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

Abstract

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 T_N = 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 μ_B 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.

1. 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) 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 Ir⁴⁺ (d⁵, S = 1/2) ions [5,6,7] and the ruthenium chloride α-RuCl₃ with honeycomb layers of low-spin Ru³⁺ (d⁵, 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-Li₂IrO₃ has also been examined for Kitaev magnetism, but its Ir⁴⁺ 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 Ir⁴⁺ ions are antiferromagnetically coupled, Figure 1b) or a stripy order (i.e., AFM zigzag chains of Ir⁴⁺ 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., Ni²⁺ ions). Each honeycomb layer consists of edge-sharing NiO₆ octahedra each containing a Ni²⁺ ion, in which every hexagon of Ni²⁺ ions has its center occupied by a Sb⁵⁺ cation to form a SbO₆ octahedron (Figure 1d). The magnetic structure of A₃Ni₂SbO₆ (A = Li, Na) in the honeycomb layers of Ni²⁺ ions is described by a zigzag order [13], similar to the one found for the S = 1/2 systems Na₂IrO₃ and α-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 of 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 A₃Ni₂SbO₆ (A = Li, Na) [13] (namely, the first NN exchanges J₁ and J₁′ as well as the third NN exchanges J₃ and J₃′, Figure 1f), and found that J₁ is AFM, J₁′ is FM, while the third NN spin exchanges J₃ and J₃′ are practically zero. Although this is consistent with a zigzag order in A₃Ni₂SbO₆ (A = Li, Na), we note that a zigzag order also occurs in the honeycomb layers of M²⁺ (M = Fe, Co, Ni) ions in the sulfides MPS₃, in which each hexagon of M²⁺ ions has its center occupied by a P₂S₆⁴⁻ ion (Figure 1d) to form MS₆ octahedra. The strongest spin exchange of NiPS₃ was found to be the third NN exchanges (J₃ and J₃′), which are strongly AFM, while the first NN spin exchanges (J₁ and J₁′) are FM and are weaker than the third NN exchanges by a factor greater than three [15]. Thus, the zigzag order of NiPS₃ 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 A₃Ni₂SbO₆ (A = Li, Na) as reported [13].

Honeycomb layers of Ni²⁺ ions similar to those of A₃Ni₂SbO₆ (A = Li, Na) are also present in A₂Ni₂TeO₆ (A = K, Li) [16,17,18,19]. Formally, A₂Ni₂TeO₆ results from A₃Ni₂SbO₆ by replacing the Sb⁵⁺ ion with a Te⁶⁺ 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 A₂Ni₂TeO₆ (A = K, Li) and found that they are very similar to those of A₃Ni₂SbO₆ (A = Li, Na), and we determined the spin exchanges (Figure 1f) of A₂Ni₂TeO₆ using the energy-mapping analysis [20,21] to show that the honeycomb layers of not only A₂Ni₂TeO₆ but also A₃Ni₂SbO₆ have a zigzag magnetic order for the same reason as found for NiPS₃. Thus, the cause for the zigzag order of not only A₂Ni₂TeO₆ (A = K, Li) but also A₃Ni₂SbO₆ (A = Li, Na) is the third NN spin exchange, which is strongly AFM, as found for NPS₃.

Both static and dynamic magnetic properties of A₂Ni₂TeO₆ (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 A₂Ni₂TeO₆ (A = K, Li) and Li₃Ni₂SbO₆ adopt a zigzag order, in which zigzag ferromagnetic chains are antiferromagnetically coupled, and that the short-range order of Li₂Ni₂TeO₆ and K₂Ni₂TeO₆ arises from the short-range order of the ferromagnetic chains along the direction perpendicular to the chains.

2. Sample Preparation and X-ray Diffraction

2.1. K₂Ni₂TeO₆

Light-green samples of K₂Ni₂TeO₆ 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 K₂Ni₂TeO₆ represents a hexagonal superlattice of the P2 type, space group P6₃/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 Ni³⁺ 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 Ni³⁺ 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.

2.2. Li₂Ni₂TeO₆

We prepared Li₂Ni₂TeO₆ from Na₂Ni₂TeO₆ 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-Li₂NiMn₂O₆ [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. Lattice parameters of Li₂Ni₂TeO₆ (Cmca) from different sources.

Source	a, Å	b, Å	c, Å	V, Å3
[16]	8.9667 (18)	5.1574 (14)	10.1878 (26)	471.1
[17]	8.9925 (4)	5.1469 (2)	10.1691 (5)	470.7
This work	8.9945 (19)	5.1488 (12)	10.1628 (18)	470.6

). Li₂Ni₂TeO₆ is not isostructural with its sodium precursor (P6₃/mcm) because small Li⁺ ions cannot be accommodated in the trigonal prismatic sites (i.e., Na⁺ ion sites), so the adjacent honeycomb (Ni₂TeO₆)²⁻ layers are shifted to provide tetrahedral interlayer sites suitable for Li⁺ ions, as found in T2-Li₂NiMn₂O₆ [23]. Unfortunately, crystal structure refinement of the T2-type Li₂Ni₂TeO₆ [17] resulted in unrealistic bond lengths and bond valence sums (BVSs) (

Table 2. Average lengths of the Li-O, Ni-O, and Te-O bonds and oxidation states of Li, Ni, Te, and O of Li₂Ni₂TeO₆ expected.

(a) Average bond lengths
Bonds	Sum of Ionic Radii [24]	X-ray Diffraction [17]	Optimized (This Work)
Li-O	1.98	2.20	2.08
Ni-O	2.07	2.13	2.07
Te-O	1.94	1.87	1.97
(b) Oxidation states from bond valence sum analysis [25]
	Expected a	X-ray Diffraction [17]	Optimized (This Work)
Li	+1	+0.68	+0.82
Ni	+2	+2.13	+1.99
Te	+6	+6.98	+5.29
O1 (16g)	−2	−2.15	−1.87
O2 (8f)	−2	−1.70	−1.72

^a From the ionic electron counting scheme.

), 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 P6₃/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 Li₂Ni₂TeO₆ [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 Li₂Ni₂TeO₆ was optimized by DFT calculations [17], but the resulting atomic positions were not reported. Thus, we optimized the crystal structure of Li₂Ni₂TeO₆ 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 Li₂Ni₂TeO₆, 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 Li₂Ni₂TeO₆ 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).

3. Results and Discussion

3.1. Magnetic Properties

3.1.1. Magnetic Susceptibility and Magnetization

The magnetic susceptibilities, χ(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–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 Li₂Ni₂TeO₆ and K₂Ni₂TeO₆ 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), are ~22.8 K for K₂Ni₂TeO₆ and ~24.4 K for Li₂Ni₂TeO₆. 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 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.

The high-temperature magnetic susceptibility can be fitted by the Curie–Weiss law plus a temperature-independent term, χ₀:

$$\chi ={\chi }_0+\frac{C}{T-\mathsf{\Theta }}$$

(1)

where Θ is the Weiss temperature, and C is the Curie constant, C = N_Aμ_eff²μ_B²/3k_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 K₂Ni₂TeO₆ and Li₂Ni₂TeO₆ were estimated to be χ₀ = −1.38 10⁻⁴ and −1.1 10⁻⁴ emu/mol, respectively, by summing the Pascal’s constants [26]. The χ₀ 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 K₂Ni₂TeO₆ and −20 K for Li₂Ni₂TeO₆, implying the presence of dominant antiferromagnetic interactions while, per formula unit (f.u.), μ_eff = 4.45 μ_B for K₂Ni₂TeO₆ and 4.47μ_B for Li₂Ni₂TeO₆. The effective g-factor obtained from ESR data (see below) is about g ≈ 2.2. The result hence well agrees with the theoretical estimate of the effective magnetic moment equal to 4.4 μ_B/f.u. for both compounds, where n is the number of Ni²⁺ ions per formula unit, assuming Ni²⁺ in a high-spin configuration (S = 1).

$${\mu }_{theor}=\sqrt{g^{2}nS\left(S+1\right){\mu }_B^2}$$

(2)

The magnetic susceptibility of A₂Ni₂TeO₆ (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:

$$\chi =\frac{2N{g}^2{\beta }^2}{3kT}\cdot \frac{1}{1+Ax+B{x}^2+C{x}^3+D{x}^4+E{x}^5+F{x}^6}$$

(3)

where x = |J|/kT, A = 4, B = 7.333, C = 7.111, D = –5.703, E = –22.281, and F = 51.737.²⁷ Fitting the χ(T) curve by Equation (3) in the range of 50–300 K yields J = −8 ± 1 K for both compounds. The magnetization isotherms, M(B), taken at 2 K (Figure 2c) demonstrate upward deviations from the linear dependences, suggesting spin–flop transitions at B_SF ≈ 4.7 T and 4.4 T for Li₂Ni₂TeO₆ and K₂Ni₂TeO₆, respectively.

Table 3. Parameters describing the magnetic subsystems of A₂Ni₂TeO₆ (A = K, Li) tellurates.

	Θ, K	μeff, μB/f.u.	Tmax, K	TN, K	J, K	BSF, T
Li2Ni2TeO6	−20 ± 1	4.47 ± 0.01	~34	24.4 ± 0.2	−8 ± 1	~4.7
K2Ni2TeO6	−13 ± 1	4.45 ± 0.01	~34	22.8 ± 0.2	−8 ± 1	~4.4

summarizes the parameters describing the magnetic subsystems of A₂Ni₂TeO₆ (A = Li, K), obtained from the magnetic susceptibility and magnetization measurements.

3.1.2. 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 Ni²⁺ ions in octahedral coordination [28] for both K₂Ni₂TeO₆ and Li₂Ni₂TeO₆ (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]:

$$\frac{dP}{dB}\propto \frac{d}{dB}\left[\frac{\Delta B}{\Delta B^2+{\left(B-B_r\right)}^2}+\frac{\Delta B}{\Delta B^2+{\left(B+B_r\right)}_2}\right]$$

(4)

where P is the power absorbed in the ESR experiment, B_r is the resonance field, and ΔB 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.

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, ΔB, 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 Ni²⁺ ions [13]. Following the same procedure, we treated ΔB(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 ΔB(T) behavior over the whole temperature range:

$$\Delta B\left(T\right)=\Delta B^{*}+A{\left[\frac{T_N^{\mathrm{ESR}}}{T-T_N^{\mathrm{ESR}}}\right]}^{\beta }+DT$$

(5)

where the first term ΔB^* describes the exchange narrowed linewidth, which is temperature-independent. 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 ΔB(T). The best fitting was obtained with ΔB^* = 290 ± 5 mT, β ≈ 1 ± 0.05, and D = 0.4 mT/K. Clearly, $T_N^{\mathrm{ESR}}$ 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 K₂Ni₂TeO₆ 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 Ni²⁺ compounds A₃Ni₂SbO₆ (A = Li, Na) [13] and Li₄NiTeO₆ [36]. According to Kawasaki–Mori–Huber theory, the temperature variation of ΔB of Li₂Ni₂TeO₆ can be described as:

$$\Delta B\left(T\right)=\Delta B^{*}+A{\left[\frac{T_N^{\mathrm{ESR}}}{T-T_N^{\mathrm{ESR}}}\right]}^{\beta }$$

(6)

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 K₂Ni₂TeO₆ and Li₂Ni₂TeO₆. Spin-dynamic parameters of the studied compounds are summarized in

Table 4. The spin-dynamic parameters in A₂Ni₂TeO₆ (A = K and Li) tellurates.

	Effective g-Factor	ΔB*, mT	D, mT/K	β
Li2Ni2TeO6	2.20 ± 0.03	217 ± 5	-	0.60 ± 0.05
K2Ni2TeO6	2.20 ± 0.03	290 ± 5	0.4	1 ± 0.05

.

3.1.3. Specific Heat

The specific heat, C_p(T), of A₂Ni₂TeO₆ (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 K₂Ni₂TeO₆ and Li₂Ni₂TeO₆ are shown in Figure 4a,b. A λ-type peak is observed at T_N for both K₂Ni₂TeO₆ and A₂Ni₂TeO₆, 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 non-magnetic system Na₂Zn₂TeO₆. 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 Li- and 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.

3.2. Spin Exchanges Leading to a Zigzag Magnetic Order

3.2.1. 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 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 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 U_eff values of 3 and 4 eV, which lead to similar trends (see below).

3.2.2. 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.

The spin exchanges of A₂Ni₂TeO₆ (A = K, Li) and Li₃Ni₂SbO₆ obtained from the energy-mapping analysis are summarized in

Table 5. Experimental Ni…Ni distances (in Å) and calculated spin exchanges (in K), obtained from DFT + U computations, of Li₂Ni₂TeO₆ and Li₃Ni₂SbO₆.

	Li2Ni2TeO6			Li3Ni2SbO6
	Ni…Ni	Ueff = 3 eV	Ueff = 4 eV	Ni…Ni	Ueff = 3 eV	Ueff = 4 eV
J1	2.961	39.57	31.41	2.983	19.6	16.1
J1′	2.999	31.39	25.05	2.995	41.7	33.0
J2	5.178	−1.93	−1.43	5.179	−0.14	−0.08
J2′	5.160	0.40	0.41	5.183	−2.5	−1.9
J3	6.019	−33.47	−26.59	5.985	−22.8	−17.7
J3′	5.949	−40.53	−31.45	5.980	−29.0	−22.8

and

Table 6. Experimental Ni…Ni distances (in Å) and calculated spin exchanges (in K), obtained from DFT + U computations, of K₂Ni₂TeO₆.

	Ni…Ni	Ueff = 3 eV	Ueff = 4 eV
J1	3.035	26.8	21.2
J2	5.256	−0.8	−0.6
J3	6.069	−41.2	−32.2

, 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.

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 Ni²⁺ ion in a NiO₆ octahedron are the e_g states, namely, the x² − y² and 3z² − r² states, which are combined out-of-phase with the 2p orbitals of the surrounding oxygen ligands. Of the two, the x² − y² 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 x² − y² 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 x² − y² 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 x² − y² states cannot be coplanar (see Supplementary Figure S6). Thus, neither the overlap integral nor the overlap density between the two x² − y² 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. 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		K2Ni2TeO6
Ueff	3 eV	4 eV	3 eV	4 eV	3 eV	4 eV
FM	0	0	0	0	0	0
AFM	6.8	5.3	3.4	2.7	0.6	0.5

, 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, T_N, which involves the ordering between the zigzag-ordered honeycomb layers, is greater for Li₂Ni₂TeO₆ than for K₂Ni₂TeO₆ (~24.4 vs. ~22.8 K).

4. 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 K₂Ni₂TeO₆ and Li₂Ni₂TeO₆, 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 K₂Ni₂TeO₆ and Li₂Ni₂TeO₆, 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 Ni²⁺ (S = 1) ions. If we were to describe the high-temperature magnetic portion of the susceptibilities of A₂Ni₂TeO₆ (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 Ni²⁺ 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 A₂Ni₂TeO₆ (A = K, Li) and Li₃Ni₂SbO₆ 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 Ni²⁺ ions are governed largely by their x² − y² 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.