Spin dynamics and spin freezing in the triangular lattice antiferromagnets FeGa${}_2$S${}_4$ and NiGa${}_2$S${}_4$

Magnetic susceptibility and muon spin relaxation ($\mu$SR) experiments have been carried out on the quasi-2D triangular-lattice spin $S=2$ antiferromagnet FeGa${}_2$S${}_4$. The $\mu$SR data indicate a sharp onset of a frozen or nearly frozen spin state at $T^{*}=31\left(2\right)$ K, twice the spin-glass-like freezing temperature $T_f=16\left(1\right)$ K. The susceptibility becomes field dependent below $T^{*}$, but no sharp anomaly is observed in any bulk property. A similar transition is observed in $\mu$SR data from the spin-1 isomorph NiGa${}_2$S${}_4$. In both compounds the dynamic muon spin relaxation rate ${\lambda }_d\left(T\right)$ above $T^{*}$ agrees well with a calculation of spin-lattice relaxation by Chubukov, Sachdev, and Senthil in the renormalized classical regime of a 2D frustrated quantum antiferromagnet. There is no firm evidence for other mechanisms. At low temperatures, ${\lambda }_d\left(T\right)$ becomes temperature independent in both compounds, indicating persistence of spin dynamics. Scaling of ${\lambda }_d\left(T\right)$ between the two compounds is observed from $\sim T_f$ to $\sim 1.5T^{*}$. Although the $\mu$SR data by themselves cannot exclude a truly static spin component below $T^{*}$, together with the susceptibility data they are consistent with a slowly fluctuating “spin gel” regime between $T_f$ and $T^{*}$. Such a regime and the absence of a divergence in ${\lambda }_d\left(T\right)$ at $T^{*}$ are features of two unconventional mechanisms: (1) binding/unbinding of $Z_2$ vortex excitations, and (2) impurity spins in a nonmagnetic spin-nematic ground state. The absence of a sharp anomaly or history dependence at $T^{*}$ in the susceptibility of FeGa${}_2$S${}_4$, and the weakness of such phenomena in NiGa${}_2$S${}_4$, strongly suggest transitions to low-temperature phases with unconventional dynamics.

Figure 1

Temperature dependence of the dc magnetic susceptibility ${\chi }_{\mathrm{dc}}$ of FeGa${}_2$S${}_4$ for various applied magnetic fields. Filled symbols: field-cooled (FC) data. Open symbols: zero-field-cooled (ZFC) data. $T_f$: low-field spin freezing temperature. $T^{*}$: onset temperature of quasistatic muon spin relaxation [cf. Fig. 3a].

Figure 2

Representative late-time muon asymmetry data (positron count asymmetry vs. time) in FeGa${}_2$S${}_4$. (a) Asymmetry vs. time at various temperatures for longitudinal field ${\mu }_0{H}_L=2.02$ mT. (b) Asymmetry vs. time for various values of ${\mu }_0{H}_L$ at temperature $T=1.7$ K. Curves: fits to Eq. (2).

Figure 3

(a) Temperature dependence of dynamic muon spin relaxation rate ${\lambda }_d$ in weak longitudinal applied field $H_L$ in FeGa${}_2$S${}_4$ and NiGa${}_2$S${}_4$. Circles: FeGa${}_2$S${}_4$ powder, ${\mu }_0{H}_L=2.02$ mT. Squares: NiGa${}_2$S${}_4$ powder, ${\mu }_0{H}_L=2.0$ mT. Triangles: NiGa${}_2$S${}_4$, mosaic of oriented single crystals, ${\mathbf{H}}_L\parallel \mathbf{c}$, ${\mu }_0{H}_L=2.0$ mT. $T_f\left({\mathrm{FeGa}}_2{\mathrm{S}}_4\right)$: freezing temperature from FC-ZFC bifurcation in ${\chi }_{\mathrm{dc}}\left(T\right)$ (Fig. 1). $T_f\left({\mathrm{NiGa}}_2{\mathrm{S}}_4\right)$: freezing temperature from onset of frequency-dependent ${\chi }_{\mathrm{ac}}\left(T\right)$ (Refs. 18 and 29). $T^{*}$: transition temperatures from $\mu$SR data. Inset: FeGa${}_2$S${}_4$, late-time asymmetry fraction ${\eta }_d$ [Eq. (1)] (filled circles) and stretching power $\beta$ [Eq. (3)] (open circles). (b) Normalized muon relaxation rate ${\lambda }_d\left(T\right)/{\lambda }_d\left(T^{*}\right)$ vs. normalized temperature $T/T^{*}$. Symbols as in (a).

Figure 4

Dependence of dynamic muon spin relaxation parameters on longitudinal field $H_L$ in FeGa${}_2$S${}_4$ and NiGa${}_2$S${}_4$ at low temperatures ($T\ll T^{*}$). Circles: FeGa${}_2$S${}_4$ powder, $T=1.7$ K. Squares: NiGa${}_2$S${}_4$ powder, $T=2.1$ K. Triangles: NiGa${}_2$S${}_4$ oriented single-crystal mosaic, $T=2.3$ K. (a) Relaxation rate ${\lambda }_d$. (b) Late-time asymmetry fraction ${\eta }_d$.

Figure 5

Dependence of ${\lambda }_d/T^3$ on inverse temperature $1/T$ in FeGa${}_2$S${}_4$ and NiGa${}_2$S${}_4$, $T⩾T^{*}$. Straight lines: fits of Eq. (4). Labeled arrows: values of $1/2\pi {\rho }_s$ (see text). Curve: fit of the $Z_2$ vortex theory[9, 10] (Sec. 3c) to data for FeGa${}_2$S${}_4$.

Figure 6

Dependence of $1/t_1{t}^3$ on $1/t$, where $1/t_1$ is the reduced dynamic relaxation rate from Ref. 49 [Eqs. (A7) and (A8)] and $t$ is the reduced temperature (see text for definitions). Solid line: renormalized-classical (RC) region. Dashed curve: quantum-critical (QC) region.