Nonlocal features of the spin-orbit exciton in Kitaev materials

Blair W. Lebert, Subin Kim, Beom Hyun Kim, Sae Hwan Chun, Diego Casa, Jaewon Choi, Stefano Agrestini, Kejin Zhou, Mirian Garcia-Fernandez, and Young-June Kim

Phys. Rev. B 108, 155122 – Published 16 October 2023

Abstract

A comparative resonant inelastic x-ray scattering (RIXS) study of three well-known Kitaev materials is presented: $α − {Li}_{2} {IrO}_{3}, {Na}_{2} {IrO}_{3}$ , and $α − {RuCl}_{3}$ . Despite similar low-energy physics, these materials show distinct electronic properties, such as the large difference in the size of the charge gap. The RIXS spectra of the spin-orbit exciton for these materials show remarkably similar three-peak features, including sharp low energy peak (peak A) as well as transitions between $j_{eff} = 1 / 2$ and $j_{eff} = 3 / 2$ states. Comparison of experimental spectra with cluster calculations reveals that the observed three-peak structure reflects the significant role that nonlocal physics plays in the electronic structure of these materials. In particular, the low-energy peak A arises from a holon-doublon pair rather than a conventional particle-hole exciton as proposed earlier. Our study suggests that while spin-orbit assisted Mott insulator is still the best description for these materials, electron itinerancy cannot be ignored when formulating low-energy Hamiltonian of these materials.

Received 1 April 2023
Revised 11 August 2023
Accepted 26 September 2023

DOI:https://doi.org/10.1103/PhysRevB.108.155122

Physics Subject Headings (PhySH)

Quantum spin liquid Quasiparticles & collective excitations Spin-orbit coupling

Kitaev model Resonant inelastic x-ray scattering

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Blair W. Lebert ¹, Subin Kim ¹, Beom Hyun Kim ^2,3, Sae Hwan Chun⁴, Diego Casa⁵, Jaewon Choi⁶, Stefano Agrestini ⁶, Kejin Zhou⁶, Mirian Garcia-Fernandez⁶, and Young-June Kim ¹

¹Department of Physics, University of Toronto, Toronto, Ontario, M5S 1A7, Canada
²Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, Republic of Korea
³Korea Institute for Advanced Study, Seoul 02455, Republic of Korea
⁴Pohang Accelerator Laboratory, Pohang, Gyeongbuk 37673, Republic of Korea
⁵Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA
⁶Diamond Light Source, Harwell Campus, Didcot OX11 0DE, United Kingdom

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Vol. 108, Iss. 15 — 15 October 2023

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Images

Figure 1
(a) In-plane reciprocal space showing hexagonal Brillouin zone (purple) and projection of $C 2 / m$ Brillouin zone (green). The $C 2 / m$ lattice is used throughout this paper and the projection of its $a^{*} = (h, 0)$ and $b^{*} = (0, k)$ axes are shown as green vectors. (b) RIXS scattering plane used for experiments and calculations. Incoming x-rays with $π$ polarization and momentum $k$ are scattered at $2 θ = 90^{\circ}$ with momentum $k^{'}$ transferring $q = k - k^{'}$ momentum to the sample (green bar). The sample is mounted with the honeycomb plane at an angle $θ$ leading to transferred in-plane momentum $q_{∥}$ . The azimuthal angle $ϕ$ rotates the sample around the $c^{*}$ axis, with $a^{*}$ ( $b^{*}$ ) in the scattering plane at $ϕ = 0^{\circ}$ ( $ϕ = 90^{\circ}$ ). (c), (d) Schematic energy diagram of two excitation processes. Filled and unfilled arrows represent electrons and holes, respectively, where up (down) arrows denote Kramer's doublet with positive (negative) eigenvalues of $j_{eff}^{z}$ . See text for details.
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Figure 2
In-plane momentum dependence of excitations in $α − {Li}_{2} {IrO}_{3}$ at 7.5 K measured with Ir $L_{3}$ -edge RIXS. The main peaks are labeled from low to high energy as A, B, and C.
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Figure 3
Experimental Ru $M_{3}$ -edge RIXS data and fits of $α − {RuCl}_{3}$ at 11 K compared to calculations. Spectra were taken with $b^{*}$ axis in scattering plane ( $ϕ = 90^{\circ}$ ) while varying $θ$ from grazing incidence [panel (a)] to grazing emission [panel (e)] with corresponding $q_{∥}$ and $θ$ shown in the top right corner of each panel. Data are shown as black circles and fits are shown as solid purple lines, with fit components as dashed purple lines: in particular peaks A, B, and C which are labeled in (c), as well as an additional higher energy excitation. RIXS calculations are shown as green lines for $Δ_{t} < 0$ . The momentum points and scattering geometry are decoupled in our calculations, therefore, since all Ru $M_{3}$ -edge measurements are close to (0,0) we calculate the RIXS spectra at (0,0) while using the experimental $θ$ angle. The momentum dependence of the extracted peak positions are plotted in (f).
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Figure 4
Comparison between experimental (black) and calculated RIXS spectra at (0,0) with $Δ_{t} > 0$ (blue) and $Δ_{t} < 0$ (green). The rows correspond to different materials and the columns to calculations with different number of sites. The energy axis has been scaled by the spin-orbit coupling ( $λ$ ) used in the calculation for each compound (Table 2). The 1-site calculated spectra's intensities have been scaled by half for visibility. The A, B, and C peaks are labeled in the middle panel.
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Figure 5
Comparison of the RIXS spectra at the room temperature and the base temperature. Note that features are sharper and background is lower at the base temperature, but overall lineshape and the peak positions are unchanged between the two temperatures.
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Figure 6
Theoretical and experimental optical conductivity spectra for (a) $α − {RuCl}_{3}$ , (b) ${Na}_{2} {IrO}_{3}$ , and (c) $α − {Li}_{2} {RuO}_{3}$ . Solid and dotted lines refer to the theoretical and experimental spectra, respectively. Theoretical spectra are calculated with physical prameters in Table 2, while experimental ones are obtained in Ref. [65] for $α − {RuCl}_{3}$ , Ref. [39] for ${Na}_{2} {IrO}_{3}$ , and Ref. [38] for $α − {Li}_{2} {Ir}_{3}$ , respectively.
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