Short-range magnetic ordering process for the triangular-lattice compound $\mathrm{Ni}{\mathrm{Ga}}_2{\mathrm{S}}_4$: A positive muon spin rotation and relaxation study

We report a study of the triangular-lattice Heisenberg magnet $\mathrm{Ni}{\mathrm{Ga}}_2{\mathrm{S}}_4$ by the positive muon spin relaxation technique. We unravel three temperature regimes: (i) below $T_c=9.2\left(2\right)\mathrm{K}$, a spontaneous static magnetic field at the muon site is observed and the spin dynamics is appreciable: the time scale of the modes we probe is $\simeq 7\mathrm{ns}$; (ii) an unconventional stretched exponential relaxation function is found for $T_c<T<T_{\text{cross}}$, where $T_{\text{cross}}=12.6\mathrm{K}$, which is a signature of a multichannel relaxation for this temperature range; and (iii) above $T_{\text{cross}}$, the relaxation is exponential as expected for a conventional compound. The transition at $T_c$ is of the continuous type. It occurs at a temperature slightly smaller than the temperature at which the specific heat displays a maximum at low temperature. This is reminiscent of the behavior expected for the Berezinskii–Kosterlitz–Thouless transition. We argue that these results reflect the presence of topological defects above $T_c$.

Figure 1

(Color online) Three zero-field $\mu \mathrm{SR}$ spectra recorded for a powder sample of $\mathrm{Ni}{\mathrm{Ga}}_2{\mathrm{S}}_4$ with the general purpose surface-muon instrument of the $\mathrm{S}\mu \mathrm{S}$ facility at the Paul Scherrer Institute, Villigen, Switzerland. The spectra probe three distinct temperature regimes. At the bottom, we display a spectrum recorded deep into the short-range ordered state. The solid line is the result of a fit to a function given in the main text. The spectrum in the middle has been taken at $11\mathrm{K}$ where the fit relaxation function is a stretched exponential (full line). For reference, the dashed line displays the best fit to an exponential relaxation. Such an exponential function is observed at $15\mathrm{K}$ (top spectrum). For clarity, the two high-temperature spectra are shifted up by 0.05 relative to the third one. Their time scale is given by the top horizontal axis, while the scale for the low-temperature spectrum is at the bottom horizontal axis: the range spanned by these two time scales differs by a factor $\sim 30$.

Figure 2

(Color online) Upper panel: thermal dependence of two parameters extracted from $\mu \mathrm{SR}$ spectra. The maximum of the static local magnetic field at the muon site $B_{\max }$ for $T<T_{\mathrm{stat}}$ is presented on the left-hand side. The relaxation rate ${\lambda }_Z$ for $T>T_{\text{cross}}$ is displayed on the right-hand side of the same panel, i.e., when the relaxation function is exponential. The full lines are results of fits described in the main text. Lower panel: zero-field magnetic specific heat divided by the temperature, $C_m∕T$, deduced from the measured specific heat using a dynamic adiabatic technique after subtraction of the lattice contribution. The full line is obtained for a linear dependence of $C_m∕T$.