Excitations in the Ordered and Paramagnetic States of Honeycomb Magnet ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$

${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ is a proposed approximate Kitaev magnet, yet its actual magnetic interactions are elusive due to a lack of knowledge on the full excitation spectrum. Here, using inelastic neutron scattering and single crystals, we determine the system’s temperature-dependent magnetic excitations over the entire Brillouin zone. Without committing to specific models, we unveil a distinct signature of the third-nearest-neighbor coupling in the spin waves, which signifies the associated distance as an emerging effective link in the ordered state. The presence of at least six nonoverlapping spin-wave branches is at odds with all models proposed to date. Above the ordering temperature, persisting dynamic correlations can be described by equal-time magnetic structure factors of a hexagonal cluster, which reveal the leading instabilities. Our result sets definitive constraints on theoretical models for ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ and provides new insight for the materialization of the Kitaev model.

Figure 1

(a) A honeycomb layer of ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$. Cyan, yellow, and red spheres represent Co, Te, and O, respectively. Solid lines indicate a 2D primitive cell. Dotted line connects a pair of third-nearest-neighbor ${\mathrm{Co}}^{2+}$ ions. The illustration is produced with vesta [54]. (b) 2D reciprocal space and hexagonal Brillouin zones. (c) Diffraction at (0, 0.5, 0) and (0.5, 0.5, 0), measured versus $T$ with $E_i=10.0\mathrm{meV}$. The two $\mathbf{Q}$ positions are indicated in (b). Inset shows a photograph of our sample. (d) Crystal-field excitations along ${\mathbf{Q}}_{2\mathrm{D}}=(-0.5K,K)$, measured at two temperatures with $E_i=52.9\mathrm{meV}$. (e) Energy distribution of intensity, after integrating the (symmetrized) data in (d) over $K\in [-1,1]$. Inset illustrates the splitting of the 12-fold degenerate atomic $L_{\mathrm{eff}}=1$, $S=3/2$ states under the influence of SOC. Arrow indicates the observed excitations in the fully localized limit.

Figure 2

(a) Low-energy spin waves along high-symmetric lines of the Brillouin zone [Fig. 1], measured with $E_i=6.1\mathrm{meV}$. Two singular signals below 1 meV are artifacts (multiple scattering). Dotted line is a fit dispersion, see text. (b) Band bottoms of spin waves along ($H$, 0), measured with $E_i=2.9\mathrm{meV}$. (c),(d) Energy cuts at (0, 0) and (0.5, 0), based on the same data as in (b). Slight difference in the peak-maximum energy is due to resolution effects. (e) Calculated spin waves using the model in Eq. (1) for comparison to (a). Inset shows the goodness of fit (${\chi }^2$) versus $J_3$ and $\mathrm{\Delta }$. White cross indicates the best-fit parameters used for the main panel. For details of the fitting, see Ref. [55].

Figure 3

(a) At least six spin-wave branches are observed at 5 K. Data are measured with $E_i=10.0\mathrm{meV}$ (lower part) and 19.4 meV (upper part). Dotted curves indicate fit dispersions [55]. (b) Brillouin-zone averaged intensity versus energy, measured with $E_i=19.4\mathrm{meV}$. Shaded area indicates background scattering (excluded from the sum-rule analysis discussed in the text). The slightly increased intensity above 12 meV at 35 K is contributed by the softened crystal field excitations (Fig. 1). (c)–(h) Spin-wave signals at selected energies.

Figure 4

(a) Energy and temperature dependence of intensity averaged over a Brillouin zone (dashed black rectangle in inset), based on data obtained at eight temperatures with $E_i=10.0\mathrm{meV}$, after subtraction against $T=290\mathrm{K}$ as background. (b) Magnetic specific heat (dots) and entropy (gray line). Shaded area indicates entropy release above $T_N$. (c),(d) Paramagnetic fluctuations measured with $E_i=10.0\mathrm{meV}$ at 63 K. (e) Local zigzag arrangements on a hexagonal unit (left), and their vector superposition forming a vortexlike pattern (right). (f),(g) Calculated $I(\mathbf{Q})$ (see text) for the two arrangements in (e).