Spin liquid behavior of a three-dimensional magnetic system Ba₃NiIr₂O₉ with $S$ = 1

Polycrystalline BNIO was synthesized by solid state synthesis route. Members of the $A_{3}MM^{\prime}_{2}$O₉ series can have site disorder between face shared, and corner shared octahedra Middey et al. (2011). It is also well known that structural disorder often jeopardizes QSL behavior, resulting in magnetic order or spin glass freezing Zhong et al. (2019). All peaks of the powder XRD pattern of BNIO (Fig. 1(e)) can be indexed and refined with 6$H$ structure having space group $P6_{3}/mmc$. The refinement also confirms that all corner-shared (face-shared) octahderal units are occupied by Ni (Ir) without any Ni-Ir site disorder. The structural parameters obtained from the refinement have been listed in SM sup . The temperature dependent XRD measurements down to 15 K also rules out any structural transition. Having confirmed that both Ni and Ir ions form triangular lattices in a-b plane without any disorder, we want to verify whether Ni has indeed $S$ = 1 state. For this purpose, XAS measurements were carried out Freeland, Van Veenendaal, and Chakhalian (2016). The comparison of Ni $L_{3,2}$ XAS line shape and energy position of BNIO, Ni²⁺O, and NdNi³⁺O₃ (Fig.  1(e)) testifies the desired +2 oxidation of Ni in present case. The octahedral crystal field of Ni²⁺ ($d^{8}$: $t_{2g}^{6}$, $e_{g}^{2}$) ensures $S$ = 1 on Ni sublattice.

Electrical measurement demonstrates insulating nature of the sample (inset of Fig.2 (a)), which can be fitted using Mott’s variable range hopping (VRH) Hill (1976) model in three-dimensions ($\rho=\rho_{o}\exp(T_{o}/T)^{1/4}$) as shown in Fig.2 (a). We also note that the insulating behavior of Ba₃ZnIr₂O₉ is well described by VRH in two-dimensions. This difference between two compounds is a manifestation of electron hopping paths along the Ni-O-Ir bonds. Our electronic structure calculations (see SM sup ) further demonstrate that the insulating state can be obtained only by considering both electron correlation and SOC, implying that BNIO is a SOC driven Mott insulator Kim et al. (2008).

The temperature dependent field cooled and zero-field cooled magnetic susceptibility does not differ from each other and also do not show any anomaly (Fig. 2(b)). This strongly implies absence of any long range magnetic ordering and spin glass behavior. We have fitted the data by a modified Cuire-Weiss law ($\chi$ = $\chi_{0}$ + $\frac{C_{W}}{T-\theta_{CW}}$) where $\chi_{0}$, $C_{W}$ and $\theta_{CW}$ represents temperature independent susceptibility contribution, Curie constant and Curie temperature, respectively. The fitting, shown as plot of 1/($\chi$-$\chi_{0}$) vs. $T$ in right axis of Fig. 2(b), results a $\theta_{CW}\sim$ -15 K. Negative values of Curie-Weiss temperature signify net antiferromagnetic interaction among the spins of BNIO. The relatively smaller value of $\theta_{CW}$ is related to the presence of multiple exchange pathways, which will be discussed in later part of this paper. The effective magnetic moment $\mu_{eff}$ (= $\sqrt{8C_{W}}$) is found to be $\sim$ 3.65 $\mu_{B}$ from the fitting. Interestingly, the effective magnetic moment of a similar compound Ba₃NiSb₂O₉, with nonmagnetic Sb⁵⁺ was reported to be around 3.54 $\mu_{B}$ Cheng et al. (2011). This gives an estimate of $g$-factor $\sim$ 2.5, similar to other Ni²⁺ based systems carlin_paramagnetism_1986 . If we assume similar value for the present compound, the effective Ir moment turns out to be 0.9 $\mu_{B}$ per dimer (= $\sqrt{\mu_{eff}^{2}-\mu_{Ni}^{2}}$) i.e. 0.64 $\mu_{B}$/Ir. This is very similar to the Ir moment (0.5 - 0.6 $\mu_{B}$) reported for Ba₃MgIr₂O₉ Nag et al. (2018a), though a nonmagnetic $J$ = 0 state is expected for Ir⁵⁺ from a pure ionic picture. Thus, our analysis highlights both Ni²⁺ and Ir⁵⁺ participate in magnetism of BNIO compound. The estimated magnetic moments from our LSDA+U+SOC calculations (shown in SM sup ) are also in good agreement with our experimental results. Fig. 2 (c) shows $M$-$H$ done at 2 K between $\pm$ 9 T. The absence of any hysteresis again confirms absence of ferromagnetism and spin glass freezing at 2 K. The presence of antiferromagnetic interaction without any long range magnetic ordering or spin glass transition strongly indicates that BNIO is a favorable candidate of QSL.

In order to further investigate the magnetic nature of the sample, we have measured specific heat ($C_{p}$) from 100 mK to 200 K. $C_{p}$ not only probes the presence/absence of any long-range magnetic ordering but also provides very crucial information about the nature of low energy excitation. The absence of any $\lambda$-anomaly (Fig. 3(a)) again confirms absence of long-range order and/or any structural transition down to 100 mK, consistent with the magnetic measurement and XRD results, respectively. For an insulating compound with magnetic spins, $C_{p}$ consists of lattice specific heat ($C_{lat}$) and magnetic specific heat ($C_{m}$). In absence of any analogous non-magnetic compound, the contribution of $C_{lat}$ has been evaluated by fitting $C_{p}$ in 30 K - 200 K range by a Debye-Einstein equation with one Debye term and two Einstein terms (details are in SM sup ) and extrapolating the fitted curve down to 100 mK. A broad peak is observed around 7 K in $C_{m}$ vs. $T$ plot (Fig. 3(a)). We can not capture this feature by Schottky anomaly, arising due to energy level splitting (see SM sup ). On the other hand, such feature is commonly observed in spin liquid materials and thus, could be considered as a signature of crossover from thermally disordered paramagnet to quantum disordered spin liquid state Okamoto et al. (2007b); Balents (2010); Li et al. (2015); Dey et al. (2017). The position of this broad peak shows negligible shifts with the application of magnetic field (shifts $\sim$ 1 K for applied field of 12 T).

At low temperature, $C_{m}$ follows power-law behavior $C_{m}$ = $\gamma T^{\alpha}$ (Fig. 3(c)). For zero field, the magnitude of coefficient $\gamma$ is 45 mJ/mol K² and the exponent $\alpha$ is 1.0$\pm$0.05 within 0.1 K - 0.6 K range. The value of $\gamma$ is very similar to other gapless spin liquid candidates: like Ba₃CuSb₂O₉ (43.4 mJ/mol K²) Zhou et al. (2011), Ba₂YIrO₆ (44 mJ/mol K²) Nag et al. (2018b), Ba₃ZnIr₂O₉ (25.9 mJ/mol K²) Nag et al. (2016), Sr₂Cu(Te_0.5W_0.5)O₆ (54.2 mJ/mol K²) Mustonen et al. (2018). Also, the linear $T$ behavior with nonzero $\gamma$ in an insulator comes due to gapless spinon excitations with a Fermi surface and has been reported in several organic and inorganic spin liquid candidates Yamashita et al. (2008, 2011); Zhou et al. (2011); Cheng et al. (2011); Mustonen et al. (2018); Clark et al. (2014); Uematsu and Kawamura (2019). Otherwise, any gapped excitation would result in an exponential dependence of $C_{m}$ on $T$. $\alpha$ becomes 2.6$\pm$0.05 within 0.8 K - 2.1 K. Interestingly, the application of an external field destroys the linear $T$ behavior of $C_{m}$. For $\mu_{0}H$ = 4 T, $\alpha$ becomes 2 for 0.15 K $\leq T\leq$ 0.50 K and 2.9 for 0.5 K $\leq T\leq$ 2.4 K. We note that $\alpha$ is found to be between 2 to 3 for several spin nematic phase Nakatsuji et al. (2005, 2007); Povarov et al. (2019); Kohama et al. (2019). Further studies are necessary to investigate the possibility of transition from spin liquid to spin nematic phase by the application of a magnetic field. The amount of released magnetic entropy ($S_{m}$) is evaluated by integrating $C_{m}/T$ w.r.t. $T$ and is shown in Fig. 3(d). For BNIO, the entropy saturates at 6.9 J/mol K for zero field measurement, which is only 75% of the total entropy expected for even a $S=1$ system [$R$ln(2$S$+1), where $R$ is universal gas constant]. The retention of a large amount of entropy at low temperature is another signature of the spin-liquid nature of BNIO, which has been reported as well for many other QSL Zhou et al. (2011); Cheng et al. (2011); Mustonen et al. (2018); Yamashita et al. (2008).

To have a further understanding of the magnetic behavior at low temperature, we have performed $\mu$SR measurements, which is a very sensitive local probe to detect a weak magnetic field, even of the order of 0.1 Oe Blundell (1999). Fig. 4(a) shows asymmetry vs time curves for zero-field (ZF) measurements at selected temperatures. Neither any oscillation nor the characteristic 1/3^rd recovery of the asymmetry is observed down to 60 mK, strongly implying the absence of long-range magnetic ordering or spin freezing. For a system with two interacting spin networks, the local magnetic field, felt by a muon at a stopping site is contributed by both magnetic sublattices. In such cases, the muon relaxation function is generally described by a product of two response functions, representing local fields from two spin networks Uemura et al. (1985). However, our such attempts considering different possible combinations of relaxation functions, including a spin glass like relaxation Morenzoni et al. (2008); Uemura et al. (1985), did not provide a satisfactory fitting of the experimental observed data (see SM sup ). This further supports the absence of spin glass freezing in present case.

Interestingly, similar to the other hexagonal Ba${}_{3}M$Ir₂O₉ Nag et al. (2016, 2018a), these asymmetry curves consist of one fast relaxing, one moderately relaxing, and one hardly relaxing components. We have fitted these curves using a combination of two dynamical relaxation functions with a common fluctuation rate $\nu$ and a Kubo-Toyabe function (KT) Hayano et al. (1979),

where $A_{1}$, $A_{2}$, $A_{3}$ are amplitudes. The static KT function, corresponding to the hardly relaxing component, accounts for the muons stopping at the silver sample holder as we find that the relaxation curve from the bare sample holder can also be described by a KT function with similar $\delta$. The dynamical relaxation, arising due to the presence of a field distribution ($\Delta H$) with a fluctuation rate ($\nu$) is represented by the Keren function $G(t,\Delta H,\nu)$ Keren (1994). The presence of two dynamical relaxations implies two inequivalent muon stopping sites, which are likely to be related with the two types of crystallographically inequivalent oxygen in hexagonal Ba${}_{3}M$Ir₂O₉ Nag et al. (2016, 2018a). The asymmetry data over a large temperature range (60 mK - 150 K) have been fitted by allowing $\nu$ to vary with $T$ and, the extracted values of $\nu$ has been shown as a function of $T$ in Fig. 4(b). The background contribution is different between measurements in dilution refrigerator and helium cryostat and, has been kept fixed for our analysis within the corresponding temperature range. The inequality $\nu>\gamma\Delta H$ ( $\gamma$ = muon gyromagnetic ratio = 2$\pi\times$135.5 Mrad s^-1 T^-1) holds for both relaxing components as $\gamma\Delta H_{1}\sim$0.425 MHz and $\gamma\Delta H_{2}\sim$ 0.09 MHz for the lowest temperature of our measurement (60 mK). This justifies the use of dynamical relaxation functions and establishes spin liquid nature of BNIO. We note that the value of $\nu$ ($\sim$ 4 MHz) at low temperature for BNIO is one order of magnitude smaller than Ba₃ZnIr₂O₉ Nag et al. (2016) and is likely to be related with the involvement of large spins on the Ni sublattice.

$\mu$-SR spectra, recorded at 100 mK in presence of an applied longitudinal field (LF) have further corroborated QSL nature of BNIO. In case of relaxation arising from a static internal field distribution with width $\Delta H_{i}$, an applied LF $\sim$ 5-10$\Delta H_{i}$ would completely suppress the relaxation. From the analysis of ZF $\mu$SR data, we found $\Delta H_{1}\sim$ 5 Gauss and $\Delta H_{2}\sim$ 1 Gauss. No such decoupling is observed in the present case in measurement up to 200 Gauss (see inset of Fig. 4(b) and SM sup ), establishing the dynamic nature of spins in BNIO down to atleast 100 mK.

To understand the underlying mechanism of the observed QSL state, we estimated the inter-atomic magnetic exchange interactions from the converged LSDA+$U$+SOC calculations using the formalism of Ref. Kvashnin et al., 2015a (see Method section and SM sup for details). As shown in Table-I, the strongest interaction is antiferromagnetic, which is between the Ir ions of the structural dimer. The strong Ir-Ni interaction further testifies three-dimensional nature of BNIO. Most importantly, Ir-Ir exchange in the a-b plane ($J_{4}$) is found to be antiferromagnetic, resulting in-plane magnetic frustration and explains the origin of the QSL behavior of the present system. However, the presence of strong Ir-Ni and Ni-Ni ferromagnetic exchange reduces the net antiferromagnetic exchange of this system, resulting a relatively lower value of negative $\theta_{CW}$. We further note that the ferromagnetic Ni-Ni exchange has been observed also in antiferromagnetic phase of analogous compound Ba₃NiRu₂O₉ Lightfoot and Battle (1990).

To summarize, our detailed measurements reveal that 6$H$ BNIO containing $S$=1 hosts a gapless spin liquid phase below 2 K. The involvement of Ir⁵⁺ and Ni²⁺ in the magnetic properties of BNIO is revealed by dc magnetic measurements, $\mu$SR experiments, and electronic structure calculation. The antiferromagnetic interaction between Ir₂O₉ dimers in a-b plane facilitates geometrical frustration driven QSL phase of BNIO. Since many Ba₃$MM_{2}^{\prime}$O9 compounds can be stabilized in 6$H$ structure, we believe this work will lead to the realization of many 3-D QSLs with large spin by judicial choice of $M$ and $M^{\prime}$.