# Fileset [BJ14949_main.pdf](https://mdr.nims.go.jp/filesets/2a8c4a05-3f94-4cc6-93a9-c1492de7aca4/download) ## Creator Muhammad Maikudi Isah, Biswajit Dalal, [Xun Kang](https://orcid.org/0000-0003-4364-6218), Dario Fiore Mosca, Ifeanyi John Onuorah, Valerio Scagnoli, Pietro Bonfà, Roberto De Renzi, [Alexei A. Belik](https://orcid.org/0000-0001-9031-2355), Cesare Franchini, [Kazunari Yamaura](https://orcid.org/0000-0003-0390-8244), Samuele Sanna ## Rights [Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/) ## Other metadata [Magnetic behavior of the

5 d^{1}

Re-based double perovskite

{Sr}_{2} {ZnReO}_{6}

](https://mdr.nims.go.jp/datasets/d6b1082b-90b8-49dc-8ae0-1b324e106c95) ## Fulltext Magnetic behavior of 5d1 Re-based double perovskite Sr2ZnReO6Muhammad Maikudi Isah ,1 Biswajit Dalal ,2, 3 Xun Kang,2 Dario Fiore Mosca ,4Ifeanyi John Onuorah ,5 Valerio Scagnoli ,6, 7 Pietro Bonfà ,5 Roberto De Renzi ,5Alexei A. Belik ,2 Cesare Franchini,4 Kazunari Yamaura ,2, 8, ∗ and Samuele Sanna 1, †1Dipartimento di Fisica e Astronomia “A. Righi”, Universitá di Bologna, I-40127 Bologna, Italy2Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan3Department of Physics, Achhruram Memorial College, Jhalda, Purulia, West Bengal, 723202, India4University of Vienna, Faculty of Physics and Center for Computational Materials Science, Vienna, Austria5Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Universitá di Parma, I-43124 Parma, Italy6Laboratory for Mesoscopic Systems, Department of Materials, ETH Zürich, Zürich, Switzerland7PSI Center for Neutron and Muon Sciences, 5232 Villigen PSI, Switzerland8Graduate School of Chemical Sciences and Engineering, Hokkaido University,North 10 West 8, Kita-ku, Sapporo, Hokkaido 060-0810, Japan(Dated: February 13, 2026)The subtle interplay between spin-orbit coupling, exchange interactions, and cation ordering can lead to exoticmagnetic states in transition-metal ions. We report a comprehensive study of the Re-based (5d1) ordered doubleperovskite oxide Sr2ZnReO6 combining synchrotron x-ray diffraction (XRD), magnetic susceptibility, muon spinrelaxation (µSR) measurements, and density functional theory (DFT) calculations. XRD reveals that Sr2ZnReO6crystallizes in the monoclinic structure (space group P21/n) at low temperature. Magnetic susceptibility dataindicate a transition below ∼ 13 K, with M–H loops showing ferromagnetic-like hysteresis and an unusuallyhigh coercive field of 23 kOe at 2 K. Zero-field µSR measurements detect static and spatially disordered internalfields below TM ≃ 12 K, consistent with a canted antiferromagnetic ground state determined by detailed DFTand force-theorem in Hubbard-I calculations. The reduced high-temperature effective moment (∼0.76 µB) andvery small static moment (≲0.2 µB) derived from µSR analysis and local-field simulations indicate a decisiverole of spin-orbit coupling. Through a combined experimental and computational approach we unambiguouslydetermine the canted antiferromagnetic order in Sr2ZnReO6, showing that a very small ordered moment coexistswith an exceptionally large coercivity. These results underscore the crucial role of spin–orbit coupling and orbitalordering, providing new insights into magnetism in 5d1 double perovskites.I. INTRODUCTIONThe magnetic and orbital ordering in strongly correlated ma-terials plays a crucial role in modern condensed matter physics.In particular, transition-metal ions give rise to a wealth of novelphysics properties through the complex interplay of charge,orbital, and spin degrees of freedom [1]. Among the mostextensively studied systems are the B-site ordered double per-ovskite (DP) oxides, with general formula A2BB′O6, where Ais an alkaline-earth or rare-earth cation and B/B′ are transition-metal ions in different oxidation states. These materials havegarnered significant interest owing to the influence of spin-orbitcoupling (SOC), electronic correlations and crystal field inter-actions on the electronic and magnetic properties that drive theemergence of exotic quantum phases [2–4] such as the Mott in-sulators [5, 6], Weyl semimetals [7–9], half metallicity [10–13]and quantum spin liquids [14].Generally, the B-site ordered DPs have been reported to crys-tallize in the cubic, monoclinic and tetragonal structures andconsist of ordered geometry of corner-shared BO6 and B′O6octahedra network, alternatively forming two interpenetratingface-centered (fcc) lattices, and the A site positioned at thevoids between the octahedra [15]. The combination of the B∗ yamaura.kazunari@nims.go.jp† s.sanna@unibo.itand B′ sites and the hosting order of the magnetic cations atthese sites are crucial determinants of the electronic and mag-netic properties, and the nature of the exchange interactions inthese compounds. When B and B′ sites host magnetic ions, theproperties are driven by the B−O−B′ mediated super-exchangeinteraction. However, when magnetic ion resides only on theB′ site, these compounds often become Mott insulators, sincethe large separation between neighbouring B′ sites reduces thebandwidth and makes the on-site Coulomb repulsion domi-nant [16]. In such a case, the magnetic interactions are definedby edge-shared network of tetrahedra in a fcc lattice and oftenexhibit geometric frustration in the presence of antiferromag-netic exchange couplings [17–19]. Further, theoretical studieson analogous DP compounds subjected to strong SOC withonly B′ magnetic site hosting either 4dn or 5dn electronic statepredict the possibility to realize a number of magnetic states in-cluding magnetocrystalline anisotropic aligned ferromagnetic(FM) and antiferromagnetic (AFM) states, spin nematic phases,multipolar orders [2, 3, 20] and also canted spin states that arestabilized by Jahn-Teller distortions, owing to the interplay ofHeisenberg, non-Heisenberg interactions, and the quadrupolarcouplings [21].In this work, we focus on the Re-based DP Sr2ZnReO6,where Sr2+, non-magnetic Zn2+, and magnetic Re6+ (5d1) ionsoccupy the A, B and B′ sites respectively. Sr2ZnReO6 belongsto a class of materials where experimental measurements re-vealed a wide variety of structural, magnetic and electronicproperties. As mentioned in Ref. [22], these class of materi-https://orcid.org/0000-0002-6615-2977https://orcid.org/0000-0001-8607-485Xhttps://orcid.org/0000-0003-2496-0455https://orcid.org/0000-0001-6358-303https://orcid.org/0000-0002-8116-8870https://orcid.org/0000-0001-6358-3037https://orcid.org/0000-0002-5015-0061https://orcid.org/0000-0001-9031-2355https://orcid.org/0000-0003-0390-8244https://orcid.org/0000-0002-4077-5076mailto:yamaura.kazunari@nims.go.jpmailto:s.sanna@unibo.it2als have no direct correlation between the magnetic groundstate and crystal symmetry. For example, Ba2ZnReO6 exhibitscanted ferromagnetic order [23], Ba2MgReO6 hosts multipolarorder [24], and Ba2YReO6 displays a spin disordered groundstate [25], all within the cubic Fm3̄m structure. The diversebehaviors observed in cubic compounds naturally motivateinvestigations of tetragonally elongated systems, which havedrawn even greater interest due to their unusual magnetic prop-erties. Initially, the isoelectronic compounds Sr2CaReO6 andSr2MgReO6 were suggested to host spin-glass state below ≈14K and ≈50 K, respectively [18, 26], but recently resonant x-raydiffraction experiments on a high-quality Sr2MgReO6 sample,have proposed a layered antiferromagnetic order at tempera-tures below ≈55 K with a propagation vector k = (0, 0, 1) [27].Early theoretical studies predicted that these structures couldexhibit antiferromagnetic ordering of magnetic octupoles [28],a phenomenon that has evaded direct experimental confirma-tion. More recently, theory has pointed to Sr2MgReO6 as aspecific candidate for realizing such ordering [29]. Much lessattention has been devoted to understanding the magnetic be-havior of Sr2ZnReO6, including the role of SOC in stabilizingits ground state. Only the existence of contrasting antiferro-magnetic features [15] and weak ferromagnetic transition [30]were detected at low temperature from magnetization mea-surements, and no magnetic order was detected by neutrondiffraction measurements [30]. The unusually small magneticmoment is probably a consequence of strong SOC in 5d1 DPs,combined with effects such as covalency, orbital ordering, andJahn-Teller distortions, all of which contribute to partial can-cellation between spin and orbital magnetism [16].In this paper, we take advantage of high-quality crystallinesamples to revise the crystallographic structure of Sr2ZnReO6using synchrotron x-ray diffraction. We then investigate themagnetization dynamics and magnetic structure through bulkmagnetization measurements and muon spin rotation and relax-ation (µSR), the latter being a highly sensitive local probe ofmagnetism on the atomic scale and ideally suited to detect thevery small magnetic moments predicted in this compound. Fi-nally, we complement our experimental findings with DFT+µand force-theorem in Hubbard-I calculations to identify andvalidate the nature of the magnetic order.The remainder of this paper is organized as follows; InSec. II we present both the experimental and computationalmethods that have been utilized in this work followed by thedescription of the obtained crystal structure from synchrotronXRD in Sec. III A. In Sec. III B, the magnetization measure-ments data are presented while the µSR measurements togetherwith the DFT calculations of the muon sites are presented inSec. III C and Sec. III C 1, respectively. The identification ofthe magnetic structure based on force-theorem in Hubbard-I approach and the validation of the structure by the muonlocal field simulation is presented in Sec. III C 2. The sum-mary is presented in Sec. IV. More details on data analysis arepresented in the supplemental material (SM) [31].II. EXPERIMENTAL AND COMPUTATIONAL METHODSPolycrystalline Sr2ZnReO6 was synthesized via solid-statereaction using SrO (prepared from 99.9% pure SrCO3, WakoPure Chemical Industries, by heating at 1300 ◦C in oxygen),ZnO (99.9%, Wako Pure Chemical Industries), and ReO3 (syn-thesized in the lab from 99.99% pure Re, Rare Metallic Co.Ltd.). The powders were mixed in an agate mortar inside anargon-filled glovebox to ensure precise stoichiometry. Themixture was sealed in a platinum capsule and subjected toisotropic compression in a multi-anvil press (CTF-MA1500P,C&T Factory, Tokyo) at 6 GPa and 1100 ◦C for 1 hour, withan 11-minute ramp to reach the target temperature. After heat-ing, the capsule was rapidly quenched to below 100 ◦C in 1minute, followed by slow pressure release over several hours.The resulting product was a dense, polycrystalline black pellet.A portion was ground for phase identification using a Mini-Flex600 x-ray diffractometer (Rigaku, Tokyo) with Cu-Kα ra-diation, and further analyzed by synchrotron XRD with a largeDebye-Scherrer camera at BL02B2, SPring-8, Japan [32, 33].The x-ray wavelength was λ = 0.65298 Å, calibrated using aCeO2 standard. Patterns were analyzed and visualized usingthe RIETAN-VENUS software package [34, 35].The direct current (dc) magnetic susceptibility (χ)of Sr2ZnReO6 was measured using a magnetometer (MPMS,Quantum Design, San Diego, CA, USA). The empty sampleholder’s magnetization was measured to account for its dia-magnetic contribution. Measurements were taken from 2−280K with a 10 kOe magnetic field under both zero-field-cooled(ZFC) and field-cooled (FC) conditions. Isothermal magneti-zation loops were recorded at various temperatures with themagnetic field swept between −70 kOe and +70 kOe underZFC conditions. Data reproducibility was confirmed by testingmultiple samples from different synthesis runs.The zero-field (ZF) µSR measurements were carried out onthe GPS spectrometer at the Paul Scherrer Institut (Switzer-land) as a function of temperature, ranging from 1.6 K toabout 70 K. In a typical ZF-µSR experiment a beam of posi-tive muons 100% spin-polarized along the beam direction isimplanted in the sample [36]. The positive muons thermalizeat interstitial sites where they act as sensitive probes of devel-opment of spontaneous magnetic ordering, precessing in thelocal magnetic field, Bµ, at the Larmor frequency ωµ = γµBµ(γµ = 2π × 135.5 MHz/T). By studying the angular distribu-tion of the positrons emitted during the muon decay process(muon lifetime τµ ≈ 2.2 µs) we measure the time evolutionof the muon-spin asymmetry A(t) = A0Pz(t), where A0 is theinitial muon asymmetry and Pz(t) the time dependent muonpolarization. The µSR experimental data were analyzed vialeast-squares optimization using MUSRFIT software [37] andMULAB, a home-built MATLAB suite.In order to reliably interpret the µSR results, it is cru-cial to accurately determine the muon stopping site, whichwe have done via DFT calculations within the DFT+µ ap-proach [38–41]. Non-spin polarized DFT calculations wereperformed using the plane wave (PW) based code QuantumESPRESSO [42]. The Perdew-Burke-Ernzerhof (PBE) [43]functional was used to estimate the exchange and correlation3term. The muon and the host atoms were modeled with ultra-soft pseudopotentials [44, 45] using 100 Ry and 900 Ry cutoffenergy for the wavefunctions and charge density respectively.The muon in the DFT+µ procedure was treated as a hydro-gen impurity in a charged 2 × 2 × 2 supercell comprising of160 host atoms and 1 muon. A 2 × 2 × 2 Monkhorst-Packk-point mesh [46] was used for the Brillouin zone integration.The structural relaxations were carried out until forces andtotal energy differences were less than 1 mRy/Bohr and 0.1mRy, respectively. All calculations were performed keepingfixed the experimental lattice parameters (monoclinic phase)reported in Sec. III A below. To resolve the magnetic structureof Sr2ZnReO6, additional calculation for local magnetic fieldsat the muon-stopping sites were performed (see SM [31]).The magnetic order was determined by calculating the inter-site exchange interactions (IEI) for the general low-energy ef-fective many-body Hamiltonian coupling multipolar momentswith total angular momentum Jeff = 3/2 and subject to a mon-oclinic crystal field (see SM [31]). The calculation involvesseveral steps. To start, the paramagnetic electronic structureof Sr2ZnReO6 is determined using the charge self-consistentdensity functional theory with dynamical mean-field theory(DFT+DMFT) [47–50] in the quasi-atomic Hubbard-I (HI) ap-proximation [51]. The DFT calculations were performed usingthe full-potential LAPW method implemented in Wien2k [52],with the SOC effect included via the standard variational treat-ment. The local density approximation (LDA) was used forthe DFT exchange-correlation potential, together with a 500k-point mesh in the full Brillouin zone and a basis cutoff ofRmtKmax = 7. The fully localized limit was adopted for thedouble-counting correction, assuming the nominal 5d shelloccupancy of 1. The Wannier orbitals representing the Re dstates were constructed from the d Kohn-Sham bands withinthe energy window [−1.36 : 5.44] eV relative to the KS Fermilevel and the full d-shell parameters were set to F0 = U = 3.2eV and JH = 0.5 eV, consistent with previous studies on d1and d2 DPs [53–55]. The calculations were performed withfixed (monoclinic) experimental lattice parameters as reportedin Sec. III A. Our DFT+HI calculation correctly reproducethe effective atomic level scheme, finding a t2g − eg crystalfield splitting of ∼4.22 eV and a SO splitting of ∼0.5 eV. TheJeff = 3/2 ground state multiplet is further split by the mono-clinic crystal field in two doublets separated by ≈0.17 eV withthe corresponding crystal field matrix, reported in the SM, ex-hibiting a strong mixing between different mJ components dueto both the monoclinic distortion and the hybridization withexcited Jeff=1/2 states. Since theoretical studies predict thatmultipolar ordering plays a central role in 5d1 DPs, the multi-polar IEI was extracted using the force-theorem in Hubbard-I(FT-HI) approach described in Ref. [56] via the open-sourceMagInt code [57]. This framework allows the computationof multipolar IEI for general lattice structures with multiplecorrelated sites and crystal field environments (see SM [31]).All quantities were evaluated in the global reference frame asdefined in Fig. 1(b).FIG. 1. (a) Synchrotron XRD pattern of Sr2ZnReO6 at room temper-ature [298(2) K]. The observed pattern is indicated by red crosses, thecalculated pattern by the green solid line, and the difference profileby the blue line. Vertical bars mark the Bragg reflection positionsfor the tetragonal (I4/m; first green ticks) and monoclinic (P21/n;second violet ticks) models. Inset: DSC heat-flow curves on heat-ing (red) and cooling (blue) measured at 10 K min−1, showing anendothermic/exothermic pair near 300 K with thermal hysteresis. (b)Crystal structure of monoclinic Sr2ZnReO6 showing the network ofthe Zn/ReO6 octahedra. The three inequivalent oxygen atoms arerepresented with spheres of different colors: O1 (red), O2 (green) andO3 (blue).III. RESULTS AND DISCUSSIONA. Crystal structureThe synchrotron powder x-ray diffraction patterns showedno detectable impurity peaks, indicating that the samplescontained only minimal impurities, if any. Previous stud-ies [15, 30] reported a tetragonal phase with the presence of2.6% to 15% of the monoclinic Sr2ZnReO6 phase at room tem-perature, likely due to variations in synthesis conditions. There-fore, we analyzed the diffractograms of the samples across atemperature range of 100 K to 300 K, using a model whichtakes into account both tetragonal (space group I4/m; no. 87)and monoclinic (space group P21/n; no. 14) phases. As il-lustrated in Fig. 1(a), the combined analysis of both modelsyielded more accurate results than using either model individu-4FIG. 2. (a) Temperature dependence of ZFC (open dot) and FC (solid red line) dc magnetic susceptibility (χ(T ); main axes) and inverse of ZFCDC magnetic susceptibility with the CW fit (solid line) ((χ − χ0)−1(T ); inset axes) in an applied magnetic field (H = 10 kOe). (b) Isothermalmagnetization (M) loop as a function of magnetic field (H) ranging from -70 to +70 kOe under ZFC mode at six different temperatures. (c)Temperature dependence of coercive field (HC ; left axes) and saturation magnetization (MS ; right axes) obtained from the ZFC M(H) loops atH = 70 kOe. The solid lines are guide to the eye unless otherwise stated.ally. Within our experimental uncertainty, no antisite disorderor significant metal/oxygen non-stoichiometry is detected (seeTable S1 note and Fig. S2 of the SM [31]). Therefore, inthe final analysis, the site occupancy of all atoms was fixedas fully occupied. Figure 1(b) shows the crystal structure ofthe monoclinic phase including the network of corner-sharedZn/ReO6 octahedra, while the quantitative details of the latticeparameters and atomic positions of both phases are tabulatedin Table S1 in the SM [31].Having established the structural framework and refinedatomic positions, we now turn to the temperature dependenceof the crystallographic phases. At 100 K, the monoclinic phasewas dominant, comprising 87% of the sample. In contrast,at 300 K, the tetragonal phase became dominant, although24% of the sample volume is in the monoclinic phase. Thephase fraction changed most markedly above 200 K duringheating, but the transition remained incomplete even at 300K. The evolution of the phase fractions shows a steep changenear 225 K superimposed on a broad transformation regimeextending to ≈300 K, consistent with a single, first-order-liketransition exhibiting extended coexistence rather than multipletransitions. The extended coexistence of tetragonal and mon-oclinic phases up to 300 K, together with the endothermic /exothermic pair of DSC exhibiting thermal hysteresis [insetto Fig. 1(a)], indicates a structural transition similar to firstorder. The ≈ 10% tetragonal remnant at low T originates fromhysteresis of the broad first-order-like transition. Magnetiza-tion data–collected on cooling–show no extraneous featuresattributable to this minority phase. We therefore refer to thetransition as first-order-like below. Our results indicate that thistransition is coupled with changes in the degree of tilting andbuckling of the bonds between the octahedrally coordinatedReO6 and ZnO6 units, in agreement with neutron diffractionstudies [30]. The thermal evolution of lattice parameters andsynchrotron XRD patterns (see Fig. S2 of the SM [31]) furthersupports the occurrence of this first-order structural transitionover a broad temperature range.B. Magnetic susceptibility and magnetizationFigure 2(a) presents the temperature dependence of the mag-netic susceptibility (χ vs T ) in an applied magnetic field of 10kOe measured on a Sr2ZnReO6 powder sample between 2 Kand 280 K. The ZFC curve shows a peak centered at Tp ∼ 10K; while the FC curve reveals a sharp increase in χ at TM ∼ 13K, implying that the transition at TM (Tp) has a magnetic origin.This can be further corroborated by the dip in the dχ/dT vsT curve at around 13 K (not shown here). In addition, the fre-quency independent nature of the ac susceptibility peaks at thetransition (see Fig. S4 of the SM [31]) indicates the absenceof a glassy magnetic state. The inverse magnetic suscepti-bility, was fitted [inset to Fig. 2(a)] in the high-temperaturerange (T > 100 K) by assuming a χ0 temperature-independentcontribution due to core diamagnetism and Van Vleck param-agnetism, plus a Curie-Weiss (CW) temperature dependence:χ = χ0+C/(T − ΘCW), where C is the Curie constant and ΘCWis the CW temperature. The fit yields C = 0.072(1) emu Oe−1K/mol, and ΘCW = −20(1) K.The paramagnetic effective moment, µeff =√8CµB =0.758(5) µB/Re is smaller than the calculated value, µcal =g√S (S + 1)µB = 1.732 µB, in the spin only (S = 1/2) limitand assuming g = 2. This discrepancy hints at the likelyroles played by the neglected orbital moment and the effects ofstrong SOC in this compound, as obtainable in several 5d1 DPs.For instance, in the cubic structured Ba2NaOsO6, the Jeff = 3/2quartet ground state has been established and a corrected g-factor arising from the effect of hybridization with the ligands,has been utilized to compute the paramagnetic effective mo-ment [58]. The extent of the hybridization is represented bya scale factor γ, that reduces the effective orbital momentumfrom the ideal Leff = −1 such that 2S +Leff , 0 and the g-factoris g = 2(1 − γ)/3. On this basis, we expect a γ value of 0.41in comparison to γ = 0.536 for Ba2NaOsO6 [58] and γ = 0.49for Ba2MgReO6 [59] to explain µeff ∼0.76 µB for Sr2ZnReO6.To further elucidate the magnetic behavior of Sr2ZnReO6,5FIG. 3. (a) ZF-µSR asymmetry time spectra at varying temperatures above and below TM . The right panel is a zoom for short acquisition timewindow. The data are fitted with the function shown in Eq. (1). The temperature dependence of the fitted (b) depolarization rate (longitudinal)λL1 and (c) the internal field Bµ of the oscillatory component, consistent with the magnetic order transition below TM . The solid red line are fitto the phenomenological function described in the text.we recorded isothermal field-dependent magnetization M(H)curves under ZFC conditions. Figure 2(b) shows the M(H)curves of Sr2ZnReO6 at selected temperatures. Notably, mag-netization curves at low temperatures (T ≲ 20 K) displayunexpected large hysteresis loops (coercive field ∼23 kOe at2 K) indicating the presence of a hard FM-like character withlow magnetic moment, whereas at T ≳ 20 K, it shows a PMbehavior. The coercive field (HC) is order of magnitude higherthan previously reported [30]. On the other hand, the negativesign of ΘCW suggests an AFM interaction which appears incon-sistent with the observed FM behavior. But then, similar neg-ative ΘCW has been measured in ferromagnetic Ba2MgReO6,Ba2ZnReO6 [22] and Ba2NaOsO6 [5, 60] compounds, andis attributed to the impact of orbital ordering in the CW be-haviour of 5d1 DPs, able to stabilize canted ferromagnetic andnoncollinear antiferromagnetic orders [16].In figure 2(c), we plot temperature dependence of coercivefield HC (left axes) and saturation magnetization MS (rightaxes). The maximum value obtained for HC is 23 kOe at 2K. HC clearly reveals two slopes as a function of temperature.The slope change occurs at about 13 K, corresponding to acoercivity smaller than the value obtained at 2 K by a factorof ∼5. As temperature increases, coercivity slowly decreasesreaching a minimum value of 0.36 kOe at 30 K. This maysuggest that there is no ordered magnetic phase present fortemperatures T ≳ 30 K. The M(H) curves remain unsaturatedeven at 70 kOe as shown in Fig. 2(b), and very small saturatedmagnetic moment ∼0.05 µB/Re at 2 K is obtained [Fig. 2(c)].To reconcile the unusually large HC with the tiny moment,we note that strong spin–orbit coupling of Re6+ (5d1) in anoncubic crystal field can produce substantial magnetocrys-talline anisotropy; microstructural pinning from strain/defectsmay also contribute. To estimate MS, we fitted the high–fieldregion of the 2 K isotherm with the Law of Approach toSaturation (LAS), M(H) = MS(1 − B/H2)+ χH, obtainingMS = 0.0566 µB/Re (see Fig. S5 of the SM [31]), consistentwith the ∼ 0.05 µB/Re value at 70 kOe. Because LAS fitsare not reliable for extracting precise anisotropy constants instrongly anisotropic magnets, we refrain from quoting K here.Additional diagnostics (e.g., Arrott/Arrott–Noakes analysisand SEM/crystallite-size estimates) will be pursued to separateintrinsic anisotropy from pinning effects. The residual slopein M(H) at high field arises from the intrinsic high–field sus-ceptibility of the monoclinic phase rather than a paramagneticimpurity. The minor tetragonal fraction detected by XRD re-mains paramagnetic and contributes negligibly to the overallmagnetization.C. µSRIn order to further probe the magnetic behaviorof Sr2ZnReO6, ZF-µSR measurements were performed. Thetime evolution of muon asymmetry is shown in Fig. 3(a) forrepresentative temperatures below and above the magnetictransition. A strongly depolarized fraction and spontaneousoscillations associated with the static ordered Re magneticmoment appears below the magnetic transition. The muonasymmetry data, A(t), were modeled by the following func-tion (solid lines in in Fig. 3) in the whole temperature range(χ2 ≈ 1.1):A(t) = AT[e−σ2T1 t22 + η cos(γµBµt)e−σ2T2 t22]++ AL[e−(λL1t)β+ ηe−λL2t] + Abkge−λbkgt.(1)The model takes into account two main muon site fractions(the two terms in the square brackets) with fixed ratio η throughthe all T range plus an additional constant contribution, Abkgwith paramagnetic character. At low T the magnetic transi-tion is captured by considering that each of the two fractionsdevelops a transverse component respect to the initial direc-tion of the muon spin Sµ (AT , Bµ ⊥ Sµ) and the longitudinalcomponent (AL, Bµ∥Sµ). The transverse component displaysa non-oscillatory Gaussian signal AT (reflecting overdampedoscillations) plus a single oscillatory damped fraction ηAT withGaussian depolarization function. The longitudinal componentrequires a fraction AL with a stretched decay function (with astretched exponential β = 0.5 temperature independent) plusa simple exponential decay ηAL. The stretched component6accounts for a multisite muon population with a distributionof depolarization rates, i.e. of correlation times, that cannotbe resolved [61]. The best global fit by keeping AT + AL con-stant yields to η = 0.24(1) consistently for both longitudinaland transverse components through the whole T range. Thisreflects a muon site population of about 80% for the site withthe longitudinal stretched and highly depolarized transversecontribution, and about 20% for the other one. Below themagnetic transition the transverse and longitudinal amplitudes(see Fig. S8 of the SM [31]) weights respectively nearly 2/3 and1/3 of the sum AT +AL, as expected for a powder sample with afull magnetic volume with static ordered moments. Above themagnetic transition only the AL component remains recoveringthe full amplitude. The static character of the magnetic state isalso confirmed by longitudinal fields (LF) µSR measurements(reported in section SIII of the SM [31]).The model also includes a background contribution whichturns out to have a constant asymmetry Abkg and a small con-stant decay rate of 0.05 µs−1 which accounts for the presenceof a paramagnetic phase and a possible additional small muonsignal coming from the cryostat and sample holder (the lat-ter being typically 1-2% of the total amplitude on GPS). Theglobal fit returns a Abkg amplitude of about 12% of the totalsignal, a value compatible with the tetragonal paramagneticfraction measured by XRD.The estimated temperature dependence of depolarizationrate λL1 and the internal magnetic field Bµ is presentedin Fig. 3(b-c) (for completeness the behavior of the other freeparameters of Eq. 1 is reported in section SIII of the SM [31]).As the temperature approaches TM a sharp increase in λL1 is ob-served, reflecting the critical fluctuations expected at the mag-netic transition. The internal magnetic field Bµ [see Fig. 3(c)]displays a very good agreement with the phenomenologicalexpression:Bµ(T ) = Bµ(0)[1 −(TT µM)α]δ(2)where Bµ(0) is the zero-temperature internal magnetic field, αis an empirical parameter controlling the saturation behav-ior, and δ is the critical exponent-like parameter [62, 63].The solid red line in Fig. 3(c) shows that Bµ(T ) follows anorder-parameter-like behavior according to Eq. 2 below TMwith α = 1.6(3) and δ = 0.38(4). The α value is small(close to the Bloch’s T 3/2 law) and suggests the presence offerromagnetic-like order [64], while the δ value agrees withthe expected δ ∼ 0.367 for a three-dimensional Heisenbergferromagnet [65]. We also estimate a slightly lower transi-tion temperature T µM = 12 K (≃ TM) if compared to the esti-mate analysis of our magnetization data, consistent with thezero-field limit probed by µSR. The internal magnetic field isBµ(0) = 17.4(3) mT, corresponding to a ground-state frequencyof νµ(0) = 2.35(4) MHz. This value is consistent with the peakof the field distribution revealed by the Fourier transform ofthe transverse component signal, discussed in the next section.TABLE I. Candidate muon stopping sites in Sr2ZnReO6 at 4e Wyckoffsite symmetries. First and second columns contain the site labels. Thepositions in fractional coordinates are reported in the third column.The fourth column indicates the energy difference with respect to thelowest energy site AI.Sites Label Fractionalcoord.∆E(meV)µ–O1AI (0.068, 0.382, 0.792) 0AII (0.409, 0.103, 0.707) 106AIII (0.853, 0.359, 0.806) 118AIV (0.213, 0.302, 0.246) 149µ–O2BI (0.716, 0.806, 0.592) 24BII (0.616, 0.698, 0.501) 73BIII (0.175, 0.043, 0.519) 128BIV (0.801, 0.308, 0.151) 175µ–O3CI (0.804, 0.103, 0.485) 32CII (0.713, 0.209, 0.592) 48CIII (0.164, 0.301, 0.850) 144CIV (0.051, 0.170, 0.993) 199FIG. 4. Zn/ReO6 (gray/cyan sphere) octahedra showing the muonstopping sites (pink spheres) bonded to the three distinct oxygen sites(red for O1, green for O2 and blue for O3). The µ–Oi label describesthe grouping of inequivalent muons bonded to a distinct O site (seeTable I). Plots are reproduced using the VESTA program [34].1. Muon sitesTo further analyze the ZF-µSR asymmetry spectra andvalidate the magnetic structure, the muon-stopping sitesin Sr2ZnReO6 were identified using the standard DFT+µ pro-tocol. The results reveal twelve candidate crystal symmetryinequivalent muon positions located in the 4e Wyckoff positionin the unit cell. In each case, the muon site forms a bond withan oxygen atom, with a µ–O bond length of ≈1 Å, as shownin Fig. 4. The positions (Table I) are grouped into µ–O1, µ–O2and µ–O3 sites with respect to the three distinct oxygen. Therelative energy differences among all sites are < 0.2 eV, indicat-ing that all candidate sites are likely to be occupied, even as themuon zero-point motion energy (typically around 0.5 eV [40])is not taken into account. Indeed, the multiple muon sites iden-tified for this compound account for the broad distribution ofthe local field in the low temperature µSR signal, rather than asingle field, consistent with the stretched multisite componentof Eq. 1. The form of this distribution includes local fieldsthat vary for the symmetry inequivalent muons closer to each7of the three distinct O2− ion and equivalent muons closer toeither the magnetic Re6+ ion or the nonmagnetic Zn2+ ion inthe octahedra network.FIG. 5. (a) Schematic illustration of the magnetic configurationshowing the canting angle ϕ with spins aligned along (011) plane andthe proposed magnetic structure from the FT-HI calculations withcanting angle ϕ = 55◦ between the two magnetic sublattices, labeledRe1 (blue plane) and Re2 (green plane). (b) Comparison betweenthe experimental ZF-µSR field distribution measured at T = 1.6 K(black line) and the calculated local field distribution ρ(B). The fitsare shown for the proposed pristine magnetic structure (“undistorted”;solid magenta line) and for the case including the effects of latticedistortions to the magnetic ions when evaluating the field at the muonsite (‘distorted”; solid yellow line). The vertical ticks of same coloris the calculated local fields at the muon sites using the magneticmoment obtained from the fit.2. Magnetic structure: FT-HI and muon local field simulationsTo identify the ground-state magnetic structure, the low-energy effective Hamiltonian was solved in the presence ofmonoclinic crystal field within the single site mean-field ap-proximation, relying on the IEI calculated from the the FT-HI approach, as described in Sec II. These calculations wereachieved using the “McPhase” package [66] together with anin-house module. Due to the quasi-atomic approximation em-ployed, the resulting gyromagnetic factor is gJ = 0 because ofthe exact compensation of spin and orbital moments, whichdoes not occur in Sr2ZnReO6. While covalency effects areimplicitly included in the DFT+HI calculation via the Wannierorbitals, calculating the orbital magnetic moment from themis a non trivial procedure [67] and lies beyond the scope ofthis work. To compute the magnetic moments, we incorpo-rated the covalency effects through the γ = 1 − 3gJ/2 = 0.41factor described previously in Sec. III B. The mean-field re-sult reveal a single second-order phase transition at ∼30 Ktowards a canted antiferromagnetic phase with propagationvector k = (0, 0, 1) and net magnetic moment of ≈0.04 µB, invery good agreement with the magnetization measurementsfrom Fig. 2(b). The overestimation of TN (in comparison to12 K of experiment), well known for mean-field approachesis consistent with the approximation used and with previousstudies [54, 55]. In Fig. 5(a), we show the calculated magneticorder including the schematic description of the observed cant-ing angle ϕ between the two Re sublattices. ϕ = 0◦ is the FMlimit, ϕ = 90◦ is the AFM limit, 0◦ < ϕ < 45◦ angles are closerto FM and thus are defined as the canted FM orders, whilethose within 45◦ < ϕ < 90◦ correspond to canted AFM orders.The obtained magnetic configuration consists of spin aligneddominantly in the (011) plane and vanishing contribution alongx direction, with magnetic moments of magnitude ≈0.06 µBand canting angle ϕ ∼ 55◦, suggesting the existence of a cantedAFM order. The result was validated against variations of theon-site interaction U, confirming the robustness of the solution(see SM [31]).To elucidate the microscopic origin of the canted AFMphase, we carried out MF calculations in which individ-ual IEI channels were selectively deactivated. The resultsshow that the experimentally observed magnetic structureemerges from the cooperative action of all active time-oddmultipolar couplings—dipole–dipole, dipole–octupole, and oc-tupole–octupole (see SM [31]), as previously found in other5d1 DP [53]. A quantitative analysis of the role of SOC andcrystal-field effects, left for a future study, is expected to pro-vide additional insights into the origin of this magnetic phase.To validate the calculated magnetic order and the magnitudeof the magnetic moment with the ZF-µSR asymmetry spec-tra, we perform local field simulations at the muon-stoppingsites [68–71]. The local field for a magnetic configuration iscalculated using the expression B ≈ Bdip +BLor, where the twoterms are the dipole field from the magnetic Re6+ ions and theLorentz field respectively [72]. The contact hyperfine contribu-tion arising from the overlap between the muon wavefunctionand the magnetic Re6+ ions is neglected, as vanishing valuesare expected since the muon positions are relatively far fromthe magnetic Re6+, which also likely host a weak moment. Thesimulated local field distribution ρ(B) for the obtained cantedAFM order is then fitted to the measured field distribution i.e.the Fourier transform power spectrum in the ZF-µSR at T = 1.6K. The fitting is discussed in more details in the SM [31]. Theresult is shown in Fig. 5(b), and the simulated field distributionwith the proposed canted AFM structure captures the overallfeatures of the experimental data including the peak in the fielddistribution ρ(B) with R2 ∼ 0.91, signifying the goodness ofthe fit, validating the obtained magnetic order. A broad dis-tribution of the local field ranging from ∼2 mT to ∼33 mT isobtained and assigned to the multiple muon sites. The staticmagnetic moment obtained from the fit is very small, 0.222(5)µB/Re for the pristine Sr2ZnReO6 structure (“undistorted”) andslightly reduced to 0.193(3) µB/Re when muon-induced latticedistortions are included (“distorted”) [Fig. 5(b)]. Taking intoaccount systematic errors in our analysis and the neglectedcontact contribution to the local field at the muon sites, a morereliable estimate for the moment at Re is ≲0.2 µB. This is stilllarger than the HI estimate, possibly because the hybridizationeffects are not explicitly taken into account in our calculations,which are known to be strong in this class of materials [73].As a consequence, it is possible that the effective moment dueto hybridization is different and this can affect both DFT+HIestimates as well as DFT+µ results through e.g. oxygen polar-ization.8In all, the obtained canted AFM order is in agreement withthe magnetization measurements and reconciles the observedvarying FM and AFM signatures in the experimental data. Thecoexistence of both the large coercive field ∼23 kOe at 2 K,indicative of FM-like behaviour with low magnetic momentand the negative sign of ΘCW suggesting an AFM interaction isattributed to the impact of orbital ordering in the CW behaviourof 5d1 DPs [16], able to stabilize the obtained canted AFMorder.IV. SUMMARYIn summary, we have investigated the crystal structure andmagnetic properties of polycrystalline Sr2ZnReO6. The crystalstructure at room temperature is defined in a tetragonal unitcell (space group I4/m) while at low temperature, Sr2ZnReO6undergoes a reduction in symmetry to monoclinic (P21/n)phase. The magnetic susceptibility data show a sharp increaseindicative of a magnetic phase transition below 13 K which iscorroborated by ZF-µSR measurements where divergence inthe temperature dependence of the relaxation rates has beenobserved at TM≃12 K, below which the existence of static butspatially disordered internal magnetic field, revealed throughthe presence of multiple muon sites has been measured. M–Hcurves indicate an unusually large hysteresis loop with coer-cive field of 23 KOe at 2 K indicative of a FM-like behavior.On the contrary the fit of the magnetic susceptibility abovethe magnetic transition show a negative Curie-Wiess constant,ΘCW = −20 K indicating AFM correlations.A canted AFM order was obtained from DFT and force-theorem in Hubbard-I calculations and validated by the sim-ulation of the muon experimental data. The obtained mag-netic configuration consists of spin aligned dominantly in the(011) plane and vanishing contribution along x direction, withmagnetic moments of magnitude ≈0.06 µB and canting angleϕ ∼ 55◦. This magnetic order reconciles the hysteretic FM-likebehaviour and the AFM character observed with the negativesign of Curie-Wiess temperature.A small static ordered magnetic moment, of the order of0.2 µB/Re, was obtained from the analysis of the muon data,which neglects hybridization effects and can be considered anupper limit. This is further reduced with respect to the paramag-netic effective moment of 0.76 µB obtained from Curie-Wiessfitting of the magnetic susceptibility measurements, and con-sistent with reduced moment obtainable in 5d1 DPs owing toeffect of SOC. To account for the reduced effective momentwith respect to the spin only 1.73 µB value, the Jeff = 3/2quartet-like ground state has been assumed together with co-valency effects on the cation orbital moment, underlining theimpact of SOC in this compound. These results provide evi-dence for the magnetic ground state of Sr2ZnReO6 and guide tothe theoretical description of the strong SOC effects envisagedfor this and similar compounds.ACKNOWLEDGMENTSThe work here presented is partly supported by project“Spin-charge-lattice coupling in relativistic Mott insulators”(ID No. 202243JHMW, CUP J53D23001350006), funded byEuropean Union–Next Generation EU project–“PNRR–M4C2,investimento 1.1–Fondo PRIN 2022”.This work was also par-tially supported by a Grant-in-Aid for Scientific Research (No.JP25K01657) from the Japan Society for the Promotion ofScience. 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Agrestini, F. Borgatti, P. Florio, J. Frassineti, D. Fiore Mosca,Q. Faure, B. Detlefs, C. J. Sahle, S. Francoual, J. Choi,M. Garcia-Fernandez, K.-J. Zhou, V. F. Mitrović, P. M. Wood-ward, G. Ghiringhelli, C. Franchini, F. Boscherini, S. Sanna, andM. Moretti Sala, Origin of Magnetism in a Supposedly Nonmag-netic Osmium Oxide, Phys. Rev. Lett. 133, 066501 (2024).https://doi.org/10.7566/JPSCP.21.011052https://doi.org/10.7566/JPSCP.21.011052https://doi.org/10.1103/PhysRevLett.133.066501 Magnetic behavior of Re-based double perovskite Abstract Introduction Experimental and Computational Methods Results and Discussion Crystal structure Magnetic susceptibility and magnetization Muon sites Magnetic structure: FT-HI and muon local field simulations Summary Acknowledgments References