Ferromagnetic fragmented state in the pyrochlore Ho$_2$Ru$_2$O$_7$

TL;DR

Ho$_2$Ru$_2$O$_7$ exhibits a two-stage magnetic order in which Ru spins first order at $T obreak \approx 95$ K into a Γ$_5$ easy-plane antiferromagnet, followed by Ho spins ordering at $T_c obreak \approx 1.55$ K into a Γ$_9$ ferromagnetic state with substantial ground-state entropy. The data support a novel ferromagnetic fragmented ground state, wherein a saturated apical spin coexists with a kagome-ice-like dipolar fragment, and where Ru moments must tilt out of their easy planes via Ho–Ho and Ho–Ru mediated interactions to induce Ho order. Across the transition, ac susceptibility reveals two well-separated dynamic processes, and diffuse neutron scattering shows evolving kagome-ice correlations consistent with fragmentation rather than a simple mosaic of domains. A minimal model and a cubic-field–driven fragmentation framework are developed to rationalize the observed partial order, residual entropy, and dynamics, highlighting the role of inter-sublattice coupling and topological constraints in establishing a fragmented ferromagnet along a body-centered cubic easy axis.

Abstract

The consecutive magnetic ordering of the Ho and Ru ions in the pyrochlore Ho2Ru2O7 and their interplay are investigated by neutron scattering, magnetic and specific heat measurements. The Ru moments order at 95 K into a $Γ_5$ easy-plane antiferromagnetic state. At 1.55 K the Ho moments order into an unusual $Γ_9$ ferromagnetic state with extensive ground state entropy and structured spin dynamics. It is shown how the internal fields with $Γ_5$ and $Γ_9$ geometry allow for two symmetry breaking transitions. The lower temperature ordering is driven by ruthenium mediated interactions between holmium moments as spin ice correlations develop. The unsaturated order is compatible with a fragmented ferromagnetic state equivalent to pyrochlore kagome ice.

Figures (15)

• Figure 1: (a) The Ho$_2$Ru$_2$O$_7$ pyrochlore lattice; Ho sublattice in blue and Ru sublattice in red. (b) Ordered spin ice structure from the $\Gamma_9$ representation (left) and two states $\psi_2$ and $\psi_3$ from which easy-plane antiferromagnetic structures belonging to the $\Gamma_5$ representation can be built (right) (See for example Refs. Wills2006Poole2007Petit2017 for the definition of these representations). (c) Ferromagnetic fragmented magnetic structure. (d) Graphical representation of the magnetic fragmentation (Equation \ref{['eqFrag']}). The green spin is the apical spin and the red spin is the majority moment of the dipolar fragment.
• Figure 2: Neutron diffractograms: (a) at 31 and 120 K showing the Ru magnetic ordering; (b) at 0.7 K showing the Ho magnetic ordering. Black lines are the Rietveld refinements performed with the Fullprof software FullProf1993, with the $\Gamma_5$ and $\Gamma_9$ representations (See Fig. \ref{['Fig_structure']}) for Ru and Ho respectively supmat. Blue lines are the differences between the fit and the data and green ticks index the Bragg peaks. The (002) peak, indicated by a blue arrow, is only present in the $\Gamma_9$ representation, and appears below the Ho transition. Dark blue and orange arrows indicate the peaks originating from the Ho$_2$O$_3$ and RuO$_2$ impurities respectively.
• Figure 3: (a) Magnetic specific heat in zero and applied magnetic fields, corrected for nuclear hyperfine contribution and for the phonon signal using a non magnetic sister compound supmat. (b) Magnetic entropy difference with the expected values for spin ice (red), monopole crystal (orange), kagome ice (green), $R\ln(2)$ (full black), $\frac{1}{4}R\ln(2)$ (dashed black). (c) In-phase part of the ac susceptibility $\chi'$ as a function of temperature, together with the magnetization to field ratio measured in a ZFC-FC protocol with a field of 100 Oe (black).
• Figure 4: (a) $\chi"$ as a function of frequency at fixed temperatures. (b) Relaxation times $\tau$ from a fit to a generalized Debye function with two characteristic times supmat. The transparency of the symbols indicate the relative amplitude of the two processes. Dotted lines are Arrhenius laws $\tau_0 \exp(E/T)$, with $\tau_{0f}=5.5 \times 10^{-7}$ s and $E_f=6$ K above $T_c$ and $\tau_{0s}=1.1 \times 10^{-5}$ s and $E_s=15$ K below.
• Figure 5: (a) Refined component $s$ as a function of temperature in the proposed fragmented structure (See text). The shaded region shows the error bars. (b) Magnetic neutron scattering intensity measured on D007@ILL between 60 mK and 120 K. Inset: Zoom in the magnetic diffuse neutron scattering above the transition temperature at 1.6 and 10 K, compared to the pattern expected for the kagome ice and spin ice scattering functions.