Hallmarks of Hund’s coupling in the Mott insulator CaRuO
Abstract
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.
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.
Results
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.
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 Hundâs 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.
Methods
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).
References
- Imada, M., Fujimori, A. & Tokura, Y. Metal-insulator transitions. Rev. Mod. Phys. 70, 1039–1263 (1998).
- Monceau, P. Electronic crystals: an experimental overview. Advances in Physics 61, 325–581 (2012).
- Hashimoto, M., Vishik, I. M., He, R.-H., Devereaux, T. P. & Shen, Z.-X. Energy gaps in high-transition-temperature cuprate superconductors. Nature Physics 10, 483–495 (2014).
- Fradkin, E., Kivelson, S. A. & Tranquada, J. M. Colloquium : Theory of intertwined orders in high temperature superconductors. Rev. Mod. Phys. 87, 457–482 (2015).
- Georges, A., de’ Medici, L. & Mravlje, J. Strong Correlations from Hund’s Coupling. Annu. Rev. Condens. Matter Phys. 4, 137 (2013).
- Kunkemöller, S. et al. Highly Anisotropic Magnon Dispersion in : Evidence for Strong Spin Orbit Coupling. Phys. Rev. Lett. 115, 247201 (2015).
- Jain, A. et al. Soft spin-amplitude fluctuations in a Mott-insulating ruthenate. http://arxiv.org/abs/1510.07011 (2015).
- Khaliullin, G. Excitonic Magnetism in Van Vleck type Mott Insulators. Phys. Rev. Lett. 111, 197201 (2013).
- Fatuzzo, C. G. et al. Spin-orbit-induced orbital excitations in and : A resonant inelastic x-ray scattering study. Phys. Rev. B 91, 155104 (2015).
- Damascelli, A. et al. Fermi Surface, Surface States, and Surface Reconstruction in . Phys. Rev. Lett. 85, 5194–5197 (2000).
- Zabolotnyy, V. B. et al. Surface and bulk electronic structure of the unconventional superconductor SrRuO : unusual splitting of the Î² band. New Journal of Physics 14, 063039 (2012).
- Puchkov, A. V. et al. Layered Ruthenium Oxides: From Band Metal to Mott Insulator. Phys. Rev. Lett. 81, 2747–2750 (1998).
- Mizokawa, T. et al. Spin-Orbit Coupling in the Mott Insulator . Phys. Rev. Lett. 87, 077202 (2001).
- Nakatsuji, S. et al. Mechanism of Hopping Transport in Disordered Mott Insulators. Phys. Rev. Lett. 93, 146401 (2004).
- Neupane, M. et al. Observation of a Novel Orbital Selective Mott Transition in . Phys. Rev. Lett. 103, 097001 (2009).
- Shimoyamada, A. et al. Strong Mass Renormalization at a Local Momentum Space in Multiorbital . Phys. Rev. Lett. 102, 086401 (2009).
- Anisimov, V., Nekrasov, I., Kondakov, D., Rice, T. & Sigrist, M. Orbital-selective Mott-insulator transition in CaSrRuO. Eur. Phys. J. B 25, 191 (2002).
- Koga, A., Kawakami, N., Rice, T. M. & Sigrist, M. Orbital-Selective Mott Transitions in the Degenerate Hubbard Model. Phys. Rev. Lett. 92, 216402 (2004).
- Liebsch, A. & Ishida, H. Subband Filling and Mott Transition in . Phys. Rev. Lett. 98, 216403 (2007).
- Gorelov, E. et al. Nature of the Mott Transition in . Phys. Rev. Lett. 104, 226401 (2010).
- Watanabe, H., Po, H. C., Vishwanath, A. & Zaletel, M. Filling constraints for spin-orbit coupled insulators in symmorphic and nonsymmorphic crystals. Proceedings of the National Academy of Sciences 112, 14551–14556 (2015).
- Park, K. T. Electronic structure calculations for layered LaSrMnO and CaRuO. Journal of Physics: Condensed Matter 13, 9231 (2001).
- Liu, G.-Q. Spin-orbit coupling induced Mott transition in CaSrRuO (). Phys. Rev. B 84, 235136 (2011).
- Acharya, S., Dey, D., Maitra, T. & Taraphder, A. The Iso-electronic Series CaSrRuO: Structural Distortion, Effective Dimensionality, Spin Fluctuations and Quantum Criticality. arXiv:1605.05215 (2016).
- Woods, L. M. Electronic structure of A comparison with the electronic structures of other ruthenates. Phys. Rev. B 62, 7833–7838 (2000).
- Georges, A., Kotliar, G., Krauth, W. & Rozenberg, M. J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13–125 (1996).
- Mravlje, J. et al. Coherence-Incoherence Crossover and the Mass-Renormalization Puzzles in . Phys. Rev. Lett. 106, 096401 (2011).
- Dang, H. T., Mravlje, J., Georges, A. & Millis, A. J. Electronic correlations, magnetism, and Hund’s rule coupling in the ruthenium perovskites and . Phys. Rev. B 91, 195149 (2015).
- Fukazawa, H., Nakatsuji, S. & Maeno, Y. Intrinsic properties of the Mott insulator CaRuO. Physica B 281, 613–614 (2000).
- Nakatsuji, S. & Maeno, Y. Synthesis and Single-Crystal Growth of CaSrRuO. Journal of Solid State Chemistry 156, 26 – 31 (2001).
- Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).
- Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758–1775 (1999).
- Kresse, G. & Hafner, J. Ab initio molecular dynamics for open-shell transition metals. Phys. Rev. B 48, 13115–13118 (1993).
- Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).
- Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
- Marzari, N. & Vanderbilt, D. Maximally localized generalized Wannier functions for composite energy bands. Phys. Rev. B 56, 12847–12865 (1997).
- Souza, I., Marzari, N. & Vanderbilt, D. Maximally localized Wannier functions for entangled energy bands. Phys. Rev. B 65, 035109 (2001).
- Franchini, C. et al. Maximally localized Wannier functions in LaMnO within PBE+ U , hybrid functionals and partially self-consistent GW: an efficient route to construct ab initio tight-binding parameters for perovskites. Journal of Physics: Condensed Matter 24, 235602 (2012).
- Aichhorn, M. et al. Dynamical mean-field theory within an augmented plane-wave framework: Assessing electronic correlations in the iron pnictide LaFeAsO. Phys. Rev. B 80, 085101 (2009).
- Aichhorn, M., Pourovskii, L. & P., S. TRIQS/DFTTools: A TRIQS application for ab initio calculations of correlated materials. Computer Physics Communications 204, 200–208 (2016).
- Parcollet, A. et al. TRIQS: A toolbox for research on interacting quantum systems. Computer Physics Communications 196, 398 – 415 (2015).
- Blaha, P., Schwarz, K., Madsen, G., Kvasnicka, D. & Luitz, J. WIEN2k: An Augmented Plane Wave Plus Local Orbitals Programfor Calculating Crystal Properties. Technische Universität Wien (2001).
- Gull, E. et al. Continuous-time Monte Carlo methods for quantum impurity models. Rev. Mod. Phys. 83, 349–404 (2011).
- Seth, P., Krivenko, I., Ferrero, M. & Parcollet, O. TRIQS/CTHYB: A continuous-time quantum Monte Carlo hybridisation expansion solver for quantum impurity problems. Computer Physics Communications 200, 274 – 284 (2016).