Spin-glass transition in the spin–orbit-entangled Jeff = 0 Mott insulating double-perovskite ruthenate

We have successfully synthesized new Ru4+ double perovskite oxides SrLaInRuO6 and SrLaGaRuO6, which are expected to be a spin–orbit coupled Jeff = 0 Mott insulating ground state. Their magnetic susceptibility is much significant than that expected for a single Ru4+ ion for which exchange coupling with other ions is negligible. Their isothermal magnetization process suggests that there are about 20 percent isolated spins. These origins would be the Ru3+/Ru5+ magnetic defects, while the regular Ru4+ sites remain nonmagnetic. Moreover, SrLaGaRuO6 shows a spin-glass-like magnetic transition at low temperatures, probably caused by isolated spins. The observed spin-glass can be interpreted by the analogy of a dilute magnetic alloy, which can be seen as a precursor to the mobile Jeff = 1 exciton as a dispersive mode as predicted.


Results
Crystal structure. Figure 1a and b shows powder XRD patterns from thus-obtained samples. All peaks are indexed to monoclinic unit cells based on the space group of P2 1 /c. The Rietveld analysis converged well with the distorted double perovskite structure shown in Fig. 1c and the structural parameters in Table 1. No deviation from the ratio of Sr:La = 1:1 was detected within the experimental error. We estimate the modified tolerance factor t m as structural stability in double perovskite using the ionic radii values, yielding t m = 0.93834 and 0.98015 in SrLaInRuO 6 and SrLaGaRuO 6 , respectively. In the condition of t m < 1, the double perovskite-type compounds should crystallize a monoclinic structure 25,26 . Our samples certainly satisfy the criterion.  www.nature.com/scientificreports/ In estimating the valence of the B-site cations at the center of the octahedral MO 6 (M = In, Ga) and RuO 6 in the two double perovskite oxides, we used the bond valence sum B V expressed by the following formula 27 , where R 0 is the empirical bonding parameter, R i is the inter-bond cation-anion distance, and N is the coordination number. The estimated B V values of In/Ga and Ru ions in SrLaInRuO 6 and SrLaGaRuO 6 are listed in Table 2, which aligns with the expected values. Figure 2a shows the temperature dependence of magnetization M/H of SrLaMRuO 6 (M = In, Ga) under an applied field of 1 T. For comparison, the M/H data of La 2 MgRuO 6 28,29 with a similar d 4 electron configuration is displayed. The magnetic response of SrLaMRuO 6 and La 2 MgRuO 6 is quite different despite the similar electronic state of the Ru 4+ single ion. When considered from a crystallographic point of view, these compounds are not expected to have strong magnetic interactions because of the significant separation of Ru 4+ ions. Therefore, it seems strange that the magnetic responses of SrLaMRuO 6 and La 2 MgRuO 6 are so different. Kotani theoretically predicted the effective magnetic moment of d n ions (n = 1 ~ 5) as a function of electron filling n, spin-orbit coupling, temperature, and ligand environment 30 . The effective magnetic moment μ eff (T) of low-spin d 4 in an octahedral environment can be expressed as follows, where x = λ/k B T, λ is the spin-orbit coupling interaction 30 , and k B is the Boltzmann constant. Thus, the magnetic susceptibility of isolated Ru 4+ ions χ calc can be expressed as follows,

Magnetism.
The dotted black curve in Fig. 2a represents the χ calc curve calculated using a value of λ = 980 cm −1 for Ru 4+ ions. The λ-value is expected to be smaller than the completely free-ion value of λ = 1400 cm −1 determined in the study used Ru 4+ complexes 31 , which origin of λ-shrinking would be covalency. Note that it is necessary to incorporate the effect of the low symmetry field in order to reproduce the susceptibility in distorted Ru 4+ double perovskites since Eq. (2) is calculated in the cubic symmetry field. The M/H data of La 2 MgRuO 6 seemingly  www.nature.com/scientificreports/ follows χ calc , while those of SrLaMRuO 6 significantly deviate from χ calc . This fact indicates that the Ru 4+ ions in SrLaMRuO 6 are not simply in the J eff = 0 ground state. Both M/H data for SrLaMRuO 6 are in the rough agreement above 250 K, but below 250 K, are greatly enhanced compared to the χ calc curve. In Ir 5+ double perovskites, in which a similar J eff = 0 ground state is expected, the M/H data of Sr 2 YIrO 6 shows almost temperature-independent behavior 21 . On the other hand, a similar enhancement in low-temperature M/H data is observed in the solid solution system Sr 2-x Ca x YIrO 6 21 . Therefore, this increase in magnetization may affect randomness, of which a mechanism will be discussed later.
Moreover, SrLaGaRuO 6 shows a magnetic anomaly at low temperatures (displayed by an arrow in Fig. 2a), contrasting with no anomaly in SrLaInRuO 6 . Figure 2b expands the low-temperature region under magnetic fields from 0.01 to 1 T. At the lowest field of 0.01 T, the M/H data exhibit an apparent thermal hysteresis between the zero-field-cooled (ZFC) and field-cooled (FC) data below T f ~ 50 K. This hysteresis is suppressed by increasing the magnetic field and is eventually merged at 7 T. This behavior is a typical feature of spin-glass transition 32 .
High-field magnetization. Figure 2c shows the isothermal magnetization M up to 60 T. The M-H curves show convex behavior upward, implying an isolated spin different from the van Vleck magnetism of Ru 4+ pseudospin J eff = 0 state. The origin of the isolated spin will be discussed later. The increase in magnetization at highfield regions is due to the van Vleck paramagnetism.

Discussion
As described above, the two novel double perovskite ruthenates SrLaInRuO 6 and SrLaGaRuO 6 are expected to show a van Vleck magnetism of Ru 4+ pseudospin J eff = 0 state. However, the observed M/H is considerably larger than a single Ru 4+ spin, indicating the deviation from the J eff = 0 state. In addition, the isothermal magnetization demonstrates the existence of an isolated spin.
A similar enhancement of magnetization has been reported in highly solid-solution double perovskite iridates Sr 2-x Ca x YIrO 6 21 . An X-ray magnetic circular dichroism (XMCD) measurement demonstrates an emergent partial charge disproportionation (PCD) of Ir 5+ → 0.5Ir 4+ + 0.5Ir 6+ due to a site-randomness 21 . In light of this result, similar Ru 3+ /Ru 5+ magnetic defects possibly occurs in SrLaInRuO 6 and SrLaGaRuO 6 due to a similar intrinsic A-site randomness.
The magnetization of isolated spin M iso follows a Brillouin function, while the van Vleck term M VV should be proportional to H. Both terms would contribute to the observed nonlinear behaviors of the isothermal M. Here, in order to separate the contributions of the isolated spins and the van Vleck term, we analyze the M data with a modified Brillouin function, where N J represents a scaling factor to account for a finite number of paramagnetic free spins, g J (~ 2) is the g-factor, μ B is the Bohr magneton, J (= 1/2 and 3/2) is the total angular momentum. For the second term, χ vv indicates the van Vleck term. The values of N and χ vv are summarized in Table 3. Provided that N 1/2 and N 3/2 are fixed to equal considering the local charge disproportionation model, the M data up to 60 T fit the Eq. (4). The best fits are shown by the dashed lines in Fig. 2c, with the fitting parameters given in Table 3. Our analysis suggests that ~ 20% of free spins (J = 1/2 and 3/2) are present. The orphan spins possibly emerged by the valence being off from tetravalent, which is no evidence from the crystal structural analysis. Although we cannot rule out other origins, these facts support that the PCD model is a good solution. As in the Ir 5+ system, the PCDgenerated isolated spins may be directly detected by XMCD measurements: it is a further issue. In addition, the van Vleck term was found to be more significant for SrLaGaRuO 6 .
The estimated van Vleck term of SrLaGaRuO 6 is larger than SrLaInRuO 6 . According to Boltzmann statistics, the van Vleck term is proportional to the concentration of J eff = 1 exciton. Therefore, the difference in χ vv between SrLaInRuO 6 and SrLaGaRuO 6 is due to the different Δ. In the theoretical prediction, a non-cubic crystal field, generated by a distortion of the RuO 6 octahedra, effectively reduces Δ 33 . Here, we introduce the bond angle variance σ, as a scale parameter of the polyhedral distortion. The σ-value in the RuO 6 octahedra can be parametrized by the following formula, where m is the number of O-Ru-O angles, φ i is the ith bond angle of the distorted coordination-polyhedra, and φ 0 is the bond angle of the coordination polyhedral with O h symmetry; φ 0 equals 90° for octahedron. Calculations www.nature.com/scientificreports/ using the atomic position parameters listed in Table 1 yield the σ-values of 7.7976° and 10.2708° for SrLaInRuO 6 and SrLaGaRuO 6 , respectively, indicating a strikingly larger non-cubic crystal field in SrLaGaRuO 6 than SrLaInRuO 6 . Therefore, the concentration of J eff = 1 exciton of SrLaGaRuO 6 should be larger than SrLaInRuO 6 , consistent with the large-small relationship of χ vv . Furthermore, it is theoretically predicted that the SE interaction between J eff = 0 reduces Δ. In SrLaInRuO 6 and SrLaGaRuO 6 , the J eff = 0 pseudospins interact via the SE interaction through Ru 4+ -O 2-M 3+ -O 2-Ru 4+ paths with M = In, Ga. Thus, it is considered that the difference in the SE interaction between these two systems arises from the filled outermost orbitals, which are 4d and 3d orbitals for SrLaInRuO 6 and SrLaGaRuO 6 , respectively. Therefore, it is reasonable that the SE magnitude is different.
Based on the results so far, it is reasonable to consider that the spin-glass transition in SrLaGaRuO 6 is due to randomly arranged isolated spins. Strangely enough, however, no spin-glass transition has been observed in SrLaInRuO 6 , where the isolated spin concentration is comparable. However, it is unlikely that all the 19% localized spins interact strongly in SrLaGaRuO 6 where the Ru-Ru distance is far apart. This fact suggests a difference in the magnitude of the interaction between randomly arranged isolated spins.
The origin of the spin-glass transition in SrLaGaRuO 6 can be inferred by analogy with dilute magnetic alloys. In dilute magnetic alloys, partially arranged magnetic atoms interact with each other via RKKY interactions. As mentioned in the introduction, the J eff = 1 excitons become a dispersive mode due to strong SE interactions 9 . In this situation, the mobile J eff = 1 exciton may behave like a conduction electron. Therefore, the interaction via a mobile J eff = 1 exciton between the free spins in a J eff = 0 magnet can be regarded as an RKKY interaction. A schematic diagram of this mechanism is shown in Fig. 3. This interaction should be proportional to the concentration of J eff = 1, which is consistent with the presence/absence of spin-glass transition. The feasibility of the spin-glass transition in the category of spin-orbit excitonic magnetism is very interesting and requires further theoretical studies. In the broad context, this finding also suggests that the several magnetic responses in J eff = 0 magnets, which have been found so far, would be explained by the generated isolated spin model. Thus, we sincerely hope that it should be carefully re-examined.

Summary
We have successfully synthesized new Ru 4+ double perovskite oxides SrLaInRuO 6 and SrLaGaRuO 6 . The temperature-dependent M/H and isothermal M data can be explained by the van Vleck magnetism of J eff = 0 states with additional isolated spins possibly generated by the Ru 3+ /Ru 5+ magnetic defects. While SrLaInRuO 6 is paramagnetic down to 2 K, SrLaGaRuO 6 shows spin-glass transition at T f ~ 50 K. We propose that the origin of spin-glass is isolated spins couple via mobile J eff = 1 excitons as an analogy of a dilute magnetic alloy. It is expected that the spin-glass transition due to the introduction of isolated spins demonstrates the existence of mobile J eff = 1 excitons as dispersive modes as predicted in spin-orbit-entangled d 4 ions.

Data availability
The datasets generated and analyzed during the current study are available from the corresponding author.
Received: 28 October 2021; Accepted: 27 January 2022 Figure 3. Schematic of the mechanism of spin-glass induced by isolated spin and J eff = 1 excitons, as an analogy of a dilute magnetic alloy. The interaction between free spins mediated by mobile J eff = 1 excitons corresponds to the RKKY interaction.