Charge ordering in Ir dimers in the ground state of Ba 5 AlIr 2 O 11
Vamshi M. Katukuri, Xingye Lu, D. E. McNally, Marcus Dantz, Vladimir N. Strocov, M. Moretti Sala, M. H. Upton, J. Terzic, G. Cao, Oleg V. Yazyev, Thorsten Schmitt
CCharge ordering in Ir dimers in the ground state of Ba AlIr O Vamshi M. Katukuri, ∗ Xingye Lu,
2, 3, † D. E. McNally, Marcus Dantz, Vladimir N. Strocov, M.Moretti Sala, M. H. Upton, J. Terzic,
6, 7
G. Cao,
6, 7
Oleg V. Yazyev, and Thorsten Schmitt ‡ Institute of Physics, ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland Swiss Light Source, Photon Science Division, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland Center for Advanced Quantum Studies and Department of Physics,Beijing Normal University, Beijing 100875, China European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble Cedex, France Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506, USA Department of Physics, University of Colorado at Boulder, Boulder, CO 80309 (Dated: February 4, 2021)It has been well established experimentally that the interplay of electronic correlations and spin-orbit interactions in Ir and Ir oxides results in insulating J eff =1/2 and J eff =0 ground states,respectively. However, in compounds where the structural dimerization of iridum ions is favourable,the direct Ir d – d hybridisation can be significant and takes a key role. Here, we investigate the effectsof direct Ir d – d hybridisation in comparison with electronic correlations and spin-orbit coupling inBa AlIr O , a compound with Ir dimers. Using a combination of ab initio many-body wave func-tion quantum chemistry calculations and resonant inelastic X-ray scattering (RIXS) experiments,we elucidate the electronic structure of Ba AlIr O . We find excellent agreement between the cal-culated and the measured spin-orbit excitations. Contrary to the expectations, the analysis of themany-body wave function shows that the two Ir (Ir and Ir ) ions in the Ir O dimer unit in thiscompound preserve their local J eff character close to 1/2 and 0, respectively. The local point groupsymmetry at each of the Ir sites assumes an important role, significantly limiting the direct d – d hybridisation. Our results emphasize that minute details in the local crystal field (CF) environmentcan lead to dramatic differences in electronic states in iridates and 5 d oxides in general. Dimerization or clustering of TM atoms is observed inmany TM compounds, e.g. in vanadium oxides [1, 2]and titanates [3] where dimers of spin singlets akinto the Peierls state in one dimension [4] are stabilizedwhen the t g orbitals of the TM d -manifold are par-tially filled. In these systems, the TM ions tend tohave a strong direct (intra-dimer) d – d overlap that re-sult in molecular-like orbitals with appreciable bonding-antibonding splitting. Consequently, the local electronicstructure depends on the intra-dimer hopping integral( t d ), intra-atomic Hund’s coupling ( J H ) and inter-atomic( U ) Coulomb interactions and electron filling of the or-bitals localized at TM clusters. Alternatively, dimeriza-tion of TM ions can also be favourable from crystallo-graphic considerations, particularly in compounds withheavy TM ions, e.g. 5 d ions, where the d orbitals aremore spread out. A number of dimerized or cluster 4 d and 5 d compounds [5–8] with intriguing properties havebeen synthesized recently. Novel physical phenomenahave been observed in these compounds, e.g. the inelas-tic X-ray scattering analogue of Young’s double slit ex-periment has been realized in Ba CeIr O [9], where themolecular orbital formation within the Ir dimers is cru-cial. In lacunar spinels Ga M X ( M =Nb, Mo, Ta andW and X =S, Se and Te), spin-orbit coupled molecular J eff states [10, 11] and topological superconductivity [12]have been proposed where molecular orbital formationwithin the tetrahedral cluster of M ions is the key.The interplay of inter-site electron hopping ( t ), J H , U and the strong atomic spin-orbit coupling (SOC)in 5 d and in some 4 d compounds result in the J eff physics [13–17]. For instance, in compounds with Ir ( d ) configuration in an octahedral environment, e.g. inSr IrO [13, 14], the strong SOC leads to completelyfilled J eff = 3/2 and half-filled J eff = 1/2 levels. Sim-ilarly, in Ba YIrO and NaIrO , the Ir ions realisea completely filled J eff = 3/2 and empty J eff = 1/2 sub-manifolds [18, 19], resulting in a non-magnetic J eff = 0ground state [20]. In dimerized systems, t d can be muchlarger and successively may play a dominant role com-pared to other local interactions, which could result inthe breakdown or a significant modification of the J eff physics. Thus, it is crucial to identify the role of thesemultiple physical interactions in Ir dimer systems to gaina better understanding of the electronic and magneticproperties of these materials.In this letter, we illustrate how subtle crystal struc-ture details are extremely important to precisely under-stand the electronic structure of 5 d compounds wherestructural dimerization or clustering is prevalent. Usingstate-of-the-art ab initio many-body electronic structuremethods in combination with high-resolution resonant in-elastic X-ray scattering (RIXS) experiments, we presenta detailed analysis of the electronic structure of Ir O dimers in Ba AlIr O and unravel the nature of elec-tronic ground and excited states of Ba AlIr O . Whilewe find an excellent agreement between the RIXS spectraand the calculated excitations, analysis of the many-body a r X i v : . [ c ond - m a t . s t r- e l ] F e b wave functions reveal a nearly complete charge separa-tion – Ir and Ir – within the dimers in the groundstate, in contrary to an earlier report of formation ofmolecular orbitals in Ba AlIr O [21]. The strong SOCof the Ir and Ir ions results in J eff = 1/2 and J eff =0 local configurations, respectively, and thus we con-clude that a localized J eff picture is more appropriatein Ba AlIr O .Ba AlIr O contains dimers composed of crystallo-graphically inequivalent Ir cations encaged in face sharingO octahedra [22, 23], see Fig. 3a and 3b, and Supple-mentary material (SM) Fig. S1 [24]. At 210 K, a lat-tice distortion is believed to lower the symmetry of thecrystal and enhance the charge disproportionation lead-ing to charge ordering that correspond to Ir and Ir valence configurations [23]. However, analysis of RIXSspectrum of Ba AlIr O using density functional theoryand model Hamiltonian calculations [21] has proposedthe formation of hybridized dimer orbitals, debunkingthe charge disproportionation phenomenon. Neverthe-less, given the complex low-symmetry crystal environ-ment and the interplay of spin and orbital degrees offreedom in Ba AlIr O , it is unclear if the dimer or-bitals are actually realised in the ground state. Results : The RIXS spectra shown in Fig. 1 was mea-sured on single crystals grown by flux method [23] at theID20 beam line of the European Synchrotron RadiationFacility (ESRF) with ∼
25 meV resolution [25] and the 27-ID-B beamline with ∼
30 meV resolution at the AdvancedPhoton Source (APS), with π polarization at a scatteringangle close to 2 θ = 90 ◦ . The incident-energy dependenceof RIXS spectra across the Ir- L edge ( E i = 11 . E i as determined from previous measure-ments on iridates such as Sr IrO and Ba YIrO [20, 26]was chosen, we find that the maximum of the resonanceis not at E i in Ba AlIr O as the precise CF environ-ment around Ir ions and the mixing of the valence statesinfluences the resonance energy. However, we see thatthe energies of the modes remain unchanged in a broadrange around E i .The features marked by A-K in Fig. 1(b) are incident-energy independent Raman modes as shown in Fig. 1(a).These modes correspond to intrinsic electronic transi-tions between various occupied and unoccupied states,and therefore provide direct information about low en-ergy electronic structure. To resolve all the Ramanmodes and determine the low energy electronic structure,we show in Figs. 1(b) and 1(c) high statistic energy spec-tra collected at E i (white dashed line in Fig. 1(a)). InFig. 1(b), several sharp Raman modes below 1 eV anda broad peak at 1.2 eV, named A to K are determinedby fitting the spectra using multiple gaussians. The sumof the fitting curves is shown as a green solid curve. InFig. 1(c), higher energy excitations up to 8 eV are shown.This spectrum is decomposed into several peaks and in- A Ei = 11.2150 keV Fitting I n t e n s i t y ( a r b . u . ) Energy Loss (eV)
Q = (23.5, 0, 2.5)11.21411.21511.21611.21711.218 I n c i d e n t E n e r g y ( k e V ) (b)(a) I n t e n s i t y ( a r b . u . ) (c) B C D E F G H I J K K Ei = 11.2150 keV Fitting
Q = (23.5, 0, 2.5)T = 20 K
L M N O P Q R
FIG. 1. (a) Incident-energy dependence of elementary ex-citations below 1.2 eV for Ba AlIr O measured around Ir L edge with Q = (23 . , , .
5) at 20 K. (b) RIXS spectra(below 1.2 eV) measured at E i = 11 .
215 keV (marked aswhite dashed line in (a)). (c) High energy excitations (1-8eV) measured with the same setup as that for (b). Green lineis a Multi-Gaussian fitting of the raw data in red open circles(with error bars). terestingly, these modes show very little momentum de-pendence (see SM Fig. S6 [24]), indicating that all ofthem correspond to local spin-orbital ( d − d ) excitationsand reflecting the low energy electronic structure.We now turn to the RIXS results measured using O- K edge (Fig. 2 and Fig. S2 in SM [24]) carried out atthe ADRESS beamline of the Swiss Light Source at thePaul Scherrer Institut, with ∼
70 meV energy resolutionfor both σ and π polarizations at a scattering angle of2 θ =130 deg. [27, 28], see SM, Fig. S1. With the presenceof strong hybridization between O 2 p orbitals and Ir 5 d orbitals, O- K RIXS is sensitive to various elementary ex-citations of iridates [29]. Figs. 2(a) and 2(b) are RIXSmaps collected at O- K edge with π and σ polarizationsat 25 ◦ grazing incidence. Besides the sharp spin-orbitalexcitations ( E ≈ . , .
57 eV) below 1 eV consistentwith those measured with Ir- L edge, two high energyexcitations at E ≈ .
26 and 2 .
71 eV have also been ob-served. Note the RIXS maps in Fig. 2 contains sub-stantial fluorescence which is absent in the results col-lected at Ir L edge (Fig. 1), indicating complex energylevels/bands of oxygen ions. In addition, significant po-larization dependence of the excitations have also beenobserved, which we attribute to the overlap between lightpolarization (electric field E ) and the different O 2 p or-bitals hybridized with different Ir 5 d orbitals (for details,see [24]).To decipher the nature of the rich excitation spectrumobserved in RIXS spectra and to examine the formationof dimer orbitals in Ba AlIr O , we performed many-body ab initio cluster-in-embedding quantum chemistry(QC) calculations, starting from the crystal structure re-ported in Ref. [22, 23]. These are based on the construc-tion of the exact wave function for the atoms in the clus-ter using configuration interaction wave function theory – TABLE I. Relative energies (eV) of the excitation levels calcu-lated at CASSCF+NEVPT2 level of theory. The first columncontains non-relativistic multiplet structure, the multipletsymbols on the left correspond to the octahedral ( O h ) sym-metry. The degeneracy of the states is split in Ba AlIr O due to the lowered symmetry in the two octahedra due to theanisotropic crystal fields, see text. Spin-orbit coupled mul-tiplet structure is shown in the second column. Note thateach state is doubly degenerate (Kramers doublet). The cor-responding peaks in the RIXS data in Fig. 1a are shown incolumn 3.CASSCF+NEVPT2 + SOC (x 2) Ir L-edge RIXS A – 0.00 0.00 0.00 T – 0.03, 0.08, 0.10 0.18 0.18 (A) A – 0.14 0.24 , 0.27 0.28 (B) 0.33 (C) T – 0.16, 0.17, 0.17 0.44 (D) E – 0.18, 0.23 0.48, 0.62 0.50 (E) E – 0.25, 0.28 0.55, 0.56, 0.58 0.56 (F) T – 0.77, 0.80, 0.94 0.60, 0.78 0.75 (G) T – 0.84, 0.86, 0.95 0.84, 0.90 0.82 (H) T – 0.86, 0.86,0.90 1.14 – 1.20 (4) 0.98 (I), 1.00 (J) A – 0.91 1.21, 1.24 1.20 (K) E – 1.00, 1.01 1.37 – 1.43 (4) 1.4 (L) A – 1.03 1.47 T – 1.10,1.10,1.12 1.48, 1.53, 1.56 E – 1.60, 1.63 1.73 – 1.80 (5) 1.77 (M) T – 1.68, 1.78, 1.81 2.02, 2.042.11, 2.17, 2.20 2.17 (N)2.592.63 – 2.77 (4) 2.70 (O) Energy loss (eV) E x c i t a t i o n e n e r g y ( e V ) π pol. δ = -40 o T = 10 K σ pol.T = 10 K I n t en s i t y ( a r b . u . ) σ , δ = -40° π , δ = -40° x 0.95 04080120160200 σ , δ = 25° π , δ = 25° x 1.25 σ , δ = 25° σ , δ = -40° x 1.12 π , δ = 25° π , δ = -40° x 0.6504080120 Energy Loss (eV)
Energy Loss (eV) (c) (d)(e) (f) c ab σπ c ab σπ c ab σ c ab πσ π (a)(b) E i = 527.6 eV FIG. 2. RIXS results of Ba AlIr O as measured at O- K edge. (a, b) Energy dependence of RIXS spectra forBa AlIr O taken around O- K edge with δ = − ◦ ( Q =(0 . , , . complete active space self-consistent field (CASSCF) andmultireference perturbation methods [30]. The calcula-tions were performed on a cluster containing one Ir O dimer unit, two neighboring AlO tetrahedra and the sur-rounding 15 Ba ions. All the calculations were per-formed using ORCA quantum chemistry program [31],see SM [24] for all the computational details.Relative energies of the multiplet structure of the Ir O dimer unit obtained from CASSCF + NEVPT2 (N-electron valence perturbation theory) [32] calculationsare shown in Table I. An active space of nine electronsin six orbitals (three t g orbitals on each iridium) wasconsidered in the CASSCF calculation which sufficientlycaptures the important static correlations (i.e. near de-generacies) in Ba AlIr O . In the NEVPT2 calculation,the correlations involving all the neighboring occupiedoxygen 2 p and iridium 5 s , 5 p orbitals as well as all the un-occupied orbitals are accounted for, accurately describingthe O-2p to Ir-d charge transfer effects and other dynamiccorrelation effects. It is important to note that the intra-(Hund’s coupling J H ) and inter-site ( U ) Coulomb inter-actions and the hybridization between different orbitalsare included in the calculation, accurate within the basis-set limit.The lowest nine quartet ( s = ) and 24 doublet ( s = )scalar relativistic states (first column in Table I) arefirst computed and then are allowed to admix via theSOC, resulting in 84 states, see the second column of Ta-ble I. It can be seen that the excitation energies obtainedfrom CASSCF+NEVPT2+SOC calculations are in ex-cellent agreement with the peaks observed in RIXS ex-periments, except for peak D . This peak is related to theelectron-hole exciton which is also observed in other iri-date materials such as Sr IrO [29, 33] and Na IrO [34].Such excitations are not considered in the current QCcalculations[35]. Further, our calculations reveal excita-tions from the t g to e g manifold starting at 3.4 eV whichcorrespond to RIXS peaks P and Q.To elucidate the origin of these excitations, we firstanalyze the scalar-relativistic multiplet structure. Whenthe two iridium ions in the dimer unit are in cubic envi-ronment ( O h symmetry), the low energy multiplet struc-ture is a result of the interaction of the ground state T g multiplet of the Ir ion [36] and the T g groundstate term of the Ir ion. In addition, the lowest T g and E g [20] singlet states contribute significantly tothe low energy spin-orbit excitations [20]. The resultingspectrum contains T g , T g , A g , E g , quartets and 11doublet terms – T g, g, g, g, g , A g, g, g , E g, g, g [37].However, in Ba AlIr O the Ir ions are enclosed in dis-torted octahedra resulting in low symmetry CFs andsplitting of the t g levels at each Ir ion [38]. Further, thesmall Ir-Ir intra-dimer distance of 2.73 ˚A in Ba AlIr O (2.698 ˚A in elemental iridium) may result in direct over-lap of the Ir d orbitals and the formation of bonding andantibonding states [39]. Consequentially, the multipletdegeneracies in the spectrum are split.To understand the formation of bonding and antibond-ing dimer orbitals in Ba AlIr O , we plot the evolutionof orbital energies as a function of Ir -Ir intra-dimer dis-tance ( d ) in Fig. 3(c). The six levels for each d correspondto the CASSCF canonical orbital [40] energies of the six t g -like orbitals in the Ir O dimer unit. The colour vari-ations of the energy levels represent the orbital compo-sitions [41] from Ir , Ir and O ions. Interestingly, for d ≥ .
73 ˚A, we find 20% and 13% hybridization for Ir d – O 2 p and Ir d – O 2 p , respectively, while there isnegligible direct Ir -Ir d -orbital hybridization. The a g orbital of Ir contains 4.5% contribution from a g orbitalof Ir and vice versa. For d = 2 .
65 ˚A, a significant di-rect Ir -Ir d -orbital hybridization is observed. We findthis hybridization increasing up to 25% for d =2 .
45 ˚A, re-sulting in large bonding – antibonding energy separation,as seen in the corresponding orbital plots in Fig. 3(d) .Note that for d = 2 .
73 ˚A orbitals with predominantly Ir character are at higher energies than those of Ir char-acter, reflecting different on-site orbital energies. Thisis a direct consequence of the difference in the valenceconfigurations of Ir and Ir ions and the Ir , d – O 2 p hybridization.The effect of the low crystal filed symmetry at twodifferent Ir ions can be estimated by computing the t g splittings, δ , at each of the Ir ions from restricted activespace (RAS) [42] calculations where the d orbital occu-pation at the other Ir ion is constrained. We find consid-erably large t g splittings of δ = 0 .
58 and δ = 0 .
60 eVfor Ir and Ir , respectively. Such large values competewith the SOC strength of ∼ . AlIr O is the double exchange A g multiplet [43],an orbitally non-degenerate high-spin quartet, with wavefunction ψ = α | d , d (cid:105) + β | d , d (cid:105) + γ | d , d (cid:105) with α = 0 . , β = 0 .
09 and γ = 0 .
02, where d ni cor- (b) da b OIr Ir AlBa (a) c ab (c) E n e r gy ( e V ) d (Å) (d) FIG. 3. Crystal structure of Ba AlIr O : (a) unit cell, (b)Ir O dimer units connected along b . d is the intra-dimerIr -Ir distance. (c) Orbital (relative) energy-level diagram ofthe six t g orbitals, shown in (d), for different d , the lowestenergy orbital is set to zero. The length of red, black andblue colours of each level are proportional to the percentagecontributions from Ir , Ir and O ions, respectively. responds to n electrons in Ir i d orbitals. The lowest T doublet state is 40 meV higher with wave functionweights α = 0 . , β = 0 .
11 and γ = 0 .
02. We furtherfind that the weight of | d , d (cid:105) configuration in all the ex-cited multiplet wave functions is greater than 95%. It isinteresting to note that excluding all the configurationsinvolving hopping of electrons from Ir to Ir and viceversa in the wave function preserve the spin-orbit spec-trum except for an overall shift ≤