Hallmarks of Hund’s coupling in the Mott insulator Ca{}_{2}RuO{}_{4}

Hallmarks of Hund’s coupling in the Mott insulator CaRuO

D. Sutter Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland    C. G. Fatuzzo Institute of Physics, École Polytechnique Fedérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland    S. Moser Advanced Light Source (ALS), Berkeley, California 94720, USA    M. Kim College de France, 75231 Paris Cedex 05, France Centre de Physique Théorique, Ecole Polytechnique, CNRS, Univ Paris-Saclay, 91128 Palaiseau, France    R. Fittipaldi CNR-SPIN, I-84084 Fisciano, Salerno, Italy Dipartimento di Fisica ”E.R. Caianiello”, Università di Salerno, I-84084 Fisciano, Salerno, Italy    A. Vecchione CNR-SPIN, I-84084 Fisciano, Salerno, Italy Dipartimento di Fisica ”E.R. Caianiello”, Università di Salerno, I-84084 Fisciano, Salerno, Italy    V. Granata CNR-SPIN, I-84084 Fisciano, Salerno, Italy Dipartimento di Fisica ”E.R. Caianiello”, Università di Salerno, I-84084 Fisciano, Salerno, Italy    Y. Sassa Department of Physics and Astronomy, Uppsala University, S-75121 Uppsala, Sweden    F. Cossalter Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland    G. Gatti Institute of Physics, École Polytechnique Fedérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland    M. Grioni Institute of Physics, École Polytechnique Fedérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland    H. M. Rønnow Institute of Physics, École Polytechnique Fedérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland    N. C. Plumb Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland    C.E. Matt Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland    M. Shi Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland    M. Hoesch Diamond Light Source, Harwell Campus, Didcot, OX11 0DE, United Kingdom    T. K. Kim Diamond Light Source, Harwell Campus, Didcot, OX11 0DE, United Kingdom    T.-R. Chang Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan    H.-T. Jeng Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan Institute of Physics, Academia Sinica, Taipei 11529, Taiwan    C. Jozwiak Advanced Light Source (ALS), Berkeley, California 94720, USA    A. Bostwick Advanced Light Source (ALS), Berkeley, California 94720, USA    E. Rotenberg Advanced Light Source (ALS), Berkeley, California 94720, USA    A. Georges College de France, 75231 Paris Cedex 05, France Centre de Physique Théorique, Ecole Polytechnique, CNRS, Univ Paris-Saclay, 91128 Palaiseau, France Department of Quantum Matter Physics, University of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva 4, Switzerland    T. Neupert Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland    J. Chang Physik-Institut, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

A paradigmatic case of multi-band Mott physics including spin-orbit and Hund’s coupling is realised in CaRuO. Progress in understanding the nature of this Mott insulating phase has been impeded by the lack of knowledge about the low-energy electronic structure. Here we provide – using angle-resolved photoemission electron spectroscopy – the band structure of the paramagnetic insulating phase of CaRuO 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 eV. Furthermore, the experimental spectra are in good agreement with electronic structure calculations performed with Dynamical Mean-Field Theory. The crystal field stabilisation of the d orbital due to c-axis contraction is shown to be important in explaining the nature of the insulating state. It is thus a combination of multiband physics, Coulomb interaction and Hund’s coupling that generates the Mott insulating state of CaRuO. These results underscore the importance of Hund’s coupling in the ruthenates and related multiband materials.

Electronic instabilities driving superconductivity, density wave orders and Mott metal-insulator transitions produce a characteristic energy scale below an onset temperature Imada et al. (1998); Monceau (2012); Hashimoto et al. (2014). 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 Fradkin et al. (2015). The challenge is then to disentangle the driving and secondary phenomena. In many Mott insulating systems, such as LaCuO and CaRuO, long-range 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 half-filled single band electron system LaCuO emerges due to a high ratio of Coulomb interaction to band width. This simple scenario does not apply to CaRuO. There, the orbital and spin degrees of freedom of the 2/3-filled (with four electrons) -manifold implies that Hund’s coupling enters as an important energy scale Georges et al. (2013). Moreover, recent studies of the antiferromagnetic ground state of CaRuO suggest that spin-orbit interaction also plays a significant role in shaping the magnetic moments, Kunkemöller et al. (2015); Jain et al. (2015); Khaliullin (2013) as well as the splitting of the states Fatuzzo et al. (2015).
Compared to SrRuO Damascelli et al. (2000); Zabolotnyy et al. (2012), which may realize a chiral -wave superconducting state, relatively little is known about the electronic band structure of CaRuO Puchkov et al. (1998). Angle integrated photoemission spectroscopy has revealed the existence of Ru-states with binding-energy 1.6 eV Mizokawa et al. (2001) – an energy scale much larger than the Mott gap  eV estimated from transport experiments Nakatsuji et al. (2004). Moreover, angle resolved photoemission spectroscopy (ARPES) experiments on CaSrRuO – the critical composition for the metal-insulator transition – have lead to contradicting interpretations Neupane et al. (2009); Shimoyamada et al. (2009) favouring or disfavouring the so-called orbital selective scenario where a Mott gap opens only on a subset of bands (Anisimov et al., 2002; Koga et al., 2004). Extending this scenario to CaRuO would imply orbital dependent Mott gaps (Koga et al., 2004). The electronic structure should thus display two Mott energy scales (one of  and another for the ,-states). A different explanation for the Mott state of CaRuO is that the -axis compression of the S-Pbca insulating phase induces a crystal field stabilisation of the  orbital, leading to half-filled , bands and completely filled  states (Liebsch & Ishida, 2007; Gorelov et al., 2010). In this case only one Mott gap on the , bands will be present with band insulating  states. The problem has defied a solution due to a lack of experimental knowledge about the low-energy electronic structure.

Figure 1: Oxygen band structure of CaRuO. Angle resolved photoemission spectroscopy spectra recorded with right-handed circularly polarised (C) 65 eV photons in the paramagnetic (150 K) insulating state of CaRuO, compared to DFT band structure calculations. Incident direction of the light is indicated by the blue arrow. (a) Constant energy map displaying the photoemission spectral weight at  eV. Solid and dashed lines mark the in-plane projected orthorhombic and tetragonal zone boundaries, respectively. with label orthorhombic zone centres. S and X label the zone corners and boundaries respectively. (b) Spectra recorded along the zone boundary [blue line in (a)]. Oxygen dominated bands are found between  eV and  eV whereas the ruthenium bands are located above  eV. (c) First principle Density Functional Theory band structure calculation. Within an arbitrary shift, indicated by the dashed line, qualitative agreement with the experiment is found for the oxygen bands.

Here we present an ARPES study of the electronic structure in the paramagnetic insulating state (at 150 K) of CaRuO. Three different bands – labeled ,  and  band – are identified and their orbital character is discussed through comparison to first principle 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 self-energy is used to describe the observed band structure on a qualitative level. Further insight is gained from Dynamical Mean-Field Theory (DMFT) calculations including Hund’s coupling and Coulomb interaction. The Hund’s coupling splits the  band allowing quantitative estimate of this parameter. The Coulomb interaction is mainly responsible for the insulating behaviour of the ,  bands. These experimental results, together with our theoretical analysis, elucidate the nature of the Mott phase of the prototypical multi-orbital Mott system CaRuO. Furthermore, they provide a natural explanation as to why previous experiments have identified different values for the energy gap.

Figure 2: Ruthenium band structure. (a)(b) Photoemission spectra recorded along the high-symmetry direction –S for incident circularly polarised light with photon energies as indicated. Blue points in (a) show the momentum distribution curve at the binding energy indicated by the horizontal dashed line. The double peak structure is attributed to the -band. (c) Energy distribution curves (EDCs) at the S-point, normalized at  eV. (d)(e) Linear light polarisation dependence along the S– direction at  eV. (f) EDCs at the momentum indicated by the vertical dashed lines. In both (c) and (f), the - and -bands are indicated by red and grey shading.


Figure 3: Band structure along high-symmetry directions (a) ARPES spectra recorded along high symmetry directions with 65  eV circularly polarised light. (b) Constant energy map at  eV. (c) DFT-derived spectra for CaRuO, upon inclusion of a Mott gap  eV acting on ,  bands and an enhanced crystal field  eV, shifting spectral weight of the  bands (for details, see method section) and plotted with spectral weight representation. (d) DMFT calculation of the spectral function, with Coulomb interaction  eV and a Hund’s coupling  eV.

Crystal and electronic structure: CaRuO is a layered perovskite, where the Mott transition coincides with a structural transition at  K, below which the -axis lattice constant is reduced. We study the paramagnetic insulating state ( K) of CaRuO with orthorhombic S-Pbca crystal structure ( Å,  Å and  Å). It is worth noting that due to this nonsymmorphic crystal structure, CaRuO could not form a Mott insulating ground state at other fillings than 1/3 and 2/3 Watanabe et al. (2015). In Fig. 1, the experimentally measured electronic structure is compared to a first-principle DFT calculation of the bare non-interacting bands. We observe two sets of states: near the Fermi level the electronic structure is comprised of Ru-dominated bands, while oxygen bands are present only for  eV. Up to an overall energy shift, good agreement between the calculated DFT and observed CaRuO oxygen band structure is found.

Figure 4: Calculated orbital band character. (a) DFT calculation of the bare band structure. - and ,-characters are indicated by blue and red colors respectively. (b) and (c) are the spectral function calculated within the DMFT approach and projected on the  and ,  orbitals respectively. The indicated energy splittings stem from a multiplet analysis in the atomic limit (d): (1) Ground state multiplet defined by the crystal field and Hund’s coupling . (2)  electron removal configurations, split by (see main text for explanation). (3) Representation of the twofold degenerate ,  electron addition and removal states, split by .

Non-dispersing 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 high-symmetry directions, is presented in Figs. 2 and 3a. In consistency with previous angle-integrated photoemission experiments Mizokawa et al. (2001), a broad and flat band is found around the binding energy  eV. However, we also observed spectral weight closer to the Fermi level ( eV), especially near the zone boundaries (see Fig. 2a,d). These two flat ruthenium bands (labeled and ) are revealed as a double peak structure in the energy distribution curves (EDCs) – 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  eV eV (see Fig. 2c). The Mott gap, defining the energy scale between lower and upper Hubbard bands, has previously been associated with an activation energy scale  eV derived from resistivity measurements Nakatsuji et al. (2004). Assuming that the Fermi level is centred approximately symmetrically between lower and upper Hubbard bands, our spectroscopic observation is consistent with the transport experiments.

Fast dispersing ruthenium bands: In addition to the flat and bands, a fast dispersing circular shaped band is observed (Fig. 3b) around the -point (zone centre) in the interval  eV eV – see Figs. 2a,b and 3a. A weaker replica of this band is furthermore found around (Fig. 3a,b). The band velocity, estimated from momentum distribution curves (MDC’s) (Fig. 2a), yields  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 varies depending on exact methodology, all band structure calculations of CaRuO find a single fast dispersing branch Park (2001); Liu (2011); Acharya et al. (2016); Woods (2000). Our DFT calculation reveals that the fast dispersing band has predominantely  character (Fig. 4a). We thus conclude that the in-plane extended -orbital is responsible for the -band. Within the DFT calculation, the  and  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 ,  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-polarisation. However, probing with 65 eV light, the spectral weight of the -band is not displaying any regularity within the plane – see supplementary Fig. S1. 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  and / character (see Fig. 4a). Including a uniform Coulomb interaction – acting equally on all orbitals – results in a single Mott gap, inconsistent with the observed flat  and  bands. Adding in a phenomenological fashion orbital dependent Mott gaps to the self-energy produces two sets of flat bands. However, it is not shifting the bottom of the fast V-shaped dispersion to the observed position. Better agreement with the observed band structure is found, when a Mott gap  eV is added to the self-energy of the ,  states and a crystal field induced downward shift  eV of the  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  band below the Fermi level.

A similar structure emerges from DMFT calculations Georges et al. (1996) including  eV and Hund’s coupling  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  and ,  orbital character (Fig. 4b,c). The -band is also present in the DMFT calculation around  eV eV. Even though it is not smoothly connected with the -band, it has in fact  character (Fig. 4b). By analysing the multiplet eigenstates and electronic transitions in the atomic limit of an isolated shell, we can provide a simple qualitative picture of both observations (Fig. 4d): (i) the energy splitting between the and bands having  orbital character, which we find to be of order , and (ii) the  and  orbital driven band splitting across the Fermi level, found to be of order . Within this framework, the atomic ground-state has a fully occupied  orbital, while the ,  orbitals are occupied by two electrons with parallel spins () and thus effectively half-filled. The Mott gap developing in the ,  doublet is thus in the atomic limit (Georges et al., 2013), 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  orbital (Fig. 4d). One of these final states (high spin) has , (corresponding pictorially to one electron in each orbital all with parallel spins), while the other (low spin) has , (corresponding to the case when the remaining electron in the  orbital has a spin opposite to those in , ). The energy difference between these two configurations is , thus accounting for the observed ARPES splitting between the two  removal peaks. Furthermore, this analysis allows to assess, from the experimental value of this splitting  eV, that the effective Hund’s coupling for the shell is of the order of  eV. This is consistent with previous theoretical work in ruthenates (Mravlje et al., 2011; Dang et al., 2015) and provides the first direct quantitative estimate of this parameter from spectroscopic experimental data. Because the high spin state is energetically favorable with respect to the low spin state (by ), it can be assigned to the band near the Fermi level, while the low spin state can be assigned to the band (See Ref. Georges et al., 2013 for a detailed description of the atomic multiplets of the Kanamori Hamiltonian). The Hund’s coupling has thus profound impact on the electronic structure of the paramagnetic insulating state of CaRuO. The fact that Hund’s coupling mainly influence the  electronic states highlights orbital differentiation as a key characteristic of the Mott transition. Moreover, our findings emphasise the importance of the crystal field stabilisation of the  orbital (Liebsch & Ishida, 2007; Gorelov et al., 2010). To further understand the interplay between and , detailed experiments through the metal-insulator transition of CaSrRuO would be of great interest.

Acknowlegdements: D.S. and J.C. acknowledge support by the Swiss National Science Foundation and Y.S. was supported by the Wenner-Gren foundation. T.R.C. and H.T.J. are supported by the Ministry of Science and Technology, National Tsing Hua University, and Academia Sinica, Taiwan. We also thank NCHC, CINC-NTU, and NCTS, Taiwan for technical support. A.G. and M.K. acknowledge the support of the European Research Council (ERC-319286 QMAC, ERC-617196 CORRELMAT) and the Swiss National Science Foundation (NCCR MARVEL). S.M. acknowledges support by the Swiss National Science Foundation (Grant No. P2ELP2-155357). 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. DE-AC02-05CH11231. M.K. and A.G. are grateful to M. Ferrero, O. Parcollet and P. Seth for discussions and support.

Authors contributions: R. F., A. V., V. G. grew and prepared the CaRuO 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, J. C., prepared and carried out the ARPES experiment. D.S, C.G.F., F.C, J.C. performed the data analysis. T.R.C, H.-T. J., 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. D.S. and C.G.F contributed equally.

Experimental: High-quality single crystals of CaRuO were grown by the flux-feeding floating-zone technique Fukazawa et al. (2000); Nakatsuji & Maeno (2001). ARPES experiments were carried out at the SIS, I05, and MAESTRO beamlines at the Swiss Light Source (SLS), the Diamond Light Source (DLS), and the Advanced Light Source (ALS). Both horizontal and vertical electron analyser geometry were used. Samples were cleaved in-situ using the top-post cleaving method. All spectra were recorded in the paramagnetic insulating phase ( K), resulting in an overall energy resolution of approximately  meV. To avoid charging effects, care was taken to ensure electronic grounding of the sample. Using silver epoxy (EPO-TEK E4110) cured just below  K (inside the s-Pbca phase – space group 61) for 12 hours, no detectable charging was observed when varying the photon flux.

DFT+LDA band structure calculations: We computed electronic structures using the projector augmented wave method Blöchl (1994); Kresse & Joubert (1999) as implemented in the VASP Kresse & Hafner (1993); Kresse & Furthmüller (1996) package within the generalized gradient approximation (GGA) Perdew et al. (1996). Experimental lattice constants ( Å,  Å and  Å) and a Monkhorst-Pack -point mesh was used in the computations with a cutoff energy of 400 eV. The spin-orbit coupling (SOC) effects are included self-consistently. In order to model Mott physics, we constructed a first-principles tight-binding model Hamiltonian, where the Bloch matrix elements were calculated by projecting onto the Wannier orbitals Marzari & Vanderbilt (1997); Souza et al. (2001), which used the VASP2WANNIER90 interface Franchini et al. (2012). We used Ru orbitals to construct Wannier functions without using the maximizing localization procedure. The resulting 24-band spin-orbit coupled model with Bloch Hamiltonian matrix reproduces well the first principle electronic structure near the Fermi energy. To model the spectral function, we added a gap with a leading divergent term to the self-energy . To the Hamiltonian we added a shift . and are projectors on the and orbitals respectively, while is the weight of the poles, mimics an enhancement crystal field. From the imaginary part of the Green’s function with the two adjustable parameters and , we obtained the spectral function 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 (Aichhorn et al., 2009) and the TRIQS library (Aichhorn et al., 2016; Parcollet et al., 2015). In the DFT part of the computation the Wien2k package (Blaha et al., 2001) was used. The LDA is used for the exchange-correlation functional. For projectors on the correlated orbital in DFT+DMFT, Wannier-like 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 insure a correct description of atomic multiplets (Georges et al., 2013). To solve the DMFT quantum impurity problem, we used the strong-coupling continuous-time Monte Carlo impurity solver (Gull et al., 2011) as implemented in the TRIQS library (Seth et al., 2016). In the and parameters of the Kanamori interaction, we used  eV and  eV which successfully explains correlated phenomena of other ruthenate such as SrRuO and RuO ( Ca, Sr) within the DFT+DMFT framework (Mravlje et al., 2011; Dang et al., 2015).

Competing financial interest:
The authors declare no competing financial interests.

Additional information Correspondence to: J. Chang (johan.chang@physik.uzh.ch) and D. Sutter (dsutter@physik.uzh.ch).


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