of the incident (scattered) photon. We used the fast collision approximation for the Ir compounds because the energy scale of the lifetime broadening of Ir ion ($\approx$ 4-5 eV) [22] is one order of magnitude larger than that of the electron-hole dynamics measured by the $j=3/2$ band width ($\approx 0.7$ eV) [7].

For the derivation of the vibronic states, Eq. (4), we performed a numerical diagonalization of the model Hamiltonian [see e.g. Refs. [14, 15, 23]]. We expanded the lattice parts of the vibronic states, Eq. (4), as

To determine the coefficients $C_{jm_{j},n_{u},n_{v};\nu}$, we calculated the Hamiltonian matrix of the JT model ($\hat{H}_{\text{SO}}+\hat{H}_{\text{JT}}$) and diagonalized it. The basis of the matrix was a set of the direct products of the $|j,m_{j}\rangle$ ($j=\frac{1}{2},\frac{3}{2}$) and the eigenstates of the harmonic oscillator part of $\hat{H}_{\text{JT}}$, $|n_{u},n_{v}\rangle$ $(n_{u},n_{v}=0,1,2,\cdots)$. We truncated our basis so that $0\leq n_{u}+n_{v}\leq 15$.

The dynamic JT Hamiltonian contains interaction parameters, $\lambda$, $\omega$, and $g$. We took $\lambda$ from the RIXS data and $\omega$ from the Raman scattering data and treated $g$ as a variable. We determined $g$ so that we can reproduce the experimental RIXS spectra.

To check the validity of the chosen parameters, we also derived all the interaction parameters by a series of post Hartree-Fock calculations as in Refs. [23, 15, 14]. We made clusters of K₂IrCl₆ and K₂IrBr₆ using the neutron diffraction data at 0.3 K for the former and 200 K for the latter, respectively [8]. Each cluster consists of three groups of atoms. The first group corresponds to K₈Ir$X_{6}$ ($X=$ Cl, Br) cluster, which we treated ab initio with atomic-natural-orbital relativistic-correlation consistent-valence quadruple-$\zeta$ polarization basis set. The second group contains the nearest 12 octahedra and the neighboring 48 K atoms, which are replaced by ab initio embedded potential [24]. The third part consists of point charges. The total charge of the entire cluster is neutral. We calculated ${}^{2}T_{2g}$ states of the clusters with the complete active space self-consistent field (CASSCF) method [25] followed by a correction within the extended multistate complete active space second-order perturbation theory (XMS-CASPT2) [26, 27], and then, the spin-orbit $j$ multiplet states with spin-orbit restricted active space state interaction (SO-RASSI) method [25] implemented in OpenMolcas [28, 29]. In these calculations, we set three $5d$ $t_{2g}$ orbitals with one hole as the active space. We determined $\lambda$ by using the SO-RASSI energy gap.

For the calculations of the vibronic coupling parameters and the frequencies, we fit the ${}^{2}T_{2g}$ adiabatic potential energy surfaces with respect to the JT deformation to the JT model, Eq. (2) [See Refs. [23, 15, 14]]. We generated several JT-deformed Cartesian coordinates of an octahedron by using $\bm{R}_{A}=\bm{R}_{A}^{(0)}+Q_{u}\left(\bm{e}_{Eu}\right)_{A}/\sqrt{M}$ and calculated the term energies $U_{\gamma}$ ($\gamma=yz,zx,xy$) at each structure. Here $\bm{R}_{A}$ are the Cartesian coordinates of ligand atom $A$, $\bm{R}^{(0)}$ indicates the experimental octahedral configurations, $\bm{e}_{Eu}$ is the JT-active $Eu$ polarization vectors of the neighboring six Cl’s or Br’s, $Q_{u}$ the mass-weighted normal coordinates ($Q_{u}=\sqrt{\hslash/\omega}q_{u}$), and $M$ the mass of ligand atoms. The gradient of $U_{xy}$ with respect to $Q_{u}$ is the vibronic coupling parameter, $V=-\partial U_{xy}/\partial Q_{u}$. The curvature of $U$ is mainly $\omega^{2}$ with a small correction from the quadratic vibronic coupling. The quadratic vibronic coupling has a similar form as the linear one in Eq. (2): the former is obtained by replacing $\hat{q}_{\gamma}$ and $g$ in the latter with the symmetrized quadratic form of $\hat{q}$ [$\hat{q}_{u}\rightarrow-\frac{1}{\sqrt{2}}(\hat{q}_{u}^{2}-\hat{q}_{v}^{2})$ and $\hat{q}_{v}\rightarrow\sqrt{2}\hat{q}_{u}\hat{q}_{v}$] and $w$, respectively (see e.g., Ref. [15]). With $V$ and $\omega$, $g$ is defined by $g=V/\sqrt{\hslash\omega^{3}}$.

In this work, we theoretically demonstrated that the dynamic JT effect determines the shape and the temperature dependence of the RIXS spectra of K₂IrCl₆. In particular, the dynamic JT effect in the $j=3/2$ multiplet states develops and splits the peak for the $j=3/2$ multiplet in the RIXS spectra. The interaction parameters used for the simulations are consistent with our ab initio interaction parameters. Thus, our results indicate that the system is cubic. The present results suggest that the RIXS spectra of spin-orbit Mott insulators with (quasi) degenerate multiple states on sites have to be interpreted fully considering the dynamic (pseudo) JT effect.