Unconventional magnetism in the 4d$^{4}$ based ($S=1$) honeycomb system Ag$_{3}$LiRu$_{2}$O$_{6}$

TL;DR

This study explores the unconventional magnetic behavior of Ag$_{3}$LiRu$_{2}$O$_{6}$, a 4d$^{4}$ honeycomb system, revealing dynamic, gapless spin correlations without long-range order and suggesting complex magnetic interactions.

Contribution

It provides the first detailed investigation of magnetic properties and underlying interactions in Ag$_{3}$LiRu$_{2}$O$_{6}$, highlighting its potential for novel magnetism.

Findings

No magnetic long-range order down to 0.4 K.

Evidence of gapless spin excitations and dynamic correlations.

Indications of spin freezing below 5 K with borderline static/dynamic magnetism.

Abstract

We have investigated the thermodynamic and local magnetic properties of the Mott insulating system Ag$_{3}$LiRu$_{2}$O$_{6}$ containing Ru$^{4+}$ (4$d$$^{4}$) for novel magnetism. The material crystallizes in a monoclinic $C2/m$ structure with RuO$_{6}$ octahedra forming an edge-shared two-dimensional honeycomb lattice with limited stacking order along the $c$-direction. The large negative Curie-Weiss temperature ($\theta_{CW}$ = -57 K) suggests antiferromagnetic interactions among Ru$^{4+}$ ions though magnetic susceptibility and heat capacity show no indication of magnetic long-range order down to 1.8 K and 0.4 K, respectively. $^{7}$Li nuclear magnetic resonance (NMR) shift follows the bulk susceptibility between 120-300 K and levels off below 120 K. Together with a power-law behavior in the temperature dependent spin-lattice relaxation rate between 0.2 and 2 K, it suggest dynamic spin correlations with gapless excitations. Electronic structure calculations suggest an $S = 1$ description of the Ru-moments and the possible importance of further neighbour interactions as also bi-quadratic and ring-exchange terms in determining the magnetic properties. Analysis of our $\mu$SR data indicates spin freezing below 5 K but the spins remain on the borderline between static and dynamic magnetism even at 20 mK.

Figures

Equations

$$P(t)=\frac{A(t)-Bgd}{A_0-Bgd}$$

$$P_{200\text{leavevmode}\text{K}}(t)=\frac{1}{3}+\frac{2}{3}\left[1-{\left({\sigma }_{Nucl}t\right)}^2\right]e^{-\frac{1}{2}{\left({\sigma }_{Nucl}t\right)}^2}$$

$$P(t)=f_{Fro}\left(\frac{2}{3}{e}^{-{\lambda }_{Fast}t}+\frac{1}{3}{e}^{-\lambda t}\right)$$

$$+\left(1-f_{Fro}\right)P_{200\text{leavevmode}\text{K}}{e}^{-\lambda t}$$

Full text

I introduction

Over the past few years, there has been a shift in focus from 3d-based systems to the exploration of 4d and 5d-based ones due to the possibility of strong spin-orbit coupling (SOC) driving exotic magnetism Okamoto et al. (2007); Kim et al. (2009); Singh et al. (2012); Choi et al. (2012); Takayama et al. (2015); Dey et al. (2012, 2013); Dey and Mahajan (2013); Borzi et al. (2007); Wu et al. (2011); de Vries et al. (2010); Kumar et al. (2016); Witczak-Krempa et al. (2014); Thompson et al. (2016); Taylor et al. (2016). The SOC is found to be very strong for 5*d-*based systems and could stabilize a Mott insulating state as also other novel phases Okamoto et al. (2007); Kim et al. (2009); Dey et al. (2012); Kumar et al. (2016); Witczak-Krempa et al. (2014); Thompson et al. (2016); Taylor et al. (2016). Issues such as the realization of the $J_{eff}$ = 1/2 state for the $d^{5}$ configuration (half-filled) Kim et al. (2009), possible realization of the Kitaev model Kitaev (2006) in $d^{5}$ Mott insulators Takayama et al. (2015); Jackeli and Khaliullin (2009), and the emergence of spin-liquid states in triangular lattice materials have been widely explored for Ir-based materials Dey et al. (2012, 2013); Kumar et al. (2016); Catuneanu et al. (2015); Becker et al. (2015).

However, a very interesting scenario could arise for materials away from half-filling; such as four electrons in the $t_{2g}$ manifold. It has been proposed by Khaliullin Khaliullin (2013) that for oxide systems containing Ru4+, Re3+, Os4+ or Ir5+, there can often be comparable values of SOC ($\lambda\sim 50-200$ meV) and superexchange energy scales ($\frac{4t^{2}}{U}\sim 50-100$ meV), which could give rise to excitonic magnetism and resultant novel phases. Recently, Meetei et al. Meetei et al. (2015) and Svoboda et al. Svoboda et al. (2017) have worked further on this and suggested the possible formation of a spin-orbital liquid even in the absence of geometric frustration. Further, theoretical/experimental attempts have been made to realize this novel magnetism in Ir5+ based perovskite NaIrO3 Bremholm et al. (2011), double perovskites Ba2YIrO6 and Sr2YIrO6 Dey et al. (2016); Hammerath et al. (2017); Chen et al. (2017); Cao et al. (2014); Corredor et al. (2017); Bhowal et al. (2015); Pajskr et al. (2016) and triple perovskite Ba3ZnIr2O9 Nag et al. (2016). However, conclusive evidence of this novel magnetism is still elusive. This implies that one should explore materials with a lower SOC and Ru4+ materials might be a good starting point. Recent theoretical studies also proposed Ru-based materials as good candidates to search for excitonic magnetism Meetei et al. (2015); Svoboda et al. (2017).

In this report, we detail the structural, bulk, and local magnetic properties of a t${}^{4}{}_{2g}$ based honeycomb system Ag3LiRu2O6 Kimber et al. (2010) using x-ray diffraction, neutron diffraction, heat capacity, muon spin rotation ($\mu$SR) and nuclear magnetic resonance (NMR) techniques. The honeycomb structure decorated with any of the $d^{4}$ ions (Ru4+, Re3+, Os4+, and Ir5+) has been proposed to manifest novel physical properties Khaliullin (2013). In Ag3LiRu2O6, structurally, Ru atoms give rise to a honeycomb geometry with the Li atom sitting at the center of the honeycomb. Our bulk data do not show any magnetic ordering down to 1.6 K inspite of strong antiferromagnetic interactions. A magnetic contribution to the specific heat is present compared to the non-magnetic analog Ag3LiTi2O6, however, without any sharp anomaly. Static susceptibility deduced from the 7Li-NMR line shift shows a plateau below 120 K and down to 150 mK. These signatures suggest that the static spin correlations are absent/frozen out in the material. The 7Li nuclear spin-lattice relaxation rate 1/$T$1 decreases with decreasing temperature $T$ without any anomaly and displays a power law ($T^{4}$) behaviour below 2 K. This is suggestive of magnetic moments remaining dynamic and the excitations being gapless. From our $\mu$SR measurements, Ag3LiRu2O9 presents a spin-glass-like ground state with a transition temperature $T_{g}$=5.5(5) K though the spins display behaviour which is at the borderline between static and dynamic even at 20 mK. Our ab-initio electronic structure calculations infer negligible SOC and point towards a ferromagnetic coupling between the three nearest neighbours of each Ru and antiferromagnetic further neighbor couplings. This, coupled with deviations from the Heisenberg model, possibly results in frustration which might drive the observed behavior.

II experimental details

The polycrystalline samples of Ag3LiRu2O6 and Ag3LiTi2O6 (nonmagnetic analog used for heat capacity analysis) were prepared in two steps. The precursor Li2RuO3 was synthesized by the solid state reaction route by firing stoichiometric amounts of Li2CO3 and Ru at $1000$∘C for $12$ hours in an alumina crucible and followed by another heating cycle at 950∘C for 24 hours after grinding the sample and mixing $10$% excess Li2CO3. The nonmagnetic Li2TiO3 was prepared by firing a stoichiometric mixture of Li2CO3 and TiO2 at $1000$∘C. Having obtained single phase samples of the precursors Li2RuO3 and Li2TiO3, high purity AgNO3 was mixed with each of the starting materials in the ratio $1:10$ to prepare the final compositions of Ag3LiRu2O6, and Ag3LiTi2O6. The crucibles containing the mixture of materials in a $1:10$ ratio were slowly heated to $300$∘C in air and held at this temperature for $6$ hours followed by cooling to room temperature. The residual AgNO3 and the reaction byproduct LiNO3 were removed by washing the materials with water. X-ray diffraction measurements on the powder samples at room temperature were performed with a Panalytical Xpert Pro diffractometer using Cu-$\mathrm{K_{\alpha}}$ radiation. Neutron diffraction data were taken on the HRPT beamline at the Paul Scherrer Institute PSI at 300 K and 1.6 K using a wavelength $\lambda=2.4586$ Å.

Magnetisation $M$ measurements as a function of applied field $H$ (0 to 90 kOe) and temperature $T$ (in the range 1.8 K to 400 K)) were performed using a Quantum Design SQUID VSM. Zero-field cooled (ZFC) and field cooled (FC) magnetisation measurements in a low field of 25 Oe were performed down to 1.8 K. The heat capacity $C_{\mathrm{p}}(T)$ was measured with a Quantum Design PPMS in various applied fields down to about 0.4 K. The $\mu$SR experiments were performed on a powder sample at the PSI. In the high temperature range (1.5 K < $T$ < 200 K) we used the General Purpose Surface-muons instrument (GPS). We mounted about 1 g of sample in a 15 mm$\times$15 mm Al envelope on a Cu fork. Therefore the sample stops the whole muon beam and we can neglect the experimental background. For the low temperature regime (20 mK < $T$ < 17.5 K) we used the Low Temperature Facility Instrument (LTF). We glued about 1 g of powder with GE-varnish on a silver plate to ensure thermal conductivity. Additionally, local probe nuclear magnetic resonance NMR measurements were performed on the 7Li nucleus in a fixed field of 93.95 kOe as also at a fixed frequency of $95$ MHz. The variation of lineshape with $T$ was measured as also that of the spin-lattice relaxation rate 1/$T_{1}$ down to 150 mK.

III results

III.1 Xrd and structural details

The diffraction patterns in Fig. 1 show a sawtooth shaped peak at low angles (see inset of Fig. 1(a) for x-ray data and Fig. 1(b) for neutron diffraction data) which is commonly known as the Warren peak and is characteristic of 2D structural order with stacking faults in the $c$-direction Warren (1941). Note that stacking faults are not uncommon in such systems, for instance, in Na2IrO3 (Ref. Choi et al. (2012)), in Li2RhO3 (Ref. Khuntia et al. (2017)) as also in $\alpha$-RuCl3 (Ref. Majumder et al. (2015)). This is not likely to affect the two-dimensional magnetic properties. The neutron diffraction data do not evidence the appearance of additional Bragg peaks down to 1.6 K. Further, absence of magnetic long-range order in Ag3LiRu2O6 is not due to stacking faults as we have, in fact, observed LRO in the structurally analogous Ag3LiMn2O6 (Ref. Kumar ). The x-ray diffraction pattern of Ag3LiRu2O6 could be successfully indexed with the monoclinic structure under space group: $C\,2/m$ (Space Group no. 12), $Z=2$, and the Rietveld refinement of the x-ray diffraction data with the FULLPROF suite Rodriguez-Carvajal (1993) yields the profile parameters R_{wp}=4.66$$\%, R_{exp}=2.72$$\%, R_{p}=3.46$$\% and $\chi^{2}=2.93$. The obtained lattice parameters $a=5.2248(9)$ Å, $b=9.0459(15)$ Å, $c=6.5101(12)$ Å and $\beta=74.480(12)\text{\textdegree}$ are in excellent agreement with the previously reported results Kimber et al. (2010).** **In Ag3LiRu2O6, Ru/Li ions coordinate with eight surrounding oxygen atoms and make RuO6/LiO6 octahedra (see Fig. 2 a). The RuO6 octahedra connect in an edge-sharing fashion and give rise to a honeycomb network and the Ru ions are best viewed as forming a two-dimensional (2D) honeycomb lattice in the $a-b$ plane (see Fig. 2 b). Note that although there is a unique Ru site, there are inequivalent O sites. As a result, there are two types of Ru-O-Ru bonds between a Ru and its three nearest neighbour Ru. As seen later, this results in two different couplings between a Ru and its nearest neighbours. The incorporation of Ag atoms into the primary material Li2RuO3 actually works as an intercalation between the Ru layers and essentially makes Ag3LiRu2O6 a 2D system.

III.2 Magnetisation

Figure 3 shows the DC susceptibility $\chi(T)$ of Ag3LiRu2O6 measured in the $T$-range $2-600$ K on a Quantum Design MPMS with the oven option. The $\chi(T)$ data do not exhibit any anomaly in the $T$-range $2-600$ K though there are hints of a plateau around 100 K. Our neutron diffraction data, see Fig. 1(b), collected down to $1.6$ K with wavelength $\lambda=2.4586$ Å do not show any evidence of a phase transition either. A fit of the data to the Curie-Weiss law ($\chi=\chi_{0}+\frac{C}{T-\theta_{CW}}$) in the $T$-range $300-600$ K gives $\chi_{0}=1.7\times 10^{-4}$cm3/mol Ru and the asymptotic Curie-Weiss temperature $\theta_{CW}=-57$ K. The negative $\theta_{CW}$ infers the presence of antiferromagnetic coupling between Ru moments. Note that if $\chi_{0}$ is not left as a free parameter but fixed to a larger value, it yields a smaller $\theta_{CW}$ and a smaller Curie constant, though with a poorer fit. Measurements to even higher temperatures would have helped obtain $\chi_{0}$ with better accuracy but the sample degrades at higher temperatures. The value of the Curie constant $C$ is about 0.88 cm3 K/mol Ru. This leads to an effective moment of 2.65 $\mu_{B}$ which is slightly smaller than the expected spin-only value (for $S$ =

1. of $2.83$ $\mu_{B}$. Magnetisation under zero field cooled (ZFC) and field cooled (FC) conditions was measured in a low field of 25 Oe. This is shown in Fig. 4. We find that there is a weak ZFC-FC bifurcation below about 3K. In another sample of Ag3LiRu2O6 (from a different batch) on which detailed $\mu$SR measurements (as also neutron diffraction) were performed, the bifurcation is greater as also at a higher temperature of about 6 K. This could arise from a fraction of moments in the sample (extrinsic or intrinsic) which freeze.

III.3 Heat capacity

Heat capacity measurements were made to probe low-energy excitations associated with possible magnetism in the sample. Whereas data were taken in a larger temperature range, Fig. 5(a) depicts the heat capacity data ($C$p) of Ag3LiRu2O6 and structurally identical (nonmagnetic) Ag3LiTi2O6 in the temperature range $0.4-40$ K. The measured heat capacity for Ag3LiRu2O6 does not show any significant dependence on the magnetic field and remains featureless in the measured $T$-range. To extract the magnetic specific heat $C_{m}$ of Ag3LiRu2O6 a procedure as in Ref. Bouvier et al. (1991) was employed. The heat capacity of nonmagnetic Ag3LiTi2O6 was measured. The ratio of the Debye temperatures of the Ru-compound and the Ti-compound $\frac{\theta_{D}(Ru)}{\theta_{D}(Ti)}$ was determined using the procedure of Ref. Bouvier et al. (1991). The temperature axis of the Ag3LiTi2O6 was multiplied by the ratio of the Debye temperatures ($\frac{\theta_{D}(Ru)}{\theta_{D}(Ti)}$ = 0.95) before the specific heat of Ag3LiTi2O6 was subtracted from the total specific heat of Ag3LiRu2O6. The magnetic specific heat $C_{m}$ thus obtained is shown in Fig. 5(b). At low-$T$, a power law behaviour is seen with an exponent of about $1.65$. The calculated entropy change ($\triangle S$) for Ag3LiRu2O6, shown in Fig. 5(c), is estimated to be about $11\%$ of $9.12$ J/K mol Ru expected for $S=1$. The power law $T$-dependence of $C_{m}$ and the large quenching of $\triangle S$ supports the realization of a highly degenerate ground state which is presumably gapless.

III.4 7Li NMR

NMR, being a local probe, is instrumental** **in identifying the change in magnetization at a local level. We performed 7Li (nuclear spin: $I=\frac{3}{2}$, gyromagnetic ratio: $\frac{\gamma}{2\pi}=16.546\,$MHz/T) NMR spectra measurements (echo integral at variable frequency) at various temperatures from $300$ K down to $150$ mK. Data were obtained in two ways: (i) in a fixed field of $H=93.954$ kOe, the echo integral was obtained as a function of frequency in the temperature range $300$ K to $80$ K and (ii) at a fixed frequency of $95$ MHz, the echo integral was obtained as a function of the field in the $T$-range $120$ K to $150$ mK. The spectra are displayed together in Fig. 6 after scaling the $x$-axis of the frequency sweep data with the gyromagnetic ratio of 7Li to obtain it in field units corresponding to the frequency of the field sweep measurements. The 7Li-NMR spectra were found to be asymmetric throughout the measured temperature range: (i) a main peak, which qualitatively displays a variation with temperature and (ii) a shoulder on the higher field side which is centered around the zero-shift position, and which remains almost unaffected in the entire temperature range (see Fig. 6). The appearance of this shoulder in 7Li-NMR powder spectra is most likely due to the anisotropy of the hyperfine field. Any significant Li/Ru antisite structural disorder is ruled out by our x-ray analysis. In the light of this anisotropy, the static susceptibility was estimated by extracting the $K_{iso}$ (powder averaged line shift) as a function of $T$ by matching the experimental 7Li-NMR spectra with the simulated one. The $K_{iso}$ follows the bulk susceptibility data, suggestive of a significant hyperfine coupling between Ru and Li atoms. The $K_{iso}$ becomes nearly $T$-independent (or perhaps weakly decreases) below about $120$ K as seen in Fig. 7(a). So the low-temperature rise in the bulk susceptibility appears to be driven by some extrinsic Curie contributions. A finite shift at the lowest temperature suggests gapless spin excitations. Note however that NMR measurements are made in a magnetic field and a finite shift could result from the closing of the gap due to the field but the zero-field heat capacity data exclude the possibility of a gap. The width of the spectrum remains unchanged below $T\approx 2$ K (see Fig. 6). The hyperfine coupling constant $A_{hf}$ follows the relation: $K_{iso}=K_{chem}+\frac{A_{hf}}{N_{A}\mu_{B}}\chi_{spin}$ ($K_{chem}$ is the chemical shift, $N_{A}$ is the Avogadro number, $\mu_{B}$ is the Bohr magneton and $\chi_{spin}$ is the bulk susceptibility). From the slope of $K_{iso}$ vs $\chi_{spin}$ plot $A_{hf}$ was found to be 2.34(13) kOe/$\mu_{B}$ with $K_{chem}=0.03(1)$%, as shown in the inset of Fig. 7(a). The value of the hyperfine coupling will turn out somewhat larger in case the fitting range is limited to higher temperatures.

The 7Li-NMR spin-lattice relaxation rate ($1/T_{1}$) measurements were performed with the saturation recovery method to study the low-energy spin dynamics or to probe the $q$-averaged dynamical susceptibility of Ag3LiRu2O6 in the temperature range $0.3-210$ K at a transmitter frequency of 95 MHz ($H\simeq 57.3$ kOe). The recovery of the longitudinal 7Li nuclear magnetisation was monitored after a saturating pulse sequence and the data in the low temperature regime are shown in Fig. 8. The recovery of the longitudinal magnetisation $m(t)$ was fit to (1-$\frac{m(t)}{m(\infty)}=Aexp(-t/T_{1L})+Bexp(-t/T_{1S})$ ) where $T_{1L}$, $T_{1S}$ are the long and short components of the relaxation time and, $A$ and $B$ are constants. The short component ($T_{1S}$) likely corresponds to an initial fast relaxation associated with spectral diffusion due to incomplete saturation of the broad line. The long component ($T_{1L}$) is expected to be the intrinsic constribution. However, both the long and the short $T$1 components, follow the same qualitative behaviour. The Fig. 7(b) illustrates the variation of $1/T_{1L}$ and $1/T_{1S}$ in the $T$ range 0.3-210 K. A gradual decrease is seen down to 30 K with a broad plateau around $2$ K which is followed by a fall-off with a $T^{4}$ power law at lower $T$. The broad maximum is not due to any spin freezing as the 7Li NMR line remains unbroadened all the way from $120$ K down to $150$ mK. The $T^{4}$ variation of $1/T_{1}$ below $2$ K suggests that the spins remain dynamic and that the excitations are gapless. The data could be fit to a gapped behaviour with a gap of about $3$ K but our heat capacity data show a power-law decrease. Such power law behavior is also seen in other spin liquid candidate materials Dey et al. (2017); Kumar et al. (2016).

III.5 $\mu$SR

As mentioned before, for the GPS experiments, the background signal can be neglected. For the LTF data (20 mK < T < 17.5 K) we need to estimate the background signal $Bgd$. At 3 K, we have data from the GPS as also the LTF beamlines. We then vary $Bgd$ for the LTF data so that the background subtracted data for LTF (at 3 K) coincides with the GPS data at 3 K. This yields $Bgd=0.014(1)$. For clarity, and to directly compare the GPS and LTF data, the raw curves are presented in terms of polarization, $P(t)$ where :

[TABLE]

We present first the zero field (ZF) experiments that we have performed from 200 K down to 20 mK (Figure 9 left). From these data, it is clear that there is a transition in the whole compound between 20 K and 1.56 K. Indeed, at high temperature, in the paramagnetic regime, the depolarization is quite slow and can be attributed to the influence of Li magnetic nuclei. We assumed that the depolarization at 200 K is of a Gaussian form due to static nuclear moments. We then fitted the 200 K data data with a Kubo-Toyabe function Hayano et al. (1979); Kubo (1981):

[TABLE]

This yields $\sigma_{Nucl}=0.156(1)$ $\mu$s*-1* which is directly linked to the nuclear field $H_{Nucl}$ via $\sigma_{Nucl}=\gamma_{\mu}H_{Nucl}$ where $\gamma_{\mu}=2\pi\times 135.5$ $\mu$s.T*-1* is the gyromagnetic factor of the muons. We found $H_{Nucl}=1.83(2)$ G which is in the usual range of nuclear field values.

At low temperatures, the depolarization is rather quick, on the 0.1 $\mu$s scale, and the depolarization at long time is close to 1/3. This is characteristic of frozen or quasistatic magnetism. Nevertheless, the lack of spontaneous oscillations could be directly linked to an absence of long range ordered magnetism. Therefore, our $\mu$SR experiment reveals a short range ordered spin-glass-like ground state or a dynamic ground state with a large distribution of fields. To have better insight on the ground state probed by $\mu$SR, we fitted the data with an equation containing two relaxing components which has been used for other spin-glass systems Aczel et al. (2016); Guo et al. (2016) :

[TABLE]

where $f_{Fro}$ is the fraction of sample in the frozen state, $\lambda_{Fast}$ accounts for the fast depolarization at short time and represents the distribution of quasistatic fields in the sample and $\lambda$ accounts for the electronic magnetism and could be linked to its fluctuations. Note that we also tried to fit the data using the dedicated spin glass function Uemura et al. (1985) as well as dynamic and static Kubo-Toyabe functions which resulted in poorer fits than with equation (3).

The results are presented in Figure 9 right. The small differences between GPS and LTF likely arise from the difficulty to fully characterize the $Bgd$ in LTF. The frozen fraction increases at 10 K to go to close to 1 below 3 K. Therefore, the transition is not due to an impurity but presents a bulk character. Further, from the depolarization rate $\lambda$ we can determine the transition temperature $T_{g}$. Indeed, $\lambda$ possesses a peak around 5.5(5) K which is characteristic of a transition to frozen magnetism. Moreover, there is a very small plateau at 0.11(1) $\mu$s*-1* which could be related to small fluctuations of the magnetism below 1 K and could be due to a quasistatic order. Further, below this temperature, $\lambda_{Fast}$ presents a plateau around 25(2) $\mu$s*-1* which is due to the distribution of the quasistatic fields. From this value, one can directly compute the field distribution $\Delta=\lambda_{Fast}/\gamma_{\mu}=290(30)$ G.

To distinguish between static and dynamic magnetism we applied several longitudinal fields in the direction of the muons beam. Indeed, in the case of static magnetism a longitudinal field which is 10 times the field distribution should decouple the muons whereas in the dynamic case a longitudinal field 50 times stronger than the field distribution is needed Yaouanc and de Reotier (2011). At 20 mK, the muons are almost fully decoupled under a field of 0.5 T ($\sim 17\times\Delta$) (Figure 10) indicating that the magnetism observed below 5 K is at the borderline between static and dynamic behavior. Therefore, due to the lack of spontaneous oscillations and the decoupling experiment, Ag3LiRu2O6 presents a spin-glass-like ground state with a transition temperature $T_{g}$=5.5(5) K based on the $\mu$SR analysis.

III.6 Electronic structure calculations

In order to obtain further insight into the possible origin of the observed magnetic behaviour, given the apparent absence of gemetric frustration, we have carried out electronic structure calculations using the full-potential linearized augmented plane wave (FP-LAPW) Madsen et al. (2001); Schwarz et al. (2002) plus local orbitals method using the WIEN2K code Blaha et al. (2001). Exchange and correlation effects are treated within generalized gradient approximation (GGA) Perdew et al. (1996) of Perdew-Burke-Ernzerhof including Hubbard $U$ Dudarev et al. (1998) and spin-orbit coupling (SOC). The double counting correction in the GGA+$U$ formalism is taken into account within around mean field approximation Czy ÅŒ ÅŒyk and Sawatzky (1994). The calculations were done with usual values of $U$ and $J$H Streltsov (2013) chosen for Ru; $U$ = 3.0 eV and Hund’s coupling ($J$H) = 0.7 eV. The calculations were also checked for various other values of $U$. In order to achieve the convergence of energy eigen values, the kinetic energy cut off was chosen to be Kmax RMT = 7.0 where RMT denotes the smallest atomic sphere radius and Kmax gives the magnitude of the largest K vector in the plane-wave expansion in the interstitial region. The Brillouin-Zone integrations were performed with 8 $\times$ 8 $\times$ 6 k-points mesh. The total energies necessary for the calculation of symmetric exchange interactions Xiang et al. (2011) were calculated using density functional theory (DFT) and projector augmented-wave (PAW) method as encoded in the Vienna ${\it ab}$ ${\it initio}$ simulation package (VASP). The kinetic energy cut off of the plane wave basis was chosen to be 600 eV and a $\Gamma$ centered 4 $\times$ 4 $\times$ 6 k-mesh has been used for Brillouin-Zone (BZ) integration.

The spin polarized density of states in the framework of GGA + $U$ with $U=3.0$ eV and $J_{H}=0.7$ eV in the ferromagnetic configuration foo is shown in Fig. 11. The system is found to be insulating with the majority Ru $t_{2g}$ spin states completely occupied while the minority $t_{2g}$ states are only partially occupied. The $e_{g}$ states for both the spin channels are completely empty. As a consequence of the monoclinic distortion promoted by Jahn-Teller active Ru 4+ ion** ** the degeneracy of the Ru $t_{2g}$ states is completely lifted which on inclusion of the Hubbard $U$ introduces a gap in the minority spin channel. The total moment is calculated to be $4\mu_{B}$ (per formula unit) with moment at Ru and O sites being $1.16\mu_{B}$ and$0.18\mu_{B}$ respectively, suggesting strong hybridization of the Ru with oxygen states. Next, we have included spin-orbit coupling in our calculation. The total moment is then calculated to be 3.97$\mu_{B}$ with spin and orbital moment at the Ru site 1.15$\mu_{B}$ and 0.03$\mu_{B}$, respectively. The small orbital moment suggests that a spin-only description is valid and neither the LS nor the jj coupling schemes should be employed. These calculations suggest that the system is far away from the $J=0$ limit and $S=1$ description of the system is more appropriate. To understand the absence of an ordered state, we calculated the first, second, and third neighbor symmetric exchange interactions mapping the density functional total energies obtained using Vienna ${\it ab}$ ${\it initio}$ simulation package (VASP) within the projector-augmented wave (PAW) method onto the Heisenberg model following the method proposed in Xiang et al. (2011). The magnitude and the sign of the symmetric exchange interactions are found to be very sensitive to the chosen configuration for calculation and the size of the simulation cell suggesting the importance of the higher order magnetic interactions (biquadratic and four-spin ring couplings) as the usual approximation of the Hubbard model reducing to the Heisenberg model in the limit of large $U$ may not be applicable here Fedorova et al. (2015). The calculations have been done (with the experimental structural parameters) for several spin configurations using simulation cells of different sizes. Our calculations reveal that the nearest neighbor exchange interactions J${}_{1}^{a}$ and J${}_{1}^{b}$ are ferromagnetic while the further neighbor interactions (such as J${}_{2}^{a}$, J${}_{2}^{b}$, J${}_{3}^{a}$ and J${}_{3}^{b}$) are antiferromagnetic. On the other hand, if we relax the structure and then calculate the couplings, the nearest and the next-nearest neighbour interations turn out to be antiferromagnetic. With a larger number of antiferromagnetic further neighbour couplings, one can still recover a negative $\theta_{CW}$ as observed experimentally. This, coupled with the ring exchange and biquadratic couplings, introduces frustration in the system and possibly drives the system away from order.

IV discussion

From the wide range of measurements that we have presented together with first principles electronic structure calculations, let us now look at things in perspective. Usual bulk susceptibility measurements did not show any signatures of long-range order down to 1.8 K. Heat capacity measurements (zero or non-zero applied field) also do not evidence any peak down to 0.4 K. 7Li NMR measurements on the same sample did not show any line broadening nor any peak in the spin lattice relaxation rate down to 0.15 K. In fact, the 7Li shift (which probes the intrinsic susceptibility), starting at room temperature, increases with decreasing temperature. This suggests that there is magnetism in the system which must come from Ru4+ ions (4$d$4). Note that we tried to fit the $T$-variation of the intrinsic susceptibility to the formula given by Kotani Kotani (1949) for the Van Vleck susceptibility of $d^{4}$ systems. This gave rise to very poor fits. The apparent agreement with the Kotani formula for the susceptibility of Ag3LiRu2O6 in Ref. Lu et al. (2018) is misleading as they seem to have considered only the $T$-region where the susceptibility is nearly constant. Also, our calculations suggest that this is a spin-only moment. With a continued decrease in temperature, the 7Li shift levels off below about 120 K and remains so down to 150 mK. . The 7Li NMR linewidth does not show any divergence either and remains constant down to 150 mK. All the above suggests that the magnetism in this system somehow gets quenched below 120 K and no order sets in. Note that the NMR measurements are in a field of about 57 kOe which corresponds to a significant energy scale in our low-$T$ regime (below 5 K or so). Let us now look at the results of low/zero field measurements. The magnetic heat capacity shows no special features other than a power law variation at low-$T$ (below about 4 K) and a broad maximum around 10 K. The low field (25 Oe) ZFC and FC magnetisation does show some bifurcation at low-$T$ which is sample dependent. The sample on which we performed the heat capacity and all the NMR measurements showed a weak ZFC-FC bifurcation (less than 10%) below about 3 K. When the same sample was measured in $\mu$SR (zero field), we found evidence of freezing around 3.5 K. On the other hand, the sample on which we did detailed $\mu$SR measurements (presented here) showed a larger ZFC-FC bifurcation and around 7 K. This sample shows a freezing of moments below about 5.5 K from $\mu$SR. The question now is how to reconcile the $\mu$SR data with the NMR, susceptibility and heat capacity data? It appears that the zero-field $\mu$SR and low-field magnetisation results are consistent with each other. For the zero field heat capacity, the lack of entropy as well as the broad peak around 10 K and the power law behavior at temperatures could perhaps arise in case of a spin glass state. The differing NMR results might come from a field effect. The 7Li NMR linewidth should have shown a critical divergence (or at least a significant increase) around the freezing temperature. The absence of this suggests that the applied field prevents the formation of static moments! Nevertheless, these results are very similar to the one obtained in another honeycomb compound, Li2RhO3 Khuntia et al. (2017). Note also that the magnetism observed by us in $\mu$SR below 5 K is at the borderline of static and dynamic. Therefore, even if the $\mu$SR experiments rule out the possibility of a quantum spin liquid ground state due to the presence of frozen magnetism below 5.5(5) K, this compound does not present a regular spin-glass behavior.

V conclusions

In summary, we have investigated the structural, thermodynamic and local magnetic properties of a honeycomb structure based novel quantum material Ag3LiRu2O6 by performing x-ray diffraction, neutron diffraction, susceptibility, heat capacity and 7Li-NMR measurements. The presence of an asymmetric peak in both x-ray and neutron diffraction profiles is suggestive of a 2D structural ordering (honeycomb) in the $a-b$ plane. The $\chi(T)$ data infers a strong antiferromagnetic coupling between the Ru moments without showing any anomaly down to 2 K, and the neutron diffraction carried out down to 1.6 K does not detect any magnetic order. Heat capacity displays a $\sim$ $T^{1.7}$-dependence at low-$T$ and the deduced entropy change was found to be highly suppressed; $\Delta S\sim 11\%$ of that for an ordered spin-one system. 7Li-NMR powder spectra measurements help extracting the instrinsic susceptibility of Ru moments and a leveling off of the NMR line shift was found for $T$ $\leq$ 120 K. Our electronic structure calculations provide a clue to the origin of the observed susceptibility saturation in this system which (in principle) is not geometrically frustrated. Our study suggests that the magnetism here is not excitonic in origin. We propose that the frustration induced by further neighbor couplings and a deviation from the simple Heisenberg model is responsible for the lack of LRO in this system. While spin freezing below about 5 K is evidenced from our zero field $\mu$SR data, from the longitudinal field decoupling experiments, the moments are at the borderline between static and dynamic even at 20 mK. It needs to be explored whether defects such as stacking faults finally drive the system to a frozen state and whether the pristine system might be a spin-liquid.

We thank Department of Science and Technology (DST), Govt. of India for financial support through the BRICS project Helimagnets. RK acknowledges CSIR (India) and IRCC (IIT Bombay) for awarding him research fellowships to carry out this research work. P. M. Ette acknowledges CSIR, INDIA for providing financial support under CSIR-SRF Fellowship (grant no. 31/52(14)2k17). AVM thanks the Alexander von Humboldt Foundation for support. Work at Augsburg was supported by the Deutsche Forschungs Gemeinschaft (DFG) through the collaborative research center TRR80 (Augsburg/Munich). A.A.Gippius acknowledges the financial support from the RFBR Grant №17-52-80036. We thank Kedar Damle for useful discussions and Dana Vieweg for technical help.