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[Ba2CaOsO6_RIXS_SI_3.pdf](https://mdr.nims.go.jp/filesets/2f6b9335-5c55-40b8-86b4-593363ce31ab/download)

## Creator

J. Okamoto, G. Shibata, Yu. S. Ponosov, [H. Hayashi](https://orcid.org/0000-0001-7787-9082), [K. Yamaura](https://orcid.org/0000-0003-0390-8244), H. Y. Huang, A. Singh, C. T. Chen, A. Tanaka, S. V. Streltsov, D. J. Huang, A. Fujimori

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[Spin-orbit-entangled state of Ba2CaOsO6 studied by O K-edge resonant inelastic X-ray scattering and Raman spectroscopy](https://mdr.nims.go.jp/datasets/e93d8291-2f12-43bf-bf45-b5ce367f1275)

## Fulltext

Supplementary Information forSpin-orbit-entangled state of Ba2CaOsO6 studiedby O K-edge resonant inelastic X-ray scatteringand Raman spectroscopyJ. Okamoto, G. Shibata, Yu. S. Ponosov, H. Hayashi, K. Yamaura, H. Y. Huang,A. Singh, C. T. Chen, A. Tanaka, S. V. Streltsov, D. J. Huang, and A. FujimoriFebruary 26, 2025This SI file includes:Supplementary Notes 1 to 2Supplementary Figures 1 to 3Supplementary Table 1Supplementary References 1 to 51Supplementary Note 1. Phonon sidebands in RIXS spectraIn the low-energy O K-edge RIXS spectrum of Ba2CaOsO6 in Fig. 4(a), subpeaks sep-arated by an interval of ∼ 80-meV are observed to accompany the quasi-elastic peak and thepeaks at Eloss ≃ 0.4 eV and 0.8 eV with intensities decreasing with energy. We attribute thesesub-peaks to phonon replicas created by the simultaneous excitation of optical phonons. Weinvestigated these multiplet excitation peaks and the phonons below energy loss of 1.5 eV re-ferring to Živković et al.’s method [1]: Three multiplet excitation peaks and phonon side bandsare fitted by Voigt functions and anti-Lorentzian functions, respectively. The background curveis a cubic function. Parameter values for the three multiplet excitations with Voigt functionsare summarized in Supplementary Table 1: peak energy ω0, Gaussian FWHM ΓG, integratedintensity, and the FWHM ratio of the Lorentzian function against the Gaussian function ΓL/ΓG.For the phonon satellites, the relative peak energies and intensities of the phonon satellitesare shown in Supplementary Figures 1(a) and (b). The FWHMs of the anti-Lorentzian functionsare set to 85 meV. The relative peak energies of the phonon satellites Eαn (α = 1st, 2nd, and 3rdmultiplet peaks) are well approximated by linear functions of phonon number n, Eαn = nEmodewith the energy of the optical phonon mode Emode = 82±8 meV. From the intensity ratio of thephonon sidebands, the second harmonic intensities are ∼ 0.3 − 0.4 of the first harmonic ones.Then, one can estimate the parameter of the electron-phonon coupling strength M/ω0 ≳ 1referring to [2].Supplementary Note 2. DFT calculations of phonon spectrumIn order to confirm the assignment of the phonon modes, we performed phonon calcula-tions by the frozen phonons method [3] within the non-magnetic DFT. Os is a heavy transition-metal atom and, therefore, it is natural to include both strong electronic correlations (U ) and the2spin-orbit coupling (SOC) via DFT+U+SOC approach. However, this method tends to stabilizemagnetic solutions and cannot directly simulate the non-Kramers many-electron Eg states withzero projected total angular momentum Jzeff = 0, which were proposed to be the ground stateof Ba2CaOsO6 in case of cubic symmetry. Therefore, in Supplementary Figure 2 we presentresults of the non-magnetic phonon calculations [3]. The results of DFT+U+SOC approachshall be discussed below.Phonon frequencies at the Γ-point were calculated by density functional perturbation the-ory (DFPT) in the DFT+U+SOC approach [4]. The same convergence criteria and parametersetup was used as in the frozen phonon calculations. On-site Hubbard repulsion parameter Uand Hund’s intra-atomic exchange (JH) were chosen to be U = 3 eV and JH = 0.5 eV, which areclose to what is used for Os ions in the literature. The crystal structure was taken from [5]. Wetested several combinations of magnetic orders [ferromagnetic and antiferromagnetic (AFM) ofA-type] and directions of the total momentum ([001], [110], [111]) and obtained that the low-est total energy corresponds to Immm structure with Os AFM-A (ferromagnetic planes) andmagnetic moments ordered in the ab-plane close [110] direction. In the relaxed structure OsO6octahedra are slightly elongated: four short 1.930 A and two long 1.945 A Os-O bonds.Results of phonon calculations are shown in Supplementary Figure 2. The experimentalspectrum agrees well with the oversimplified non-magnetic DFT calculation (except for the Egmode). However, theoretical results for the cubic structure do not indicate any additional A1gphonons at 720 cm−1 seen in the experiment.Account of the spin-orbit coupling and Coulomb correlations by non-magnetic DFT+U+SOC(U − JH = 2.5 eV) calculations of phonon frequencies was performed only at the Γ-point us-ing DFPT. It improves the position of the Eg band and yields ωA1g = 787 cm−1 (experiment:796.5 cm−1), ωEg = 525 cm−1 (experiment: 495 cm−1), ωT2g = 355 cm−1 (experiment: 375cm−1), and ωT2g = 105 cm−1 (experiment: 101.5 cm−1). However, as explained above the3undistorted cubic structure turns out unstable in GGA+U+SOC therefore there appear imagi-nary acoustic and low-frequency optical modes. The DFPT calculations demonstrate that theEg phonon mode splits, but the splitting does not exceed 30 cm−1, so one cannot attribute twoexperimentally observed peaks at 495 and 720 cm−1 to Eg phonon split due to the Jahn-Tellereffect. Nevertheless, experimental peak at 495 cm−1 is extremely broad and therefore the split-ting of this line can remain unnoticed. Unexpected appearance of the extra A1g mode at 720cm−1 can be due to a non-negligible disorder in B sites. Indeed folding of the Brillouin zone,e.g. when U → Γ, would lead to the appearance of an additional (weak) mode at ∼90 meV.Supplementary References[1] I. Živković, et al., Dynamic Jahn-Teller effect in the strong spin-orbit coupling regime, Nat.Commun. 15, 8587 (2024).[2] L. J. P. Ament, M. van Veenendaal, J. van den Brink, Determining the electron-phononcoupling strength from resonant inelastic X-ray scattering at transition metal L-edges, Eu-rophys. Lett. 95, 27008 (2011).[3] A. Togo, First-principles phonon calculations with phonopy and phono3py, J. Phys. Soc.Jpn. 92, 012001 (2023).[4] S. L. Dudarev, S. Y. Savrasov, C. J. Humphreys, A. P. Sutton, Electron-energy-loss spectraand the structural stability of nickel oxide: An LSDA+U study, Phys. Rev. B 57, 1505-1509(1998).[5] K. Yamamura, M. Wakeshima, Y. Hinatsu, Structural phase transition and magnetic proper-ties of double perovskites Ba2CaMO6 (M = W, Re, Os), J. Solid State Chem. 179, 605-612(2006).4(b)(a)Supplementary Figure 1: Line-shape analysis of the phonon side bands in the RIXS spectrumin Fig. 4(a), (a) Relative energies of the phonon satellites. (b) Relative intensities of the phononsatellites. 1st: Jeff = 2 → 2 multiplet excitation, 2nd: Jeff = 2 → 0, 1 excitation, 3rd:Jeff = 2 → 2 excitation.Supplementary Table 1: Parameter values of the Voigt functions for the three multiplet excita-tion peaks.Parameter 1st 2nd 3rdPeak energy ω0 (eV) 0.0 0.397 0.788FWHM of Gaussian ΓG (meV) 32.4 45.2 45.0Integrated intensity (arb. units) 216.6 145.1 36.0FWHM ratio ΓL/ΓG 0.85 0.54 1.285Supplementary Figure 2: Phonon dispersions (together with characters) as calculated by non-magnetic DFT.6Supplementary Figure 3: Powder X-ray diffraction of the polycrystalline Ba2CaOsO6 sample.7