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

Abstract

We have successfully synthesized new Ru⁴⁺ double perovskite oxides SrLaInRuO₆ and SrLaGaRuO₆, which are expected to be a spin–orbit coupled J_eff = 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 J_eff = 1 exciton as a dispersive mode as predicted.

Introduction

Recent material investigations have revealed novel phenomena driven by spin–orbit coupling (SOC)^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}. The strong SOC splits a t_2g 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 J_eff = 1/2 pseudospin. Such a spin–orbit-entanglement gives rise to unconventional interaction among pseudospins. In the case of J_eff = 1/2 pseudospins, the bond-dependent Ising interactions, which have been called the Kitaev interaction in recent years, are realized¹. Such a realization of the SOC pseudospin state was first observed in layered perovskite Sr₂IrO₄². Consequently, it is theoretically predicted that a honeycomb lattice magnet with J_eff = 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^3,4,5,6.

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, J_eff = 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^7,8,9,10. In order to realize such a state, the exchange interaction must overcome a critical value sufficient to exceed the energy gap Δ between the J_eff = 0 and J_eff = 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 system¹³. 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^{14,15,16,17,18,19,20,21}. 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 SOEC²².

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⁺ (J_eff = 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 T_f ~ 50 K. We discuss the origin of the observed spin-glass transition from the analogy of the dilute magnetic alloy from the viewpoint of J_eff = 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 software²³. 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 t_m as structural stability in double perovskite using the ionic radii values, yielding t_m = 0.93834 and 0.98015 in SrLaInRuO₆ and SrLaGaRuO₆, 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.

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 visualization²⁴.

Table 1 Crystallographic parameters for SrLaInRuO₆ and SrLaGaRuO₆ (both space group: P2₁/c) determined from powder X-ray diffraction experiments.

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 B_V expressed by the following formula²⁷,

$${B}_{V}={\sum }_{i}^{N}\mathrm{exp}\left(\frac{{R}_{0}-{R}_{i}}{0.37}\right)$$

(1)

where R₀ 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 SrLaInRuO₆ and SrLaGaRuO₆ are listed in Table 2, which aligns with the expected values.

Table 2 Calculation of the bond valence sum (B_V) for the octahedron in SrLaInRuO₆ and SrLaGaRuO₆.

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 dⁿ ions (n = 1 ~ 5) as a function of electron filling n, spin–orbit coupling, temperature, and ligand environment³⁰. The effective magnetic moment μ_eff (T) of low-spin d⁴ in an octahedral environment 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.

$$\begin{array}{c}{{\mu }_{\mathrm{eff}}}^{2}=\frac{3\left[24+\left(x/2-9\right){e}^{-x/2}+\left(5x/2-15\right){e}^{-3x/2}\right]}{x\left[1+3{e}^{-x/2 }+5{e}^{-3x/2}\right]}\end{array}$$

(2)

where x = λ/k_BT, λ is the spin–orbit coupling interaction³⁰, and k_B is the Boltzmann constant. Thus, the magnetic susceptibility of isolated Ru⁴⁺ ions χ_calc can be expressed as follows,

$$\begin{array}{c}{\chi }_{\mathrm{calc}}=\frac{N{{\mu }_{\mathrm{eff}}}^{2}}{3{k}_{B}T}\end{array}$$

(3)

The dotted black curve in Fig. 2a represents the χ_calc 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⁺ complexes³¹, 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 χ_calc, while those of SrLaMRuO₆ significantly deviate from χ_calc. This fact indicates that the Ru⁴⁺ ions in SrLaMRuO₆ are not simply in the J_eff = 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 χ_calc curve. In Ir⁵⁺ double perovskites, in which a similar J_eff = 0 ground state is expected, the M/H data of Sr₂YIrO₆ shows almost temperature-independent behavior²¹. On the other hand, a similar enhancement in low-temperature M/H data is observed in the solid solution system Sr_2-xCa_xYIrO₆²¹. 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 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 transition³².

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 J_eff = 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 J_eff = 0 state. However, the observed M/H is considerably larger than a single Ru⁴⁺ 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-xCa_xYIrO₆²¹. 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-randomness²¹. 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 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,

$$M={M}_{\mathrm{iso}}+{M}_{\mathrm{vv}}={\sum }_{J}{N}_{J}{g}_{J}{\mu }_{B}J\left\{\frac{2J+1}{2J}\mathrm{coth}\left(\frac{2J+1}{2J}\frac{{g}_{J}{\mu }_{B}JH}{{k}_{B}T}\right)-\frac{1}{2J}\mathrm{coth}\left(\frac{1}{2J}\frac{{g}_{J}{\mu }_{B}JH}{{k}_{B}T}\right)\right\}+{\chi }_{\mathrm{vv}}H$$

(4)

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 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 N_1/2 and N_3/2 are fixed to equal.

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 J_eff = 1 exciton. Therefore, the difference in χ_vv 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 Δ³³. 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,

$$\sigma =\sqrt{{\sum }_{i=1}^{12}\frac{{\left({\varphi }_{i}-{\varphi }_{0}\right)}^{2}}{m-1}}$$

(5)

where m is the number of O–Ru–O angles, φ_i is the ith bond angle of the distorted coordination-polyhedra, and φ₀ is the bond angle of the coordination polyhedral with O_h 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 J_eff = 1 exciton of SrLaGaRuO₆ should be larger than SrLaInRuO₆, 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 SrLaInRuO₆ and SrLaGaRuO₆, the J_eff = 0 pseudospins interact via the SE interaction through Ru⁴⁺–O^2–M³⁺–O^2–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 J_eff = 1 excitons become a dispersive mode due to strong SE interactions⁹. 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.

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.

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 J_eff = 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 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 d⁴ ions.