Inhomogeneous dynamic state in the double trillium lattice antiferromagnet KBaFe$_2$(PO$_4$)$_3$

Abstract

The three-dimensional (3D) magnet KBaFe$_2$(PO$_4$)$_3$ hosts a double-trillium lattice of Fe$^{3+}$ (spin, $S=5/2$) ions offering a prototypical platform to study the frustration induced effects in 3D. Through magnetization, specific heat, $^{31}$P nuclear magnetic resonance (NMR), and muon spin relaxation ($μ$SR) experiments, supported by first principles calculations, we uncover an unconventional ground state. Despite strong antiferromagnetic interactions with a large Curie-Weiss temperature $θ_{\rm CW} = -70(2)$ K, no magnetic long-range order is observed down to 30 mK. Below $T^{\ast}\simeq 3.5$ K, the NMR linewidth becomes nearly field-independent and the spin-spin relaxation rate $1/T_2$ saturates, accompanied by an inhomogeneous distribution of transverse nuclear magnetization $M_{xy}$. The latter indicates the emergence of short-range dynamical correlations, which was further corroborated by a robust and field-insensitive broad maximum in specific heat. In $μ$SR, we detect neither a static internal field nor spin freezing; instead the relaxation remains dynamic and is best described by two coexisting dynamic relaxation channels: a dominant fast (sporadic) channel and a slower Markovian component. Their differing weights and fluctuation rates suggest microscopic inhomogeneity in spin dynamics. Altogether, KBaFe$_2$(PO$_4$)$_3$ exemplifies a rare high-spin stochiometric 3D antiferromagnet that evades ordering and instead fosters a mosaic of spin dynamics driven by strong geometric frustration intrinsic to the trillium lattice.

Figures (3)

• Figure 1: (a) A coupled trillium unit of Fe$^{3+}$ ions, connected via PO$_4$ tetrahedra. (b) $\chi_{\rm dc}$ vs $T$ measured under ZFC and FC conditions in $\mu_0 H = 0.01$ T and in other applied fields. The solid line represents the classical Monte-Carlo simulation using the optimized exchange parameters from Table. \ref{['tab:exchange']}. Inset: CW fit to $\chi_{\rm dc}(T)$. (c) Left $y$-axis: $C_{\rm p}$ vs $T$ for KBFPO and its non-magnetic analogue KBaIn$_2$(PO$_4$)$_3$ measured in zero-field and right $y$-axis: magnetic entropy $\Delta S_{\rm m}(T)$ calculated from $C_{\rm mag}(T)$. (d) $C_{\rm p}(T)$ measured in different applied fields. Inset: $C_{\rm mag}$ vs $T^2$ plot fitted with a straight line at low-$T$s.
• Figure 2: (a) Temperature dependent field-sweep $^{31}$P NMR spectra measured at 126.6 MHz. The vertical dashed line marks the Larmor field. Inset: temperature-dependent signal intensity multiplied by temperature with the $T_2$ corrections. (b) Temperature dependence of the $^{31}$P NMR shift ($\mathcal{K}$) measured at 126.6 MHz, overlaid with $\chi_{\rm dc}$ measured at 7 T. Inset: $\mathcal{K}$ vs $\chi_{\rm dc}$. (c) Full width at half maximum ($\Delta B$) 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: $\Delta B$ as a function of the NMR frequency. (d) Temperature dependence of $1/T_1$ at different frequencies, together with the temperature dependence of $\chi_{\rm dc}T$ measured at 2 T. (e) $1/T_2$ 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.
• Figure 3: (a) ZF muon spin asymmetry [$A_{\rm ZF}(t)$] at different temperatures. Left panel: magnified short-time behavior ($t \le 0.6\,\mu$s) and right panel: long-time tail ($t \ge 0.6\,\mu$s). Solid lines are fits using Eq. \ref{['eq:empirical']}. (b) Muon-spin asymmetry $A_{\rm LF}(t)$ measured under different LF at $T\simeq0.1$ K. Left panel: enlarged short-time scale; right panel: long-time tail. The dashed lines represent simulated depolarization curves using Eq. \ref{['eq:composite']}. Solid lines are the fits using Eq. \ref{['eq:sporadic']}. (c) Temperature dependence of the ZF muon-spin relaxation rates ($\lambda_1$ and $\lambda_2$). Inset: temperature dependence of the stretching exponent $\beta$. (d) Temperature dependence of the fractional weight $\alpha$. Inset: LF dependence of the contribution $\phi$ from sporadic field fluctuations.