Coexistence of random singlets and disordered Kitaev spin liquid in ${\mathrm{H}}_3{\mathrm{LiIr}}_2{\mathrm{O}}_6$

We combine static magnetic susceptibility $\chi \left(T\right)$, muon-spin relaxation, and ${}^1\mathrm{H}$ nuclear magnetic resonance measurements to explore the spin dynamics in the disordered-induced quantum spin liquid candidate ${\mathrm{H}}_3{\mathrm{LiIr}}_2{\mathrm{O}}_6$. Inverse Laplace transform analysis of the ${}^1\mathrm{H}$ spin-lattice relaxation rate $1/T_1$ enables us to identify two characteristic temperatures $T_{\mathrm{g}}=110\mathrm{K}$ and $T^{*}=26\mathrm{K}$. Below $T_{\mathrm{g}}$, a slower $1/T_1^{\mathrm{slow}}$ component dictated by gapped excitations with a spin gap ${\mathrm{\Delta }}_h=30–38\mathrm{K}$ evolves distinctly from a faster $1/T_1^{\mathrm{fast}}$ component pertaining to gapless excitations. Furthermore, we observe a sub-Curie divergent $\chi \left(T\right)\propto T^{-0.68}$, a power-law dependent $1/T_1^{\mathrm{fast}}\propto T^{1.4}$, and a weakly activated $1/T_1^{\mathrm{slow}}\propto exp(-{\mathrm{\Delta }}_l/k_{\mathrm{B}}T)$ (${\mathrm{\Delta }}_l=3–6\mathrm{K}$) below $T^{*}=26\mathrm{K}$. All these features suggest the coexistence of a disordered spin-liquid state and spin singlets with spatially distributed gaps.

Figure 1

(a) Log-log plot of the static magnetic susceptibility $\chi \left(T\right)$ versus temperature measured at ${\mu }_{0}H=1\mathrm{T}\left|\right|ab$. The solid line denotes a sub-Curie increase $T^{-0.68}$ below $T^{*}=26\mathrm{K}$ (vertical arrow). (b) Scaling of $H^{0.66}\chi \left(T\right)$ with $T/H$. (c) Log-log scaled plot of $T^{-0.4}M\left(H\right)$ versus $H/T$. (d) Thermal evolution of the Raman spectra in parallel polarization. (e) Representative muon spin polarization at zero field with the fitted curves as described in the main text. (f) Temperature dependence of the two muon spin relaxation rates ${\lambda }_1\left(T\right)$ and ${\lambda }_2\left(T\right)$.

Figure 2

(a) Temperature dependence of ${}^1\mathrm{H}$ NMR spectra in an applied magnetic field of ${\mu }_{0}H=1\mathrm{T}$ parallel to the honeycomb $ab$ plane. (b) Thermal evolution of the full width at half maximum of the ${}^1\mathrm{H}$ NMR spectra. (c) Knight shift $K\left(T\right)$ vs temperature for ${\mu }_{0}H=1$, 1.5, and 2 T.

Figure 3

(a) Stretched (dashed lines) and ILT (solid lines) fits of normalized $M\left(t\right)$ at selected temperatures. (b) Temperature dependence of the stretched exponent. (c) Distribution function $P(1/T_1)$ as a function of temperature. The symbols mark two components of $1/T_1$: $1/T_1^{\mathrm{slow}}$ (pink circles) and $1/T_1^{\mathrm{fast}}$ (green squares). The dash-dotted and dashed lines at 40 K and the dotted lines at 100 K are representative of two Gaussian fits. (d) The color contour plot of $P(1/T_1)$ in the temperature-$1/T_1$ plane. (e) Temperature dependence of $1/T_1^{\mathrm{slow}}$ (pink circles), $1/T_1^{\mathrm{fast}}$ (green squares), $1/T_1^{\mathrm{str}}$ (open triangles), and the center of gravity (COG; stars). (f) Temperature dependence of the spin-spin relaxation rate $1/T_2$. The pink line is a guide to the eye. Pale green shadings denote a glassy transition of the hydrogen hopping at $T_{\mathrm{g}}=110\mathrm{K}$.

Figure 4

(a) $1/T_1^{\mathrm{slow}}$ as a function of inverse temperature $1/T$. The solid and dashed lines are fits to an activation behavior in two different temperature windows. The inset plots the lower- and the higher-$T$ spin gap as a function of an applied field. (b) Log-log plot of $1/T_1^{\mathrm{fast}}$ vs temperature at selected fields. The dashed line is a power-law fit. (c) Scaling of $H^{0.3}/T_1^{\mathrm{fast}}$ with $H/T$ on a double logarithmic scale. (d) Log-log scaled plot of $H^{0.5}/T_1^{\mathrm{str}}$ versus $H/T$.

Figure 5

(a) X-ray diffraction patterns of $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ and ${\mathrm{H}}_3{\mathrm{LiIr}}_2{\mathrm{O}}_6$ at room temperature. Peaks are indexed with Miller (00l) indices. (b) Raman spectra of $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ and ${\mathrm{H}}_3{\mathrm{LiIr}}_2{\mathrm{O}}_6$ at room temperature. (c) Polarized Raman spectra of ${\mathrm{H}}_3{\mathrm{LiIr}}_2{\mathrm{O}}_6$ at room temperature in (xx) and (xy) polarizations.

Figure 6

(a) Temperature dependence of the ZF-$\mu \mathrm{SR}$ fitting parameters. Fast- and slow-muon spin relaxation rates ${\lambda }_1$ and ${\lambda }_2$ vs temperature in a semilog scale. (b) Temperature dependence of the mean value of the Gaussian-broadened Gaussian distribution ${\mathrm{\Delta }}_0$. (c) Ratio of the distribution width to the distribution mean value. (d) Thermal evolution of the fast-relaxing fraction $f_1$.

Figure 7

LF dependence of $\mu \mathrm{SR}$ spectra at $T=0.3\mathrm{K}$ in representative fields. The solid curves indicate the fittings to the data as described in the text. The internal field/frequency distribution is evaluated to be $\mathrm{\Delta }=11.08\left(3\right)\mathrm{MHz}$.