Spin-Liquid State in the $S=1/2$ Hyperkagome Antiferromagnet ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$

A spinel related oxide, ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$, was found to have a three dimensional network of corner shared ${\mathrm{Ir}}^{4+}$ ($t_{2g}{}^5$) triangles. This gives rise to an antiferromagnetically coupled $S=1/2$ spin system formed on a geometrically frustrated hyperkagome lattice. Magnetization $M$ and magnetic specific heat $C_m$ data showed the absence of long range magnetic ordering at least down to 2 K. The large $C_m$ at low temperatures is independent of applied magnetic field up to 12 T, in striking parallel to the behavior seen in triangular and kagome antiferromagnets reported to have a spin-liquid ground state. These results strongly suggest that the ground state of ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$ is a three dimensional manifestation of a spin liquid.

Figure 1

(a) Crystal structure of ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$ with the space group $P{4}_{1}32$. Among the three Na sites, only Na1 site is shown for clarity. Black and gray octahedra represent ${\mathrm{IrO}}_6$ and $\mathrm{Na}{\mathrm{O}}_6$, respectively. The spheres inside the octahedra represent Ir and Na atoms and oxygens occupy all the corners. (b) The x-ray diffraction pattern of ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$ at room temperature. The crosses indicate the raw data and the solid line indicates the spectrum calculated based on the refinement using $P{4}_{1}32$. (c) and (d) Hyperkagome Ir and Na sublattice derived from the structure of ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$ with the space group $P{4}_{1}32$ (c) and $P{4}_{3}32$ (d). These two structures with different chirality are indistinguishable by conventional x-ray diffraction, giving the identical result in refinement.

Figure 2

Temperature dependence of the inverse magnetic susceptibility ${\chi }^{-1}$ under 1 T (a), magnetic specific heat $C_m$ divided by temperature $T$ (b) and magnetic entropy $S_m$ (c) of polycrystalline ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$. To estimate $C_m$, data for ${\mathrm{Na}}_4{\mathrm{Sn}}_3{\mathrm{O}}_8$ is used as a reference of the lattice contribution. Inset: (a) Temperature dependence of magnetic susceptibility $\chi$ of ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$ in various fields up to 5 T. For clarity, the curves are shifted by 3, 2, and $1\times {10}^{-4}\mathrm{emu}/\mathrm{mol}$ Ir for 0.01, 0.1, and 1 T data, respectively. (b) $C_m/T$ vs $T$ of ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$ in various fields up to 12 T. Broken lines indicate $C_m$ proportional to $T^2$ and $T^3$, respectively.

Figure 3

Comparison of the normalized magnetic specific heat of ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$ with those of other frustrated antiferromagnets ${\mathrm{SrCr}}_{9p}{\mathrm{Ga}}_{12-9p}{\mathrm{O}}_{19}$ ($p=0.98$) [6] and ${\mathrm{NiGa}}_2{\mathrm{S}}_4$ [8]. $M$ in the unit of vertical axis denotes magnetic element Ir, Cr and Ni for ${\mathrm{Na}}_4{\mathrm{Ir}}_3{\mathrm{O}}_8$, ${\mathrm{SrCr}}_{9p}{\mathrm{Ga}}_{12-9p}{\mathrm{O}}_{19}$ and ${\mathrm{NiGa}}_2{\mathrm{S}}_4$, respectively. Temperature $T$ is normalized by the Curie-Weiss constant ${\theta }_W$ for comparison.

Figure 4

(a) Temperature dependence of the magnetic susceptibility $\chi (T)$ of polycrystalline ${\mathrm{Na}}_4({\mathrm{Ir}}_{1-x}{\mathrm{Ti}}_x{)}_3{\mathrm{O}}_8$ ($x=0.1$) in magnetic fields up to 5 T, compared to the data of $x=0$ under 0.01 T. (b) Magnetic specific heat $C_m$ of the $x=0.1$ sample plotted as $C_m/T$ vs $T$. The broken line indicates $C_m/T$ of $x=0$ under zero field. inset: Orphan spin Curie constant $C^{*}$ of ${\mathrm{Na}}_4({\mathrm{Ir}}_{1-x}{\mathrm{Ti}}_x{)}_3{\mathrm{O}}_8(0\le x\le 0.3)$, defined by Schiffer and Daruka [15].