Short-range Spin Freezing State in the Double Trillium Lattice Spin-Liquid Candidate KSrFe$_2$(PO$_4$)$_3$ Revealed via $^{31}$P NMR

Abstract

A comprehensive $^{31}$P nuclear magnetic resonance (NMR) study, combined with thermodynamic measurements and first-principle band-structure calculations, has been conducted to explore the ground state of the $S = 5/2$ double trillium lattice antiferromagnet KSrFe$_2$(PO$_4$)$_3$. Our experimental results indicate that the magnetic ground state is neither a conventional three-dimensional (3D) long-range order (LRO) nor a pure gapless spin-liquid state, as conjectured previously [Boya et al., APL Mater. 10, 101103 (2022)]. Specifically, the observation of a nearly field-independent NMR linewidth below $T^{*}$ = (3.5 $\pm$ 0.4) K, and a significant enhancement of spin-spin relaxation rate $1/T_2$ below $2T^{*}$ (where $T^{*}$ is the characteristic temperature identified from the magnetic susceptibility), indicate a complex magnetic ground state where spin freezing coexists with persistent dynamics. Furthermore, we argue that the lack of magnetic LRO and the persistence of strong magnetic fluctuations in KSrFe$_2$(PO$_4$)$_3$ are unlikely to originate from intersite K/Sr disorder, rather arise due to intrinsic magnetic frustration. Our findings position KSrFe$_2$(PO$_4$)$_3$ into a broader family of geometrically frustrated magnets characterized by coexisting spin freezing and pronounced antiferromagnetic fluctuations, marking it as a promising platform for investigating exotic phenomena in 3D frustrated magnets.

Figures (3)

• Figure 1: (a) Left $y$-axis: $\chi_{\rm dc}$ vs $T$ measured under ZFC and FC conditions in $\mu_0 H = 0.01$ T and in other applied fields. $T^*$ denotes the characteristic temperature. The solid line represents the classical Monte-Carlo simulation for the optimized exchange parameters from Table \ref{['tab:exchange']}. Right $y$-axis: $C_{\rm p}$ vs $T$ measured in different magnetic fields. (b) Temperature-dependent $\chi^{\prime}_{\rm ac}$ measured at a fixed ac field of $H_{\rm ac}= 10$ Oe in different frequencies. The data at different frequencies are vertically offset for clarity. Inset: a partial view of the coupled trillium lattice of Fe$^{3+}$ ions, connected via PO$_4$ tetrahedra.
• Figure 2: (a) Temperature dependence of the field-swept $^{31}$P NMR spectra measured at 126.6 MHz. The vertical dashed line marks the Larmor field. The inset shows the temperature dependence of the signal intensity multiplied by temperature with the $T_2$ corrections. (b) Temperature dependence of the $^{31}$P NMR shift ($K$) measured at 126.6 MHz, overlaid with the magnetic susceptibility data measured at 7 T. The inset shows $K$ vs $\chi_{\rm dc}$. (c) Full width at half maximum ($\Delta H$) of the $^{31}$P NMR line as a function of temperature, measured at three different frequencies. The solid line represents the magnetization ($M$) vs $T$ plot for $\mu_0 H = 7$ T. Inset: Variation of $\Delta H$ as a function of the NMR frequency, above and below $T^{*}$.
• Figure 3: (a) Temperature dependence of $1/T_1$ at different frequencies, together with the temperature dependence of $\chi_{\rm dc}T$ measured at 2 T. The black dashed curve represents the temperature dependence of $1/T_1$ reported for the typical conventional 3D AFM Na$_3$Fe(PO$_4$)$_2$ with a Néel temperature of $T_{\rm N}$$\simeq$ 10.9 K from Ref. Devi015803. (b) The $^{31}$P $1/T_2$ as a function of temperature measured at various frequencies. The red solid line represents the fit discussed in the text, with $T^{*} = 3.5$ K. The orange dashed curve is included as a visual guide. The black dashed curve represents the temperature dependence of $1/T_2$ for Na$_3$Fe(PO$_4$)$_2$Devi015803.