Abstract
Hundâs rule coupling (J) has attracted much attention recently for its role in the description of the novel quantum phases of multi-orbital materials. Depending on the orbital occupancy, J can lead to various intriguing phases. However, experimental confirmation of the orbital occupancy dependency has been difficult as controlling the orbital degrees of freedom normally accompanies chemical inhomogeneities. Here, we demonstrate a method to investigate the role of orbital occupancy in J related phenomena without inducing inhomogeneities. By growing SrRuO3 monolayers on various substrates with symmetry-preserving interlayers, we gradually tune the crystal field splitting and thus the orbital degeneracy of the Ru t2g orbitals. It effectively varies the orbital occupancies of two-dimensional (2D) ruthenates. Via in-situ angle-resolved photoemission spectroscopy, we observe a progressive metal-insulator transition (MIT). It is found that the MIT occurs with orbital differentiation: concurrent opening of a band insulating gap in the dxy band and a Mott gap in the dxz/yz bands. Our study provides an effective experimental method for investigation of orbital-selective phenomena in multi-orbital materials.
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Introduction
Mott physics with strong electron correlations due to on-site Coulomb repulsion (U) have been a central paradigm in condensed matter1,2. Recently, Hundâs metal, a new type of strongly correlated material, was proposed, for which Hundâs rule coupling (J) dominantly drives the electron correlations rather than U3,4,5,6,7,8,9,10,11,12. In such a system, the roles of J are highly dependent on the number of electrons/orbitals, so a variety of physical phenomena depend strongly on the orbital occupancy. In particular, rich phases with orbital differentiations have been suggested theoretically for Hundâs correlated system3,13,14. Since the energy scale of J is smaller than that of U for most multi-orbital materials, the associated phase transitions can be achieved by controlling an energy scale much smaller than that in Mott physics3,13,14.
Despite extensive theoretical interest3,6,7,10,13,14,15,16,17,18, observations of Hund-driven phase transitions have remained challenging because direct control of the J value is experimentally difficult as it is usually determined by atomic physics. On the other hand, an alternative but indirect approach to control orbital occupancy is possible via doping and/or substitution with different chemical elements19,20. During such experiments, however, numerous defects are easily formed due to the random distribution of chemical elements. These inhomogeneity problems place serious limitations on experimental investigations of Hundâs systems. Therefore, precise control of orbital occupancy without random chemical distribution is the key for experimental investigation of phase transitions in Hundâs systems.
We propose that crystal field splitting can be controlled with a suitable experimental approach to observation of phase transitions in Hundâs systems with negligible impurity problems. In cubic perovskite oxides (a-lattice constant (a) = c-lattice constant (c)), the five d-orbitals of transition metal elements are split into t2g and eg levels due to the oxygen octahedral environment (Fig. 1a). When the oxygen octahedron becomes distorted, the orbitals experience further tetragonal crystal field splitting (Ît). For instance, the three t2g orbitals split into dxy and dxz/yz levels if the oxygen octahedron is elongated or compressed. Such variation of Ît can tune the orbital occupancy of the d-orbitals without chemical doping in partially-filled d-electron systems21. There have been previous experimental studies that demonstrated the manipulation of orbital polarization22,23,24. They include the charge transfer at the cuprate-manganate interfaces22, metal-insulator transition in VO223, and nickelate-cuprate heterostructures24. However, the impact of J in such orbital polarization changes has not been considered. Here, our objective is to explore the interplay between Ît and J by controlling the orbital polarization in widely accepted Hundâs system.
In this study, we investigated tuning of orbital occupancy in SrRuO3 (SRO) ultrathin films by artificially controlling the crystal field splitting. It should be noted that Sr2RuO4 is well-known as Hundâs metal, so 2D limit of SRO can provide insight on the Hundâs physics7. In the bulk SRO, Ît is small, and the nearly degenerate dxy/xz/yz orbitals are partially-filled with 4 electrons (3-orbital/4-electron system). As shown in Fig. 1b, with an increase in Ît, the dxy (dxz/yz) band moves downward (upward), leading to a redistribution of electrons among the three t2g orbitals. When the energy level of the dxy band is lower than the Fermi energy (EF), the dxy (dxz/yz) band becomes fully-filled (half-filled). Then, SRO will behave effectively as a 2-orbital/2-electron system. This crystal field splitting effect (i.e. with negligible U/D) makes dxy level lower than that of dxz/yz. When U/D is large (D is the half-bandwidth), the dxz/yz bands can further experience a Mott transition14, as schematically shown in the last band configuration of Fig. 1b. The relative position of dxz/yz can be a good indication of how U and the crystal field play roles in determining electronic structures in of SRO.
For a comprehensive understanding of the orbital-selective phase transitions, we should pay much attention to the roles of J14. Figure 1c schematically displays how the phase transitions occur in SRO for two different J values. With Jâ=â0, it can be challenging to detect the Mott transition induced by the crystal field because it only happens in a small range of U. On the other hand, J facilitates such transition in a much wider interaction range. Therefore, the system with a sizable J experiences orbital-selective phase transitions as Ît varies. For small U/D, increasing Ît induces a transition from a metallic state into another metallic state with a gap in the dxy band while, for a large U/D, Ît causes a phase transition from a Hundâs metal to an insulating state with band (dxy) + Mott (dxz/yz) gaps. Note that for small Ît, the critical U value for opening the Mott gaps (Uc) increases when we take J into account (dashed arrow (1) in Fig. 1c). On the other hand, with large Ît, Uc decreases with sizable J (dashed arrow (2)). This variable nature of J makes the phase transition with orbital differentiation intriguing3,14. Therefore, our experimental approach with control of Ît can provide a way to systematically explore the Hundâs physics.
Results
Symmetry-preserving strain engineering in 2D SRO
In this study, we observed an orbital-selective phase transition in an SRO monolayer with in-situ angle-resolved photoemission spectroscopy (ARPES). Thick SRO films (three-dimensional systems) did not exhibit metal-insulator transitions as shown in Supplementary Fig. 1. As stated earlier, the metal to band+Mott insulator transition is feasible with small control of Ît and large U/D (red arrow in Fig. 1c). To study such effects more clearly, we investigated SRO monolayer systems, which contain a two-dimensional (2D) RuO2 layer. When the thickness of the SRO film decreases and approaches the monolayer limit (2D), electron hopping along the z-direction diminishes25,26,27,28,29. The bandwidth of the dxz/yz orbitals can be further reduced by orbital-selective quantum confinement effects30.
We used strain engineering to compress the oxygen octahedra along the out-of-plane direction in SRO monolayers. We used pulsed laser deposition to grow SRO layers on five different substrates, (LaAlO3)0.3(Sr2TaAlO6)0.7(001) [LSAT(001)], SrTiO3(001) [STO(001)], Sr2(Al,Ga)TaO6(001) [SAGT(001)], KTaO3(001) [KTO(001)], and PrScO3(110) [PSO(110)]. They are known to have (pseudocubic) lattice constants of 3.868, 3.905, 3.931, 3.989, and 4.02âà , respectively. Since the pseudocubic lattice constant of bulk SRO is 3.923âà , the substrates impart â1.4%, â0.5%, +0.2%, +1.7%, and +2.5% epitaxial strain on the SRO monolayers. For convenience, we have indicated compressive (tensile) strain using a minus (plus) sign. One should note that these substrates have different oxygen octahedral rotation (OOR) patterns, so each SRO monolayer grown on the above-mentioned substrates could have a different OOR pattern31,32,33. Such undesirable occurrence of the complex structural modifications could hinder our systematic investigation of the orbital-selective phase transitions.
To suppress the structural complications of OOR, we developed a symmetry-preserving strain engineering technique. As shown in Fig. 2a, 10 unit cells (u.c.) of the SrTiO3 (STO) layer were inserted between the SRO monolayer and substrates. This preserves the OOR pattern of the SRO monolayer while applying different epitaxial strains. Figure 2b, c show low-energy electron diffraction (LEED) results for the SRO monolayers under both compressive and tensile strains at 6âK. All samples showed LEED diffraction peaks at (mâ+â0.5, nâ+â0.5) (m, n: integers), indicating the existence of OOR along the out-of-plane axis. On the other hand, no sample showed (mâ+â0.5, n) and (m, nâ+â0.5) peaks, indicating that OOR did not occur along the in-plane axis. These LEED diffraction results indicate that all SRO monolayers on the substrates used exhibited OOR with a0a0c-crystal symmetry (Supplementary Fig. 2).
The atomic arrangements of the SRO monolayers were confirmed by scanning transmission electron microscopy (STEM). We covered the heterostructures with a 10-u.c. STO layer to protect the SRO layer from possible damage during STEM measurements. Figure 2d and Supplementary Fig. 3 show low-magnification STEM images in high-angle annular dark-field (HAADF) mode. High-magnification STEM images were acquired in the HAADF and annular bright-field (ABF) modes (Fig. 2e, f, respectively). The SRO monolayer did not show OOR along the in-plane axis, consistent with the LEED results. The analysis of lattice constants from the STEM results shows coherent strain state (Supplementary Fig. 4). Although most areas showed abrupt interfaces with the SRO single layer, we observed a few regions for which thickness inhomogeneities (i.e., 0 or 2âu.c. thickness of SRO) were observed (Supplementary Fig. 3). These inhomogeneous regions usually occurred near step terraces34,35, which can break the continuity of the SRO monolayer for transport measurements. Instead, we used optical and in situ ARPES measurements to obtain reliable area-averaged responses. Possibilities for contributions from the surface states of STO and SRO layers to the spectroscopic results are discussed in Supplementary Note 1.
Orbital occupancy changes
To investigate the orbital occupancy changes, we explored the interatomic optical transitions between Ru4+ ions via ellipsometric spectroscopy. Upon absorption of a photon, an electron can hop from one Ru4+ ion to a nearest-neighbor ion (d4â+âd4âââd3â+âd5 transition). The matrix element of such an interatomic transition can be significantly large due to hybridization between the Ru d â O p orbitals. Note that this interatomic transition can occur only between the same t2g orbitals due to the orbital geometry (Fig. 3a). Specifically, the transition from dxz to dxy (or dyz) will be small, given that there is only a small overlap between the corresponding orbitals. Figure 3b, c display the allowed interatomic d4â+âd4âââd3â+âd5 transition in SRO36. With a small ât, the three t2g orbitals are equally filled with four electrons. Then, interatomic transitions can occur at two photon energies (UâââJ and Uâ+âJ) (Supplementary Fig. 5)36. With a large ât, the dxy orbital becomes fully occupied, and the U - J transition cannot occur.
We obtained an optical spectrum of the SRO monolayer under a strain of â1.4% (Fig. 3d). The spectrum exhibits three peaks (AâC). The A and B peaks are assigned to interatomic d-d transitions with energy positions at U â J and Uâ+âJ, respectively. Peak C reflects a charge transfer transition from O 2p to Ru t2g37. We fitted the spectrum (solid circles) with Lorentzian oscillators (dashed lines). The obtained peak positions were in good agreement with those previously reported for ruthenates36,37. The peak position difference between A and B is 2âJ; J was thus ~0.6âeV36,38. The extracted U values from the optical results could be underestimated due to formation of excitons during optical processes37. The presence of both U â J and Uâ+âJ peaks suggested that the compressively strained SRO monolayer contained partially-filled orbitals, which is similar to the situation for bulk SRO (Fig. 3b).
When the tensile strain is applied, a significant spectral weight (SW) change occurs, indicating strain-induced electron redistribution of the t2g orbitals (Fig. 3dâf). The change in SW is plotted in Fig. 3g. As the tensile strain increases, the SW of peak A decreases and that of peak B increases. For the tensile-strained SRO monolayer (+0.2%), peak A nearly disappears, suggesting that one of the bands (i.e., dxy) becomes fully-filled. Dynamical mean-field theory (DMFT) calculations with Jâ=âU/6 (Uâ=â2.7âeV, Jâ=â0.45âeV) also reveals that the filling of dxy (dxz/dyz) orbital increases (decreases) with an increase in strain (Fig. 3h). When the strain is +0.2% or higher, dxy is fully filled while dxz/yz is half-filled (Supplementary Fig. 6). The control of orbital polarization in the Hund system can lead to an orbital-selective phase transition. Note that existence of such orbital polarization alludes to an insignificant role of the spin-orbit coupling in the orbital dependent Mott transition in the ruthenate films39.
Control of electronic structures with orbital differentiation
The 2D metallic phase in the â1.4% strained SRO monolayer was studied by in-situ ARPES measurement. As SRO monolayers have 2D atomic arrangements (Fig. 2dâf), the electronic structure should be similar to that of Sr2RuO4 (a well-established quasi-2D system)40,41,42. Figure 4a shows the low-temperature constant energy map at EF for SRO monolayer with â1.4% strain. The experimentally obtained Fermi surface (FS) is similar to the schematic FS of Sr2RuO4 (shown as solid lines). Therefore, we hereafter follow the notation generally used for Sr2RuO440,41,42, in which the three bands at EF are labeled α, β, and γ. The orbital characters of the α and β bands are dxz/yz, and that of γ is dxy.
Energy maps of the SRO monolayers show the strain-induced MIT. We measured constant energy maps for SRO monolayers at EF under epitaxial strains of â1.4, â0.5, and +0.2%. Compressively strained SRO monolayers (â1.4 and â0.5%) exhibit a nonzero density of states (DOS) at EF, indicating that they are metallic (Fig. 4a, b). In contrast, tensile-strained SRO monolayers (+0.2%) have nearly zero DOS at EF, showing that they are in an insulating state (Fig. 4c). These strain-dependent DOSs at EF in the SRO monolayer indicate that MIT occurs at a strain between â0.5% and +0.2%, which agrees well with the optical spectra of the SRO monolayer (Fig. 3g, h).
To elucidate the mechanism of the MIT, we focused on strain-dependent E-k dispersion along the ÎâM line indicated by the black arrow in Fig. 4a (Fig. 4d). Two major features with epitaxial strain dependence (â1.4, â0.5, +0.2, +1.7, and +2.5%) are observed. First, the energy level of the band near M/2 (red dashed line) moves smoothly to higher binding energies with increasing tensile strain. Second, the spectral weight (SW) near Î (blue dashed line) at Eâ=âEF â 1.6âeV appears abruptly at a strain of +0.2% and is enhanced above +0.2% tensile strain.
Orbital characters of the bands in SRO monolayers were identified by examining constant energy maps below EF for SRO films under aâ+â0.2% strain. Figure 5a shows an energy distribution curve (EDC) from the ÎâM line of the SRO monolayer (+0.2%). The constant-energy maps at EâEFâ=ââ 0.5âeV (red dashed line) and â 1.6âeV (blue dashed line) were acquired for two experimental geometries, as shown in Fig. 5b. Note that the photoemission intensity for the dxy orbital should vary with respect to the azimuthal angle Ï, whereas that for the dxz/yz orbital should change little with respect to Ï (Supplementary Note 2). At EFâââ0.5âeV, we observe a significant difference in the photoemission intensity between Ïâ=â0° and 45°. In particular, the intensity along the ÎâM line with Ïâ=â45° (Fig. 5d) was much lower than that with Ïâ=â0° (Fig. 5c). On the other hand, at EFâââ1.6âeV, the photoemission intensity at Ïâ=â0° and 45° show little difference (Fig. 5e, f). These observations can be understood by considering the matrix element43. Therefore, we conclude that the orbital characteristics of near EFâââ0.5âeV band is dominantly of dxy while that for the band near EFâââ1.6âeV is dominantly of dxz/yz.
This orbital differentiation can be used to explain the band structure evolution in Fig. 4d, which confirms the rationale illustrated in Fig. 1b. As the tensile strain is applied, the dxy band near M/2 moves steadily toward the high binding energy side and becomes a band insulator at the MIT (red dashed line). We also observe that near Π(dxz/yz) spectral weight suddenly appears close to the MIT (blue dashed line). As dxz/yz becomes half-filled with tensile strain (Fig. 3h), the band near Πshould be the lower Hubbard band (LHB). Therefore, the MIT occurrs with orbital differentiation. The two gaps open nearly simultaneously at the MIT: a band insulating gap for dxy and a Mott gap for dxz/yz38,44. More details on the energy positions of LHB and UHB are discussed in Supplementary Note 3.
Careful analysis of the ARPES spectra also reveals how ât can indeed be tuned via our symmetry-preserving strain engineering technique. EDCs of the ÎâM cut for different strain values are shown in Supplementary Fig. 7. Peaks are seen at energies from EFâââ0.8 to â 0.4âeV (from dxy) and â 1.6âeV (from the LHBs of dxz/yz). The peak position of the dxy band (ζ) was obtained via Gaussian fitting. The ζ value (black circles) increases with the tensile strain, consistent with the density functional theory (DFT) (pink squares). Since the LHB energy varies little with the strain, the change in the ζ value should be from the change in ât. Therefore, the systematic ζ change in the ARPES indicates that ât increases with an increase in tensile strain, as expected from Fig. 1b.
We also tried to change the orbital occupancy by dosing potassium (K) on the SRO monolayer surface. K atoms donate electrons without significantly perturbing the SRO monolayer structure. The coverage of the K layer was controlled from 0.0 monolayer (ML) to 1.0âML. Supplementary Fig. 8 shows K-coverage-dependent ARPES data for the SRO monolayer in a band (dxy)+Mott(dxz/yz) insulating state with +2.5% strain. With electron doping, the DOS for the LHBs (dxz/yz) decreases while the DOS near EF increases. This indicates that the correlation-induced Mott gap collapses as the system moves away from the half-filled dxz/yz. However, the SRO monolayer with 1.0 ML K-coverage is still insulating (almost zero DOS at EF), possibly due to strong Anderson localization effects in the 2D system45.
Discussion
In summary, we demonstrated tuning of the crystal field of a SRO monolayer with symmetry-preserving strain engineering. Given the nature of 2D materials, the modulated ât induces compulsory changes in the orbital occupancy, which results in a dramatic orbital-selective phase transition. Remarkably, a simultaneous MIT with orbital differentiation was observed via in situ ARPES measurements, in which one of the bands becomes band insulating while the other opens a Mott gap. Note that J and crystal field energy scales compete each other in terms of the orbital polarization. Therefore, the observed orbital-selective phase transition induced by the crystal field tuning in Hundâs system will provide an insight on the role of J.
Our symmetry-preserving strain engineering technique can be applied to studies of various multi-orbital physics in numerous transition metal oxide monolayers46,47. According to recent theoretical and experimental studies3,4,5,6,7,8,9,10,11,12, Hundâs physics is crucial in the description of novel quantum phenomena such as unconventional superconductivity and magnetism as well as potential device applications. However, impurity problems and complex structural distortions in materials have hindered manipulation and observation of Hundâs physics, such as orbital-selective Mott phases. In this context, our strategy can be used extensively for precise control of materials with simplified structures and negligible inhomogeneity problems. Thus, our work provides a way to investigate novel phenomena in multi-orbital systems and promote multi-orbital-based device applications.
Methods
Sample preparation
STO and SRO epitaxial layers were grown on various substrates via PLD. The targets were single-crystalline STO and polycrystalline SRO. A KrF excimer laser (λâ=â248ânm; coherent) was operated with a repetition frequency of 2âHz. For deposition of the STO and SRO layers, the laser energies were 1.0 and 2.0âJâcmâ2, respectively. The deposition temperature was 700â°C, and the oxygen pressure was 100 mTorr. The film thickness was controlled at the atomic scale by monitoring reflection high-energy electron diffraction patterns (Supplementary Fig. 9).
Scanning transmission electron microscopy
An electron-transparent STEM specimen was prepared via focused ion-beam milling (Helios 650 FIB; FEI) and further thinned via focused Ar-ion milling (NanoMill 1040; Fischione). Cross-sectional STEM images were acquired at room temperature using an instrument corrected for spherical aberration (Themis Z; Thermo Fisher Scientific Inc.) and equipped with a high-brightness Schottky-field emission gun (operating at an electron acceleration voltage of 300âkV). The semiconvergence angle of the electron probe was 17.9 mrad. The collection semiangle for ABF was 10â21 mrad.
In situ angle-resolved photoemission spectroscopy
ARPES measurements were performed using home-built laboratory equipment comprising an analyzer (Scienta DA30) and discharge lamp (Fermion instrument). He-Iα (hvâ=â21.2âeV) light partially polarized in a linear vertical direction (70%) was used. As all of the substrates used in this study were insulators, ARPES measurements could be hampered by charging effects arising from the sample geometry (Fig. 2a). Thus, we inserted a conducting layer of 4-u.c.-thick SRO between the substrate and 10âu.c. STO layer25. The heterostructures are fully strained to substrates as shown in Supplementary Fig. 10. The STO layer decoupled the electronic structure of the topmost SRO monolayer from that of the inserted SRO conducting layer25. After growth, the SRO heterostructures were transferred to an ultrahigh vacuum (UHV; <1.0âÃâ10â10 Torr) environment without air exposure and annealed at 570â°C for 20âmin immediately before measuring ARPES. This annealing process makes the surface quality of the sample suitable for ARPES measurements (Supplementary Figs. 11, 15). It was previously shown that oxygen vacancy effects are negligible for SRO films annealed under the optimized condition25.
Low-energy electron diffraction
All LEED data were collected using a SPECS ErLEED 1000-A. The base pressure of the UHV chamber was maintained below 8.0âÃâ10â11 Torr. To match the results of the ARPES and LEED measurements, sample preparation before LEED measurements was identical to that before ARPES (preannealing of 570â°C for 20âmin at <1. 0âÃâ10â10 Torr). The UHV chamber for LEED measurements was connected to the chambers used for ARPES measurements and sample growth; the sample was not exposed to air before the LEED measurements.
Optical spectroscopy
Optical spectroscopic characterization was performed using a spectroscopic ellipsometer (M-2000 DI; J.A. Woollam Co.). The reflectance of the bare substrate, STO (10âu.c.)/substrate, and SRO (1âu.c.)/STO (10âu.c.)/substrate were measured separately, and the optical conductivity was extracted for each layer (Supplementary Fig. 12). It was challenging to obtain the optical spectra of samples under higher tensile strain (+1.7 and +2.5%). The as-received KTO(001) substrate had 3â4 nm-deep surface holes. Although ARPES measurements were not seriously affected by such holes, obtaining reliable optical data from spectroscopic ellipsometry was challenging. In addition, as PSO(110) (+2.5%) single crystalline substrates have orthorhombic structures, it was quite difficult to subtract the strongly anisotropic responses.
Density functional theory and dynamic mean-field theory calculations
We carried out density functional theory (DFT) calculations within the PerdewâBurkeâErnzerhof exchange-correlation functional revised for solids using VASP code48. For the lattice constant, the experimental value of the corresponding substrate material was used for each strain configuration. We used a 600âeV plane-wave cutoff energy and 6âÃâ6âÃâ1 k-points for all DFT calculations. The internal atomic positions of the three layers closest to the surface were fully relaxed until the maximum force was below 5 meVà â1. Maximally localized Wannier functions49 for t2g bands were constructed for the tight-binding model to be used in DMFT calculations. We performed a single-site DMFT calculation on top of the Wannier Hamiltonian with a continuous-time QMC hybridization-expansion solver implemented in TRIQS/CTHYB. Supplementary Fig. 13, Fig. 14 show the detailed band structure obtained from the Wannierization for the t2g orbitals and the electron hopping term of dxy-dxy and dxz-dxz (or dyz-dyz).
Data availability
Source data are provided with this paper. Other data is available from the authors upon request. Source data are provided with this paper.
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Acknowledgements
All authors are grateful for the major support provided by the Research Center Program of the Institute for Basic Science of Korea (grant no. IBS-R009-D1). S.H. and C.K. acknowledge the supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2022R1A3B1077234). C. Sohn was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (Creative Materials Discovery Program-No.2017M3D1A1040834) and the Ministry of Science and ICT (2020R1C1C1008734). S.-S.B.L. was supported by the National Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2022R1F1A107452211). S. Lee and M. Kim acknowledge financial support from the Korean government through the National Research Foundation (grant no. 2017R1A2B3011629). Cs-corrected STEM was performed at the Research Institute of Advanced Materials (RIAM) of Seoul National University.
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E.K.K. and T.W.N. conceived and designed the project. E.K.K. fabricated and characterized the films via optical spectroscopy. S.H. characterized the films via ARPES and LEED. C.S., J. Son., J. Song, and T.W.N. contributed to the optical spectroscopy data analysis. S.H., D.K., and Y.K. contributed to the ARPES measurements. S.L. and M.K. performed STEM and analyzed the data. S.H., B.S., J.R.K., and C.K. contributed to the analysis of the ARPES results. C.H.K. S.S.B.L. contributed to the DFT and DMFT calculations. E.K.K., S.H., C.H.K., C.K., and T.W.N. wrote the paper with contributions and feedback from all authors. T.W.N. initiated the study and was responsible for the overall research direction. E.K.K. and S.H. contributed equally to this work.
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Ko, E.K., Hahn, S., Sohn, C. et al. Tuning orbital-selective phase transitions in a two-dimensional Hundâs correlated system. Nat Commun 14, 3572 (2023). https://doi.org/10.1038/s41467-023-39188-9
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DOI: https://doi.org/10.1038/s41467-023-39188-9