Spin-orbit coupled systems in the "atomic" limit: rhenates, osmates, iridates
Arun Paramekanti, David J. Singh, Bo Yuan, Diego Casa, Ayman Said, Young-June Kim, Andrew D. Christianson
SSpin-orbit coupled systems in the “atomic” limit: rhenates, osmates, iridates
Arun Paramekanti,
1, 2
David J. Singh, Bo Yuan, Diego Casa, Ayman Said, Young-June Kim, and A. D. Christianson
5, 6, 7 Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada ∗ Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211-7010, USA † Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN-37831, USA Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN-37831, USA Department of Physics & Astronomy, University of Tennessee, Knoxville, TN-37966, USA (Dated: June 12, 2018)Motivated by RIXS experiments on a wide range of complex heavy oxides, including rhenates,osmates, and iridates, we discuss the theory of RIXS for site-localized t g orbital systems with strongspin-orbit coupling. For such systems, we present exact diagonalization results for the spectrum atdifferent electron fillings, showing that it accesses “single-particle” and “multi-particle” excitations.This leads to a simple picture for the energies and intensities of the RIXS spectra in Mott insulatorssuch as double perovskites which feature highly localized electrons, and yields estimates of thespin-orbit coupling and Hund’s coupling in correlated 5 d oxides. We present new higher resolutionRIXS data at the Re-L edge in Ba YReO which finds a previously unresolved peak splitting,providing further confirmation of our theoretical predictions. Using ab initio electronic structurecalculations on Ba M ReO (with M =Re, Os, Ir) we show that while the atomic limit yields areasonable effective Hamiltonian description of the experimental observations, effects such as t g - e g interactions and hybridization with oxygen are important. Our ab initio estimate for the strength ofthe intersite exchange coupling shows that, compared to the d systems, the exchange is one or twoorders of magnitude weaker in the d and d materials, which may partly explain the suppressionof long-range magnetic order in the latter compounds. As a way to interpolate between the site-localized picture and our electronic structure band calculations, we discuss the spin-orbital levelsof the M O cluster. This suggests a possible role for intracluster excitons in Ba YIrO which maylead to a weak breakdown of the atomic J eff = 0 picture and to small magnetic moments. INTRODUCTION
In recent years, much attention has been paid to com-plex oxides of heavy transition elements where electroniccorrelations become comparable to the spin-orbit cou-pling (SOC) λ . This provides a new route to realizingexotic quantum ground states.[1] A large part of this ef-fort has been focussed on the Ir iridates with a 5 d configuration, corresponding to a single hole in the t g orbitals. [2–8] At this filling, the physics is that of a half-filled j eff = 1 / j eff arising from the coupling of the spin to the effectiveorbital angular momentum (cid:96) eff = 1 of the t g triplet.Interest in the spin-orbit coupled materials stems fromthe possibility of realizing analogues of high-temperaturesuperconductivity upon electron doping, and exotic mag-netic phases such as Kitaev spin liquids and topologicalsemimetals.[9–19] Currently, there is an effort to exploreother complex oxides, such as osmates and rhenates, aswell as iridates with different valence states, which maylead to further exotic phenomena at different electronfillings.[20–28] An important step in this programme is toelucidate the ‘atomic’ interactions which govern the localphysics, which then feeds into understanding how suchlocal degrees of freedom interact and organize at longerlength scales. Here, we discuss this step in the context of double perovskite materials using a theoretical analysisof resonant inelastic X-ray scattering (RIXS), exact diag-onalization studies of the single-site problem with SOCat different electron fillings (d , d , d ), and complemen-tary ab initio electronic structure calculations. a RIXS has proven to be a particularly valuable tool toexplore spin and orbital excitations, and there has beenextensive experimental [29–36] and theoretical work [37–51] in this area (see Ref. 52 for a review). In this paper,we discuss the theory of RIXS for systems with highlylocalized t g electronic states at various fillings. Us-ing exact diagonalization (ED) calculations of the RIXSspectrum, we show that we can quantitatively extractthe spin-orbit and Hund’s couplings, and explain boththe energies and the spectral intensities observed in ex-periments on rhenates, osmates, and iridates. We also a This manuscript has been authored by UT-Battelle, LLC un-der Contract No. DE-AC05-00OR22725 with the U.S. Depart-ment of Energy. The United States Government retains and thepublisher, by accepting the article for publication, acknowledgesthat the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce thepublished form of this manuscript, or allow others to do so, forUnited States Government purposes. The Department of En-ergy will provide public access to these results of federally spon-sored research in accordance with the DOE Public Access Plan(http://energy.gov/downloads/doe-public-access-plan). a r X i v : . [ c ond - m a t . s t r- e l ] J un present new experimental high resolution RIXS resultson Ba YReO at the Re L edge which finds a peaksplitting in the spectrum, in perfect agreement with ourtheoretical predictions. This splitting was not resolvedin previous experiments at the Re L edge. This paperthus extends and generalizes previous well-known workon RIXS for 5 d iridates,[43] and provides a useful com-panion to a recent study of the RIXS operator in t g spin-orbital systems.[53]Additionally, in order to complement this effectiveHamiltonian study, we have carried out ab initio elec-tronic structure calculations for the cubic double per-ovskites Ba YReO , Ba YOsO , and Ba YIrO . Thispermits us to understand material-to-material variationsof these effective parameters across the 5 d oxides, and toshow that t g - e g interactions and hybridization with theligand ions (oxygen) play a key role when we attempt toconnect the parameters of the effective Hamiltonian witha more microscopic theory. Furthermore, our ab initio estimates for the strength of the magnetic exchange cou-pling between moments in these materials shows that,compared to the osmates, the exchange is one or two or-ders of magnitude weaker in the rhenates and iridates.This could explain the robust magnetic long range orderobserved in the osmates, which should be contrasted withweak ordering tendencies in the latter compounds.Based on our ab initio calculations, hybridization ofthe transition metal ion with the surrounding oxygen oc-tahedral cage plays an important role in complex 5 d ox-ides. This leads us to examine the spin-orbital states onthe M O metal-oxygen octahedra, which could be usefulin future studies of the effect of extended interactions onsuch clusters as a way to bridge the gap between ED andDFT results on Ba YIrO . RIXS FOR HIGHLY LOCALIZED STATES
The Kramers-Heisenberg expression [37–41, 52] for thetwo-photon RIXS scattering cross section is given by d σd Ω dE in = E out E in (cid:88) f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n (cid:104) f | T † | n (cid:105)(cid:104) n | T | g (cid:105) E g − E n + E in + i Γ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × δ ( E g − E f + E in − E out ) . (1)Here, g, n, f refer to ground (initial) state, intermediatestate, and final state, respectively with corresponding en-ergies E g , E n , E f , and Γ n is the inverse lifetime of theintermediate state. E in = ¯ hω in and E out = ¯ hω out arethe incoming and outgoing photon energies, and the δ -function enforces energy conservation. Within the dipoleapproximation for the photon field, the transition is in-duced by the dipole operator T ∼ ˆ (cid:15) · r , where ˆ (cid:15) denotes thephoton polarization, which we label as ˆ (cid:15) in for the incom-ing photon which excites from the ground state (whichenters in the (cid:104) n | T | g (cid:105) matrix element above), and ˆ (cid:15) out for the outgoing photon which de-excites into the final state(which enters in the (cid:104) f | T † | n (cid:105) matrix element above). Onresonance, with E out ≈ E in (since the energy transfer E f − E g (cid:28) E in , E out ), the cross section simplifies to d σd Ω dE in ≈ A (cid:88) f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n (cid:104) f | T † | n (cid:105)(cid:104) n | T | g (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × δ ( E g − E f + E in − E out ) (2)where the prefactor A = (cid:12)(cid:12)(cid:12) E g − ¯ E n + E in + i ¯Γ n (cid:12)(cid:12)(cid:12) , with¯ E n , ¯Γ n being the average energy and inverse lifetime ofthe intermediate states. This approximation, which isvalid for short core-hole lifetime,[38, 39] allows us to ig-nore intermediate state interactions between the core-hole and other electrons. We show below that the re-sulting spectra are in good agreement with experiments,providing a phenomenological justification for this ap-proximation.RIXS excites an electron from a highly spin-orbit cou-pled core level into the relevant d -orbitals; here, we focuson excitation into the t g states. This leads to an in-termediate state with a core-hole and an added electronin the t g orbitals. These intermediate states decay onthe timescale of the core-hole lifetime ∼ / Γ n , leavingthe original t g electrons in a final excited spin-orbitalstate. We can thus consider simplified transition matrixelements [24] (cid:104) n | T | g (cid:105) = ˆ (cid:15) α in (cid:104) n | p † βσ d † αβσ | g (cid:105) (3) (cid:104) f | T † | n (cid:105) = ˆ (cid:15) µ out (cid:104) f | d µνσ (cid:48) p νσ (cid:48) | n (cid:105) . (4)Here p † ασ creates a 2 P core-hole in orbital α ( α = p x , p y , p z ) with spin σ , while d † αβσ creates a d -electronin the t g orbital ( d yz , d zx , d xy ) with spin σ , and we haverestricted attention to parity-allowed nonzero dipole ma-trix elements. Using this, we arrive at the following ex-pression for the RIXS cross-section: d σd Ω dE in ∝ (cid:88) f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n ˆ (cid:15) µ out ˆ (cid:15) α in (cid:104) f | d µνσ (cid:48) p νσ (cid:48) | n (cid:105)(cid:104) n | p † βσ d † αβσ | g (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × δ ( E g − E f + E in − E out ) . (5)The core-level part of the process consists of excit-ing a single core-hole from the core-vacuum and de-exciting back into the vacuum, with intermediate statesfor the core-hole being 2 P / ( L edge) or 2 P / ( L edge). Let us denote the corresponding P -matrix el-ements as M Jνσ (cid:48) ; βσ with J = 1 / , /
2. This leads to( d σ/d Ω dE in ) ∝ I J ( ω ), where I J ( ω ) ≡ (cid:88) f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n ˆ (cid:15) µ out ˆ (cid:15) α in M Jνσ (cid:48) ; βσ (cid:104) f | d µνσ (cid:48) | n (cid:105)(cid:104) n | d † αβσ | g (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × δ ( E g − E f +¯ hω ) , (6) j eff =3/2j eff =1/2 (b)(a) (d)(c) FIG. 1. (Color online) Schematic picture of the L edge inelastic RIXS processes (for the 5d osmates), with dashed (purple) linesindicating photon-induced transitions and solid (green) lines indicating interaction-induced transitions. (a): Noninteractingcase, where the two-photon process consists of a core electron getting excited into the j eff = 1 / j eff = 3 / λ/
2. (b),(c),(d): Interactingcase, where the local Hund’s coupling leads to visible multi-particle excitations. Within perturbation theory, interactions canscatter electrons into higher energy states as depicted by the solid (green) lines for the (b) initial, (c) intermediate, and (d)final states. For J H < λ , this would lead to 2-particle peaks near 3 λ but with suppressed intensity ∼ ( J H /λ ) . with ¯ hω = E in − E out being the photon energy loss. Wecontinue to use the notation g, n, f , for ground, interme-diate, and final states, but henceforth these will refer toonly the t g states.The matrix M J for the core hole for the L and L edges is given by M J =1 / = 13 (1 − (cid:126)L · (cid:126)S ) (7) M J =3 / = 13 (2 + (cid:126)L · (cid:126)S ) (8)where L, S refer to the oxygen 2 P hole orbital- and spinangular momentum operators. Labelling the P -states as( µσ ) = ( p x ↑ , p y ↑ , p z ↑ , p x ↓ , p y ↓ , p z ↓ ), we can explicitlywrite out M J =1 / = 13 − i − i − i i
00 0 1 1 i
00 0 − i − i − i (9)and M J =3 / = − M J =1 / . The sum over all intermedi-ate d -states in Eq. 6 can be done, which leads to I J ( ω ) ≡ (cid:88) f (cid:12)(cid:12)(cid:12) ˆ (cid:15) µ out ˆ (cid:15) α in M Jνσ (cid:48) ; βσ (cid:104) f | d µνσ (cid:48) d † αβσ | g (cid:105) (cid:12)(cid:12)(cid:12) × δ ( E g − E f +¯ hω ) . (10)Below, we will discuss a physical picture for the excita-tions, before turning to exact diagonalization results.Note that everywhere below, we will work with the sin-gle particle spin and orbital basis states for the t g elec-trons. However, when interactions are absent or weak compared with SOC, we will refer to the j eff = 1 / , / PHYSICAL PICTURE OF EXCITATIONS
In the absence of electron-electron interactions, theRIXS process is schematically illustrated in Fig. 1(a).Here, we depict spin-orbit split t g levels, having an ef-fective (single-particle) angular momentum states, witha low energy j eff = 3 / j eff = 1 / λ/ λ is the spin-orbit coupling. We consider a filling corre-sponding to a 5d configuration (e.g., osmates), and de-pict the photon-induced transitions by dashed (purple)lines. For the L edge, the incoming photon excites acore electron from 2 P / into the higher energy j eff = 1 / j eff = 3 / single peak at ¯ hω = 3 λ/ t g manifold, given by H int = U (cid:88) (cid:96) n (cid:96) ) : − J H (cid:88) (cid:96)<(cid:96) (cid:48) n (cid:96) n (cid:96) (cid:48) − J H (cid:88) (cid:96)<(cid:96) (cid:48) (cid:126)S (cid:96) · (cid:126)S (cid:96) (cid:48) + J H (cid:88) (cid:96) (cid:54) = (cid:96) (cid:48) d † (cid:96) ↑ d † (cid:96) ↓ d (cid:96) (cid:48) ↓ d (cid:96) (cid:48) ↑ (11)with :: denoting normal ordering. Here the various termsin the Kanamori interaction are: (i) the total “charg-ing energy” to change the electron number at a site, (ii)the difference term between interorbital and intraorbitalcharge repulsion, (iii) the Hund’s exchange between spinsin different orbitals, and (iv) singlet pair hopping betweenorbitals. While the first term is governed by the “Hub-bard U ”, the latter three interactions are all set by theHund’s coupling. Since RIXS is a number conserving pro-cess, and the intermediate state of the t g orbitals playsno role in the expression in Eq. 10, the charge repulsion U plays no role in determining I J ( ω ). The interactionsrelevant to RIXS are therefore parameterized by a singleenergy: the Hund’s coupling J H .Such interaction effects will lead to multiple peaks inthe RIXS spectrum, deviating from the single-particle ex-pectation. At the perturbative level, this stems from tworeasons. First, many-body effects will split the degenera-cies associated with the single-particle states; this willsplit the peak at 3 λ/ J H . Second, interactions canperturbatively excite electrons into higher energy single-particle states. This is shown in Figs. 1(b-d), where inter-actions excite electrons between j eff = 3 / j eff = 1 / two electronsexcited across the spin-orbit gap. Such ‘multi-particle’excitations will produce a second set of peaks around anenergy ∼ λ . For small J H /λ , these secondary peakswill have an intensity ∼ ( J H /λ ) . This is in addition toany suppression of matrix elements arising from quantumnumbers (i.e., selection rules).Below, we will use the expression in Eq. 10, and presentresults from a (non-perturbative) numerical computationusing exact diagonalization for the g, n, f states of the t g orbitals. While we have presented preliminary results forthe case of the iridates in previous work, we focus hereon other fillings, which are also relevant to the osmatesand rhenates. EXACT DIAGONALIZATION RESULTSMode energies
We have used the t g orbital basis with SOC andthe Kanamori interaction, and numerically computed theeigenstates and the RIXS intensity from Eq. 10 using ex-act diagonalization. The projection to the t g levels isjustified by the large crystal field splitting as seen in theab initio results (discussed below). We consider differentfillings d , d , d , and compare the resulting energies inthe spectrum to the experimental results [54–57] for var-ious 5 d Mott insulating oxides: (i) Ba YReO ( d rhen-ate), (ii) Ca LiOsO and Ba YOsO ( d osmates), and(iii) Sr YIrO , Sr GdIrO , and Ba YIrO ( d iridates).A best fit of the excitation energies to previously pub-lished experimental spectra allows us to extract the SOCstrength λ and the Hund’s coupling J H . The results are summarized in Table I, where we also show the excitationenergies from theory and experiments. The agreement isgood, showing that the projection to the t g sector yieldsan effective description of the RIXS data. Material λ J H Peak 1 Peak 2 Peak 3 Peak 4Ba YReO Ex [ ] [ ] (ref.55,this) Th 0.380 0.260 0.41 0.47 0.89 1.83Ba YOsO Ex (ref.54) Th 0.335 0.275 0.75 0.91 1.46 1.71Sr YIrO Ex ∼ (ref.55) Th 0.425 0.250 0.41 0.64 1.31 2.06Ba YIrO Ex (ref.56) Th 0.385 0.230 0.37 0.58 1.19 1.88TABLE I. Table showing the optimal λ, J H (in eV) in dif-ferent materials deduced from fitting the theoretical calcula-tions to experimental RIXS excitation energies. We presentthe comparison of the observed (from Refs. 54–56) RIXS peakenergies (in eV) (top, ‘Ex’, in italics) with the correspondingtheoretical values (bottom, ‘Th’) for the optimal parameterset computed here. For Ba YReO , peaks L edgeRIXS, but are resolved here using the L edge (see text andFig. 3). For Sr YIrO , peak ∼ YIrO .Results for Ca LiOsO (Ref. 54) and Sr GdIrO (Ref. 55)are nearly identical to Ba YOsO and Sr YIrO respectively. Spectral intensities
RIXS experiments are typically carried out in the ‘hor-izontal geometry’ where the in-photon polarization liesin the scattering plane, with the scattering angle to be2 θ = 90 ◦ , so that ˆ (cid:15) in · ˆ (cid:15) out = 0. Fig. 2 shows the spectrumcomputed for this scattering geometry. Here, we averageover ˆ (cid:15) out with ˆ (cid:15) in · ˆ (cid:15) out = 0. The precise incident po-larization direction does not matter since the results arerotationally invariant (so single crystals and powder sam-ples should yield the same result in this ‘atomic limit’).We have chosen experimentally relevant values for theresolution, with a full width at half maximum (FWHM)of 100meV (rhenates, L edge), 150meV (osmates, L edge), and 40meV (iridates, L edge). In all cases, thetwo lower energy peaks (energies < ∼ J eff = 0 groundstate are most robust to interaction effects, and exhibitnegligible intensity for two-particle excitations. d Rhenate λ = 0.380 eVJ H = 0.260 eV I / ( ω ) ( e V - ) (a) h ω (eV) d Osmate λ = 0.335 eVJ H = 0.275 eV (b) I / ( ω ) ( e V - ) h ω (eV) d Iridate λ = 0.425 eVJ H = 0.250 eV (c) I / ( ω ) ( e V - ) h ω (eV) FIG. 2. (Color online) Theoretically computed RIXS spectrum for (a) d rhenates ( L edge), (b) d osmates ( L edge), and (c) d iridates ( L edge) broadened by instrumental resolution. We have used shown best fit values for λ (SOC) and J H (Hund’scoupling). In (a), dashed arrows indicate two closely spaced peaks, which are not resolved in L edge RIXS; see Fig. 3 for new L edge RIXS which detects this splitting. In all panels, solid arrows indicate ‘two-particle’ transitions due to Hund’s coupling. L edge RIXS for rhenates RIXS measurements at Re L3 edge (E i =10.537 keV)were carried out at the 27ID-B beam line at AdvancedPhoton Source. The same polycrystalline sample ofBa YReO used in Ref. 51 was used. The beamwas monochromatized by Si(111) double-crystal and aSi(119) channel-cut secondary crystal. A sphericallydiced Si(119) analyser with 2m radius of curvature wasused to achieve an overall energy resolution of 60meV.A horizontal scattering geometry with scattering angle2 θ = 90 ◦ was used to minimize elastic background. Themeasurement was carried out at room temperature. Thismeasurement allows us to resolve the splitting betweentwo low energy peaks at ¯ hω = 0 . . . . Discussion
Our model Hamiltonian in the t g spin-orbital sectorprovides a good description of the RIXS data, with com-parable strengths of the SOC λ and Hund’s coupling J H .Thus, the site-localized limit provides a good startingpoint to understand these double perovskites. However, astrict projection of the physics to t g orbitals completelyignores the e g states. Furthermore, the d -orbitals of thetransition metal ions are expected to hybridize with theneighboring oxygens. Such effects could be important inrelating the t g model Hamiltonian parameters to a moremicroscopic description.For instance, Table I shows a small, but systematic, difference between the RIXS peak energies in polycrys-talline cubic Ba YIrO from Ref. 56, and those reportedon single crystals of Sr YIrO and Sr GdIrO in Ref. 55.This must be attributed to the different size of Ba ioncompared with Sr, which leads to slight differences inbond lengths and angles of the IrO octahedra, suggest-ing that Ir-O hybridization might lead to renormalizationof λ and J H . Table I also shows that the inferred SOC λ for Os is smaller than both Re and Ir, while the corre-sponding Hund’s coupling is slightly larger. Again, sucha non-monotonic trend across the 5 d series reflects howthe two microscopic effects discussed above might renor-malize the parameters of the effective Hamiltonian. Sucheffects may be phenomenologically accounted for by go-ing beyond the Kanamori Hamiltonian, for instance bymodifying the coupling strengths appearing in Eq. 11 asdone in Ref. 54 for the osmates. Below, we use electronicstructure calculations to provide an ab initio perspective. FIG. 3. (Color online) RIXS intensity as a function of energytransfer ¯ hω in Ba YReO . The RIXS spectrum was obtainednear Re L edge with incident energy, E i = 10 . θ =90 ◦ was used to minimize elasticbackground. The two indicated peaks at ¯ hω = 0 . . L edge measurements. AB INITIO ELECTRONIC STRUCTURECALCULATIONS
We carried out density functional theory (DFT) cal-culations for the cubic double perovskites Ba YReO ,Ba YOsO , and Ba YIrO . From a structural viewpoint,these cubic double perovskites can be regarded as con-sisting of clusters of the metal atom M with the sixO ions comprising the ( M O ) − octahedral cage, sep-arated by Ba and Y ions that maintain the charge bal-ance. We used the generalized gradient approximation(GGA) of Perdew, Burke and Ernzerhof (PBE) [58], withthe addition of an on-site Coulomb repulsion using thePBE+U method ( U = 4 eV) in the so-called fully local-ized limit, and the general potential linearized augmentedplanewave (LAPW) method [59] as implemented in theWIEN2k code. [60] In this method U is a parameterapplied in order to mimic the effects of Coulomb correla-tions [61]. This value is applied in the LAPW method tothe d-orbitals within an LAPW sphere. Typical valuesfor transition metal oxides range from 4 eV to 8 eV. Inthe present case, where we deal with a multi-orbital 5dmaterial, low values are likely to be more physical. Wefind that 3 eV is inadequate to give an insulating gap forall the compounds studied, while experimental resistiv-ity data suggests insulating character. We choose 4 eVbecause this is adequate to give an insulating gap in allcompounds at least for an AFM state. Further detailsof the DFT calculations are given in Appendix A. Notethat in our ab initio electronic structure calculations, werely on experimental lattice parameters to fix atomic po-sitions because they are well established for these mate-rials and are without doubt more accurate than can beobtained from DFT.For all three materials, we have studied 5 d momentsarranged in a type-I antiferromagnetic (AFM-I) patternand a ferromagnetic (FM) pattern. The calculated DOSfor FM order in different compounds are given in theAppendix. For FM order, the DOS peaks are generallybroader, leading to incomplete gapping for the Ir andRe compounds. We also considered non-spin-polarizedsolutions which, however, are not energetically favoredfor any of the compounds studied even with U = 0 eV.This argues against explanations for the lack of observedmagnetic ordering in the Ir and Re compounds that relyon the absence of moments. Crystal field splitting
Fig. 6 shows the metal d projection of the calculateddensity of states (DOS) for all three compounds, includ-ing the spin projections, in the AFM-I state. We seethat all compounds show a very large crystal field split-ting, with t g states, which are near the Fermi energy( E = 0), being well separated from the e g -like states at ≈ ± e g -like states correspond to strongly hy-bridized bonding and anti-bonding combinations of 5 de g states and O 2 p states arising from a significant σ -overlap. This large crystal-field splitting is consistentwith RIXS data.[54–56] Since the top of the t g DOS isseparated from the bottom of the e g -like DOS by ≈ t g orbitals, as we have dis-cussed above, is appropriate to understand the RIXSspectra for energy transfers ¯ hω < ∼ M O ) − cluster, and a study of different magnetic or-dering patterns, reveals interesting physics beyond theatomic limit. Spin and orbital moments
For U = 4eV, we find that all three compounds areinsulating and show local moment behavior in the sensethat the spin and orbital moments on the metal site arepractically identical for the AFM and FM orders. A sum-mary of the moments is given in Table II.We start with a discussion of the spin-moment. As seenfrom the values of M spin , the total spin in the unit cellin the FM pattern, SOC only weakly reduces the totalspin-moments from the nominal values of 2 µ B /atom forRe and Ir, and 3 µ B for Os (based on electron count in anisolated t g shell). However, the moments as quantifiedby the part residing in the metal sphere, m spin , are only ≈ / ∼ M O ) − cluster is on the O site.(Within our DFT calculations for Ba YIrO , decreasing U leads to moment reduction on the Ir site, which maybring it in closer alignment with susceptibility measure-ments. However, we find that this also leads to a metallicDOS, in apparent contradiction with transport data. Wereturn to this issue later.) N ( E ) E (eV)IrOsReMaj. N ( E ) E (eV)IrOsReMin.
FIG. 4. (Color online) 5 d projections of the electronic densityof states onto the Ir, Os, and Re LAPW spheres of majorityand minority spin character on a per ion basis. The Os valuesare offset and the energy zero is at the highest occupied state. It is also interesting to note that while the Re and
TABLE II. Calculated spin and orbital moments m (in µ B )in the transition metal LAPW sphere for AFM and FMarrangements (see text) from DFT calculations with U =4 eV. M spin is the total spin moment per formula unit including allatoms for the FM case. ∆ E = E FM − E AFM is the energydifference (per formula unit) between FM and AFM states.Material AFM FM ∆ Em spin m orb m spin m orb M spin (meV)Ba YReO YOsO YIrO Os compounds have orbital moments in accord with the ionic
Hund’s rule (i.e., opposite to the spin moment),this is not the case for the Ir compound. This is a conse-quence of the strong crystal field splitting noted above,between the t g and e g orbitals. This reversal of the or-bital moment for the Ir compound does not follow thethird Hund’s rule for a free ion, but does follow theHund’s rule if one considers the t g orbital as an inde-pendent shell (i.e. as an effective p level, which is nowmore than half full for the Ir compound).Finally, we turn to the issue of why DFT finds large in-duced O moments in these compounds. The explanationlies in an indirect exchange mechanism where the on-site Hunds exchange coupling couples the t g momentsto produce an exchange splitting of the e g d orbitals,which occur both in the e g upper crystal field level andat the bottom of the O 2 p bands as seen from Fig. 6.This leads to the spin dependent hybridization of the e g orbitals, and therefore a magnetization of the nominallyO 2 p derived bands.We note that the LAPW method divides space intonon-overlapping spheres centered at the atoms and a re-maining interstitial space. The O spheres in our calcu-lation are necessarily small due to this non-overlappingrequirement, and therefore the moment in these spheresunderestimates the O contributions, but is expected tobe roughly proportional to them. The contribution fromthe LAPW spheres of the six O around a given transitionmetal atom are 0 . µ B , 0 . µ B , and 0 . µ B , for the Re,Os and Ir compounds, respectively. Note that the totalof the O and transition metal atoms is not the total mo-ment due to the interstitial, and also that even with agap, the spin-orbit interaction reduces the total spin mo-ments from nominal integer values that may be expectedfrom the band filling. Magnetic ordering
Our DFT calculations yield magnetic ground statesin all three compounds, in that the AFM-I structuregives lower energy than a non-magnetic case. This re-sult is robust against changes in the parameter U , and in particular also holds for U =0. However, we find thatthe exchange interaction between ( M O ) − clusters, asquantified by the AFM-FM energy difference, while al-ways antiferromagnetic, is one to two orders of magnitudesmaller in the Re and Ir compounds as compared to theOs compound. This may be important for explaining ex-periments showing evidence for the presence of momentsin the Ir and Re compounds, but without the robust longrange order observed in the Os compound.For Ba YOsO , the magnetic structure has been ex-perimentally determined [62] to be type-I AFM orderbelow a N´eel temperature T N = 69K. Indeed our re-sults show that AFM-I order leads to a lower energythan FM order. Based on the energy difference ∆ E = E F M − E AF M = 54 . CW > ∼ YReO , we find a much smaller energy dif-ference ∆ E = 3 . YReO is reported to show a glassy magnetic groundstate possibly without long range order and without evi-dence in thermodynamics or susceptibility for a fluctuat-ing state.[63–65] We suppose that an AFM-I state maybe the true ground state if a perfectly chemically orderedsample could be made, with the observed glassy state re-sulting from low levels of disorder. Oxygen vacancies, ifpresent in large quantities, might also provide a sourceof disorder affecting ordering.The results on Ba YIrO are still controversial,[56, 57,66–70] with experimental reports of magnetism being at-tributed to impurities or to weakly fluctuating ∼ . µ B moments whose origin is unclear. Previous electronicstructure calculations and model studies[26, 28, 66, 68,71] reach somewhat contradictory conclusions based onwhether one starts from a band picture or an atomicpicture. From our calculations with U = 4eV, we finda significant moment ∼ . µ B on the (IrO ) − cluster,but with ∆ E = 0 . U , but we verified that it remainsmuch smaller than in the Os compound for different val-ues. At some values of U (e.g. U = 3eV) the ferromag-netic order can even have lower energy than the AF-Iorder. Assuming that the experimentally measured mo-ments in Ba YIrO are indeed intrinsic, our calculationscould help to understand why these moments may notorder down to very low temperatures.We note that the very small energy difference betweenferromagnetic and antiferromagnetic orderings meansthat the inter-site exchange couplings are small, whichis the reason for inferring weak magnetic interactions.As noted previously [72, 73], in 4d and 5d double per-ovskites, oxygen takes a substantial spin polarizationleading to effective MO octahedral magnetic clusters.These interact through the O atoms so that the con-tact and distances between O in different octahedra isimportant for the exchange. This suggests a sensitiv-ity to structure. It will be of interest to experimen-tally explore strain and pressure effects on magnetic orderin these compounds especially to better understand thenon-ordered states of the Re and Ir compounds; this is atopic for future investigation. DISCUSSION
Our ED results show that the RIXS excitations inall the 5 d double perovskites are well described by theatomic limit picture. In this limit, the d rhenates and d osmates should support local moments, which is con-sistent with our complementary electronic structure cal-culations. In addition, our ab initio estimates for theexchange interaction strength is consistent with exper-iments which find robust magnetic order in Ba YOsO as opposed to Ba YReO . However, the ED and DFTcalculations are in disagreement for the ground state ofBa YIrO . While the atomic limit ground state in EDis a J eff = 0 singlet, our DFT results indicate that the d iridates should show a significant local moment in theinsulating phase.Within our DFT calculations on Ba YIrO , we findthat decreasing U leads to a smaller moment on theIr site. This could partially bridge the gap with ED,and may bring the moment in closer alignment to thatinferred from susceptibility measurements,[66, 67] How-ever, the resulting state then becomes metallic whichseems to be at odds with the apparently insulatingresistivity,[68] unless we ascribe this to disorder inducedlocalization. Assuming that the insulating transport isintrinsic and due to interactions, we are led to con-clude that quantum spin-orbital fluctuations and dynam-ical self-energy effects beyond DFT must be crucial inBa YIrO . Including these may lead to one of two out-comes. (i) This could stabilize a Mott insulator withsmall moments which are weakly coupled, which couldexplain both the susceptibility and transport data, show-ing that going beyond the simple atomic limit is impor-tant. (ii) Alternatively, it might stabilize the J eff = 0state as in our ED study; the measured magnetism mustthen be attributed to defects.[68, 70]At the same time, in order to understand potentiallyhow the atomic limit picture might weakly break downin Ba YIrO , it is useful to study an isolated IrO oc-tahedral cluster which allows for some degree of electrondelocalization in the Mott insulating phase. Within aperfect octahedral cage, the e g orbitals of Ir will each hy-bridize with one appropriate symmetry combination ofthe p σ oxygen orbitals. Similarly, each t g orbital canhybridize with only one symmetry combination of the p π orbitals. Figs. 5(a),(b) present an illustrative level Ir e g [4]Ir t [6] O p σ [12]O p π [24] Ir-O e g -p σ [4]Ir-O t -p π [6] O p σ [8]O p π [18]O-Ir p π -t [6]O-Ir p σ -e g [4] Non-BondedO-states t σ t π (a) (b) Ir-O (j eff =1/2) O-Ir (j eff =1/2)O-Ir (j eff =3/2)
Non-BondedO-states
O-Ir p σ -e g O p σ O p π Ir-O (j eff =3/2) (c)
Ir-O e g -p σ FIG. 5. (Color online) Schematic single-particle level diagramfor Ir and O orbitals in the IrO cluster of Ba YIrO , withfilled, partially filled, and empty boxes indicating electron fill-ing. (a) Ir and O orbitals showing small crystal field splittingin the absence of hybridization; dashed black lines show inter-site hybridization via hopping matrix elements t σ , t π . Num-bers indicate degeneracies of the orbitals including spin. (b)Hybridization leads to a large splitting of the σ -hybridized or-bitals, and to a smaller splitting of the π -hybridized orbitals,together with a set of unbonded O orbitals. The notationIr-O or O-Ir indicates respectively that the Ir or O statesare the dominant contribution to the hybridized wavefunc-tion. (c) SOC (thin blue lines) on Ir leads to a splitting ofthe hybridized Ir-O and O-Ir t g states. Nominally filled andunfilled levels are indicated by filled and empty boxes in thisfinal schematic, which would lead to a J eff = 0 insulator. Ar-rows shows possible impact of interactions which might causea partial occupancy of the j eff = 1 / scheme where we have shown how the σ -hybridizationleads to Ir-O e g levels which are strongly split, while thesmaller π -hybridization of the t g states with a subset ofO p π , leaving a residual set of non-bonded O levels. Forhybridized states, we have used the notation Ir-O andO-Ir to respectively depict states which are dominantlyIr versus dominantly O. Here, the numbers indicate thelevel degeneracy (including spin). Incorporating SOC, asshown in Fig. 5(c), leads to hybridized j eff = 1 / , / j eff = 3 / j eff = 1 / e g states are unfilled,leading to a J eff = 0 ground state. This level schemeis consistent with the ‘atomic’ limit, with the effectiveSOC, as determined from the separation between the Ir-O j eff = 3 / / YIrO is a Slater insulator with AFM order,electron itinerancy is expected to lead to a mixing ofthe j eff = 1 / , / > ∼ YIrO is a Mott insulator, with the weak momentsfound in experiments indeed being intrinsic, it is clearthat the tiny Ir-Ir superexchange, which is far smallerthan SOC, cannot destabilize the J eff = 0 singlet and leadto these moments. So the ‘exciton condensation’ mech-anism for J eff = 0 Mott insulators studied in Refs. 22and 24 cannot be operative here. However, the origin ofthe moments could arise from Ir-O interactions within anIrO cluster. In this case, our level scheme suggests somepotential intrinsic mechanisms to explain the magneticmoments reported in the Mott insulator. For instance,electron interactions might lead to a partial depletion ofthe non-bonded O levels just below the Fermi level andpartial occupation of the j eff = 1 / J eff = 0picture for the Mott insulator, resulting in the formationof weak moments. This state may be schematically rep-resented as | Ir (cid:105)| O nb no − hole (cid:105) + (cid:15) | Ir (cid:105)| O nb − hole (cid:105) ,where ‘O nb ’ denotes non-bonded oxygen orbitals. Al-ternatively, interactions which generalize the rotation-ally invariant Kanamori form may lead to mixing of theform | J eff = 0 (cid:105) + (cid:15) | J eff = 1 (cid:105) on the Ir site since theIr-O wavefunctions are not strictly ionic but representstates hybridized with oxygen. In analogy with previouswork,[22, 24] we may term such states as ‘intracluster ex-citons’. Such excitons would be dispersionless, with ex-tremely weak coupling between clusters leading to pos-sible weak long-range magnetic order. Further studiesof such a cluster Hamiltonian would be valuable in ex-ploring this scenario, since it is unclear if these intrinsicexplanations for the observed moments can also simulta-neously be as successful at describing the RIXS data asour present model. SUMMARY
We have shown that the theory of RIXS yields modeenergies and spectral intensities for 5 d complex oxides atdifferent fillings which are in good agreement with ex-periments, leading to estimates of SOC and Hund’s cou-pling. Our work provides a natural interpretation of thelow energy peaks as single-particle excitations across thespin-orbit gap, which are split by Hund’s interaction, andthe higher energy peaks as emerging from two-particleexcitations across the spin-orbit gap which also leads toa lower intensity. We note that recent work on 4 d and5 d oxides suggests that t g - e g interactions might becomeimportant for certain parameter regimes,[27] thus goingbeyond the approximation of projecting to the t g or-bitals. Our ab initio calculations show that the e g statesmight also enter the picture differently, via strong hy-bridization with ligand oxygens. Our electronic struc-ture calculations allow us to extract the exchange inter-actions, from which we deduce that Ba YOsO should show robust AFM ordering, but that Ba YReO andBa YIrO have very weak exchange interactions whichwould strongly suppress magnetic ordering. Finally, ourelectronic structure and atomic ED calculations lead usto a model for the M O cluster which may suggest a dis-tinct mechanism for generating intrinsic weak magneticmoments in Ba YIrO .AP, BY, and YJ were supported by the Natural Sci-ences and Engineering Research Council of Canada.ADC was supported by the U.S. DOE, Office of Science,Basic Energy Sciences, Materials Sciences and Engineer-ing Division ∗ [email protected] † [email protected][1] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Ba-lents, Annual Review of Condensed Matter Physics , An-nual Review of Condensed Matter Physics , 57 (2014).[2] B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park,C. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. 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The DFT calculations were carried out with the gener-alized gradient approximation (GGA) of Perdew, Burkeand Ernzerhof (PBE) [58] and the general potential lin-earized augmented planewave (LAPW) method [59] asimplemented in the WIEN2k code. [60] The LAPWsphere radii were 2.1 bohr for Ir, Os, Re and Y, 2.5 bohrfor Ba and 1.55 bohr for O. We used the standard LAPWbasis set plus local orbitals for the semicore states. Withthe PBE GGA, including magnetism, we obtain a semi-conducting gap for Ba YOsO , reflecting the exchangesplit t g crystal field level of this d system, but we donot obtain a gap in either Ba YReO or Ba YIrO , evenwith magnetic order and spin orbit coupling. Experimen-tal data (e.g. specific heat) imply that Ba YReO is non-metallic. Experimental data is less clear for Ba YIrO but it is presumed to be non-metallic based on transportdata. Accordingly, we show electronic structures with the PBE+U method, with the choice U =4 eV. This is suffi-cient to open a gap in both Ba YReO and Ba YIrO .We find that with U =3 eV, a gap is opened in Ba YReO but not Ba YIrO , with the assumed magnetic orderingpattern. For U =4 eV, and the assumed antiferromag-netic order we obtain gaps of 0.31 eV, 1.21 eV and 0.12eV, for the Re, Os and Ir compounds, respectively. Wenote that the selected value of U is higher than that usedby Bhowal et al. [71] in a prior study of Iridates, whereU = 2 eV was employed. In our calculations we find thatneither Ba YIrO nor Ba YReO is insulating for U =2 eV. From an experimental point of view it is not fullyestablished whether Ba YIrO is a true insulator, butresistivity data points to such a state.For the structure we used the experimentally de-termined lattice parameters, a =8.3395 ˚A, [64] forBa YReO and a =8.34383 ˚A, for Ba YOsO , [54] and a =8.3387 ˚A, for Ba YIrO . [74] We relaxed the free in-ternal parameter associated with the O position usingthe PBE GGA. Since bonding and moment formationare inter-related, we allowed the formation of ferromag-net moments in these relaxations, i.e. the FM order.The resulting structures have Ba at (0.25,0.25,0.25) and(0.75,0.75,0.75), Y at (0,0.0), Re/Os at (0.5,0.5,0.5) andO at (0,0, z O ) and equivalent positions, with z O =0.2642,0.2639 and 0.2636 for the Re, Os and Ir compounds, re-spectively. We used this structure for calculating elec-tronic and magnetic properties as discussed below. Allcalculations included spin-orbit coupling, except for thestructure relaxation.The majority and minority DOS for the optimal AFMstate have been presented in the main text. Below, weplot the corresponding DOS for the FM state. N ( E ) E (eV)IrOsReMaj. N ( E ) E (eV)IrOsReMin.
FIG. 6. (Color online) 5 dd