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.

Introduction

Recent material investigations have revealed novel phenomena driven by spin–orbit coupling (SOC)[1-20]. The strong SOC splits a t2g band into J = 3/2 and J = 1/2 bands, which results in the realization of the SOC Mott insulating state. For example, the Ir4+ ions with the d5 electron configuration have an effective orbital moment L = 1, resulting in a Jeff = 1/2 pseudospin. Such a spin–orbit-entanglement gives rise to unconventional interaction among pseudospins. In the case of Jeff = 1/2 pseudospins, the bond-dependent Ising interactions, which have been called the Kitaev interaction in recent years, are realized[1]. Such a realization of the SOC pseudospin state was first observed in layered perovskite Sr2IrO4[2]. Consequently, it is theoretically predicted that a honeycomb lattice magnet with Jeff = 1/2 pseudospins is a promising host of a quantum spin liquid (QSL). In the realistic compounds, RuCl3 and H3LiIr2O6 exhibit the Kitaev QSL behavior[3-6].

In contrast, the d4 electron system (Ru4+, Os4+, and Ir5+) has not been attracted much attention due to an absence of local moments in the ionic ground state. However, Jeff = 1 excitations become dispersive modes in a crystal due to moderate superexchange (SE) interactions. These mobile spin–orbit excitons may condense in this situation, which results in a magnetically ordered state[7-10]. In order to realize such a state, the exchange interaction must overcome a critical value sufficient to exceed the energy gap Δ between the Jeff = 0 and Jeff = 1[7,11,12]. Such a condensed state, for which the physicists conceive a terminology—spin–orbit-exciton condensation (SOEC), is analogous to magnon condensation phenomena in a spin dimer system[13]. One factor differentiating the d4 system from the spin-dimer one is the anisotropy of the strong exchange interaction, which originates from the strong spin–orbit interaction. Therefore, it is expected that a novel condensed phase will be realized.

Although theoretical studies have been enormously advanced to search for anomalous phenomena driven by SOEC, experimental studies have not been carried out due to the lack of model materials. This situation is because the energy scale of Δ is too large; the 5d4 (Ir5+ and Os4+) compounds should be typically nonmagnetic. Indeed, the weak magnetic anomalies observed in some Ir5+ double perovskites are better explained by the Ir4+ and Ir6+ magnetic defects rather than SOEC[14-21]. Therefore, a SOEC seems to be much less feasible for 5d compounds. On the other hand, SOEC is more likely realizable in 4d4 compounds such as Ru4+, where SOC is smaller than Ir5+ and is comparable to SE. Moreover, the SOC vs. SE competition can be tuned by a lattice control. Therefore, Ru4+ double perovskites would be good model-compounds for a realization of SOEC[22].

We report the magnetic properties of novel double perovskites SrLaInRuO6 and SrLaGaRuO6 with Ru4+ ion. These deviate significantly from the single-spin susceptibility expected for Ru4+ (Jeff = 0) ions, even though the distance between the magnetic ions is sufficiently large. Furthermore, the magnetization process up to 60 T demonstrates the presence of about 20 percent isolated spins. These behaviors can be explained as originating from the Ru3+/Ru5+ magnetic defects. Moreover, only SrLaGaRuO6 shows a spin-glass transition at Tf ~ 50 K. We discuss the origin of the observed spin-glass transition from the analogy of the dilute magnetic alloy from the viewpoint of Jeff = 0 physics in SOC Mott insulators.

Experimental methods

Polycrystalline samples of SrLaMRuO6 (M = In, Ga) were synthesized by the conventional solid-state reaction from stoichiometric mixtures of SrCO3, La2O3, M2O3 (M = In, Ga), and RuO2. The obtained samples were characterized by powder X-ray diffraction (XRD) measurements using a diffractometer with Cu Kα radiation. The cell and crystal structure parameters were refined using the Rietveld method using rietan-fp version 2.16 software[23]. The temperature dependence of the magnetization was measured using the magnetic property measurement system (MPMS; Quantum Design) equipped in the Institute for Solid State Physics at the University of Tokyo. Magnetization curves up to 60 T were measured using an induction method with a multilayer pulsed magnet at the Institute for Solid State Physics at the University of Tokyo.

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 P21/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 tm as structural stability in double perovskite using the ionic radii values, yielding tm = 0.93834 and 0.98015 in SrLaInRuO6 and SrLaGaRuO6, respectively. In the condition of tm < 1, the double perovskite-type compounds should crystallize a monoclinic structure[25,26]. Our samples certainly satisfy the criterion.

Figure 1

Powder X-ray diffraction patterns of (a) SrLaInRuO6 and (b) SrLaGaRuO6. The observed intensities (red), calculated intensities (black), and their differences (blue) are shown. Vertical bars (green) indicate the positions of the Bragg reflections. (c) Crystal structure of SrLaMRuO6 (M = In, Ga) obtained from crystal structure parameters refined using the Rietveld method. The vesta program is used for visualization[24].

Table 1

Crystallographic parameters for SrLaInRuO6 and SrLaGaRuO6 (both space group: P21/c) determined from powder X-ray diffraction experiments.

atom	site	x	y	z	B (Å)
SrLaInRuO6
Sr/La	4e	0.5060 (7)	0.5346 (3)	0.252 (1)	0.23 (4)
In	2c	0	1/2	0	1.3 (3)
Ru	2b	1/2	0	0	1.3 (3)
O1	4e	0.229 (7)	0.222 (7)	0.992 (6)	2.7 (2)
O2	4e	0.324 (7)	0.704 (6)	0.963 (4)	2.7 (2)
O3	4e	0.405 (5)	0.988 (2)	0.222 (5)	2.7 (2)
SrLaGaRuO6
Sr/La	4e	0.4999 (3)	0.498 (3)	0.2492 (9)	0.10 (4)
Ga	2c	0	1/2	0	0.97 (7)
Ru	2b	1/2	0	0	0.97 (7)
O1	4e	0.24 (1)	0.241 (7)	1.007 (6)	2.2 (2)
O2	4e	0.328 (8)	0.676 (5)	1.028 (3)	2.2 (2)
O3	4e	0.50 (2)	0.99 (2)	0.250 (4)	2.2 (2)

The obtained lattice parameters are a = 5.6976 (4), b = 5.7427 (4), c = 8.0669 (5) Å, and β = 90.05 (1)º for SrLaInRuO6, and a = 5.5077 (4), b = 5.5538(3), c = 7.8210 (3) Å, and β = 90.586 (3)º for SrLaGaRuO6. B is the atomic displacement parameter.

In estimating the valence of the B-site cations at the center of the octahedral MO6 (M = In, Ga) and RuO6 in the two double perovskite oxides, we used the bond valence sum BV expressed by the following formula[27],where R0 is the empirical bonding parameter, Ri is the inter-bond cation–anion distance, and N is the coordination number. The estimated BV values of In/Ga and Ru ions in SrLaInRuO6 and SrLaGaRuO6 are listed in Table 2, which aligns with the expected values.

Table 2

Calculation of the bond valence sum (BV) for the octahedron in SrLaInRuO6 and SrLaGaRuO6.

Formula	cation	Bond valence sum
SrLaInRuO6	In3+	+ 2.838
	Ru4+	+ 4.352
SrLaGaRuO6	Ga3+	+ 2.944
	Ru4+	+ 4.056

Magnetism

Figure 2a shows the temperature dependence of magnetization M/H of SrLaMRuO6 (M = In, Ga) under an applied field of 1 T. For comparison, the M/H data of La2MgRuO6[28,29] with a similar d4 electron configuration is displayed. The magnetic response of SrLaMRuO6 and La2MgRuO6 is quite different despite the similar electronic state of the Ru4+ 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 Ru4+ ions. Therefore, it seems strange that the magnetic responses of SrLaMRuO6 and La2MgRuO6 are so different. Kotani theoretically predicted the effective magnetic moment of d ions (n = 1 ~ 5) as a function of electron filling n, spin–orbit coupling, temperature, and ligand environment[30]. The effective magnetic moment μeff (T) of low-spin d4 in an octahedral environment can be expressed as follows,where x = λ/kBT, λ is the spin–orbit coupling interaction[30], and kB is the Boltzmann constant. Thus, the magnetic susceptibility of isolated Ru4+ ions χcalc can be expressed as follows,

Figure 2

(a) Temperature dependence of magnetic susceptibility M/H of SrLaMRuO6 (M = In, Ga) and La2MgRuO6[28,29] under a magnetic field of 1 T. In this figure, only the results of field cooling (FC) data are shown. The black dotted curve shows the free-spin magnetization of Ru4+ ions calculated as described in the text. (b) The M/H curves of SrLaGaRuO6 under several magnetic fields. In each field, measurements were conducted upon heating after zero-field cooling (ZFC, open circles) and then upon cooling (FC, closed circles). (c) The isothermal magnetization of SrLaInRuO6 and SrLaGaRuO6 under a magnetic field up to 60 T at 4.2 K. The black dashed lines represent the best fit by Eq. (4) described in the text.

The dotted black curve in Fig. 2a represents the χcalc curve calculated using a value of λ = 980 cm−1 for Ru4+ 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 Ru4+ 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 Ru4+ double perovskites since Eq. (2) is calculated in the cubic symmetry field. The M/H data of La2MgRuO6 seemingly follows χcalc, while those of SrLaMRuO6 significantly deviate from χcalc. This fact indicates that the Ru4+ ions in SrLaMRuO6 are not simply in the Jeff = 0 ground state.

Both M/H data for SrLaMRuO6 are in the rough agreement above 250 K, but below 250 K, are greatly enhanced compared to the χcalc curve. In Ir5+ double perovskites, in which a similar Jeff = 0 ground state is expected, the M/H data of Sr2YIrO6 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 Sr2-CaYIrO6[21]. Therefore, this increase in magnetization may affect randomness, of which a mechanism will be discussed later.

Moreover, SrLaGaRuO6 shows a magnetic anomaly at low temperatures (displayed by an arrow in Fig. 2a), contrasting with no anomaly in SrLaInRuO6. 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 Tf ~ 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 Ru4+ pseudospin Jeff = 0 state. The origin of the isolated spin will be discussed later. The increase in magnetization at high-field regions is due to the van Vleck paramagnetism.

Discussion

As described above, the two novel double perovskite ruthenates SrLaInRuO6 and SrLaGaRuO6 are expected to show a van Vleck magnetism of Ru4+ pseudospin Jeff = 0 state. However, the observed M/H is considerably larger than a single Ru4+ spin, indicating the deviation from the Jeff = 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 Sr2-CaYIrO6[21]. An X-ray magnetic circular dichroism (XMCD) measurement demonstrates an emergent partial charge disproportionation (PCD) of Ir5+ → 0.5Ir4+ + 0.5Ir6+ due to a site-randomness[21]. In light of this result, similar Ru3+/Ru5+ magnetic defects possibly occurs in SrLaInRuO6 and SrLaGaRuO6 due to a similar intrinsic A-site randomness.

The magnetization of isolated spin Miso follows a Brillouin function, while the van Vleck term MVV 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 represents a scaling factor to account for a finite number of paramagnetic free spins, gJ (~ 2) is the g-factor, μB is the Bohr magneton, J (= 1/2 and 3/2) is the total angular momentum. For the second term, χvv indicates the van Vleck term. The values of N and χvv are summarized in Table 3. Provided that N1/2 and N3/2 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 Ir5+ system, the PCD-generated 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 SrLaGaRuO6.

Table 3

Results of fits to the isothermal M using the model described in the text. The parameters N1/2 and N3/2 are fixed to equal.

	N_1/2 (Ru5+)	N_3/2 (Ru3+)	χvv (cm3/mol Oe)
SrLaInRuO6	0.1088 (3)	0.1088 (3)	0.00869 (2)
SrLaGaRuO6	0.0937 (21)	0.0937 (21)	0.01503 (2)

The estimated van Vleck term of SrLaGaRuO6 is larger than SrLaInRuO6. According to Boltzmann statistics, the van Vleck term is proportional to the concentration of Jeff = 1 exciton. Therefore, the difference in χvv between SrLaInRuO6 and SrLaGaRuO6 is due to the different Δ. In the theoretical prediction, a non-cubic crystal field, generated by a distortion of the RuO6 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 RuO6 octahedra can be parametrized by the following formula,where m is the number of O–Ru–O angles, φ is the ith bond angle of the distorted coordination-polyhedra, and φ0 is the bond angle of the coordination polyhedral with Oh symmetry; φ0 equals 90° for octahedron. Calculations using the atomic position parameters listed in Table 1 yield the σ-values of 7.7976° and 10.2708° for SrLaInRuO6 and SrLaGaRuO6, respectively, indicating a strikingly larger non-cubic crystal field in SrLaGaRuO6 than SrLaInRuO6. Therefore, the concentration of Jeff = 1 exciton of SrLaGaRuO6 should be larger than SrLaInRuO6, consistent with the large-small relationship of χvv.

Furthermore, it is theoretically predicted that the SE interaction between Jeff = 0 reduces Δ. In SrLaInRuO6 and SrLaGaRuO6, the Jeff = 0 pseudospins interact via the SE interaction through Ru4+–O2–M3+–O2–Ru4+ 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 SrLaInRuO6 and SrLaGaRuO6, 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 SrLaGaRuO6 is due to randomly arranged isolated spins. Strangely enough, however, no spin-glass transition has been observed in SrLaInRuO6, where the isolated spin concentration is comparable. However, it is unlikely that all the 19% localized spins interact strongly in SrLaGaRuO6 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 SrLaGaRuO6 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 Jeff = 1 excitons become a dispersive mode due to strong SE interactions[9]. In this situation, the mobile Jeff = 1 exciton may behave like a conduction electron. Therefore, the interaction via a mobile Jeff = 1 exciton between the free spins in a Jeff = 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 Jeff = 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 Jeff = 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.

Figure 3

Schematic of the mechanism of spin-glass induced by isolated spin and Jeff = 1 excitons, as an analogy of a dilute magnetic alloy. The interaction between free spins mediated by mobile Jeff = 1 excitons corresponds to the RKKY interaction.

Summary

We have successfully synthesized new Ru4+ double perovskite oxides SrLaInRuO6 and SrLaGaRuO6. The temperature-dependent M/H and isothermal M data can be explained by the van Vleck magnetism of Jeff = 0 states with additional isolated spins possibly generated by the Ru3+/Ru5+ magnetic defects. While SrLaInRuO6 is paramagnetic down to 2 K, SrLaGaRuO6 shows spin-glass transition at Tf ~ 50 K. We propose that the origin of spin-glass is isolated spins couple via mobile Jeff = 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 Jeff = 1 excitons as dispersive modes as predicted in spin–orbit-entangled d4 ions.