Structural, thermodynamic, and local probe investigations of a honeycomb material Ag$_{3}$LiMn$_{2}$O$_{6}$

TL;DR

This study combines structural, thermodynamic, and local probe techniques with ab-initio calculations to investigate the magnetic properties and ordering phenomena in the honeycomb material Ag$_{3}$LiMn$_{2}$O$_{6}$, revealing an antiferromagnetic ground state.

Contribution

It provides a comprehensive multi-technique analysis of Ag$_{3}$LiMn$_{2}$O$_{6}$, including experimental magnetic and local probes and theoretical calculations, to elucidate its magnetic interactions and ordering behavior.

Findings

Magnetic ordering occurs near 48 K.

Short-range magnetic correlations exist above the ordering temperature.

The system exhibits a predominantly antiferromagnetic ground state.

Abstract

The system Ag[Li$_{1/3}$Mn$_{2/3}$]O$_{2}$ belongs to a quaternary 3R-delafossite family and crystallizes in a monoclinic symmetry with space group $C\,2/m$ and the magnetic Mn$^{4+}$($S=3/2$) ions form a honeycomb network in the $ab$-plane. An anomaly around 50 K and the presence of antiferromagnetic (AFM) coupling (Curie-Weiss temperature $\theta_{CW}\sim-51$ K) were inferred from our magnetic susceptibility data. The magnetic specific heat clearly manifests the onset of magnetic ordering in the vicinity of 48\,K and the recovered magnetic entropy, above the ordering temperature, falls short of the expected value, implying the presence of short-range magnetic correlations. The (ESR) line broadening on approaching the ordering temperature $T_{{\rm N}}$ could be described in terms of a Berezinski-Kosterlitz-Thouless (BKT) scenario with $T_{{\rm KT}}=40(1)$ K. $^{7}$Li NMR line-shift probed as a function of temperature tracks the static susceptibility (K$_{iso}$) of magnetically coupled Mn$^{4+}$ ions. The $^{7}$Li spin-lattice relaxation rate (1/$T$$_{1}$) exhibits a sharp decrease below about 50 K. Combining our bulk and local probe measurements, we establish the presence of an ordered ground state for the honeycomb system Ag$_{3}$LiMn$_{2}$O$_{6}$.Our ab-initio electronic structure calculations suggest that in the $ab$-plane, the nearest neighbor (NN) exchange interaction is strong and AFM, while the next NN and the third NN exchange interactions are FM and AFM respectively. In the absence of any frustration the system is expected to exhibit long-range, AFM order, in agreement with experiment.

Figures

Tables

Table 1. Table 1: The structural parameters obtained after Rietveld refinement of the x-ray diffraction data collected at a wavelength of λ = 1.5418 Å for Ag 3 LiMn 2 O 6 under the space group C ​  2 / m 𝐶 2 𝑚 C\,2/m at 300 K. The obtained lattice constants are a = 5.0690 ​ ( 7 ) 𝑎 5.0690 7 a=5.0690(7)\, Å , b = 8.7863 ​ ( 11 ) 𝑏 8.7863 11 b=8.7863(11)\, Å and c = 6.3755 ​ ( 9 ) 𝑐 6.3755 9 c=6.3755(9)\, Å with β = 74.938 ​ ( 11 ) ∘ 𝛽 74.938 superscript 11 \beta=74.938(11)^{\circ} .

Atoms	site	x	y	z	Biso	Occ
Ag1	2d	0	0.5	0.5	0.52	1
Ag2	4h	0.5	0.3271(4)	0.5	0.52	1
Li	2a	0	0	0	0.49	1
Mn	4g	0	0.6648(11)	0	0.45	1
O1	8j	0.4270(23)	0.3471(17)	0.8420(18)	0.36	1
O2	4i	0.1459(45)	0.5	0.1393(36)	0.36	1

Table 2. Table 2: Variation of the magnetic moments (in the GGA + U calculation) on the various ions for different ordering arrangements and their corresponding energies are tabulated here.

Magnetic	Total Moment	Moment/Mn	Moment/O	Gap	$\Delta$E/f.u.
Configuration	/f.u. in ${\mu }_B$	in ${\mu }_B$	in ${\mu }_B$	in eV	in meV
AFM-1	0.0	2.88	0.0	1.54	0.0
AFM-2	0.0	2.91	0,$\pm$ 0.01	1.33	36.6
AFM-3	0.0	2.89	0,$\pm$ 0.02	1.47	24.6
FM	6.0	2.95	-0.01	0.99	35.5

Equations

$$\Delta H(T)=\Delta H_0+A{\left(\frac{T_{\mathrm{N}}}{T-T_{\mathrm{N}}}\right)}^{\beta }$$

$$\Delta H=A\exp (3b/(T/T_{\mathrm{KT}}-1{)}^{0.5})+\Delta H_0$$

$$\frac{T_{\mathrm{N}}-T_{\mathrm{KT}}}{T_{\mathrm{KT}}}=\frac{4b^2}{[\ln (J/J^{\prime }){]}^2}$$

$$(1-m(t)/m(0))=Aexp(-x/T_{1L})+Bexp(-x/T{}_{1S})$$

$$E=-J_{12}{S}_1\cdot S_2+S_1\cdot h_1+S_2\cdot h_2+E_{all}+E_0$$

$$J_{12}=-\frac{E_{\uparrow \uparrow }+E_{\downarrow \downarrow }-E_{\uparrow \downarrow }-E_{\downarrow \uparrow }}{4S^2}$$

Full text

I Introduction

Materials based on the delafossite structure with the general chemical formula ABO2 exhibit interesting physical properties (Shannon et al., 1971; Kawazoe et al., 1997; Wawrzyńska et al., 2008; Poienar et al., 2009; Maignan et al., 2009). In general, A and B-sites in ABO2 represent monovalent and trivalent cations, respectively, having a linear and octahedral environment of oxygen atoms. In this case, the B-site which is responsible for magnetism in delafossite materials forms an edge-shared triangular lattice or a honeycomb lattice when the system crystallizes in hexagonal/rhombohedral or monoclinic space groups, respectively.

There exist a variety of honeycomb materials (Marquardt et al., 2006) and in recent years attempts have been made to synthesize delafossite materials with tetravalent ions at the B-site (honeycomb lattice), which are known as 3$R$-delafossites. A few examples of 3$R$-delafossites are Ag(Li1/3Ru2/3)O2 (Kimber et al., 2010; Ramesha et al., 2011; Kumar et al., 2019), Ag(Li1/3Rh2/3)O2 (Todorova et al., 2011) and Ag(Li1/3Ir2/3)O2 (Todorova et al., 2011).** **We would like to mention here that Ag insertion into the primary structure of Li2MO3 (M = Ru, Rh or Ir) results in its placement between consecutive metal layers which essentially reduces the inter-layer connectivity and thus makes the materials highly two-dimensional (2D) in nature.

As per some recent theoretical studies (Khaliullin, 2013; Meetei et al., 2015; Svoboda et al., 2017) novel ground state properties are expected for 4$d$/5$d$ materials depending upon the variation of superexchange energy scale, Hund’s coupling ($J$H) and spin-orbit coupling (SOC). In particular, honeycomb lattice based 5$d$ materials are at the forefront of current experimental and theoretical research because of the possibility of stabilizing the Heisenberg-Kitaev Hamiltonian and a rich phase diagram upon variation of the exchange couplings is envisaged (Trebst, 2017). On the other hand, a different scenario might emerge while dealing with 3$d$ transition metal ions, where SOC is much weaker than the other energy scales, namely $U$ (on-site Coulomb repulsion) and $J$H. For instance, it has recently been shown by Wei et al. (Wei et al., 2011) that the 2D honeycomb lattice based Affleck-Kennedy-Lieb-Tasaki (AKLT) state with $S=3/2$ is a prototype for the physical realization of the measurement based quantum computation (MBQC). According to Maciej et al. (Koch-Janusz et al., 2015) the multiorbital insulators in the framework of Hubbard models with nearest-neighbor hopping on a honeycomb lattice could even lead to the $S=3/2$ AKLT state. In light of the aforementioned proposals, it is worth investigating the physical properties of a honeycomb lattice with $S=3/2$, with the intention of exploring some novel ground state properties.

Herein, we report for the first time the sample preparation, structural, and physical properties of a 3$d$ transition metal oxide (delafossite) Ag(Li1/3Mn2/3)O2. In this compound, the Mn4+ ions ($S=3/2$) form a 2D honeycomb lattice with Li-ions positioned at the center of each honeycomb unit. The material was structurally characterized by a combination of x-ray and neutron diffraction measurements, along with magnetization, specific heat, and electron spin resonance (ESR). In addition, 7Li nuclear magnetic resonance (NMR) spectra and spin-lattice relaxation rate measurements were performed. Structural characterization done with x-ray and neutron diffraction suggest a superstructure (honeycomb) formed by magnetic Mn4+ions in the crystallographic $ab$-plane. Interestingly, the same asymmetric peak, seen in the paramagnetic region, is found to get more intense on lowering the temperature, as observed in our neutron diffraction studies. Magnetization data show an anomaly in the vicinity of 50 K and antiferromagnetic coupling is inferred from our susceptibility analysis. Another thermodynamic measurement, specific heat, also confirms this anomaly and locates the magnetic ordering at 47 K. The integrated intensities of the electron spin resonance ESR absorption lines mimic the bulk magnetic susceptibility. The ESR measurements evidence a critical broadening as a function of temperature with a transition temperature $T_{{\rm N}}$ = 45 K. Further, the line broadening on approaching $T_{{\rm N}}$ may be alternatively described in terms of a Berezinski-Kosterlitz-Thouless (BKT) scenario with $T_{{\rm KT}}=40(1)$ K. The static susceptibility (free from defects/impurity) tracked using local probe 7Li NMR spectra measurements also reproduce the anomaly observed in bulk susceptibility measurements, while the 7Li spin-lattice relaxation rate does not show any sharp anomaly around the 50 K transition. This is likely due to the symmetric location of the 7Li with respect to the magnetic Mn4+ ions, giving rise to a filtering of antiferromagnetic fluctuations. Our experimental results corroborate the establishment of a magnetically ordered ground state for the 3$d$-based system Ag(Li1/3Mn2/3)O2 which is also supported by first principles electronic structure calculation.

II Experimental and Computational details

The polycrystalline samples of the quaternary 3R-delafossite oxide Ag[Li1/3Mn2/3]O2 were prepared by a combination of sol-gel and ion-exchange methods. First, the starting material Li2MnO3 was prepared by sol-gel route and then fired at 500°C for 6 hours. After confirming by x-ray diffraction that Li2MnO3 was single phase, AgNO3 was mixed with Li2MnO3 in the ratio 1 : 3. The resultant mixture was slowly heated to 300°C and held for 6 hours following which the desired material Ag[Li1/3Mn2/3]O2 was obtained by removing the residual byproduct LiNO3 by washing the mixture with water.

The room temperature powder x-ray diffraction (xrd) measurements were performed using a PANalytical Xpert Pro x-ray diffractometer with Cu-Kα radiation ($\lambda$= 1.5418 Å). Neutron diffraction data were taken on the DMC beamline at the Paul Scherrer Institute PSI using a wavelength $\lambda=2.4586$ Å. Microstructural investigations were performed with a CM 200 Philips transmission electron microscope (TEM) operating at 200 kV. Magnetization $M$ measurements in the temperature range 2-400 K as a function of applied field $H$ were performed on a Quantum Design SQUID VSM with the powder sample loaded in a capsule and for measurements in the 400-630 K range, the high temperature option of the Quantum Design VSM was used. The heat capacity measurements were carried out on a Quantum design PPMS using the thermal-relaxation method. ESR was measured in an ELEXSYS E500 spectrometer (Bruker) at X-band frequency of 9.4 GHz in a magnetic field of about $H$ = 18 kOe. The spectrometer was equipped with a helium gas-flow cryostat ESR 900 (Oxford Instruments) operating in the temperature range $T=$ 4 - 300 K. The polycrystalline samples were immersed in paraffin in suprasil quartz glass tubes and mounted in the cavity. ESR detects the microwave absorption due to magnetic dipolar transitions induced between the Zeeman levels of the sample as a function of the external magnetic field. Due to the lock-in amplification technique by field modulation in ESR, one records the field derivative of the absorption spectra. The 7Li-NMR measurements (spectra and spin-lattice relaxation rate 1/$T_{1}$) were performed both in the fixed field (93.9543 kOe) and swept field ($\nu=95$ MHz) mode to gain further insights into the intrinsic static susceptibility of Mn-moments in Ag3LiMn2O6 by measuring the line shift as a function of temperature.

All the electronic structure calculations based on DFT presented in this paper were carried out in the planewave basis within generalized gradient approximation (GGA)** (Perdew et al., 1996)** of the Perdew-Burke-Ernzerhof exchange correlation supplemented with Hubbard U as encoded in the Vienna** ${\it ab}$-${\it initio}$ simulation package (VASP) (Kresse and Hafner, 1993; Kresse and Furthmüller, 1996) with projector augmented wave potentials (Blöchl, 1994; Kresse and Joubert, 1999). The calculations are done with usual values of U and Hund’s coupling ($\mathrm{\mathrm{J}}_{\mathrm{H}}$) chosen for Mn with Ueff  ($\equiv$U- $\mathrm{\mathrm{J}}_{\mathrm{H}}$) = 3.0 eV in the Dudarev scheme (Dudarev et al., 1998). In order to achieve convergence of energy eigenvalues, the kinetic energy cut off of the plane wave basis was chosen to be 600 eV. The Brillouin-Zone integrations are performed with 8 $\times$ 4 $\times$ 6 Monkhorst grid of $k$-points. The exchange paths were identified by calculating hopping parameters by constructing the Wannier function using the VASP2WANNIER and the WANNIER90 codes (Mostofi et al., 2014). In addition, to get the minimum energy structure, symmetry protected ionic relaxation was been carried out using the conjugate-gradient algorithm until Hellman-Feynman forces on each atom were less than the tolerance value of 0.01 eV/Å. **

III results and discussion

We will now present the results of our various bulk and local probe measurements on Ag3LiMn2O6.

III.1 Structure analysis

Figure 1 depicts the x-ray diffraction pattern for Ag3LiMn2O6 at 300 K. The x-ray diffraction pattern for Ag3LiMn2O6 was found to be similar to the isostructural material Ag3LiRu2O6 (Kimber et al., 2010) of the quaternary 3R-delafossite family. A peak corresponding to a small amount of Ag is seen in the xrd pattern. An asymmetric reflection (a superstructure peak) was also seen around 2$\theta\approx$ 21° (see the inset of Fig. 1). This asymmetric peak (more prominent in neutron diffraction), commonly known as the Warren peak (Warren, 1941), arises as a consequence of the irregular stacking sequences of layers in a structure. In the present case, Mn4+ ions are found to form 2D honeycomb layers in the $ab$-plane and the irregular stacking pattern results from the stacking faults which then limit the correlation length in the crystallographic $c$-direction. The x-ray diffraction peaks for Ag3LiMn2O6 were found to be broader than those of Ag3LiRu2O6 and the particle size determined from the Scherrer formula is estimated to be about 20 nm. We then performed microstructure analysis from the TEM and selected area electron diffraction (SAED) images depicted in Fig. 1. SAED analysis shows that the continuous ring patterns (see inset of 1(c)) from our polycrystalline sample could be well indexed with the hexagonal lattice of Ag3LiMn2O6. Our TEM analysis showed that the mean size of nanoparticles was between 10-20 nm and the nanoparticles appeared to be nearly spherical in morphology. This value is in good agreement with results obtained from XRD. The low-temperature synthesis route of the starting material Li2MnO3 (see experimental section) could possibly be the reason for the nano crystallinity of this material.

In order to extract information about the unit cell parameters and atomic positions, the x-ray diffraction pattern of Ag3LiMn2O6 was refined under the FullProf Suite program (Carvajal, 1990) using the structural parameters of Ag3LiRu2O6 as an initial model. All the Bragg reflections obtained for Ag3LiMn2O6 could be successfully indexed with a monoclinic space group $C\,2/m$ and the refined atomic coordinates and lattice constants are listed in Table 1. Because of the particle size being in the nanometer range, evident from the broadened x-ray peaks and TEM images, microstructure parameters were also taken into account while refining the crystal structure and a quadratic form of strain formation under Laue class $mmm$ was considered. The Rietveld refinement quality factors expressed by $R_{wp}$, $R_{exp}$, $R_{p}$ and $\chi^{2}$ have the values 2.89%, 1.82%, 2.22% and 2.52, respectively. The crystal structure of Ag3LiMn2O6 based on x-ray diffraction refinement is shown in Fig. 2. The MnO6 octahedra form an edge-sharing, 2D, honeycomb network in the crystallographic $ab$-plane and the LiO6 octahdera sit at the center of the honeycomb network, see Figs. 2(b) and (c). Intercalated Ag atoms go in-between the consecutive honeycomb layers.** **

In our neutron diffraction data (see the inset of Fig. 3), an asymmetric peak is seen to emerge below about 50 K while all the other peaks are nearly unchanged. This must be from the ordering transition which is evident in other measurements such as magnetic susceptibility, heat capacity, etc. which are detailed in the following sections.

III.2 Magnetization

Figure 4 depicts the dc susceptibility ($M$/$H$) of Ag3LiMn2O6 measured in the temperature range $2-630$ K with an applied field of 30 kOe. The susceptibility shows a gradual increase on lowering the temperature before exhibiting a well rounded anomaly at around 50 K following which it exhibits an upturn. The anomaly observed at 50 K might be a signature of magnetic order, while the low-$T$ increase of susceptibility could be partly due to some extrinsic contributions and/or defects (see ESR/NMR results later in the paper). The Curie-Weiss fit ($\chi=\chi_{0}+C/(T-\theta_{CW})$) to the susceptibility data in the temperature range $240-630$ K yields: temperature independent susceptibility $\chi_{0}=-7.835\times 10^{-4}$ cm3/mol Mn, the Curie-Weiss temperature $\theta_{CW}$ $\sim$ -51(1) K, indicative of antiferromagnetic interactions among Mn4+ moments, and a Curie constant $C=2.495\pm 0.014$ cm3 K/mole Mn or an effective paramagnetic moment$\sim 4.46\pm 0.33\,\mu_{B}$. The electronic configuration of Mn4+ is 3$d$3 and hence the expected effective moment (considering $g=2$ as obtained from electron spin resonance discussed later) is 3.87 $\mu_{B}$. Note that the value of $\chi_{0}$ can not be determined very accurately as the susceptibility has still not leveled off even above 600 K. Consequently, there is some uncertainty in the determination of the Curie constant and $\theta_{CW}$. Further, 10-20 nm size grains are present in our sample. Defects on the surfaces of nanoparticles could also stabilize a moment and contribute to the observed value. A higher than expected effective moment was also observed in CaMnO3 which increased with oxygen depletion. (Zeng et al., 1999)

III.3 Specific heat

The specific heat of the cold pressed powder sample of Ag3LiMn2O6 was measured and an anomaly was also noticed there as was observed in our magnetization data. Figure 5(a) depicts the temperature variation of specific heat $C_{p}(T)$ of Ag3LiMn2O6 in the temperature range $1.8-200$ K in 0 and 90 kOe magnetic field. The $C_{p}(T)$ data distinctly show the presence of an anomaly around $48$ K, which is found to be insensitive to the applied magnetic field. The magnetic specific heat contribution ($C_{m}$) in Ag3LiMn2O6 was obtained by subtracting the lattice specific heat $C_{lattice}$. To estimate the $C_{lattice}$ of Ag3LiMn2O6, an isostructural material Ag3LiTi2O6 was prepared and its specific-heat was measured in zero magnetic field and prior to subtracting this from the $C_{p}(T)$ data of Ag3LiMn2O6 it was scaled with the $C_{p}(T)$ data of Ag3LiMn2O6. Initially, Bouvier scaling (Bouvier et al., 1991) was used to scale the lattice specific heat of Ag3LiTi2O6, which gives a correction factor ($\frac{\theta_{D}(Mn)}{\theta_{D}(Ti)}$ = 0.988) to the temperature axis, where $\theta_{D}(Mn)$ and $\theta_{D}(Ti)$ are the Debye temperatures for the Mn and Ti compounds, respectively. However, this does not appear to reliably estimate the lattice specific heat of Ag3LiMn2O6. In fact, it was found to exceed the total specific heat of Ag3LiMn2O6. We then manually matched the specific-heat of nonmagnetic sample, in the high temperature region, by rescaling its temperature axis by multiplying it by 1.085. The magnetic specific heat $C_{m}$ is plotted in Fig. 5(b). The $C_{m}$ data clearly show a $\lambda$-like anomaly at 47 K. The experimental magnetic entropy change $\Delta S$, estimated from the $C_{m}-T$ curve, is about 80$\pm 5$% of the theoretical value of 11.52 J/Mn mol K for spin $S=3/2$, as can be seen in Figure 5(c). The transition seen at 47 K approximately accounts for 55% entropy change while nearly the rest of the entropy is recovered above 70 K. The recovery of a significant amount of entropy above the ordering temperature is probably suggestive of the presence of short-range magnetic correlations. One should recall that the intensity of asymmetric peak seen in neutron diffraction measurements (see inset of Fig. 3) also does not immediately collapse to the peak recorded at 300 K, indicating again that magnetic correlations survive at least up to 60 K and are in-line with our specific heat analysis.

III.4 ESR Results

To investigate the correlated magnetism originating as a result of interacting Mn-moments, arranged in a honeycomb geometry, ESR measurements were performed as a function of temperature. In order to study the spin dynamics of Ag3LiMn2O6 and to get the evolution of the corresponding ESR parameters with temperature, the ESR line shape was analyzed. Typical ESR spectra of a powder sample of Ag3LiMn2O6 are shown in Fig. 6. All spectra consist of a single exchange-narrowed resonance line — i.e., any line splitting or inhomogeneous broadening is averaged out by the isotropic exchange interaction. The resonance line is well described by a Lorentz shape at resonance field $H_{{\rm res}}$ with half width at half maximum $\Delta H$ including the counter resonance at $-H_{{\rm res}}$ and a small contribution of dispersion (D) to the absorption (A) given by the (D/A) ratio in case of large line width as described in Ref. (Joshi and Bhat, 2004). The results of ESR line shape fitting for some representative spectra are shown by solid lines in Fig. 6.

The ESR line is related to the manganese exchange coupled system. The temperature dependencies of the resonance field, the linewidth, the intensity, and the inverse intensity are summarized in Fig. 7. Starting with the resonance field (see Fig. 7 (a))in the upper left frame, we obtain a $g$ value of $g=h\nu/\mu_{B}H_{{\rm res}}=1.99$ at high temperature which is typical for Mn4+ ions (spin $S=3/2$) in an octahedral ligand field with its half-filled $t_{2g}$ triplet. On decreasing temperature, the $g$-factor remains approximately constant down to a weak anomaly at $T_{{\rm N}}=48$ K and then decreases on further cooling followed by a divergence on approaching zero temperature.

Focusing on the integrated intensity and its inverse representation shown in the frames on the right hand side in Fig. 7, the data above $T_{{\rm N}}$ are perfectly described by a Curie-Weiss law $I=C_{1}/(T-\theta)$ with a Weiss temperature $\theta=-33$ K. Below $T_{{\rm N}}$ the temperature dependence of the intensity is well approximated by a pure Curie law $I=C_{2}/T$ (dashed line). As the integrated intensity corresponds to the spin susceptibility, it turns out from the comparison of the prefactors $C_{1}$ and $C_{2}$ that below $T_{{\rm N}}$ only 4% of the spins contribute to the ESR signal, i.e. weakly interacting spins, which are not involved in the antiferromagnetic order e.g. at the surface of the powder grains or at defect sites.

Turning finally to Fig. 7 (b), the linewidth is found to depend strongly on temperature, indicating two different spin-dynamic regimes above and below $T_{{\rm N}}$. Starting from a value of about 200 Oe at high temperature, the linewidth is found to increase with decreasing temperature, diverging close to the antiferromagnetic phase transition with values larger than 1 kOe. On further cooling, the linewidth recovers to about 800 Oe but diverges again with values above 3 kOe below 4 K.

The broadening of the ESR line on approaching $T_{{\rm N}}$ may be treated in terms of critical behavior due to slowing down of spin fluctuations in the vicinity of an order-disorder transition. This causes the divergence of the spin correlation length, which in turn affects the spin-spin relaxation time of exchange narrowed ESR lines resulting in the critical broadening, given by

[TABLE]

where the first term $\Delta H_{0}$ describes the limiting temperature-independent linewidth for the exchange narrowed regime, while the second term reflects the critical behavior with the critical exponent $\beta$. The dashed and the solid lines on the left lower panel in Fig. 7 represent a least-squares-fit of the linewidth data. The best fitting was obtained with the parameters $\Delta H_{0}=221(2)$ Oe, $A=13(2)$ kOe, $T_{N}=45(1)$ K and $\beta=1.37(5)$. The value of $T_{{\rm N}}$ obtained here is close to that from our heat capacity data. Below $T_{{\rm N}}$, the residual ESR signal diverges approximately with a power law $T^{-1.5}$.

Given the fact that the critical exponent of the divergence above $T_{{\rm N}}$ is significantly smaller than the expected value of 2.6 for the 2D Heisenberg antiferromagnetic model(Benner and Boucher, 1990), the line broadening on approaching $T_{{\rm N}}$ may be alternatively described in terms of a Berezinski-Kosterlitz-Thouless (BKT) scenario like in the case of the honeycomb system BaNi2V2O8 [Ref. (Heinrich et al., 2003)] with spin $S=1$ Ni2+. Indeed the temperature dependence of the linewidth is very well approximated by the expression

[TABLE]

with the Kosterlitz-Thouless temperature $T_{{\rm KT}}=40(1)$ K, the parameter $b=0.83(6)$, the prefactor $A=11(2)$ Oe and the residual linewidth $\Delta H_{0}=196(5)$ Oe, as shown in the lower left frame of Fig. 7 The BKT scenario indicates the spin-spin relaxation via magnetic vortices, governed by the vortex correlation length which diverges at $T_{{\rm KT}}$ due to vortex-antivortex pairing. Originally this topological phase transition was derived for the XY model by Berezinskii (Berezinskii, 1971) and by Kosterlitz and Thouless (Kosterlitz and Thouless, 1973). But later on it was shown (Cuccoli et al., 2003) that already a weak anisotropy is enough to provide a BKT scenario. Concerning 2D spin $S=3/2$ antiferromagnets a BKT type scenario was reported for the triangular lattice antiferromagnets $A$CrO2 with $A=$ H, Li, Na, Cu Ag, Pd (Hemmida et al., 2011; Alexander et al., 2007). In those Heisenberg antiferromagnets the frustration of the antiferromagnetic couplings gives rise to so called $Z_{2}$ vortices (Kawamura and Miyashita, 1984). Returning to the present Mn system, the obtained fit parameters are comparable to those found in BaNi2V2O8. The parameter $b<\pi/2$ is in the range of theoretically sound values. The Kosterlitz-Thouless temperature of about $0.83T_{{\rm N}}$ is typical for quasi-2D-antiferromagnets, where the 3D antiferromagnetic order masks the Kosterlitz-Thouless transition. Using the relation(Bramwell and Holdsworth, 1993)

[TABLE]

derived for quasi 2D antiferromagnets (exchange constant $J$) with weak planar anisotropy and inter-plane coupling $J^{\prime}$ we obtain a ratio $J/J^{\prime}\approx 40$, if we use the experimental value of $b$, (or $\approx 800$, if we use $b=\pi/2$). Appearance of LRO close to $\theta_{CW}$ in spite of the 2D nature might result from a renormalisation of $T_{\rm N}$ due to a large in-plane AF correlation length. (Chakravarty et al., 1988)

III.5 NMR Results

In order to develop a microscopic understanding of the magnetic properties, 7Li ($I$ $=3/2$, $\frac{\gamma}{2\pi}=16.546$ MHz/T) nuclear magnetic resonance (NMR) measurements were carried out on polycrystalline Ag3LiMn2O6.

The obtained 7Li-NMR spectra in the entire measured temperature range display a shoulder along with the main line, see Fig. 8. The main line is found to be broadened and shifted to the lower field side as a function of temperature with respect to the 7Li-NMR line measured for the isostructural diamagnetic sample Ag3LiTi2O6, while the shoulder remained almost unshifted on lowering the temperature. The asymmetry (shoulder) in spectra could result from the anisotropy of hyperfine coupling. It must be noticed that, this asymmetry is most likely not from any chemical disorder between Li/Ru as our x-ray diffraction refinement results discard this possibility. Surprisingly, the asymmetry in spectra was also seen in the isostructural material Ag3LiRu2O6 [Ref. (Kumar et al., 2019)]. Interestingly, the hydrogenated analogue of Ag3LiIr2O6, i.e. H3LiIr2O6 does not show significant asymmetry in 1H NMR line shape (Kitagawa et al., 2018).

We analyzed the powder averaged 7Li NMR spectra of Ag3LiMn2O6, similar to the previously studied material Ag3LiRu2O6, by fitting the spectra to a combination of the anisotropic shift parameters $K_{iso}$ and $K_{aniso}$ . A few representative simulated patterns are shown in Fig. 9. The extracted $K_{iso}$ from 7Li NMR spectra in the temperature range $300-3$ K as a function of temperature is shown in Fig. 10. The $K_{iso}$ data follow the bulk susceptibility data down to about 50 K and show an anomaly around the ordering temperature. The low-$T$ deviation of the bulk susceptibility from the NMR shift could be the result of some extrinsic impurity contribution or intrinsic defects. The hyperfine coupling constant ($A_{hf}$) and the chemical shift obtained from the $K$-$\chi$ plot (inset of Fig. 10) yield 1.45 $\pm$ 0.04 kOe/$\mu_{B}$ and 0.009(6)%, respectively.

The 7Li NMR spin-lattice relaxation rate ($1/T_{1}$) measurements were performed with the intention to study the low-energy spin dynamics or to probe the $q$-averaged dynamical susceptibility of Ag3LiMn2O6 in the temperature range $2-300$ K at the transmitter frequency 95 MHz. The saturation recovery method was employed to measure the $1/T_{1}$ data. In order to deduce the spin-lattice relaxation time from the measured data, the data were fitted to a combination of two exponential decays given by the equation:

[TABLE]

where $T_{1L}$ and $T_{1S}$ are the long and short components of spin-lattice relaxation time with $A$ and $B$ being constants.

Figure 11(a) depicts a plot for 7Li nuclear magnetization saturation recoveries at selected temperatures for Ag3LiMn2O6. A two-component spin-lattice relaxation was also observed in Ag3LiRu2O6. This could result from an incomplete saturation of the NMR line leading to spectral diffusion and an initial fast recovery. The long component of the spin-lattice relaxation rate ($1/T_{1L}$) for Ag3LiMn2O6 is illustrated in Fig. 11(b). The $1/T_{1L}$ data in the temperature range $300-60$ K do not show any variation as a function of temperature, however, below about 60 K $1/T_{1L}$ starts to deviate from this behavior. At the antiferromagnetic ordering temperature, the $1/T_{1L}$ data should have also exhibited a distinct anomaly, but no peak was seen and a smooth decrease was noticed in $1/T_{1L}$ data in the temperature range $2-50$ K. The absence of the signature of any AFM order in the $1/T_{1L}$ data most likely results from a cancellation of the antiferromagnetic fluctuations at the Li position because of its symmetric position in a honeycomb network of Mn atoms. So, because of the 2D symmetric arrangement of Mn atoms around Li atom, which sits in the middle of honeycomb lattice, the $1/T_{1L}$ data measured for 7Li nuclei will not sense fluctuations in the hyperfine field perpendicular to the applied field. Thus one, in principle, does not expect to see any sharp kink or any critical divergence in the $1/T_{1L}$ data. This finding further strengthens the idea of a symmetric crystallographic position of Li atom, 2a (0, 0, 0), with respect to its surrounding Mn atoms in a unit cell and also an absence of Li/Mn site disorder. The $1/T_{1L}$ becomes very long with decrease in temperature as static order develops and there is absence of any fluctuations.

III.6 Electronic structure calculation

In order to identify the dominant exchange paths and the relevant spin Hamiltonian of the system we have performed first principles electronic structure calculation using VASP. The MnO6 octahedral units that host magnetism form an edge shared honeycomb geometry in the $a-b$ plane with Li ions at the center of the honeycomb. Upon relaxation, the distances of three nearest neighbor (NN) Mn atoms become 2.926 Å and 2.932 Å with co-ordination two and one respectively. The Mn-O-Mn angles in the respective NN paths are 102.33*∘* and 102.61*∘. The MnO6* octahedra have a monoclinic distortion and hence the Mn-O bond-lengths are unequal (1.876 Å to 1.878 Å) as also the Mn-O-Mn bond angles (77.68*∘* to 95.05*∘*).

In order to understand the basic electronic structure, we have first carried out non-spin polarized calculations enforcing the spin degeneracy. The octahedral crystal field breaks the degeneracy of 5 $d$ states of Mn atoms into triply degenerate t2g and doubly degenerate eg states which get further split due to monoclinic distortion maintaining crystal field splitting between t2g and eg states to be 2.5 eV. The calculations reveal that the Mn t2g states are half filled consistent with the Mn4+ ($d$3) configuration resulting in a metallic solution.

Next we have performed spin polarized calculations with FM arrangement of Mn spins within the GGA approximation. A plot of the spin polarized density of states (DOS) for the FM configuration shown in Fig.12(a) reveals that in the majority spin channel the Mn-d t2g states are completely filled and the minority t2g states are complete empty with an exchange splitting of about 0.62 eV. The eg states in both the spin channels are completely empty. In the FM calculation, the total moment per formula unit, containing 2 Mn atoms, is calculated to be 6.0 $\mu_{B}$ which further supports the 4$+$ charge state of Mn and also consistent with experimentally calculated value of $\mu_{eff}$ (4.46 $\pm$ 0.33 $\mu_{B}$). FM calculation gives magnetic moment per Mn site to be 2.68 $\mu_{B}$ as the rest of the moment lies in the ligand sites (0.04 $\mu_{B}$/ O) due to substantial hybridization. Inclusion of Coulomb correlation (U) further increases the exchange splitting (0.99 eV) and localizes the $d$ orbitals which essentially increase the moment of Mn (2.95 $\mu_{B}$/ Mn).

In order to identify the nature of the ground state we have calculated the energies of different possible collinear magnetic configurations namely FM, AFM-1, AFM-2 and AFM-3. The results are shown in Table-II. In the AFM-1 configuration, all nearest neighbor interactions (J1 and J2) are antiferromagnetic similar to the ground state AFM solution of a bipartite lattice. In the AFM-2 configuration, among the two types of NN interactions, J1 is AFM while J2 is ferromagnetic. On the contrary the AFM-3 configuration has FM J1 and AFM J2 interactions. From Table 2 it is evident that AFM-1 is the lowest energy configuration with NN interactions antiferromagnetic.

Now for a quantitative estimate of Mn inter-site exchange strengths, we have calculated symmetric exchange interactions with "Four State" method based on the total energy of the system with few collinear spin alignments. If the magnetism in the system is fully described by the Heisenberg Hamiltonian, the energy for such a spin pair can be written as follows (Xiang et al., 2011):

[TABLE]

where $J_{ij}$ is the symmetric exchange coupling along bond which connects spin pair $i$ and $j$. $h_{1}$ = $-\sum\limits_{i\neq 1,2}J_{1i}S_{i}$ and $h_{2}$ = $-\sum\limits_{i\neq 1,2}J_{2i}S_{i}$ and $E_{all}$ = $-\sum\limits_{i,j\neq 1,2}J_{ij}S_{i}\cdot S_{i}$ and $E_{0}$ contains all other non-magnetic energy contributions. The second (third) term in Eqn. 4 corresponds to the coupling of the spin 1 (2) with all other spins in the unit cell except spin 2 (1), $E_{all}$ characterizes the exchange couplings between all spins in the unit cell apart from spins 1 and 2. The exchange interaction strength between site 1 and 2 obtained with total energy of four collinear spin alignments (such as ($1_{\uparrow}$ $2_{\uparrow}$), ($1_{\uparrow}$ $2_{\downarrow}$), ($1_{\downarrow}$ $2_{\uparrow}$), ($1_{\downarrow}$ $2_{\downarrow}$)) has the expression (Xiang et al., 2011)

[TABLE]

The first (second) suffix of energy ($E$) tells the spin state of site 1 (2). The obtained symmetric exchange interactions are J1= -2.59 meV (AFM), J2= -2.28 meV (AFM), J3 = 0.54 meV (FM), J4= 0.69 meV (FM), J5= -0.16 meV (AFM) and J6= -0.15 meV (AFM) and the respective exchange paths are shown in 13(c). The possible mechanism of spin conserved exchange couplings for the NN AFM exchange interactions (J1 and J2) and FM second neighbors (J3 and J4) are shown in Fig.12(b). For NN AFM alignments, exchange splitting and inter-site hoppings are the key parameters while for the FM arrangement, spins have to overcome the crystal field splitting with nearest neighbor superexchange hopping to obey the Hund’s coupling. The strongest nearest neighbor (NN) interactions along with the finite (though small) further neighbor interactions results in the long ranged AFM ordering in the system.

The Wannier function plots in Fig. 13(a) and (b) clearly shows that the NN interactions via Mn-O-Mn superexchange path are dominant. The substantial deviation of Mn-O-Mn angle(s) ($\sim$ 102*∘) from $\frac{\pi}{2}$ along paths J1(2)* favor an antiferromagnetic alignment in accordance with Goodenough-Kanamori-Anderson rules (Anderson, 1950; Goodenough, 1955, 1958; Kanamori, 1959), which is consistent with our results. To check the strength of inter-layer coupling along $c$ direction we have fixed the static intra-plane magnetic arrangement to AFM-1 and applied the above mentioned "Four state" method to calculate the interlayer exchange J7. The interplanar coupling strength is estimated to be FM in nature with magnitude 0.14 meV. Theoretically calculated $\theta_{CW}$ from these exchange strengths turns out to be $-54.8$K which is very close to experimentally obtained value, $-51$K. The ratio of inter-planar and NN intra-planar interaction ($\frac{J_{NN}}{J_{inter}}$) is nearly 19.0 which suggests that the magnetic network in primarily of 2D in nature.

IV Conclusions

In summary, a new honeycomb material Ag3LiMn2O6 has been studied using x-ray diffraction, neutron diffraction, magnetization, specific heat, ESR and NMR measurements and first principles calculations. An asymmetric peak, signature of superstructure, is seen in both the x-ray and neutron diffraction and it is found to grow in intensity below about 50 K before saturating at lower temperatures. This suggests magnetic ordering of the honeycomb lattice. The susceptibility measurements carried out for Ag3LiMn2O6 show an anomaly around 50 K and the presence of antiferromagnetic interactions (\theta_{CW}$$\sim-51 K) among Mn moments. The neutron diffraction data measured down to 2 K clearly show the onset of magnetic ordering below 50 K, in agreement with the anomaly observed in the susceptibility data of Ag3LiMn2O6. The heat capacity measurements further support long-range magnetic order in Ag3LiMn2O6 by exhibiting a sharp peak in the measured specific heat around 47 K. The entropy change inferred from the heat capacity data suggests that the system needs to be heated to nearly 1.6 times the ordering temperature to recover the full entropy. This suggests that 2D magnetic correlations start building up at high temperature; on cooling, a fraction of the entropy is already lost when the sample locks into LRO around 50 K. Our local probe 7Li NMR spectra measurements done on the powder samples, which basically measure the static susceptibility also support thermodynamic measurements by exhibiting a clear anomaly in the measured 7Li NMR shift. The 7Li spin-lattice relaxation rate, which is a measure of the $q$-averaged dynamical susceptibility, does not show a peak as observed in other measurements presumably because of the cancellation of antiferromagnetic fluctuations at the center (Li-site) of the hexagon also implying that the structure remains that of a regular honeycomb with Li sitting at the centers of the honeycomb network. However, 7Li $1/T_{1}$ too shows a sharp decrease of nearly four orders of magnitude below about 50 K indicating the quenching of magnetic fluctuations due to the onset of magnetic order. Taken together, our thermodynamic and 7Li NMR shift measurements evidence the emergence of an ordered ground in the 3$d$ honeycomb material Ag3LiMn2O6. Our experimental results are corroborated by first principles electronic structure calculations. Our $ab$ $initio$ calculations find that the NN interactions J1(2) are antiferromagnetic. The further neighbor interactions also do not give rise to any frustration. Thus, in the case of a dominant Heisenberg term (as might be expected in this case), and even a small inter-planar coupling (as per our calculations) the honeycomb system displays long-range order as seen here. The manifestation of 2D effects in Ag3LiMn2O6 is seen from the analysis of the $T$-dependence of the ESR linewidth above the transition temperature. We obtained a Kosterlitz-Thouless temperature of about $0.83T_{{\rm N}}$ which is typical for quasi-2D-antiferromagnets. The weak interplanar coupling is sufficient to lock the system into 3D order which then masks the Kosterlitz-Thouless transition.

V Acknowledgments

This work is partially based on experiments performed at the Swiss spallation neutron source SINQ, Paul Scherrer Institute, Villigen, Switzerland. We thank Department of Science and Technology (DST), Govt. of India for financial support through the BRICS project Helimagnets. ID thanks DST, Govt. of India and TRC for financial support. R. Kumar acknowledges CSIR, India and IRCC, IIT Bombay for awarding him fellowships for the completion of this work. P. M. Ette acknowledges CSIR, INDIA for providing financial support under CSIR-SRF Fellowship (grant no. 31/52(14)2k17). AVM would like to thank the Alexander von Humboldt Foundation for financial support during the stay at the University of Augsburg. The work of RE were done within the framework of fundamental research AAAA-A18-118030690040-8 of FRC Kazan Scientific Center of RAS. A.A.Gippius acknowledges the financial support from the RFBR Grant No. 17-52-80036. Additionally, we kindly acknowledge support from the German Research Society (DFG) via TRR80 (Augsburg, Munich).