Spin-orbit coupling control of anisotropy, ground state and frustration in 5d² Sr₂MgOsO₆

Abstract

The influence of spin-orbit coupling (SOC) on the physical properties of the 5d² system Sr₂MgOsO₆ is probed via a combination of magnetometry, specific heat measurements, elastic and inelastic neutron scattering and density functional theory calculations. Although a significant degree of frustration is expected, we find that Sr₂MgOsO₆ orders in a type I antiferromagnetic structure at the remarkably high temperature of 108 K. The measurements presented allow for the first accurate quantification of the size of the magnetic moment in a 5d² system of 0.60(2) μ_B–a significantly reduced moment from the expected value for such a system. Furthermore, significant anisotropy is identified via a spin excitation gap and we confirm by first principles calculations that SOC not only provides the magnetocrystalline anisotropy, but also plays a crucial role in determining both the ground state magnetic order and the size of the local moment in this compound. Through comparison to Sr₂ScOsO₆, it is demonstrated that SOC-induced anisotropy has the ability to relieve frustration in 5d² systems relative to their 5d³ counterparts, providing an explanation of the high T_N found in Sr₂MgOsO₆.

Introduction

There is a great deal of interest in materials with inherently frustrated magnetic exchange interactions and the resulting unusual magnetic ground states^1,2. One class of materials currently under investigation in this context is the double perovskites, formula A₂BB′O₆, where A and B are non-magnetic cations and B′ is a magnetic 4d or 5d transition metal cation³. The resulting network of B′ cations forms a quasi-face-centered cubic (fcc) type lattice where antiferromagnetic interactions are highly frustrated as each B′ cation has twelve nearest neighbor B′ cations³. Particularly complex magnetic states can arise in frustrated systems due to the presence of significant spin-orbit coupling (SOC) that typically accompanies cations of the 4d and 5d transition metals. It’s imperative to understand the role SOC plays in such materials due to its role in lifting the orbital degeneracy and enhancing multipolar interactions, leading to rich phase diagrams^4,5,6,7.

In the case of double perovskites containing a single magnetic cation with a d³ electronic configuration, the relatively large S = 3/2 spins are expected to have a quenched orbital contribution resulting in classical behavior⁶. While experimental results indicate that SOC is allowed in the d³ configuration for 4d or 5d cations, the reduction in the magnetic moment due to SOC is small relative to the effects of covalency, which is large for cations with high oxidation states^8,9. Double perovskites with 4d³ or 5d³ ions tend to exhibit frustration indexes (|Θ|/T_N) ranging from 4 to 14 and adopt a type I antiferromagnetic structure upon ordering^{8,10,11,12,13}, although incommensurate behavior has also been reported^14,15.

In the case of the much smaller S = 1/2 d¹ configuration, SOC is expected to play a significant role and antiferromagnetic, ferromagnetic and quadrupolar order have all been predicted as a result^5,16. These predictions appear to be in line with measurements of antiferromagnetic order in Ba₂LiOsO₆ and ferromagnetic order in Ba₂NaOsO₆^17,18. However, there are additional examples such as spin glass Sr₂MgReO₆¹⁹ and spin singlet Ba₂YMoO₆^20,21 which do not easily fit into this framework. Investigation of magnetism due to the d⁴ configuration has also begun, such as in A₂BIrO₆ (A = Sr, Ba; B = Sc, In, Y), where questions have arisen concerning the strength of SOC and the magnetism of the resulting ground state^22,23,24,25.

In this work, we investigate the influence of SOC on the magnetic state for the intermediate 5d² S = 1 configuration, by studying the double perovskite Sr₂MgOsO₆^26,27,28. Theoretical work⁶ has predicted a rich phase diagram with seven different phases/regions for the present 5d² S = 1 scenario, however there has been difficulty in sorting known materials in this context. Ba₂YReO₆, Sr₂YReO₆ and Ca₂MgOsO₆ appear to be spin glasses, which have not been predicted^28,29,30, La₂LiReO₆ and Sr₂InReO₆ host non-predicted spin singlet states^29,30, while μSR experiments indicate that cubic Ba₂CaOsO₆ orders antiferromagnetically, though with moments too small for detection in the reported neutron scattering experiment³¹. By combining magnetization, specific heat and neutron scattering measurements with first principles calculations for Sr₂MgOsO_6, we are able to provide insight into the nature of frustration and influence of SOC in this material.

Sr₂MgOsO₆ orders at 108 K with a type I antiferromagnetic structure, shown in Fig. 1, a results that was previously predicted in a model considering the influence of SOC⁶. The Os⁶⁺ moments of 0.60(2) μ_B are significantly reduced from the 2 μ_B spin only value expected for an S = 1 ion. Density functional theory (DFT) confirms that this substantial reduction in moment occurs through a combination of both covalency and SOC and furthermore predicts that SOC-induced anisotropy is essential in the selection of the magnetic ground state. The presence of this anisotropy is experimentally confirmed by the observation of a spin gap in the magnetic excitation spectrum via inelastic neutron scattering. Sr₂MgOsO₆ is therefore a rare example of a compound where the Néel order, rather than just the anisotropy, is set by the spin-orbit interaction. Furthermore, we find that SOC-induced anisotropy is responsible for the reduced magnetic frustration in Sr₂MgOsO₆ relative to d³ double perovskites, therefore explaining the enhanced T_N in Sr₂MgOsO₆.

Figure 1

The crystal and magnetic structure of Sr₂MgOsO₆ with Mg and Os shown as grey and red spheres located within octahedra of the same color with O ions positioned at the corners.

Sr cations are omitted for clarity. The magnetic moment on Os⁶⁺ is shown as a black arrow and while shown along the b axis, is known only to be in the a-b plane.

Experimental

Powder samples of Sr₂MgOsO₆ were synthesized by grinding stoichiometric amounts of SrO₂, MgO, Os and OsO₂ together using a mortar and pestle according to the following chemical equation:

Ground mixtures of up to 3 g were contained in high-density alumina tubes and sealed in evacuated silica ampoules (approximate volume 40 mL with 3 mm thick walls) for heatings of 48 hours at 1000 °C in a box furnace located within a fumehood. This was followed by regrinding and identical reheating for an additional two cycles. Larger sample sizes were produced by synthesizing multiple aliquots which would be ground together in the intermittent grindings and redistributed for the subsequent heatings. Powdered Sr₂MgWO₆ samples were synthesized in air following the procedure outlined in the literature³².

The temperature dependence of the magnetization of Sr₂MgOsO₆ powders was measured using a Quantum Design MPMS SQUID magnetometer. Data were collected over the temperature range 2.5 to 400 K under zero-field-cooled (ZFC) and field-cooled conditions (FC) in an applied field of 10 kOe. Powders were contained in gel capsules and mounted in straws for insertion into the device for measurement. An analogous data set was collected using an empty sample mount and subtracted from the temperature dependent magnetization data of Sr₂MgOsO₆ in order to remove the background response.

Powders of Sr₂MgOsO₆ and Sr₂MgWO₆ were cold pressed and sintered at their synthesis temperatures overnight (in an evacuated ampoule for Sr₂MgOsO₆) to prepare polycrystalline pellets. The specific heat measurements were conducted on the pellets mounted with Apiezon grease using a Quantum Design PPMS instrument using a relaxation technique.

Laboratory x-ray powder diffraction measurements were conducted at room temperature on a Bruker D8 Advance equipped with a Ge (111) monochromator and a Cu radiation source. Time of flight neutron powder diffraction (NPD) measurements were conducted on Sr₂MgOsO₆ at Oak Ridge National Laboratory’s (ORNL) Spallation Neutron Source (SNS) on the POWGEN beamline³³ using a sample size of 1.359 g. Data were collected at 10, 50 and 300 K using the POWGEN Automatic Changer (PAC) environment. Separate data sets with the bank 2 and bank 7 chopper settings corresponding to respective d-spacing ranges of 0.2760–3.0906 Å and 2.2076–10.3019 Å were collected at each temperature. Data were analyzed using the Rietveld method as implemented in the GSAS EXPGUI software package^34,35. Additional NPD data were collected at High Flux Isotope Reactor (HFIR) facility at ORNL on the triple-axis spectrometer HB-1A using a sample size of 11 g. The sample was sealed under a He atmosphere into a cylindrical can made of aluminum with an inner diameter 0.6 cm. Data were collected at a constant wavelength of λ = 2.37Å using collimation of 40′-40′-40′-80′. The data were analyzed using the Rietveld refinement suite FULLPROF³⁶ and the magnetic form factor for Os⁶⁺ from ref. 37 was assumed.

Inelastic neutron scattering experiments were performed on an 11 g sample of Sr₂MgOsO₆ and on a 16.5 g sample of Sr₂ScOsO₆ that was previously examined in ref. 38. Measurements were performed on the SEQUOIA chopper spectrometer at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory (ORNL). The samples were sealed in aluminum cans and an identical empty Al can was measured as a background. A closed-cycle refrigerator was used to reach temperatures between 6 K and 125 K and measurements were performed using an incident neutron energy 20 meV. Empty-can measurements were subtracted from the data sets, which were then normalized by a factor m_f.u./m_s, where m_f.u. is the formula unit mass and m_s is the sample mass for each of Sr₂MgOsO₆ and Sr₂ScOsO₆. The presented magnetic scattering intensity is therefore per Os ion.

First principles calculations were performed using the generalized gradient approximation (GGA) of Perdew, Burke and Ernzerhof (PBE)³⁹ with the general potential linearized augmented planewave (LAPW) method⁴⁰ as implemented in the WIEN2k code⁴¹. We used the experimental 10 K crystal structure and highly converged basis sets, including local orbitals with LAPW sphere radii of 2.0 bohr for the metal atoms and 1.55 bohr for O. We did calculations both with the PBE-GGA itself and with an additional Coulomb repulsion parameter U = 3 eV in the PBE + U approach with the fully localized limit double counting. The key difference between these treatments is that metallic behavior is predicted with PBE while an insulating gap is obtained with U = 3 eV, consistent with experimental reports for the resistivity of polycrystalline samples²⁸. We focus on results that do not depend on U.

Results

Sr₂MgOsO₆ crystallizes in the tetragonal I4/m space group as previously reported^27,28 and shown in Fig. 1, which is common to a number of other Sr₂BOsO₆ compositions^{42,43,44,45,46}. The I4/m space group is associated with the a⁰a⁰c⁻ Glazer tilt system, where out of phase tilting occurs about the c-axis⁴⁷. Rietveld refinement of the x-ray powder diffraction data did not indicate any disorder between Mg and Os cations, a typical result considering the charge difference of 4+ between the anticipated oxidation states of Mg²⁺ and Os^6+ 3. Refinements also did not indicate any loss of Os during synthesis within experimental error.

The results of the refinement of neutron powder diffraction data on Sr₂MgOsO₆ from POWGEN are given in Table 1 and the refined pattern at 10 K is given in Fig. 2. The average Os−O bond length is typical of recent results for octahedrally coordinated Os⁶⁺ in the double perovskite structure^42,46,48. Both the MgO₆ and OsO₆ octahedra are slightly elongated, with two slightly longer and four slightly shorter M−O bonds. For the d² cation Os⁶⁺, this is the expected Jahn-Teller distortion. The unit cell is tetragonally distorted as compared to the cubic cell, with a c-axis which is greater than √2 of the a-axis. The tetragonal distortion is enhanced at lower temperatures, with a c/√2a ratio of 1.0068 at 300 K and 1.0206 at 10 K. The distortion manifests as a combination of enhanced tetragonal elongation of the octahedra and octahedral tilting as evidenced by a reduced Mg−O−Os bond angle. No structural phase transition or change in symmetry occurs within the temperature range studied. The short Os−O bond lengths compared to the d³ osmates reflect the contraction and increased covalency that can be anticipated as the oxidation state is increased.

Table 1 Neutron powder diffraction parameters obtained from Rietveld refinement for Sr₂MgOsO₆ at 10, 50 and 300 K.

Figure 2

Refined neutron powder diffraction pattern of Sr₂MgOsO₆ at 10 K.

Black symbols, red curves and blue curves correspond to the observed data, the calculated patterns and the difference curve, respectively. The black hashes correspond to the nuclear peak positions of the compound. The inset shows low Q neutron powder diffraction data on Sr₂MgOsO₆ from POWGEN at 300, 50 and 10 K. Arrows indicate the position of the potential magnetic reflections associated with Type I AFM order.

The temperature dependence of the magnetization of Sr₂MgOsO₆ is given in Fig. 3a. A clear cusp corresponding to an antiferromagnetic transition occurs with a maximum at 108 K in both the FC and ZFC data sets, in approximate agreement with previous reports^27,28, while the Fisher heat capacity, d(χT)/dT, shown in Fig. 3b, indicates a maximum value at 102 K– a lower T_N from this approach is typical of double perovskites³¹. A divergence of the FC and ZFC data at 15 K is noted in Fig. 3a which is absent in previous reports^27,28 despite being present in numerous independent samples and measurements by the present authors. A Curie-Weiss fit was conducted in the temperature range 250 to 400 K, which resulted in an effective moment of 1.88 μ_B, in close agreement with reported values and a Weiss constant of Θ = −269 K, in approximate agreement with reported values^27,28. The effective paramagnetic moment is substantially reduced from the theoretical spin-only result of 2.83 μ_B for S = 1, indicating that the influence of spin-orbit coupling is significant for Os⁶⁺. The ratio between the Weiss constant and ordering temperature T_N yields a relatively low frustration index, (|Θ|/T_N), of 2.5.

Figure 3

(a) The temperature dependence the field-cooled (red filled circles) and zero-field-cooled (blue open circles) magnetization under an applied field of Sr₂MgOsO₆ plotted against the left axis. The inverse data (black circles) is plotted against the right axis with a red line indicating the higher temperature Curie-Weiss fitting. (b) The Fisher heat capacity of Sr₂MgOsO₆ derived from the data in panel (a).

The specific heat of Sr₂MgOsO₆ is shown in Fig. 4, with a clear second-order type anomaly positioned at 108 K indicating that the transition is likely due to long-range antiferromagnetic order. Much like the cusp in magnetic susceptibility, this anomaly is significantly broadened, a possible indication of magnetic frustration in the system. An isostructural nonmagnetic material with similar mass, Sr₂MgWO₆, was measured to approximate the nonmagnetic lattice contributions to the specific heat. The solid line, also shown in Fig. 4(a), represents the specific heat data of Sr₂MgWO₆ scaled by mass. The difference of these two data sets, plotted as C_mag/T and shown in Fig. 4b, corresponds to the magnetic component of the specific heat in Sr₂MgOsO₆. Clearly, there is a large peak at the antiferromagnetic transition, but there is also a significant tail up to temperatures much higher than T_N, indicating persistent magnetic fluctuations. The magnetic specific heat C_mag/T is integrated to obtain the magnetic entropy, plotted against the right axis in Fig. 4b. Analysis of the magnetic entropy over the entire temperature range results in S_mag ~ 11.2 J/mol K, which is in between the theoretical values for a simple L-S scheme for a d² cation with an expected total spin J = 2 (S_mag = 13.38 J/mol K) and a spin-only scenario with S = 1 (S_mag = 9.134 J/mol K). These results clearly contrast from a similar analysis recently conducted on 3d⁸ S = 1 Sr₂NiWO₆⁴⁹, where a spin-only analysis in a nominally orbitally quenched material resulted in good agreement with the experimentally determined magnetic entropy. Therefore, we conclude that the magnetic entropy of Sr₂MgOsO₆ is significantly impacted by the orbital contribution to the magnetic moment in this compound.

Figure 4

(a) The temperature dependence of the specific heat of Sr₂MgOsO₆ (blue) and diamagnetic Sr₂MgWO₆ (red) which is used to approximate the lattice specific heat of S₂MgOsO₆. (b) The magnetic specific heat (black) of Sr₂MgOsO₆ taken by subtracting the scaled specific heat of Sr₂MgWO₆ and magnetic entropy (red) obtained by integration.

In order to obtain a microscopic insight into the ordered magnetic structure, we examined the low Q region of the neutron powder diffraction data collected on the POWGEN instrument, shown as the inset of Fig. 2. However, no apparent magnetic reflections were observed arising below the ordering temperature. The specific location of the anticipated peaks associated with the common type I antiferromagnetic order are highlighted in the inset. This is similar to the case of 5d² Ba₂CaOsO₆, where NPD data from the C₂ diffractometer at the Canadian Neutron Beam Centre at Chalk River National Laboratories did not yield any observable magnetic reflections despite evidence of long range magnetic order from muon spin relaxation experiments³¹. In that study, it was determined that the ordered moments must be less than an estimated detection limit of 0.7 μ_B per Os⁶⁺.

In order to search for the presence of weak magnetic reflections, additional neutron powder diffraction data was collected on the HB-1A beamline using an 11 g sample, nearly 10 times the mass measured on POWGEN and with the sample contained in an aluminum can to minimize incoherent scattering. HB-1A was utilized for this particular investigation because of its excellent signal-to-noise ratio, arising from the combined use of a double-bounce monochromator and an analyzer. Below T_N, two magnetic reflections previously anticipated due to type I antiferromagnetic order were observed, see Fig. 5. The diffraction pattern was analyzed using constant structural values as determined from POWGEN, but varying the background and instrument-dependent parameters, resulting in a good fit to the data, Fig. 5a. Extra peaks are visible which are due to the aluminum sample can scattering and a small, unidentified impurity phase is visible in this sample, as indicated in Fig. 5, which is present at all temperatures. The magnetic structure refinement yielded Os⁶⁺ moments of 0.60(2) μ_B, which are aligned within the a-b plane. The resulting value is just below the proposed detection limit from the case of Ba₂CaOsO₆³¹. The temperature dependence of the Q(001) = 0.78 Å⁻¹ peak is shown in Fig. 5c. A power-law curve was fit to the data to extract T_N and confirms the T_N = 108(2) K transition temperature associated with this magnetic peak. This is a remarkably high Néel temperature for a double perovskite with B = Mg, since the Os⁶⁺ ions are on a quasi-fcc lattice and are separated by more than 5.5 Å.

Figure 5

(a) Neutron powder diffraction data collected on Sr₂MgOsO₆ using the HB-1A beamline at 125 K. (b) A close view of the low Q data at 125 K (red open circles) and 6 K (blue filled circles) with a Rietveld fitting to the 6 K data (green line) as described in the text. (c) The temperature dependence of the Q(001) = 0.78 Å⁻¹ magnetic reflection with a power-law fitting to the data as described in the text.

For our DFT calculations, we considered different magnetic orders including ferromagnetic, the observed type I order and checkerboard antiferromagnetic order in the basal a-b plane of the tetragonal cell. We find that PBE-GGA calculations without SOC predict an incorrect magnetic order, specifically a ferromagnetic ground state. Only when spin-orbit coupling is included do we obtain the correct type I order as the lowest energy state. This conclusion is robust, as when spin orbit coupling is included we obtain type I order independent of the moment direction and for both PBE and PBE + U calculations. It follows that Sr₂MgOsO₆ is a rare example of a material where the Néel order itself, rather than just the anisotropy is set by the spin-orbit interaction–a reflection of the strong SOC.

Also for both PBE and PBE + U calculations with SOC we find the lowest energy spin direction to be along the tetragonal <100> direction in the tetragonal I4/m cell. The anisotropy is sizable and increases with U. For the PBE calculations we find that the <110> direction is disfavored by 1 meV/Os, while the <001> direction is disfavored by 5 meV/Os. The easy axis and plane do not depend on U. This anisotropy and the symmetry breaking due to the tetragonal lattice no doubt partly explain the relatively high ordering temperature on a dilute fcc-like lattice. The other needed ingredient in obtaining the ordering is the intersite exchange interaction. The value of this coupling depends on the choice of U and as such we cannot directly predict the precise magnitude of this coupling. However, as seen from the energy differences, regardless of the choice of U the correct ground state is predicted, i.e. there is sufficient intersite exchange coupling. We find that the ferromagnetic ordered state is 31 meV above the ground state in the PBE + U calculation and 168 meV above the ground state without U.

Turning to the moment size, based on integration within the LAPW spheres (2.0 bohr for Os) we obtain moments that are strongly reduced from the nominal values due both to covalency and SOC. The total Os moment in the PBE calculation is 0.48 μ_B consisting of a spin moment of 0.77 μ_B and an orbital moment of −0.29 μ_B. In the probably more realistic PBE + U calculation we obtain 0.57 μ_B, from a spin moment of 1.07 μ_B and an orbital moment of −0.50 μ_B, in close agreement with the experimental result of 0.60(2) μ_B from NPD. We also find sizable moments on the O ions, which is a result of strong covalency. This can be seen in the density of states, shown for the ground state with PBE + U in Fig. 6. The O 2p bands extend from −7.4 eV to −1.1 eV relative to the valence band maximum. The region from the bottom to −5.1 eV comprise O 2p–Os e_g σ bonding states and one can see very strong Os character in this region reflecting the covalency. The Os t_2g states, which are the active orbitals here, extend from −0.5 eV to + 1.8 eV and are split by U to give a gap of 0.22 eV. The t_2g band width is similar in the PBE calculation. The Os e_g states extend from 4.3 eV to 6.3 eV. The large crystal field splitting is another reflection of very strong covalency.

Figure 6

PBE + U (U = 3 eV) density of states and projection onto Os for the ground state antiferromagnetic order, including SOC.

A consequence of this covalency is the presence of sizable moments on the O sites. In the double perovskite structure each O has only one Os nearest neighbor. The O moments are parallel to those of the neighboring Os regardless of the treatment (PBE or PBE + U), the magnetic order or the inclusion of SOC. While Os⁶⁺ d orbitals can be regarded as largely inside a 2 bohr LAPW sphere, this is not the case for O²⁻ p orbitals with a 1.55 bohr sphere. In order to estimate the O contribution, we turn to the ferromagnetic case. In the PBE + U calculation the total spin moment in the unit cell is 1.93 μ_B per formula unit, while the Os spin moment is only 1.14 μ_B. Thus ~40% of the spin moment is distributed over the six neighboring O ions. This moment will be active in specific heat and susceptibility data, but not in refinements of neutron diffraction data as the moment is spread across neighboring oxide ions and bonds. The O moments provide an explanation for the sizable intersite exchange. Although the double perovskite lattice can be regarded from a Zintl perspective as touching (OsO₆)⁶⁻ anions held together by interstitial Mg²⁺ and Sr²⁺ cations, the fact that the O atoms on the exterior of these contacting polyanions carry sizable moments provides a mechanism for the intersite exchange.

Having shown that SOC has a significant influence on the moment size observed by neutron scattering, we anticipate that SOC may have a major effect on the magnetic dynamics in Sr₂MgOsO₆. Confirmation of this can be found by examining the inelastic neutron scattering spectra of both Sr₂MgOsO₆ and Sr₂ScOsO₆ shown in Fig. 7. We compare Sr₂MgOsO₆ to Sr₂ScOsO₆ because Sr₂ScOsO₆ also shows high-T_N type I AFM order with T_N = 92 K, but has Os⁵⁺ 5d³ ions which are expected to show significantly reduced SOC due to a S = 3/2 state. In both materials we observe scattering emanating from the type I antiferromagnetic wavevector at Q ≈ 0.8 Å⁻¹, Fig. 7. We identify the development of a spin gap in both materials at low temperatures—compare Fig. 7(a–c) for Sr₂MgOsO₆ and Fig. 7(b–d) for Sr₂ScOsO₆. The Sr₂MgOsO₆ spectrum at 6 K is remarkably similar to that of Sr₂ScOsO₆, in which the gap has been extensively characterized³⁸, see Fig. 7(a,b), respectively. This suggests that the physical mechanisms controlling each system are more similar than previously predicted⁶, with SOC having influence in both materials.

Figure 7

Neutron scattering intensity maps, measured with E_i = 20 meV, showing the gap below T_N for (a) Sr₂MgOsO₆ and (b) Sr₂ScOsO₆ which closes above T_N in (c,d) respectively.

The similar size of the gaps in Sr₂MgOsO₆ and Sr₂ScOsO₆ does, however, support a picture of SOC having stronger influence in Sr₂MgOsO_6. The microscopic mechanism by which SOC typically produces the spin gap is associated with either exchange anisotropy or single-ion anisotropy (or a combination)^10,15,38,50. For either mechanism, for a fixed strength of SOC the magnitude of the gap observed by neutron scattering scales with the magnetic moment size. Therefore, as Sr₂MgOsO₆ has a smaller magnetic moment, the similarity of observed gap to that in Sr₂ScOsO₆ demonstrates that SOC is stronger in Sr₂MgOsO₆ resulting in a comparable gap.

Despite the similarity in the spectra at 6 K, above T_N the intensity of the observed scattering is very different between the two compounds, see Fig. 7(c,d). To examine the temperature dependence further, we present in Fig. 8(a) the integrated intensity of the scattering for the range 0.7 < Q < 1 Å⁻¹ and 3 < E < 12 meV. For both materials the integrated intensity in this region increases with temperature, because of both the modification of the scattering due to the closing of the gap and the Bose thermal population factor. The integrated intensity per Os ion for each of the samples is similar at low temperatures, with a slightly higher value for Sr₂ScOsO₆, as expected for the larger spin system. However, as the magnetic transition temperatures are approached the integrated intensities diverge dramatically. This difference in fluctuation intensity above T_N is indicative of the level of frustration in each system - a strong signal implies strong correlations despite the absence of long range magnetic order.

Figure 8

(a) Comparison of the evolution of normalized per Os integrated neutron scattering intensity of Sr₂MgOsO₆ and Sr₂ScOsO₆ and (b) the temperature dependence of the scattering within the gap of Sr₂MgOsO₆, converted to χ″ as in ref. 10.

In Fig. 8(b) we present the temperature dependence of the scattering at very low energies, i.e. within the gap at low temperatures. The data is converted to χ″(T) for the fixed range 0.7 < Q < 1 Å⁻¹ and 3 < E < 5 meV following the method described in ref. 10, in which the lowest temperature data set has been subtracted as a background and a Bose factor correction has been applied. This confirms that the reduction in the scattering at low energies is far beyond what would be expected due to thermal population, thereby confirming the opening of a gap at low temperature.

Discussion

A phase diagram depicting seven potential magnetic ground states, including three potential antiferromagnetic configurations, has been proposed for double perovskites with a single magnetic 5d² cation⁶. Through a combination of reduced paramagnetic effective moment, an enhanced magnetic entropy from a spin-only scenario, a substantially reduced moment refined from neutron diffraction and a significant spin-excitation gap observed in neutron spectroscopy, we have unequivocally shown that SOC plays a major role in the magnetic behavior of the Os⁶⁺ 5d² cation in Sr₂MgOsO₆. Despite this, we have shown that the ground state remains in the “AFM100” region of the phase diagram predicted in ref. 6, similar to many 4d³ and 5d³ double perovskites^{8,10,11,12,13}. The predicted influence of SOC on the d² state would inherently infer anisotropic interactions on the Os⁶⁺ ions, with the strong Os−O hybridization in Sr₂MgOsO₆ ensuring that the anisotropy has significant influence on the collective properties. We have confirmed anisotropy is present in Sr₂MgOsO₆ via observation of the spin gap in the magnetic excitation spectrum, Fig. 7a.

The moment observed by neutron diffraction of 0.60(2) μ_B is considerably reduced (70%) from the expected high-field spin-only value of 2 μ_B. Covalency has been shown to play a significant role in the reduction of the moment in the case of 4d³ and 5d³ transition metal oxides, resulting in 37–47% reductions of the 5d³ Os⁵⁺ 3 μ_B moment to 1.6 to 1.9 μ_B^{8,9,10,11,12,13} and we expect a similar effect in the Os⁶⁺ 5d² case. Here, however, the magnetization and specific heat analyses strongly suggests that an orbital contribution is also important, consistent with the recent x-ray magnetic circular dichroism (XMCD) study of Os⁶⁺ in the related material Ca₂CoOsO₆⁴⁸. Our DFT results confirm that both SOC and covalency together cause the major reduction in the observed spin-moment, predicting a 47% reduction from covalency and a further 25% reduction from SOC, consistent with the total 70% reduction we observed experimentally. The result is a moment which is challenging to observe with standard neutron diffraction instrumentation, but via a high-flux, low-background experiment we were able to determine both the ground state and the moment size - it would be interesting to revisit Ba₂CaOsO₆ on a similar instrument in order to conclusively place the magnetic ground state among those known and predicted.

For 5d² double perovskites the type I AFM structure was anticipated via theory including strong SOC in ref. 6. The predicted influence of SOC on the d² state would inherently infer anisotropy, with the strong Os−O hybridization in Sr₂MgOsO₆ ensuring that the anisotropy has significant influence on the collective properties. Via inelastic neutron scattering we have indeed observed anisotropy in Sr₂MgOsO₆ via the spin gap.

It is interesting to investigate the comparison between 5d² and 5d³ systems. Sr₂MgOsO₆ orders at a higher temperature than other double perovskites with a single magnetic ion²⁸. While 4d³ and 5d³ double perovskites like Sr₂ScOsO₆ have larger magnetic moments, which should yield stronger interactions and higher ordering temperatures, they also have significantly larger frustration indices ranging from 4 to 14^{8,10,11,12,13} in comparison to 2.5 in Sr₂MgOsO₆. The similarity of the QE-space dependence and the intensity of the excitation spectra of Sr₂MgOsO₆ and Sr₂ScOsO₆ at 6 K, shown in Fig. 7, indicates that similar interaction mechanisms are responsible for the collective properties in each. However, Fig. 8a shows that the intensity of fluctuations in Sr₂ScOsO₆ above T_N is far greater than the intensity in Sr₂MgOsO₆ above T_N. This implies strong correlations persist in Sr₂ScOsO₆ in the absence of long range magnetic order–a hallmark of frustration. While the tetragonal symmetry of Sr₂MgOsO₆ may play some role in reducing the geometric frustration, it is less distorted than monoclinic Sr₂ScOsO₆, which has angles that deviate more substantially from 180° ⁸.

The question, therefore, is why is the frustration relieved in Sr₂MgOsO₆ compared to Sr₂ScOsO₆? The strength of hybridization plays a major role in determining the strength of the interactions, but does not directly explain relief of frustration. Similarly, the unit cell volume and B/B′ site disorder affect interaction strengths and all these mechanisms likely contribute to differences in T_Ns amongst the AFM d² DPs Sr₂MgOsO₆ (|Θ|/T_N = 2.5) [this work], Ba₂CaOsO₆ (|Θ|/T_N = 3.1)⁵¹ and Ca₂CaOsO₆ (|Θ|/T_N = 3.0)⁵², but in all of these materials the frustration index is low compared to d³ DP Sr₂ScOsO₆. Our DFT results reveal that the SOC induced anisotropy leads to selection of the magnetic ground state, as also found experimentally for Sr₂ScOsO₆³⁸, therefore stronger anisotropy promotes a more robust ground state. Given the observation of the large spin gap in Sr₂MgOsO₆ and the multiple sources of evidence we have presented for stronger SOC in this d² material, we conclude that SOC induced anisotropy is the dominant factor in the relieved frustration in Sr₂MgOsO₆.

This conclusion is supported by an earlier theoretical prediction that T_N should vary with the strength of anisotropy in such systems, due to reduced competition between possible ground states⁵³. The anisotropy term is proportional to the gap size (probed by INS) divided by the moment size (probed by neutron diffraction). Given that the gap is of the same magnitude in Sr₂MgOsO₆ and Sr₂ScOsO₆, Fig. 7, but the moment is less than half the size in Sr₂MgOsO₆, we conclude that the anisotropy is indeed much larger in Sr₂MgOsO₆. Therefore, the results presented demonstrate the relief of frustration via SOC induced anisotropy and represent the experimental demonstration of the evolution of T_N with SOC.

Conclusion

The double perovskite osmate Sr₂MgOsO₆ has been synthesized and characterized by magnetometry, specific heat measurements and elastic and inelastic neutron scattering. The combined results demonstrate that spin-orbit coupling is essential to describe the magnetic properties of the system. The Os⁶⁺ moments order antiferromagnetically at 108 K in a type I configuration on the Os fcc sublattice as theoretically predicted⁶. For the first time in such a 5d² material, refinements of neutron powder diffraction data yields a moment of 0.60(2) μ_B per Os cation which is significantly reduced due to the combined effects of SOC and covalency. Through comparison of inelastic neutron spectra, it is shown that SOC induced anisotropy has the ability to relieve frustration in d² systems, relative to analogous d³ materials which have systematically higher frustration indices, promoting high magnetic transition temperatures.