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## 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
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[Magnetic behavior of the Re-based double perovskite ](https://mdr.nims.go.jp/datasets/d6b1082b-90b8-49dc-8ae0-1b324e106c95)
## Fulltext
Supplemental Material for −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, JapanSI. CRYSTAL STRUCTUREIn order to assign the correct space group symmetry for Sr2ZnReO6 we tested all the space groups expected for an ordereddouble perovskite structure, monoclinic and tetragonal. Combined analysis of both monoclinic and tetragonal model yield moreaccurate results than using either model individually, with monoclinic and tetragonal phase dominant at low and high temperature,respectively. We attempted to analyze the present diffraction data obtained at low temperatures with two monoclinic structuralmodels, i.e. I2/m (no. 12; Glazer notation a0b−b− [S1]) and P21/n (no. 14; Glazer notation a+a−a− [S1]). At 150 K, there islittle difference between the results of P21/n model (Rwp = 2.7%, Rp = 2.0%, χ2 = 3.10) and that of I2/m model (Rwp = 2.6%,Rp = 1.9%, χ2 = 3.02). It should be pointed out that P21/n is the space group most frequently found for the symmetry of themonoclinic perovskites. Therefore, we concluded that Sr2ZnReO6 has the structure with P21/n at low temperature. At roomtemperature, the best-fit refinement of the dominant phase was achieved with I4/m space group (Glazer notation a0b0c− [S1]).The refined structural parameters are shown in Tab. S1. The conventional unit cell of Sr2ZnReO6 at 150 K and room temperatureis shown in Fig. S1.The temperature dependence of the lattice parameters of Sr2ZnReO6 is shown in Fig. S2(a-c) while the temperature evolutionof selected diffraction peaks is shown in Fig. S2(d).TABLE S1. Crystallographic parameters for Sr2ZnReO6 at 150 K and room temperature [298(2) K]. From left to right: Atomic labels, Wyckoffsymbols (“Wyck.”), Site occupancies (“Occ.”), fractional atomic coordinates (x/a, y/b, z/c), and isotropic thermal parameters (Biso).monoclinic at 150 K: spacegroup P21/n (no. 14) tetragonal at RT: spacegroup I4/m (no. 87)Atom Wyck. Occ. x/a y/a z/a Biso (Å2)a Atom Wyck. Occ. x/a y/a z/a Biso (Å2)aSr 4e 1.0 0.989(1) 0.0131(8) 0.2508(5) 0.20(4) Sr 4d 1.0 0 0.5 0.25 0.56(3)Zn 2c 1.0 0.5 0 0.5 0.01(5) Zn 2a 1.0 0 0 0 0.06(6)Re 2d 1.0 0.5 0 0 0.27(2) Re 2b 1.0 0 0 0.5 0.31(2)O1 4e 1.0 0.044(3) 0.538(5) 0.253(2) 0.3 O1 4e 1.0 0 0 0.256(1) 0.1(1)O2 4e 1.0 0.703(3) 0.278(3) 0.00(3) 0.3 O2 8h 1.0 0.286(2) 0.232(1) 0 0.1(1)O3 4e 1.0 0.224(3) 0.200(3) 0.00(3) 0.3a = 5.61949(5) Å, b = 5.58271(5) Å, c = 7.89335(7) Å a = 5.57423(2) Å, b = a, c = 7.9925(7) Åβ = 90◦; V = 247.630(4) Å3 β = 90◦; V = 248.553(3) Å3Rwp = 2.7%, Rp = 2.0%; χ2 = 3.10 Rwp = 2.39%, Rp = 1.7%; χ2 = 5.69Note: Additional occupancy refinements at 150 K yielded 1.000(8) Zn+ 0.000 Re on the Zn site and 0.999(5) Re+ 0.001 Zn on the Re site. Oxygen occupanciesrefined to g(O1) = 0.984(17), g(O2) = 1.05(16), and g(O3) = 1.05(16). These results indicate no resolvable antisite disorder or oxygen non-stoichiometry withinthe experimental uncertainty (≤ 2–3 %).a Biso of oxygen atoms were refined together with constraint to yield same value∗ yamaura.kazunari@nims.go.jp† s.sanna@unibo.ithttps://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.it2FIG. S1. The crystal structure of Sr2ZnReO6 is depicted in (a) the monoclinic phase at 150 K and (b) the tetragonal phase at room temperature,showing ReO6 and ZnO6 octahedra. Green, black, cyan, and red balls denote Sr, Zn, Re, and O, respectively.FIG. S2. (a–c) Temperature dependence of the lattice parameters and unit cell volume for both tetragonal and monoclinic Sr2ZnReO6. Thelowercase letters “t” and “m” indicate the parameters for the tetragonal and monoclinic phases, respectively. Error bars are omitted for clarity, asthey are smaller than the markers. (d) An enlarged view of a section of the synchrotron XRD pattern of Sr2ZnReO6. Measurements were firsttaken at 100 K and then during the heating process up to room temperature [298(2) K].TABLE S2. Comparison of single- and two-phase Rietveld refinements of Sr2ZnReO6 over identical data ranges.T (K) Model Phases Npar Rwp (%) Rp (%)150 Single P21/n 62 3.50 2.30150 Two P21/n + I4/m 78 2.70 2.00Phase fraction (monoclinic) at 150 K (two-phase): fm = 0.87(3)298 Single I4/m 60 3.06 1.94298 Two I4/m + P21/n 75 2.39 1.70Phase fraction (tetragonal) at 298 K (two-phase): ft = 0.76(4)Note: Npar denotes the number of refined (free) parameters in the Rietveld model. All refinements used the same instrument profile and background functions. Fitsinitialized from different structural and peak-shape parameters converged to the same final solution, confirming refinement robustness.3FIG. S3. Antisite–sensitive region of the synchrotron XRD pattern of Sr2ZnReO6 at T = 150 K and λ = 0.65298 Å. (a) Refinement withfull Zn/Re order (P21/n): observed data (black symbols), calculated profile (red line), and difference (blue line). (b) Test refinement with fullZn/Re disorder, showing a pronounced misfit in intensity and peak shape for the (101)/(1̄01)/(011) features near 2θ ≈ 8.2◦. The comparisondemonstrates the refinement sensitivity to Zn/Re ordering.SII. SUSCEPTIBILITY AND MAGNETIZATION DATAAs shown in Figure S4 both the real [Fig. S4(a)] and imaginary [Fig. S4(b)] components of the ac susceptibility data revealsharp peaks at ∼13 K, consistent with TM observed for the dc susceptibility data. Furthermore, the inset of Figure S4(a) shows thefrequency independent nature of the observed peaks, which hints of the absence of any glassy magnetic transition.FIG. S4. Temperature dependence of the (a) real [χ′] and (b) imaginary [χ′′] component of the ac susceptibility of Sr2ZnReO6with ac field ofHac = 5 Oe and at 2, 7, 110, 300 and 500 Hz frequency. The inset show the enlarged view of the real component of the ac susceptibility below35 K.4FIG. S5. Law-of-approach-to-saturation (LAS) fit to the high-field region of the M(H) curve for Sr2ZnReO6 at T = 2 K. The open green circlesrepresent the experimental data and the red solid line shows the fitted curve using M(H) = MS(1−B/H2)+χH. The fit yields MS = 0.0566 µB/Re,slightly higher than the measured value at 70 kOe. The LAS model provides a qualitative estimate of MS but is not reliable for extracting K instrongly anisotropic systems.TABLE S3. Curie–Weiss (CW) fits to χ(T ) of Sr2ZnReO6 for different temperature windows. Units: χ0 in 10−4 emu Oe−1 mol−1-Re; C in emuOe−1 K mol−1-Re; ΘCW in K; µeff in µB/Re.Window (K) χ0 ΘCW (K) C / µeff100–280 −1.37 ± 0.03 −20(1) 0.072(1) / 0.758(5)120–280 −1.31 ± 0.03 −24(1) 0.073(1) / 0.764(1)150–280 −1.33 ± 0.06 −25(4) 0.074(2) / 0.769(4)Note: µeff =√8C µB/Re. Fits include χ0.FIG. S6. Curie–Weiss (CW) fits to the magnetic susceptibility of Sr2ZnReO6 over three temperature intervals: (a) 100–280 K, (b) 120–280K, and (c) 150–280 K. Circles denote experimental data and solid lines represent CW fits including a temperature-independent term χ0. Theresulting χ0, Curie constant (C), Weiss temperature (ΘCW), and effective moment (µeff) are summarized in Table S3. The close similarity ofparameters across different fitting ranges demonstrates the robustness of the Curie–Weiss analysis.5Sensitivity to Re valence impurities. A dilute fraction x of impurity moments modifies the effective moment as µ2eff ≈(1 − x)µ26+ + x µ2imp, where µ6+ is the moment for Re6+ sites and µimp for the impurity valence. Using µ6+ ≈ 0.76 µB/Re fromTable S3: (i) for Re5+ (5d2, µimp≈2.83 µB), keeping µeff within ∼0.01 µB of the measured value requires x ≲ 0.3%; (ii) for Re7+(5d0, µimp≈0), a ∼0.01 µB decrease would need x ≳ 2.5%. Thus, the CW fits are incompatible with any sizable Re5+/Re7+ fraction;our synthesis/stoichiometry analysis agrees with this bound.FIG. S7. Normalized Fourier transform of the ZF-µSR time spectra measured at T = 1.6 K [see the main manuscript Fig. 3]. The inset is azoom view of Fourier transform below 100 mT.SIII. µSR DATAFigure S7 shows the Fourier-transform of the ZF-µSR asymmetry data at T = 1.6 K used to obtain the probability distributionof the muon local field described in Sec. SIV.Figure S8 shows the temperature dependence of extracted parameters from fit using Eq. (1) described in the main manuscript.The µSR spectra were also measured under longitudinal fields (LF) of µ0HLF = 10, 20, 50, 100, and 200 mT collected at T =1.6 K shown in Fig. S9. With increasing LF, a field of about 200 mT is enough to have a constant asymmetry (full recovery of themuon tail). Since this value is about 10 times the value of the internal field Bµ, the magnetism has static character [S2].SIV. DIPOLAR SIMULATIONS AND CANTED SPIN MODELSThe total local field at the muon site is approximated by the following contributions: B ≈ Bdip + BLor, where Bdip given by;Bdip =µ04πNm∑j=1[3r j(m · r j)r5j−mjr3j], (S1)is the long range dipolar field contribution which is obtained here in real space using the Lorentz method for Nm magneticmoments within the Lorentz sphere and BLor is the Lorentz contribution of magnetic moments outside the Lorentz sphere [S3].Fit procedure: For each magnetic structure, to fit the magnitude of computed local field (B) at N = 48 (at the eight muonpositions including at their symmetry equivalent positions i.e 48 positions in all) muon sites to the experimental distribution, firstthe computed local field distribution as a function of Re moments is approximated as:p(B,mRe) =N∑i=1δ(B − mReBi). (S2)Such that the Fourier-transform distribution of the ZF-µSR experimental spectra at T = 1.6 K can be approximated as a convolutionwith Gaussian broadening g define as:6FIG. S8. Temperature dependence of fit parameters obtain from the fit of ZF-µSR function Eq. (1) described in the main manuscript.FIG. S9. LF dependence of µSR spectra at 1.6 K.p̃(B,mRe, σ) = (p ∗ g)(B) :=∫ ∞−∞p(τ,m)g(B − τ, σ)dτ. (S3)yielding the fitting function ρρ(B; m, σ, A) = Ap̃(B,m, σ) + Abg, (S4)where m, σ, A are the magnitude of the Re ordered moment, width and amplitude of the distribution respectively. Abg is a constantbackground to offset the zero of the distribution.7FIG. S10. (a) Comparison between experimental ZF-µSR field distribution measured at T = 1.6 K (black line) and fit of the calculated localfield distribution ρ(B) using the proposed magnetic structure (solid colored lines) for the undistorted (solid magenta colored line) and the resultsof points (i)–(iii). The inset shows the calculated fit parameters. The vertical ticks of same color is the calculated local fields at the muon sitesusing the magnetic moment obtained from the fit. (b) Radial displacements of the Sr, Zn, Re and O atoms from their equilibrium positions as afunction of their distances from a representative muon site AII in Sr2ZnReO6 (see Table I of the main text).A. Sensitivity of the calculated magnetic momentIn Fig. S10 we show the calculated ρ(B) in the undistorted monoclinic lattice and it sensitivity against the following conditions:(i) the contact hyperfine field contribution, for which we assumed values corresponding to 10% of the computed dipolarcontribution for each muon site in the monoclinic lattice;(ii) the change in lattice parameters as a function of temperature, where, based on x-ray diffraction results [Fig. S2], weassumed a maximum variation of 1% in the monoclinic lattice (δa). between 150 K and 1.6 K; and(iii) the effects of the DFT computed muon-induced distortions on the monoclinic lattice.The results of points (i)–(iii) are presented in the Fig. S10(a), which includes the representative fits and a table containing thekey fit parameters. As expected, the computed magnetic moments are robust and vary only slightly with respect to the quantitiesconsidered. The most significant effect arises from the muon-induced distortion with obtained moment of 0.193 µB, as shown inthe inset of the figure. As shown in the Fig. S10, the implantation of the muon does not lead to significant distortion of the hostcarrying magnetic ion Re6+ from their equilibrium position, as the maximum displacement remains below 0.1 Å [Fig. S10(b)]which is the most important for the reported fit. Overall, this check confirms that the estimation of the ordered moment to be≲0.2 µB.SV. FORCE-THEOREM IN HUBBARD-IA. Calculation of Intersite Exchange InteractionsWe evaluate the intersite exchange interactions (IEI) in Sr2ZnReO6 using the force-theorem in the Hubbard-I (FT-HI) methodderived in Ref. [S4] as publicly available in the GitHub repository [S5]. We incorporate both the electron-mediated IEI and themonoclinic crystal field contribution. The IEI Hamiltonian describing the interactions between multipolar moments with a definedtotal angular momentum jeff = 3/2, is expressed asH =∑⟨i j⟩VQQ′KK′ (i j)OQK(i)OQ′K′ ( j), (S5)where OQK(i) is the multipolar operator acting on the jeff-multiplet on site i of the rank K = 0...2J and projection Q = −K...K [S6].The first summation (i j) runs over the Re-Re bonds and VQQ′KK′ (i j) represent the corresponding IEI.The FT-HI method is based on a converged electronic paramagnetic structure calculation in the quasi atomic (Hubbard-I)approximation. At its core, it considers small symmetry-breaking fluctuations of the density matrix of the ground-state multiplet8occurring simultaneously at two interacting (neighboring) magnetic (Re) sites, i and j. The IEI VQQ′KK′ (i j) is then computedby analyzing the response of the DFT+DMFT grand potential to these two-site fluctuations. The approach shares conceptualsimilarities with other force-theorem approaches developed for symmetry-broken magnetic states [S7, S8], but is designed for thesymmetry-unbroken paramagnetic regime, accessible in DFT+DMFT. In the case of Sr2ZnReO6, we follow closely previousapplication to other jeff = 3/2 double perovskites [S9, S10]. Further details are provided in the Supplemental Material ofRef. [S11], while the full derivation is available in Ref. [S4].B. Crystal Field in Je f f = 3/2 basisWe report the crystal-field matrix HMM′ for the Re sites, as computed in the global reference frame and expressed in the|J = 3/2,mJ⟩ basis (see Table S4) followingHMM′ =∑Γ⟨JM|Γ⟩EΓ⟨Γ|JM′⟩ (S6)where |Γ⟩ are the ground state multiplet eigenstates of the atomic Hamiltonian. Because of the monoclinic symmetry of thelattice, we do not rotate the basis into a local octahedral frame: the tilt of the apical oxygen atoms prevents the definition of anorthonormal Cartesian frame aligned with the oxygens in the ab plane.The strong mixing between states with different mJ values arises not only from the use of the global reference frame, but alsofrom significant hybridization between the Jeff=3/2 ground-state manifold and the excited Jeff=1/2 states, as consequence of thelarge crystal-field splitting of the multiplet.M Real Imaginary-3/2 44.4561 -67.9360 -15.2325 0 0 29.8418 1.3137 0-1/2 -67.9360 129.1068 0 -15.2324 -29.8418 0 0 1.31371/2 -15.2325 0 129.1062 67.9358 -1.3137 0 0 -29.84193/2 0 -15.2324 67.9358 44.4572 0 -1.3137 29.8419 0TABLE S4. Crystal Field matrix of Re in the monoclinic crystal field of Sr2ZnReO6C. Origin of the canted AFM phaseTo identify which multipolar channels drive the canted antiferromagnetic phase, we performed a series of mean-field (MF)calculations in which specific intersite exchange interactions were selectively switched off.The results are summarized in Table S5. They show that the canting angle originates from a cooperative interplay among alltime-odd multipolar couplings—dipole–dipole, quadrupole–quadrupole, dipole–octupole, and octupole–octupole. The correctcanting angle and magnetic ground state cannot be reproduced by any single interaction channel alone, but only when all areincluded. This behavior is consistent with findings for other 5d1 double perovskites such as Ba2NaOsO6 [S11].Canting Angle vs Active IEIIEI: All dd dd + qq do + oo oo qqϕ (degrees) 55 28 28 20 20 no dipolesTABLE S5. Canting angle (ϕ) dependence on IEI inSr2ZnReO6. The different interactions are reported as follows: Dipole-Dipole (dd),Dipole-Octupole (do), Quadrupole-Quadrupole (qq) and Octupole-Octupole (oo)D. Canting angle dependence vs UWe investigated the robustness of the canted antiferromagnetic phase by tuning the U parameter in DFT+HI steps of 0.25 eVfrom U = 3.0 eV to U = 3.5 eV. The results are shown in Table S6. We find for each value the same canted AFM configuration,with similar values of the canting angle. The small difference can be a consequence of the precision of the MF. We recall, that thevalues of the magnetic moments are very small ∼ 0.06µB, and therefore, small variation of components are expected.9Canting Angle vs UU (eV) 3.0 3.25 3.5ϕ (degrees) 56 55 60TABLE S6. Canting angle (ϕ) dependence on IEI in Sr2ZnReO6. The different interactions are reported as follows: Dipole-Dipole (dd),Dipole-Octupole (do), Quadrupole-Quadrupole (qq) and Octupole-Octupole (oo)[S1] P. M. Woodward, Octahedral Tilting in Perovskites. I. Geometrical Considerations, Acta Crystallographica Section B 53, 32 (1997).[S2] A. Yaouanc and P. Dalmas de Réotier, Muon Spin Rotation, Relaxation, and Resonance: Applications to Condensed Matter (OxfordUniversity Press, Oxford, 2011).[S3] P. Bonfà, I. J. Onuorah, and R. D. Renzi, Introduction and a Quick Look at MUESR, the Magnetic Structure and mUon Embedding SiteRefinement Suite, in Proceedings of the 14th International Conference on Muon Spin Rotation, Relaxation and Resonance (µSR2017),Vol. 21 (2018).[S4] L. V. Pourovskii, Two-site fluctuations and multipolar intersite exchange interactions in strongly correlated systems, Phys. Rev. B 94,115117 (2016).[S5] L. V. Pourovskii and D. Fiore Mosca, MagInt, https://github.com/MagInteract/MagInt.[S6] P. Santini, S. Carretta, G. Amoretti, R. Caciuffo, N. Magnani, and G. H. Lander, Multipolar interactions in f -electron systems: Theparadigm of actinide dioxides, Rev. Mod. Phys. 81, 807 (2009).[S7] A. Liechtenstein, M. Katsnelson, V. Antropov, and V. Gubanov, Local spin density functional approach to the theory of exchangeinteractions in ferromagnetic metals and alloys , Journal of Magnetism and Magnetic Materials 67, 65 (1987).[S8] M. I. Katsnelson and A. I. Lichtenstein, First-principles calculations of magnetic interactions in correlated systems, Phys. Rev. B 61,8906 (2000).[S9] D. Fiore Mosca, C. Franchini, and L. V. Pourovskii, Interplay of superexchange and vibronic effects in the hidden order of Ba2MgReO6from first principles, Phys. Rev. B 110, L201101 (2024).[S10] D. Fiore Mosca and L. V. Pourovskii, Antiferro octupolar order in the 5d1 double perovskite Sr2MgReO6 and its spectroscopic signatures,Phys. Rev. Res. 7, L032016 (2025).[S11] D. Fiore Mosca, L. V. Pourovskii, B. H. Kim, P. Liu, S. Sanna, F. Boscherini, S. Khmelevskyi, and C. Franchini, Interplay betweenmultipolar spin interactions, Jahn-Teller effect, and electronic correlation in a Jeff =32 insulator, Phys. Rev. B 103, 104401 (2021).https://doi.org/10.1107/S0108768196010713https://doi.org/10.7566/JPSCP.21.011052https://doi.org/10.1103/PhysRevB.94.115117https://doi.org/10.1103/PhysRevB.94.115117https://github.com/MagInteract/MagInthttps://doi.org/10.1103/RevModPhys.81.807https://doi.org/10.1016/0304-8853(87)90721-9https://doi.org/10.1103/PhysRevB.61.8906https://doi.org/10.1103/PhysRevB.61.8906https://doi.org/10.1103/PhysRevB.110.L201101https://doi.org/10.1103/tvp5-mpy9https://doi.org/10.1103/PhysRevB.103.104401 Supplemental Material for - Magnetic behavior of Re-based double perovskite Crystal structure Susceptibility and magnetization data data Dipolar simulations and canted spin models Sensitivity of the calculated magnetic moment Force-Theorem in Hubbard-I Calculation of Intersite Exchange Interactions Crystal Field in Jeff = 3/2 basis Origin of the canted AFM phase Canting angle dependence vs U References