Magnetic Properties of A2Ni2TeO6 (A = K, Li): Zigzag Order in the Honeycomb Layers of Ni2+ Ions Induced by First and Third Nearest-Neighbor Spin Exchanges

The static and dynamic magnetic properties and the specific heat of K2Ni2TeO6 and Li2Ni2TeO6 were examined and it was found that they undergo a long-range ordering at TN = 22.8 and 24.4 K, respectively, but exhibit a strong short-range order. At high temperature, the magnetic susceptibilities of K2Ni2TeO6 and Li2Ni2TeO6 are described by a Curie–Weiss law, with Curie-Weiss temperatures Θ of approximately −13 and −20 K, respectively, leading to the effective magnetic moment of about 4.46 ± 0.01 μB per formula unit, as expected for Ni2+ (S = 1) ions. In the paramagnetic region, the ESR spectra of K2Ni2TeO6 and Li2Ni2TeO6 show a single Lorentzian-shaped line characterized by the isotropic effective g-factor, g = 2.19 ± 0.01. The energy-mapping analysis shows that the honeycomb layers of A2Ni2TeO6 (A = K, Li) and Li3Ni2SbO6 adopt a zigzag order, in which zigzag ferromagnetic chains are antiferromagnetically coupled, because the third nearest-neighbor spin exchanges are strongly antiferromagnetic while the first nearest-neighbor spin exchanges are strongly ferromagnetic, and that adjacent zigzag-ordered honeycomb layers prefer to be ferromagnetically coupled. The short-range order of the zigzag-ordered honeycomb lattices of K2Ni2TeO6 and Li2Ni2TeO6 is equivalent to that of an antiferromagnetic uniform chain, and is related to the short-range order of the ferromagnetic chains along the direction perpendicular to the chains.


Introduction
Compounds with honeycomb layers of magnetic ions attracted much attention in the field of low-dimensional magnetism [1], owing in part to Kitaev's conjecture [2] that they can have gapped and gapless liquid-like ground states in a certain range of spin exchange parameters. It has been elusive to find honeycomb-layered ( Figure 1a) magnetic systems confirming Kitaev's conjecture because of the inevitable interlayer interactions, which can lead to a three-dimensional (3D) long-range ordering. In real honeycomb-layered materials, anisotropic in-plane Kitaev interactions compete with isotropic Heisenberg interactions. The ordering of magnetic moments in honeycomb layers can be zigzag (Figure 1b (Figure 1c) [3]. Kitaev's conjecture was originally analyzed for the spin S = 1/2 case, but it has subsequently been extended to S > 1/2 systems (in particular, S = 1), although these systems are not exactly solvable [4]. It was shown that the bond-dependent Kitaev interaction model can be realized in two-dimensional (2D) Mott insulators if there exists strong Hund coupling of electrons at the cation sites and strong spin-orbit coupling at the anion sites [4]. The iridate Na 2 IrO 3 with honeycomb layers of low-spin Ir 4+ (d 5 , S = 1/2) ions [5][6][7] and the ruthenium chloride α-RuCl 3 with honeycomb layers of low-spin Ru 3+ (d 5 , S = 1/2) ions [8][9][10] were examined as the S = 1/2 systems that can capture the basics of Kitaev magnetism. The iridate β-Li 2 IrO 3 has also been examined for Kitaev magnetism, but its Ir 4+ ions form a 3D framework rather than honeycomb layers [11,12]. The magnetic structure in the honeycomb layers of Na 2 IrO 3 is consistent with a zigzag order (i.e., FM zigzag chains of Ir 4+ ions are antiferromagnetically coupled, Figure 1b) or a stripy order (i.e., AFM zigzag chains of Ir 4+ ions are ferromagnetically coupled, Figure 1c) [6]. The magnetic structure of α-RuCl 3 is more consistent with a zigzag order than with a stripy order [9]. The layered-phase A 3 Ni 2 SbO 6 (A = Li, Na) [13,14] consists of honeycomb layers of S = 1 ions (i.e., Ni 2+ ions). Each honeycomb layer consists of edge-sharing NiO 6 octahedra each containing a Ni 2+ ion, in which every hexagon of Ni 2+ ions has its center occupied by a Sb 5+ cation to form a SbO 6 octahedron ( Figure 1d). The magnetic structure of A 3 Ni 2 SbO 6 (A = Li, Na) in the honeycomb layers of Ni 2+ ions is described by a zigzag order [13], similar to the one found for the S = 1/2 systems Na 2 IrO 3 and α-RuCl 3 .
Materials 2022, 15, x FOR PEER REVIEW 2 of 14 or stripy (Figure 1c) [3]. Kitaev's conjecture was originally analyzed for the spin S = 1/2 case, but it has subsequently been extended to S > 1/2 systems (in particular, S = 1), although these systems are not exactly solvable [4]. It was shown that the bond-dependent Kitaev interaction model can be realized in two-dimensional (2D) Mott insulators if there exists strong Hund coupling of electrons at the cation sites and strong spin-orbit coupling at the anion sites [4]. The iridate Na2IrO3 with honeycomb layers of low-spin Ir 4+ (d 5 , S = 1/2) ions [5][6][7] and the ruthenium chloride α-RuCl3 with honeycomb layers of low-spin Ru 3+ (d 5 , S = 1/2) ions [8][9][10] were examined as the S = 1/2 systems that can capture the basics of Kitaev magnetism. The iridate -Li2IrO3 has also been examined for Kitaev magnetism, but its Ir 4+ ions form a 3D framework rather than honeycomb layers [11,12]. The magnetic structure in the honeycomb layers of Na2IrO3 is consistent with a zigzag order (i.e., FM zigzag chains of Ir 4+ ions are antiferromagnetically coupled, Figure 1b) or a stripy order (i.e., AFM zigzag chains of Ir 4+ ions are ferromagnetically coupled, Figure 1c) [6].
The magnetic structure of α-RuCl3 is more consistent with a zigzag order than with a stripy order [9]. The layered-phase A3Ni2SbO6 (A = Li, Na) [13,14] consists of honeycomb layers of S = 1 ions (i.e., Ni 2+ ions). Each honeycomb layer consists of edge-sharing NiO6 octahedra each containing a Ni 2+ ion, in which every hexagon of Ni 2+ ions has its center occupied by a Sb 5+ cation to form a SbO6 octahedron ( Figure 1d). The magnetic structure of A3Ni2SbO6 (A = Li, Na) in the honeycomb layers of Ni 2+ ions is described by a zigzag order [13], similar to the one found for the S = 1/2 systems Na2IrO3 and -RuCl3. So far, it is not clear why the honeycomb layers of these magnetic systems form a zigzag order, namely, why they adopt FM zigzag chains that are antiferromagnetically coupled. To understand the cause for this ordering, it is necessary to know the spin exchanges between the first, second, and third nearest-neighbor (NN) magnetic ions, which are J1, J2, and J3 depicted in Figure 1e, respectively, when a honeycomb layer is made up (d) A honeycomb layer of magnetic ions (solid blue circles) in which the center of every hexagon of magnetic ions is occupied by a cation (e.g., Na + in Na 2 IrO 3 , Sb 5+ in A 3 Ni 2 SbO 6 , Te 6+ in A 2 Ni 2 TeO 6 , (P-P) 8+ dimer in NiPS 3 ). (e) First NN spin exchange (J 1 ), second NN spin exchange (J 2 ), and third NN spin exchange (J 3 ) in a honeycomb layer made up of regular hexagons of magnetic ions. (f) First NN spin exchange (J 1 and J 1 ), second NN spin exchange (J 2 and J 2 ), and third NN spin exchange (J 3 and J 3 ) in a honeycomb layer made up of slightly distorted hexagons of magnetic ions (e.g., 2-fold rotational symmetry). regular hexagons of magnetic ions. However, when these hexagons are lower in symmetry (e.g., 2-fold rotational symmetry), the first, second, and third NN spin exchanges are each split into two different ones, as depicted in Figure 1f. Zvereva et al. evaluated three spin exchanges of A 3 Ni 2 SbO 6 (A = Li, Na) [13] (namely, the first NN exchanges J 1 and J 1 as well as the third NN exchanges J 3 and J 3 , Figure 1f), and found that J 1 is AFM, J 1 is FM, while the third NN spin exchanges J 3 and J 3 are practically zero. Although this is consistent with a zigzag order in A 3 Ni 2 SbO 6 (A = Li, Na), we note that a zigzag order also occurs in the honeycomb layers of M 2+ (M = Fe, Co, Ni) ions in the sulfides MPS 3 , in which each hexagon of M 2+ ions has its center occupied by a P 2 S 6 4− ion ( Figure 1d) to form MS 6 octahedra. The strongest spin exchange of NiPS 3 was found to be the third NN exchanges (J 3 and J 3 ), which are strongly AFM, while the first NN spin exchanges (J 1 and J 1 ) are FM and are weaker than the third NN exchanges by a factor greater than three [15]. Thus, the zigzag order of NiPS 3 is caused by the strongly AFM third NN spin exchanges. This finding makes it necessary to check if the third NN spin exchanges vanish in A 3 Ni 2 SbO 6 (A = Li, Na) as reported [13].
Honeycomb layers of Ni 2+ ions similar to those of A 3 Ni 2 SbO 6 (A = Li, Na) are also present in A 2 Ni 2 TeO 6 (A = K, Li) [16][17][18][19]. Formally, A 2 Ni 2 TeO 6 results from A 3 Ni 2 SbO 6 by replacing the Sb 5+ ion with a Te 6+ ion, by removing one cation A + to satisfy the charge balance requirement, and by changing the layer stacking mode leading to trigonal prismatic and tetrahedral coordination for K + and Li + , respectively. In the present work, we probed the static and dynamic magnetic properties of A 2 Ni 2 TeO 6 (A = K, Li) and found that they are very similar to those of A 3 Ni 2 SbO 6 (A = Li, Na), and we determined the spin exchanges (Figure 1f) of A 2 Ni 2 TeO 6 using the energy-mapping analysis [20,21] to show that the honeycomb layers of not only A 2 Ni 2 TeO 6 but also A 3 Ni 2 SbO 6 have a zigzag magnetic order for the same reason as found for NiPS 3 . Thus, the cause for the zigzag order of not only A 2 Ni 2 TeO 6 (A = K, Li) but also A 3 Ni 2 SbO 6 (A = Li, Na) is the third NN spin exchange, which is strongly AFM, as found for NPS 3 .
Both static and dynamic magnetic properties of A 2 Ni 2 TeO 6 (A = K, Li) evidence the behavior inherent in a magnet with reduced dimensionality, and the long-range order in these compounds is preceded by a short-range order. These observations are fully supported by specific heat measurements, which reveal sharp λ-type singularities establishing the occurrence of a Néel order. We show that the honeycomb layers of A 2 Ni 2 TeO 6 (A = K, Li) and Li 3 Ni 2 SbO 6 adopt a zigzag order, in which zigzag ferromagnetic chains are antiferromagnetically coupled, and that the short-range order of Li 2 Ni 2 TeO 6 and K 2 Ni 2 TeO 6 arises from the short-range order of the ferromagnetic chains along the direction perpendicular to the chains.

K 2 Ni 2 TeO 6
Light-green samples of K 2 Ni 2 TeO 6 were prepared by conventional solid-state synthesis. Since the material is highly hygroscopic, it was necessary to protect it from the atmospheric moisture (see the Supplementary Materials for details). In agreement with the previous data [18], the XRD pattern (Supplementary Figure S1) shows that our sample of K 2 Ni 2 TeO 6 represents a hexagonal superlattice of the P2 type, space group P6 3 /mcm, with only weak extra reflections. Least squares refinements resulted in the lattice parameters a = 5.258(3), c = 12.417(1) Å, and c/a = 2.362, which are consistent with the literature data [18]. The small differences in the absolute values are systematic and may hence be due to uncertainties in our refined sample displacement and/or in the wavelengths used. Thus, it is important to compare the axial ratios. Masese et al. [18] reported a slightly higher c/a ratio of 2.370, together with the dark green color of their sample, which suggests the presence of some Ni 3+ ions, presumably due to potassium deficiency. Chemical analysis of our light-green sample by reverse redox titration [22] yielded the oxidation state of 2.01 ± 0.01 for Ni, thus confirming the stoichiometry. In contrast, our potassium-deficient samples were black, containing a considerable amount of Ni 3+ ions and exhibiting much larger c/a ratios. This is a common feature of the present class of structures: the oxidation of an octahedral cation results in the a-axis contraction, and the loss of the interlayer alkali ions, in the c-axis expansion. However, the deviation from stoichiometry in the reported structure [18] is evidently small and the reported structure is quite reasonable, so we did not attempt a re-refinement.

Li 2 Ni 2 TeO 6
We prepared Li 2 Ni 2 TeO 6 from Na 2 Ni 2 TeO 6 by an ion-exchange reaction [16] (see the Supplementary Materials for details). Previously, the XRD powder pattern (Supplementary Figure S2) was indexed in the space group Cmca by analogy with T2-Li 2 NiMn 2 O 6 [23]. The same preparation route and the same space group were adopted by Grundish et al. [17], and the orthorhombic lattice parameters are in reasonable agreement (Table 1). Li 2 Ni 2 TeO 6 is not isostructural with its sodium precursor (P6 3 /mcm) because small Li + ions cannot be accommodated in the trigonal prismatic sites (i.e., Na + ion sites), so the adjacent honeycomb (Ni 2 TeO 6 ) 2− layers are shifted to provide tetrahedral interlayer sites suitable for Li + ions, as found in T2-Li 2 NiMn 2 O 6 [23]. Unfortunately, crystal structure refinement of the T2-type Li 2 Ni 2 TeO 6 [17] resulted in unrealistic bond lengths and bond valence sums (BVSs) ( Table 2), and our refinement results were not better. The reasons for this might be: (i) Admixture of foreign phase(s): the zoomed-in view of the XRD pattern ( Figure 5 in [17]) reveals a strong unindexed reflection at 2Θ ≈ 17.53 • . Visually, it might seem to be the α2 component of the strongest reflection (2Θ ≈ 17.38 • ), but their separation is three times larger than the doublet separation. Much weaker shoulders are seen in our patterns as well ( Figure S2). (ii) The powder patterns are diffuse in nature arising from the stacking faults due to the layer gliding, which is induced by the ion-exchange transforming the space group of the crystal structure from P6 3 /mcm to Cmca. (iii) X-ray diffraction measurements have a low sensitivity to the positions of light atoms (Li and O) in the presence of heavy elements (Te and Ni). Thus, in the X-ray crystal structure reported for Li 2 Ni 2 TeO 6 [17], some Li, O1, and O2 positions are known only in three decimal places. Thus, it is important to have more accurate atomic positions. The crystal structure of Li 2 Ni 2 TeO 6 was optimized by DFT calculations [17], but the resulting atomic positions were not reported. Thus, we optimized the crystal structure of Li 2 Ni 2 TeO 6 by DFT + U calculations with U eff = 4 eV, and the optimized atomic positions are summarized in Supplementary Table S1. The average Li-O, Ni-O, and Te-O bond lengths of Li 2 Ni 2 TeO 6 , determined from the X-ray diffraction and our DFT + U optimized structures, are compared in Table 2, and so are the oxidations states for the Li, Ni, Te, and O atoms of Li 2 Ni 2 TeO 6 obtained from bond valence sum (BVS) calculations based on the two crystal structures. According to the optimized crystal structure, the oxidation states of Li and Ni are very close to those expected from the ionic electron counting scheme (+1 and +2, respectively). In contrast, the oxidation state of Te is considerably smaller than expected from the ionic electron counting scheme (i.e., +5.29 vs. +6), while that of O is considerably higher than expected (i.e., −1.72/−1.87 vs. −2).

Magnetic Susceptibility and Magnetization
The magnetic susceptibilities, χ(T), of Li 2 Ni 2 TeO 6 and K 2 Ni 2 TeO 6 were measured at B = 0.1 T in the temperature range of 2-300 K using a Quantum Design PPMS-9T system, and their isothermal magnetizations, M(B), using a Quantum Design MPMS-7T SQUID-VSM magnetometer under an external field up to 7 T at various temperatures after cooling the sample at zero magnetic field. Results of zero-field-cooled (ZFC) and field-cooled (FC) measurements carried out for Li 2 Ni 2 TeO 6 and K 2 Ni 2 TeO 6 as well as magnetization curves are shown in Figure 2a-c. A small divergence between the results of ZFC and FC measurements indicates the presence of a modest spin disorder, due most likely to impurity-related effects. On lowering the temperature, the magnetic susceptibilities of both Li 2 Ni 2 TeO 6 and K 2 Ni 2 TeO 6 exhibit a broad maximum, χ max , at T max ≈ 34 K, then the value drops by about one third of χ max . The values of the Néel temperatures, T N , deduced as the temperature, T, at which the derivative dχ(T)/dT shows a maximum (not shown), arẽ 22.8 K for K 2 Ni 2 TeO 6 and~24.4 K for Li 2 Ni 2 TeO 6 . These T N values are considerably lower than T max , signaling the presence of strong short-range correlations. We note that AFM uniform chains exhibit a broad magnetic susceptibility maximum due to their short-range magnetic order. The occurrence of a magnetic susceptibility maximum in A 2 Ni 2 TeO 6 (A = Li, K) and A 3 Ni 2 SbO 6 (A = Li, Na) [13,14] suggests that their magnetic properties possess a one-dimensional (1D) character, although their honeycomb magnetic lattices are 2D in nature. The high-temperature magnetic susceptibility can be fitted by the Curie-Weiss law plus a temperature-independent term, χ 0 : where Θ is the Weiss temperature, and C is the Curie constant, C = N A µ eff 2 µ B 2 /3k B (µ eff is the effective magnetic moment, while N A , µ B , and k B are Avogadro's number, Bohr magneton, and Boltzmann constant, respectively). The diamagnetic contributions of K 2 Ni 2 TeO 6 and Li 2 Ni 2 TeO 6 were estimated to be χ 0 = −1.38 10 −4 and −1.1 10 −4 emu/mol, respectively, by summing the Pascal's constants [26]. The χ 0 values were fixed to reduce the number of variable parameters during the fitting analysis. From this analysis, it was found that Θ = −13 K for K 2 Ni 2 TeO 6 and −20 K for Li 2 Ni 2 TeO 6 , implying the presence of dominant ions per formula unit, assuming Ni 2+ in a high-spin configuration (S = 1).
The magnetic susceptibility of A 2 Ni 2 TeO 6 (A = K, Li) can be analyzed on the basis of the high-temperature series expansion (HTSE) approach for a 2D planar honeycomb lattice using the Rushbrook and Wood model [27]. Then, the χ(T) curve in the paramagnetic region can be described by:  Table 3 summarizes the parameters describing the magnetic subsystems of A 2 Ni 2 TeO 6 (A = Li, K), obtained from the magnetic susceptibility and magnetization measurements.

Magnetic susceptibility and magnetization
The magnetic susceptibilities, (T), of Li2Ni2TeO6 and K2Ni2TeO6 were measured at B = 0.1 T in the temperature range of 2-300 K using a Quantum Design PPMS-9T system, and their isothermal magnetizations, M(B), using a Quantum Design MPMS-7T SQUID-VSM magnetometer under an external field up to 7 T at various temperatures after cooling the sample at zero magnetic field. Results of zero-field-cooled (ZFC) and field-cooled (FC) measurements carried out for Li2Ni2TeO6 and K2Ni2TeO6 as well as magnetization curves are shown in Figure 2a- These TN values are considerably lower than Tmax, signaling the presence of strong short-range correlations. We note that AFM uniform chains exhibit a broad magnetic susceptibility maximum due to their short-range magnetic order. The occurrence of a magnetic susceptibility maximum in A2Ni2TeO6 (A = Li, K) and A3Ni2SbO6 (A = Li, Na) [13,14] suggests that their magnetic properties possess a one-dimensional (1D) character, although their honeycomb magnetic lattices are 2D in nature.

Electron Spin Resonance
Electron spin resonance (ESR) studies were carried out using an X-band ESR spectrometer CMS 8400 (ADANI) (f ≈ 9.4 GHz, B ≤ 0.7 T), equipped with a low-temperature mount, operating in the range of T = 6-300 K. The effective g-factors were calculated using an external reference for the resonance field, i.e., BDPA (a,g-bisdiphenylene-b-phenylallyl), for which g et = 2.00359. The ESR data in the paramagnetic phase (T > T N ) show a single broad Lorentzian line-shape ascribable to Ni 2+ ions in octahedral coordination [28] for both K 2 Ni 2 TeO 6 and Li 2 Ni 2 TeO 6 (Supplementary Figure S10). The main ESR parameters (effective g-factor, the ESR linewidth, and the integral ESR intensity) were deduced by fitting the experimental spectra with the Lorentzian profile [29]: where P is the power absorbed in the ESR experiment, B r is the resonance field, and ∆B is the linewidth. The integral ESR intensity, χ ESR , which is proportional to the number of magnetic spins, was estimated by double integration of the first derivative ESR spectrum, dP/dB. Evidently, the temperature dependence of χ ESR (T) follows the Curie-Weiss relationship and agrees well with the static magnetic susceptibility χ(T), as shown in the upper panels of Figure 3. (4) where P is the power absorbed in the ESR experiment, Br is the resonance field, and ΔB is the linewidth. The integral ESR intensity, ESR, which is proportional to the number of magnetic spins, was estimated by double integration of the first derivative ESR spectrum, dP/dB. Evidently, the temperature dependence of ESR(T) follows the Curie-Weiss relationship and agrees well with the static magnetic susceptibility χ(T), as shown in the upper panels of Figure 3.
The average effective g-factor of 2.20  0.03 remains almost temperature-independent in the paramagnetic phase down to ~100 K (lower panels of Figure 3), and then the visible shift of the resonant field to higher magnetic fields begins upon approaching the Néel temperature from above. This behavior implies the presence of strong short-range correlations at temperatures noticeably higher than TN, which is characteristic of the systems with spin frustration and low dimensionality [29]. The linewidth, B, of K2Ni2TeO6 shows three different dynamic regimes: It decreases weakly and almost linearly on lowering the temperature down to ~150 K, then remains constant down to ~100 K. Upon a further decrease in the temperature, the absorption line broadens significantly and the ESR signal vanishes in the vicinity of the Néel temperature, indicating the opening of an energy gap for resonance excitations, e.g., due to the occurrence of a long-range order. Similar spin dynamics were observed recently for A3Ni2SbO6 (A = Li, Na) with a honeycomb lattice of Ni 2+ ions [13]. Following the same procedure, we treated B(T) in the frame of the critical broadening model using the modified Huber's formula [30][31][32][33] with the third linear term to account for the B(T) behavior over the whole temperature range: where the first term B * describes the exchange narrowed linewidth, which is temperature-independent. The second term describes the critical behavior, with TN ESR as the temperature of the order-disorder transition and  as the critical exponent. The third term The average effective g-factor of 2.20 ± 0.03 remains almost temperature-independent in the paramagnetic phase down to~100 K (lower panels of Figure 3), and then the visible shift of the resonant field to higher magnetic fields begins upon approaching the Néel temperature from above. This behavior implies the presence of strong short-range correlations at temperatures noticeably higher than T N , which is characteristic of the systems with spin frustration and low dimensionality [29].
The linewidth, ∆B, of K 2 Ni 2 TeO 6 shows three different dynamic regimes: It decreases weakly and almost linearly on lowering the temperature down to~150 K, then remains constant down to~100 K. Upon a further decrease in the temperature, the absorption line broadens significantly and the ESR signal vanishes in the vicinity of the Néel temperature, indicating the opening of an energy gap for resonance excitations, e.g., due to the occurrence of a long-range order. Similar spin dynamics were observed recently for A 3 Ni 2 SbO 6 (A = Li, Na) with a honeycomb lattice of Ni 2+ ions [13]. Following the same procedure, we treated ∆B(T) in the frame of the critical broadening model using the modified Huber's formula [30][31][32][33] with the third linear term to account for the ∆B(T) behavior over the whole temperature range: where the first term ∆B * describes the exchange narrowed linewidth, which is temperatureindependent. The second term describes the critical behavior, with T N ESR as the temperature of the order-disorder transition and β as the critical exponent. The third term relates to the temperature-linear spin-lattice relaxation term. The solid line on the lower panel of Figure 3a represents a least-squares-fitting of ∆B(T). The best fitting was obtained with ∆B * = 290 ± 5 mT, β ≈ 1 ± 0.05, and D = 0.4 mT/K. Clearly, T ESR N is in good agreement with T N . According to Kawasaki's approach [25,26], the absolute value of the critical exponent can be expressed as β = [(7 + η)ν/2 − 2(1 − ζ)], where ν describes the divergence of the spin-correlation length, η is a critical exponent for the divergence of static correlations, and ζ reflects the divergence of the specific heat. Using the values η = ζ = 0 and ν = 2/3 for 3D antiferromagnets in the Heisenberg model, β becomes 1/3. Thus, the value of β ≈ 1 extracted for K 2 Ni 2 TeO 6 is noticeably higher than 1/3 but is still below the value expected for pure 2D antiferromagnets (i.e., β ≈ 3/2) [34,35], but it is quite comparable to the β reported for other related quasi-2D Ni 2+ compounds A 3 Ni 2 SbO 6 (A = Li, Na) [13] and Li 4 NiTeO 6 [36]. According to Kawasaki-Mori-Huber theory, the temperature variation of ∆B of Li 2 Ni 2 TeO 6 can be described as: The best agreement by the least square method was obtained with the following parameters: ∆B* = 217 ± 5 mT and β = 0.60 ± 0.05. Thus, the analysis of spin dynamics supports the picture of rather a 2D character of magnetic correlations for both K 2 Ni 2 TeO 6 and Li 2 Ni 2 TeO 6 . Spin-dynamic parameters of the studied compounds are summarized in Table 4.

Specific Heat
The specific heat, C p (T), of A 2 Ni 2 TeO 6 (A = Li, K) has been measured using a relaxation method of a Quantum Design PPMS-9T. The data were collected at the zero magnetic field as well as under applied fields of 3, 6, and 9 T in the temperature range of 2-70 K. The C p (T) vs. T plots for K 2 Ni 2 TeO 6 and Li 2 Ni 2 TeO 6 are shown in Figure 4a,b. A λ-type peak is observed at T N for both K 2 Ni 2 TeO 6 and A 2 Ni 2 TeO 6 , which clearly shows an occurrence of long-range antiferromagnetic order. These ordering temperatures, T N , coincide with the temperatures at which the peaks of the d(χT)/dT vs. T plot occur (i.e., the Fisher specific heat) [37,38], which is characteristic of low-dimensional antiferromagnets with strong short-range correlations. The positions of T N slightly shift toward the lower temperatures in the applied external magnetic field, as shown in the lower insets in Figure 4a,b, which is typical for antiferromagnetic compounds. To examine the magnetic contribution to specific heat in the title compounds, the C p (T) curve has been measured for the isostructural nonmagnetic system Na 2 Zn 2 TeO 6 . Application of the scaling procedure [39] allows extracting magnetic specific heat, C m (T), as shown in the upper insets in Figure 4a,b. According to these data, the magnetic entropy, S m , released below T N is 6.74 and 5.71 J/mol·K for Liand K-compounds, respectively. These values are to be compared with the thermodynamic limit S m = nRln(2S + 1) = 18.27 J/mol·K at n = 2 and S = 1, meaning that the dominant part of the magnetic entropy is released above T N .

Computational details
To extract the values of the spin exchanges in A2Ni2TeO6 (A = K, Li) and Li3Ni2SbO6 (A = Li, Na), we carried out spin-polarized DFT calculations by using the frozen-core projector augmented plane wave met [40,41], encoded in the Vienna ab initio simulation package [42], and the PBE exchange-correlation functionals [43]. The electron correlation associated with the 3d states of Ni was taken into consideration by performing the DFT + U calculations [44] with the effective on-site repulsion U eff = U − J on magnetic ions. All our DFT calculations used the plane wave cutoff energy of 450 eV and the threshold of 10 −6 eV for self-consistent-field energy convergence. To relax the atom positions, DFT + U calculations (with Ueff = 4 eV) were performed using a set of (4  6  4) k-points with the criterion of 5  10 −4 eV/Å for the relaxation of the atom positions. Our DFT + U calculations employed a set of (4  4  4) k-points for Li2Ni2TeO6, (5  5  3) k-points for K2Ni2TeO6, and (4  2  4) k-points for Li3Ni2SbO12. As a representative example for the A3Ni2SbO6 (A = Li, Na) family, we examined Li3Ni2SbO12 because Li3Ni2SbO12 is isostructural and isoelectronic with Na3Ni2SbO12, and because A3Ni2SbO6 (A = Li, Na) has already been studied [13]. In our DFT + U calculations, we employed the Ueff values of 3 and 4 eV, which lead to similar trends (see below).

Spin Exchanges and Zigzag Order
The first, second, and third NN intralayer spin exchanges to consider for Li2Ni2TeO6 and Li3Ni2SbO6 are presented in Figure 5a, and those for K2Ni2TeO6 in Figure 5b. We evaluate these spin exchanges by using the energy-mapping analysis based on DFT calculations [15,20,21]. The three spin exchanges of K2Ni2TeO6 (Figure 5b) were determined using the four ordered spin states of Supplementary Figure S3. Similarly, the six intralayer spin exchanges of Li2Ni2TeO6 and Li3Ni2SbO6 (Figure 5a) were determined using the seven ordered spin states of Supplementary Figures S4 and S5, respectively. In the energy-mapping analysis, we determined the relative energies of the ordered spin states by DFT + U calculations (Supplementary Tables S2-S4), expressed the energies of these states in terms of the spin exchanges (Supplementary Tables S5-S7), and finally mapped the relative energies of the DFT + U calculations to the corresponding relative energies expressed in terms of the spin exchanges to find the values of the spin exchanges. . Temperature dependences of the specific heat in Li 2 Ni 2 TeO 6 (a) and K 2 Ni 2 TeO 6 (b). The data for Na 2 Zn 2 TeO 6 are shown by solid lines. The dashed lines represent the reference curves obtained through the scaling procedure [39]. Lower insets represent C p (T) curves measured at various magnetic fields. Upper insets represent temperature dependences of magnetic specific heat, C m .

Computational Details
To extract the values of the spin exchanges in A 2 Ni 2 TeO 6 (A = K, Li) and Li 3 Ni 2 SbO 6 (A = Li, Na), we carried out spin-polarized DFT calculations by using the frozen-core projector augmented plane wave met [40,41], encoded in the Vienna ab initio simulation package [42], and the PBE exchange-correlation functionals [43]. The electron correlation associated with the 3d states of Ni was taken into consideration by performing the DFT + U calculations [44] with the effective on-site repulsion U eff = U − J on magnetic ions. All our DFT calculations used the plane wave cutoff energy of 450 eV and the threshold of 10 −6 eV for self-consistent-field energy convergence. To relax the atom positions, DFT + U calculations (with U eff = 4 eV) were performed using a set of (4 × 6 × 4) k-points with the criterion of 5 × 10 −4 eV/Å for the relaxation of the atom positions. Our DFT + U calculations employed a set of (4 × 4 × 4) k-points for Li 2 Ni 2 TeO 6 , (5 × 5 × 3) k-points for K 2 Ni 2 TeO 6 , and (4 × 2 × 4) k-points for Li 3 Ni 2 SbO 12 . As a representative example for the A 3 Ni 2 SbO 6 (A = Li, Na) family, we examined Li 3 Ni 2 SbO 12 because Li 3 Ni 2 SbO 12 is isostructural and isoelectronic with Na 3 Ni 2 SbO 12 , and because A 3 Ni 2 SbO 6 (A = Li, Na) has already been studied [13]. In our DFT + U calculations, we employed the U eff values of 3 and 4 eV, which lead to similar trends (see below).

Spin Exchanges and Zigzag Order
The first, second, and third NN intralayer spin exchanges to consider for Li 2 Ni 2 TeO 6 and Li 3 Ni 2 SbO 6 are presented in Figure 5a, and those for K 2 Ni 2 TeO 6 in Figure 5b. We evaluate these spin exchanges by using the energy-mapping analysis based on DFT calculations [15,20,21]. The three spin exchanges of K 2 Ni 2 TeO 6 ( Figure 5b) were determined using the four ordered spin states of Supplementary Figure S3. Similarly, the six intralayer spin exchanges of Li 2 Ni 2 TeO 6 and Li 3 Ni 2 SbO 6 ( Figure 5a) were determined using the seven ordered spin states of Supplementary Figures S4 and S5, respectively. In the energymapping analysis, we determined the relative energies of the ordered spin states by DFT + U calculations (Supplementary Tables S2-S4), expressed the energies of these states in terms of the spin exchanges (Supplementary Tables S5-S7), and finally mapped the relative energies of the DFT + U calculations to the corresponding relative energies expressed in terms of the spin exchanges to find the values of the spin exchanges. The spin exchanges of A2Ni2TeO6 (A = K, Li) and Li3Ni2SbO6 obtained from the energy-mapping analysis are summarized in Tables 5 and 6, from which we note the following trends: (1) the first NN spin exchanges are all strongly FM, (2) the second NN spin exchanges are all negligible, and (3) the third NN spin exchanges are all strongly AFM. The magnetic order that accommodates these three factors is a zigzag order, as depicted in Figure 1b, which is what has been experimentally observed for A2Ni2TeO6 (A = Li, K) and A3Ni2SbO6 (A = Li, Na) [13,14]. To understand the occurrence of a short-range magnetic order in these materials, one might consider every FM chain of their zigzag-ordered honeycomb lattices as a pseudo-spin unit. Then, each zigzag-ordered honeycomb lattice is equivalent to an AFM uniform chain, so the short-range order of the zigzag-ordered honeycomb lattice becomes equivalent to that of an AFM uniform chain. Namely, the short-range order in the zigzag-ordered honeycomb lattices of A2Ni2TeO6 (A = Li, K) and A3Ni2SbO6 (A = Li, Na) is associated with short-range ordering of the FM chains along the direction perpendicular to the chains.  The spin exchanges of A 2 Ni 2 TeO 6 (A = K, Li) and Li 3 Ni 2 SbO 6 obtained from the energy-mapping analysis are summarized in Tables 5 and 6, from which we note the following trends: (1) the first NN spin exchanges are all strongly FM, (2) the second NN spin exchanges are all negligible, and (3) the third NN spin exchanges are all strongly AFM. The magnetic order that accommodates these three factors is a zigzag order, as depicted in Figure 1b, which is what has been experimentally observed for A 2 Ni 2 TeO 6 (A = Li, K) and A 3 Ni 2 SbO 6 (A = Li, Na) [13,14]. To understand the occurrence of a short-range magnetic order in these materials, one might consider every FM chain of their zigzagordered honeycomb lattices as a pseudo-spin unit. Then, each zigzag-ordered honeycomb lattice is equivalent to an AFM uniform chain, so the short-range order of the zigzagordered honeycomb lattice becomes equivalent to that of an AFM uniform chain. Namely, the short-range order in the zigzag-ordered honeycomb lattices of A 2 Ni 2 TeO 6 (A = Li, K) and A 3 Ni 2 SbO 6 (A = Li, Na) is associated with short-range ordering of the FM chains along the direction perpendicular to the chains. It is of interest to examine why the zigzag order arises in the honeycomb lattices of A 2 Ni 2 TeO 6 (A = Li, K) and A 3 Ni 2 SbO 6 (A = Li, Na). The magnetic orbitals of the Ni 2+ ion in a NiO 6 octahedron are the e g states, namely, the x 2 − y 2 and 3z 2 − r 2 states, which are combined out-of-phase with the 2p orbitals of the surrounding oxygen ligands. Of the two, the x 2 − y 2 state can have a substantial interaction in the first and third NN exchange paths, as depicted in Figure 6a,b, respectively, because the NiO 4 square planes containing these orbitals can be coplanar. The first NN exchange consists of two Ni-O-Ni paths, in which the two x 2 − y 2 states have their p-orbitals orthogonally arranged at the shared oxygen atoms (Figure 6a). Thus, between the two magnetic orbitals, the overlap integral is zero while the overlap density is nonzero. As a result, the first NN exchange becomes FM [15,20,21,45]. The third NN exchange consists of two Ni-O···O-Ni paths, in which the p-orbital tails of the two x 2 − y 2 states are arranged such that the overlap integral is nonzero while the overlap density is practically zero. As a result, the third NN exchange becomes AFM. For the second NN exchange, the NiO 4 square planes containing the x 2 − y 2 states cannot be coplanar (see Supplementary Figure S6). Thus, neither the overlap integral nor the overlap density between the two x 2 − y 2 states can be substantial, so the second NN exchange is weak. It is of interest to examine why the zigzag order arises in the honeycomb lattices of A2Ni2TeO6 (A = Li, K) and A3Ni2SbO6 (A = Li, Na). The magnetic orbitals of the Ni 2+ ion in a NiO6 octahedron are the eg states, namely, the x 2 − y 2 and 3z 2 − r 2 states, which are combined out-of-phase with the 2p orbitals of the surrounding oxygen ligands. Of the two, the x 2 − y 2 state can have a substantial interaction in the first and third NN exchange paths, as depicted in Figure 6a,b, respectively, because the NiO4 square planes containing these orbitals can be coplanar. The first NN exchange consists of two Ni-O-Ni paths, in which the two x 2 − y 2 states have their p-orbitals orthogonally arranged at the shared oxygen atoms (Figure 6a). Thus, between the two magnetic orbitals, the overlap integral is zero while the overlap density is nonzero. As a result, the first NN exchange becomes FM [15,20,21,45]. The third NN exchange consists of two Ni-OO-Ni paths, in which the porbital tails of the two x 2 − y 2 states are arranged such that the overlap integral is nonzero while the overlap density is practically zero. As a result, the third NN exchange becomes AFM. For the second NN exchange, the NiO4 square planes containing the x 2 − y 2 states cannot be coplanar (see Supplementary Figure S6). Thus, neither the overlap integral nor the overlap density between the two x 2 − y 2 states can be substantial, so the second NN exchange is weak. Finally, we examined the effective interlayer spin exchanges in A2Ni2TeO6 (A = K, Li) and Li3Ni2SbO6. What matters in a long-range magnetic ordering in these systems at low temperature is whether the zigzag-ordered honeycomb layers become ferromagnetically or antiferromagnetically ordered (Supplementary Figures S7-S9). Results of these calculations are summarized in Table 7, which predicts that the honeycomb layers should be ferromagnetically coupled. This is in agreement with the experiment for Li3Ni2SbO6 [13]. The preference for the FM interlayer coupling is much stronger for Li2Ni2TeO6 than for K2Ni2TeO6. This reflects that the interlayer distance is shorter for Li2Ni2TeO6, which strengthens the interlayer interaction. This result is consistent with the observation that the long-range ordering temperature, TN, which involves the ordering between the zigzagordered honeycomb layers, is greater for Li2Ni2TeO6 than for K2Ni2TeO6 (~24.4 vs. ~22.8 K). Table 7. Relative energies (in K per formula unit), calculated from DFT + U computations, of the FM and AFM arrangements between the zigzag-ordered layers in A2Ni2TeO6 (A = K, Li) and Li3Ni2SbO6.

Li2Ni2TeO6
Li3Ni2SbO6 Finally, we examined the effective interlayer spin exchanges in A 2 Ni 2 TeO 6 (A = K, Li) and Li 3 Ni 2 SbO 6 . What matters in a long-range magnetic ordering in these systems at low temperature is whether the zigzag-ordered honeycomb layers become ferromagnetically or antiferromagnetically ordered (Supplementary Figures S7-S9). Results of these calculations are summarized in Table 7, which predicts that the honeycomb layers should be ferromagnetically coupled. This is in agreement with the experiment for Li 3 Ni 2 SbO 6 [13]. The preference for the FM interlayer coupling is much stronger for Li 2 Ni 2 TeO 6 than for K 2 Ni 2 TeO 6 . This reflects that the interlayer distance is shorter for Li 2 Ni 2 TeO 6 , which strengthens the interlayer interaction. This result is consistent with the observation that the long-range ordering temperature, T N , which involves the ordering between the zigzag-ordered honeycomb layers, is greater for Li 2 Ni 2 TeO 6 than for K 2 Ni 2 TeO 6 (~24.4 vs.~22.8 K). Table 7. Relative energies (in K per formula unit), calculated from DFT + U computations, of the FM and AFM arrangements between the zigzag-ordered layers in A 2 Ni 2 TeO 6 (A = K, Li) and Li 3 Ni 2 SbO 6 .

Concluding Remarks
The static magnetic susceptibility along with specific heat data showed the onset of antiferromagnetic order at T N ≈ 22.8 and 24.4 K for K 2 Ni 2 TeO 6 and Li 2 Ni 2 TeO 6 , respectively, which is preceded by a short-range order. The high-temperature magnetic susceptibility data exhibited Curie-Weiss behavior, with Weiss temperatures Θ of approximately −13 and −20 K for K 2 Ni 2 TeO 6 and Li 2 Ni 2 TeO 6 , respectively. The effective magnetic moment was estimated to be about 4.46 µ B per formula unit and agrees with the theoretical value expected for Ni 2+ (S = 1) ions. If we were to describe the high-temperature magnetic portion of the susceptibilities of A 2 Ni 2 TeO 6 (A = K, Li), they can be described by a honeycomb with spin exchange J ≈ −8 ± 1 K. ESR spectra in the paramagnetic phase showed a single Lorentzian-shaped line, which was attributed to Ni 2+ ions at octahedral sites, which were characterized by the isotropic effective g-factor 2.20 ± 0.01. In addition, our ESR data indicated an extended region of short-range order correlations, typical of low-dimensional or frustrated magnets. The intralayer spin exchanges evaluated for A 2 Ni 2 TeO 6 (A = K, Li) and Li 3 Ni 2 SbO 6 showed that the honeycomb layers of these magnets adopted a zigzag order, largely because the third nearest-neighbor spin exchanges are strongly antiferromagnetic and because the first nearest-neighbor spin exchanges are strongly ferromagnetic. This finding arises largely from the fact that the spin exchanges between adjacent Ni 2+ ions are governed largely by their x 2 − y 2 magnetic orbitals. Adjacent zigzag-ordered honeycomb layers prefer to be ferromagnetically than antiferromagnetically coupled. The short-range order of the zigzag-ordered honeycomb lattice is equivalent to that of an antiferromagnetic uniform chain, and arises from the short-range ordering of the ferromagnetic chains along the direction perpendicular to the zigzag chains.