Magnetization density distribution of Sr 2 IrO 4 : Deviation from a local j eff =1/2 picture
Jaehong Jeong, Benjamin Lenz, Arsen Gukasov, Xavier Fabreges, Andrew Sazonov, Vladimir Hutanu, Alex Louat, Dalila Bounoua, Cyril Martins, Silke Biermann, Véronique Brouet, Yvan Sidis, Philippe Bourges
MMagnetization density distribution of Sr IrO : Deviation from alocal j eff = 1 / picture Jaehong Jeong, ∗ Benjamin Lenz, Arsen Goukasov, Xavier Fabr`eges, AndrewSazonov, Vladimir Hutanu, Alex Louat, Dalila Bounoua, Cyril Martins, Silke Biermann, V´eronique Brouet, Yvan Sidis, and Philippe Bourges † Laboratoire L´eon Brillouin, CEA-CNRS,CEA-Saclay, F-91191 Gif sur Yvette, France Centre de Physique Th´eorique (CPHT),Ecole Polytechnique, CNRS, Universit´e Paris-Saclay,Route de Saclay, 91128 Palaiseau Cedex, France Institute of Crystallography, RWTH Aachen University and J¨ulich Centre for NeutronScience (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), 85747 Garching, Germany Laboratoire de Physique des Solides, Universit´e Paris-Sud,Universit´e Paris-Saclay, 91405 Orsay, France Laboratoire de Chimie et Physique Quantiques,UMR 5626, Universit´e Paul Sabatier,118 route de Narbonne, 31400 Toulouse, France Laboratoire de Physique des Solides,Universit´e Paris-Sud, UMR 8502, 91405 Orsay, France a r X i v : . [ c ond - m a t . s t r- e l ] J a n bstract d iridium oxides are of huge interest due to the potential for new quantum states driven bystrong spin-orbit coupling. The strontium iridate Sr IrO is particularly in the spotlight becauseof the novel j eff = 1 / t g orbitalswith nearly equal population, which stabilizes an unconventional Mott insulating state. Here,we report an anisotropic and aspherical magnetization density distribution measured by polarizedneutron diffraction in a magnetic field up to 5 T at 4 K, which strongly deviates from a local j eff = 1 / xy orbital contri-bution is dominant. Theoretical considerations based on a momentum-dependent composition ofthe j eff = 1 / j eff = 1 / IrO possesses a tetragonal structure with I /acd space group, in which the IrO octahedra are rotated by ≈ ◦ around the c -axis with an opposite phase for the neighboringIr ions [1, 2] and it orders antiferromagnetically below T N ≈
230 K [3]. Strong spin-orbitcoupling (SOC) stabilizes an unconventional Mott insulating ground state, which is com-monly described by a spin-orbital product state within a so-called j eff = 1 / d electrons at the Ir (5 d ) ions occupy the t g states with an effective angularmomentum l eff = 1, which are split by the relatively large SOC into a j eff = 1 / j eff = 3 / j eff = 1 / j eff = 1 / t g band (Fig. 1a): (cid:12)(cid:12)(cid:12)(cid:12) j eff = 12 , ± (cid:29) = 1 √ | xy, ± σ (cid:105) ± | yz, ∓ σ (cid:105) + i | xz, ∓ σ (cid:105) ) . (1)While resonant and inelastic X-ray scattering [4, 7] gave credit to a description in termsof J eff = 1 / j eff = 1 / t g band is split into three Kramers doubletstates, which consist of the mixing between j eff = 1 / j eff = 3 / d and O 2 p orbitals, which seems to benatural for a large spatial extent of 5 d orbitals, has been proposed to account for a largereduction of the ordered magnetic moment [3] as well as for AFM exchange interactionsbetween the nearest-neighboring Ir ions and for the canted magnetic moments followingthe octahedral rotations [5, 12]. The strong hybridization of the d -orbitals with the p -orbitals of the ligand oxygen is reminiscent of K IrCl [13] and the isostructural ruthenateCa . Sr . RuO [14], where similar covalency effects have been reported. In Sr IrO , recentmuon spin relaxation measurements have suggested the formation of oxygen moments [15],and charge redistribution between adjacent IrO and SrO layers has been revealed usingelectron spin resonance measurement [16]. Further, unusual magnetic multipoles have beenproposed to be observed by neutron diffraction [17] and recently a hidden magnetic orderhaving the same symmetry as a loop-current state has been observed by polarized neutrondiffraction [18].The magnetic moments of Ir ions are confined in the ab -plane and track the staggered3 b)(a) ȁ ۧ𝑥𝑦, +𝜎 ȁ ۧ𝑦𝑧, −𝜎 + ȁ ۧ𝑥𝑧, −𝜎 ห ൿ𝑗 eff = ൗ1 2 , + ൗ1 2 = ൗ + ȁ ۧ𝑦𝑧, −𝜎 + ȁ ۧ𝑥𝑧, −𝜎 ab z = c /8IrO1SrO2a bc H FIG. 1:
The j eff = 1 / state and uniform magnetization of Sr IrO . (a) The electronand spin density distributions for the ideal j eff = 1 / , m j = 1 / t g orbitals with mixed spin states. The red and blue colors represent spin-upand spin-down states, respectively. (b) The magnetization vs temperature curve under H = 1 T( H// [110]). It exhibits a weak ferromagnetic moment inherited from the AF-II order transition [2]at ≈
235 K. The inset shows the crystal and magnetic structure of Sr IrO for an applied magneticfield along H// [110]. octahedral rotation in an − + + − sequence along the c -axis in the unit cell [2]. Owingto this canted AFM structure, each IrO layer has a weak ferromagnetic moment alongthe principal crystallographic axis in the ab -plane at zero magnetic field. This WFM iscompensated due to the − + + − stacking sequence whereas, in a magnetic field higher than H c ≈ . ab -plane [2, 3], a net homogeneous WFM moment appears inthe plane (inset of Fig. 1b) above the metamagnetic transition. Remarkably, this WFMmoment follows the direction of applied magnetic field in the ab -plane [19–21] and attains asaturation value of ≈ µ B /Ir in the field of 1 T [19]. In the current experiment, a uniformmagnetic field ( H ) upto 5 T has been applied along the vertical direction (Fig. 2a). The IrO octahedral rotation generates two additional terms in the simple Heisenberg-type magneticHamiltonian [21]: J z and Dzyaloshinskii-Moriya terms, which restrict the angle betweenadjacent pseudospins to π + 2 α with the octahedral rotation angle α [5] (the situation isshown in the inset of Fig. 2b for a field applied along the [110] direction). However, it doesnot break the in-plane rotational symmetry as the pseudospins are free to rotate in the planewhile keeping the same canting angle between them. Therefore, under the applied magnetic4eld, the WFM moment does not interlock with the rotation of IrO octahedra in contrastwith the AFM staggered moment at zero field.The existence of this WFM allows us to probe the magnetization density distribution incrystals by polarized neutron diffraction (PND). This technique is unique because it providesdirect information about the 3-dimensional distribution of the magnetization throughout theunit cell, which in turn allows for a determination of the symmetry of occupied orbitals. Thismethod has been successfully used in the study of FM ruthenate Ca . Sr . RuO , isostruc-tural to Sr IrO , where an anomalously high spin density at the oxygen site and the xy character of the Ru d -orbitals have been reported [14].The typical experimental setup for PND, shown in Fig. 2a, consists of a neutron polarizer,a flipping device that reverses the incident neutron polarization, a magnet and a detector.The sample is magnetized by a magnetic field applied along the vertical axis and scatteringintensities of Bragg reflections for the two opposite states (spin-up and spin-down) of theincident polarization are measured. They are used to calculate the so-called flipping ratio,allowing access to the Fourier components of the magnetization density, as R PND = I ↑ I ↓ = F N + 2 p sin αF N F M + sin αF M F N − pe sin αF N F M + sin αF M , (2)where F N is the nuclear structure factor and F M is the magnetic structure factor. p and e are the polarization efficiency of the polarizer and flipper, respectively, and α is the anglebetween the scattering vector and the magnetization (see Supplemental section 2).The flipping ratios R PND of more than 280 ( hkl ) reflections were measured in the weaklyferromagnetic state above the metamagnetic transition at 2 K for two magnetic field orienta-tions, H (cid:107) [010] and H (cid:107) [¯110] (well above the critical field H c ≈ . F M were directly obtained from the measured flip-ping ratios by using Eq. (2) and known nuclear structure factors F N . For convenience, theamplitudes are given in Bohr magnetons, normalized by the number of Ir atoms (8) in theunit cell, and taken in absolute values to remove alternating signs of the phase factor. Theamplitude, F M (0), is imposed in agreement with the saturation moment (0.08 µ B /Ir) givenby the uniform magnetization measurement [19].In the dipole approximation , F M ( Q ) is usually described by a smooth decreasing functionof Q , the magnetic form factor, corresponding to a linear combination of radial integrals5 a) (b) sample detectorflipperpolarizer 𝐼 ↑ ~ 𝐹 𝑁 + 𝐹 𝑀 offon 𝐼 ↓ ~ 𝐹 𝑁 − 𝐹 𝑀 𝐌 𝒛 𝐤 𝑖 𝐐 = 𝐤 𝑖 − 𝐤 𝑓 𝐻 ∥ 𝑧 𝐤 𝑓 Neutron spin (c) (d)
FIG. 2:
Polarized neutron diffraction setup and measured neutron magnetic structurefactor of Sr IrO . (a) The experimental setup for a polarized neutron diffraction experiment.The arrows at the bottom denote a spin polarization of neutrons. The vertical direction correspondsto either the [010] or [¯110] crystallographic direction for each sample orientation (see supplementalsections 2 and 3). (b)
The magnetic structure factor of all measured momentum transfer Q withthe theoretical radial integrals (cid:104) j n (cid:105) for isolated Ir ions. A series of reflections along the (0 , , l ) arehighlighted: (0 , , n ) in blue squares, (2 , , n ) in green diamonds, (1 , , n + 2) in red up-triangles,(2 , , n + 1) in purple down-triangles, and (2 , , n ) in black left-triangles. The (4 , , , , , ,
0) are also presented in black right-triangles, and the rest in grey circles. Measured andfitted magnetic structure factors | F M ( Q ) | for (c) the optimized MEM result and (d) optimizedmultipole expansion result. calculated from the electronic radial wave function. Instead in Fig. 2b, one observes a largedistribution of the measured structure factor indicating unusually large anisotropy. Thatlarge anisotropy is explained by a predominance of xy -orbital as shown below using thereconstruction of the magnetization density in real space. The theoretical radial integrals6 j n (cid:105) for an isolated Ir ion [22] are also shown in Fig. 2b for comparison. We recall that (cid:104) j (cid:105) describes a spherical form factor of the magnetic moment, while (cid:104) j (cid:105) , (cid:104) j (cid:105) and higher-order integrals are needed to describe the departures from spherical symmetry. As seenfrom Fig. 2 except for the (0 , , l ) reflections, decreasing gradually with increasing Q , themajority of reflections strongly deviate from any expected smooth curve. Moreover, whilethe (0 , , n ), (2 , , n ) and (2 , , n ) reflections are close to the (cid:104) j (cid:105) curve in a small Q region,the (1 , , n + 2) and (2 , , n + 1) reflections deviate from it quite strongly. This indicates anaspherical magnetization density, which is typical of ions with one or two unpaired electronsin the d -orbitals [13, 23, 24]. In addition, one can see that high- Q reflections like (4 , , , ,
0) and (4 , ,
0) ones show anomalously large values.Next, a real space visualization has been performed by a reconstruction of the magne-tization density, using two different very well-established and widely used approaches; amodel-free maximum entropy method (MEM) [25] and a quantitative refinement using themultipole expansion of the density function [26]. Both techniques have advantages and lim-its and should be employed where they are the most efficient. Typically, no assumptionis made for the initial magnetization distribution in MEM whereas the d -orbitals shape isconstrained in the multipole expansion.Since the crystal structure is centrosymmetric, the magnetization density can be directlyreconstructed from the measured magnetic structure factors by MEM [25]. Fig. 3a-d, showsthe 3-dimensional magnetization density reconstructed by using a conventional flat densityprior. A positive magnetization density in red color denotes a magnetic moment densityparallel to the applied magnetic field and a negative one in blue is antiparallel. There arethree key features to be noted in the figure. First, the magnetization density at Ir siteshas four positive density lobes directed along the a , b axes, corresponding to a dominantpositive magnetization density of d xy orbital symmetry (Fig. 3b). The two other compo-nents of the effective j eff = 1 / d yz and d xz , which would form an axiallysymmetric doughnut-shaped density above and below the xy plane (see Fig. 1a), does notappear as seen in Fig. 3c,d. Thus the WFM density originates predominantly from the xy orbital (a schematic illustration of the magnetic components in this situation is given in thesupplemental file, Fig. S4, in contrast with the local j eff = 1 / octahedra. Third, contrary to the expectation of strong7 a) (b) (c) (d)(e) (f) (g) (h) FIG. 3:
Magnetization density distribution reconstructed by MEM and multipole ex-pansion refinement.
3D magnetization density distribution on the z = c/ (a) the MEM and (e) multipole expansion model refinement. Isosurfaces encompassing 30%,50% and 70% of the volume density are plotted with a desceding opacitiy according to their iso-values. Red and blue surfaces denote positive and negative magnetizations, repectively. The solidsquare and dotted lines denote the unit cell and Ir-O bonds, respectively. Sliced density contourmaps at (b,f ) ( x, y, c/ (c,g) ( x, a/ , z ) and (d,h) ( a/ x, x, z ) are also shown for both meth-ods. The contour step is 0.04 and 0.08 µ B / ˚A for (b-d) and (f-h) , respectively. The blacks arrowcorrespond to the Ir-O bonding directions. iridium oxygen ligand hybridization, no visible induced magnetization density appears atthe oxygen sites. Actually, no significative polarization dependence has been found in any ofdozens measured (2 , , n + 1) reflections where oxygen atoms contribute. This is in contrastwith the isostructural 4 d compound Ca . Sr . RuO , where ∼
20% of the magnetic momentis transferred to the in-plane O sites [14]. However, one can notice the presence of a negativemagnetic density mostly along the Ir-O direction existing between the large positive lobes.In fact, a significant negative magnetization density as large as half of the net moment isessential for a better description in the MEM analysis (see Supplemental section 4).To confirm the symmetry found by MEM, multipole expansion model was perfomed foran alternative refinement of the WFM density. It is composed of radial and angular parts:8later-type radial wave functions and real spherical harmonic density functions (see Sup-plemental section 5). In Fig. 3e-h, the magnetization density distribution with the bestrefinement is shown. The main positive magnetization density lobes located between thelocal x - and y -axis appear clearly, which corresponds to the d xy symmetry. Therefore, themultipole expansion model fully confirms the d xy symmetry fround by MEM. A benefit ofthe multipole method is to determine the contribution of all five d -orbitals to the magnetiza-tion. Using the orbital-multipole relations [26], the magnetic moments on each orbitals wereobtained as: +0 . − . − .
035 and − . µ B /Ir for d xy , d yz/xz , d z and d x − y , respec-tively. Thus a positive d xy and to a lesser extent a negative d x − y orbital are dominant inthe refinement (the latter effect is minor in the MEM method), while the d yz/xz orbitals arebarely populated. Interestingly, the admixture of d x − y character to the j eff = 1 / R w ∼ .
09) compared to the model with a single radial exponent ( R w ∼ .
18) (see Supplemental section 5). It confirms the anomalously large spatial extent ofthe magnetization density of Ir found by the MEM analysis. To appreciate the relevanceof the obtained magnetization maps, we calculate the magnetic structure factors from theoptimized MEM and multipoles results. By plotting them along with the measured ones inFig. 2c (for MEM) and Fig. 2d (for multipoles), one sees that the calculated densities withMEM reproduce better the experimental data.The predominant d xy -orbital WFM moment can be understood from a modelization ofSr IrO based on a spin-orbit generalization of the multi-orbital Heisenberg model, (seesupplemental section 6). Key to our proposed effective low-energy model is the observationthat the hole in the t g -manifold resides in a k -dependent effective α = 1 / ϕ k ,α . We thereby account explicitly for the strong k -dependence of t g components in the j eff states revealed both by ab initio calculations [27, 28] and photoemission experiments [29].In terms of Fourier-transformed spin operators of such a hole, s i,α , the model is given by aHamiltonian of the form H = (cid:88) i,j,α,β s i,α J iα,jβ s j,β , (3)9here J iα,jβ denotes the tensor of spin interactions in real-space. Away from half-filling, asimilar t − J model in orbital space has been derived for iron pnictides [30].Solving such a model is beyond the scope of this work. However, to get a qualitative ideaof the underlying physics, we proceed by making a few further assumptions. First, if wesuppose that we only have to retain the diagonal terms of J , Eq.(3) decomposes into a sumof three Heisenberg models, one for each of the three t g components. Let us furthermoreassume that we can consider each component separately. In this case, the spin exchange ofthe xz − and yz − components of J is essentially described by quasi-1D Heisenberg chainsin x − and y − direction, suggesting an antiferromagnetic alignment at low temperature. Incontrast, the xy − component is characterized by longer-ranged exchange of nearest neighbor( J ) and next-nearest neighbor exchange ( J ). It is well described by the J − J Heisenbergmodel on the square lattice, which has a quantum disordered singlet ground state at zerotemperature for 0 . (cid:46) J /J (cid:46) . J /J ≈ / xy − component disordered.As a consequence, in this picture the xy magnetic component aligns much easier alongan external magnetic field than the antiferromagnetically ordered xz − and yz − components,which are less susceptible to such a perturbation. Projecting the magnetization density onto t g components hence reveals a predominant xy -character, which follows the field directionin accordance with the measurements. It should be make clear that the proposed picture isstill consistent with the resonant X-ray scattering data[3] observed at the antiferromagneticBragg points (1 , , L ). This results from the k -dependence of the proposed electronic state[27, 29]. Recently, an alternative interpretation of our PND results has been proposed interms of spin anapole correlations [32].In summary, using PND we have evidenced a magnetization density distribution inSr IrO that is inconsistent with the naive local j eff = 1 / Q ,which indicate an aspherical magnetization density distribution with a significant orbitalcontribution. Real space visualization exhibits a dominant d xy orbital character with highlyelongated lobes of Ir magnetization densities towards the next Ir atoms. Although a strong d - p hybridization is expected in Sr IrO , the magnetization density at the ligand oxygensites is barely present. Our results elucidate that the ground state of Sr IrO substantially10eviates from the commonly accepted local j eff = 1 / t g or-bitals. Rather, the hole resides in an orbital that results from a strongly non-local (thatis, k -dependent) superposition of Wannier functions of t g character. These considerationsgive an additional twist to the exotic properties of Sr IrO and the possibilities of modelingthem as well as to the relationship to superconducting copper oxides.We acknowledge supports from the projects NirvAna (contract ANR-14-OHRI-0010) andSOCRATE (ANR-15-CE30-0009-01) of the French Agence Nationale de la Recherche (ANR),by the Investissement dAvenir LabEx PALM (GrantNo. ANR-10-LABX-0039-PALM) andby the European Research Council under grant agreement CorrelMat-617196 for financialsupport. J.J. was supported by an Incoming CEA fellowship from the CEA-EnhancedEurotalents program, co-funded by FP7 Marie-Sklodowska-Curie COFUND program (GrantAgreement 600382). We acknowledge computing time at IDRIS/GENCI Orsay (Project No.t2017091393). We are grateful to the CPHT computer support team. We thank StephenBackes and Hong Jiang for useful discussions and for sharing with us their cRPA data priorto publication [S. Backes, H. Jiang et al., unpublished.]. Instrument POLI at Maier-LeibnitzZentrum (MLZ) Garching is operated in cooperation between RWTH Aachen University andForschungszentrum J¨ulich GmbH (J¨ulich-Aachen Research Alliance JARA). We thank PrS.V. Lovesey for valuable comments on the manuscript and J. Porras for scientific discussions. ∗ [email protected] † [email protected][1] Q. Huang, J. Soubeyroux, O. Chmaissem, I. N. Sora, A. Santoro, R. J. Cava, J. J. Krajewski,and W. F. Peck, Journal of Solid State Chemistry , 355 (1994).[2] F. Ye, S. Chi, B. C. Chakoumakos, J. A. Fernandez-Baca, T. Qi, and G. Cao, Physical ReviewB , 140406(R) (2013).[3] 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. 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High-quality single crystals of Sr IrO were grown at Laboratoire de Physique des Solidesat Orsay using a self-flux method in the platinum crucibles. The temperature dependence ofthe uniform magnetization was measured using a SQUID under a magnetic field of H = 1 Talong the [110] (Fig. S1 a). Sr IrO undergoes a canted antiferromagnetic (AFM) transition.The ordered AFM moment if found to be about 0.21 µ B /Ir or about 0.36 µ B /Ir by singlecrystal neutron diffraction. We here discussed the weak in-plane ferromagnetic moment dueto the moment canting (see Fig 1.b in the main text). In presence of the magnetic fieldabove H > . T ≈
235 K, wherethe derivative d
M / d T diverges. The saturation weak ferromagnetic (WFM) moment is ≈ µ B /Ir at T = 10 K. It is well consistent with the reported value for single crystalsgrown under the best conditions . For the polarized neutron diffraction measurements,which require a large amount of the sample, 50 single crystals of a small size (typically < × × . ) were coaligned within a diameter of 10 mm on an aluminum plate (Fig. S1 b). mm (a) (b) FIG. S1 . (a) Temperature dependence of the uniform magnetization and its derivative measuredby a SQUID. (b) Coaligned 50 single crystals on an aluminum plate. I. POLARIZED NEUTRON DIFFRACTION
Polarized neutron diffraction (PND) measurements were performed on three different po-larized neutron diffractometers, and at the
ORPH ´EE reactor (LLB CEA Saclay) ,and
POLI at FRM-II (Maier-Leibnitz Zentrum Garching). Thermal and hot polarized neu-tron wavelengths were used with λ =1.4 ˚A (6T2), 1.15 ˚A (POLI) and 0.84 ˚A (5C1). Avertical uniform magnetic field of 5 T on 5C1/6T2 at LLB and 2.2 T on POLI at FRMIIhas been applied at the sample position. As the weak ferromagnetic component saturatesabove ∼ IrO .Incoming neutrons were polarized by a Heusler alloy monochromator, a supermirror ben-der or a He filter. The direction of incident neutron polarization – spin-up or spin-down– was chosen by a polarization flipper. The incident spin-polarized neutrons are scatteredby the nuclei and by the local magnetization, which are denoted by nuclear and magneticscattering, respectively. The total neutron cross section for a non-chiral system is given by σ = F N F ∗ N + M ⊥ · M ∗⊥ + P i · ( M ⊥ F ∗ N + M ∗⊥ F N ) , (S1)where F N is the nuclear structure factor, M ⊥ is the magnetic interaction vector and P i is the incident neutron polarization vector. The magnetic interaction vector is given as M ⊥ = ˆ Q × F M ( Q ) × ˆ Q with the scattering vector Q and F M ( Q ) the magnetic structurefactor, i.e. , the Fourier transform of the magnetic moment distribution. Therefore, onlymagnetic components perpendicular to the scattering vector participate in the magneticscattering. The last magnetic-nuclear interference term appears only for a polarized beam( | P i | > q = 0. It corresponds to the magnetic response at the Brillouinzone center, Γ point that would differ from the magnetic response at the aniferromagneticpropagation wave vector. In all the manuscript, we are only speaking about the weak fer-romagnetic component of the moment. This method is well established for paramagneticand ferromagnetic (FM) systems. It gave access, e.g., to the 3 d -orbital population in fer-romagnetic insulator YTiO . While in conventional spin density studies either a positiveor a negative spin component can be present at a given ion, in the iridates thanks to thespin-orbit coupling both positive and negative densities can coexist at the same Ir site (Fig.1.a of the manuscript) . To access this intra-atomic variation of magnetization density high16esolution polarized hot neutron diffraction data are needed. Here we have performed PND(i) to establish the symmetry of the Ir 5 d orbitals occupied by unpaired electrons and (ii)to check the presence of unpaired electron density on the oxygen ligand.In case of PND where incoming neutron polarization is directed parallel or antiparallel toan external magnetic field, one measures the intensity of the scattered neutrons for the twostates of the flipper. We note that no final polarization analysis is performed. Therefore,the measured intensity, obtained for each state of the flipper, is written as a function of thereal structure factors, I ↑ = F N + M ⊥ + 2 P i F N M z ⊥ I ↓ = F N + M ⊥ − P i F N M z ⊥ , (S2)where M ⊥ is now the projection of the magnetic interaction vector M ⊥ along the polariza-tion vector P i with P i and P i being the incident polarization for spin-up and spin-down,respectively. The PND measurement is thus only sensitive to the uniform magnetizationalong the applied magnetic field, which is the vertical z -direction in our case. Using P i = p , P i = pe , M ⊥ = sin αF M and M z ⊥ = sin αF M where α is the angle between the magneticinteraction vector and the scattering vector, the flipping ratio is given as Eq. (2) in the maintext with the polarization efficiency p of the polarizer and the flipping efficiency e of theflipper. Note that I ↑ ∼ | F N + F M | and I ↓ ∼ | F N − F M | when P i ≈ α ≈ π for thehorizontal scattering plane.It is well known that extinction effects might have a crucial influence in the polarizedneutron data treatment procedure. Therefore, the Becker-Coppens model was used to treatfor the extinction in our refinement. We found that the introduction of extinction correctionshad no beneficial effect on the refinement. This is likely due to the very small thickness ofthe crystallites constituting the sample and the rather short neutron wavelengths, 0.84 -1.4˚A, used in the experiment. The same last argument is also valid for the multiple scattering.Moreover we have performed in total 7 different experiments with two sample orientations,using three different wavelengths 0.84, 1.18 and 1.4 ˚A. Hence in each of the experiments themultiple scattering contributions was different. However, the extracted magnetic structuresamplitudes were found in agreement within error bars after merging equivalents with theredundancy factor 5. 17 II. DATA TREATMENT WITH TWO MAGNETIC FIELD ORIENTATIONS
The flipping ratios were measured at a number of Bragg reflections with two differentsample orientations with respect to the applied magnetic field: H k [010] and H k [¯110]. Themagnetic structure factors at common reflections such as (0 , , l ), (2 , , l ) and (2 , , l ) areshown for each orientation in Fig. S2 . At large Q , Bragg reflection (4,0,0) was measuredonly with H k [010] while (4,2,0) and (4,4,0) only with H k [¯110].It is important to notice that the scaling for both orientations is the same and it isconsistent to macroscopic SQUID measurements, which show the same saturation magneticmoment for a magnetic field along the two directions. It also supports an isotropic WFMmoment, which follows the applied magnetic field . We therefore have combined andtaken an average of two data sets without an additional scaling. a b FIG. S2 . Measured magnetic structure factors with a magnetic field along (a) H k [010] and (b) H k [¯110]. V. ANALYSIS USING THE MAXIMUM ENTROPY METHOD
We have tested the maximum entropy method (MEM) by changing the initial positiveand negative magnetic moments. The resulting positive and negative magnetic moments arelimited by the initial values, thus they are starting points for fitting and also serve as upperbounds. The difference between positive and negative moments was set to the net unit cellmagnetic moment, 0 . µ B /u.c. (8 times 0 . µ B /Ir). As seen in Fig. S3 a, the agreementfactor defined by R w = P ( | F obs − F calc | /σ F ) / P ( | F obs | /σ F ) is poor ( ≈ R w ≈
8% for a significant negative momentabove 0 . µ B . Initial values larger than 0 . µ B , however, do not improve the agreementfactor any more and the resulting negative moment either does not follow the initial value.Instead, the both positive and negative densities spread out so the density distribution is onlydispersed while the spatial features are unchanged as shown in Fig. S3 d,g. Therefore, wehave chosen +0 . µ B and − . µ B as optimized initial positive and negative moments, andthe resulting moments are +0 . µ B (+0 . µ B /Ir) and − . µ B ( − . µ B /Ir), respectively.In the figure 3a and 3b of the manuscript, the WFM density originates predominantlyfrom the xy orbital, at variance with a naive j eff = 1 / S4 aschematic illustration of the magnetization for both the local j eff =1/2 model and a pre-dominant xy orbital for the WFM moment. One also remark that positive density lobesare very strongly elongated, in such a way that some magnetization density is delocalizedwell beyond of the IrO octahedra. It is supported by very large spatial extent of the t g orbitals (reaching the nearest neighbouring Ir atoms) found by core-to-core RIXS and abinitio calculation . It also could give a support to a direct Ir-Ir exchange mechanism, viaelectron hopping between the neighboring ions.19 g)(b) (d)(f)(c)(e) (a) FIG. S3 . (a) Agreement factor R w and resulting negative moment per unit cell versus the initialvalue for negative moment for the MEM analysis. (b-d) The 3D magnetization density distributionon the z = c/ − − − µ B per unit cell. Isosurfaces encompassing 30%, 50% and 70%of the volume density are plotted with a descending opacity according to their isovalues. Redand blue surfaces denote positive and negative magnetizations, respectively. The solid square anddotted lines denote the unit cell and Ir-O bonds, respectively. (e-g) Sliced density contour mapsat ( x, y, c/
8) for the given initial negative moments. The contour step is 0.04 µ B / ˚A . The blacksarrow correspond to the Ir-O bonding directions tilted by 11 ◦ from the crystallogaphic tetragonalaxes. b)(a) Direction of in-planemagnetic moment
H H xyyz/xz xyyz/xz HH FIG. S4 . (a) Magnetization density distribution for the naive local j eff =1/2 picture. The xy , yz and xz magnetic components have the same pseudospin moment (black arrows) at each Ir site.The magnetic moments at adjacent sites are aligned antiferromagnetically, perpendicular to theapplied magentic field H, and the canted FM moment (orange arrows) along H appears in both xy and yz/xz components. (b) Magnetization density distribution for the proposed k -dependent j eff =1/2 picture. The magnetic moment in the xy component is aligned along H, while one inthe yz/xz component is perfectly antiparallel and perpendicular to H . The WFM moment solelycomes from the xy component. . ANALYSIS USING A MULTIPOLE EXPANSION MODEL The magnetization density can be modeled by a superposition of spherical harmonicdensity functions, known as the multipolar expansion : m ( r ) = X atoms ∞ X l =0 R n ( r ) l X m = − l p l,m y l,m ( θ, φ ) , (S3)where R n ( r ) are radial wave functions, y l,m ( θ, φ ) are angular density functions, and p l,m arepopulation coefficients. Simple normalized, Slater-type nodeless radial wave functions R n ( r )are defined as: R n ( r ) = s (2 ζ ) n +1 (2 n )! r n − exp( − ζr ) , with n = 5 chosen for the 5 d -orbitals. The angular density functions y l,m ( θ, φ ) are thereal-valued spherical harmonic functions: y l,m ( θ, φ ) = N l,m P | m | l (cos θ ) cos mφ for m > | m | φ for m < , where P ml are the associated Legendre polynomials and N l,m are the normalization factors satisfying Z | y l,m | dΩ = l = 02 for l > . Note that these angular density functions are distinct from the angular wave functions, Y ml ,which are complex-valued and have a different normalization condition, R | Y ml | dΩ = 1.The magnetic form factors corresponding to the model magnetization density in Eq. (S3)become f ( Q ) = ∞ X l =0 φ l ( Q ) l X m = − l p l,m y l,m ( θ k , φ k )with φ l ( Q ) = 4 πi l Z ∞ R n ( r ) j l ( Qr ) r dr, where j l is a spherical Bessel function of order l . Then the least-squares refinement with themeasured F M was done using the mplsq program . For the refinement, the total magneticmoment is constrained: µ Ir + 2 µ O1 + 2 µ O2 = 0 . µ B /Ir.22 a) (b) FIG. S5 . (a) Multipole density functions allowed by 4 /mmm symmetry. Red and blue surfacedenote positive and negative density, respectively. (b) Density distribution of d orbitals constructedby a linear combination of multipole density functions. The multipole density functions allowed by the point group D h (4 /mmm ) are shownin Fig. S5 a. Because the spherical harmonic functions constitute a complete set in thespherical harmonic point group, the orbital density distribution given by a square of wavefunction | Y ml | must be a linear combination of spherical harmonic density functions y l,m .The orbital-multipole relations for the point group D h are defined as p ml | Y ml | = p i,j y i,j with p p ± p p − = .
200 1 .
039 1 .
396 0 . .
200 0 . − .
931 0 . . − .
039 0 .
233 1 . . − .
039 0 . − . p , p , p , p , . The d orbital densities reconstructed using the spherical density functions are also shown inFig. S5 b. Using this orbital-multipole relations, the orbital populations were obtained fromthe fitted density function populations.In order to describe the large spatial extent of the positive magnetization density, theradial wave function was also examined. It is clear as shown in Fig. S6 f that the theoreticalradial function with the radial exponent ζ = 3 .
74 is not fit well ( R w ≈ ζ = 2 . R w ≈ S6 a. Therefore, we introduced an additional radial exponent. The radial wave functionwith two radial exponents provides significantly better agreement ( R w ≈ S6 h.Further, between the positive lobes, negative density lobes occurs as well which are morepronounced in the multipole refinement than in the MEM results (see the Fig. 3 of the mainmansucript). They are about 60% of positive ones and surprisingly have d x − y symmetry,requiring an admixing of the e g orbital to the ground state. Ii is worth to remind that theMEM method is model-free whereas in the multipole expansion method, we have necessarillyto start with certain multipoles corresponding to the d -orbitals. The d x − y contributioncould be overestimated due to insufficient modeling in the multipole description. However,the main result in the MEM - dominant xy and no yz/xz - is well reproduced by themultipole expansion method. We tried various models (see Fig. S6 ), and the given result isstill the best fit.In order to examine a possible magnetization density expected from a strong d - p hy-bridization at the oxygen site, the refinements with a spherical magnetization density atthe O1 and O2 sites were also performed, but contrary to expectations, no evidence for theexistence of the oxygen moment was found. The agreement factor did not become betterand also the refined magnetization density at the O sites is statistically negligible.We have also examined to fit the data with actual orbital density functions instead ofmultipole density functions. When all possible 5 d orbitals are allowed, it obviously givesthe same result ( R w ≈ S7 a.In addition, as it is known to consist mostly of t g orbitals, the model fit allowing only d xy , d yz , d xz orbitals was also performed. As shown in Fig. S7 b, the negative magneticdensity with d x − y symmetry disappears as expected, but the agreement becomes twiceworse ( R w ≈ y angular density function, is essential to describe our measured data.24 c) (d) (e) (f) (g) (h) (a) (b) FIG. S6 . (a) Agreement factor R w versus the radial exponent ζ for the multipole expansion modelwith a single radial exponent. The agreement factor decreases by ∼
20% at ζ ’ ζ = 3.74. (b) Agreement factor R w with two radial exponents ζ and ζ . Radial wave functions R ( r ) and radial distribution functions r R ( r ) with (c) thetheoretical radial exponent for isolated Ir atoms, (d) an optimized single radial exponent, and (e)double radial exponents. (f-h) Reconstructed magnetization density distributions with differentradial wave functions in (c-e). a) (b) FIG. S7 . Magnetization density distribution obtained by orbital density model fit with (a) all five d orbitals and (b) only three t g orbitals. I. THEORETICAL MODELING OF Sr IrO In order to understand the puzzling magnetization density distribution of Sr IrO wepropose a modelization based on a multi-orbital generalization of the Heisenberg-model.The Heisenberg model can be understood as a lowest-order expansion in the ratio betweenhopping and Coulomb interaction around a localized electronic state in a half-filled periodiclattice system. Here, this simple philosophy becomes more subtle due to the spin-orbitalentangled nature of the localized hole state.Let us denote by ϕ k ,α = P ( l,σ ) S ( k ) α, ( lσ ) χ k ( lσ ) a Wannier representation of the t g mani-fold chosen such that the hole resides fully in the upmost (two-fold degenerate) state, whichwe will denote in the following as ϕ k , / ,m j , m j = ± /
2. Here, χ k ( lσ ) denotes the Fouriertransform of a tensor product of a Wannier function of dominant cubic harmonic character l and a spin state σ . Importantly, the transformation S between the cubic harmonic basisand the effective α = 1 / α = 3 / k -dependent. This means that itdoes not only deviate from the standard isotropic √ : √ : √ composition of t g orbitals,but in particular does so in a momentum-dependent fashion, in accordance with ab initiocalculations and recent photoemission experiments .In real space, the hole has a representation by creation and annihilation operators d ( † ) i, / ,m j = X i , ( l,σ ) S ( R i − R i ) ( † )(1 / m j ) , ( lσ ) d ( † ) i ( lσ ) . (S4)The Heisenberg-model construction then proceeds in this basis to second order in the hoppingyielding a Hamiltonian of the form H = X n, ˜ n X i ,i ,j ,j X l,l , ˜ l, ˜ l ˜ J ( n, ˜ n )( i ,i ,l,l ) , ( j ,j , ˜ l, ˜ l ) S l,l ( n ) i ,i S ˜ l, ˜ l (˜ n ) j ,j , (S5)where i , i , j , j denote sites on the lattice, l, l , ˜ l, ˜ l orbitals on these sites in a basis of cubicharmonics and n, ˜ n the index of Pauli spin matrices, τ ( n ) σσ . Here, we defined the generalizedspin operator S l,l ( n ) i ,i = X σ,σ d † i lσ τ ( n ) σσ d i l σ . (S6)This cumbersome object is a direct generalization to spin-orbit space of the multi-orbital t - J -model that has been derived for the iron pnictides , considered in the special case ofhalf-filling. 27n the following, we inspect the diagonal terms in site and orbital space of the tensor ˜J H = X n, ˜ n X i ,j S ( n ) i ,l ˜ J ( n, ˜ n ) i j ,l S (˜ n ) j ,l . (S7)If we further focus on the case n = ˜ n , these terms amount to a slightly modified versionof a standard Heisenberg term, where the exchange matrix ˜J is a spatial modulation ofthe bare exchange in t g space. This modulation is due to the k -dependent nature of thetransformation S that defines the operators d ( † ) . By assuming an antiferromagnetic orderingof the moments in α = 1 / J values are reduced by a factor ∼ . (cid:15) t x t y t U d xz -0.34 0.31 0.05 -0.01 2.1 d yz -0.34 0.05 0.31 -0.01 2.1 R i − R j ± e x ± e y ± e x ± e y (cid:15) t t t t U d xy -0.55 0.26 0.15 0.06 0.05 0.6 R i − R j ± e x , ± e y ± e x ± e y ± e x ± e y ± e x ± e y TABLE I: Real-space tight-binding parameterization of the t g hopping part and cRPAvalues for the Coulomb repulsion U of the model. The corresponding vectors between sites i and j are listed as R i − R j . All values are in eV.We are turning now to the exchange interactions in the t g space, which are given forthe xz character of nearest neighbors in x − direction by J ( xz, x ) = 4 t x /U xz = 0 .
18 eV. Dueto symmetry, they are the same for the yz character in y − direction ( J ( yz, y ) ), but further J iα,jα elements with α ∈ { xz, yz } are negligibly small. Here, we used hopping parameters t λ , which were obtained from a tight-binding fit of the t g manifold, and on-site interactions U calculated within the constrained random phase approximation (cRPA) , both shownin Table I. For the xy character we derive similarly J ( xy, = 4 t /U xy = 0 .
47 eV. However,due to the extended nature of the xy component, the next-nearest neighbor contribution J ( xy, = 4 t /U xy = 0 .
16 eV is rather large and only longer-ranged J i xy,j xy can be ignored.It should be noted that if rescaled by the aforementioned factor ∼ , these parameters are28n good agreement with values extracted from magnon dispersions using resonant inelasticx-ray spectroscopy .If we assume that we only have to retain these diagonal terms, the three components of J are modeled differently. The xz and yz components are each described by a one-dimensionalHeisenberg chain in x - and y -direction respectively, and hence order antiferromagnetically.However, the α = xy component is described by the J - J -Heisenberg model on the squarelattice with J /J ∼ /
3. This model is known to have a quantum disordered singletground state at zero temperature for 0 . . J /J . .
6, where quantum fluctuations preventlong-range ordering of the spins into an antiferromagnetic configuration . With the ratiocalculated above, in the present case the model is close to this state, which means thatat finite temperatures small thermal fluctuations can be sufficient to destroy any putativeantiferromagnetic zero temperature ground state and lead to a spin-disordered ground stateinstead.Therefore, the d xy spin components align along an external magnetic field, whereas thequasi-1D AF ordered spin components are less susceptible and give a weak response. As aresult, the xy -component of the magnetization density follows the applied field and leads toa predominant xy -character as seen in the measurements.In the reasoning above, the k -dependent nature of the effective Wannier states is essential.If one assumes a given fixed composition of the j eff orbital instead and formulates pseudospin-1 / , but theapplication of a magnetic field does not change its composition and cannot explain thepredominant xy -character of the magnetization density. Interestingly, ab initio calculations(DFT+SOC) revealed that the overall ( k -averaged) t g orbital composition of such a local j eff = 1 / √ : √ : √ and, if anything, has a smaller contribution of the xy -orbital . One should also note thatthe magnetic field strength of 5 T is too small to explain why the WFM moment of a local j eff state does not interlock with the octahedral rotation, but follows the direction of appliedmagnetic field instead. To capture the magnetization density distribution, it is thereforenecessary to extend the standard description in terms of a static t g composition of the29 eff = 1 / k -dependent Wannier description ϕ k ,α as proposed here. ∗ [email protected] † [email protected] F. Ye, S. Chi, B. C. Chakoumakos, J. A. Fernandez-Baca, T. Qi, and G. Cao, Physical ReviewB , 140406(R) (2013). C. Dhital, T. Hogan, Z. Yamani, C. de la Cruz, X. Chen, S. Khadka, Z. Ren, and S. D. Wilson,Physical Review B , 144405 (2013). N. H. Sung, H. Gretarsson, D. Proepper, J. Porras, M. Le Tacon, A. V. Boris, B. Keimer, andB. J. Kim, Philosophical Magazine , 413 (2016). A. Gukasov, A. Goujon, J.-L. Meuriot, C. Person, G. Exil, and G. Koskas, Physica B: CondensedMatter , 131 (2007). A. Gukasov, S. Rodrigues, J.-L. Meuriot, T. Robillard, A. Sazonov, B. Gillon, A. Laverdunt,F. Prunes, and F. Coneggo, Physics Procedia , 150 (2013). J. Akimitsu, H. Ichikawa, N. Eguchi, T. Miyano, M. Nishi, and K. Kakurai, Journal of thePhysical Society of Japan , 3475 (2001). M. Moretti Sala, S. Boseggia, D. F. McMorrow, and G. Monaco, Physical Review Letters ,026403 (2014). L. Fruchter, D. Colson, and V. Brouet, Journal of Physics: Condensed Matter , 126003(2016). M. Nauman, Y. Hong, T. Hussain, M. S. Seo, S. Y. Park, N. Lee, Y. J. Choi, W. Kang, andY. Jo, Physical Review B , 155102 (2017). J. Porras, J. Bertinshaw, H. Liu, G. Khaliullin, N. H. Sung, J.-W. Kim, S. Francoual, P. Steffens,G. Deng, M. Moretti Sala, A. Efimenko, A. Said, D. Casa, X. Huang, T. Gog, J. Kim, B. Keimer,and B. J. Kim, Physical Review B , 085125 (2019). S. Agrestini, C.-Y. Kuo, M. Moretti Sala, Z. Hu, D. Kasinathan, K.-T. Ko, P. Glatzel, M. Rossi,J.-D. Cafun, K. O. Kvashnina, A. Matsumoto, T. Takayama, H. Takagi, L. H. Tjeng, and M. W.Haverkort, Physical Review B , 205123 (2017). N. K. Hansen and P. Coppens, Acta Crystallographica Section A , 909 (1978). P. Coppens,
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