Magnetic phase diagram and possible Kitaev-like behavior of honeycomb-lattice antimonate Na₃Co₂SbO₆ 

The specific heat data at various external magnetic fields are presented in Fig. 2. As it was found earlier [22] the temperature dependence $C_{p}(T)$ in zero magnetic field demonstrates a distinct $\lambda$-type anomaly, associated with an onset of the magnetic order at $T_{N}=6.7$ K. Under magnetic field ($B\leq$ 1 T) the singularity at $T_{N}$ somewhat broadens and shifts to lower temperatures (inset in Fig. 2a). Higher fields lead to progressive weakening and eventual fading of the anomaly, which finally transforms to a broad hump; its position shifts to higher temperatures with the field (Fig.  2b).

In order to analyze the nature of the magnetic phase transition in Na₃Co₂SbO₆ at various external fields and evaluate corresponding contribution to the specific heat we used data for its non-magnetic isostructural analogue Li₃Zn₂SbO₆. The correction to this contribution for Na₃Co₂SbO₆  has been made taking into account the difference between the molar masses for each type of atom in the compound (Na-Li and Co-Zn).  [25] The magnetic contribution to the specific heat was determined by subtracting the lattice contribution using the data for Li₃Zn₂SbO₆ (Fig. 2b). Analysis of the low-temperature part of $C_{m}(T)$ was performed in the terms of the spin-wave (SW) theory [26]

where $C{\rm{}_{SW}}$ is low-temperature specific heat due to spin-wave excitations, $d$ stands for the dimensionality of the magnetic lattice and $n$ is defined as the exponent in the dispersion relation $\omega(q)\sim q^{n}$. The analysis has shown that below $T_{N}$ the C_m(T) dependence follows satisfactorily $T^{3}$ – law at low fields $B<$ 1 T (see inset in Fig. 3b). This result implies a presence of antiferromagnetic magnons consistent with 3D antiferromagnetic ordering at the low temperatures in this field range. The same procedure for fields 1 T $<B\leq$ 2 T reveals $d$ = 2, indicating the predominance of 2D AFM exchanges in the studied compound. It is noteworthy that $B\approx\$1 T corresponds to the metamagnetic transition observed in  [22; 16]. In the vicinity of quantum critical point, temperature dependence of magnetic specific heat obeys power law $C_{m}$ $\sim$ $T^{2}$ [27]. It is worth mentioning that the low-temperature magnetic specific heat in Na₃Co₂SbO₆ for fields $B=2$ T and higher can be described by the same model with the spin gap as was used for $\alpha$-RuCl₃ [27]

In order to estimate the energy gap, $C_{m}(T)$ was fitted by  (2), the results are shown by dashed curves on Fig. 2b.

In accordance with previously reported data [22], the static susceptibility $\chi=M/B$ of Na₃Co₂SbO₆ (Fig. 3) demonstrates an antiferromagnetic behavior with low-field Weiss temperature $\Theta\approx-10$ K. For low external fields $B<$ 1.5 T, magnetic susceptibility passes through a maximum with decrease in temperature and then drops. From the maximum $\chi(T)$, one can find the Néel temperature $T_{N}$ = 7.6 K at 0.05 T and it decreases with increase in magnetic field. The fit of field dependence of T_N (Fig. 3 insert) by expression $T_{N}\sim(1-B/B_{C})^{\textit{z}\nu}$ with z$\nu$ = 0.25 $\pm$ 0.013 yields the field of the suppression of 3D AFM order $B_{C}=1.33\pm 0.002$ T. The critical exponent z$\nu$ is consistent with the magnetization and neutron scattering data for $\alpha$-RuCl₃ [28]. At the same time, the NMR data (see below) indicate some features typical for the AFM transition even above 1.3 T. Apparently, the strong anisotropy of the magnetic properties entails a difference in the field of AFM order suppression for different orientations of the powder crystallites. The M/H(T) data for some directions of single crystals [17; 15] show that the magnitude of the peak at a field parallel to the honeycomb plane is much larger than at perpendicular direction. Therefore, the Néel temperature in the powder sample is determined mainly by crystallites of this orientation. In addition, there is a strong anisotropy even in the plane  [17]. Thus, below we will assume that the AFM order exists in the fields below $B^{*}\approx$ 2 T. The $M/B(T)$ data obtained at $B\geqslant 2.7$ T indicate the saturation developed in the studied sample.

To study the local properties of the magnetic spin system, we performed NMR measurements on the powder sample of Na₃Co₂SbO₆ on the ²³Na nuclei which have a spin $I=3/2$, the gyromagnetic ratio $\gamma_{n}$ = 11.2685 MHz/T, and the quadruple moment Q = 0.104 10^-28 m². The measurements were carried out at several frequencies 10.7, 13.512, 22.52, 30.33, 45.04 and 86.714 MHz in the corresponding field ranges since the effect of an external field on local magnetic characteristics is of particular interest. Temperature transformation of NMR spectra in some of field ranges is presented in Fig. 4.

At temperatures of the order of 100 K and above, a classical powder spectrum with quadrupole shoulders can be recorded. As the temperature decreases, the line broadens and shifts towards lower fields. The low temperature spectrum at $B<B^{*}$ has a flat top but does not have a stepwise rectangular shape that is typical for antiferromagnets [29]. This can be understood assuming the fact that the total width of the spectrum at low T is comparable to the quadrupole splitting and the overlap of the main line and quadrupole satellites smears the shape of the spectrum resulting in a bell-like profile. The same reason makes the modelling of the line powder profile technically impossible and one cannot resolve the positions of the spectral contributions from the parallel and perpendicular orientations of the crystallites.

The dynamics of the electron spin system determines the process of nuclear spin-lattice relaxation $T_{1}^{-1}$ (see Fig.  5, left panels). The relaxation rate at external fields $B\leq B^{*}$ exhibits a sharp peak at $T<6$ K indicating the magnetic phase transition T_N. The temperature of this peak decreases with increasing field, similarly to the Néel temperature determined from the static magnetic susceptibility data. Relaxation in the nearest upper vicinity of T_N can be described in terms of the critical exponent

where $\textit{p}=\nu(\textit{z}-\eta)$  [30]. The obtained critical parameters are p = 0.328 $\pm$ 0.013, $T_{N}=5.51\pm 0.08$ K for 1.2 T and p = 0.319 $\pm$ 0.031, $T_{N}=2.57\pm 0.07$ K for 2 T. Such a small value of p is typical for XY-triangular and honeycomb lattices [31; 32] and can be estimated as close to the calculated p = 0.32 obtained assuming $\nu\approx$ 0.5 [33] and z = 0.64 [34]. It confirms the results of paragraph  III.1: in-plane 2D spin correlations at $B>$ 1 T play a key role in the magnetism of Na₃Co₂SbO₆ including onset of a three-dimensional AFM order.

With a further increase of the field, a less pronounced but clearly observed maximum appears on the temperature dependence of the relaxation rate. Its position shifts to the higher $T$ with the field and corresponds to the plateau-like behavior of temperature dependence of static susceptibility taken in the same fields. While the electron spin fluctuations slow down with temperature, the nuclear spin-lattice relaxation depends on correlation time as  [35]

where $\tau_{c}$ is the correlation time and $\omega_{0}$ is the Larmor frequency. Below the maximum of this dependence, $\tau_{c}>1/\omega_{0}$. This allows attributing this maximum $T_{sat}$ to the establishment of a quasi-static saturated phase induced by the external field. To characterize the spin dynamics at low temperatures, we tried to estimate the magnon gap $\Delta_{m}$ using a formula for a gapped three-magnon process [36]

where $n=2$. At $B>B^{*}$, such an estimation gives a gap of the order of 4 – 16 K, which grows with the field (see Fig.  6). At the same time, fitting of the $T_{1}^{-1}$(T) curve below 2.5 T, i.e., in the AFM state, results in an extremely low value of the gap, and the data can be described by a power law $T_{1}^{-1}\sim T^{3}$ typical for three-magnon scattering in the antiferromagnets at temperatures above the energy gap in the spin wave spectrum. This is consistent with the small anisotropy gap value $<$ 2 K determined from the field of metamagnetic transition  [22; 16]. It should be noted that the number of experimental points does not allow making a reliable quantitative analysis; therefore, the presented conclusions should be only considered as an estimation.

Above $T_{N}$ or the assumed saturation temperature $T_{sat}$, the temperature dependence of the relaxation rate in the high fields contains a wide hump. It is noteworthy that this hump firstly appears at 2 T, which is somewhat lower than $B^{*}$. This feature can be interpreted as the so-called “low-dimensional maximum”, which characterizes the slowdown of 2D spin correlations in two-dimensional compounds. Given the possible Kitaev-Heisenberg physics in the compound under study, it is tempting to apply as a working hypothesis the approach developed to describe the Kitaev behavior of nuclear spin-lattice relaxation for $\alpha$-RuCl₃ [37; 38]. This theoretical model [39] correlates the boundaries of this hump with the magnitude of the Kitaev exchange and propose the extended exponential function

We evaluate the spin gap assuming $n=1$ (see Fig. 5, right panel) as $\sim 8$ K at 2 T and $\sim 45$ K at 7.7 T which is comparable to the values obtained for $\alpha$-RuCl₃. Certainly given a relatively small change of $T_{1}^{-1}$ in the considered temperature range the so obtained values of the gap should be considered as estimates. Nevertheless they appeared consistent with the gap values obtained from the specific heat (see the discussion below).

The shift of the NMR line can be defined as $K$ = (($B_{L}-B_{res})/B_{L})\times$100%, where $B_{res}$ is the field of the spectral magnitude maximum and $B_{L}$ = $\omega_{Larmor}/\gamma_{n}$. Its temperature-dependent part corresponds to the local static susceptibility and in paramagnetic state it is proportional to the bulk susceptibility $K$ = $A\chi_{bulk}$, where A is the hyperfine tensor. The hyperfine interaction constants $A_{hf}$ taken from the linear part of the $K$ ($\chi_{bulk})$ dependencies (see Fig. 7) in each external field are found to be almost equal and the average value $A_{hf}$ is determined as 0.25$\pm$0.014 kOe/$\mu_{B}$. This value is comparable to that observed in Na₂Co₂TeO₆ [44] and is significantly smaller than $A_{hf}$ for copper  [45] and nickel  [32] honeycomb compounds. The difference in the electronic configurations and orbital occupancy for cobalt ions seems to be responsible for the smaller orbital overlap and the transfer of spin density to sodium ions. As a result, despite the larger bulk susceptibility in cobalt compounds compared to nickel or copper ones (see also  [46] for tellurates), the increase in the local hyperfine field on sodium nuclei is insignificant. This leads to a remarkably larger decoupling of the honeycomb planes and to a more pronounced two-dimensionality of the strongest exchanges in the cobalt compound as a result.

The NMR is well known probe of local fields, susceptibility and correlations and we use it to establish the field and temperature limits of correlated regions. The line shift is proportional to the local field which is static on the NMR timescale $K\sim\langle H_{hf}\rangle_{t}\sim A_{hf}\langle S\rangle_{t}$. In most cases when the fluctuations of electron spins $S$ are much faster than inversed NMR frequency, time averaging can be done at zero frequency, i.e. $K\sim\chi_{loc}(0)$. In specific situations, for example in some low-dimensional systems, where the correlation time is comparable to the NMR time scale in the extended temperature range, the result of time averaging differs from that at zero frequency and NMR shift deviates from bulk static susceptibility.

Slight deviation of $K(\chi_{bulk})$ dependence from linear at fields $B<B^{*}$ indicates the slowdown of cobalt spin fluctuations at low temperatures. Development of antiferromagnetic correlations with big correlation time suppressing the local field at the sodium positions located between the cobalt planes. At $B>B^{*}$ the local static susceptibility does not deviate from the bulk one anymore.

The spin-lattice relaxation can be considered as a direct probe of the local dynamic susceptibility of the electron spin subsystem [47]

where $A_{\perp}({\bf q})$ is the q-dependent hyperfine constant, q is the wave vector and $\omega_{L}$ is the Larmor frequency. In the paramagnetic regime $(T_{1}T)^{-1}$ is proportional to the static bulk susceptibility. The sharp deviation of the local dynamic susceptibility from the static bulk one occurs below 10 K in the entire range of fields (see Fig. 7), marking the critical slowing down of the electron spin fluctuations in the vicinity of the transition to the static state. In fields above $B^{*}$, the local dynamic susceptibility deviates from the static one already at 20 - 50 K, marking the development of dynamic correlations at a relatively low-frequency part of the spin fluctuation spectra.

The AC susceptibility at zero external field exhibits a sharp peak at the Néel temperature. Above $B^{*}$, the critical low-temperature anomaly is absent, but a wide hump is observed, which shifts to higher temperatures with increasing field. At any fields, the $\chi’(T)$ curves obtained at different frequencies do not exhibit any frequency dependence (see Fig. 8). This rules out that the hump is associated with any kind of conventional glassy transition. The $\chi’$ hump is not accompanied by an anomaly of $\chi^{\prime\prime}$ and therefore one cannot attribute it to the crossover to the field-induced ferromagnetic order, since at ferro- or ferrimagnetic transition or crossover the imaginary component is present [48]. On the other hand, this broad maximum shifts to higher temperatures with the applied DC field and it does not allow us to attribute it to the development of two-dimensional antiferromagnetic correlations in cobalt planes. It is remarkable that a pronounced anomaly in this temperature range is absent in the temperature dependence of both the bulk and the local static susceptibility, but it is observed in the dynamic susceptibility in both the kHz and MHz (NMR relaxation) ranges indicating the dynamic nature of the emerging magnetic state. It should be noted that the temperature of the AC susceptibility hump maximum corresponds in order of magnitude to the values of the Kitaev-Heisenberg gap obtained from nuclear relaxation analysis.