Thermal conductivity across the metal-insulator transition in the single-crystalline hyperkagome antiferromagnet ${\mathrm{Na}}_{3+x}{\mathrm{Ir}}_3{\mathrm{O}}_8$

The hyperkagome antiferromagnet ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$ represents the first genuine candidate for the realization of a three-dimensional quantum spin liquid. It can also be doped towards a metallic state, thus offering a rare opportunity to explore the nature of the metal-insulator transition in correlated, frustrated magnets. Here, we report thermodynamic and transport measurements in both metallic and weakly insulating single crystals down to 150 mK. While in the metallic sample the phonon thermal conductivity $\left({\kappa }^{\mathrm{ph}}\right)$ is almost in the boundary scattering regime, in the insulating sample, we find a large reduction ${\kappa }^{\mathrm{ph}}$ over a very wide temperature range. This result can be ascribed to the scattering of phonons off the gapless magnetic excitations that are seen in the low-temperature specific heat. This works highlights the peculiarity of the metal-insulator transition in ${\mathrm{Na}}_{3+x}{\mathrm{Ir}}_3{\mathrm{O}}_8$ and demonstrates the importance of the coupling between lattice and spin degrees of freedom in the presence of strong spin-orbit coupling.

Figure 1

(a) Electrical resistivity $\left(\rho \right)$ and (b) magnetic susceptibility $\left(\chi \right)$ of ${\mathrm{Na}}_{3+x}{\mathrm{Ir}}_3{\mathrm{O}}_8$ single crystals as a function of temperature for $x=0.6$ (red open circles) and $x=0.0$ (blue open circles). In this and all subsequent figures, the metal-like $(x=0)$ and insulating $(x=0.6)$ samples are plotted using blue open and red open circles, respectively.

Figure 2

Zero-field thermal conductivity of ${\mathrm{Na}}_{3+x}{\mathrm{Ir}}_3{\mathrm{O}}_8$ as a function of temperature: (a) between 10 and 300 K and (b) between 150 mK and 6 K plotted on a log-log scale.

Figure 3

Zero-field thermal conductivity of ${\mathrm{Na}}_{3+x}{\mathrm{Ir}}_3{\mathrm{O}}_8$ between 150 mK and 6 K plotted on a log-log scale (a) ${\kappa }_{\mathrm{ins}}/T$ as a function of temperature for the insulating sample (0 T, red open circles; 5 T, black closed circles). The black dashed line is a single-component fit to $\frac{{\kappa }_{\mathrm{ins}}}{T}=b{T}^{\beta }$ with $b=0.026 {\mathrm{W}\mathrm{K}}^{-(2+\beta )} {\text{m}}^{-1}$ and $\beta =1.1$. (b) ${\kappa }_{\mathrm{met}}/T$ as a function of $T^2$ for the metallic sample (0 T, blue open circles). The black dashed line is a fit to $\frac{{\kappa }_{\mathrm{met}}}{T}=\frac{{\kappa }^{\mathrm{el}}}{T}+a{T}^{\alpha }$, where $\frac{{\kappa }^{\mathrm{el}}}{T}=\frac{L_0}{{\rho }_{\mathrm{met}}}=4.1\times {10}^{-3}{\mathrm{W}\mathrm{K}}^{-2} {\text{m}}^{-1},a=0.61 {\mathrm{W}\mathrm{K}}^{-(2+\alpha )} {\text{m}}^{-1}$, and $\alpha =1.7$.

Figure 4

${\kappa }_{\mathrm{met}}/T$ as a function of $T^2$ for the metallic sample (0 T, blue open circles, and 6 T, brown open squares). The blue dash line is a fit to $\frac{{\kappa }_{\mathrm{met}}}{T}=\frac{L_0}{{\rho }_{\mathrm{met}}}+a{T}^{\alpha }$, where $\frac{L_0}{{\rho }_{\mathrm{met}}}=4.1\times {10}^{-3}{\mathrm{W}\mathrm{K}}^{-2} {\text{m}}^{-1},a=0.61 {\mathrm{W}\mathrm{K}}^{-(2+\alpha )} {\text{m}}^{-1}$, and $\alpha =1.7$. The blue line is a fit to $\frac{{\kappa }_{\mathrm{met}}}{T}=a_0+a^{*}{T}^2$, where $a_0=1.4\times {10}^{-2}{\mathrm{W}\mathrm{K}}^{-2} {\text{m}}^{-1}$ and $a^{*}=0.76 {\mathrm{W}\mathrm{K}}^{-4} {\text{m}}^{-1}$.

Figure 5

(a) Temperature dependence of the specific heat $C_{\mathrm{ins}}\left(T\right)$ of insulating ${\mathrm{Na}}_{3+x}{\mathrm{Ir}}_3{\mathrm{O}}_8$ in magnetic fields of 0 (red), 6 (orange), and 12 T (black open circles) plotted on a log-log scale. (b) Low-$T$ magnetic specific heat of the insulating sample $\left(C_{\mathrm{ins}}\right)$ divided by $T$ as function of $T$. The black line is a fit of $C_{\mathrm{ins}}\left(T\right)={\gamma }_{\mathrm{ins}}T+{\alpha }_{\mathrm{ins}}{T}^2+{\beta }_{\mathrm{ins}}{T}^3$ from 0.5 to 5 K. The values of ${\gamma }_{\mathrm{ins}},{\alpha }_{\mathrm{ins}}$ and ${\beta }_{\mathrm{ins}}$ are given in the text. (c) $C_{\mathrm{met}}/T$ as a function of $T^2$. The black dotted line is a fit to $C_{\mathrm{met}}\left(T\right)={\gamma }_{\mathrm{met}}T+{\alpha }_{\mathrm{met}}{T}^3$ where ${\gamma }_{\mathrm{met}}$ and ${\alpha }_{\mathrm{met}}$ are given in the text.

Figure 6

Annealing study on insulating crystals of ${\mathrm{Na}}_{3+x}{\mathrm{Ir}}_3{\mathrm{O}}_8$. The crystal was initially insulating (top curve). It was repeatedly heat-treated between $300{}^{\circ }\mathrm{C}$ and $500{}^{\circ }\mathrm{C}$ for 12 hours in flowing oxygen and measured between each treatment. Electrical contacts (50-$\mu \mathrm{m}$ gold wire attached with Dupont 4929 silver epoxy) were kept on the crystal to ensure that the relative error in the absolute magnitude of the resistivity was minimized between measurements.

Figure 7

(a) Comparison of the magnetic specific heat divided by temperature as function of the temperature from Okamoto et al. (red points) [5] and measured on the single-crystal ${\mathrm{Na}}_{3.6}{\mathrm{Ir}}_3{\mathrm{O}}_8$ [black points]. (b) ${\chi }^{-1}$ as a function of the temperature for the single-crystal ${\mathrm{Na}}_{3.6}{\mathrm{Ir}}_3{\mathrm{O}}_8$