Valence bond glass state in the 4d¹ fcc antiferromagnet Ba₂LuMoO₆

Abstract

B-site ordered 4d¹ and 5d¹ double perovskites have a number of potential exotic ground states including multipolar order, quantum spin liquids and valence bond glass states. These arise from the complex interactions of spin-orbital entangled J_eff = 3/2 pseudospins on the geometrically frustrated fcc lattice. The 4d¹ Mo⁵⁺ perovskite Ba₂YMoO₆ has been suggested to have a valence bond glass ground state. Here we report on the low temperature properties of powder samples of isostructural Ba₂LuMoO₆: the only other known cubic 4d¹ perovskite with one magnetic cation. Our muon spectroscopy experiments show that magnetism in this material remains dynamic down to 60 mK without any spin freezing or magnetic order. A singlet-triplet excitation with a gap of Δ = 28 meV is observed in inelastic neutron scattering. These results are interpreted as a disordered valence bond glass ground state similar to Ba₂YMoO₆. Our results highlight the differences of the 4d¹ double perovskites in comparison to cubic 5d¹ analogues, which have both magnetic and multipolar order.

Introduction

Recently, 4d and 5d transition metal compounds have been widely investigated as potential hosts for exotic quantum states of matter^1,2,3,4. The origin of the rich physics of these heavy transition metal compounds lies in the interplay of strong electron correlation effects (Hubbard U) and significant spin-orbit coupling λ1. This can result in entanglement of the spin and orbital degrees of freedom. An important example of this is Sr₂IrO₄, in which the 5d⁵ Ir⁴⁺ cations have a spin-orbital entangled J_eff = 1/2 state leading to insulating behavior⁵. Moreover, honeycomb lattice iridates and α-RuCl₃ with J_eff = 1/2 pseudospins have emerged as possible realizations of Kitaev’s exactly solvable quantum spin liquid model^3,6. While these 4d⁵ and 5d⁵ J_eff = 1/2 systems have garnered the most attention, rich physics are also observed in other 4d and 5d materials². The 5d⁵ and 4d⁵ low-spin systems have one hole on the t_2g orbitals. A mirror of this situation occurs in 4d¹ and 5d¹ compounds with one electron on the t_2g orbitals.

Cubic A₂B’B”O₆ double perovskites, where the only magnetic cation is a 4d¹ or a 5d¹ transition metal on the B” site, have a variety of potential unusual ground states^1,2. The structure, as shown in Fig. 1a, consists of corner-sharing octahedra, where the B-site cations alternate in a rocksalt-type order forming an fcc lattice of the B” d¹ cations⁷. The octahedral crystal field splits the d-orbitals into six t_2g states and four e_g states. When the crystal field splitting is large enough to prevent t_2g-e_g mixing, the t_2g states can be described as having effective orbital angular momentum of L_eff = 1. Spin-orbit coupling, as observed in 4d and 5d transition metal compounds, further splits the t_2g orbitals into spin-orbital entangled J = L + S states: a J_eff = 3/2 quartet ground state and a J_eff = 1/2 doublet excited state (Fig. 1b)⁸. The J_eff = 3/2 ground state is nominally nonmagnetic with M = 2S–L = 0 as the spin and orbital moments oppose and cancel out. In real compounds, the orbital moment is reduced by hybridization with oxygen leading to a small overall moment². Conversely, in the excited J_eff = 1/2 state the spin and orbital moments add up to a larger moment.

Fig. 1: Structure and orbital splitting.

a The double perovskite structure of Ba₂LuMoO₆. Ba, Lu, Mo, and O are represented by the green, gray, blue, and red spheres, respectively. The magnetic Mo⁵⁺ (4d¹) cations in blue form an undistorted fcc lattice. b Scheme of the orbital splitting for a 4d¹ or a 5d¹ cation adapted from refs. ^5,8. The octahedral crystal field splits the d orbitals into t_2g and e_g states. The t_2g states have an effective orbital angular momentum L_eff = 1. Spin-orbit coupling further splits the six-fold degenerate t_2g states into a J_eff = 3/2 quartet ground state and a J_eff = 1/2 doublet excited state. Note that for 4d⁵ and 5d⁵ cations the situation is reversed and J_eff = 1/2 is the ground state while J_eff = 3/2 is the excited state.

The J_eff = 3/2 pseudospins and their complex interactions on the geometrically frustrated fcc lattice of d¹ double perovskites give rise to rich physics in these materials. This can enable bond-directional exchange on the fcc lattice similar to Kitaev interactions on the honeycomb lattice^9,10,11. Theoretical models of the d¹ fcc systems predict exotic ground states including spin liquid states^9,10,12, multipolar order¹² and valence bond glass states¹³. In terms of materials, the 5d¹ double perovskites Ba₂NaOsO₆^{14,15,16,17,18,19,20} and Ba₂MgReO₆^21,22,23,24 have been widely investigated for possible multipolar order. While both compounds have magnetically ordered ground states, there is indirect evidence of quadrupolar ordering above the magnetic ordering transition.

The most studied 4d¹ double perovskite is Ba₂YMoO₆^{25,26,27,28,29,30}. It has an fcc lattice of 4d¹ Mo⁵⁺ cations with a J_eff = 3/2 ground state. No structural distortions from cubic Fm$\bar 3$m symmetry are observed down to 3 K³¹, although a Jahn–Teller distortion would be expected in d¹ compounds even in the presence of strong spin-orbit coupling³². Despite strong antiferromagnetic interactions with Θ_CW = −160 K, muon spin rotation and relaxation measurements show that Ba₂YMoO₆ does not magnetically order, although a partial spin glass transition occurs at 1.3 K²⁹. Nuclear magnetic resonance (NMR) and specific heat measurements suggested the presence of spin singlets in the ground state^26,31. Inelastic neutron scattering measurements revealed a gapped singlet-triplet excitation with Δ = 28 meV²⁸.

These experimental results for Ba₂YMoO₆ have been interpreted as a valence bond glass (VBG) ground state, where nonmagnetic spin singlets gradually form in a disordered fashion as the temperature is decreased, while some orphan spins remain paramagnetic^13,26,29. Ba₂YMoO₆ has also been suggested to be a spin liquid candidate^9,10. The proposed spin liquid has specific power law scaling in susceptibility and heat capacity, which could be used to distinguish it from a valence bond glass state¹⁰. Both interpretations can explain the observed inelastic neutron scattering data^10,13, while the valence bond glass better explains the glassy behavior observed in AC susceptibility around 50 K^13,26. However, the partial spin glass transition at 1.3 K in Ba₂YMoO₆ is not expected for the valence bond glass nor the spin liquid. This raises the question whether a better d¹ candidate material for either of these unusual ground states could be found.

Ba₂LuMoO₆ is the only other known cubic 4d¹ double perovskite with one magnetic cation, but its ground state is not known³³. Like Ba₂YMoO₆, it retains the Fm$\bar 3$m symmetry with an fcc lattice of 4d¹ Mo⁵⁺ cations down to 2 K³³. In this article, we report on the low-temperature properties and possible ground states of Ba₂LuMoO₆. Our muon spin rotation and relaxation measurements reveal a lack of magnetic order or spin freezing down to 60 mK. Inelastic neutron scattering measurements show a gapped magnetic excitation with Δ = 28 meV, which is interpreted as a singlet-triplet excitation. The presence of both spin singlets and dynamic magnetism is interpreted as a valence bond glass state similar to Ba₂YMoO₆, but without freezing of orphan spins. Our work highlights the differences between the cubic 4d¹ double perovskites Ba₂LuMoO₆ and Ba₂YMoO₆ and their 5d¹ analogues Ba₂NaOsO₆ and Ba₂MgReO₆, which have both magnetic and quadrupolar order.

Results

X-ray diffraction

Phase purity and crystal structure of Ba₂LuMoO₆ samples were investigated using X-ray powder diffraction. The sample was phase pure without any impurity peaks in the diffraction pattern (Supplementary Fig. 1). The structure of Ba₂LuMoO₆ was found to be Fm$\bar 3$m in excellent agreement with previous neutron diffraction results³³ (Supplementary Table 1). The most common type of structural disorder observed in double perovskites is antisite disorder of the B-sites⁷. Our refined site occupancies are 0.99(1) Mo and 0.01(1) Lu on the Mo-site with 0.99(1) Lu and 0.01(1) Mo on the Lu-site. This shows Ba₂LuMoO₆ is a highly ordered double perovskite without significant antisite disorder similar to other Mo⁵⁺ double perovskites^25,27.

Magnetic susceptibility

The magnetic properties of Ba₂LuMoO₆ were investigated using an MPMS3 superconducting quantum interference device (SQUID) magnetometer. Magnetic susceptibility of Ba₂LuMoO₆ (Fig. 2a) exhibits typical paramagnetic behaviour. The zero-field cooled (ZFC) and field cooled (FC) curves overlap, therefore only the former is shown. The inverse magnetic susceptibility reveals two regions: a high-temperature Curie-Weiss region as expected and an additional low-temperature linear region. The change in slope between these two regions occurs at ≈50 K. Curie-Weiss fits to the high-temperature region between 200 and 300 K yielded a Curie-Weiss constant of Θ_CW = −114(1) K, indicating significant antiferromagnetic interactions between Mo⁵⁺ cations. The effective paramagnetic moment of Mo⁵⁺ was found to be μ_eff = 1.32(1) μ_B, which is lower than the 1.73 μ_B expected for a S = 1/2 cation. This reduced moment could be evidence of an unquenched orbital moment as expected in the J_eff = 3/2 state or strong quantum fluctuations^12,14. The low-temperature inverse susceptibility was fitted in the range 2–20 K yielding Θ_CW = −1.9(2) K and μ_eff = 0.67(1) μ_B.

Fig. 2: Magnetic susceptibility.

a DC magnetic susceptibility (black) and inverse molar magnetic susceptibility (red) of Ba₂LuMoO₆ as a function of temperature. Only the ZFC data is shown as the ZFC and FC curves overlap. The magnetic susceptibility appears paramagnetic with no transitions observed down to 2 K. In the inverse susceptibility two linear temperature regimes are observed: one at high temperatures (as expected) and one at low temperatures. b Field-dependent magnetization curve at 2 K. The M(H) curve has an S shape without any hysteresis as expected for paramagnetic materials at low temperatures.

The magnetic behaviour of Ba₂LuMoO₆ is very similar to Ba₂YMoO₆., which also has two linear Curie-Weiss regions in the inverse susceptibility. The overall antiferromagnetic interactions in Ba₂LuMoO₆ are somewhat weaker than in Ba₂YMoO₆ with Curie-Weiss constants Θ_CW = −114 K and Θ_CW = −160 K, respectively²⁶. The high-temperature effective paramagnetic moments are around 1.4 μ_B in both compounds. The low-temperature susceptibility is similar in both Ba₂LuMoO₆ and Ba₂YMoO₆, and the change in slope occurs around 50 K for both compounds. This feature in the inverse susceptibility of Ba₂YMoO₆ and related Ba_2-xSr_xYMoO₆ phases has been interpreted as the gradual formation of valence bond singlets^26,27. It should be noted that the change in inverse susceptibility occurs near 50 K²⁶, but spin singlets in Ba₂YMoO₆ were observed up to 125 K in inelastic neutron scattering²⁸.

Another possible explanation for the magnetic susceptibility is that at low temperatures the Mo⁵⁺ cations are in the J_eff = 3/2 ground state with a low moment, but at higher temperatures some electrons are excited to the high-moment J_eff = 1/2 state. This is supported by the low-temperature effective moments of ≈0.6–0.7 μ_B, which are very similar to those of 5d¹ double perovskites known to have a J_eff = 3/2 ground state. On the other hand, this excitation gap is far too large for thermal excitations at relevant temperatures as the spin-orbit coupling constant λ = 128 meV for Mo⁵⁺ corresponds to 1500 K^31,34. Recently, it has been proposed³⁵ that a dynamical Jahn-Teller effect could mix the J_eff = 3/2 and J_eff = 1/2 states explaining the observed change in the effective paramagnetic moment. Nevertheless, it is clear that magnetic susceptibility alone is insufficient evidence for a valence bond glass state.

The field-dependent magnetization M(H) curve at 2 K is shown in Fig. 2b. The curve has an S shape without any hysteresis and relatively low magnetization of less than 0.1 μ_B/Mo at 50 kOe. The field-dependent data is in qualitative agreement with the Brillouin function behavior expected of paramagnets at low temperatures. The low-temperature M(H) data are similar to those reported for Ba₂YMoO₆²⁶.

Specific heat

The specific heat of Ba₂LuMoO₆ was measured using a thermal relaxation method in order to investigate possible phase transitions such as magnetic ordering (Fig. 3). A small increase in C_p/T is observed at low temperatures, similar to Ba₂YMoO₆²⁶. The specific heat data for Ba₂LuMoO₆ does not contain any sharp lambda anomalies expected for magnetic ordering transitions. Despite the small magnetic moment of the 4d¹ and 5d¹ systems, lambda anomalies have been observed in magnetically ordered double perovskites such as La₂LiMoO₆ and Ba₂MgReO₆^22,36. Ba₂MgReO₆ also has a quadrupolar ordering transition T_q above T_N, which can be detected in the specific heat as a broader peak²². We do not observe any such peak in the specific heat data for Ba₂LuMoO₆. This suggests that there are no magnetic transitions in Ba₂LuMoO₆ at least down to 2 K.

Fig. 3: Specific heat.

Zero-field specific heat of Ba₂LuMoO₆ as function of temperature (black line) and the estimated lattice specific heat from Debye fits above 90 K (red dash line). No lambda anomalies indicative of magnetic order are observed down to 2 K. Moreover, there is no broad peak in specific heat associated with possible quadrupolar order in related materials. Inset: Estimated magnetic specific heat of Ba₂LuMoO₆. A broad maximum is observed around 30 K.

The specific heat of a magnetic compound consists of its magnetic specific heat and the lattice contribution to specific heat. The lattice contribution was estimated by fitting two Debye functions to the high-temperature data above 90 K with Debye temperatures 275 K and 956 K. The same approach has been previously used for Ba₂MgReO₆²². The estimated magnetic specific heat of Ba₂LuMoO₆ is shown in the Fig. 3 inset. The magnetic specific heat has a broad maximum around 30 K. No clear magnetic transition is observed. Integrating up to 90 K results in a magnetic entropy of 5.5 J K⁻¹ mol⁻¹. This is 48% of the expected entropy for a J_eff = 3/2 system, but close to that of an S = 1/2 system. It is common for magnetic materials that do not order to retain significant spin entropy at low temperatures³⁷. Moreover, the magnetic specific heat and integrated entropy both have significant uncertainties due to the difficulty of estimating and subtracting the lattice contribution. For these reasons, we cannot confidently determine the spin state of the 4d¹ Mo cation based on the specific heat.

Muon spin rotation and relaxation

Muon spin rotation and relaxation (μSR) experiments were performed to investigate the magnetic ground state of Ba₂LuMoO₆. Muons are a highly sensitive local probe of both static and dynamic magnetism³⁸. In a magnetically ordered material below T_N, spontaneous oscillations in the measured positron asymmetry in zero-field (ZF) μSR arise from the precession of the muon spin in the static local field. These are observed in the magnetically ordered related Mo⁵⁺ double perovskite La₂LiMoO₆³¹. In a spin glass, the muon spins feel a distribution of static local fields below the freezing transition. For any type of static magnetism, the asymmetry should return to 1/3^rd of the initial asymmetry at high counting times³⁸.

We do not observe any signatures of magnetic ordering or spin freezing in Ba₂LuMoO₆ in the ZF-μSR data down to 60 mK (Fig. 4a). Instead, simple exponential relaxation behavior, typical of dynamic magnetic systems is observed. The time resolution at pulsed muon sources such as ISIS is not always sufficient to detect oscillations in magnetically ordered materials. However, if Ba₂LuMoO₆ was magnetically ordered, the static local fields would still strongly depolarize the muon spins leading to a drop in the initial asymmetry below T_N³⁸. This is not observed. We also do not observe the 1/3^rd tail typical of ordered materials or spin glasses. The lack of any magnetic transitions or spin freezing is consistent with both the proposed valence bond glass and spin liquid states. In a valence bond glass, the dynamic magnetism would be related to the paramagnetic orphan spins. The main component of the VBG state, the disordered spin singlets, cannot be observed by μSR as they are nonmagnetic.

Fig. 4: Muon spin rotation and relaxation (μSR).

a Zero-field μSR of Ba₂LuMoO₆ at different temperatures. Magnetism in Ba₂LuMoO₆ remains dynamic down to 60 mK as no magnetic transitions or spin freezing are observed. Muon spin relaxation of the sample can be described with a simple exponential Ae^-λt. b Muon spin rotation and relaxation rate λ as a function of temperature. Upon cooling below 1.5 K, λ starts to rapidly increase. This shows that the internal dynamic magnetic fields slow down. Below 300 mK the relaxation rate stays constant, which indicates that the ground state is dynamic. c Longitudinal-field muon spin rotation and relaxation of Ba₂LuMoO₆ at 100 mK. Decoupling the muon spins from the internal magnetic fields required a field of over 1000 G, confirming that they are related to electronic and not nuclear spins. Error bars in a and c represent one standard deviation, whereas the error bars in b represent standard error.

The zero-field μSR between 60 mK and 4 K could be fitted using a simple exponential relaxation function:

$$A\left( t \right) = A_{{{{\mathrm{exp}}}}}{{{\mathrm{e}}}}^{ - \lambda t} + A_{{{{\mathrm{flat}}}}}$$

(1)

where A is the total asymmetry, A_exp is the asymmetry of the relaxing component, λ is the muon spin relaxation rate and A_flat is a flat background term from the silver sample holder. This flat background term was unusually high due to sample settling in the sample holder during measurement. The fitted muon spin relaxation rate λ is plotted as a function of temperature in Fig. 4b. The relaxation rate is inversely proportional to the fluctuation frequency of the dynamic magnetic fields. The relaxation rate starts to increase as the temperature is lowered below 2 K. This shows that the dynamic magnetic fields in Ba₂LuMoO₆ slow down as the temperature decreases as expected. Below 300 mK, the relaxation rate plateaus and stays constant down to at least 60 mK. This plateau in relaxation rate is a common feature of quantum spin liquid candidates such as Zn_xCu_4−x(OH)₆Cl₂³⁹, Zn_xCu_4−x(OH)₆Cl₂⁴⁰ and Sr₂CuTe_0.5W_0.5O₆^41,42. If Ba₂LuMoO₆ had a spin glass or a magnetic transition, the relaxation rate would peak at the transition temperature and no plateau would be observed. The plateau in muon spin relaxation, in addition to the lack of 1/3^rd tail or drop in initial asymmetry down to 60 mK, confirms that Ba₂LuMoO₆ has a dynamic magnetic ground state ruling out magnetic order or spin freezing.

Longitudinal field (LF) μSR was measured at 100 mK in fields up to 2000 G (Fig. 4c). In LF-μSR, as the applied longitudinal field is increased, the muon spins start to decouple from the internal magnetic fields. This decoupling is observed as an increasing flat background, and the asymmetry becomes entirely flat when the muon spins are fully decoupled from the internal fields. Weak nuclear fields can be completely decoupled with low fields of 20–50 G. In Ba₂LuMoO₆, fully decoupling the muon spins required fields larger than 1000 G. This shows that the dynamic magnetism observed in the ZF-μSR measurements is of electronic origin and not due to nuclear magnetic moments.

The muon spin rotation and relaxation measurements reveal significant differences between Ba₂LuMoO₆ and Ba₂YMoO₆. For Ba₂LuMoO₆, the ZF-μSR data can be described with exponential relaxation, similar to many quantum spin liquid candidates. This dynamic magnetism is also consistent with the paramagnetic orphan spins expected in a valence bond glass state. In the case of Ba₂YMoO₆, two separate muon environments were proposed: a non-magnetic one with weak exponential relaxation and a magnetic one described with a stretched exponential²⁹. The nonmagnetic muon environment in Ba₂YMoO₆ is related to the spin singlets, while the magnetic environment is related to paramagnetic orphan spins. A transition in the magnetic environment was observed at 1.3 K, which was interpreted as freezing of the orphan spins in Ba₂YMoO₆. The absence of this spin freezing makes Ba₂LuMoO₆ a more promising valence bond glass candidate than Ba₂YMoO₆.

Inelastic neutron scattering

Magnetic excitations in Ba₂LuMoO₆ were investigated using inelastic neutron scattering. The nonmagnetic spin singlets expected in a valence bond glass state can be probed using this technique. The inelastic scattering from the sample was weak, and features in the spectra were only visible after subtraction of Bose-factor corrected high-temperature data. The spectra measured at 7.5 K with incident energy E_i = 70 meV are shown in Fig. 5a. The sample spectra are featureless except for a gapped excitation at 28 meV. Note that the feature observed up to 10 meV in this figure is the coherent and incoherent elastic scattering, which is broad in energy transfer on the intensity scale necessary to see this excitation. The excitation at 28 meV gets weaker with increasing |Q | suggesting that it is magnetic in origin. The energy of this excitation is mostly |Q | independent, and it can be interpreted as a weakly dispersing singlet-triplet excitation^28,43. This indicates that Ba₂LuMoO₆ has a singlet ground state with a singlet-triplet gap of Δ = 28 meV. This is consistent with the singlet-triplet gap measured for Ba₂YMoO₆²⁸.

Fig. 5: Inelastic neutron scattering of Ba₂LuMoO₆.

Inelastic neutron scattering intensity maps measured with E_i = 70 meV at a 7.5 K, b 32 K, c 75 K and d 100 K as function of energy transfer and |Q |, corrected for background by subtracting Bose-factor corrected 200 K data. A weakly dispersing, gapped singlet-triplet excitation is observed at 28 meV in the 7.5 K and 32 K data. At higher temperatures singlets are no longer observed. e Cuts of the E_i = 70 meV Bose-factor corrected inelastic neutron scattering data showing the integrated intensity for 0 < |Q| < 2 Å⁻¹ as function of energy transfer. The singlet-triplet excitation centered at 28 meV decreases in intensity with increasing temperature, but no other features are observed. f Cut of Bose-factor corrected 7.5 K E_i = 30 meV data minus the Bose-factor corrected 200 K E_i = 30 meV data showing additional excitation features at 11 and 17 meV. Error bars represent one standard deviation.

The singlet-triplet excitation is clearly observed also in the 32 K data (Fig. 5b), showing that the spin singlets persist up to at least this temperature. At 75 K (Fig. 5c) and 100 K (Fig. 5d) the excitation is already difficult to observe. Figure 5e shows the integrated scattering intensity for 0 < |Q| < 2 Å⁻¹ as a function of energy transfer at different temperatures. The excitation at 28 meV becomes weaker as temperature increases, which is also consistent with it being of magnetic origin. If the feature at 28 meV was related to a crystal field excitation of Mo⁵⁺, it would not decay this fast with increasing temperature as 28 meV corresponds to over 300 K. Paramagnetic scattering within the gap is not visible in the 70 meV incident energy data, but in the 30 meV data at 7.5 K there is evidence of a far weaker continuum of states peaked at 11 and 17 meV (Fig. 5f), similar to those observed in Ba₂YMoO₆²⁸.

Discussion

What is the ground state of Ba₂LuMoO₆? The proposed ground states of a d¹ double perovskite are different types of magnetic order, spin glass, multipolar order, valence bond glass and spin liquid. A previous neutron diffraction study suggests the material is not magnetically ordered down to 2 K, since no magnetic Bragg peaks were observed³³. However, as the intensity of the magnetic scattering is proportional to the square of the magnetic moment (<1 μ_B), establishing the presence or absence of magnetic order using neutron diffraction is challenging in the d¹ double perovskites. Our muon spin rotation and relaxation results conclusively rule out magnetic order or spin freezing down to 60 mK. It is unlikely that the ground state involves multipolar order, as the double perovskites with quadrupolar order also develop magnetic order at low temperatures^{14,16,20,21,22,23}. Moreover, the quadrupolar ordering transition can be observed in the specific heat data as a broad peak, which is not present for Ba₂LuMoO₆. Therefore, the main candidate ground states are a valence bond glass state or a spin liquid.

Ba₂LuMoO₆ has several properties that support a valence bond glass state. First, the magnetic susceptibility has two Curie-Weiss regions, where the low-temperature region is consistent with the pseudo-gap expected of a valence bond glass state^26,44. The muon results support a dynamic magnetic ground state, which is consistent with the presence of dynamic orphan spins as expected for a VBG. Finally, we observe a singlet-triplet excitation in inelastic neutron scattering experiments as expected of the dimer singlets of a VBG. Orbital excitations of the valence bond glass state can also explain the weak in-gap scattering in Ba₂LuMoO₆¹³. The presence of all these features makes a valence bond glass state the natural explanation for the ground state of Ba₂LuMoO₆¹³.

The absence of any magnetic order or spin freezing in the muon experiments could also be interpreted as a spin liquid state. Natori et al.^9,10 have proposed a chiral spin-orbital liquid as a possible ground state for a d¹ double perovskite. The simulated inelastic neutron scattering spectra of such a phase consists of a single broad excitation at a defined energy¹⁰, which is also what we observe for Ba₂LuMoO₆. However, it does not explain the weak in-gap inelastic scattering. The proposed gapless spin liquid has a vanishing density of states, which results in a vanishing magnetic susceptibility (χ ∝ T^1/2) and specific heat (C_p ∝ T^3/2) at low temperatures⁹. This is inconsistent with our results for Ba₂LuMoO₆: in fact, we observe a 1/T relation for the low-temperature magnetic susceptibility. While we cannot completely rule out the possibility of some type of a spin liquid state in Ba₂LuMoO₆, it does not appear to be the proposed chiral spin liquid state.

Finally, quenched disorder in quantum spin systems can induce a random singlet ground state, which has properties similar to those of spin liquids. Random singlet states have mainly been studied on two-dimensional lattices^{45,46,47,48,49,50} and the pyrochlore lattice^51,52,53, and the authors are unaware of any theoretical predictions of such state on the fcc lattice of Ba₂LuMoO₆. A key requirement for a random-singlet state is the presence of disorder. The main type of structural disorder in double perovskites is antisite disorder between the two B-sites⁷. Rietveld refinement of our laboratory x-ray diffraction data revealed 1(1)% antisite disorder between Lu³⁺ and Mo⁵⁺, whereas a previous neutron diffraction study on Ba₂LuMoO₆ did not reveal any antisite disorder³³. This is comparable to Ba₂YMoO₆, where 3% antisite disorder was detected by NMR³¹. Whether this would be sufficient disorder to form a random singlet state in the fcc lattice is an open question. The lack of magnetic ordering or spin freezing down to 60 mK is consistent with a random singlet state, as is the upturn in magnetic susceptibility at low temperatures. In terms of inelastic neutron scattering, we do not observe significant scattering at the |Q | positions of the magnetic Bragg peaks of the ordered parent phases as is the case in the Cu²⁺ double perovskite Sr₂CuTe_1-xW_xO₆^54,55,56, which has been proposed to have a random singlet ground state^48,49. Cuts of the elastic line (Supplementary Fig. 2) and various energy ranges (Supplementary Figs. 3–5) confirm such scattering is not present for Ba₂LuMoO₆.

It should be noted that the main theory papers on valence bond glass and spin liquid ground states on fcc lattices assume a J_eff = 3/2 state on the Mo⁵⁺, whereas a random singlet state could likely arise in a S = 1/2 scenario as well. While the lower than expected paramagnetic moment suggests the presence of a partially unquenched orbital moment as opposed to a pure S = 1/2 state, we are unable to convincingly establish a J_eff = 3/2 state based on the experimental data. Given that a theoretical framework for understanding all the physical properties of Ba₂LuMoO₆ exists, the valence bond glass state of J_eff = 3/2 pseudospins, the authors have chosen to use this interpretation.

The natural comparison for Ba₂LuMoO₆ is Ba₂YMoO₆, which has also been suggested to be a valence bond glass. Both compounds are cubic Mo⁵⁺ double perovskites with similar lattice parameters, where a static Jahn-Teller distortion from cubic symmetry is not observed even at 2 K^27,33. The absence of anomalous oxide displacements, as manifested in anisotropic oxide displacements³³, also rule out any dynamic or disordered local breaking of the Mo⁵⁺ ligand crystal field in Ba₂LuMoO₆. The magnetic susceptibilities of Ba₂LuMoO₆ and Ba₂YMoO₆ are very similar: two Curie-Weiss regions are observed and interpreted as related to a pseudo-gap of the VBG state. In inelastic neutron scattering, both compounds have a singlet-triplet excitation with the same energy of 28 meV. The main difference is in the muon spin relaxation responses. In Ba₂LuMoO₆, we do not observe any static magnetism down to 60 mK. The muon spin relaxation rate of the dynamic low-temperature state is very high, corresponding to exceedingly slow field fluctuations. In comparison, the orphan spins of Ba₂YMoO₆ form a dilute spin glass at T_g= 1.3 K, and the muon spin relaxation rate is much lower than in Ba₂LuMoO₆. The lack of this spin glass transition makes Ba₂LuMoO₆ a more promising system for investigating exotic ground states of the d¹ double perovskites.

Recently, the cubic double perovskite Ba₂Y_xWO₆ has been proposed to be a W⁵⁺ 5d¹ system with a possible valence bond glass state⁵⁷. Muon spin relaxation experiments revealed the lack of magnetic ordering or spin freezing down to 26 mK⁵⁷. A broad maximum was observed in the specific heat similar to Ba₂YMoO₆ and Ba₂LuMoO₆, and magnetic susceptibility suggested the presence of random magnetism⁵⁷. However, this material is known to have a high number of Y³⁺ vacancies^58,59, which were not considered in the study⁵⁷. The oxidation state of the tungsten cation depends on the yttrium stoichiometry: if x = 2/3, the compound contains only nonmagnetic W⁶⁺. If x = 1, all tungsten in the material is W⁵⁺ and it becomes a true 5d¹ fcc antiferromagnet. Previous neutron diffraction studies put the solubility limit of Ba₂Y_xWO₆ at 2/3 ≤ x ≤ 0.78⁶⁰. This means that at most 1/3^rd of tungsten cations are magnetic W⁵⁺, and the fcc lattice is significantly diluted. Thus, the lack of magnetic ordering in muon experiments⁵⁷ is likely due to the low proportion of magnetic W⁵⁺ cations in the material. This is supported by the very low effective paramagnetic moment observed at high temperatures⁵⁷. The specific heat data is identical to previous data on Ba₂Y_0.78WO₆, and the resulting high magnetic entropy can be attributed to a poor lattice match⁶⁰. Due to the uncertainty around stoichiometry, tungsten oxidation state and magnetic dilution in Ba₂Y_xWO₆, we will not include it in the following discussion.

We present a comparison of the magnetic properties of related d¹ double perovskites in Table 1. Ba₂LuMoO₆ and Ba₂YMoO₆ are the only cubic Mo⁵⁺ double perovskites with one magnetic cation. Sr₂YMoO₆ adopts a monoclinic P2₁/n structure due to the smaller size of the A-site cation²⁷. A ferromagnetic transition is observed in the magnetic susceptibility at 8 K, but the saturation magnetization is only 0.12 μ_B per Mo. Magnetic Bragg peaks were not observed by neutron diffraction, but this could be due to the very weak magnetic scattering from such small moments. La₂LiMoO₆ is also monoclinic with a Type I antiferromagnetic ground state and an unusually low ordered moment of 0.32 μ_B³⁶. This is likely due to the orbital moment, which in a d¹ compound will oppose the spin moment due to spin-orbit coupling. Sr₂ScMoO₆ is a tetragonal double perovskite with a high degree of B-site disorder between Mo⁵⁺ and Sc³⁺⁶¹. The magnetic susceptibility does not follow the Curie-Weiss law and the magnetic ground state is not known.

Table 1 Magnetic properties of double perovskites, where the only magnetic cation is a d¹ cation (Mo⁵⁺ 4d¹, Re⁶⁺ 5d¹ or Os⁷⁺ 5d¹) on the B”-site. Adapted and expanded from refs. ^12,21.

The unusual properties of the d¹ double perovskites are linked to the spin-orbital entangled J_eff = 3/2 pseudospins and their complex interactions. However, structural distortions from cubic symmetry can lift the degeneracy of these states and quench the orbital moment. It is therefore important to compare the cubic 4d¹ Ba₂LuMoO₆ double perovskite to the cubic 5d¹ analogues, that also retain the J_eff = 3/2 pseudospins. Four such cubic compounds are known, where the only magnetic cation is a 5d¹ cation. Ba₂MgReO₆ has canted antiferromagnetic order below T_N = 18 K. More importantly, there is growing evidence of quadrupolar ordering in Ba₂MgReO₆ at T_q = 33 K based on specific heat, X-ray and neutron diffraction and resonant inelastic X-ray scattering experiments^21,22,23,24. The quadrupolar ordering is associated with a subtle structural transition from cubic to tetragonal symmetry^23,24. Ba₂ZnReO₆ is similar to Ba₂MgReO₆ with a magnetic ordering transition at 16 K and a broad bump in specific heat at 33 K, which could be related to quadrupolar ordering^21,62,63. However, the type of magnetic ordering is different: Ba₂ZnReO₆ is a canted ferromagnet⁶², while Ba₂MgReO₆ is a canted antiferromagnet²³. Ba₂NaOsO₆ is also a canted ferromagnet but with a lower transition temperature of T_C = 7 K. The magnetic order is thought to be related to orbital ordering due to a quadrupolar transition at T_q = 9.5 K^{14,15,16,17,18,19,20}. Ba₂LiOsO₆ is antiferromagnetic below T_N = 8 K with some disorder in the static fields based on muon experiments^15,64.

The cubic 4d¹ double perovskites Ba₂LuMoO₆ and Ba₂YMoO₆ have very similar properties, but they differ greatly from the 5d¹ analogues that all involve static magnetism, with some compounds also having likely multipolar order. The origin of this difference between 4d¹ and 5d¹ double perovskites is not well understood. In general, 5d systems have stronger spin-orbit coupling λ and weaker on-site coulombic repulsion U in comparison to 4d systems¹. The differences in the strength of the spin-orbit coupling λ could explain the manifest difference in ground states between the 4d¹ and 5d¹ double perovskites. Romhányi, Balents and Jackeli¹³ have proposed a microscopic model for the d¹ double perovskites. In this model, a weaker λ favors a disordered dimer-singlet phase as observed in 4d¹ Ba₂LuMoO₆ and Ba₂YMoO₆, whereas a stronger λ favors magnetically ordered phases found in the 5d¹ analogues Ba₂MgReO₆ and Ba₂NaOsO₆. However, questions remain over the role of the Jahn-Teller effects, which are not observed in the 4d¹ systems, but are thought to drive the quadrupolar ordering in the 5d¹ systems².

In conclusion, our muon and neutron experiments support a valence bond glass ground state in the 4d¹ fcc antiferromagnet Ba₂LuMoO₆. Inelastic neutron scattering experiments confirmed the presence of spin singlets with a singlet-triplet gap of Δ = 28 meV. In addition to the nonmagnetic singlets, muon spin rotation and relaxation measurements confirm the presence of dynamic electronic spins down to 60 mK. These results are interpreted as a valence bond glass state, where spin singlet dimers form in a disordered manner and some of the leftover orphan spins remain paramagnetic¹³. However, a quantum spin liquid¹⁰ or a disorder-induced random-singlet state cannot be ruled out. The properties of Ba₂LuMoO₆ are very similar to those of isostructural Ba₂YMoO₆, which has been proposed to be a valence bond glass. The lack of any spin freezing at low temperatures makes Ba₂LuMoO₆ a more promising candidate for such a ground state than Ba₂YMoO₆, in which the orphan spins freeze into a spin glass at 1.3 K²⁹. The behavior of the cubic 4d¹ double perovskites Ba₂LuMoO₆ and Ba₂YMoO₆ is fundamentally different to that of the 5d¹ analogues Ba₂MgReO₆, Ba₂ZnReO₆ and Ba₂NaOsO₆, which have both magnetic and multipolar order.

Methods

Sample synthesis and purity

Polycrystalline powder samples of Ba₂LuMoO₆ were prepared by a solid-state reaction method. Stoichiometric amounts of BaCO₃ (Alfa Aesar 99.997%), Lu₂O₃ (Alfa Aesar 99.99%, pre-treated at 800 °C), and MoO₃ (Alfa Aesar 99.998%) were mixed, pelletized and calcined at 800 °C in air for 24 h. The pellets were reground and fired at 1250 °C in flowing 5% H₂/N₂ gas (60 cm³/min) for 96 h with intermittent grindings. Phase purity of the samples was investigated using X-ray powder diffraction. The data were collected with a Panalytical X’Pert 3 Powder diffractometer (Cu K_α radiation). Rietveld refinement⁶⁵ was carried out using FULLPROF⁶⁶ and the crystal structure was visualized using VESTA⁶⁷.

Bulk properties

Magnetic properties were measured with a Quantum Design MPMS3 SQUID magnetometer. 100 mg of sample powder was enclosed in a gelatin capsule and placed in a plastic straw. The sample shape was estimated as a cylinder of 5 mm diameter and 2 mm length. This corresponds to a moment artefact of 1.072 in a DC measurement on the MPMS3, and all measured moments were divided by this factor. The temperature-dependent magnetization was measured in a field of 1000 Oe from 2 to 300 K in zero-field cool (ZFC) and field cool (FC) modes. Field-dependent magnetization was measured at 2 K between −50,000 and 50,000 Oe. The specific heat of Ba₂LuMoO₆ was measured on a Quantum Design Physical Property Measurement System. Sample and silver powders were mixed in a 1:1 ratio and pressed into pellets. Specific heat was then measured on pellet pieces of ≈20 mg mass. The silver contribution was subtracted from the measured specific heat using data from a pressed pellet of pure silver powder.

Muon spectroscopy

Muon spin rotation and relaxation (μSR) experiments were carried out at the MUSR instrument at the ISIS Neutron and Muon Source. Approximately 3 g of sample powder was attached to a silver sample holder with GE varnish. The sample was cooled in a dilution fridge. Zero-field, transverse field and longitudinal field measurements were carried out between 60 mK and 4 K. The collected data is available online⁶⁸.

Inelastic neutron scattering

Inelastic neutron scattering was measured on an 8 g Ba₂LuMoO₆ sample on the MERLIN spectrometer at ISIS Neutron and Muon Source. The sample powder was contained in a cylindrical aluminium can to give an annular geometry. Measurements were performed with incident energies of 70 and 30 meV at temperatures between 7 and 200 K. The data were reduced using standard Mantid reduction procedures⁶⁹. The inelastic neutron scattering data is available online⁷⁰.