Persistent spin dynamics in magnetically ordered honeycomb-lattice cobalt oxides

In the quest to find quantum spin liquids, layered cobalt oxides ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ and ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$ have been proposed as promising candidates for approximating the Kitaev honeycomb-lattice model. Yet, their suitability has been thrown into question due to observed long-range magnetic order at low temperatures and indications of easy-plane, rather than Kitaev-type, spin anisotropy. Here, we use muon-spin relaxation to reveal an unexpected picture: Contrary to the anticipated static nature of the long-range order, the systems show prevalent spin dynamics with a spatially uneven distribution and varied correlation times. This underlines that the magnetic ground states cannot be solely described by the long-range order, suggesting a significant role of quantum fluctuations. Our findings not only shed light on the complex physics of these systems but also underscore the need for a refined approach in the search for realizable quantum spin liquids.

Figure 1

(a), (b) Crystal structures of ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ and ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$, respectively, along with calculated muon stoppage sites indicated by small spheres near the cobalt atoms. (c), (d) Magnetic susceptibility of ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ and ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$, measured on single crystals with in-plane magnetic fields of 5000 and 1000 Oe, respectively. The data are plotted together with specific heat data measured in zero field to demonstrate $T_{\mathrm{N}}$ and the homogeneity of the samples.

Figure 2

(a) $\mu \mathrm{SR}$ spectra of NCTO measured far above $T_{\mathrm{N}}$. A longitudinal field (LF) of 0.01 T decouples the nuclear dipolar fields. (b) Near-zero-field $\mu \mathrm{SR}$ spectra measured at selected temperatures. (c) Demonstration of stretched-exponential fitting of the data obtained in near-zero field and just above $T_{\mathrm{N}}$. (d) Lowest-temperature spectra under various LFs, revealing a dual static and dynamic nature of the relaxation.

Figure 3

$\mu \mathrm{SR}$ spectra of ${\mathrm{Na}}_2{\mathrm{Co}}_2{\mathrm{TeO}}_6$ (NCTO) under various longitudinal fields (LFs) at (a) $T=27.9$ K, (b) 26.6 K, (c) 19.2 K, and (d) 13.7 K.

Figure 4

(a) $\mu \mathrm{SR}$ spectra of ${\mathrm{Na}}_3{\mathrm{Co}}_2{\mathrm{SbO}}_6$ (NCSO) measured far above $T_{\mathrm{N}}$. A longitudinal field (LF) of 0.01 T decouples the nuclear dipolar fields. (b) Near-zero-field $\mu \mathrm{SR}$ spectra measured at selected temperatures. (c) Demonstration of stretched-exponential fitting of the data obtained in near-zero field and just above $T_{\mathrm{N}}$. (d) Lowest-temperature spectra under various LFs, revealing a dual static and dynamic nature of the relaxation.

Figure 5

$\mu \mathrm{SR}$ spectra of NCSO measured using (a) the helium-4 cryostat and (b) the helium-3 cryostat. The ZF spectra clearly show that the background level for the helium-4 cryostat is negligible and that for the helium-3 cryostat is about 0.025. All the $\mu \mathrm{SR}$ data of NCSO discussed in the paper were measured by the helium-3 cryostat except for that in (a).

Figure 6

(a) Muon-spin asymmetry in the nominal $t=0$ limit measured by our relatively poor time resolution. (b) Muon relaxation rate as a function of temperature. The relaxation is primarily caused by dynamic internal fields because longitudinal fields up to 0.4 T have little effect on it [Fig. 2]. (c) Stretched exponent used in our fits to the time spectra (see text). Below $T_{\mathrm{N}}$, the data are consistent with a stable exponent and are therefore fit to a common value. Arrows indicate $T_{\mathrm{N}}$ of the two systems.

Figure 7

Volume fraction of dynamic [$A_1/(A_1+A_2)$] and static phases [$A_2/(A_1+A_2)$] as determined from the fit values of $A_1$ and $A_2$ according to Eq. (1). $A_2=0$ has been assumed in the fits above $T_{\mathrm{N}}$.