Magnetic structure, excitations and short-range order in honeycomb Na₂Ni₂TeO₆

Na₂Ni₂TeO₆ powder samples for the present study were synthesized using a standard solid state reaction [13, 16]. NaCO₃, NiO, TeO₂ (4N or higher purity) purchased from Sigma Aldrich ⁷ were used for the synthesis. The chemical and crystalline quality of the synthesized samples of Na₂Ni₂TeO₆ were estimated by taking powder x-ray diffraction (PXRD) using a PAN'lytical Empyrean diffractometer with Cu K_α radiation (λ = 1.5406 Å). Synchrotron x-ray diffraction experiments were performed at the beamline 11-BM, APS. Rietveld refinements [20] using Fullprof suite of programs [21] were performed for structural and phase analysis. Inductively coupled plasma—mass spectroscopy was performed to obtain average chemical composition of the synthesized samples. Solid samples were dissolved in a 1:1:0.3 mixture of H₂O:HNO₃:HCl with heating. The resulting solutions were diluted and analyzed on a PerkinElmer NexION 300X ICP-MS. Calibration and internal standards were prepared from high purity standards single- and multi-element stock solutions. Internal standards Be, Sc, In, Tl were used for normalization. DC magnetic susceptibility of the samples were recorded using quantum design physical property measurement system (PPMS) equipped with a vibrating sample magnetometer option. AC and DC magnetic susceptibility of the samples were additionally recorded using quantum MPMS magnetometer. The hydrostatic pressure was applied using the beryllium-copper pressure cell with Daphne 7373 oil as a pressure transmitting medium. The pressure value was determined from the measurement of the superconducting transition temperature of a Pb sensor placed inside the cell. Specific heat was measured using the relaxation time technique in the PPMS from quantum design. The specific heat of Na₂Zn₂TeO₆ was measured to serve as a phonon analogue for Na₂Ni₂TeO₆. Neutron powder diffraction experiments on Na₂Ni₂TeO₆ were performed using the BT-1 diffractometer at the NIST Center for Neutron Research, Gaithersburg using a wavelength λ = 2.0772 Å using 60' collimation and the Ge(311) monochromator. Powder sample of Na₂Ni₂TeO₆ weighing 3.034 g was filled in a vanadium sample canister containing helium exchange gas loaded at room temperature. Neutron powder diffraction patterns were collected at 295 K, 25 K and at 5 K. Diffraction data was analyzed using Fullprof suite of programs. Inelastic neutron scattering experiments were conducted on the triple-axis spectrometer (TRIAX) at University of Missouri Research Reactor. Elastic scattering experiments were performed at λ = 2.36 Å using the TRIAX spectrometer. For the inelastic scattering experiments, the spectrometer was operated in the fixed final energy mode set to E_f = 14.7 meV. Collimation settings of 60'–60'–40'–40' were used to limit the divergence of the beam and higher order contamination was filtered out using a pyrolytic graphite filter in the scattered beam. 6 g of Na₂Ni₂TeO₆ powder was filled in an aluminum can in helium exchange gas environment for the experiment. Elastic and inelastic scans were performed at 5 K, 20 K, 35 K 150 K and 290 K. Data reduction steps were performed through the software DAVE [22]. The spin wave calculations were performed using the SpinW code [23] and the representation analysis of magnetic structures were performed using the software SARAh [24].

The electron calculations are carried out within DFT using the pseudo-atomic orbital based [25] OpenMX code [26]. We employ the Perdew–Burke–Ernzerhof generalized gradient approximation [27] for exchange–correlation functional and the Dudarev's rotationally invariant DFT + U formalism [28, 29] to account for the localized 3d states in Ni. In this approach, the effective on-site Coulomb U and exchange parameters J do not enter separately but only the difference U_eff ≡ U − J is meaningful. The core electrons are replaced with norm-conserving pseudopotential [30, 31] with energy cutoff 300 Ry. The 9 × 9 × 5 k-point mesh was used for BZ integration. The exchange coupling constants J_i were calculated using Green's function method [32] implemented in OpenMX 3.9 [33]. This approach allows the direct calculation of exchange coupling parameter between any pair of magnetic sites for any inter-pair distance.

Powder samples of Na₂Ni₂TeO₆ and Na₂Zn₂TeO₆ used in the present study were prepared using standard solid state chemistry methods and the phase/chemical purity were confirmed using PXRD at a laboratory source. Our samples presented similar crystal structure parameters as reported by previous research groups [11, 16], viz, P6₃/mcm space group for Na₂Ni₂TeO₆ and P6₂2 for Na₂Zn₂TeO₆. During the synthesis of the Na₂Ni₂TeO₆ we noticed that the color of powders varied from pale to dark green in different synthesis batches even though the heating protocols were identical. The reported values of T_N for this compound seem to be strongly material-dependent, presenting a variation of about 10 K; from 25 K to 34 K [18, 34]. A magnetic phase transition is observed in Na₂Ni₂TeO₆ at T_N ≈ 27 K, where a broad peak is present in the magnetic susceptibility, χ(T) [13, 16, 18]. Broad features seen in magnetic susceptibility are often related to quasi-2D magnetic correlations especially when the crystal structure of the compound supports a layered arrangement of magnetic atoms. In Na₂Ni₂TeO₆, initial reports of the magnetic structure supports a low dimensional zig-zag structure [16, 18, 19]. Generally, this allows the application of high temperature series expansion models to the magnetic susceptibility. In the case of Na₂Ni₂TeO₆, such approaches have lead to the estimation of a mean-field exchange parameter, J/k ≈ −10.8 K [18]. Previous theoretical estimates of exchange parameters for Na₂Ni₂TeO₆ suggest FM first-neighbour and AFM next-nearest neighbour exchange which are in-plane, and additionally, a weak interplane AFM coupling [16].

The macroscopic magnetic properties of the present sample of Na₂Ni₂TeO₆ conforms with the existing reports showing a phase transition at T_N = 27 K [16, 18]. Figure 2(a) shows the dc magnetic susceptibility χ_DC(T) of Na₂Ni₂TeO₆ measured at 500 Oe and under the application of ambient pressure and 1.2 GPa external pressure, describing a broad peak centered at around 30 K. χ_DC(T) measured at 50 kOe under similar conditions did not produce any marked changes. We also have measured the magnetic susceptibility of Na₂Ni₂TeO₆ under the application of up to 80 kOe (top left inset of (a)), however the broad peak at T_N is not largely altered. The compound also displays such robustness in external pressure up to 1.2 GPa as can be seen from (a). There is no frequency-dependence of magnetic susceptibility as can be seen from the AC response, χ'(T) in the bottom inset of (a), measured at five different frequencies. These measurements on our sample of Na₂Ni₂TeO₆ confirms the T_N and highlights the field- and pressure-robustness without signatures of glassy magnetism. Curie–Weiss fit of the inverse magnetic susceptibility was administered in the temperature range 200 K–300 K. Effective paramagnetic moment, μ_eff = 2.2(2) μ_B/Ni²⁺ and Curie–Weiss temperature, θ_cw = −10 K are obtained from the fit. The effective moment is comparable to the value reported recently, but the Curie–Weiss temperature shows a slight reduction [18]. The magnetization isotherms at 2 K, 10 K, 25 K and 60 K are plotted in the top right inset of (a). No sign of strong ferromagnetism is seen.

Zoom In Zoom Out Reset image size

In order to characterize the magnetic phase transition reported for Na₂Ni₂TeO₆, we measured the specific heat, C_p(T), figure 2(b). Plotted in the left inset of (b) is C_p/T versus T, highlighting the sharp peak at T_N ≈ 27 K confirming the transition temperature. The other prominent feature in the C_p/T plot is the very broad feature at about 100 K. Oxides with some sort of short-range order have been reported to show similar features in specific heat [35–37]. In the case of CuMn spin glasses or Fe₂TiO₅ [35, 36], the observation of the broad feature in specific heat at high temperature is correlated to the short-range order inherent to the spin glass state. In contrast, in systems like Ca₃Co₂O₆, spin-state crossover leading to a Schottky-like contribution to the total specific heat is cited as the reason [37]. In the case of Na₂Ni₂TeO₆, this is attributed to the short-range ordered magnetic domains due to different ordering wave vectors. The right inset is the magnetic entropy, S_m, obtained by integrating C_diff/T. The maximum entropy released close to the T_N is about 10 J mol⁻¹ K⁻¹ which is diminished compared to the 2R ln(2S + 1) = 18.3 J mol⁻¹ K⁻¹. In general, the observed values are comparable to those found in earlier reports [13, 18]. The magnetic entropy of Na₂Ni₂TeO₆ support a scenario of low-dimensionality and short-range correlations [38].

The effective paramagnetic moment, μ_eff = 2.23 μ_B/Ni atom and Curie–Weiss temperature, θ_p = −9.7(2) K are estimated from Curie–Weiss analysis of χ_DC(T), where the negative sign for θ_p suggesting predominant AFM spin correlations. Na₂Ni₂TeO₆ single crystals are reported to have μ_eff = 3.446 μ_B/Ni atom, attributing the increased value compared to spin-only moment of Ni, to possible spin–orbit effects [13]. Our powder samples yield a value closer to the magnetic moment of S = 1 Ni²⁺ (2.82 μ_B). An independent verification of the T_N of the present Na₂Ni₂TeO₆ samples was attained through the temperature dependence of specific heat, C_p(T) (not shown), which confirmed the T_N from magnetic susceptibility.

The magnetic properties of Na₂Ni₂TeO₆, as pointed out in reference [16], are insensitive to the occupancy of sodium atoms for a given concentration. Therefore, in our computations following the convention in reference [16], we focus on the case where only Na2 sites (see table 2) are 100% occupied. Our theoretical results for exchange coupling constants as functions of on-site Coulomb potential U_eff and exchange parameter J_i are summarized in table 1. The data show that, across the whole range of U_eff, Ni favors a weak FM coupling with its first-nearest neighbours (nn) (i.e. a small positive J₁), and a strong (weak) AFM coupling with the third-nearest neighbour 3NN (2NN). This suggests that the qualitative behavior and the ground state magnetic ordering are not sensitive to the empirical U_eff value. The dominant term J₃, which is missing in the previous work, is particularly crucial in forming stable zigzag AF ordering as it only connects between two adjacent zigzag lines. The J₁–J₂–J₃ configuration has been used previously to stabilize a zigzag AF ordered phase in Na₂Co₂TeO₆ [10]. Figures 3(a) and (b) are respectively the plots of J₃, the dominant component, and T_N, estimated from table 1 based on mean-field theory, as functions of U_eff. Although both J₃ and T_N vary with U–J value, T_N is generally in a good agreement with the experimental data [16, 18] within a reasonable range of U_eff. The exchange constants for U_eff = 4 eV; J₁ (0.025 meV ≡ 0.29 K), J₂ (−0.136 meV ≡ 1.57 K), J₃ (−2.250 ≡ 26 K) are in a broad range inclusive of the T_N and the θ_cw. Figure 3(c) shows the total energy as a function of the angle (θ) between the spin quantization axis and z-axis in the presence of SOC for U_eff = 6. The simple sin²(θ) dependence that we find is the signature of uniaxial anisotropy and the trend suggests an in-plane easy-axis.

Zoom In Zoom Out Reset image size

The neutron diffraction pattern of Na₂Ni₂TeO₆ collected at 295 K was refined (see [39]) to obtain the lattice parameters and the atomic coordinates given in table 2. The room temperature crystal structure of Na₂ M₂TeO₆ (M = Ni) is reported in hexagonal P6₃/mcm (No. 193) space group with lattice parameters a = 5.2042(5) Å and c = 11.1383(5) Å, which is different from the M = Co, Zn, Mg compounds adopting P6₃22 (No. 182) [12]. Based on Rietveld analysis of the neutron diffraction data on our samples at T = 295 K, the hexagonal P6₃/mcm structure is confirmed. Comparing the 5 K diffractogram, presented in figures 4(a) and (b), with that at 295 K, major magnetic reflections are identified at ≈13° and 36°. DC and AC magnetic susceptibility of our Na₂Ni₂TeO₆ sample showed a broad anomaly centered at approximately 30 K, while specific heat data showed a sharp peak at close to 27 K thereby suggesting that the T_N is ≈27 K. In order to determine the magnetic structure of Na₂Ni₂TeO₆ below the phase transition temperature, we performed diffraction experiments below T_N.

Zoom In Zoom Out Reset image size

The diffraction pattern of Na₂Ni₂TeO₆ at T = 5 K (circles) shown in figures 4(a) and (b) are analyzed using different models in this paper since the previously reported models in literature did not provide a complete description of the observed pattern. Two propagation vectors, commensurate ${ \overrightarrow {k}}_{\mathrm{c}}\left(0.5\enspace 0\enspace 0.5\right)$ and incommensurate ${ \overrightarrow {k}}_{\text{ic}}\left(0.47\enspace 0.44\enspace 0.28\right)$, with the same T_N are suggested for Na₂Ni₂TeO₆ [16, 18, 19]. In our analysis, the commensurate wave vector ${ \overrightarrow {k}}_{\mathrm{c}}\left(0.5\enspace 0\enspace 0\right)$ was suggested by the k-search utility within Fullprof as well as by the Monte Carlo search within SARAh. Using the above-mentioned commensurate wave-vector, the magnetic structure of Na₂Ni₂TeO₆ is refined using representation analysis methods, which suggests eight irreducible representations, Γ_mag = Γ₁ + Γ₂ + Γ₃ + Γ₄ + Γ₅ + Γ₆ + Γ₇ + Γ₈ [18]. The irreducible representation, Γ₅ offered the best fit to the experimental data at 5 K. The red solid line in figure 4(a) is the Rietveld fit using a model (model 1, table 3) that combines P6₃/mcm nuclear structure; and magnetic structure representations Γ₅ and Γ₁ described by the respective propagation vectors, ${ \overrightarrow {k}}_{\mathrm{c}}\left(0.5\enspace 0\enspace 0\right)$ and ${ \overrightarrow {k}}_{\text{ic}}\left(0.47\enspace 0.44\enspace 0.28\right)$. The inset of (a) provides an enlarged view of the low-angle region to show the faithfulness of the fits. It can be seen that the diffraction pattern has broad peak shaped around 14° and 36° indicating the short-range ordered magnetic contribution, which is not completely captured by the model-based fit.

Table 3. A description of the different models used for the analysis of the neutron diffraction pattern at T = 5 K. The parameters signifying the quality of fit, R_p, R_wp, R_B, R_mag and χ² are tabulated to provide a quantitative comparison between the models used in the study.

Model	Phases	Rp	Rwp	RB	Rmag	χ2
Model 1	P63/mcm + ${ \overrightarrow {k}}_{\mathrm{c}}\left(0.5\enspace 0\enspace 0\right)$ + ${ \overrightarrow {k}}_{\text{ic}}\left(0.47\enspace 0.44\enspace 0.25\right)$	6.3	8.4	5.6	35.6, 23.9	7.3
Model 2	P63/mcm + ${ \overrightarrow {k}}_{\mathrm{c}}\left(0.5\enspace 0\enspace 0\right)$	6.5	8.9	5.5	80.5	8.4
Model 3	P63/mcm + ${ \overrightarrow {k}}_{\mathrm{c}}\left(0.5\enspace 0\enspace 0\right)$ + ${ \overrightarrow {k}}_{\mathrm{c}}\left(0.5\enspace 0\enspace 0.5\right)$	6.3	8.5	5.2	66.2, 54.3	7.7
Model 4	P63/mcm + ${ \overrightarrow {k}}_{\mathrm{c}}\left(0.5\enspace 0\enspace 0\right)$ + ${ \overrightarrow {k}}_{\text{ic}}\left(0.5\enspace 0\enspace 0.5\right)$ + s.r.o.	6.0	8.2	4.9	42.2, 39.2, 27.4	6.8

In order to obtain the best magnetic structure description for Na₂Ni₂TeO₆, we performed different model-based Rietveld analysis of the 5 K data. The results are presented in figures 4(c)–(e). In (c), only one magnetic phase corresponding to ${ \overrightarrow {k}}_{\mathrm{c}}\left(0.5\enspace 0\enspace 0\right)$ was incorporated in addition to the nuclear structure (model 2, table 3). Evidently, the profile fit is not faithful as indicated by a high value of χ² = 8.4. Adding a second magnetic phase corresponding to ${ \overrightarrow {k}}_{\mathrm{c}}\left(0.5\enspace 0\enspace 0.5\right)$ improves the fit with a reduced χ² = 7.7, however, the broad features corresponding to the short-range ordered phase are still under-represented, see figure 4(d). This is the model 3 described in table 3. A reasonable representation of the experimental data is obtained by including an additional magnetic phase with Lorentzian peak shape to the model of (d) in order to account for the short-range order. This is represented in figure 4(e) with the corresponding χ² = 6.8 (model 4, table 3). Though model 4 allows a better fit of the observed peaks at 5 K, it is noted that this model has the shortcoming of the possibility to over-parameterize the fit. However, in section 3.4 we show experimental support for the presence of short-range correlations arising from magnetic clusters, which argues against an over-parameterization of the magnetic structure model. A plot of the Rietveld refinement as per this model in the full-range of 2θ is provided in figure 4(b).

The neutron diffraction pattern at T = 5 K from TRIAX using λ = 2.36 Å is presented in figure 4(f) along with the Rietveld fit using the same multi-phase model described above. Additional magnetic Bragg peaks at high-d region were not observed by using longer wavelength neutrons λ = 2.36 Å and λ = 4.36 Å. A schematic representation of the long-range ordered magnetic structure of Ni²⁺ in Na₂Ni₂TeO₆ as per the wave vector ${ \overrightarrow {k}}_{\mathrm{c}}\left(0.5\enspace 0\enspace 0.5\right)$ is shown in figure 4(g). If we consider a single honeycomb layer formed by the Ni atoms, then the estimated magnetic structure implies that the three nearest magnetic moments are aligned parallel to each other while the other three are aligned opposite resulting in an almost quasi two-dimensional zig-zag arrangement. The magnetic moment of Ni²⁺ obtained by refining the neutron diffraction pattern at T = 5 K is 1.32(4) μ_B/fu. The value reported by Kurbakov et al is 1.72(1) μ_B/Ni, at 1.7 K [18]. 3d⁸ Ni²⁺ in an octahedral crystal field environment will give rise to a spin-only magnetic moment corresponding to 2.7 μ_B/Ni. The short-range ordered magnetic regions in Na₂Ni₂TeO₆ resulting from the incommensurate ordered phase contributes to the reduced value of ordered magnetic moment; along with structural inhomogeneities arising from stacking faults and non-stoichiometric Na regions. Honeycomb magnets similar to Na₂Ni₂TeO₆ have been observed to possess similar commensurate propagation vectors to describe their long-range ordered magnetic structure. For instance, Na₂Co₂TeO₆, although having a different space group, reports a magnetic structure with a commensurate propagation vector of $\left(0.5\enspace 0\enspace 0\right)$ [10].

The magnetic structure of Na₂Ni₂TeO₆ is described using a multi-phase model consisting of a commensurate and an incommensurate phase by one of the initial reports on the magnetic structure of Na₂Ni₂TeO₆ [16], and subsequent reported have supported that picture [18, 19]. According to our analysis, several similar ${ \overrightarrow {k}}_{\mathrm{c}}$ and their combinations reproduce the commensurate phase described of Na₂Ni₂TeO₆. For example, a model consisting of two commensurate phases $\left(0.5\enspace 0\enspace 0.5\right)$ and $\left(0.5\enspace 0\enspace 0\right)$ can describe the observed diffraction pattern at T = 5 K, see the models described in table 3. We note that a model without including the incommensurate wave vector, ${ \overrightarrow {k}}_{\text{ic}}\left(0.47\enspace 0.44\enspace 0.25\right)$, offers the best fit as per our detailed analysis. At the same time, our model includes the short-range ordered magnetism which is unequivocally supported by different studies including polarized neutron scattering [16, 18, 19]. The diffuse nature of the short-range magnetic correlations is brought out through the polarized neutron scattering study [19]. For completeness, the refined patterns and the structural parameters obtained using T = 25 K data are given in figure S2 [40] and table S1 [41] respectively.

Figure 5(a) shows the temperature evolution of intensities of the Bragg peaks corresponding to Q = (0.5 0 0.5), (0.5 0 0), (0.5 0 1). The solid lines in the figure corresponds to the curve-fits using power law expression, $I\propto {(1-T/{T}_{\text{N}})}^{\beta }$. The fit leads to T_N = 27.4(3) K, 28.0(2) K and 28.1(4) K and corresponding β values, 0.36(3), 0.37(3) and 0.46(1). These values are comparable to those reported [16]. Figure 5(b) shows the intensity scan at 30 K and 5 K in the 2θ region where the commensurate and incommensurate peaks are reported to be observed [16]. At 30 K, which is above the T_N, we do not observe intensities to the Bragg peaks corresponding to Q = (0.5 0 0.5), (0.5 0 0), (0.5 0 1). At 5 K, a rather broad peak is observed very similar to the reported case earlier [16]. Our results thus align with the previous cases where the commensurate and the incommensurate peaks are inferred to have the same T_N. The indication of the short-range order we obtained from the specific heat data (inset, figure 2(b)) is not clearly absorbed in the data presented in figure 5. We observe that the broadened peak at (0.5 0 0.5) vanishes at the same temperature as the peak corresponding to the long-range ordered peak. However, short segments of 1D zig-zag magnetic chains would result in a scattering signal that would be too diffuse to detect in our measurements, yet still contribute to the specific heat feature.

Zoom In Zoom Out Reset image size

With this background of the magnetic structure, we now turn to the magnetic excitations in Na₂Ni₂TeO₆. The magnetic excitations from the zig-zag ground state of Na₂Ni₂TeO₆ are mapped out using inelastic neutron scattering on powder samples at TRIAX spectrometer. The plot of energy, E (meV), versus momentum transfer, Q (Å⁻¹), at T = 5 K is shown in figure 6(a). The experimental data at 5 K is shown here after subtracting the paramagnetic response at 150 K (see, figure S4 [42]) and after applying a correction of Bose factor. The scattered intensity is shown color coded. Below the T_N, a magnetic excitation is observed near 5 meV in the T = 5 K data. The theory of FM or AFMc spin wave excitations in spin systems have been developed for the general case, as well as for specific spin arrangements [43–46]. The calculated spin wave spectra of Na₂Ni₂TeO₆ under the approximation of linear spin wave theory [23] is presented in figure 6(b). The J values obtained through the DFT computations (as per table 1) were used to simulate spin wave spectra using the following Hamiltonian,

$$\begin{equation*}\mathcal{H}={J}_{1}\sum\limits _{\langle i,j\rangle }{\enspace S}_{i}\cdot {S}_{j}+{J}_{2}\sum\limits _{\langle i,j\rangle }{\enspace S}_{i}\cdot {S}_{j}+{J}_{3}\sum\limits _{\langle i,j\rangle }{\enspace S}_{i}\cdot {S}_{j}.\end{equation*}$$

Anisotropy or Zeeman terms which can often be accommodated in the spin wave Hamiltonian were not used in the present case. The 'flat-like' excitation band at ≈5 meV is reproduced. Low dimensional magnetic lattices are being studied recently owing to the possibility of realizing flatbands in the magnon spectrum which can then lead to dissipation-less spin transport and associated magnon Hall effect, for example seen in kagome magnets [47]. In other words, one could expect to find a magnon insulator, similar to a topological insulator. Re-investigating simpler magnetic systems in the very recent past has shown motivating results [48].

Zoom In Zoom Out Reset image size

Constant-Q scans were recorded at different temperatures; as was the temperature dependence of intensity for Q = 0.75 Å⁻¹. The inelastic scans are plotted in figures 7(a)–(d). The E-scan at T = 35 K for the Q-values of 0.75 Å⁻¹, 0.89 Å⁻¹ and 1 Å⁻¹, are presented in figure 7(a). The energy scans at T = 35 K show an almost monotonic decrease in intensity as a function of energy, however, are suggestive of non-zero spin fluctuations at T > T_N. A well-defined asymmetric broad peak at ≈4.5 meV is present in the inelastic scattered intensity at 5 K and 20 K as shown in (b) and (c) respectively. The peak shapes are fit to standard profile functions incorporating a Bose term to account for the temperature. An additional Lorentzian term is used in some cases to account for additional features, for example, as seen in the case of Q = 2.09 Å⁻¹ data in (b). At T = 5 K the excitation peak that is present at E = 4.52(6) meV has a full-width-at-half-maximum, FWHM = 1.87(7) meV centered at Q = 0.89 Å⁻¹. With increasing temperature, the FWHM at 20 K broadens to 3.05(7) meV for Q = 0.89 Å⁻¹.

Zoom In Zoom Out Reset image size

The temperature-dependence of intensity for three different Q's, Q = 0.7 Å⁻¹, 0.75 Å⁻¹ and 0.89 Å⁻¹, was observed to follow a power-law behavior, (1 − T/T_N)^2β . For the set of three q-values, we obtained values for the exponent β as 0.27 ± 0.04, 0.21 ± 0.02 and 0.24 ± 0.06 respectively, with the corresponding T_N determined at 26.4 ± 0.4 K, 27 K and 26 K (not shown). The constant-Q (2.09 Å⁻¹) scan at T = 5 K shows two major peaks, centered at 4.8(3) meV and 8.6(5) meV, as can be seen in panel (b) of figure 7. The intensity observed at three different temperatures, 5 K, 20 K and 35 K at Q = 0.75 Å⁻¹ are plotted together in figure 4(c). The Q-scans at T = 5 K for three different energy values at (5 meV), below (2 meV) and above (8 meV) the magnon excitation as seen in figure 6(b) are shown in (d).

In the following paragraph we discuss the main results in the context of previous reports on the same compound, in order to highlight the distinctiveness of the present work. The major findings of the present study are (i) that the magnetic order is described purely by commensurate wavevectors, (ii) establishing the existence of static short-range order below T_N and—through the convergence of several experimental techniques—fluctuations associated with incipient short-range order extending far above T_N, and (iii) first report of the inelastic neutron spectrum that reveals a spin-wave excitation near 5 meV that is well described by our DFT analysis, which points to a strong J₃ term in the J₁–J₂–J₃ model. In one of the first detailed scattering studies on this compound, Karna et al [16] reported the magnetic structure of Na₂Ni₂TeO₆ by using a model involving a commensurate (0.5 0 0), and an incommensurate propagation vector (0.47 0.44 0.28). Our analysis shows that a reliable Rietveld fit to the experimental neutron diffraction pattern can be obtained by assuming only commensurate propagation vectors and an additional term to account for the broad features arising from short-range ordering of magnetic clusters. Nuclear magnetic resonance study on Na₂Ni₂TeO₆ pointed to the fact that the broad maximum in magnetic susceptibility (near T_N) is due to a low-dimensional short-range correlation developing in the honeycomb lattice [49]. An extreme condition of no long-range magnetic order is realized in the honeycomb antiferromagnet, Bi₃Mn₄O₁₂(NO₃), experimentally verified using magnetic susceptibility and muon spin resonance spectroscopy [50]. The case of Na₂Ni₂TeO₆ may be visualized as an intermediate case where a commensurate magnetic order coexist with short-range magnetism from clusters. Our specific heat data and inverse magnetic susceptibility supports this picture. Korshunov et al, have given experimental validation of diffuse magnetic scattering present in Na₂Ni₂TeO₆ through polarization analysis [19], however the results they presented are at 30 K which is only marginally above the transition temperature, T_N = 27 K. Diffraction studies by Kurbakov et al [18] claim a single ordering wavevector of (0.5 0 0); however, we have clearly demonstrated that this does not provide a sufficient description of our new diffraction data obtained both in traditional energy integrated mode (BT-1) as well as energy resolved mode (TRIAX). This points to unsettled composition variation in the study of this material, but we note here that our study is the only one to date that directly address the issue of composition variation via ICP-MS and magnetization characterizations as outlined in section 3.5. Further, we incorporate the short-range order directly and concurrently in our Rietveld refinement, rather than to treat them separately as was done in the Kurbakov study. Our detailed analysis brings out the short-range order combined with two commensurate propagation vectors and establishes the connection with features observed in specific heat. In addition, we report the spin wave excitation at 5 meV in this compound, which is the first to the best of our knowledge. We note that a traditional two-axis diffraction configuration was used in the Korshunov study that does not distinguish static magnetic order from low lying short range correlations as we observe in the current study in the range above T_N. In figure 8 the Q–T map at constant E = 1.5 meV is shown in the left panel, which is obtained by subtracting the T = 290 K data from a series of temperature dependent Q-scans. The effect of superparamagnetic ordering from incipient short-range order is observed at Q ≈ (0.5 0 0.5). On the right panel, the integrated intensity as a function of temperature is presented. The vertical solid line in the figure marks the location of the T_N. Appreciable intensity is observed at temperatures above the T_N which can clearly be attributed to superparamagnetic scattering from incipient AFM clusters. The experimental proof of short-range correlations from magnetic clusters support the magnetic structure model used in the present study, involving only commensurate ordering vectors and a short-range component. An experimental signature of low-dimensionality and short-range spin fluctuations is seen in our specific heat data, see section 3.1.

Zoom In Zoom Out Reset image size

Na ions in Na₂Ni₂TeO₆ is reported to adopt a particular chiral ordering [16]. This special ordering of Na between the honeycomb layers of Ni could influence the physical properties of Na₂Ni₂TeO₆. In order to understand if the Na content has any role in the magnetic properties in general, we prepared three different batches of Na₂Ni₂TeO₆ powders: S₁ (the batch ofsamples used for most of the measurements reported here) and S₂ with the nominal stoichiometry Na₂Ni₂TeO₆ (both prepared under identical conditions) and S₄ with a 2% reduction in Na content. We then have carried out compositional, structural and magnetic characterization of these three samples using ICP-MS, synchrotron x-ray diffraction and magnetometry. The ICP-MS results are shown in table 4 indicating no significant departures in the amount of Na in the synthesized samples from the nominal values.

The samples S₁, S₂ and S₄ were studied using high-resolution synchrotron x-ray diffraction at 11-BM. The diffraction data obtained at 295 K and the analysis results are presented in the supplemental information [51, 52]. The chemical formulae determined from refining the occupancies of metal atoms showed that the nominally stoichiometric samples did not present a large deviation from the starting stoichiometry. However the S₄ sample showed a reduced Na content which was as expected. We note that a slight discrepancy in stoichiometry of S₄ between the ICP-MS and the diffraction results may be sample dependent.

Shown in figure 9 are the magnetic susceptibility curves of S₁, S₂ and S₄ along with a Curie–Weiss fit of S₄ (right inset) and the magnetization isotherms (left inset). It is seen that slight variation in Na-content does not influence the transition temperature of Na₂Ni₂TeO₆, which are determined to lie with in 27 ± 1 K for S₁, S₂ and S₄. Curie–Weiss analysis of S₁, S₂ and S₄ yielded a value of 2.3 μ_B/Ni which is comparable to the value obtained for the Na₂Ni₂TeO₆ sample that was used for neutron diffraction study shown in this work. Increasing the Na-content beyond ±5% leads to the appearance of impurity phases like Ni₃TeO₆.

Zoom In Zoom Out Reset image size