Spin-trimer antiferromagnetism in ${\mathrm{La}}_4{\mathrm{Cu}}_3\mathrm{Mo}{\mathrm{O}}_{12}$

${\mathrm{La}}_4{\mathrm{Cu}}_3\mathrm{Mo}{\mathrm{O}}_{12}$ is a cluster antiferromagnet where copper spin-$1∕2$ trimers form a network of strongly coupled spin trimers. The magnetic properties of this material have been examined using magnetic neutron scattering. At low temperatures, excitations from the ground state are observed at $7.5\left(3\right)\mathrm{meV}$ and $132.5\left(5\right)\mathrm{meV}$. An additional peak in the neutron scattering spectrum, which appears at $125.0\left(5\right)\mathrm{meV}$ on heating, is ascribed to a transition between excited states. The wave vector and temperature dependence of the inelastic magnetic scattering cross section are consistent with intratrimer transitions. Magnetic neutron diffraction reveals antiferromagnetic order below $T_N=2.6\mathrm{K}$ with a wave vector $\left(\frac{1}{2}00\right)$. The ordered magnetic structure is described as intertrimer order where spin correlations within trimers are controlled by the strong intratrimer interactions. Combining the information derived from elastic and inelastic magnetic neutron scattering with group theoretical analysis, a consistent set of intratrimer interactions and ordered magnetic structures is derived. The experiment provides a simple worked example of magnetism associated with interatomic composite degrees of freedom in the extreme quantum limit.

Figure 1

${\mathrm{Cu}}_3\mathrm{Mo}{\mathrm{O}}_4$ plane of ${\mathrm{La}}_4{\mathrm{Cu}}_3\mathrm{Mo}{\mathrm{O}}_{12}$. Actual coordinates for copper atoms are given in the caption to Table . The solid lines represent the triangle clusters. $J_1$, $J_2$, and $J_3$ are the intratriangle couplings; $J$’s are the weak intertriangle couplings that are assumed to be antiferromagnetic and of similar magnitude as those in the mean-field analysis of Wessel and Haas (Ref. 9). The arrows illustrate one plane of the ${\tau }_2$ magnetic structure with $\psi =0$ listed in Table .

Figure 2

Temperature dependence of the $\left(\frac{1}{2}00\right)$ magnetic Bragg peak intensity. The inset shows the wave-vector dependence of the difference between elastic magnetic neutron scattering at $T=0.4\mathrm{K}$ and $T=10\mathrm{K}$. The energy resolution was $1.3\mathrm{meV}$ and the wave-vector resolution is indicated by the solid line in the inset.

Figure 3

$\hslash \omega$ dependence of the normalized $Q$-averaged neutron scattering intensity $\tilde{I}(Q,\hslash \omega )$ at various temeratures and in two different energy ranges. The open circles are raw experimental data. The open triangles are the phonon background subtracted data in (a)–(c) and the excess scattering above the background of the sloped Gaussian fit at $10\mathrm{K}$ for (e) and (f). The solid circles and solid triangles in (a) are total scattering and magnetic scattering intensities, respectively, from the polarized neutron measurement. A single scale factor was applied to the polarized data for best agreement with the unpolarized time-of-flight data.

Figure 4

Temperature dependence of parameters characterizing high- and low-energy magnetic excitations in ${\mathrm{La}}_4{\mathrm{Cu}}_3\mathrm{Mo}{\mathrm{O}}_{12}$. Frame (a) shows the integrated intensities for the $132.5\text{−}\mathrm{meV}$ (open circles), $125\text{−}\mathrm{meV}$ (triangles), and $7.5\text{−}\mathrm{meV}$ (solid circles) modes. Frame (b) shows the peak positions and frame (c) shows the intrinsic half width at half maximum relaxation rate for each mode. Lines in frame (a) were calculated from the trimer model.

Figure 5

Wave-vector dependence of the energy-integrated intensity for the $132.5\text{−}\mathrm{meV}$ (circles) and $7.5\text{−}\mathrm{meV}$ excitations (triangles) in ${\mathrm{La}}_4{\mathrm{Cu}}_3\mathrm{Mo}{\mathrm{O}}_{12}$. Solid lines show the calculated $Q$ dependence of the neutron scattering cross section from Eqs. (A6, A7) with no adjustable parameters.

Figure 6

Zero-temperature magnetic phase diagram of the AFM spin-$1∕2$ trimer square lattice with a weak intertriangle coupling $J^{\prime }=0.01J_3$. The excitation energies observed in the present experiment imply that the ratios of exchange constants lie on the ellipse shown in the center of the figure. The observation of $\left(\frac{1}{2}00\right)$ magnetic order also helps to constrain possible values of the exchange constants.

Figure 7

(a) Expectation value for the magnetic moment along a small applied field and at $T=0$ for the three copper ions in a trimer as a function of index angle $\psi$. $\psi$ is the azimuthal angle spanning intratrimer exchange interactions that are consistent with a $7.5\text{−}\mathrm{meV}$ doublet-doublet transition (see Fig. 6). $J_1∕J_3$ is equal to $J_2∕J_3$ at $\psi =0$ and $\pi$. (b) Elastic magnetic neutron scattering cross section for the $\left(\frac{1}{2}00\right)$ Bragg peak per crystal unit cell calculated as a function of $\psi$ when the spins are along the $\mathbf{c}$ direction and adopt the magnetic structure associated with the ${\tau }_4$ irreducible representation. The dashed line shows the actual measured magnetic Bragg intensity.

Figure 8

Combination of index angle $\psi$ and the spin orientation angle $\varphi$ for magnetic structures with irreducible representation ${\tau }_2$ (solid line) and ${\tau }_3$ (dashed line) that yield a $\left(\frac{1}{2}00\right)$ magnetic Bragg intensity consistent with the experiment. Parameters in the hatched areas stabilize the $\left(\frac{1}{2}00\right)$ magnetic structure for weak intertrimer interactions (Ref. 9).