Vanishing ordered moment in the frustrated triangular lattice antiferromagnet CuNdO$_2$

TL;DR

This study shows that CuNdO$_2$ hosts a triangular lattice of Nd$^{3+}$ moments with strong Ising anisotropy, ordering antiferromagnetically below $T_N=0.78$ K. Crystal-field analysis reveals a dominant out-of-plane dipole moment forming a Kramers doublet, while the in-plane component is extremely small, explaining the absence of observable magnetic Bragg peaks despite bulk evidence of order. The magnetic ground state is interpreted as a 120$^ obreaks$ in-plane antiferromagnet with a tiny net moment, well described by an XXZ Hamiltonian with $J=0.2$ meV and $\epsilon=0.5$, which also accounts for the nearly dispersionless spin-wave spectrum. Overall, the work illustrates how strong single-ion anisotropy and geometric frustration can cooperate in rare-earth triangular magnets to yield long-range order with vanishing dipolar moments and provides a quantitative framework for similar systems.

Abstract

We investigate the magnetic ground state of CuNdO$_2$, which is a delafossite with a triangular lattice of magnetic Nd$^{3+}$ ions that are well separated by non-magnetic Cu spacer layers. From inelastic neutron scattering measurements of the crystal electric field, we determine the strong Ising character of the pseudo-spin 1/2 Nd$^{3+}$ moments. Magnetic susceptibility and heat capacity measurements reveal the onset of long-range antiferromagnetic order at $T_N=0.78$ K. While the magnetic transition is definitively observed with muon spin relaxation, accompanied by the formation of a weakly dispersing spin wave excitation, no dipole-ordered moment is detected with neutron diffraction. We show that the apparent absence of a dipolar ordered moment is a consequence of the dominant Ising character of the antiferromagnetically coupled Nd$^{3+}$ moments, which experience extreme frustration on the triangular lattice. Consequently, the frustration in CuNdO$_2$ is relieved through in-plane ordering of the substantially smaller perpendicular component of the Nd$^{3+}$ moments into a 120\textdegree\ structure, with a nearly vanishing ordered moment.

Figures (4)

• Figure 1: Onset of long-range magnetic order in the triangular lattice antiferromagnet CuNdO$_2$(a) The $R\overline{3}m$ crystal structure of CuNdO$_2$ is composed of alternating, staggered triangular layers of magnetic Nd$^{3+}$ separated by non-magnetic Cu$^{1+}$. (b) The magnetic susceptibility of CuNdO$_2$, measured in an $\mu_0H = 0.05$ T field, is well described by the Curie-Weiss law between 200 and 400 K (shown in the inset), where the filled symbols show the susceptibility and open symbols mark the inverse susceptibility. A sharp cusp at $T_N = 0.78$ K marks an antiferromagnetic ordering transition. (c) The same transition is also detected in heat capacity measurements and the computed entropy release approaches $R\ln{(2)}$ (shown in the inset), as expected for a crystal field ground state doublet.
• Figure 2: The crystal field ground state of Nd$\mathbf{^{3+}}$ in CuNdO$_2$ is an effective spin-$\mathbf{1/2}$ with strong Ising anisotropy. Inelastic neutron scattering measurements performed at (a,b)$T=5$ K and (c,d)$T= 100$ K with $E_i=45$ and 150 meV, respectively. The CEF excitations from the ground state ($\Delta_1$ to $\Delta_4$) and those arising from thermally populated levels ($\Delta_{1\rightarrow2}$, $\Delta_{1\rightarrow3}$, $\Delta_{2\rightarrow4}$, and $\Delta_{1\rightarrow4}$) are indicated by the dashed lines. These experimentally observed CEF levels were used to determine the CEF Hamiltonian, the result of which is shown at (e)$T=5$ K and (f)$T=100$ K. The positions of unmatched intensity at 25, 40, and 55 meV correspond to intense phonon modes. (g) The paramagnetic susceptibility of CuNdO$_2$ computed with the fitted CEF Hamiltonian, showing good agreement with the experimental data.
• Figure 3: Absence of magnetic Bragg peaks in CuNdO$_2$ despite clear bulk ordering observed with $\mu$SR.(a) Neutron diffraction measurements on CuNdO$_2$, plotted on a log intensity scale, above (1 K) and below (0.25 K) the ordering temperature $T_N = 0.78$ K. (b) Difference of the elastic intensity above and below $T_N$ confirming the absence of any magnetic Bragg peaks within the signal-to-noise ratio of the experiment. (c) Rietveld refinement of the neutron diffraction pattern of CuNdO$_2$ collected at 0.25 K, showing excellent agreement ($\chi^2=1.02$) with a purely structural description. The refined phases include Al, originating from the sample can, and minor Cu and Cu$_2$O impurities, which are residual from the synthesis. (d) Representative muon decay asymmetry spectra at varying temperatures above and below $T_N=0.78$ K. The data are fitted with the function shown in Eq. (\ref{['muon-fcn']}). Error bars represent one standard deviation. The temperature dependence of the fitted (e) internal field of the oscillatory component and (f) relaxation rate of the fast (filled symbols) and slow (open symbols) components, consistent with the magnetic transition at $T_N=0.78$ K marked by the dashed line.
• Figure 4: Weakly dispersive low-temperature spin excitations in CuNdO$_2$. Inelastic neutron scattering measurements of CuNdO$_2$ acquired at (a)$T=4$ K and (b)$T=0.05$ K using neutrons with incident energies of $E_i=3.6~$meV ($\lambda = 4.8$$\hbox{\normalfont\AA}$), revealing the formation of a nearly dispersionless collective spin excitation below $T_N$. (c) Temperature dependence of the $Q$-integrated $4.8~\hbox{\normalfont\AA}$ inelastic neutron scattering spectra. Error bars represent one standard deviation. The inset shows the temperature dependence of the integrated scattered intensity associated with the 0.27(2) meV inelastic neutron scattering mode. (d) Inelastic neutron scattering spectra of CuNdO$_2$ collected with $E_i=2.27$ meV ($\lambda=6$$\hbox{\normalfont\AA}$) incident neutrons energy at $0.05$ K. A background measurement acquired at 4 K was subtracted from this data set. (e) Powder-averaged linear spin-wave calculation for CuNdO$_2$ assuming a 120° antiferromagnetic order, which is energetically stabilized by a XXZ spin Hamiltonian with exchange constant $J=0.2$ meV and an anisotropic exchange coefficient $\epsilon=0.5$.