Spin dynamics and spin freezing behavior in the two-dimensional antiferromagnet $\mathrm{Ni}{\mathrm{Ga}}_2{\mathrm{S}}_4$ revealed by Ga-NMR, NQR and $\mu \mathrm{SR}$ measurements

We have performed ${}^{69,71}\mathrm{Ga}$ nuclear magnetic resonance (NMR), nuclear quadrupole resonance (NQR), and muon spin rotation and resonance on the quasi-two-dimensional antiferromagnet $\mathrm{Ni}{\mathrm{Ga}}_2{\mathrm{S}}_4$, in order to investigate its spin dynamics and magnetic state at low temperatures. Although there exists only one crystallographic site for Ga in $\mathrm{Ni}{\mathrm{Ga}}_2{\mathrm{S}}_4$, we found two distinct Ga signals by NMR and NQR. The origin of the two Ga signals is not fully understood, but possibly due to stacking faults along the $c$ axis which induce additional broad Ga NMR and NQR signals with different local symmetries. We found the spin freezing occurring at $T_{\mathrm{f}}$, at which the specific heat shows a maximum, from a clear divergent behavior of the nuclear spin-lattice relaxation rate $1∕T_1$ and nuclear spin-spin relaxation rate $1∕T_2$ measured by Ga-NQR as well as the muon spin relaxation rate $\lambda$. The main sharp NQR peaks exhibit a stronger tendency of divergence, compared with the weak broader spectral peaks, indicating that the spin freezing is intrinsic in $\mathrm{Ni}{\mathrm{Ga}}_2{\mathrm{S}}_4$. The behavior of these relaxation rates strongly suggests that the Ni spin fluctuations slow down towards $T_{\mathrm{f}}$, and the temperature range of the divergence is anomalously wider than that in a conventional magnetic ordering. A broad structureless spectrum and multicomponent $T_1$ were observed below $2\mathrm{K}$, indicating that a static magnetic state with incommensurate magnetic correlations or inhomogeneously distributed moments is realized at low temperatures. However, the wide temperature region between $2\mathrm{K}$ and $T_{\mathrm{f}}$, where the NQR signal was not observed, suggests that the Ni spins do not freeze immediately below $T_{\mathrm{f}}$, but keep fluctuating down to $2\mathrm{K}$ with the MHz frequency range. Below $0.5\mathrm{K}$, all components of $1∕T_1$ follow a $T^3$ behavior. We also found that $1∕T_1$ and $1∕T_2$ show the same temperature dependence above $T_{\mathrm{f}}$ but different temperature dependence below $0.8\mathrm{K}$. These results suggest that the spin dynamics is isotropic above $T_{\mathrm{f}}$, which is characteristic of the Heisenberg spin system, and becomes anisotropic below $0.8\mathrm{K}$.

Figure 1

(Color online) Crystal structure of $\mathrm{Ni}{\mathrm{Ga}}_2{\mathrm{S}}_4$. The lattice constants are $a=3.624\mathrm{\AA }$, $c=11.996\mathrm{\AA }$.

Figure 2

(a) ${}^{69,71}\mathrm{Ga}\text{−}\mathrm{NMR}$ spectra from single-crystal $\mathrm{Ni}{\mathrm{Ga}}_2{\mathrm{S}}_4$ at $40\mathrm{K}$ (curve in upper part), $17\mathrm{K}$ (gray area), and $1.6\mathrm{K}$ (lower part). NMR intensity normalization, i.e., taking into account difference in the $T_2$ relaxation rate ($T_2$ correction) has not been made. The resonant frequency is $93.5\mathrm{MHz}$ and the magnetic field is applied along the $c$ axis. The time between first and second pulses to observe spin echo is $30\mu \sec$. The ${}^{63,65}\mathrm{Cu}$ signals are from an NMR coil. (b) ${}^{69,71}\mathrm{Ga}\text{−}\mathrm{NQR}$ spectra after $T_2$ correction at 40 and $1.5\mathrm{K}$ using polycrystalline samples. NQR intensity is normalized by temperature.

Figure 3

Temperature dependence of the Knight shift $K$ for ${}^{71}\mathrm{Ga}\left(1\right)\left(●\right)$ and ${}^{71}\mathrm{Ga}\left(2\right)\left(○\right)$ sites. The dotted curves in the main panel give the temperature dependence of susceptibility $\chi$ normalized to $K$ above $100\mathrm{K}$. Inset: $K$ vs $\chi$ plot for the two ${}^{71}\mathrm{Ga}$ sites. Dotted lines are the linear fits between 70 and $200\mathrm{K}$.

Figure 4

(a) Recovery curves $m\left(t\right)$ derived from the ${}^{69}\mathrm{Ga}$ nuclear magnetization at two Ga sites [${}^{69}\mathrm{Ga}\left(1\right)\left(●\right)$ and ${}^{69}\mathrm{Ga}\left(2\right)\left(○\right)$] by changing the time interval between a saturation pulse and first pulse. Single component of $T_1$ was derived at two sites. (b) Decay curve $I\left(2\tau \right)$ from the ${}^{69}\mathrm{Ga}$ nuclear magnetization obtained by changing the duration time $\tau$ between first and second pulses. The decay curves are consistently fitted by an exponential function $[I\left(2\tau \right)\propto \exp (-2\tau ∕T_2)]$.

Figure 5

(a) Relaxation curves $m\left(t\right)$ plotted at $200\mathrm{mK}$ against $t$ (bottom axis, black circles) and $\sqrt{t}$ (top axis, open circles), where $t$ is the time between the saturation pulse and the first spin-echo pulse. (b) Several recovery curves at several temperatures far below $T_{\mathrm{f}}$ plotted against $t{T}^3$, where $T$ is the temperature. The inset of Fig. 5b shows the same recovery curves plotted against $t$.

Figure 6

Ga-NQR decay curves $I\left(2\tau \right)$ of $\mathrm{Ni}{\mathrm{Ga}}_2{\mathrm{S}}_4$ are plotted against $2\tau T$ at various temperatures far below $T_{\mathrm{f}}$. The inset shows the same decay curves plotted against $t$.

Figure 7

(a) Temperature dependence of nuclear spin-lattice relaxation rate $1∕T_1$. $1∕T_1$ was measured at ${}^{69}\mathrm{Ga}\left(1\right)\left(●\right)$, ${}^{69}\mathrm{Ga}\left(2\right)\left(○\right)$ site above $T_{\mathrm{f}}$. Below $1\mathrm{K}$, $1∕{\tilde{T}}_1$ was derived from the fitting of $m\left(t\right)$ measured at four frequencies in a range between 9.5 and $13\mathrm{MHz}$. Solid squares give $1∕{\tilde{T}}_1$ measured at $1.08\mathrm{T}$ with resonant frequency of $11.5\mathrm{MHz}$. $1∕T_1$ for NMR is normalized for the difference in the matrix elements (see text). (b) Temperature dependence of nuclear spin-spin relaxation rate $1∕T_2$. Below $2.5\mathrm{K}$, $1∕T_2$ was measured at $12\mathrm{MHz}$ in the broad NQR spectrum affected by the inhomogeneous internal fields. Decay curves below $2.5\mathrm{K}$ are consistently fitted by a single exponential function down to a lowest temperature as shown in Fig. 6 although $1∕T_1$ is widely distributed as shown in Fig. 5b. (c) Temperature dependence of integrated intensities of spin echo after $T_2$ correction. Below $2.5\mathrm{K}$, we assume that $T_2$ is independent of frequency.

Figure 8

Temperature dependence of the NQR relaxation rate $1∕T_1$ and $1∕\left(4T_2\right)$ at ${}^{69}\mathrm{Ga}\left(1\right)$ and ${}^{69}\mathrm{Ga}\left(2\right)$ sites.

Figure 9

Zero-field $\mu \mathrm{SR}$ asymmetry relaxation functions $G\left(t\right)$ in $\mathrm{Ni}{\mathrm{Ga}}_2{\mathrm{S}}_4$ at various temperatures. The dotted curves are fits to the power exponential form $G\left(t\right)=A\exp [-{\left(\lambda t\right)}^{\beta }]$.

Figure 10

Temperature dependence of (a) $\mu \mathrm{SR}$ relaxation rate $\left(\lambda \right)$, (b) exponent $\beta$, and (c) asymmetry $A$ derived from the fitting of zero-field asymmetry by $G\left(t\right)=A\exp [-{\left(\lambda t\right)}^{\beta }]$.

Figure 11

Temperature dependence of $1∕T_1$ and $1∕\left(4T_2\right)$ of the Ga(1) NQR spectra, and $\mu \mathrm{SR}$ relaxation rate $\lambda$. The inset shows the plot of $1∕T_1$ and $1∕\left(4T_2\right)$ against $(T-T_{\mathrm{f}})∕T_{\mathrm{f}}$.

Figure 12

Temperature dependence of the characteristic energy of the spin fluctuations $\Gamma \left(T\right)∕k_{\mathrm{B}}$ (see text). The inset shows the plot of $\Gamma \left(T\right)∕k_{\mathrm{B}}$ against $1∕T$.

Figure 13

${}^{69,71}\mathrm{Ga}\text{−}\mathrm{NQR}$ spectra at $40\mathrm{K}$ (upper panel) and at $1.5\mathrm{K}$ (bottom panel). The broad spectrum in the bottom figure is approximately reproduced by the presence of inhomogeneous internal fields pointing along the $c$ axis. The distribution of the internal fields is shown in the inset of the bottom figure.

Figure 14

Temperature dependence of $1∕T_1$ calculated by the 3D and 2D spin-wave models are compared with the experimental data (see text).