Kitaev interactions in the Co honeycomb antiferromagnets ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$ and ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$

${\mathrm{Co}}^{2+}$ ions in an octahedral crystal field stabilize a $j_{\mathrm{eff}}=1/2$ ground state with an orbital degree of freedom and have been recently put forward for realizing Kitaev interactions, a prediction we have tested by investigating spin dynamics in two cobalt honeycomb lattice compounds, ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ and ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$, using inelastic neutron scattering. We used linear spin wave theory to show that the magnetic spectra can be reproduced with a spin Hamiltonian including a dominant Kitaev nearest-neighbor interaction, weaker Heisenberg interactions up to the third neighbor, and bond-dependent off-diagonal exchange interactions. Beyond the Kitaev interaction that alone would induce a quantum spin liquid state, the presence of these additional couplings is responsible for the zigzag-type long-range magnetic ordering observed at low temperature in both compounds. These results provide evidence for the realization of Kitaev-type coupling in cobalt-based materials, despite hosting a weaker spin-orbit coupling than their $4d$ and $5d$ counterparts.

Figure 1

(a) Honeycomb layer with the ${\mathrm{Co}}^{2+}$ ions in edge-sharing octahedra. The zigzag magnetic structure within the plane stabilized in ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ is shown (blue arrows), as well as the first to third nearest-neighbor Heisenberg interactions. The red, blue, and green lines indicate the three nearest-neighbor bonds associated to the local $x$, $y$, and $z$ axes in the Kitaev model. (b) Electronic configuration of $3d^7{\mathrm{Co}}^{2+}$ in an octahedral environment and representation of the $j_{\mathrm{eff}}$ = 1/2 ground state, and excited $j_{\mathrm{eff}}$ = 3/2 and $j_{\mathrm{eff}}$ = 5/2 manifolds under the action of crystal field and spin-orbit coupling.

Figure 2

[(a), (b)] $\mathrm{S}(\mathbf{Q}$, $\omega$) at $T=1.5$ K measured on MACS in ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ and ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$, respectively. [(c), (d)] Spin-wave calculations using the K-H Hamiltonian with the parameter values reported in Table 2. [(e), (f)] Spin-wave calculations using the $J_1\text{−}{J}_2\text{−}{J}_3$ Heisenberg model as described in the text and in Fig. 8.

Figure 3

(a) Color map showing the $Q$ dependence of the spin-orbit excitations between the $j_{\mathrm{eff}}$ = 1/2 and $j_{\mathrm{eff}}$ = 3/2 manifolds measured at $T=2$ K for ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$. [(b), (c)] View of the slightly distorted ${\mathrm{CoO}}_6$ octahedra in ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$ and ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$, respectively. [(d), (e)] Constant-$Q$ scans showing the spin-orbit excitations between the $j_{\mathrm{eff}}$ = 1/2 and $j_{\mathrm{eff}}$ = 3/2 manifolds in ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$ and ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$, respectively. The straight lines indicate the instrumental resolution and the red lines are Gaussian fits. The additional peaks are phonon modes

Figure 4

Momentum dependence of the energy integrated inelastic intensity of ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ (left) and ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$ (right).

Figure 5

Magnetic phase diagrams calculated for ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ (left panels) and ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$ (right panels) for different pairs of parameters. The blue, yellow, and orange regions correspond to ferromagnetic, zigzag, and incommensurate magnetic structures while the pink areas correspond to a degenerate ground state. The black stars show the best-fit solution.

Figure 6

Dependence of the spin-wave calculation with respect to variation of the six parameters involved in the K-H model for ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$. The panel (a) corresponds to the chosen solution that best reproduces the experimental data. Each panel [(a)–(f)] corresponds to a set of parameters, which are represented in the phase diagram (right panels). The blue, yellow, and orange regions correspond to ferromagnetic, zigzag, and incommensurate magnetic structures while the pink areas correspond to a degenerate ground state.

Figure 7

Dependence of the spin-wave calculation with respect to variation of the six parameters involved in the K-H model for ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$. The panel (a) corresponds to the chosen solution that best reproduces the experimental data. Each panel [(a)–(f)] corresponds to a set of parameters, which are represented in the phase diagram (right panels). The blue, yellow, and orange regions correspond to ferromagnetic, zigzag, and incommensurate magnetic structures, respectively.

Figure 8

Dependence of the spin-wave calculations with respect to variation of the first, second, and third nearest-neighbor couplings involved in the Heisenberg $XXZ$ model, for a honeycomb lattice system. The sets of parameters are represented in the phase diagram (bottom right panel). No interplane coupling was considered in these calculations. Panels (a) and (b) are the best-fit solutions for ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ and ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$, respectively. The colors in the phase diagrams refer to the different possible magnetic ground states identified in Ref. [35].

Figure 9

[(a), (b)] $S$(Q,$\omega$) at $T=1.5$ K measured on MACS for ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ and ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$, respectively. Comparison of the Kitaev-Heisenberg model and the $J_1\text{−}{J}_2\text{−}{J}_3$ model with the data through constant-$Q$ cuts [(c)–(f)] and constant-energy cuts [(g), (h)].

Figure 10

Temperature dependence of the magnetic excitations in ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$.