Thermodynamics and heat transport of quantum spin liquid candidates NaYbS₂ and NaYbSe₂

Figure 1 shows the crystal structure of NaYbS₂ and NaYbSe₂, in which the magnetic Yb³⁺ ions form ideal triangular lattice layers that are separated by the non-magnetic Na$Ch_{6}$ octahedra along the $c$ axis. It is notable that the Yb$Ch_{6}$ octahedra edge-shared with Na$Ch_{6}$ octahedra and are weakly distorted. The tilt of the YbSe₆ octahedra is smaller than that of YbS₆ octahedra, which is related to the larger radius of Se^2- and leads to the difference of intralayer interactions between NaYbSe₂ and NaYbS₂. Due to the large difference in ionic size between Na⁺ and Yb³⁺, the anti-site disorder is much smaller than that of the Yb-based QSL candidate YbMgGaO₄ with Mg²⁺/Ga³⁺ site disorder NC4949 . The site disorder in NaYbS₂ and NaYbSe₂ has been well studied. For NaYbS₂, the results of X-ray diffraction (XRD), scanning electron microscopy (SEM), electron spin resonance (ESR), and nuclear magnetic resonance (NMR) all evidenced an absence of inherent structural distortions PRB220409 ; QuantumFrontiers13 . For NaYbSe₂, the previous literatures have reported the absence of intrinsic structural disorder by using XRD, ESR, NMR, and neutron scattering measurements PRB224417 ; Scheie . An exceptional result is that a small amount of Yb occupying the Na site was probed by using single-crystal XRD and the inductively coupled plasma measurements PRX021044 . The structural and compositional characterization of these single crystals have been reported in some of our previous papers CPL35 ; PhysRevB184419 ; SciChinaPhysMechAstron67 . However, we have not carried out deeper experimental characterization of the site disorder in the samples used in the present work.

Figure 2(a) shows the field dependence of ac susceptibility $\chi^{\prime}(B)$ measured at selected temperatures with $B\parallel a$ for NaYbS₂. At $T=$ 30 mK, the $\chi^{\prime}(B)$ curve displays three weak peaks around 3.5, 6.5, and 12.5 T, respectively, and approaches saturation around 13.5 T. With increasing temperature, these peaks become weaker and broader. These transition fields are consistent with the previously reported results, indicating the magnetic transitions induced by the external magnetic field QuantumFrontiers13 . For NaYbSe₂, at $T=$ 20 mK, the $\chi^{\prime}(B)$ curve exhibits much sharper peaks around 3.5, 6.5, and 12 T with $B\parallel a$. With increasing temperature, these peaks become broader and shift to lower magnetic field, as shown in Fig. 2(b). The overall behavior of $\chi^{\prime}(B)$ curves are similar to those of NaYbS₂. The remarkable difference is that there is a deep valley at 3.5 $\sim$ 6.5 T for NaYbSe₂, which corresponds to the 1/3$M_{s}$ plateau ($M_{s}$ is the saturation magnetization), as reported by Ranjith et al. PRB224417 , and indicates an up-up-down phase. This has been predicted by the mean-field theory and further confirms an easy-plane anisotropy in NaYbSe₂ CondensMatter69 . Therefore, the present ac susceptibility data are basically consistent with the previous magnetization studies. In passing, we explain a bit about the difference of ac susceptibility data between NaYbS₂ and NaYbSe₂. The ac susceptibility measurement was performed using home-made ac coils PRB064401 . Different coils lead to different background of ac signal. The data of NaYbS₂ and NaYbSe₂ were obtained on two different coils and the coil for NaYbS₂ has large background. This is the main reason for that there is no so clear valley feature of NaYbSe₂ data.

While completing the present work, we noticed a recent study on NaYbSe₂ by Scheie et al. that reported the ultralow-temperature ac susceptibility for not only $B\parallel a$ but also $B\parallel c$ Scheie . It can be seen that our susceptibility data are in good consistent with theirs. It should be pointed out that their data for $B\parallel c$ show a very broad and weak peak-like feature at $\sim$ 7 T, which behaved significantly different from those for $B\parallel a$.

To characterize the thermodynamics of NaYbS₂ and NaYbSe₂ single crystals, the low-temperature specific heat measurements were performed down to 60 mK with external magnetic field along the $c$ axis. As shown in Fig. 3, the zero-field specific heat is almost identical for NaYbS₂ and NaYbSe₂, and there is no signature of long-range magnetic order. The upturn at ultralow temperatures is originated from the nuclear Schottky contribution, which can be simply described by $C_{\rm{nuc}}\propto T^{2}$ PRB224417 ; PRX021044 . The plot $C_{p}/T$ vs $T$ displays a broad peak between 0.2 and 1.5 K, which is a common behavior observed in some other QSL candidates, such as YbMgGaO₄ and NaYbO₂ NatPhys15 ; SciRep5 . After subtracting the estimated nuclear Schottky contributions, the obtained magnetic specific heat $C_{\rm{mag}}$ show a nearly linear temperature dependence at low temperatures, which indicates a gapless QSL ground state with a constant density of states. All these results are reasonably consistent with the previous reports PhysRevB184419 ; QuantumFrontiers13 ; PRX021044 . Moreover, the constant density of states at low energies can be explained by the spinon Fermi surface in QSL. Although the linear-$T$ specific heat and the constant density of states can be understood in terms of the random field effect in the spin glass, we think this is unlikely for our systems here. There does not exist any signature of spin freezing transition in our measurement, and thus the spin glassy features are absent. The Yb³⁺ ions in these systems provide the Kramers doublet whose degeneracy is protected by the time reversal symmetry. Unlike the non-Kramers doublets for which the random fields can be obtained by the crystal disorders, the non-magnetic disorders cannot generate the random fields for the Kramers doublets. Thus, we think the spinon Fermi surface state is more compatible with the low-temperature specific heat behaviors.

Figures 4(a) and 4(b) show the magnetic field dependencies of specific heat of NaYbS₂ and NaYbSe₂ at selected temperatures with external magnetic field along the $c$ axis, respectively. There are some similarities between the data of these two materials: (i) there is a weak peak at low fields, which shifts to higher magnetic field with increasing temperature and is related to the nuclear Schottky anomaly. (ii) With increasing magnetic field, some field-induced transitions appear while the overall trend is decreasing. There are also certain differences between these two materials. At $T=$ 200 mK, the $C_{p}(B)/C_{p}(0)$ of NaYbS₂ first decreases and then increases with increasing magnetic field, exhibiting a broad valley and a kink around 4 and 10 T, respectively. However, at $T=$ 200 mK, the $C_{p}(B)/C_{p}(0)$ curve of NaYbSe₂ exhibits a valley and a broad peak around 3 and 9 T, respectively; and the specific heat continues to decrease at high field, which is contrast to the case of NaYbS₂. In addition, for NaYbSe₂, the low-field valley becomes weaker and shifts to higher magnetic field with increasing temperature. It is likely that these intermediate-field behaviors are correlated with the broad and weak peak-like feature of ac susceptibility data Scheie . It should be pointed out that the nuclear Schottky term can also be strongly affected by the magnetic field; in particular, at very low temperatures it can be strongly suppressed upon increasing field.

Figure 5(a) shows the temperature dependence of thermal conductivity for NaYbS₂ single crystal in zero field and in 14 T magnetic field along the $c$ axis or the $a$ axis. As one can see, the zero-field $\kappa(T)$ curve exhibits a $T^{1.75}$ power law behavior at very low temperatures, which distinctly deviates from the standard phonon transport behavior ($T^{3}$-dependence) Berman . This deviation is apparently due to the strong phonon scattering by magnetic excitations. In addition, the direction of the external magnetic field has a significantly different effect on changing thermal conductivity. To be more specific, the 14 T field along the $c$ axis slightly suppresses the thermal conductivity and the $\kappa(T)$ curve roughly obeys a $T^{1.65}$ dependence. On the contrary, the thermal conductivity is strongly enhanced for 14 T $\parallel a$ and displays a rough $T^{2.7}$ dependence at low temperatures, which is quite close to the typical behavior of phonon thermal conductivity in the boundary scattering limit. The previous studies on the magnetization of NaYbS₂ single crystal revealed that 14 T along the $a$ axis can nearly fully polarize the Yb³⁺ spins, while the spin polarization field is much higher for the $c$ direction QuantumFrontiers13 ; PRB220409 . Therefore, under the 14 T $c$-axis field the spins are still in a fluctuating state and can strongly scatter phonons, while the 14 T in-plane field can almost smear out the magnetic scattering on the phonons. Figure 5(b) shows the ultralow-temperature thermal conductivity at zero field. We fit the data by using a formula $\kappa/T=$ $\kappa_{0}/T$ + $bT^{\alpha-1}$ Science328 ; NC4216 ; NC4949 , in which the two terms represent contributions from the itinerant fermionic excitations and phonons, respectively. The fitting gives $\kappa_{0}/T$ = 0 and $\alpha=$ 1.75. The zero residual term implies the absence of the itinerant fermionic excitations in NaYbS₂.

Figures 6 and 7 show the $\kappa(B)$ isotherms for NaYbS₂ single crystal with $B\parallel c$ and $B\parallel a$, respectively. It is notable that the field dependence of $\kappa$ are significantly different between the $c$-axis and in-plane fields. For $B\parallel c$, the $\kappa$ is always suppressed by magnetic field and the suppression can reach $\sim$ 40$\%$ at 14 T without saturation. At very low temperatures of 100 and 151 mK, the $\kappa(B)/\kappa(0)$ curves show a shallow and broad dip at $\sim$ 5 T and an anomaly around 9 T, which disappear with increasing temperature and is consistent with the anomalies in the specific heat. In contrast, the in-plane field induces much more complex $\kappa(B)$ behavior with strong enhancement of $\kappa$ at higher fields. These indicate the strong anisotropic behavior between the $c$-axis and in-plane fields. At very low temperatures, upon increasing field the $\kappa$ firstly decreases slightly ($\sim$ 10 %) and then increases non-monotonically. The $\kappa(B)/\kappa(0)$ curves display a shallow valley around 2 T, a broad peak between 3 and 6 T, and a sharp increase around 12.5 T, accompanied with a saturation above 13.5 T. At higher temperatures, the low-field broad peak weakens and the 14 T-field enhancement can reach $\sim$ 10 times without saturation. The previous magnetization and torque measurements indicated that there are four transition fields around 3.3, 6.1, 10.2, and 14.8 T for $B\parallel a$ at $T=$ 0.8 K QuantumFrontiers13 . Therefore, the characteristic fields at $\kappa(B)/\kappa(0)$ isotherms are roughly consistent with these transition fields by magnetic measurements. The broad peak is related to the up-up-down phase and nearly disappears at $T=$ 850 mK. The sharp increase of $\kappa$ is associated with a phase transition from the up-up-down phase to the oblique phase. Moreover, the saturation magnetic field is reached around 13.5 T at low temperatures. At higher temperatures above 151 mK, the $\kappa(B)/\kappa(0)$ curves may saturate at fields higher than 14 T, which is consistent with higher saturation field of magnetization QuantumFrontiers13 .

Figure 8(a) shows the temperature dependence of thermal conductivity for NaYbSe₂ single crystal in zero field and in 14 T magnetic field along the $c$ axis or the $a$ axis. In zero field, the $\kappa(T)$ curve exhibits a rough $T^{1.9}$ dependence, which also deviates from the standard $T^{3}$ behavior of phonon thermal conductivity at the boundary scattering limit Berman and indicates strong phonon scattering. Moreover, we observed a negligibly small $\kappa_{0}/T$ in the $\kappa/T$ vs $T^{0.9}$ plot, as shown in Fig. 8(b), which also points to the absence of mobile fermionic excitations. The different responses are observed for applying 14 T field along the $c$ or $a$ axis. The $\kappa(T)$ curve displays a $T^{2.65}$ dependence for 14 T field along the $a$ axis, which is slightly weaker than the standard $T^{3}$ behavior and may be due to the specular reflections at the sample surfaces or the remaining spin-phonon scattering. For 14 T field along the $c$ axis, the $\kappa(T)$ curve displays shows a $T^{1.7}$ dependence and even weaker temperature dependence below 100 mK, which also indicates the strong phonons scattering by the magnetic excitations.

The $\kappa(B)/\kappa(0)$ isotherms of NaYbSe₂ single crystal for $B\parallel c$ and $B\parallel a$ are shown in Figs. 9 and 10, respectively. It can be seen that the $\kappa(B)/\kappa(0)$ curves exhibit a broad valley at low fields followed by an enhancement at high fields for $B\parallel c$. The $\kappa(B)/\kappa(0)$ for $B\parallel a$ behaves very similar to that of NaYbS₂, and there is a broad peak around 4 $\sim$ 6 T, corresponding to the up-up-down spin arrangement, which is consistent with 1/3$M_{s}$ plateau. Moreover, a shallow dip can be observed around 12 T, which is a field-induced magnetic transition, and the $\kappa(B)$ curves finally saturate at 13 T for $T=$ 100 and 151 mK. With increasing temperature, the broad peak becomes invisible and the shallow dip gradually disappears. All these critical fields are consistent with the previous magnetization results PRB224417 .