Abstract
A paradigmatic case of multiband Mott physics including spinorbit and Hund’s coupling is realized in Ca_{2}RuO_{4}. Progress in understanding the nature of this Mott insulating phase has been impeded by the lack of knowledge about the lowenergy electronic structure. Here we provide—using angleresolved photoemission electron spectroscopy—the band structure of the paramagnetic insulating phase of Ca_{2}RuO_{4} and show how it features several distinct energy scales. Comparison to a simple analysis of atomic multiplets provides a quantitative estimate of the Hund’s coupling J=0.4 eV. Furthermore, the experimental spectra are in good agreement with electronic structure calculations performed with Dynamical MeanField Theory. The crystal field stabilization of the d_{xy} orbital due to caxis contraction is shown to be essential to explain the insulating phase. These results underscore the importance of multiband physics, Coulomb interaction and Hund’s coupling that together generate the Mott insulating state of Ca_{2}RuO_{4}.
Introduction
Electronic instabilities driving superconductivity, density wave orders and Mott metal–insulator transitions produce a characteristic energy scale below an onset temperature^{1,2,3}. Typically, this energy scale manifests itself as a gap in the electronic band structure around the Fermi level. Correlated electron systems have a tendency for avalanches, where one instability triggers or facilitates another^{4}. The challenge is then to disentangle the driving and secondary phenomena. In many Mott insulating systems, such as La_{2}CuO_{4} and Ca_{2}RuO_{4}, longrange magnetic order appears as a secondary effect. In such cases, the energy scale associated with the Mott transition is much larger than that of magnetism. The Mott physics of the halffilled singleband 3d electron system La_{2}CuO_{4} emerges due to a high ratio of Coulomb interaction to band width. This simple scenario does not apply to Ca_{2}RuO_{4}. There the orbital and spin degrees of freedom of the 2/3filled (with four electrons) t_{2g}manifold implies that Hund’s coupling enters as an important energy scale^{5}. Moreover, recent studies of the antiferromagnetic ground state of Ca_{2}RuO_{4} suggest that spin–orbit interaction also plays a significant role in shaping the magnetic moments^{6,7,8}, as well as the splitting of the t_{2g} states^{9}.
Compared to Sr_{2}RuO_{4} (refs 10, 11), which may realize a chiral pwave superconducting state, relatively little is known about the electronic band structure of Ca_{2}RuO_{4} (ref. 12). Angle integrated photoemission spectroscopy has revealed the existence of Ru states with binding energy 1.6 eV (ref. 13)—an energy scale much larger than the Mott gap ∼0.4 eV estimated from transport experiments^{14}. Moreover, angleresolved photoemission spectroscopy (ARPES) experiments on Ca_{1.8}Sr_{0.2}RuO_{4}—the critical composition for the metal–insulator transition—have led to contradicting interpretations^{15,16} favouring or disfavouring the socalled orbitalselective scenario where a Mott gap opens only on a subset of bands^{17,18}. Extending this scenario to Ca_{2}RuO_{4} would imply orbitaldependent Mott gaps^{18}. The electronic structure should thus display two Mott energy scales (one of d_{xy} and another for the d_{xz}, d_{yz} states). A different explanation for the Mott state of Ca_{2}RuO_{4} is that the caxis compression of the SPbca insulating phase induces a crystal field stabilization of the d_{xy} orbital, leading to halffilled d_{xz}, d_{yz} bands and completely filled d_{xy} states^{19,20}. In this case, only one Mott gap on the d_{xz}, d_{yz} bands will be present with band insulating d_{xy} states. The problem has defied a solution due to a lack of experimental knowledge about the lowenergy electronic structure.
Here we present an ARPES study of the electronic structure in the paramagnetic insulating state (at 150 K) of Ca_{2}RuO_{4}. Three different bands—labelled , and band—are identified and their orbital character is discussed through comparison to firstprinciple Density Functional Theory (DFT) band structure calculations. The observed band structure is incompatible with a single insulating energy scale acting uniformly on all orbitals. A phenomenological Green’s function incorporating an enhanced crystal field and a spectral gap in the selfenergy is used to describe the observed band structure on a qualitative level. Further insight is gained from Dynamical MeanField Theory (DMFT) calculations including Hund’s coupling and Coulomb interaction. The Hund’s coupling splits the d_{xy} band allowing quantitative estimate of this parameter. The Coulomb interaction is mainly responsible for the insulating behaviour of the d_{xz}, d_{yz} bands. The experimental results, together with our theoretical analysis, clarify the origin of the Mott phase in the multiorbital system Ca_{2}RuO_{4}. Furthermore, they provide a natural explanation as to why previous experiments have identified different values for the energy gap.
Results
Crystal and electronic structure
Ca_{2}RuO_{4} is a layered perovskite, where the Mott transition coincides with a structural transition at T_{s}∼350 K, below which the caxis lattice constant is reduced. We study the paramagnetic insulating state (T=150 K) of Ca_{2}RuO_{4} with orthorhombic SPbca crystal structure (a=5.39 Å, b=5.59 Å and c=11.77 Å). It is worth noting that due to this nonsymmorphic crystal structure, Ca_{2}RuO_{4} could not form a Mott insulating ground state at other fillings than 1/3 and 2/3 (ref. 21). In Fig. 1, the experimentally measured electronic structure is compared to a firstprinciple DFT calculation of the bare noninteracting bands. We observe two sets of states: near the Fermi level the electronic structure is comprised of Rudominated bands, while oxygen bands are present only for =E−E_{F}<−2.5 eV. Up to an overall energy shift, good agreement between the calculated DFT and observed Ca_{2}RuO_{4} oxygen band structure is found.
Nondispersing ruthenium bands
The structure of the ruthenium bands near the Fermi level is the main topic of this paper, as these are the states influenced by Mott physics. A compilation of ARPES spectra, recorded along highsymmetry directions, is presented in Figs 2 and 3a. In consistency with previous angleintegrated photoemission experiments^{13}, a broad and flat band is found around the binding energy =−1.7 eV. However, we also observe spectral weight closer to the Fermi level (∼−0.8±0.2 eV), especially near the zone boundaries (see Fig. 2a,d). These two flat ruthenium bands (labelled and ) are revealed as a double peak structure in the energy distribution curves—Fig. 2c,f. Between the band and the Fermi level, the spectral weight is suppressed. In fact, complete suppression of spectral weight is found for −0.2 eV<<0 eV (see Fig. 2c). This energy scale is in reasonable agreement with the activation energy ∼0.4 eV extracted from resistivity experiments^{14}.
Fast dispersing ruthenium bands
In addition to the flat and bands, a fast dispersing circularshaped band is observed (Fig. 3b) around the Γpoint (zone centre) in the interval −2.5 eV<<−2 eV—see Figs 2a,b and 3a. A weaker replica of this band is furthermore found around Γ_{2} (Fig. 3a,b). The band velocity, estimated from momentum distribution curves (Fig. 2a), yields v=(2.6±0.4) eV Å. As this band, which we label , disperses away from the zone centre, it merges with the most intense flat band. From the data, it is difficult to conclude with certainty whether the band disperses between the and bands. As this feature is weak in the spectra recorded with 78 eV photons (Fig. 2b), it makes sense to label and as distinct bands.
Orbital band character
Next we discuss the orbital character of the observed bands. As a first step, comparison to the band structure calculations is made. Although details can vary depending on exact methodology, all band structure calculations of Ca_{2}RuO_{4} find a single fast dispersing branch^{22,23,24,25}. Our DFT calculation reveals that the fast dispersing band has predominantly d_{xy} character (Fig. 4a). We thus conclude that the inplane extended d_{xy} orbital is responsible for the band. Within the DFT calculation, the d_{xz} and d_{yz} bare bands are relatively flat throughout the entire zone. This is also the characteristic of the observed band. It is thus natural to assign a dominant d_{xz}, d_{yz} contribution to this band. The orbital character of the band is not obviously derived from comparisons to DFT calculations. In principle, photoemission matrix element effects carry information about orbital symmetries. As shown in Fig. 2, the band displays strong matrix element effects as a function of photon energy and photon polarization. However, probing with 65 eV light, the spectral weight of the band is not displaying any regularity within the (k_{x}, k_{y}) plane—see Supplementary Fig. 1. The contrast between linear horizontal and vertical light therefore vary strongly with momentum. This fact precludes any simple conclusions based on matrix element effects.
Discussion
Having explored the orbital character of the electronic states, we discuss the band structure in a more general context. Bare band structure calculations, not including Coulomb interaction, find that states at the Fermi level have d_{xy} and d_{xz}/d_{yz} character (see Fig. 4a). Including a uniform Coulomb interaction U results in a single Mott gap acting equally on all orbitals. Generally, this produces one single flat band inconsistent with the observation of two distinct flat bands ( and ). Adding in a phenomenological manner orbitaldependent Mott gaps to the selfenergy produces two sets of flat bands. For example, one can introduce Δ_{xy}=0.2 eV and Δ_{xz,yz}=1.5 eV to mimic the and bands. However, such Mott gaps are not shifting the bottom of the fast Vshaped dispersion to the observed position. Better agreement with the observed band structure is found, when a Mott gap Δ_{xz,yz}=1.55 eV is added to the selfenergy of the d_{xz}, d_{yz} states and a crystal fieldinduced downward shift Δ_{CF}=0.6 eV of the d_{xy} states is introduced. As shown in Fig. 3c, this reproduces two flat bands and simultaneously positions correctly the fast dispersing band. From the fact that the bottom of the band is observed well below the band, we conclude that an—interaction enhanced—crystal field splitting is shifting the d_{xy} band below the Fermi level.
A similar structure emerges from DMFT calculations^{26} including U=2.3 eV and Hund’s coupling J=0.4 eV. The obtained spectral function (Fig. 3d) is generally in good agreement with the experimental observations (Fig. 3a). Both the and bands are reproduced with the previously assigned d_{xy} and d_{xz}, d_{yz} orbital character (Fig. 4b,c). The band is also present in the DMFT calculation around −0.5 eV<<0 eV. Even though it is not smoothly connected with the band, it has in fact d_{xy} character (Fig. 4b). By analysing the multiplet eigenstates (Fig. 4d) and electronic transitions in the atomic limit of an isolated t_{2g} shell, we can provide a simple qualitative picture of both observations: (i) the energy splitting between the and bands having d_{xy} orbital character, which we find to be of order 3J, and (ii) the d_{xz} and d_{yz} orbitaldriven band splitting across the Fermi level, found to be of order U+J. Within this framework, the atomic ground state has a fully occupied d_{xy} orbital, while the d_{xz}, d_{yz} orbitals are occupied by two electrons with parallel spins (S=1) and thus effectively halffilled. The Mott gap developing in the d_{xz}, d_{yz} doublet is thus U+J in the atomic limit^{5}, corresponding to the electronic transition where one electron is either removed from this doublet or added to this doublet (leading to a double occupancy). In contrast, there are two possible atomic configuration that can be reached when removing one electron out of the fully filled d_{xy} orbital (Fig. 4d). One of these final states (high spin) has S=3/2, L=0 (corresponding pictorially to one electron in each orbital all with parallel spins), while the other (low spin) has S=1/2, L=2 (corresponding to the case when the remaining electron in the d_{xy} orbital has a spin opposite to those in d_{xz}, d_{yz}). The energy difference between these two configurations is 3J, thus accounting for the observed ARPES splitting between the two d_{xy} removal peaks. Furthermore, this analysis allows to assess, from the experimental value of this splitting ∼1.2 eV, that the effective Hund’s coupling for the t_{2g} shell is of the order of 0.4 eV. This is consistent with previous theoretical work in ruthenates^{27,28} and provides a direct quantitative experimental estimate of this parameter. Because the high spin state is energetically favourable with respect to the low spin state (by ∼3J), it can be assigned to the band near the Fermi level, while the low spin state can be assigned to the band (See ref. 5 for a detailed description of the atomic multiplets of the t_{2g} Kanamori Hamiltonian). The Hund’s coupling has thus profound impact on the electronic structure of the paramagnetic insulating state of Ca_{2}RuO_{4}. The fact that Hund’s coupling mainly influence the d_{xy} electronic states highlights orbital differentiation as a key characteristic of the Mott transition. Moreover, our findings emphasize the importance of the crystal field stabilization of the d_{xy} orbital^{19,20}. To further understand the interplay between U and J, detailed experiments through the metal–insulator transition of Ca_{2−x}Sr_{x}RuO_{4} would be of great interest.
Methods
Experimental
Highquality single crystals of Ca_{2}RuO_{4} were grown by the fluxfeeding floatingzone technique^{29,30}. ARPES experiments were carried out at the SIS, I05 and MAESTRO beamlines at the Swiss Light Source, the Diamond Light Source and the Advanced Light Source. Both horizontal and vertical electron analyser geometry were used. Samples were cleaved in situ using the toppost cleaving method. All spectra were recorded in the paramagnetic insulating phase (T=150 K), resulting in an overall energy resolution of approximately 50 meV. To avoid charging effects, care was taken to ensure electronic grounding of the sample. Using silver epoxy (EPOTEK E4110) cured just below T=350 K (inside the SPbca phase—space group 61) for 12 h, no detectable charging was observed when varying the photon flux.
DFT band structure calculations
We computed electronic structures using the projector augmented wave method^{31,32} as implemented in the VASP^{33,34} package within the generalized gradient approximation^{35}. Experimental lattice constants (a=5.39 Å, b=5.59 Å and c=11.77 Å) and a 12 × 10 × 4 MonkhorstPack kpoint mesh was used in the computations with a cutoff energy of 400 eV. The spin–orbit coupling effects are included selfconsistently. In order to model Mott physics, we constructed a firstprinciples tightbinding model Hamiltonian, where the Bloch matrix elements were calculated by projecting onto the Wannier orbitals^{36,37}, which used the VASP2WANNIER90 interface^{38}. We used Ru t_{2g} orbitals to construct Wannier functions without using the maximizing localization procedure. The resulting 24band spin–orbit coupled model with Bloch Hamiltonian matrix reproduces well the firstprinciple electronic structure near the Fermi energy. To model the spectral function, we added a gap with a leading divergent 1/ω term to the selfenergy . To the Hamiltonian, we added a shift . and are projectors on the d_{xy} and d_{xz}, d_{yz} orbitals, respectively, while Δ_{xz,yz} is the weight of the poles and Δ_{CF} mimics an enhanced crystal field. From the imaginary part of the Green’s function with the two adjustable parameters Δ_{CF} and Δ_{xz,yz}, we obtained the spectral function A(k, ω) by taking the trace over all orbital and spin degrees of freedom.
DFT+DMFT band structure calculations
We calculate the electronic structure within DFT+DMFT using the full potential implementation^{39} and the TRIQS library^{40,41}. In the DFT part of the computation, the Wien2k package^{42} was used. The localdensity approximation (LDA) is used for the exchangecorrelation functional. For projectors on the correlated t_{2g} orbital in DFT+DMFT, Wannierlike t_{2g} orbitals are constructed out of Kohn–Sham bands within the energy window (−2, 1) eV with respect to the Fermi energy. We use the full rotationally invariant Kanamori interaction in order to ensure a correct description of atomic multiplets^{5}. To solve the DMFT quantum impurity problem, we used the strongcoupling continuoustime Monte Carlo impurity solver^{43} as implemented in the TRIQS library^{44}. In the U and J parameters of the Kanamori interaction, we used U=2.3 eV and J=0.4 eV, which successfully explains the correlated phenomena of other ruthenate such as Sr_{2}RuO_{4} and ARuO_{3} (A=Ca, Sr) within the DFT+DMFT framework^{27,28}.
Data availability
All relevant data are available from the authors.
Additional information
How to cite this article: Sutter, D. et al. Hallmarks of Hunds coupling in the Mott insulator Ca_{2}RuO_{4}. Nat. Commun. 8:15176 doi: 10.1038/ncomms15176 (2017).
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Acknowledgements
D.S., J.C., C.G.F. and H.M.R. acknowledge support by the Swiss National Science Foundation and its Sinergia network MPBH. Y.S. is supported by the WennerGren foundation. T.R.C. and H.T.J. are supported by the Ministry of Science and Technology, National Tsing Hua University, National Cheng Kung University and Academia Sinica, Taiwan. T.R.C. and H.T.J. also thank NCHC, CINCNTU and NCTS, Taiwan for technical support. A.G. and M.K. acknowledge the support of the European Research Council (ERC319286 QMAC, ERC617196 CORRELMAT) and the Swiss National Science Foundation (NCCR MARVEL). S.M. acknowledges support by the Swiss National Science Foundation (Grant No. P2ELP2155357). This work was performed at the SIS, I05 and MAESTRO beamlines at the Swiss Light Source, Diamond Light Source and Advanced Light Source, respectively. We acknowledge Diamond Light Source for time on beamline I05 under proposal SI14617 and SI12926 and thank all the beamline staff for technical support. The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DEAC0205CH11231. M.K. and A.G. are grateful to M. Ferrero, O. Parcollet and P. Seth for discussions and support.
Author information
Author notes
 D. Sutter
 & C. G. Fatuzzo
These authors contributed equally to this work.
Affiliations
PhysikInstitut, Universität Zürich, Winterthurerstrasse 190, Zürich CH8057, Switzerland
 D. Sutter
 , F. Cossalter
 , T. Neupert
 & J. Chang
Institute of Physics, École Polytechnique Fedérale de Lausanne (EPFL), Lausanne CH1015, Switzerland
 C. G. Fatuzzo
 , G. Gatti
 , M. Grioni
 & H. M. Rønnow
Advanced Light Source (ALS), Berkeley, California 94720, USA
 S. Moser
 , C. Jozwiak
 , A. Bostwick
 & E. Rotenberg
College de France, Paris Cedex 05 75231, France
 M. Kim
 & A. Georges
Centre de Physique Théorique, Ecole Polytechnique, CNRS, Univ ParisSaclay, Palaiseau 91128, France
 M. Kim
 & A. Georges
CNRSPIN, Fisciano, Salerno I84084, Italy
 R. Fittipaldi
 , A. Vecchione
 & V. Granata
Dipartimento di Fisica ‘E.R. Caianiello’, Università di Salerno, Fisciano, Salerno I84084, Italy
 R. Fittipaldi
 , A. Vecchione
 & V. Granata
Department of Physics and Astronomy, Uppsala University, Uppsala S75121, Sweden
 Y. Sassa
Swiss Light Source, Paul Scherrer Institut, Villigen PSI CH5232, Switzerland
 N. C. Plumb
 , C. E. Matt
 & M. Shi
Diamond Light Source, Harwell Campus, Didcot OX11 0DE, UK
 M. Hoesch
 & T. K. Kim
Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan
 TR Chang
 & HT Jeng
Department of Physics, National Cheng Kung University, Tainan 701, Taiwan
 TR Chang
Institute of Physics, Academia Sinica, Taipei 11529, Taiwan
 HT Jeng
Department of Quantum Matter Physics, University of Geneva, Geneva 4 1211, Switzerland
 A. Georges
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Contributions
R.F., A.V. and V.G. grew and prepared the Ca_{2}RuO_{4} single crystals. D.S., C.G.F., M.S., F.C., Y.S., G.G., M.G., H.M.R., N.C.P., C.E.M., M.S., M.H., T.K.K. and J.C., prepared and carried out the ARPES experiment. D.S., C.G.F., F.C. and J.C. performed the data analysis. T.R.C., H.T.J. and T.N. made the DFT band structure calculations. M.K. and A.G. performed and analysed the DMFT calculations. All authors contributed to the manuscript.
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